Fractal space, cosmic strings and spontaneous symmetry breaking

Chaos, Solitons and Fractals 12 (2001) 1±48

www.elsevier.nl/locate/chaos

Fractal space, cosmic strings and spontaneous symmetry breaking J. Argyris a, C. Ciubotariu b,*, H.-G. Matuttis a a

Institute for Computer Applications (ICA I), University of Stuttgart, Pfa€enwaldring 27, D-70569 Stuttgart, Germany b Physics Department, Technical University of Iasi, Strada Dacia 9, O.P. Iasi 3, RO-6600 Iasi, Romania Accepted 17 August 1999

Abstract The present paper is conceived within the framework of El Naschie's fractal-Cantorian program and proposes to develop a model of the fractal properties of spacetime. We show that, starting from the most fundamental level of elementary particles and rising up to the largest scale structure of the Universe, the fractal signature of spacetime is imprinted onto matter and ®elds via the common concept for all scales emanating from the physical spacetime vacuum ¯uctuations. The fractal structure of matter, ®eld and spacetime (i.e. the nature and the Universe) possesses a universal character and can encompass also the well-known geometric structures of spacetime as Riemannian curvature and torsion and includes also, deviations from Newtonian or Einsteinian gravity (e.g. the R ossler conjecture). The leitmotiv of the paper is generated by cosmic strings as a fractal evidence of cosmic structures which are directly related to physical properties of a vacuum state of matter (VSM). We present also some physical aspects of a spontaneous breaking of symmetry and the Higgs mechanism in their relation with cosmic string phenomenology. Superconducting cosmic strings and the presence of cosmic inhomogeneities can induce to cosmic Josephson junctions (weak links) along a cosmic string or in connection with a cosmic string (self) interactions and thus some intermittency routes to a cosmic chaos can be explored. The key aspect of fractals in physics and of fractal geometry is to understand why nature gives rise to fractal structures. Our present answer is: because a fractal structure is a manifestation of the universality of self-organisation processes, as a result of a sequence of spontaneous symmetry breaking (SSB). Our conclusion is that it is very dicult to prescribe a certain type of fractal within an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter will decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not, but consider on the other hand a space far removed from usual matter, or a space containing isolated small particles in which a very low density matter can exist. Very low density matter might be in¯uenced by a fractal structure of space, for example in the sense that it is subject also to ¯uctuations structured by random fractals. Di€raction and di€usion experiments in an empty space and very low density matter could provide evidence of a fractal structure of space. However, at very high (Planck) densities, and a spacetime in which ¯uctuations represent also the source of matter and ®elds (which is very resonable within the context of a quantum gravity), we can assert that Einstein's dream of geometrising physics and El Naschie's hope to prove the fractalisation (or Cantorisation) of spacetime are fully realised. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction The present paper appertains to El Naschie's Cambridge Program (involving a fractalisation or Cantorisation of physics, spacetime and phase space manifolds) and can be considered as a development (within the new context of a universal and unitary concept of fractals) of the classical Felix Klein's Erlanger program on the geometrisation of physics. A geometry is de®ned by a mathematical structure and a group of symmetry transformations that preserve the structure. For example, Riemannian geometry studies the quantities that remain invariant under general coordinate transformations or isometries of a metric tensor. *

Corresponding author. Tel.: +40-32-232 229. E-mail address: [email protected] (C. Ciubotariu).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 6 1 - 7

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J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

The geometry of the phase space of Hamiltonian systems is symplectic or complex 1 geometry, because it is the symplectic structure which is invariant under canonical transformations which describe the time evolution of the system. The evolution of a Hamiltonian system preserves the action in closed loops (which is the Poincare ®rst integral invariant) and, thus, the area with respect to the symplectic structure x (the twoform dqi ^dpi ) is also preserved. 2 Just as in general relativity physics in spacetime is invariant under transformations that preserve the metric, physics in phase space is invariant under canonical transformations which preserve the Poisson bracket (or canonical Hamilton equations). The arena on which the dynamics take place is a Poisson or symplectic manifold which, in fact, represents a manifold of states with a Poisson bracket de®ned on it. In this context, a Poisson bracket is a bilinear map from pairs of functions to functions. This map endows the space of functions with a Lie algebra de®ned by the usual operations: bilinearity, anti-symmetry, Jacobi's identity, derivation properties, etc. There are few situations in which we can `see' the physical structure and the texture of spacetime or phase space. Usually, a fractal structure refers to matter (and, possibly, also to ®elds) but the problem remains if the space itself has a fractal structure. Anyhow, matter resides in space and time and thus a fractal matter may include also a fractal structure of space and time. How can we improve our understanding of these aspects? The concepts of scaling, renormalisation and universality are often invoked at this point. We can formalise these heuristic concepts by introducing a `space' which can be in®nitely dimensional, and whose points represent physical systems or, more exactly, statistical states of systems. Of course, at the most elementary (or fundamental) level, physical systems, space and time (and thus, the entire Universe) are statistical, ``intrinsically and irreducibly probabilistic in nature'' [1]. In order to study the fractal character of such a system, we ``step back'' from the system (i.e. we observe it from a greater distance), and examine all behaviour as if it were occurring on a smaller scale [2]. This is equivalent to the consideration of photographs of the ocean's surface taken at di€erent heights, or to looking at the system through a ®ner and ®ner microscope. In other words, the operation of stepping back or rescaling extracts one distribution and yields us another one. The rescaling is usually called renormalisation. The renormalised system represents an element of this in®nitely dimensional space, but is associated with sequentially different values of parameters. One de®nes in each case a class of model systems corresponding to a scale parameter l, and assumes that all these models possess the same physical properties, so that once more the physical quantities of di€erent systems are scaled and mutually related via some functions of l. If we introduce a mapping in a suitable parameter space (which can also be considered as a scale transformation), these functions of l can be determined from a set of di€erential equations de®ning a group of scale transformations which form, precisely, the renormalisation group (RG). The RG technique represents a good approach to physical systems which possess many scale lengths as it happens for a system having a fractal structure. The operating mode is the following: we start, for example, at the atomic level and integrate or average out the ¯uctuations and a transformation of all physical quantities including the Hamiltonian is made to a higher level. We repeat this procedure succesively to increasingly larger scales until all ¯uctuations have been averaged out. From this point of view, the RG does not actually represent a group, but rather a semigroup because there is no inverse transformation [3]. The subject of the present work is rather an unconventional one and examines the fractal structure of spacetime and matter at their fundamental level; we do not step back but actually step ahead to perceive more clearly the physical system (i.e. we look at it from a smaller distance). We are not interested now in the averaging procedure of ¯uctuations, but just in the opposite procedure, i.e. we want to know what an observer does see when he penetrates and lives in the world of (fundamental) ¯uctuations in vacuum spacetime, at small and large scales. For example, the fractal character of primeval density ¯uctuations, as compared to a homogeneous case, can lead to fractal properties for primordial seeds of astrophysical objects and also to 1 In order to eliminate a confusion with complex analysis, Herman Weyl considered the Latin roots com and plex and converted them into their Greek equivalents sym and plectic. 2 In the simple case of a one-dimensional (1D) harmonic oscillator, the action over one cycle is precisely the ratio energy/frequency which is an adiabatic invariant for the oscillator. Of course, the action over a closed orbit is given by the area enclosed by the orbit in the (q, p) phase plane, i.e. is given by the line integral of the canonical one-form h ˆ pi dqi around the loop or, via the Stokes' theorem, the surface integral of the two-form dqi dpi over the 2D surface enclosed by the loop.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

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the seeds of a fractal aggregation of matter (i.e. the scaling behaviour of growth process). This sequence of fractal properties is propagated as a wave of a ( fractal ) correlation from the ®rst vacuum ¯uctuations to the density perturbation and also further to various stages and scales of the growth process. In the context of astronomical observations, cosmologists have expressed an opinion that what we call a wave of (fractal) correlation acts only locally (on scales less than 50 million light years) and generates a cosmic fractal with a fractal dimension of about Dfractal ˆ 1:2. At larger scales, this fractal-correlation wave is damped out to a spatial homogeneity and isotropicity which are in agreement with Einstein's cosmological principle. The local fractal pattern manifests itself through the tendency of galaxies to trace linear sheet-like distributions which contain topological defects as walls and cosmic strings. Some fractal patterns can be observed even in regions which are apparently domains of noise. This is similar to the apparition of some fractal manifestations, for example, within the phenomenon of a critical opalescence when a normally clear substance becomes milky white due to ¯uctuations (on all scales) which do scatter all wavelengths of light. The present work refers also to fractal patterns arising in the domain of vacuum quantum ¯uctuations. The result can be extended to the so-called zitterbewegung in stochastic electrodynamics (SED) which enlarge the domain of traditional classical electrodynamics [4]. In SED (which is in essence a Lorentz type theory of electrons), which entails instead of the usual null homogeneous solution in an electromagnetic ®eld, a non-zero homogeneous solution which corresponds to a stochastic background of an electromagnetic ®eld that is randomly ¯uctuating, homogeneously distributed and isotropic in space but also homogeneous and isotropic for all inertial frames of reference. We are tempted to identify this stochastic electromagnetic background (SEB) with the cosmic microwave relic background radiation (CMB) which has the structure of a (thermal) Planck black-body radiation and, thus, has a (fractal) pattern in the phase space of variables (spectral-energy density, frequency, temperature). Furthermore, the CMB is not Lorentz invariant; it rather provides an absolute (cosmic, preferred ) frame of reference like the old ether, because it possesses a dipole anisotropy (caused by the Doppler shift) corresponding, for example, to the motion of the Earth with respect to the reference frame in which the radiation is isotropic. After subtracting the Earth's motion around the Sun, the Sun's motion around the Galactic Centre and the velocity of our Galaxy (Milky Way) with respect to the centroid of the Local Group, the dipole anisotropy yields the velocity (600 km sÿ1 ) of the Local Group 3 with respect to the cosmic frame of reference. The Standard Model in cosmology refers to the Big Bang Model (`a localised explosion in space') and assumes that there was a `big event' which triggered o€ the expansion of the universe. `Standard cosmologists' do not like so much these ideas about explosion ‡ expansion because, at very large scales, the Universe appears as homogeneous and we cannot localise a preferred centre that might have been the origin of an explosion. As regards the expansion, of course, we can consider it as a success that there appears some evidence from the theory of the origin of the light elements at a time when the mean distance between particles was ten orders of magnitude smaller than it is now. However, for the time being, we have no concept of how to ®nd an objective physical evidence for a real expansion of the Universe. The cosmological redshift may also be explained, for example, by an ageing of photons due to their long-time (and long-distance) interaction with the intergalactic matter in the course of their huge travel from their distant-source galaxies. Furthermore, the in¯ation scenario is not yet tested and cannot yet be a part of a standard cosmological model.

3 We recollect that our Solar System is near the edge of the Milky Way (a giant galaxy of stars of a spiral ± not elliptical ± nebulae form), at about 8 kpc from the centre. We are close to the central plane of the disk of the Milky Way. The Andromeda Nebula (also denoted as M31, in its place within the Messier list of bright nebulae; M31 contains the celebrated Cepheid variable stars in the constellation Cepheus) is the nearest large spiral galaxy outside the Milky Way. The spectrum of the Andromeda nebula is blueshifted, corresponding to a velocity of approach of the centres of the Milky Way and the M31 of about 100 km sÿ1 . This is an exotic exception to the expansion rule that galaxy spectra are redshifted. Of course, redshifts and blueshifts are produced by the di€erences in the gravitational potential energy. The blueshift and their small separation seem to indicate that the two galaxies are orbiting in a gravitationally bound system, called the Local Group. The other galaxies of this group are the (dwarf) satellites of the Milky way and of M31, such as the Magellanic Clouds and the Triangular Nebula M33 [5]. In turn, the Local Group of galaxies is on the outer edge of the Local Supercluster. The expansion of the Universe means that the distance between a pair of distant galaxies is increasing with time, which indicates that the galaxies are receding from each other. Of course, a gravitationally bound system (such as the Local Group) is not expanding and, in contrast to this, the gravitational force tends to collect the astrophysical objects into increasingly more massive systems that break away from the general expansion in order to form a hierarchy (fractal) of clusters. The homogeneous expansion of the universe refers to clusters far enough apart so that the local irregularities inside clusters can be neglected.

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J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

The open questions (and models) are more numerous in cosmology than in a standard well-elaborated physical theory, even if our intuition on the formation and structure of the Universe is based on `terrestrial' (small-scale) physical theories. The problem of an original localised big bang explosion but now (i.e. presently) without a site (or centre) of location can be explained if we adopt a fractal structure of spacetime. In a fractal space, the original big bang explosion plays the role of a primary source which generates a fractal travelling wave in the sense of a fractal Huygens' principle in agreement with which each point of non-di€erentiability of the fractal space becomes a new source of a wave. Also, in¯ation can be modelled by a fractal ¯ow of cosmic time. We note that the fractals that arise in nature can, generally, be modelled by a Cantor set. For instance, a classic Cantor set (remove the middle third of a segment recursively) corresponds to a periodic orbit as we continuously increase the scale. A fat Cantor set (e.g., remove middle third, then middle ninth, then middle 27th, etc.) asymptotically approaches the stable ®xed point represented by a solid line. This type of fractal occurs as the set of parameter values at which hump-map orbits are chaotic following criticality. An undernourished Cantor set (e.g., remove middle third, middle square root of 3rd, middle cube root of 3rd, etc.) approaches the stable ®xed point representing a single point in empty space. Very recently, El Naschie [6] drew attention to a connection between the KAM theorem of Hamiltonian systems, the signature of four manifolds, the 26D superstring spaces and the dimensions of the so-called 'nuclear spacetime' of quantum physics, E…1† . With respect to this result, we note that the set is obtained by removing the rationals from the unit interval and separating at each point the two sides around each rational p=q by a distance of 1=q3 scales in accordance with the terms of a continued fraction expansion of the point. The KAM tori reside in phase space like the points in this Cantor set within an arbitrary perturbation of a Hamiltonian system away from criticality. Furthermore, Liouville numbers which are easy to approximate by rationals approach a ®xed point representing an isolated point in empty space; numbers which are hard to approximate by rationals (like the Diophantine numbers) asymptote to a ®xed point on a line segment. Other numbers jump around in a complex manner and lead to chaotic renormalisation orbits. In essence, the purpose of this paper is to demonstrate that a fractal model of the Universe can be developed on the basis of a fractal structure of spacetime, matter and ®elds. The plan of the paper is as follows. In Section 2 we present the principal arguments which show that indeed space can be considered as an independent entity like matter and ®elds. Here we obtain a new derivation of the Unruh formula on the basis of Einstein's principle of equivalence. Section 3 presents the role of scalar ®elds in gauge theories. In Section 4 we summarise the principal features of the Standard Model in particle physics and cosmology. The universe represents a unique laboratory for investigating the uni®cation of fundamental forces. In Section 5 we present the fractal character of large-scale structures in the Universe as determined by topological defects in which the key-operators are cosmic strings. The possible e€ects of cosmic strings in an astrophysical space and the observationally veri®able signatures are indicated. It is also argued that a cosmic string is in fact a fractal string. In Section 6 we explain that even at very large scales for which the distribution of galaxies appears to adopt a homogeneous rule, the fractal structure can be still invoked in the form of a poly-fractal which represents a statistically uniform space distribution of bounded fractallites. In Section 7 we discuss the R ossler conjecture on gravity within the context of a fractal Universe. If the fractal behaviour of the Universe is real, gravity alone (in its Newtonian or Einsteinian forms) cannot be adopted to explain it. Some form of biased di€usion 4 may be appropriate here. In Section 8 we include a proposal for an experiment in order to demonstrate that indeed space has a fractal structure. Also, the concept of a cascade of spaces is introduced in this section. Sections 9±15 refer to the SSB, Higgs mechanism (HM) and Aharonov±Bohm (AB) e€ect which are essential to the understanding of topological defects. 2. Conventional understanting of (physical) space and a new con®rmation of Einstein's principle of equivalence In all of the present paper, we emphasise the physical aspects and signi®cance of the vacuum state (the ground state of the Hamiltonian) because it represents in fact the `empty space' (or, simply, space) in our 4

The term biased di€usion includes every transport phenomenon in which a force is superimposed on a random particle motion.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

5

conventional understanding of space, matter (substance) and ®elds, In quantum theories, (physical) vacuum has a `microscopic' structure and is considered to be a state (or a form) of matter (vacuum-like state of matter (VSM)) which macroscopically possesses the properties of an empty (void) space. From a microscopic point of view, the structure of VSM is represented by quantum ¯uctuations of all ®elds (electromagnetic, gravitational, etc,) or a zeropoint radiation ®eld. These ¯uctuations (which are not ®nite real ®elds, 5 because in VSM the averaged macroscopic ®eld is zero for any ®eld, and for this reason one asserts that quantum vacuum ¯uctuations of ®elds are equivalent to virtual particles or virtual quanta of these ®elds) exist even when there are no excitations (particles, radiation) present in VSM. In other words, within VSM, virtual particle-antiparticle pairs are created and annihilated continuously. This `perpetual-motion' vacuum pervades (or is identical with) the whole space and is similar to (or generates) the zero-point ¯uctuations within any material system at temperature T ˆ 0. Such ¯uctuations are, for example, the cause that helium remains (super) ¯uid even at the lowest temperatures. A fractal structure of VSM may be responsible for such a phenomenon. Other physical e€ects determined by VSM are as follows. (1) Casimir e€ect ˆ An e€ect due to a change of boundary conditions in VSM. If conducting plates are introduced into VSM, 6 the electromagnetic-vacuum ¯uctuations around the plates are in¯uenced by the plates and this leads to (Casimir) forces between the plates. (2) The Hamiltonian depends in relativistic theories (and, hereby, the VSM) on external ®elds, and on the choice of spacetime coordinates or perhaps, more exactly, on the system of reference. We distinguish the following phenomena which appertain to quantum ®eld theory (QFT) in curved spacetime 7 or accelerated frames of reference, and which correspond to some well-known celebrated e€ects. Vacuum polarisation and Spontaneous pair creation. The vacuum ¯uctuations of charged ®elds are in¯uenced by electric ®elds. In the case when the electric (or electromagnetic) ®eld is weak, we have merely an electrical polarisation of VSM in the sense that pairs of virtual charged particles (e.g., electrons and positrons) remain virtual (unobservable at a macroscopoic scale) and form virtual electric dipoles. If the external electromagnetic ®eld (photons) becomes suciently strong, the VSM becomes unstable and virtual pairs of charged particles become real in the sense that we can detect now the presence of electrons and positrons born from VSM (from empty space!?). This is the process of pair creation. In a way this phenomenon (creation of pairs when high-energy photons `collide with (and are stopped by) points in empty space') is similar to the situation when high-energy physicists started using accelerators and discovered a lot of new particles (leptons, mesons, barions) which were created from the kinetic energy of accelerated particles, when they were suddenly stopped. The question is: What are the points of VSM which can stop photons? If VSM has a fractal structure such points can be the points of non-di€erentiability which de®ne a fractal. Hawking radiation or black hole evaporation as a ®rst quantum gravity e€ect. Strong gravitational ®elds (of a black hole, for instance) may also produce similar e€ects on the pair creation. Any black hole has an event horizon 8 which `clothes' the singularity inside the black hole in agreement with the principle of cosmic censorship. 9 In other words, the gravitational ®eld of a black hole is so strong that the light rays inside the horizon are attracted backwards to the singularity. As shown by Hawking, the VSM around a black hole is unstable and can emit spontaneously energy quanta. Indeed, let us consider that a (virtual) particle-antiparticle pair is created near the event horizon. If one member of the pair having negative (total) energy falls into the black hole before annihilation, the other one may reach in®nity (and thus

5

However, a SED has been developed on the hypothesis of the existence of these ¯uctuations as real classical random ®elds as opposed to the virtual ®eld postulated by quantum ®eld theory [7]. 6 This means that the boundary conditions de®ning VSM are changed. 7 The usual measure theory of quantum mechanics describes phenomena within a microscopic scale since it refers to experiments performed in nuclear and subnuclear physics. In the context of QFT in curved or `accelerated' spacetime we deal with macroscopic e€ects, like Hawking and Unruh e€ects, where local coordinate systems have, however, a macroscopic character [8]. 8 Event horizons (or `one-way membranes') are the light-like surfaces dividing light rays that reach in®nity from those that do not. 9 In generic gravitational collapses the resulting spacetime singularities are hidden from the view of distant observers (principle of cosmic censorship). In other words, naked singularities do not exist. However, this principle applies only to generic spacetimes, where it is assumed that Einstein's equations hold in combination with some reasonable equation of state for matter. Otherwise, naked singularities or points at in®nity may occur.

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J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

becomes a real particle) with positive energy, if it has the appropriate (outward-pointing) initial momentum 10. The spectrum of such particles (Hawking radiation) is exactly a thermal spectrum, with the temperature of the black hole (for the particular case of a Schwarzschild black hole) given by the Bekenstein±Hawking formula TBH ˆ

hc3 1:2  1023 ˆ K: 8pkGM M…kg†

…1†

The power radiated by a black hole via Hawking's radiation is given by Stefan's law dMc2 ˆ rSB T 4 ABH ; dt

…2†

where rSB is the Stefan±Boltzmann constant and ABH ˆ

16pG2 M 2 ˆ 4pR2BH c4

…3†

is the horizon area. We observe that a black hole formed by the collapse of a star with M ˆ 2M  1030 kg possesses a temperature of T  10ÿ7 K and a radiative power of PBK  10ÿ26 W. However, a smaller black hole of mass M ˆ 1012 kg will be found to possess a temperature of T  1:2  1011 K and a radiative power of PBK  7:9  109 W. Thus Hawking's evaporation is only signi®cant for small black holes (miniholes), but these could have been formed early in the initial evolution of the Universe and hence ought to have been evaporated in the intervening aeons. Hawking radiation e€ect is a quantum e€ect, which does not ®t with the classical point of view because an object (the black hole) at a temperature TBH transfers energy to one at a lower temperature (i.e. to VSM). From a classical point of view, things can only go into a black hole, but nothing can go out. Usually one asserts that Hawking radiation provides a mechanism for the black hole to radiate energy into the vacuum [9]. In fact, nothing is obtained from the black hole itself. We consider that Hawking radiation is a VSM radiation achieved with the help of the strong gravitational ®eld of a black hole. Without the VSM around it, a black hole cannot evaporate and the Hawking radiation cannot exist [10]. Thus, the existence of a Hawking radiation would be a direct evidence that spacetime has the structure of a random fractal described by quantum vacuum ¯uctuations. Cosmic strings can also generate a Hawkingtype radiation due to their conical spacetime. Unruh e€ect. In a 4D tensorial formulation of physical laws, both `®ctitious' (inertial) forces and gravitational forces appear as Christo€el symbols in the equations of motion. This represents in fact the mathematical expression of Einstein's principle of local equivalence between gravitational and inertial forces. An immediate consequence of this equivalence is that one can consider the Hawking emission even in a ¯at (Minkowski) VSM when quantum ®elds are described in terms of accelerated observers. Indeed, there is a close similarity between the Hawking radiation and the presence of radiation quanta (Unruh e€ect) in an accelerated ¯at vacuum, i.e. a vacuum referred to an accelerated frame of reference. Most of the authors who study this radiation which appears as a result of an `acceleration of space' apply, for the sake of simplicity, the accelerated Rindler coordinates (s, n, instead of Minkowski coordinates (t, x)): 11

10

We emphasise that Hawking radiation (photons, neutrinos, gravitons, etc.) may be explained in a similar way to the particle creation and vacuum polarisation by the electric ®eld of charged particles. Now the role of the electric ®eld is played by the strong gravitational ®eld of a black hole. If a pair of particle-antiparticle is created in a highly non-uniform gravitational ®eld and the horizon happens to pass between the pair of particles as they separate, and in such a way that one (e.g. the particle) goes up and the other (e.g. the antiparticle) goes down, then they might experience great diculty in getting back together again for an annihilation. The strong tidal forces will tend to pull them apart and from a state of virtual particles they become real particles. We recollect that gravity manifests itself only as an attractive force between two bodies. However, if the bodies are subject to an external non-uniform gravitational ®eld, the tidal forces may yield a growing separation of the two bodies, as in the case of repulsive forces. Thus, a particleantiparticle pair is ``separated'' by the action of an exterior electric ®eld (vacuum polarisation and pair creation) but it can be also separated by strong tidal ®elds (Hawking radiation). 11 Rindler coordinates can be also de®ned by t ˆ …exp…an†=a† sinh as, t ˆ x ˆ …exp…an†=a† cosh as, where a is some positive constant.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

x0 ˆ t ˆ n sinh s;

7

…4†

1

x ˆ x ˆ n cosh s;

…5†

2

02

…6†

3

03

…7†

x ˆx ; x ˆx ;

which lead to the simplest example of a metric with a horizon (see Fig. 1). Of course, a true physical theory must be covariant in the sense (of General Relativity) that the studied phenomena must be considered with respect to arbitrary systems of references (coordinates).Unruh e€ect involves, in fact, a detector which is accelerated through Minkowski VSM and detects radiation with a thermal spectrum corresponding to a (vacuum) temperature given by the following equation: TU ˆ

had ; 2pck

…8†

where ad is the acceleration of the detector. We noted above that the Hawking temperature is very small for black holes of astronomical size. Similarly, the Unruh temperature is very small for usual accelerations. For example, even for a small temperature of T ˆ 1 K the Unruh corresponding acceleration is ad ˆ 2:4  1020 m sÿ2 . What practical detector can resist such a huge acceleration? However, some particles (photons, neutrinos, etc.) which are emitted by the nuclei and atoms may experience such huge acceleration in the ®rst moment of their birth, and thus they can be used as Unruh detectors. Sometimes physicists wonder if the Unruh e€ect has to remain a purely theoretical e€ect, or whether the vacuum temperature may be observed in real experiments. Finally, we o€er a new result, namely a new veri®cation of Einstein's principle of equivalence at the level of quantum ®eld theory in curved spacetimes. The local (Newtonian) gravitational acceleration at a point on the event horizon of a black hole is given by aˆG

M R2BH

…9†

If we substitute this acceleration in the Unruh formula (8) we obtain TU ˆ

h hc3   ˆ TBH ; 2pG…M=R2BH †ck 8pkGM

…10†

which coincides precisely with the Bekenstein±Hawking formula for the temperature of a black hole. This unexpected result should create the possibility to prove the existence of a Hawking-type radiation even under laboratory conditions and this might lead also to a new source of energy. This new result will be analysed and developed in a separate paper where we intend to compare this phenomenon with the `old problem' of radiation of electric charges with respect to accelerated observers [12]. 3. Some fancy language in the physics of gauge ®elds Fundamental advances are achieved if the `innocent' conventional Lagrangian approach is `desecrated' by adopting a negative mass term: 2

…mass† ˆ m2 ˆ ÿl2 ;

l2 > 0:

…11†

Starting with a simple real scalar (Higgs) ®eld /, vacuum is no more trivial, / ˆ 0 is no more the minimum energy state but / ˆ /0 is (see Fig. 2). Thus the unique (trivial) vacuum (/ ˆ 0) is split into two distinct vacua (two distinct vacuum expectation values, VEVs). The Lagrangian is symmetric with respect to / ! ÿ/ (Z2 symmetry), but neither of the vacua respects this. In other words, the vacua are degenerate. When the vacuum does not respect a symmetry of the Lagrangian, the symmetry is said to be spontaneously broken. This particular situation is an example of a spontaneously broken discrete symmetry. If we step back and look at the two vacua, which one is more physical than the other or are they the same? Does the physical system pick /0 or ÿ/0 ? How about nearby systems?

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Fig. 1. Acelerated Rindler coordinates (s, n). For x > jtj, n > 0 (I region or RRW ± right Rindler wedge), for ÿx > jtj, n < 0 (II region or LRW ± left Rindler wedge). The ¯at (Minkowski) space is `hot' with respect to the `Rindler accelerated space' in which the ground state of the Hamiltonian is de®ned in either the RRW or in the LRW. An accelerated point detector, following a path with n ®xed is heated due to interactions with VSM [11]. Fig. 2. Higgs ®eld of an in®nite domain wall in the x±y plane at z ˆ 0 (continuous line curve). It is possible to have one region in which the VEV h/i ˆ /0 holds and another nearby one in which we have h/i ˆ ÿ/0 . If so, in between there must be a domain of false vacuum where h/i ˆ 0. This is formed by the domain wall (continuous line curve). Domain walls of this type can occur whenever SSB results in two or more adjacent domains having di€erent vacuum states. The analogy with magnetic domains is quite close. If such walls appear on a cosmic scale, their thickness d would be negligible with respect to their area, so locally such walls have a sheet-like con®guration. The energy density as a function of z (dashed thick line curve) has the maximum value at / ˆ 0 and minimum values at / ˆ /0 . We note here some similarities with an anomalous dispersion of waves at points where the absorption of energy is maximum. Possibly, this type of dispersion can also invoke an SSB. A curved wall of mean radius Rw  d experiences a force per unit area of pw  r=Rw and, thus, isolated, closed walls or kinks in in®nite walls will collapse, losing their energy by the radiation of particles. However, walls that stretch right across the Universe (horizon) can survive and cause a large local density ¯uctuation and, possibly, an anisotropy of the microwave background radiation.

If two neighbouring systems end up choosing opposite ones, there will be a (domain) wall (of false vacuum where h/i ˆ 0) separating them (Fig. 2). Why do we go curting so much with scalar ®elds? Why do we need them? A complete answer for these questions has yet to come. However, what we know now with certitude is that scalar ®elds are used `to assign a mass to the elementary particles'. We always have in mind that particles were born with a mass. On the other hand, the only thing that the scalar ®eld does is to change the energy density of the vacuum. SSB ®lls the vacuum with a (constant) ®eld /0 . Everybody knows that when we construct ®eld theories of interactions, they must satisfy some local gauge principle. Maxwell's theory results merely by demanding the following: A person in Washington (W) may choose a particular phase for the wave function of an electron, while somebody in Prishtina (P) may make another choice. If the physical theories constructed in W and P have to be the same (i.e. invariant to a local choice of `gauge'), it can be achieved only at the price of introducing a ®eld, which for the simplest case turns out to be electromagnetic ®eld. A local gauge invariance represents the invariance of a theory (e.g., Schr odinger equation) under the (gauge) group of transformations w ! eih…x† w. If h is independent of x, it is called a global U(1) group. It is Abelian, i.e. di€erent transformations of the group commute with each other, and it is one-dimensional, i.e. the transformations are speci®ed by one parameter h. In other words, U(1) is the group of unitary transformations (the group of all 11 matrices or complex numbers of modulus 1) in one dimension and the phase-factor function eih forms a 1D representation of it. Electromagnetism is called an abelian gauge U(1) theory and contains only one kind of charge, the electrical charge. The interaction (mediating) particle (photon) has no charge, and equations are linear in vacuum. The four-potential Aa does not appear as a fundamental physical

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9

quantity. The global U(1) gauge invariance is also called gauge invariance of the ®rst kind and is associated with charge conservation (oa J a ˆ 0). In the case of global gauge invariance the phase h is not measurable and can be chosen arbitrarily, but once chosen it must be adhered to for all times and points in space. Electrodynamics is also locally (abelian, local U(1)) gauge invariant (or it possesses a local gauge symmetry, or gauge symmetry of the second kind in which h ˆ h…x†) because all derivatives occur in special combinations, Da ˆ oa ÿ ieAa . These combinations express the minimal coupling of the scalar and the gauge ®eld and are called `covariant derivatives', where Aa is the four-vector potential of the photon (a gauge boson, the simplest example of a gauge ®eld). Generally, gauge ®elds mean vector particles which mediate the interactions. The generalisation to non-Abelian transformation is simple in the global case but fairly complex in the local case. The simplest non-Abelian global invariance is manifested by the isospin where the ®elds are grouped into multiplets of the form 0 1 /1 B /2 C B C B
C C …12† /ˆB B
C; B C @
A /n which constitutes a basis for the representations of the isospin group SU(2) involving rotation in an isospin (internal) space. The gauge transformation is speci®ed by three parameters h ˆ …h1 ; h2 ; h3 † and the three (non-commuting) generators (n  n matrices), L ˆ Lj …j ˆ 1; 2; 3†, of the SU(2) transformations. In the particular case of an isodublet, for example a proton and a neutron, n ˆ 2, and L ˆ 12s, where sj are the Pauli matrices. The number N of generators is 3 for SU(2) (weak interactions), 8 for SU(3) (strong interactions or quantum chromodynamics, QCD), etc., and for each case of global gauge invariance of the Lagrangian, one can, as for the Abelian (electromagnetic) case, show the existence of N conserved currents. Thus, in the non-Abelian case there exist several kinds of charges, mediating particles have charges, equations are non-linear even in vacuum, and Aa is fundamental. A much more complex and subtle case is represented by the non-Abelian local gauge invariance in the context of Yang±Mills theories. The ®rst generalisation of SU(2) (or of any group with a ®nite number of generators) to locally gauge invariant Lagrangians is due to Yang and Mills [13]. The idea is to introduce as many vector (gauge) ®elds Aja …x† which are the analogues of the photon ®eld Aa , as is necessary in order to construct a `covariant derivative' and a Lagrangian which is invariant under local gauge transformations. A major new feature of Yang±Mills ®elds is that, unlike the photon case, the non-Abelian gauge ®elds Aja …x† are self-coupled through non-linear terms which appear in the Lagrangian. We parenthetically mention that a new class of solutions to the classical Yang±Mills ®eld equations, the so-called (Polyakov) instantons, may have an important e€ect in determining the structure of the vacuum in a quantum ®eld theory. Because there is a one-to-one correspondence between the dimension of the group and the number of massless gauge ®elds Aja …x† that are necessary to satisfy gauge invariance, and because the only known massles vector boson is the photon, it would appear as if non-Abelian local gauge symmetries represent an elegant theory which has very little to do with physical reality. In other words, Yang±Mills theories with non-Abelian local gauge invariance have a high degree of symmetry and may be renormalizable, but have an apparently incurable `defect': they contain a large number of massless gauge vector bosons which are not found in nature. However, there appeared a `miraculous solution' to this problem: SSB which in the case of elementary particles leads to the Goldstone theorem and the Higgs phenomenon. At the beginning of this section we noted that there exist dynamical systems in which the ground state does not possess the same properties as the Lagrangian. When this happens, one ®nds that there appear massless scalar bosons, the so-called Goldstone bosons (GBs). For every broken generator in an SSB there exists a GB (Goldstone theorem, or Goldstone phenomenon). In other words, appearance of a massless scalar boson is a `natural' consequence of the spontaneous breakdown of a continuous symmetry (e.g. the group SU(2) of rotations in a `phase' plane). This Goldstone phenomenon seems to be useless for our e€ort because we are looking to attach masses to massless particles and ended up creating new massless particles. At this point there appeared the Higgs mechanism. In a local gauge theory involving massless vector ®elds

10

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and scalar ®elds, the would-be GBs disappear, but reappear disguised as the longitudinal mode of the vector ®elds, which thereupon behave like massive vector bosons with, for example, three spin degrees of freedom. Thus, the unwanted massless vector bosons of the gauge theory are replaced by heavy vector mesons in agreement with, for example, the phenomenology of the weak interactions. This is the celebrated Higgs mechanism. With the exception of the photon all the vector particles (mediating interactions) have to be given masses, sometimes large ones (W  , Z 0  100 GeV, 100 times heavier than a hydrogen atom). 12 In other words, Higgs mechanism expresses the following `physical miracle': A cooperation between the quanta of the gauge ®eld (W  , Z 0 ; . . .) and the massless GB leads to: · the disappearance of the unwanted GB, · the gauge ®eld quanta acquire a mass. ``Kill two birds with one stone: The massless 'photon' ate the Goldstone and became fat'' said Mahajan [15]. Of course, the Higgs mechanism can be applied in Abelian as well as non-Abelian cases. SSB represents a ubiquitous phenomenon. For instance, a pencil balanced on its tip shows a rotational symmetry. It looks the same from every side. However, when it falls it must do in some particular direction, breaking the symmetry. Possibly, the masses of the W  and Z 0 (and, also, of the electron) are generated through an analogous mechanism. We can imagine that there are `pencils' or a `®eld of pencils' (i.e. Higgs ®elds, HFs) throughout (vacuum) space. 13 These pencils are coupled together, so that they all tend to fall in the same direction of the (internal) `phase space'. The presence of pencils (HFs) in vacuum space in¯uences waves (particles) travelling through it. The waves (particles) have of course a direction of motion in (con®guration, i.e. every day) space, but they have also a `direction' in a `phase space'. In some `phase directions', waves have to `move' the pencils (HFs) too, so that they become more `sluggish'. These waves (particles) represent the W  and Z 0 particles. Of course, there also exist `pure pencil waves' in vacuum space, waves in the pencils alone, where they are bouncing up and down. In the context of a Higgs ®eld, these waves represent the Higgs particles (Higgs bosons). Hence, Higgs ®eld and Higgs particle lead to new features in the understanding of a fractal structure in vacuum space. In other words, if we try to elaborate a theory of elementary particles and their interactions, the simplest approach is to consider that initially all masses of fundamental particles are zero. Then we assume that the whole of space is permeated and ®lled by a Higgs ®eld. If particles moving through the Higgs ®eld interact with it, they acquire a mass. This (Higgs mechanism) is similar to the action of viscous forces on particles moving through a viscous ¯uid: the larger the interaction of the particles with the ®eld, the more mass they appear to acquire. The Higgs boson represents the particle associated with the Higgs ®eld. The Higgs boson has only mass, and no other characteristics, such as electric charge, that distinguish particles from empty space. This picture coexists with a quantum mechanical approach describing creation and annihilation of elementary particles, as observed in accelerators. Higgs ®eld is a non-zero ®eld which is, however, associated with the lowest energy state of this ®eld (in empty space). Thus, Higgs ®eld represents a collection of zero-energy Higgs particles. The mass (or inertia or resistance to change in motion) of a particle arises from its being `grabbed' by Higgs particles when we try to move it. Unfortunately, the Higgs particle has not yet been observed. Today we can only say that if it exists, it must have a mass greater than 80 GeV=c2 . Finally, we note that SSB is related also to the phase transitions (PT). We shall show in subsequent sections how these analogies SSB±PT are contained in the evolution of the structure of ordinary matter in the Universe. We mention only here that SSB involves a transition from an ordered state to one which is less

12 These particles (W  , Z 0 ) have been discovered at CERN in 1983. However, the origin of their masses remains mysterious. The best proposal is the `Higgs mechanism', but, unfortunately, this aspect of the theory remains untested [14]. 13 Physicists conceive also the Higgs ®eld being `switched on', pervading all of space and endowing it with a `grain' like that of a plank of wood. The direction of the grain is undetectable, and only becomes important when the Higgs' interactions with other particles are taken into account. For instance, particles called vector bosons can travel with the grain, in which case they move easily for large distances and may be observed as photons; or against the grain, in which case their e€ective range is much shorter; these particles are W  or Z 0 particles. When particles such as electrons or quarks travel through the grain, they are constantly ¯ipped `head-over-heels'. This forces them to move more slowly than their natural speed, that of light; this induces a heaviness them. Possibly, the Higgs ®eld is responsible for endowing all the matter we know with mass [16]. At this point we observe a strong similarity between `Higgs' principle' and Mach principle.

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ordered. The superconductivity state emerges because a Cooper pair formation destroys the symmetry. When the temperature is raised, the condensate (Cooper pairs) shrinks to zero and the symmetric state is restored. At very high temperatures, the symmetry between weak-electric and strong interactions is restored. 4. ``Standard Model'' in particle physics and ``Standard Model'' in cosmology In the past ®fteen years it has become clear that there exists a developing interrelation between elementary particle physics and cosmology. The Standard Model 14 of particle physics has a great success in explaining the uni®cation of the weak and electromagnetic interactions at energies above 250 GeV ( 1016 K). However, Grand Uni®ed Theories (GUTs) which attempt to unify further the strong and electroweak interactions require higher and higher energies of the order of 1015 GeV. Such high energies may have arisen in the early Universe, just 10ÿ35 s after the Big Bang. Thus, for the time being, the Universe represents a unique laboratory for investigating the uni®cation of fundamental interactions [17] (see, also Fig. 3). In the context of cosmology, the Standard Hot Big Bang Model or Standard Model of an Expanding World appears to describe correctly (i.e. in agreement with astronomical observations) the history of the Universe from 10ÿ2 s after the Big Bang when the temperature was 10 MeV until today (i.e. approximately 15 billion years later) when the temperature is 2.75 K ( 10ÿ10 MeV). Once the photons have cooled, so that most have an energy very much less that the ionisation energy of hydrogen (13.6 eV  1:6  105 K), the latter will be unable to absorb them and thus matter will become transparent to radiation. This situation de®nes the so-called decoupling temperature, Td  3  103 K:

…13†

Following this state the photons continue to cool during the expansion of the Universe. However, since the photons do not interact (e€ectively) with matter, any cosmological redshift preserves their pure black-body (Planck pro®le) spectrum at all wavelengths. This radiation background has now the maximum of intensity in the microwave region and involves presently a temperature of only Tpresent ˆ 2:75  0:05 K. Thus, it has been reasonable to state that the 2.75 K background is the leftover (remnant) of the Big Bang, it is a relic or fossil radiation record of the Universe from the time when matter and radiation decoupled (100,000 years after the Big Bang). In other words, this cosmic microwave background radiation (CMBR ˆ CMB) represents a faint glow remaining from the period when the primordial plasma of the Big Bang condensed to hydrogen and helium gas. This period, when CMBR was scattered for the last time and the Universe was very homogeneous, is known as the recombination period. According to the usual cosmic chronology, before the recombination period, a transition from a chaotic state to a more uniform situation could have occurred at t  tequiv , near the equivalence between the matter and radiation densities. The number density of relic photons was found to be very large, nc;relic  400 cmÿ3 , larger by two orders of magnitude than any other form of background electromagnetic radiation, and very much larger than the average baryon number density of nB  10ÿ7 cmÿ3 . However, the present photon energy density, qc;relic  5  10ÿ34 gm cmÿ3 , represents only about 10ÿ5 of the average (critical) density of the universe (qc  10ÿ29 gm cmÿ3 ) and is very much smaller than the current qB (mB ˆ mp  1:7  10ÿ24 gm). If we ignore radiation from all the localised hot patches in the Universe, the relic radiation displays an intensity and temperature which are very nearly homogeneous and isotropic, and with no identi®able source [18]. The only observed anisotropy (detected in 1977) is a so-called dipole anisotropy generated by the Doppler shift due to the motion of the Earth through the radiation, with respect to the last scattering

14

Even if the Standard Model represents the particle physicists' current picture of the fundamental structure of matter, it incorporates a major obstacle generated by the application of QFTs. These theories predict that all physically observable quantities such as charge, mass, etc., are in®nite in the sense that they allow any given particle to spend part of its life as another particle-which can, in turn, sometimes exist as still another particles, and so on, ad in®nitum. The resulting loop corrections (e.g., the vacuum polarisation correction involving an e‡ eÿ loop) add up to divergent values for the particle's charge, mass, etc. However, for example, quantum electrodynamics (QED), via its gauge-invariant property which ensures charge conservation, is considered a renormalizable theory in the sense that the divergences can be rescaled so that the physical quantities become ®nite.

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Fig. 3. Deterioration (at lower energies or temperatures) and restoration at high temperatures in the state of symmetry lead to dynamical changes, the evolution of the Universe and the diversity of the physical world.

surface (lss) 15. From this point of view, the relic background radiation provides an absolute frame of reference. At this point, we parenthetically note that an anisotropy detected in CMBR was also interpreted by a kind of a new ether drift [19]. The incredible isotropy of the Universe is demonstrated also by the deviation in the temperature, DT K 3  10ÿ5 ; …14† T on any angular scale greater than a few seconds of arc of sky. Besides the dipole, anisotropy has been detected by the Cosmic Background Explorer (COBE) satellite, also at a very low level, DT =T  10ÿ5 , and at large angular scales h P 7 . In particular, a quadrupole anisotropy, corresponding to an angle h ˆ 90 , appears very low. Hence, the large-scale Universe appears to be spatially very smooth, homogeneous and isotropic (Einstein's cosmological principle). However, this assertion is challenged because when we look at the night sky we see no evidence for isotropy and uniformity but rather we see lots of structure. Thus, a question arises: If the Universe started o€ as isotropic as is recorded by the relic radiation, what were the causes which generated this structure? Astronomical observations (e.g., with the Hubble Space Telescope, Infrared Astronomical SatelliteIRAS, European Space Agency's Infrared Space Observatory-ISO, etc.) display that, within the framework of the large-scale structure (LSS) of the universe, galaxies and clusters of galaxies appear to be located on the apparent surface of huge (empty) void bubble-like structures nested together, forming a froth-like structure. This structure suggests that there might originally have been an underlying domain structure with domain boundaries that existed in the distant past. This represents a reasonable idea, for example, because there have been two (or more) periods of in¯ation. First in¯ation produced the spectrum of small-amplitude ¯uctuations on large scales, and the second generated the large-amplitude ¯uctuations on smaller scales that have caused the formation of the galaxies themselves. Presently, voids seem to be the most prominent characteristic of the LSS of the universe and a common feature of the galaxy distribution [20,21]. It is important to emphasise that not only do large voids exist, but they occur frequently and form a closely packed (compact) network of voids ®lling the entire observable universe. Voids have a diameter up to 50 Mpc (150 million light-years) 16 and are not completely empty. Some (faint or void) galaxies may exist within the voids. Galaxies are located at the intersections of these void bubbles, along ®laments which have a length up to 200±300 million light years. On such scales the Universe cannot be considered as smooth. In modern astrophysical terms, one can assert that the essential presence in the LSS of the Universe are voids, i.e. under-dense regions (generally ellipsoidal in shape) surrounded by walls and ®laments which are over-dense regions. Generally, the walls are thin 2D structures characterised by a high density of galaxies. Walls may be considered as boundaries between voids. Over large scales, the walls appear to be coherent, but on a small-scale they are subject to clustering, are not homogeneous and contain small breaches. Galaxies within walls are called wall galaxies. Other (non-wall or random) galaxies are called ®eld galaxies and they appear in the regions where the galaxy distribution is sparse. Of course, a faint (void) galaxy

15

CMB photons arriving at any given moment on the Earth, began their travels at approximately the same time and distance from the Earth. Thus, their starting points form a sphere, called the last scattering surface, with the Earth at the centre. 16 1 megaparsec ˆ 1 Mpc  3.26 light years.

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represents a sub-population of a ®eld galaxy. Walls and ®laments form the so-called wall-®lament skeleton and contain most of the galaxies. Thus, almost all ( 95±96%) of the galaxies are contained in the same single skeleton structure. Most of these over-dense regions (walls and ®laments) are narrower than 10 hÿ1 0 Mpc, where h0 is in the range 1=2 to 1 and incorporates the observational uncertainties. Large voids occupy approximately 50% of the volume, while walls occupy less than 25% of the volume. 5. Large scale structures and cosmic strings What is the physical mechanism which can explain the large-scale structures of the Universe? This question is presently still under debate and represents precisely the main subject of our work. A suggestion is that the voids are formed gravitationally from primordial under-dense regions; voids grow gravitationally and large voids in the galaxy distribution correspond to real voids in the matter distribution [22]. In this model, the observed bulk motions of voids involve complex non-linear e€ects in the evolution of the underlying matter distribution which may also a€ect the galaxy distribution because the voids are delineated by galaxies. However, the bubble-like structures are generated by processes that occurred within the ®rst moments of creation, few 10ÿ35 s after the Big Bang. This is the reason why many cosmologists consider that there has not been enough time for a gravitational ®eld alone to attract galaxies into these special locations. Possibly, this observed structure where the galaxies are clustered along ®laments and walls of bubbles with large void regions in between suggests that galaxy formation might not have resulted simply from the collapse of random density ¯uctuations but has been triggered by some underlying pattern. Indeed, the earliest moments of creation represent also the era in which the cosmic or cosmological strings and domain walls could have been born. Can they be considered as candidate seeds for the observed structure of the Universe? Are ®laments and wall galaxies simply e€ects of condensations along cosmic strings or domain walls? We recall that uni®cation of the four (gauge) interactions (forces) of nature is mediated by phase transitions (PT) during which a particular symmetry, which previously uni®ed the interactions at a high temperature (energy), is broken when the temperature (or energy) decreases below a critical value of Tc (or Ec ), leaving behind separate (``distinct'') interactions. For example, the critical energy for electroweak interactions is Ec ˆ 250 GeV. Above this energy there exists only one unique electroweak interaction, but below this energy there occur two distinct interactions: electromagnetic and, respectively, weak. Similarly, Ec ˆ 1015 GeV is a critical energy for the GUTs scale. Above this energy, the electroweak and strong interactions are uni®ed by a special symmetry, whereas below this energy the symmetry is broken (i.e. it becomes a defect as in condensed matter physics) and the two interactions appear as di€erent. At the cosmological level, as the Universe expands and cools, the GUT transitions may generate lowenergy regions of space (with separate interactions) trapped in the old high-energy phase (where the interactions are still uni®ed), surrounded by the new low energy or broken phases (defects). Such topological defects (produced in a transition from a symmetric to a broken-symmetry phase) can be of dimension zero (point-like topological defects or monopoles), of dimension 1 (string-like or linear de€ects or cosmic strings), and of dimension 2 (sheet-like defects or domain walls). Cosmic strings appear to be key-candidates for seeds leading to the formation of structure in the Universe. They are very thin (with diameters  10ÿ29 cm) and possess a huge tension ( 1036 Newtons). The total mass of string per unit length, qs , is about 1022 g cmÿ1 , or 107 solar masses per light-year. 17 For topological reasons, cosmic strings must be either in®nite or closed in an open universe. 18 However, there exist some theories in which a string can terminate at a magnetic monopole.

17 Unlike cosmic strings, in our 4D world, a 10D (super)string might be as small as 10ÿ20 the size of an atomic nucleus. The superstring theory asserts that the remaining 10 ÿ 4 ˆ 6 extra-spatial dimensions are shrunken or compacti®ed. The ``ripples'' along the compacti®ed dimensions de®ne the excitations of new entities, D-branes which can encode more and more information from fundamental particles to black holes. 18 In a closed universe all strings are closed loops and the role of in®nite strings is played by the loops with the same order of size as the entire Universe.

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In®nite strings contribute approximately 80% of the total string density. At birth (i.e. some 10ÿ35 s after the Big Bang) cosmic strings occurred copiously in a random arrangement as a tangled mesh (a `web network') in which the greatest part of them represents long in®nite strings winding their paths across the Universe, and the rest are closed loops. During the expansion of the Universe, the long cosmic strings, under the action of tension, move with high velocities (aproximately the light velocity) and frequently intersect each other, stretch out and chop o€ string loops (i.e. closed strings). Because of the random motion and high velocity of strings there occur lots of intercommutation processes (i.e. an interchange of their positions or change partners, Fig. 4a) and (closed) string loops (see Fig. 4c±e). An intersection of a long string with a loop may also be interpreted as a `change partner' (see Fig. 4b) as well as a breaking of a bifurcation of the Arnold type [62]. String loops may also chop themselves up further or reconnect onto the long strings. The e€ect of this process is that the initial large density of in®nite strings diminishes as closed strings are formed. The high tension of in®nite strings deforms the spacetime around them, transforming a ¯at spacetime into a conical spacetime (see the following sections), the string passing through the vertex of the cone. This distortion of space conditions particles to move with the string and to be attracted behind it into wakes. It has been suggested that matter attracted into such wakes could be the origin of the LSS apparent in the Universe. Indeed, in these wakes, where there exists now an excess of matter, the gravitational forces bind particles to one another leading to more and more massive bodies (galaxies, clusters, etc.). Matter, both luminous and dark, which is attracted into these wakes, leaves behind empty spaces (bubble voids) (see Fig. 5). Also, oscillating string loops can operate as seeds for galaxies and clusters of galaxies. The conventional (classical) scenario of galaxy formation assumes that the density ¯uctuations arise in the form of waves with random phases, while in the string scenario they are essentially non-Gaussian. String loops have sizes much smaller than those of the galaxies formed around them. Even if a string loop leads to a small density ¯uctuation on the galactic scale, in its immediate neighbourhood it generates a large density perturbation. This causes an accretion of matter around the loops and formation of massive compact objects, which can

Fig. 4. (a) `Change partners' by intercommutation of long strings. (b) `Change partners' by intercommutation of long and closed strings. This process may also be interpreted as a breaking of bifurcations (or of a symmetry) formed by an intersection of two strings. (c) Closed loop formation by intercommuting strings. (d) Pair of strings intercommuting at two points and forming closed loops. (e) Loops can be formed by self-intersection of individual strings. Such processes are important, since loops eventually loose their energy through radiation and `save' the Universe from a domination by strings. (f) A domain wall intercommutes with a wall bounded by strings (cross section).

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be identi®ed as quasars 19 or active galactic nuclei (AGN). On larger scales, rare giant string loops can be the cause of the observed cluster±cluster correlations. What we exposed above represents a short summary of the fascinating scenario of cosmic strings and LSS of the Universe. But, however attractive this theoretical model may be, it is of little use if it does not lead to some observational con®rmations. Fortunately, cosmic strings, even if they occurred at a very early stage of the Universe (some 10ÿ35 s after the Big Bang), may yield the following observationally veri®able signatures. · The e€ect of a long cosmic string which passes through an object, e.g. an observer. We can assume that the velocity of the cosmic string vs  c. The passage of a cosmic string produces no immediate redistribution of material in the observer, because space round the string is ¯at (but with a wedge of opening-de®cit angle U  8pGqs =c2 taken out) and no tidal force exists outside a very narrow coherence length of the string. However, the de®cit angle U leaves the two sides of the observer approaching with the velocity va ˆ vs U  1 km sÿ1 . This is very dangerous to an observer (an apparatus or a person). · Cosmic strings oscillate rapidly, approximately with the speed of light and, as a consequence, the string loops radiate gravitational waves. This radiation is superposed on a stochastic gravitational wave background having a scale-invariant spectrum [24] Xg ˆ

p x dwg  10ÿ4 Gqs ; qc dx

…15†

where wg is the energy density of gravitational waves and qs is the total mass of string per unit length. This equation applies within a wide range of frequencies, 10ÿ2 yearsÿ1 K x K 105 sÿ1 :

…16†

String loops radiate, shrink and possibly disappear. The detection of such a gravitational radiation could con®rm the existence of cosmic strings. For example, gravitational waves of a string would produce apparent ¯uctuations in the observed frequencies of pulsars. · Because a cosmic string distorts strongly spacetime, it may act as a gravitational lens (see Fig. 6). A string acting between Earth (observer) and a faraway galaxy will generate two images of the galaxy due to the bending of light rays passing on either side of the string. The angular separation between the images is given by a ˆ 8pGqs

L sin h; D‡L

…17†

where L and D are the distances from the string to the galaxy and to the observer, respectively, and h is the angle between the string and the line observer-galaxy. A chain of pairs of images of a line of galaxies would be a strong signature of a string, indicating the lens is long and thin, and not pointlike. In the case of a point-like gravitational lens, one obtains a pair of images or an Einstein ring (see Fig. 6a). · Cosmic strings should provide a very convincing evidence of their existence by a breaking up of the homogeneity (uniformity) and isotropy of the CMB. Indeed, a string moving rapidly through relic radiation would heat up slightly the microwave in the wake of the string, and cool it in front of the string. The relic radiation would appear roughly 10ÿ4 of a degree hotter on one side of the string than on the other. Such a small jump may soon be detectable. Photons are blue shifted in the wake. In other words, the background temperature should display step-like discontinuities on curves on the sky. Maps of the relic radiation 19 Generally, a quasar, which shines 100 times brighter than the galaxy itself, represents, possibly, a massive black hole (BH), and matter falling into the hole (accretion of mass) causes the quasar to shine. In other words, a quasar may be featured in a small BH ! large BH transition. It is a `small' astrophysical object if we consider that, for example, a black hole of a million solar masses could be only as large as the sun. However, quasars are the most luminous objects we know about in the Universe. They act like ¯ashlights illuminating the material between us and the farthest parts of the Universe. It appears that there is no quasar in our own galaxy, but possibly a small dead quasar. At the centre of our Milky Way galaxy (M ˆ 7  1011 M and R ˆ 15 kpc) it appears that there exists a BH with m  3  106 M , as indicated by infrared and radiofrequency observations [23]. Energetic considerations suggest that the lifetime, Tquasar , of a quasar is shorter than 109 years and thus the maximum distance is  cTquasar 6 2000 Mpc. Quasars mark a stellar appearance on photographic plates. The ®rst members of this class of astrophysical objects were discovered through the optical identi®cation of extragalactic radio sources and hence the origin of the name `quasi-stellar radio source' abbreviated to quasar.

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Fig. 5. An imaginative picture of the evolution of a cosmic string network in a box `cut' from the event horizon of the visible Universe. It appears that there exists a large number of small loops and few in®nite strings stretching across the box. Fig. 6. Gravitational lensing: (a) A point-like (or spherical mass) lens M generates an Einstein ring of images if the observer, the lensing mass, as also the source (e.g., a quasar) are on the same axis of symmetry. (b) An open string (or a large closed loop) generates by lensing a chain of pairs of images of a line of galaxies.

should display temperature jumps on a line tracing the position and shape of the (otherwise invisible) string. The temperature ¯uctuation is given by dT  8pGqs vs ; T

…18†

where vs  1 (in c-units) is the velocity of the string. Present observational limits indicate that Gqs K 10ÿ5 . · A new direction of research in the ®eld of cosmic strings was launched by Witten who introduced the socalled superconducting current-carrying strings [25]. Initially, it appeared that cosmic strings can manifest themselves only through their gravitational ®eld. However, in some GUT models cosmic strings can behave as superconducting wires and can display also very strong electromagnetic interaction with, for instance, a magnetized cosmic plasma. It is estimated that very large curents up to 1031 A can be carried by superconducting cosmic strings [26]. The symmetry group of the false vacuum inside the string is di€erent from the symmetry group of the true vacuum outside. Thus, an electromagnetic gauge invariance can be broken inside the string, and spontaneously broken gauge invariance leads to a state of superconductivity. When a string is moving through plasma, it is subject to a frictional force which leads to an energy loss. Most of this energy is spent to heat up the plasma, but a small fraction of it is radiated away by ultra-relativistic electrons accelerated at the shock front. 20 This synchroton radiation can make the cosmic string visible in the radiowave range. · Vibrations of current-carrying cosmic strings generate extremely powerful bursts of electromagnetic radiation. The pressure of such an intense radiation could push away and heat up the surrounding matter generating expanding spherical shells of gas, and blow cosmic bubble-like voids in space. This would explain why galaxies look as if they were forming around the edges of cosmic bubbles. Such huge bursts of electromagnetic energy could also be detected as bursts of X-rays which appear to emanate from ring 20 Charged particles are de¯ected by the strong magnetic ®eld of the string, and a shock front will be formed at a distance from the string where the magnetic pressure is balanced by the dynamical pressure of the plasma.

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17

structures (around the string) far away in space. When these rings collapse, they can release bursts of elementary particles, which can explain the origin of the highest-energy cosmic rays. On the other hand, on the low-energy level, it is possible that the line-like radio sources that have been detected at the centre of the Milky Way could have been generated from low-energy cosmic strings. · A suciently large string loop may end up by undergoing a runaway collapse to form a blackhole. · Even if cosmic strings (as linear topological defects of the vacuum) were formed at the GUT scale (i.e. 1015 GeV), their density would be too low to generate a cosmic catastrophe. This situation can arise in the case of Goto±Nambu type strings, which radiate away their energy and ultimately disappear. However, the generation of stable superconducting currents in strings can lead to a stabilisation of string loops. Indeed, the (timelike or spacelike) current breaks the Lorentz invariance along the string worldsheet and gives rise to a rotation. The centrifugal e€ect of this rotation may balance the tension and stop the contraction so that the string loop becomes a stable equilibrium con®guration known as a vorton. The energy of a non-conducting string distribution decays like that of a radiating gas, while a distribution of relic vortons manifests itself like usual matter [27]. Because of its stability, the vorton distribution may dominate the Universe. · It is interesting to note that (superconducting) vortons may have also been formed close to the electroweak phase transition (250 GeV) [28] and thus such cosmic vortons might eventually be generated in the laboratory. If discovered through astronomical observations, cosmic strings should provide a clear evidence for GUTs. Cosmic string is a subject of considerable interest to physicists and mathematicians. Not only are they fascinating on their own right, but there is also a growing belief that they may play a fundamental role in cosmology. What we exposed above can be called a cosmic string scenario of the LSS of the Universe. On the other hand, the idea that the galaxy space distribution can be a pure scale-invariant fractal or self-similar distribution raises an increasing interest in modern cosmology [29±31]. In the context of astronomy, the original name for a fractal distribution is an (unbounded) clustering hierarchy in which stars group into galaxies, galaxies (on a smaller scale) into subclusters, subclusters into clusters, clusters (at a larger scale) into superclusters, and so upon extension to very large scales. Which of these two scenarios (string and fractal scenarios) have been chosen? This fundamental question is one of the subjects of the present work. Our answer is based on the Ord-Nottale-El Naschie conjecture: spacetime is a fundamental fractal entity which imprints fractal properties on matter and ®elds. Thus, both (cosmic string and, respectively, hierarchical clustering) views of the Universe are suitable to describe Nature. We illustrate this idea by introducing the novel concept of a fractal string which uni®es the two point of views. In El Naschie's approach [32], fractal spacetime and Cantorian spacetime are intimately connected if not identical. A fractal path in Cantorian spacetime has always a fractal Hausdor€ dimension of dH ˆ 2 and can be considered as a 2D projection of a fractal string (in 3D space) which may represent a cosmic string (see Fig. 7). This could explain the fractal space distribution of galaxies along cosmic strings. Thus, we note that observations of galaxy±galaxy and cluster±cluster correlations as well as other large-scale structures can be associated with a (limited) fractal with a dimension of D  1:2 [31]. It is reasonable to assume that cosmic strings provide the (density) inhomogeneities that are needed to seed (to initiate) the formation of galaxies. Thus, we have to assume that the observed patterns and structures are formed through a growth of the aggregation process. The fractal dimension D  1:2 represents a manifestation of the fact that the observed fractal should have emerged from 2D sheet-like objects such as domain walls and string wakes. Finally, following so many possible e€ects of cosmic strings in astrophysical space, we express our optimistic point of view on the existence of these entities. Possibly, cosmic strings are closer to reality than superstrings. At this point, there occurs the following question: What is the relation, if anything, between cosmic strings and ( fundamental ) superstrings? At ®rst sight, no relation. However, in some models based on twistor space, cosmic strings appear as spacetime structures corresponding to holomorphic twistor curves. In a way, both the relativistic (super)string and the cosmic string appear as di€erent solutions to the Nambu± Goto equation which generalises the relativistic description of a free, point particle (see, for instance, Ref. [33]). Intrinsically, this does not really lead to a formally more profound relation between superstrings and cosmic strings. However, by virtue of the unity of the Universe, there must be some physical connections. In a way, superstrings in elementary particle physics possess a similar role to that played by cosmic strings in the

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large-scale structures of the Universe. We note that, as is suggested, a double galaxy may be the cosmic analogue of a hydrogen atom, assuming that quantum mechanics rules apply to both [34]. In¯ation will decrease the density of cosmic strings and, thus, these strings have an important cosmological role only if they are formed at the end of, or after, the in¯ationary period. However, in the very early Universe, while the temperature was suciently high for the generation of fundamental (super) strings by quantum e€ects, the rate of generation of superstrings may have compensated the decrease of their density due to the expansion. Thus, there would have been a constant string density and an exponentially in¯ating universe until the temperature fell below a critical value. So, superstrings can be the cause of in¯ation. It emerges from this section that the concept of a spacetime continuum is not so suitable on a sub-microscopic level as also on a cosmic level. Possibly, spacetime is really a fractal (with a fractional number of dimensions), or a lattice, modelled on ®elds and matter. It is interesting also to note Penrose's twistor theory which asserts that there are no spacetime points as such, but merely systems of lines (strings? fractal curves?) whose intersections are equivalent to points in the classical theory but which fail to intersect at all when quantum ¯uctuations are involved [35]. 6. Fractal Universe against (or in agreement with) the cosmological principle As we have noted, the space distribution of galaxies might be described as a clustering hierarchy or, equivalently, as a fractal. Usually, a quantitative measure of the galaxy (statistical) distribution is the autocorrelation function, or its transform, the power spectrum. In the case of a point process, the autocorrelation function is de®ned by a two-point galaxy±galaxy position correlation function, ngg …r†, which is implied through the joint probability of ®nding galaxies in each of the volume dV1 and dV2 at a separation r, dP ˆ n2g ‰1 ‡ ngg …r†ŠdV1 dV2 ;

…19†

where ng is the mean galaxy number density. ngg …r†, for small separation r up to  10 Mpc, is close to a power law [30], r  r ÿc ; c ˆ 1:77  0:04  1:8; …20† ˆ bgg …L† ngg L L where the correlation amplitude bgg …L† unity for (all) galaxies, L the average separation of objects in the catalogue being examined, and c  1:8 is the index of the power law correlation. This equation marked the beginning of quantitative attempts to understand the large-scale structure of the Universe. Generally, a power-law form of spatial correlations indicates that the average density of an aggregate (i.e. a random cluster formed by the irreversible aggregation of `particles') decreases inde®nitely as its size increases [36]. The clustering length (correlation length) L is given by L  0:002  RHubble ˆ 0:002  1026 m; 26

…21† 21

where RHubble  10 m represents the Hubble radius (or length) of the Universe. The correlation length L can also be de®ned as the distance at which the density of galaxies is on average twice the mean number density and is given by L  5H0 Mpc:

…22†

Here H0 is the Hubble constant in units of 100 km sÿ1 Mpcÿ1 . The two-point correlation function for clusters of galaxies, ncc …r† is described by the same equation (20), with the same exponent in the power law, but with an amplitude bcc  0:35 which is constant for all clusters of galaxies. The larger correlation for the galaxies is probably an indication of a gravitational clustering. For 10 K L K 100 Mpc, b…L† is nearly constant and the current best ®t value is b  0:26. 21

The Hubble radius represents the distance at which the cosmological expansion velocity equals the velocity of light and is of the order of the particle horizon beyond which we do not see a causal connection. We mention that inside a sphere with a Hubble radius there exist 1011 galaxies, that is approximately as many as the number of stars in a typical galaxy which has a radius of  1021 m.

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19

From a practical point of view, the two-point correlation function n…r† measures the clustering in excess (ngg …r† > 0) or in defect (ngg …r† < 0) to a Poisson distribution for which ngg …r† ˆ 0. The associated correlation integral for a fractal set of galaxies, C…r†, represents the average number of galaxies within a sphere of radius r of a given galaxy and is proportional to rD2 , where D2 is the (two-point) correlation dimension. For example, for a uniform space distribution, C…r† is proportional to the volume of the sphere, and thus D2 ˆ 3. In the case of a cluster of galaxies, noting that ncc …r† displays a power-law behaviour, within the range for which ncc …r†  1, the correlation integral yields D2 ˆ 1:2. Furthermore, it is well known that a power law correlation function with index 1.8 corresponds in 3D space to a fractal with fractal dimension D ˆ 3 ÿ c  1:2 which is the same as D2 . From all the discussions we presented above, there emanate the following conclusions: · The near constant behaviour of b for clusters indicates that the clustering process can be scale invariant and has, thereby, a fractal structure. However, it is not a true (pure) fractal because it has not a power law behaviour on a much larger scale. At scales larger than 100 Mpc, the data are rather poor so that the power law correlation is not evident, and at very large scales the galaxy distribution displays something akin to isotropy with homogeneity and smoothness. Thus the phrase `fractal Universe' seems to be too excessive. Perhaps a bounded or limited fractal or a truncated fractal or a segment of a scale-invariant clustering hierarchy or a fractal on a small scale (a fractallite, which is similar to a crystallite) is a more suitable choice to characterise the galaxy distribution. · Of course, it would be attractive to describe the entire Universe by a pure (mono) fractal because the boundary conditions of both, the Universe and a pure fractal, are that they have no boundary. However, this idea may not be lost since locally the galaxy distribution looks like a fractallite. But on much larger scales (r  L), because the fractallites have di€erent orientations of the `fractallographic axes', the galaxy distribution looks like a `poly-fractal', consistent with a homogeneous random process. In other words, the large-scale galaxy distribution can be described by a statistically uniform space distribution of bounded fractallites, each fractallite having a diameter  L. This is roughly equivalent to many Rayleigh-Levy walks (Levy ¯ights) started at randomly chosen sites. Another way to visualise this is to imagine that the galaxies are concentrated in clumps, or cluster balls which are localised in space independently and uniformly at random. If this distribution is averaged through a large smoothing length, it appears smooth but if the resolution is improved to a scale comparable to the cluster ball size, it appears to have a fractal structure. · At this point we advance towards quantum cosmology. It is well known that random walks possess fractal trajectories, and ensembles of (classical) particles moving on such trajectories are described by a Schr odinger type equation [37]. Is this equation identical with the Wheeler±DeWitt equation in quantum cosmology? We shall see. Anyhow, a fractal universe can be described by a fractal wave function of the universe. And what, if anything, does this fractal wave function have in common with Hawking's proposal for a wave function of the Universe as a solution of the Wheeler±DeWitt equation? Generally, the argument of the Wheeler±DeWitt wave function represents an entire three-geometry, which is characterised by a set of functions, the metric. What are the arguments for a fractal-wave function? All these questions are, for the time being, open. The answer will provide new (quantitative) details of the fractal structure of the Universe. · As an aside, we point out that as a possible exception to the traditional picture of the Universe as either fractal or homogeneous, it is possible to assert that an intermediate stage exists in the form of so-called quasifractals for which no exact self-similarity and self-anity can be de®ned. · What is the natural truncation of the fractal at large scales? In other words, what is the physical motivation for this growth process? The answers to these question can be formulated from a crystallographic point of view. The formation of a large-scale structure is determined, ®rst, by the primordial seeds or ¯uctuations (density perturbations) and, in what follows, by an aggregation of matter to the seed (growth process). The correlation of seeds (cosmic strings, for instance, which should have a fractal structure imposed by the fractal character of spacetime) and the scaling behaviour of growth processes can determine the fractal strucure as we observe it today. We can invoke here three points towards an explanation. (1) The random walk model [38]. In this model (a variant proposed ®rst by Mandelbrot) the galaxies are placed at each step of a random walk and, consequently, the correlation between seeds determines the fractal distribution of cosmic fractallites. Quantitatively, the fractal dimension D2 is involved in the probability distribution, P …l†, of the random walk. Indeed, for a random walk with step size l0 ,

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 P …l† ˆ

0

D2 lD0 2 =lD2 ‡1

if l < l0 ; if l > l0 :

 …23†

However, this model is merely a phenomenological computational tool than a physical mechanism describing the growth process. (2) The random Gaussian ¯uctuation model [39]. In this model, the seeds are randomly distributed in a Gaussian way. Furthermore, if the amplitude of the ¯uctuations is scale invariant, the two-point correlation on a small scale (K10 Mpc) can be established. (3) Di€usion-limited aggregation (DLA). Generally, matter has a stochastic motion in space until it is gravitationally bound by seeds to form fractal clumps. The growth rate of a clump is determined by the di€using ¯ux of matter onto the seed, and is modelled by DLA. The underlying physical idea is that the aggregate grows by absorbing particles whilst undertaking a random walk in a Dg -dimensional growth space. Furthermore, the growth process is limited by the di€usion of particles onto the aggregate. This di€usion can sink below the expansion rate of the universe. The growth space is de®ned by all possible trajectories of the stochastic motion of clumps of matter. Of course, the overal space is 3D, but the growth space of the stochastic motion is not necessarily 3D. Since D2 P Dg ÿ 1 [31], it emerges that the observed fractallite dimension D2 ˆ 1:2 implies a constraint leading to Dg 6 2:2. In other words, the agregate is not compact since the fractal dimension is not equal to that of space. This result is very important because it indicates that the growth space can involve 2D sheet-like topologiocal defects that can be seeds for LSS. As possible sheet-like defects we refer to: wakes of cosmic strings, light domain walls, superconducting strings (the explosive model), the pancake model, and collapsing textures. If the random motion of matter is not constrained (by a fractal structure), the growth space is 3D, that is Dg ˆ 3. 7. Fractal Universe and R ossler's conjecture What about the gravitational ®eld in the context of the fractal Universe? Recently, R ossler [40] noted that the fractal pictures of the distribution of cosmic matter are still fresh enough to allow to pose radically new questions. R ossler refers especially to the necessity of a new solution of the Einstein equations which incorporates the e€ect of the fractal distribution of matter. 22 How is it formally possible to accomplish this formidable task? First, we mention that we have to deal with the interior Einstein's equations and thus we need an equation of state for the cosmic material which, in the simplest case, can be of the form q ˆ q…p†:

…24†

Even in the `simple' case of a spherically symmetric distribution, the interior Einstein's equations for a static metric of the form (in geometrised units), ds2 ˆ A…r† dt2 ÿ B…r† dr2 ÿ r2 …dh2 ‡ sin2 h d/2 † ˆ A…r† dt2 ÿ

1 dr2 ÿ r2 …dh2 ‡ sin2 h d/2 †; 1 ÿ a…r†=r

…25†

and for an isotropic pressure, lead to the Oppenheimer±Volkov equations [18], dp …p ‡ q†‰a…r† ‡ 8pr3 pŠ ˆ ÿ ; dr 2r‰r ÿ a…r†Š A0 ‰a…r† ‡ 8pr3 pŠ ˆ ; A r‰r ÿ a…r†Š da…r† ˆ 8pr2 q; dr a…0† ˆ 0; p…0† ˆ p0 …initial conditions†:

…26† …27† …28† …29†

22 We parenthetically note that a very interesting problem might be also to consider how Einstein's equations have to be modi®ed in order to take into account that space time is not only curved but possesses also a fractal structure.

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21

We ®rst choose an equation of state (24) which yields the recipe of the interior of a galactic (or aggregating) object (GO) in terms of the physical density and pressure. We also have to choose a value for the central pressure p0 . Subsequently, we have to integrate the di€erential equations outwards from r ˆ 0 until r ˆ rs (the surface of GO) where the pressure drops to zero (ps ˆ 0). For r > rs , pexterior ˆ 0 and qexterior ˆ 0, and hence a…r† ˆ a…rs † ˆ 2M ˆ constant:

…30†

Thus, outside the surface of the GO, there applies the empty (curved) Schwarzschild spacetime. For the time being, very few exact solutions of Oppenheimer±Volkov equations (with more or less realistic equations of state) have been established. For example, in the case of an ultra-sti€ equation of state (q ˆconstant, incompressibility), 8pr3 8prs3 q ˆ 2m…r†; a…rs † ˆ q ˆ 2M: …31† a…r† ˆ 3 3 Hence, the complete solution of Oppenheimer±Volkov equations (that is the metric (25)) is given by [18] 9 8    2 > > 3 8prs2 1 8pr2 > = < q ÿ q 1ÿ 1ÿ if r < rs ; > 3 3 2 2 A…r† ˆ 3 > > > > : 1 ÿ 8prs q  1 ÿ 2M if r > rs ; ; 3r r 9 8 1 1 > > >  if r < rs ; > > > > > 2m…r† > > 8pr2 > > = <1ÿ q 1ÿ r 3 B…r† ˆ 1 1 > > >  if r > rs : > > > > > 3 2M > > 8pr > > s ; :1ÿ q 1ÿ r 3r

…32†

…33†

The expression a…r†=r possesses for the interior region of a GO the signi®cance of what R ossler calls the Schwarzschild density …m…r†=r† which is proportional to the Newtonian gravitational binding energy per unit mass. We note that this quantity (in the internal part of GO) grows with the distance r. Apparently this situation seems to be simple and a normal one. However, R ossler observes that it can describe also a similar situation for a fractal distribution of matter. Particularly, he suggests that the Riemannian curvature of spacetime in the Einsteinian (gravitational) sense might be a manifestation of a fractal structure of spacetime. Furthermore, ¯uctuations in the scalar curvature can act as a source of torsion of spacetime and even in the absence of (material) spin, torsion does not vanish. Thus, torsion of spacetime, which can be also interpreted as a fractal pattern, is self-generated by the geometry of spacetime and, practically, by matter-induced ¯uctuations in the geometry. It is interesting that the consequences of torsion on spacetime appear to lead to a Yukawa type of additional contribution to the Newtonian potential:  m1 m2  1 ‡ aeÿr=k ; …34† V …r† ˆ ÿGa r where a  ÿ7  10ÿ3 for k  200 m [41]. What is the connection between fractal trajectories and Riemannian geodesics of the associated curved spacetime? The answer is not simple because a fractal structure is a joint result of matter, ®elds and spacetime, and Einstein's equations are usually written and solved only for simple schemes of matter (¯uids) and ®elds (e.g., an electromagnetic ®eld) with simple symmetries in their distribution. However, for the time being we are interested in the fundamental and philosophical aspects of these questions. A D2 ˆ 1 (line-like) uniform mass distribution possesses a constant Schwarzschild density. However, if D2 > 1, for example D2 ˆ 1:2, the Schwarzschild density grows with distance and the physical situation is

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similar to that described by an interior Einstein's solution). On the other hand, the mass MGO increases with its linear size R according to 23 MGO  RD2 :

…35†

If the fractal is grown in two dimensions, it results that the (proper) density, qGO 

MGO  RD2 ÿ2 ˆ Rÿ0:8 ; R2

…36†

decreases as Rÿ0:8 , and again this is similar to the decrease of the density inside the GO as far as the surface of the GO is reached. Of course, many other similarities in the competition between general relativity results and fractal approach of the universe may occur by integrating the interior equations numerically for more realistic equations of state. Possibly, the two approaches (general relativistic and fractal, respectively) lead to similar results when special (exotic, fractal) equations of state are used for fractal distribution of matter. Anyhow, we cannot look for an exact solution to Einstein equations which describe a fractal distribution because we face a `double non-linearity', one appertains to Einstein's equations themselves and the other one to the fractal matter and ®eld distribution. Finally, we summarise some important ideas referring to gravitation, fractal structure and growth processes. (1) A pure fractal cannot explain the structure of the universe because of the isotropy of microwave radiation and the relatively uniform distribution of astrophysical objects at large distances. (2) Even if initially Einstein himself believed that an unbounded clustering hierarchy (or, in Mandelbrot's terms, an unbounded fractal) represents a logical structure for the development of the Universe, he then rejected this idea which violates Mach's principle 24 and prefered instead a homogeneous universe. Also, the idea of a homogeneous universe was supported by Hubble's observations which indicated that clustering of the kind observed in the Local Supercluster fades into a homogeneous structure when averaged over larger length scales [30]. However, our hypothesis of a poly-fractal (or fractallite) structure of the Universe is in agreement with this large-scale homogeneity. (3) Cosmic fractallites developed out of a gravitational instability of the expanding universe. In the framework of general relativistic cosmology, a small density ¯uctuation dq=q grows as a power of time evolved since the beginning of the Universe (i.e. from the start of expansion at the Big Bang). This unconventional variable is used because in general relativity and cosmology there is no characteristic length or time to permit another choice. The places where dq=q > 0 represent the seeds for a growth process and become regions with a dense matter. This region forms a gravitationally bound clump, possibly a galaxy, or a cluster of galaxies, or a supercluster, which breaks away from the general expansion of the Universe. These cosmic objects constitute a clustering hierarchy (cosmic fractal) as a result of the gravitational instability. The sequence of creation of such a fractal starts with primeval mass ¯uctuations, as from homogeneity, which can be approximated by a random Gaussian process having a weak-steep power spectrum. Then the rms ¯uctuation in mass as averaged through a domain of size R increases with decreasing R, and thus smaller mass objects may break away from the initial general expansion, and form new objects that are later incorporated in larger objects. (4) A very interesting idea would be to assume that the primeval density ¯uctuations (from homogeneity) possess a (random) fractal structure. This leads also to the suggestion of the universality of a fractal structure of vacuum ¯uctuations, of zero point energy of quantised ®elds and, thus, also of vacuum

23 As noted above, the correlation integral C…R†  RD2 measures the average number of galaxies within a `sphere' of radius R. If, instead of this average , we are interested in the number of neighbours inside the sphere centred, for instance, on the Earth, M…R†, this is given by the mass-radius (co)relation, M…R†  RD , where now the `fractal dimension' is, in general, the dimension of the space considered. The mass-radius (co)relation is not so accurate as C…R† which considers all galaxies in the sample as possible centres, but has the advantage that the measure of D can be extended to larger scales, because the redshift measurements are usually performed by an observer on the Earth. 24 In Einstein's interpretation, Mach's principle asserts that inertial frames can be de®ned in a consistent way across all space because there is about as much mass everywhere to de®ne inertial reaction by a gravitational ®eld, via the equivalence principle.

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23

ÿ
spacetime itself, They all have a fractal structure. The zero point energy 12hx is associated with the random (vacuum) ¯uctuations (Zitterbewegung, 25 Casimir e€ect) of physical ®elds (e.g., electric and magnetic ®elds) even at the lowest, vacuum state. For example, the vacuum in QED is considered as a `ferment' (or a `catalyst') of activity with large uncertainties in the strength of the electromagnetic ®eld. These uncertainties generate a quantum noise in quantum optical systems such as lasers. At this point, it is important to emphasise that the universality of the fractal structure might be experimentally proved if the quantum noise has the structure of a random fractal, as for example the Brownian motion has. This hypothesis is in agreement with Nelson's hypothesis of a universal Brownian motion [43]. We assert in this sense that any (micro)particle in empty space, or let us say the ether, is subject to a Brownian-like motion. In a way, we are tempted to model the (stochastic) ¯uctuations in vacuum by a random walk of a (Brownian) particle. We take into account that a random ®eld can be expressed formally as a sum of monochromatic components with Fourier amplitudes that are distributed according to a probabilistic law. The scale of ¯uctuations in space and time is then related to the width of the power spectrum of the Fourier amplitudes. However, in the description of the stochastic phenomenon in terms of the random Fourier components, no length or time scale can be de®ned in order to describe the random ¯uctuations. As regards the analogy ¯uctuations-random walk-Brownian movement, we note that this analogy appears to be correct, except that in the speci®c problem of a Brownian motion (as de®ned by the Langevin equation) the distribution of velocities is Gaussian. (5) It has been observed that galaxies tend to be arranged on ¯at or curved sheets (sheet-like distributions or walls), some of which are broader than the clustering length L. The existence of empty regions (also larger than L) and of sheets of galaxies is not contradictory. Furthermore, there exists a tendency for the bounding walls of empty regions to be smooth and, thus, the galaxies tend to de®ne linear structures (strings) on edges of walls. We parenthetically note that the 2D sheet-like space chosen by the (natural) distribution of galaxies is similar to the 2D space chosen by anyons (i.e. neither bosons, nor fermions) which explain the fractional quantum Hall e€ect, high Tc superconductivity, etc. in condensed matter physics. It seems that, indeed, in the regions where both the post-Newtonian and the weak ®eld approximation may be applied, there occurs a gravitational anyonisation [44]. Also, these 2D sheets may be a result of some space ®lling fractal trajectories. We emphasise that in a particle-cluster aggregation, the particle trajectory plays a very important role. It is the trajectory that determines the density of the structure and thus the fractal dimension. For example, a Brownian motion with fractal dimension Df ˆ 2 is more e€ective than a linear trajectory with Df ˆ 1 in preventing particles from penetrating into the voids of the structure; such behaviour leads to structures with a lower fractal dimension. Can we consider that this phenomenon is a result of an action of a `Conservation of void' or a `Void exclusion principle'? We recollect that Pauli exclusion principle asserts that no two fermions can be in the same state, but in our case we observe that the fractal structure forbids also some voids to be ®lled with particles. Furthermore, at the more elementary level, the sheet-like tendency of arrangement of matter is explained also by the fact that quantum ¯uctuations (e.g. the Schr odinger zitterbewegung) are area-like and not linelike and this agrees with the fractal character of quantum (Feynman) paths [32]. (6) Generally, fractal geometry allows us to consider irregularities as intrinsic entities. If spacetime structure has such fractal irregularities, a new question arises, which is also inspired by the R ossler conjecture: How the `exact laws' (e.g., Lorentz SO…3; 1†symmetry, Newton and Coulomb laws, etc.) are modi®ed by the fractal structure of a vacuum spacetime and the vacuum ¯uctuations? We o€er some examples. · A deviation from the 1=r Newtonian gravitational potential is possible if the dimension of space in which we live is not exactly an integer D ˆ 3. For example, assuming the validity of Gauss' law in an arbitrary

25 Particularly, in the Dirac theory, whilst the average velocity of the electron is less than c (velocity of light), the instantaneous velocity is always c. It emerges from this that the motion of the electron consists of a highly oscillatory component, superimposed on the average motion. Schr odinger called this oscillatory motion Zitterbewegung [42]. The amplitude of this oscillation is of the order of the Compton wavelength of the electron. We note that the Zitterbewegung is not an observable motion, because any attempt to determine the position of the electron to a better accuracy than a Compton wavelength must defeat its purpose by the creation of pairs of electron±positron. Intuitively, a vector (a line-like object or a string) represents a little perturbation (or oscillation) of a point. In a way, we can assert that the Zitterbewegung is a manifestation of the string character of the Dirac electron because the latter is a type of a relativistic (Nambu) string whose end points move with the speed of light.

24

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(integer or fractionary) D 6ˆ 3 of spacelike (non-compacti®ed) dimensions, the gravitational potential is modi®ed and can be written as [45]  ÿ r 1 ; …37† V …r†  r0 r where  ˆ D ÿ 3 and r0 is a phenomenological parameter. Comparing the perihelion shift of planetary orbits, induced by an  6ˆ 0, with astronomical observations, one obtains jj K 10ÿ9 :

…38†

It is important to stress that a similar modi®cation of Newton's potential can also be obtained if the SO…3; 1† gauge symmetry of gravity is broken, even if D ˆ 3 holds exactly. In this case,  parametrises the deviation from an exact local Lorentz invariance, and the Gauss law (as a consequence of Poisson's equation) is no longer satis®ed. Furthermore, we note that Gauss' theorem is true only for a simply connected domain or a cotractable domain. In such regions any closed curve can be continuously contracted to a point without having to leave the region. A domain which is not simply connected is multiply connected. The same thing can be asserted on the theorem of Birkho€. Thus, if the fractal topology implies multiply-connected domains, it is clear that a fractally structured spacetime can lead to a non-Newtonian (non-Einsteinian) gravitation. We parenthetically note that the Cantorian spacetime (as elaborated by El Naschie [46]) is a multiply-connected spacetime. · Vacuum ¯uctuations involve the creation of virtual pairs of particle±antiparticle and, thereby, lead to a vacuum polarisation. The uncertainty principle permits the existence of such pairs of particles, each of

Fig. 7. A fractal string (in 3D space) can experience a 2D projection of the form of a sheet. This sheet can also have a fractal structure in the sense that it is in fact, for example, a space ®lling (ergodic) Peano-like curve. Thus, the fractal structure of a cosmic string is imprinted on its wake which has also a fractal structure. This `universality' of fractal structure is caused, in our opinion, by the intrinsic fractal structure of spacetime itself. Fig. 8. Feynman diagram (here a line represents the free propagation of a particle, whilst the vertices 1 and 2 represent the interaction between particles, the structure of which is controlled by the form of the interaction term in the Lagrangian) corresponding to the virtual e‡ eÿ creation in a vacuum electromagnetic photon ®eld. The loop of these virtual particles (at the centre of ®gure) represents the origin of the in®nity in QFT of vacuum ¯uctuations. The more loops there are in a diagram, the more virulent is the in®nity. The Compton time scale tC ˆ  h=2m0 c2 is interpreted by El Naschie [48] as the time at which fractal time becomes manifest. Furthermore, this fractal time is considered as the the source of quantum Zitterbewegung and of inertial-gravitational masses of virtual particles. Extrapolating, we can assert that even the Hawking black hole evaporation and gravitation itself originate through the fractal structure of spacetime.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

25

mass m0 , for periods of t K tC ˆ h=2m0 c2 (as in Fig. 8).This phenomenon can be treated as a small perturbation of physical spacetime and thus as a seed for the development of new structures.Vacuum polarisation e€ects produce a short-range logarithmic modi®cation to the Coulomb potential [47]. For r  ctC h=m0 c  KC (the electron's Compton wavelength), the modi®ed Coulomb potential can be written as   qq0 2a r 0:998 ÿ ln ; …39† VC …r† ˆ 4pr 3p KC 1 is the ®ne-structure constant and the other symbols have their usual signi®cance. The e€ect where a ˆ 137 of the logarithmic correction to the Coulomb inverse square law is to produce an anomalously increasing force as the separation of charges becomes smaller. This e€ect tends to con®rm the Feynman±El Naschie hypothesis that gravity (attractive, negative or repulsive gravity, and ®fth force, sixth, and all that forces) may have its origins in dipole±dipole interactions (Van der Waals-type forces) generated by fractal± vacuum ¯uctuations. A physical explanation of the logarithmic term in Eq. (39) may be given as follows. For example, a negative charge polarises the vacuum surrounding it into a neutral cloud of electrons and positrons. Some of the negative charges are repelled to in®nity leaving the negative polarising charge surrounded by a closely bound (r  ctC ˆ KC ) positive-charge density. Thus, an approaching test charge will penetrate the positive-charge cloud and detect more and more of the original negative charge causing the Coulombian force to deviate from a perfect inverse square law. In a similar way, we can assume that there exists a gravitational physical vacuum (graviton ®eld) in which the ¯uctations are represented by masson-antimasson pairs (m‡ mÿ ). The presence of a (positive) mass polarises gravitationally the vacuum into massons (m‡ particles) and antimassons (mÿ particles). The massons and antimassons are related to the graviton ®eld (see Fig. 9) in the same way as the electron and positron are related to the photon ®eld in QED (see Fig. 8).

8. Fractal signatures in topological defects and fractal skeleton Presently, the elementary particle interactions are described by a GUT with an `original' gauge group G which represents an `exact' symmetry at a corresponding high energy. If the energy decreases, the physical system is involved into a sequence of spontaneous symmetry breakings (an SSB ˆ a phase transition) which can be modelled by a transition to other types of symmetry groups, G ! H !


! SU…3†  SU…2†  U…1† ! SU…3†  U…1†em ;

…40†

where H ( G) is the unbroken subgroup of G which incorporates all elements of G which leave invariant the VEV of the participating ®elds. The manifold of the equivalent vacuum states, M, can be identi®ed with the quotient space (quotient group) M ˆ G=H: 26 G and H are the symmetry groups before and after the transition, respectively. We parenthetically note that spontaneously broken symmetries can be restored through suciently high energies (temperatures) [49]. In the framework of a Hot Big Bang Cosmology (Standard Model in Cosmology) the series (40) of phase transitions in the early Universe (each of them having a critical temperature Tc associated with the corresponding SSB scale) can generate topologically stable defects (called sometimes topological singularities) which are contained, for example, in the following concepts: · vacuum domain wall, · strings, · monopoles, · hybrid topological walls bounded by strings, · monopoles connected by strings.

26 Quotient group, G=H ˆ The set of cosets G=H endowed with a group structure by a suitable de®nition of the product of two cosets. H should be a normal subgroup, i.e. a subgroup whose right and left cosets coincide (gHgÿ1 ˆ H or gH ˆ Hg 8 g 2 G).

26

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

Fig. 9. Virtual m‡ mÿ creation in the (gravitational) graviton ®eld. Fig. 10. Domains of space with di€erent directions of spontaneous symmetry breaking, i.e. with di€erent directions of a Higgs ®eld / in 2D internal symmetry space (/1 , /2 ). The topological defects (TDs) occur in between these domains. At the point in the middle of the system, j/j ˆ 0 and thus V …/† is not at the minimum. Thus, the defects occur when regions of a symmetric phase su€er a transition to di€erent broken-symmetry states. In a similar situation, when a liquid crystallises, di€erent regions may begin to crystallise with di€erent orientations of the crystallographic axes (i.e. there occurs a polycrystal formed by crystallites). The domains of di€erent crystal orientation (i.e. crystallites) grow, coalesce and, for energetical reasons, they smooth the misalignment (of the crystallographic axes) along their boundaries. The smoothing cannot be perfect and thus localised defects remain.

All these topological defects (excluding monopoles) are macroscopic. They result from a non-trivial mixing of the spatial and gauge degrees of freedom arising from the spontaneous breakdown of grand unifying symmetries. Also, they represent forms of trapped energy formed at the phase transition. As we noted, the trapped energy, or in the cosmic string case, perturbations caused by dynamics of the trapped energy, serve as a gravitational seed for the accretion of matter which generates a large-scale structure [50]. If a gauge symmetry is broken at a corresponding energy (temperature) scale, as in the GUTs, through the Higgs mechanism (see Section 3 and the following sections) the direction of the non-gauge-singlet Higgs ®eld / in the (internal /i ) symmetry space can only be correlated for l K dH . 27 Thus, generally, the direction of the Higgs ®eld will be di€erent in causally unconnected domains (see Fig. 10). Between these domains there must be regions where VEV is h/i ˆ 0, instead of h/i ˆ k which minimises the potential energy. Just within these regions there occur the topological defects (TDs). Topologically stable domain walls and monopoles appear to be too exotic and `disastrous' for actual cosmological models. These hybrid structures are merely temporary con®gurations and considered as inecient formations from a cosmological point of view. However, if we consider that space has an independent fractal structure (see Ref. [33]) these temporary structures may have a fundamental signi®cance. As we noted in Section 3, cosmic strings seem to ®t very well in cosmology. They can generate density ¯uctuations which can explain the formation of galaxies. Let us consider the case of walls bounded by strings which are described by the sequence of SSBs, G ! K  Z2 ! K;

…41†

where the ®rst transition generates strings and the second phase transition (a discrete symmetry breaking, the symmetry group Z2 has only two elements, Z2 : / ! ÿ/) generates domain walls. The strings get attached through domain walls at the second symmetry breaking. We assume that the resulting vacuum manifold M ˆ G=K has a trivial topology (i.e. M is simply connected and thus contains no non-contractible loops). It emerges from this that there are no topologically stable defects after the second symmetry breaking; furthermore, the walls can develop holes bounded by strings.

27

dH represents the distance of the horizon (Hubble radius), i.e. of the farthest objects that can be observed at a time t. Because of the expansion, the `amount of Universe' we can see increases with time. This must be so as long as the Universe expands with a velocity v < c. If the Universe expands with a velocity v ˆ c, the amount of Universe we can observe would remain constant. Presently, dH  c=H0  1026 m, where 1=H0  1017 s is the actual age of the universe. At these largest scales now visible, the Universe still appears very homogeneous and isotropic.

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27

11 Fig. 11. A cross-section of a wall-string system (a `Cantorian cosmic labyrinth' where walls are `vertically sustained' by cosmic strings). In other words, this image is obtained by intersecting a system of walls bounded by strings with a plane. This can also be considered as a 2D version of a string-monopole system. Finite strings can exist only if they connect monopoles and antimonopoles. They can break producing new pairs of monopole-antimonopole at the ends of new ®nite strings. Fig. 12. (a) Topological instability of domain walls bounded by strings. Walls are split into pieces as a result of their shrinking. (b) Domain walls emit gravitational radiation like a spinning rod.

A numerical simulation of the 2-phase transition described by Eq. (41) displays that the wall-string system represents in fact one in®nite cluster containing approximately 90% of the total wall area and string length. The intersection of this cluster with a plane yields a large number of short pieces which appear like a Cantorian-like structure (see Fig. 11). Domain walls in this model are not topologically stable. Indeed, the tension in a string of curvature radius R, Fstring  qs =R, is greater that the wall tension rwall , for R < qs =rwall . Walls tend to minimise their area and thus, isolated closed walls will contract and disappear. Also, small irregularities on a large domain wall will be damped out (smoothed). Domain walls contract and lead strings to mutual intersections and intercommutations. The walls connecting the strings are thus divided into pieces of size l  qs =rwall (see Fig. 12a). A piece of wall of size l and mass M  rwall l2 vibrates at a typical (fundamental) frequency of x  1=l. The power of gravitational waves emitted by such a wall is (in h ˆ c ˆ 1 units) dEg  GM 2 l4 x6 : dt

…42†

It is interesting to note that this equation is similar to the classical formula for the gravitational power emitted by a rod with moment of inertia I ˆ 13Ml2 , rotating with an angular frequency of x (see Fig. 12b) [51], dErod 32G 2 6 ˆ 5 I x: dt 5c

…43†

The rate of emission of gravitational energy is related to the rate dng =dt at which gravitons with frequency 2x are emitted by 28 dErod dng ˆ 2hx : dt dt

…44†

28 If a mechanical system system rotates with frequency x, it may emit an energy Erod during a cycle in the form of gravitational waves with a frequency of 2x which represents the frequency of a quadrupole radiation.

28

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The number of gravitons emitted in one period is Ng ˆ

2p dng 32 pG 2 4 32 pG 2 4 4 ˆ M lx: I x ˆ x dt 5  hc5 45  hc

…45†

It emerges from this that domain walls bounded by strings radiate away their gravitational energy and the whole system decays. At this point we pose the following question: If indeed (empty) space does possess a fractal structure in the sense that, for example, particles are (self-) con®ned to move on pre-existing fractal trajectories, following the decay or evaporation of a material fractal structure, there remains a fractal skeleton which represents an empty fractal space structure. Thus, a structured material formation can represent, in fact, a manifestation of a pre-existing fractal structure of space (which we call fractal skeleton). How can we put into evidence this fractal skeleton of nature? We believe that we have two ways to do this. First, we can use the astronomical observations and, secondly, we can conceive a `laboratory experiment'. The simplest test of fractal signatures in cosmology is to look at the arrangement (distribution) of cosmic objects (galaxies, clusters, quasars, etc.). If they reside in (statistically) regular cosmic patterns (for instance, in fractal formations or lattices) we can assert that the Universe contains cosmic fractals or cosmic crystals. However, the ®nding of such patterns is very dicult, because the images of galaxies su€er a continuous modi®cation. Astrophysicists must recognise a galaxy even if it su€ered changes of its form and shifts of position relative to other galaxies. Over the past 25 years, astronomers have looked for and found no similar images among galaxies within one billion light-years of the earth [52]. Also, astronomers have looked for patterns among quasars because these objects can be seen from large distances. They identi®ed some groups of four or more quasars but the results may not be statistically signi®cant. Furthermore, within the framework of a cosmic crystallography, if galaxy images repeat themselves periodically or in agreement with a law, a histogram of all galaxy-to-galaxy distances could display peaks at certain distances, and thus a particular pattern may be con®rmed. So far no pattern has as yet been observed, but this may be due to a lack of data on galaxies farther away than two billion light-years. Possibly, the American±Japanese collaboration SDSS (Sloan Digital Sky Survey) will realise a larger set of data for a 3D map of the Universe. On the other hand, in the context of a cosmic topology (i.e. spacetime global structure which is involved in the large scale distribution of matter in the Universe) [53], it is interesting to observe the global e€ect of a CMB. This radiation is remarkably homogeneous, but there are slight undulations (¯uctuations, perturbations) discovered in 1991 by the COBE satellite. These `undulations' illustrate density variations in the early universe, which then seeded the accretion of matter to form stars and galaxies. It is important to note that these ¯uctuations can also be related to the density perturbations produced by the string motion as we have shown above. In any case, according to the widely admitted gravitational instability scenario, galaxies and other cosmic structures result from the collapse of initially small density ¯uctuations. Just at this point we can realise the role of the fractal structure of space. Small density ¯uctuations (or isolated particles) are constrained to follow the fractal trajectories of the fractal skeleton (because this is precisely the structure of space) and constitute the seeds that lead to the actual structure of the Universe. However, as the mass and density increase, the in¯uence on the fractal skeleton of space becomes even more intense. Locally, the fractal skeleton is deformed but, from a global point of view, the matter is condensed on a fractal trajectory of empty space. Why is empty space (self-) organised as a fractalised structure? We are not in a position to give an answer to this question because it appears that the fractal structure of space represents a primary concept. It emerges from this that, on the level of an Earth laboratory, any experiment which proves the wave property of a particle (i.e. wave-particle duality does also prove the existence of a fractal skeleton of space. Indeed, it is easy to show that a particle con®ned to move on a fractal trajectory in space (e.g., a Peano± Moore curve) corresponds to a wavepacket which is governed by the uncertainty principle and a de Broglie relation (see, for instance, Ref. [37]). Completing this section we would like to observe that it is possible to conceive an experiment which can prove directly the fractal structure of space. It is well known that in ordinary optical gratings the slits are evenly distributed forming a 1D lattice or a multiple rectangular slit. However, if the slits possess a fractal

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

29

distribution (e.g., as in the case of a Cantor set type grating or, simply, we can use any fractal object 29), the resulting interference patterns display a self-similar structure and the same scaling property as those of a fractal grating [54]. The question is indeed: What type of fractal does nature choose for a fractal skeleton of space (`fabric' of space) which is devoid of matter and ®elds? Of course, the answer cannot refer to a particular type of an exact ideal fractal as e.g., a von Koch snow¯ake curve. A scheme presents itself by the path of a particle in quantum mechanics, which has a fractal curve and can be characterised by a Hausdor€ dimension which jumps from D ˆ 1 at large length-scales to D ˆ 2 at small length-scales, the transition occurring approximately on the de Broglie wavelength-scale [55]. It appears that such a Hausdor€ dimension of D ˆ 2 plays a special role in physics. Indeed, it is the fractal dimension of a Brownian motion (a Markov±Wiener process) which is re¯ected also on the corresponding motion in a Nelson stochastic quantum mechanics [43,57]. Furthermore, random walks also display fractal trajectories with a Hausdor€ dimension of 2 [58]. At this point, El Naschie [32,46] went a step further and developed his ideas in a very interesting and original way. The starting point emanates from the recollection that spacetime in Einstein's special theory of relativity is real but non-material and independent of anything else, exactly as in Newton's physics. However, in order to explain the universal presence of gravity, El Naschie conjectured that, generally, spacetime represents a Cantorian-fractal structure, which is raised to the rank of an independent entity like any matter or ®eld. The main property of spacetime is its general presence as an arena for all physical processes. Of course, the presence of matter and ®elds does in¯uence the properties of Cantorian spacetime. Gravitation reproduces an e€ect of a fractal ¯uctuation or a kind of a Zitterbewegung or quiver motion of this Cantorian spacetime. Fractal ¯uctuations generate within the bodies (embedded in spacetime) polarisations and dipole moments which can reside at the origin of the universal (gravitational) attraction between bodies, for example, this may happen through a Van der Waals force mechanism. Our opinion is that it is very dicult to prescribe a certain type of fractal to an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter should decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not but we can conceive a space far away from matter, or a space containing isolated small particles or a very low density matter. Very low density matter might be in¯uenced by a fractal structure of space, for example this may induce ¯uctuations structured in random fractals. Di€raction and di€usion experiments with empty space and very low density matter could display evidence of a fractal structure of space. However, at very high (Planck) densities, if spacetime with its ¯uctuations represents also the source of matter and ®elds (which is very resonable in the context of a quantum gravity), we can assert that, indeed, Einstein's dream of geometrising nature is fully accomplished on a Planck scale. Following this philosophical±physical discussion, we postulate that empty space as well as matter and ®elds have, each of them, their proper fractal features. At very low densities of matter (i.e. in the presence of only a few of quantum mechanical particles), space imprints its fractal characteristcs to the motion of particles. Even if at this level of our reasoning the issues appear to be somewhat clear, a new question is challanging us: If space has indeed a fractal structure, does this mean that it is organised like usual matter and that it requires a new underlying background entity, i.e. a new space? In other words, a structured space demands a new space, but this new space cannot be ideal (i.e. an absolute space, without any structure, any sign on it) and thus it also requires a new space and so on ad in®nitum. There exist in fact a cascade of spaces. Brie¯y, the endowment of our usual space (let us call it ®rst space ˆ space 1) with a physical structure is equivalent to a process of materialising space and, thus, to accept the existence of a new (second) space as an arena of the (®rst) material space. The coordinates of space 1 become internal variables in space 2. For the time being we have some suggestions only on the ®rst space. However, some signatures of space 2 seem to manifest themselves whenever we deal with the texture (fabric) of space. Indeed, on the quantum gravity (Planck) level there arises the necessity of using extra-dimensions in order to describe

29

For example, when the Koch curve is drawn on paper, one has to photograph the curve and to use the photographic negative as the fractal object for di€raction experiments. So, the transparent fractal curve in a negative ®lm represents an aperture in di€raction experiments [56].

30

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the quantum space foam. On the cosmic level, the expansion of the Universe also implies a change of the texture of space and thus the necessity of de®ning the reference space 2 which can be seen as an absolute space until we start observing its ¯uctuations. Furthermore, on the quantum mechanical level, a particle displays a wave-particle duality because it represents a test particle for the space structure. From this point of view we can interpret the wave-particle duality as a manifestation of the (fractal) properties of space rather than as a property of quantum mechanical particles. Now we can see a new interpretation of the Heisenberg uncertainty principle: Simultaneous measurements of position and momentum of a particle are subject to the uncertainty principle limitations because the `laboratory table' (i.e. space) in which the particle resides and moves on is not an ideal `smooth ¯at table' (i.e. an absolute ideal space) but rather a `table with irregularities' (i.e. a space with a fractal structure). Big particles possess no wave properties in the same sense as we observe that `big wheels are insensitive to the small irregularities of a road'. Lowvelocity and `small-wheel' particles (with small radius) will describe very exactly the irregularities in space (i.e. the exact position in space) but their momentum will be drastically a€ected. High velocity particles will `¯y above' the fractal-space irregularities and their momentum is hardly (if not at all) in¯uenced. These results do not depend on the observer action which tries to measure the position amd momentum components. Thus, the assumption that spacetime has a fractal strucure may lead to a new interpretation of the Heisenberg uncertainty principle which asserts, for example, that a measurement of the position of a particle by a microscope results in a corresponding uncertainty in the particle momentum as it recoils after interaction with the illuminating radiation. Of course, this being true, a fractal structure of spacetime may introduce new (supplementary) uncertainties because of the fractal geometry. 9. No particle (vacuum) state, spontaneous symmetry breaking and Higgs mechanism We consider ®rst a single real scalar ®eld /…x† with the Lagrangian density   o/ 1 o/ o/ ÿ V1 …/†; L1 /; a ˆ T1 ÿ V1 ˆ ox 2 oxa oxa

…46†

where the ®rst term (T) represents the `kinetic' energy and the second term (V1 ), the potential energy. In the particular case of a free scalar (i.e. spin ˆ 0) particle of mass m0 , the potential V1 …/† is quadratic in /, 1 V1 …/† ˆ m20 /2 : 2

…47†

The Euler±Lagrange equation (of motion) satis®ed by a ®eld / is given by o o oV1 oV1  2 / ‡ ˆ 0; /‡ a o/ o/ oxa ox

…48†

which in the case of free (particle) states becomes precisely the Klein±Gordon equation, 2 / ‡ m20 / ˆ 0;

…49†

where we have introduced the d'Alembert operator 2 

o o o2 ˆ 2 2 ÿ r2 : a oxa ox c ot

…50†

By de®nition, a no-particle or vacuum (ground ) state is the state in which there are no particles (i.e. all occupation numbers are zero), and occurs when [59] oV1 ˆ 0: o/

…51†

Of course, for free particle states, the vacuum state occurs when / ˆ 0. However, generally, in the case of interactions between ®elds (or particles), or self-interactons, V …/† contains higher-order terms in /, and

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

31

Eqs. (48)±(51) have, in a vacuum state, solutions / ˆconstant. We parenthetically note that the vacuum state is de®ned by zero energy, not zero ®elds [60]. We distinguish two possibilities: · Eq. (51) has only one solution (no `false' vacuum). Since the energy must be bounded on the lowest level, this unique (vacuum) solution corresponds to the minimum of the potential. · Eq. (51) has several (vacuum) solutions which correspond to several minima of potential. The lowest minimum de®nes the ultimate vacuum state of this scalar world. If the scalar world resides in a local minimum which has a higher value of the potential, then there occurs the possibility of a tunneling to a lower value. If there exist several such minima which have the same value for the potential, the vacuum is called degenerate. We shall see that such a situation may generate a SSB (breakdown) of the Lagrangian. We consider further a generalisation to the case of N real scalar ®elds (/i ; i ˆ 1; . . . ; N ) modelled by the Lagrangian   o/i 1 o/i o/i ÿ VN …/†; …52† LN / i ; a ˆ T N ÿ V N ˆ ox 2 oxa oxa where 1 1 2 VN :ˆ f0 …/i /i † ÿ l2 …/i /i † 4 2

…53†

is the potential energy term which is taken to have both quadratic and quartic terms (i in Eqs. (52) and (53) is summed over). The last (mass) term in the Lagrangian has been included in order to generate spontaneous symmetry breaking; here f0 is the bare self-interaction constant of the scalar ®eld. The Higgs mechanism induces some components of the scalar ®eld to become massive (i.e. to convert to massive bosons, see Section 3), while other components remain massless (and may be identi®ed with photons). In other words, the Higgs mechanism permits converting some of the massless particles into massive ones. We note that because the Lagrangian (52) contains only the `squared-length' of vectors / in the /i (symmetry) space, we deal with a theory which is invariant under the O(N) (orthogonal) group of rotation in the N-dimensional space. This group mixes up the ®elds with each other and possesses 12N …N ÿ 1† generators. For the time being we shall not insist on interpreting l as a mass and we consider what happens in the case of l2 being < 0 or > 0. A conventional (classical, normal) mass term in the Lagrangian (52) has negative l2 ˆ ÿm20 < 0, and f0 must be positive so that the energy is bounded from below (see the curve (1) in Fig. 13a). Equations oV ˆ0 o/i

…54†

f0 …/i /i †/j ÿ l2 /j ˆ 0

…55†

yield

which have a unique (vacuum) solution, /i ˆ 0:

…56†

The case l2 > 0 (negative mass squared in the Lagrangian) represents an unconventional choice of sign for the mass term and this is what induces developments. Now the potential V is called the Higgs potential and 2 the ®eld / with a very special kind of scalar self-interaction ‰14 f0 …/i /i † Š is called a Higgs scalar. The Higgs potential [see the curve (2) in Fig. 13a] no longer possesses a minimum at /i ˆ 0 (in which case it has now a maximum), but rather at any non-zero value of /i for which /i /i ˆ

l2 ˆ k2 f0

…57†

which de®nes the `eigenvalues' of vacuum of the Higgs theory. We observe that the unique minimum of the classical potential at /i ˆ 0 for positive mass squared (in the Lagrangian) has now been replaced by a whole

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J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

subspace of minimum action ( potential energy) de®ned by Eq. (57). Furthermore, because of the O(N) symmetry, Eq. (57) only determines the length of the vector /, its direction remaining arbitrary. Hence, the Higgs vacuum state is in®nitely degenerate since any direction yields a vacuum state of the same energy (see Fig. 13b). Clearly the degeneracy is closely associated, in the present case, with the symmetry of the potential. We see that l2 ˆ 0 is the critical transition point between the symmetric solution and the degenerate vacuum state case. However, even if the parameter k (see Eq. (57)) is invariant under the O(N) group, any particular solution corresponds to a vector pointing in a certain (given) direction and thus no longer experiences the O(N) invariance [59]. This causes the SSB. At this point we note a very important idea, namely, a certain direction may be physically (practically) realised only through a vectorial physical ®eld or a directional perturbation. Practically, a vector (a direction) can be generated by a small perturbation towards a point. Thus, the complex process of SSB may be considered as a simple lift of degeneracy by a perturbation. 10. Some analogies There exist many intuitive examples with an appearance of an order parameter which is accompanied by a reduction (or a breaking) in the symmetry of the system. These examples re¯ect also the universal character of the SSB [61,62]. It is well known that, generally, the part of dynamics that yields information on the bifurcation branches takes place in a phase space of reduced dimensionality. The reduction of dynamics takes place in certain classes of bifurcations leading to space symmetry breaking in spatially distributed systems. The early universe existed in a highly energetic yet symmetric state. Like a (highly unstable) supersaturated solution, it too was trapped on top of a hill of energy between two energy valleys (see Fig. 14 where the Mexican hat-ball analogy is illustrated). This hill of energy presents a high degree of symmetry in which all the interactions (forces) of nature are uni®ed. However, if a single crystallite (a `seed of crystal', which represents a small perturbation) is dropped into the supersaturated solution, the entire quantity of solution begins to crystallise in an almost explosive fashion.

Fig. 13. (a) The potential (53) with f0 > 0 and di€erent signs of l2 . Curve (1) corresponds to l2 < 0, and curve (2) to l2 > 0 (Higgs potential). (b) Potential of a (degenerate) Higgs scalar ®eld for N ˆ 2. Its minima lie on a ring. Fig. 14. Mexican hat-ball analogy of spontaneous symmetry breaking: (a) Symmetric but unstable. (b) Stable by breaking symmtry. Of course, the case (b) where the ball resides in the left or the right valley is less symmetrical than it is midway between them. (c) In a way, a bifurcation is equivalent to a symmetry breaking phenomenon. This bifurcation diagram displays how a state variable X is a€ected when the control parameter, T, varies. A unique (stable) solution s (the so-called thermodynamic branch in which the system is capable of damping internal ¯uctuations or external disturbances) becomes unstable at Tc . At this value of T, new branches of solutions, s1 ; s2 , which are stable, are generated. Beyond the critical value Tc , the states on this branch become unstable, the e€ect of ¯uctuations or of small external perturbations is no longer damped, and the system acts like an ampli®er: it evolves away from the reference state and moves to a new regime as a result of the symmetry breaking. The new regime may involve the formation of cosmic strings, the state of convection in the case of the Benard experiment, etc. The two bifurcation regimes coalesce at T ˆ Tc but become di€erent for T 6ˆ Tc . In the crucial moment of transition (vicinity of T ˆ Tc ), the system has to perform a critical choice (branches s1 or s2 which, in the Benard problem, are associated with the appearance of a right- or a left-handed cell in a certain region of space; also, the same situation can arise in the case of vortex type topological defects, vortons). ``Only chance will decide, via the dynamics of ¯uctuations. The system will scan the `ground', will make a few attempts, perhaps unsuccessfully at the beginning, and ®nally a particular ¯uctuation will take over. By stabilising, the system will become a historical object in the sense that its subsequent evolution will depend on this critical choice'' [63]. There occurs here an interplay, a competition, between chance and constraint, between ¯uctuations and irreversibility.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

33

A crystal rises out from all the liquid, and it becomes within seconds a solid crystalline mass. In other words, suddenly the unstable solution crystallises and its symmetry is broken. In this way the energy decreases to a stable level. Something similar may well have happened long before the ®rst second of the Big Bang creation of the Universe. The result of the symmetry breaking of the Universe was the separation of the gluon (strong), electromagnetic, weak, and gravitational interactions and the appearance of elementary particles with their unique pattern of masses [64]. When a supersaturated solution crystallises, the various crystallites grow rapidly and bump into each other so that it is possible that there exist domains of a supersaturated solution trapped by various linelike or `spaghetti-like' (topological) defects and boundaries between the crystallites. The apparently homogeneous crystalline solid may contain within its defects tiny regions of high-energy, supersaturated solution. Something analogous is also believed to have happened during the `crystallisation' of the very early Universe. Thus, within our Universe it is possible that there exist trapped domains (topological defects) of extremely high energy which were generated at the initial highly symmetric state of the universe. Let us compare the `scalar degeneracy' with the degeneracy which arises when analysing the energy eigenvalues of a particle in a central (spherically symmetric) Coulomb ®eld. We recall that when two or more quantum states have the same value of the energy (eigenvalue) we say that they are degenerate. Generally, in the case of degeneracy, even the squared moduli of the wave functions associated with the individual states do not have a direct physical signi®cance. As an example we consider the energy and eigenfunctions of the hydrogen atom. If a measurement of the energy of the atom displays a principal quantum number n ˆ 2, this means that the atom is in one of the four degenerate states: wnlm :ˆ w200 ;

…58†

wnlm :ˆ w210 ;

…59†

wnlm :ˆ w21‡1 ;

…60†

wnlm :ˆ w21ÿ1

…61†

or in some linear combination of them [65]. A measurement of the (orbital) angular momentum must yield the eigenvalue p L ˆ l…l ‡ 1† h; …62† where l is either 0 or 1, but which one of these is unpredictable. If, for instance, it happens that l ˆ 1, the value of m will be analogously unpredictable unless the z component eigenvalue, Lz ˆ ml h, is also measured. However, once all three mesurements have been made, the eigenfunctions of the hydrogen atom are completely determined and the results of further measurements of any of these physical quantities are predictable. The knowledge of the angular-momentum eigenvalues and eigenfunctions may clarify the physical signi®cance of the degeneracy found in the energy states of the hydrogen atom. Indeed, the spherically symmetric (central-®eld) Hamiltonian can be written as h2 1 o 1 ^2 ‡ V …r† ‡ L; H^ ˆ ÿ 2m0 r2 2m0 r2 or where L^2 ˆ ÿh2



1 o sin h oh



o sin h oh



 1 o2 ‡ 2 : sin h ou2

…63†

…64†

The last term in Eq. (63) represents the quantum-mechanical centrifugal operator similar to the centrifugal o do not depend on r, it follows term in the classical expression for the total energy. Since L^2 and L^z ˆ ÿihou 2 that H^ commutes with both L^ and L^z . Thus, the total energy eigenfunctions are in this case also eigenfunctions of the (square) total angular momentum and of one of its components. Energy levels corresponding to the same value of l, but di€erent values of ml , are degenerate and thus the angular part of the wave function may be a linear combination of the eigenfunctions corresponding to these various values of

34

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

ml . In other words, the spatial orientation of the angular momentum vector, ~ L, is completely unknown. Hence, no component of ~ L can be measured unless the (spherical ) symmetry of the system is broken, for example by applying a magnetic ®eld or, generally, a perturbation. A perturbation lifts the degeneracy of the energy levels, leading to additional structure (i.e. ®ne structure) in the line spectrum. If we apply a magnetic ®eld, ~ B k Oz, to an atom in a p-orbital (i.e. l ˆ 1) state, the threefold degeneracy (ml ˆ ÿ1; 0; 1) is lifted and spectral lines resulting from a quantum transition between this state and a non-degenerate s-orbital (l ˆ 0) state are split (Zeeman e€ect) into a triplet (see Fig. 15a), the angular frequency di€erence (Larmor frequency) between neighbouring lines becoming equal to xL ˆ

eB eB …cgs units†: …SI units† ˆ 2m0 2m0 c

…65†

However, if this experiment is performed with a one-electron atom such as hydrogen, a di€erent result is observed (see Fig. 15b). For strong ®elds (i.e. in the case of the normal Zeeman e€ect) the observed spectra correspond to the 2p level having been split into four sublevels instead of three, whereby the 1s level, which was expected to remain single, is split into two sublevels. Since the latter (1s) state possesses no (orbital) angular momentum associated with the motion of the electron in the ®eld of the nucleus, the observed ®ne structure must arise because of the intrinsic or spin angular momentum ~ S. In the l ˆ 0 state the spin can have two orientations, whereas in the l ˆ 1 state the orbital and spin angular momenta couple (spin-orbit coupling) and generate the four sublevels observed. In the case of the weak-®eld (anomalous) Zeeman e€ect, the spin-orbit coupling (corresponding to ~ J ˆ~ L ‡~ S) is much greater than the energy of interaction between the atom and the applied ®eld.

11. Higgs mechanism and Goldstone bosons We now return to the problem of scalar ®elds. As we demonstrated above if one thinks of the /i as the components of a vector U, then the minimum of the potential ®xes the length of the vector, but leaves its direction arbitrary. Let us consider a vacuum (ground) state which corresponds to a particular solution (57). We choose the axes in the /i space so that this ground (vacuum) state is de®ned by 0 1 0 1 /1 0 B .. C B .. C B C .C …66† U…x†vacuum ˆ B . C ˆ B @ /N ÿ1 A @ 0 A k /N which is invariant under the group O(N ÿ 1) that does not mix the Nth ®eld with the others. All the other vacuum states (con®gurations) can be obtained from this by O(N) transformations. We consider a perturbation of the form 0 1 0 1 1 0 d/1 0 0 . .. B .. C B C B .. C B C . C CˆB …67† dU…x† ˆ B . C ˆ B @ A @ 0 A: @ d/N ÿ1 A 0 H…x† /N …x† ÿ k d/N The expansion of the Lagrangian (52) to an order of …d/N †2 turns out to be LN …/k ; d/N † ˆ

1 o/k o/k 1 od/N od/N 2 3 ‡ ÿ l2 …d/N † ‡ O‰/3k ; …d/N † Š; 2 oxa oxa 2 oxa oxa

…68†

where we maintained the original ®eld /k for k ˆ 1; . . . ; N ÿ 1, and O‰/3k ; …d/N †3 Š contains terms cubic and higher in the ®elds. From the formal structure of this Lagrangian it emerges that it describes …N ÿ 1† massless scalar ®elds /k which possess a global O(N ÿ 1) symmetry, and a single scalar Higgs ®eld H of mass mH given by

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

m2H ˆ 2l2 > 0:

35

…69†

Thus, only one ®eld achieves a genuine mass (by the Higgs mechanism) whereas the other …N ÿ 1† scalar ®elds remain massless. The (boson) particles associated with the massless ®elds are usually called GBs. We emphasise that the residual or surviving symmetry group O(N ÿ 1) has 12 …N ÿ 1†…N ÿ 2† generators and is a non-trivial subgroup of the original group O(N). The di€erence between the number of generators of O(N) and of O(N ÿ 1) is exactly N ÿ 1 and represents the number of `broken generators' which are equal to the number of GBs [13]. This result is an example of the general Goldstone theorem which does not appertain speci®cally to the particular group O(N): For every broken generator in a SSB (or for any spontaneously broken continuous symmetry) there exists a (Goldstone) massless scalar boson or massless excitation. From a physical point of view, the various equivalent vacuum states di€er by the number of GBs of zero energy and momentum that they contain. We can also assert that SSB introduces its own massless bosons which can be understood as being excitations along the symmetry directions in which the potential is unchanged. 3 We note that the neglected higher-order terms in O‰/3k ; …d/N † Š show that there are also complex interactions between these massless and massive ®elds. SSB may be de®ned as a situation in which the Lagrangian of a system possesses a symmetry (namely, in our case, O(N) symmetry) but the equilibrium state (66) does not have the same symmetry (namely, in our case, U…x†vacuum possesses O(N ÿ 1) symmetry). In other words, a `spontaneously broken symmetry' occurs when the solutions of a problem (e.g. for the ground state of a system) are not symmetric even if the original Lagrangian is exactly symmetric. 12. Some examples of systems with SSB A very well-known classical example is the case of a ferromagnet. Here the Hamiltonian is rotationally invariant but the ground state is not, because in it the spins are all aligned along a de®nite, however arbitrary, direction. Thus, there exist in®nitely many ground (vacuum) states. We recall that at high temperatures (T > TCurie ) the orientations of the atomic spin magnetic moments are random and rapidly ¯uctuating. There is no correlation between the spin directions of any one atom with that of its neighbours; hereby the average spin magnetisation vanishes, hM S i ˆ 0. The system is unmagnetised and symmetric because all directions are equivalent (see Fig. 16a). If the ferromagnet is cooled below the (critical) Curie temperature Tc , it is more suitable, from an energetic and stability point of view, for all the spin magnetic moments to be lined up. Thus, a phase transition

Fig. 15. (a) Zeeman e€ect without consideration of a spin. (b) Normal Zeeman e€ect as generated by the presence of the spin. Fig. 16. (a) The directions of the atomic spin magnetic moments in a ferromagnet are random for T > TCurie ˆ Tc , and thus hM S i ˆ 0. (b) For T < TCurie the atomic spin magnetic moments are all aligned and thus there occurs a preferred direction of magnetisation and hM S i ˆ M S 6ˆ 0 within a magnetic domain. Di€erent magnetic domains are separated by domain walls which are illustrated by broken lines.

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J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

``from disorder to order'', i.e. from an unmagnetised state (hM S i ˆ 0) to a magnetised state (M S 6ˆ 0), takes place in conjunction with the release of a latent heat energy. This represents an example of a spontaneous MS of the resulting spontaneous (i.e. breaking of symmetry. It is important to emphasise that the direction jM Sj in the absence of any applied magnetic ®eld) magnetisation M S cannot be predicted. This depends on the accidental correlations between the spin directions during the cooling process. Once the transition is achieved, all the spin magnetic moments are correlated over a ®nite distance which determines the size of the magnetic domain. It emerges that the phase transition can be quantitatively characterised by the variation of the spin correlation length from zero (unmagnetised state) to the domain size (magnetised state). When a ferromagnetic material incorporates two or more domains (or, for another physical situation, e.g. a liquid coexists with its vapour), there exists a narrow region (called domain wall or interface) between the two phases in which the magnetisation (or density in the case of the liquid±vapour interface) varies very quickly. Generally, the way in which the magnetisation (spontaneous or through the impact of an external magnetic ®eld H) of a ferromagnetic sample depends on the temperature and the applied magnetic ®eld, M…T ; H†, is complex and it is necessary to take into account, for instance, the motion of domain walls, which leads to a hysteresis. We note that all critical phenomena (i.e. phenomena in the critical region which characterises the neghbourhood of Tc ), have a universal character in the sense that they are independent of the detailed microscopic structure of the system considered. For example, in the neighbourhood of the Curie temperature, the spontaneous magnetisation varies as MS / …Tc ÿ T †b

…70†

where b  13 (small di€erences occur from one ferromagnet to another). Also, in the critical region, the di€erence in density between the liquid and vapour is given by a similar equation, b

qliquid ÿ qvapour / …Tc ÿ T † ;

…71†

where now Tc represents the critical temperature for the liquid±vapour system. From a statistical point of view, the universality of critical phenomena can be understood in terms of the correlation length. For example, in the case of a ferromagnet (crystal) sample we have to consider corelations between the directions of spins at di€erent locations. The sample possesses a net magnetisation M if, on average, all the spins tend to be aligned in the same direction. At the ith lattice site, the ¯uctuations of a spin away from its average are given by dsi ˆ si ÿ hsi i:

…72†

The correlation between ¯uctuations at two di€erent positions, ri and rj within the sample lattice, is de®ned by the correlation function, G…ri ÿ rj † ˆ h…si ÿ hsi i†
…sj ÿ hsj i†i ˆ hsi
sj i ÿ hsi i
hsj i:

…73†

We may assume that only short-range forces act between spins and it is expected that the correlation function decays to zero at a large distance. Thus, it is reasonable to postulate the following form for this function:   jri ÿ rj j ; …74† G…ri ÿ rj † / exp ÿ n where n is a characteristic distance called the correlation length [66] which depends, in general, also on temperature and on the (applied) external ®eld. In the case of a spontaneous magnetisation, in the critical region, the correlation length is given by n  n0

1 ; jT ÿ Tc jm

…75†

where m  0:6±0.7 is a critical exponent of the correlation-length. We emphasise that the correlation length diverges at the critical point and thus, within the critical region the correlation length increases. The divergence of the correlation length is the key to the universality of critical phenomena. It emerges that in the

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

37

critical region ¯uctuations are strongly correlated over large distances and this explains why they, and the critical properties which depend on them, are independent of the detailed speci®cation of the forces which act over small distances. In order to understand the jump from a small correlation length (corresponding to a usual state) to a large correlation length (corresponding to a critical region) we recall that the correlation function can be tested experimentally through a scattering process. In the case of a ferromagnet, the scattering of neutrons is in¯uenced by magnetic ®elds (spin correlations). In the case of a liquid the scattering of light depends on density correlations. In the critical region, the correlation length is large and the scattered waves from particles widely separated in the sample are coherent. This leads to a strong scattering process which is visible to the naked eye in a liquid in its critical region. Indeed, a normally transparent liquid becomes foggy and milky in the critical region. This is the well-known phenomenon of critical opalescence. Another example of a system which experiences a spontaneous symmetry breaking arises at the buckling of a rod under an axial pressure (load). The resulting equations are symmetric under rotations about the axis of the rod. However, the system buckles in a particular, albeit arbitrary, direction, and there exist, in fact, in®nitely many states for the buckled rod. In these examples with a ferromagnetic sample or a buckling rod (or a falling pencil, see Section 3) the nonsymmetric phases correspond to a lower energy than the symmetric ones. The original symmetry of the Hamiltonian is hidden. It appears only in our impossibility to predict in which direction the spins will align in a magnetic domain or the direction in which the rod will bend. However, the original symmetry is present since all non-symmetric solutions are equivalent and can be deduced from one another by a symmetry operation. On the other hand, in any of the examples considered above there exists a critical point, that is a critical value of some physical quantity, either temperature or external force, in which the generation of a SSB takes place. Beyond the critical point the vacuum becomes degenerate and the symmetric solution unstable. These properties (a symmetric original Lagrangian, in®nitely many non-symmetric unpredictible solutions, the existence of a critical point) are characteristic of all examples of SSB. We ®nally note that a similar phenomenon may arise in relativistic ®eld theories as applied from the level of elementary particles to that of the Universe. Symmetry breaking may be responsible for the di€erent strengths of fundamental interactions. Also, the universe may be compared with a ferromagnet which possesses many domains in which the symmetry is broken in di€erent ways. 13. Spontaneously broken global symmetry for a neutral scalar ®eld Cosmic strings can occur from SSB whenever the manifold of equivalent vacuum states is multi-connected or in theories with global symmetry. In one of the precedent sections we considered a Lagrangian LN invariant under O(N), constructed from N real scalar ®elds. In the case of m20 < 0 (so m0 cannot be considered as the mass of /i ) we have ultimately obtained (N ÿ 1) massless (Goldstone) ®elds and a single scalar Higgs ®eld /N whose mass mH has been generated spontaneously. In this section we consider ®rst a complex neutral scalar ®eld, /…x†, de®ned by the following (classical) Lagrangian Ln ˆ

o/ o/ ÿ l2c // ÿ f0 …// †2 : oxa oxa

…76†

We note that in contrast to the previous sections and for the sake of simplicity, we changed the sign of the term incorporating the mass. The Lagrangian Ln is invariant under the (abelian) group U(1) of global phase (gauge) transformations /…x† ! /0 …x† ˆ eih

…77†

where h is an arbitrary constant. Further analysis of this case is similar to that presented in the previous 2 section. For l2c < 0, the minimum of V …// † ˆ l2c // ‡ f0 …// † occurs at // ˆ

k2c ÿl2 ÿ c 2 2f0

…78†

38

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

within a complex / plane. In fact, there exists a whole ring of minima of radius pkc2 (see Fig. 13 where we have now to replace /1 and /2 by R/ and, respectively, I/. Of course, / ˆ 0 represents an unstable point and any expression of /, kc /vacuum ˆ p eiH ; 2

…79†

which satis®es Eq. (78) and yields a true vacuum state. H is an arbitrary real number. There are in®nitely many vacuum states but every one is not symmetric in the sense that it may be broken by the gauge transformation (77). In other words, given a ground state, all the other vacuum states can be deduced from this via U(1) transformations. Also, all points on the ring of minima are equivalent because they can all be generated from any one by a gauge (77). Let us now consider a point of minimum on the real axis, which yields a (non-vacuum) state given by 1 /…x† ˆ p ‰kc ‡ A…x† ‡ iB…x†Š; 2

…80†

where A, B are real functions and A ˆ B ˆ 0 de®ne the vacuum state. According to Eq. (76), the Lagrangian is now given by Ln ˆ

ÿ
1ÿ
2 1 oA oA 1 oB oB ‡ ÿ f0 k2c A2 ÿ f0 kc A A2 ‡ B2 ÿ A2 ‡ B2 ; a a 2 oxa ox 2 oxa ox 4

…81†

where we ignored some inessential constant terms. We can consider now (81) as a quantum Lagrangian for the ®elds A…x† and B…x†. We observe that it does not contain any mass term for the B…x† ®eld but there is a normal mass term, 1 f0 k2c A2 ˆ m2A A2 2

…82†

for the ®eld A…x†, where m2A ˆ 2f0 k2c :

…83†

Possibly, the gauge transformed ®eld (80) does not modify the physical process described by the original Lagrangian (76), if the problem is solved exactly. However, if a perturbation is considered within the quantum theory, the physical situation may be di€erent. For instance, in (80) we can consider the kinetic energy and mass terms as those of the unperturbed Lagrangian Ln0 , but we cannot do so in the original Lagrangian (76) because of its negative mass term (l2c < 0). Thus, we started with a Lagrangian for a massless ®eld / and, ®nally, we obtained a massless ®eld B…x† and a ®eld A…x† with a spontaneously generated mass. 14. Spontaneously broken local symmetry for a charged scalar ®eld In the case of a global gauge invariance, the phase h is not a measurable quantity and can be chosen arbitrarily, but once chosen it must be maintained for all the events (x; t) in spacetime. A more powerful gauge symmetry is represented by the local gauge symmetry in which one can ®x the phase locally but also di€erently at di€erent sites. In other words, the transformations can depend upon the spacetime point at which the ®eld is acting. A local gauge transformation is given by /…x† ! /0 …x† ˆ eÿih…x† ;

…84†

where h…x† is an arbitrary function of x. This transformation induces a complex (and non-invariant) transformation of gradients:

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

39

  o/…x† o/0 …x† o oh…x† o ! ˆ eÿih…x† /…x†: ÿi /…x† 6ˆ eÿih…x† oxa oxa oxa oxa oxa

…85†

This means that terms involving gradients (e.g., in the case of a Lagrangian) are not invariant under a local gauge transformation. However, the local gauge invariance (or local gauge symmetry or gauge symmetry of the second kind) can be attained if we introduce, for example, a gauge boson ®eld, Aa …x†, which can be associated with photons as is the case in QED. Thus, here the electromagnetic ®eld is seen to be a ®eld which is associated with a local gauge invariance. The vector potential Aa …x† of the photon is considered as a simple example of a gauge ®eld. This way of analysing the problem represents somewhat the reverse of the usual approach in classical electrodynamics where one starts with the existence of an electromagnetic ®eld Aa and then one observes the gauge invariance. In a way, the gauge invariance in classical electrodynamics appears to be an odd property demonstrating that the four-vector Aa possesses too many degrees of freedom. Indeed, in quantum theory this property is essential. Equations of electrodynamics are locally gauge invariant because all derivatives appear in the special expression (combination) Da de®ned by o ÿ ieAa ; …86† Da :ˆ oxa called (gauge) covariant derivative which has the property that under a local gauge transformation it evolves as Da /…x† ! eÿih…x† Da /…x†:

…87†

The constant e could be any number, but in the covariant derivative (86) it plays the role of a coupling constant between the ®elds Aa and /. The fact that only combinations of the form (86) are arising in the present theory is equivalent to the assertion that we apply this approximation to a minimal coupling. We note that the gauge covariant derivative satis®es Eq. (87) only if Aa …x† possesses the property Aa …x† ! A0 …x†a ˆ Aa …x† ÿ

1 oh…x† ; e oxa

…88†

the second term being introduced in order to cancel the additional and unwanted term, oh…x†=oxa , in Eq. (85). In classical electrodynamics it suces to apply only the components of the electric and magnetic ®elds strength (E; B) or, equivalently, the electromagnetic ®eld tensor (or ®eld strength tensor) Fab :ˆ

oAb oAa ÿ  oa A b ÿ o b A a ; oxa oxb

…89†

which is itself invariant under gauge transformation (88) and thus the photon kinetic energy is gauge invariant if constructed with Fab . We can write 1 Lphoton ˆ ÿ Fab …x†F ab …x†; 4

…90†

where the factor ÿ14 was included in order to ensure that the Euler±Lagrange equations coincide with Maxwell's equations. A mass term could only be of the form 1 Lmass ˆ ÿ m2photon Aa …x†Aa …x†; 2

…91†

which is not gauge invariant unless mphoton ˆ 0 or [67] Aa …x†Aa …x† ˆ 0: Hence, electrodynamics is locally gauge invariant if the photon mass is zero.

30

…92† 30

We emphasise again that the gauge invariance plays an important role in proving that the theory is renormalizable (see Section 2).

40

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

The Lagrangian for a charged scalar ®eld can be written as 1 2 L ˆ ÿ Fab F ab ‡ gab ‰…oa ‡ ieAa †/ Š‰…ob ÿ ieAb †/Š ÿ l2 // ÿ k…// † 4

…93†

and is invariant under the local Abelian gauge transformation U(1). From the expression of the Lagrangian we observe that Aa is a massless gauge boson. As usually, we look for a minimum in the potential, and ®nd one for k > 0. If l2 < 0, there exists a ring of degenerate ground states. If we assume now that 1 /…x† ˆ p ‰v ‡ n…x† ‡ iv…x†Š; 2

v /vacuum ˆ p 2

…94†

and set

r ÿl2 ; vˆ k

…95†

the Lagrangian (93) becomes 1 1 1 1 1 L ˆ ÿ Fab F ab ‡ e2 v2 Aa Aa ‡ …oa n†2 ‡ …oa v†2 ÿ …2kv2 †n2 ÿ evAa oa v ‡


4 2 2 2 2

…96†

We observe that there appears unexpectedly a term which contains Aa Aa . From a quantum mechanical point of view, this term means that the gauge ®eld Aa has acquired a mass and, at the same time, the gauge invariance is preserved since we started from a gauge invariant Lagrangian (93). We observe also that the Lagrangian (96) describes now the interaction of a massive vector ®eld Aa and two scalar ®elds: the massive n ®eld (the Higgs boson) and the massless v ®eld (the GB). 31 In general, the Higgs mechanism shows that for each vector gauge ®eld that becomes massive we require one complex scalar ®eld, a part of which (involving the GB) becomes unphysical and disappears, leaving one real scalar ®eld, the Higgs boson. We consider however that the GB reappears as the longitudinal mode of the vector ®eld. 15. Scattering of charged particles o€ a cosmic string In classical electrodynamics the 4-vector potential Aa ˆ …A0 ; A† is considered merely an auxiliary quantity from which the electromagnetic ®eld tensor may be derived. Its gauge transformation (88) does not a€ect the physics of the electrodynamic phenomena. However, the Aharonov±Bohm e€ect [68±71] indicates that the vector potential A does have a quantum mechanical signi®cance. In order to understand this e€ect we consider an Aharonov±Bohm arrangement of a two-slit electron di€raction experiment (Fig. 17). We recall that the Hamiltonian for a free non-relativistic electron in the presence of the magnetic vector potential A (and A0 ˆ 0, E ˆ 0) is given by Hˆ

1 2 …p ÿ eA† 2m0

…97†

31 It is interesting to consider the `evolution' of the degrees of freedom in the two versions of the Lagrangian. In Eq. (93) there is one massless vector ®eld (two degrees of freedom corresponding to the two independent transverse modes) and one complex scalar ®eld (two degrees of freedom). In the Lagrangian (96) we have one massive vector ®eld (three degrees of freedom ± the longitudinal mode is now operational) and two real scalar ®elds (two degrees). It appears that we have gained an extra degree of freedom. However, this is only apparent, because by virtue of the gauge invariance we can choose a particular U-gauge in which v does not appear (in other words, the GB v is `gauged-away').

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

41

and thus the Lagrangian becomes Lˆp


oH 1 ÿ H ˆ m0 x_ 2 ‡ ex_
A: op 2

…98†

Hence, the wave function of the free electron, w…x† ˆ w0 eip
x ;

…99†

becomes, in the presence of a magnetic ®eld, w…x† ˆ w0 ei…p
xÿeA
x† :

…100†

It emerges from this expression that the phase is modi®ed by ÿeA
x. In order to determine the total change of phase along a trajectory we apply the formulation of the path integral as used in quantum mechanics. Thus, the change of phase of the wave function along path 1 (see Fig. 17) is proportional to the classical action [73] Z

Z

Q



Q



1 _ ˆ m0 x_ 2 ‡ eA
x_ dtL‰x…t†; xŠ dt S1 ˆ 2 P ;1 P ;1



2ph …d0 ‡ d1 † ‡ e k

Z

Q

P ;1

dx
A:

…101†

Similarly, the change of phase along path 2 is given by 2ph …d0 ‡ d2 † ‡ e S2  k

Z

Q

P ;2

dx
A:

…102†

Consequently, even if the particle, from a classical point of view, is not moving in an electromagnetic ®eld, there arises, however, a supplementary di€erence (Aharonov±Bohm e€ect) in the phases of those parts of the wave function that travel through di€erent slits. This phase di€erence is due to the existence of a nonvanishing vector potential and is given by DhAB

e ˆ h

Z

Q

e dx
A ÿ h  P ;2

Z

Q

e dx
A ˆ h  P ;1

I A
dx:

…103†

Applying the Stokes' theorem, we deduce DhAB

e ˆ h

Z

e $  A
dS ˆ h

Z

e e B
dS ˆ U ˆ BpR2 ; h  h

…104†

where U is the total magnetic ¯ux passing through the solenoid (whisker). We note that the phase di€erence is independent of the position on the screen. Thus, the usual total wave pattern resulting from the passage of electrons through this system will be shifted up or down (depending on the direction of the magnetic ®eld in the solenoid) without any change in its form. This represents the Aharonov±Bohm e€ect which has been con®rmed experimentally [74,75]. We note that from a classical point of view the whisker is a sort of dark matter which may be detected only if we hit it directly. However, from a quantum-mechanical point of view, we can detect the whisker because its external gauge potential changes the phases of electrons passing by. Thus, the gauge vector potential possesses, by its closed line integral, an observable quantum mechanical signi®cance. The Aharonov±Bohm e€ect raises some theoretical questions related to the fact that it reproduces a 2D situation in which the electron moves in a multiply connected space due to the presence of a cylinder of in®nite length that contains a magnetic ®eld [76]. In essence, the Aharonov±Bohm e€ect may also be considered as a di€raction of electrons on a thin (magnetic) cylinder (see Fig. 18a), having no screen with slits. From a topological point of view, the case shown in Fig. 18a does not represent the most general possible one since the electrons can adopt other paths which appertain to di€erent classes of homotopy. For

42

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

Fig. 17. The Aharonov±Bohm two slit electron di€raction experiment. An in®nite solenoid is inserted between the two trajectories of the electron corresponding to the two slits. Since the solenoid is in®nite and very thin, there is no magnetic ®eld outside the solenoid. However, there is a nontrivial nonvanishing gauge vector potential A in this region. We can ensure practically, under certain conditions, the growth of iron crystals which generate very long, microscopically thin ®laments called whiskers [72]. A purely static magnetic ®eld is concentrated into a whisker; the outside ®eld being extremely small can be neglected. A similar situation arises in the case of an interaction of cosmic strings with matter. Fig. 18. The Aharonov±Bohm e€ect simulated by a 2D di€raction of electrons on a thin magnetic cylinder of in®nite length: (a) The case of simple paths; (b) Paths arising through the nth class of homotopy.

example, they may move around the cylinder (see Fig. 18b). 32 A relativistic Aharonov±Bohm e€ect may be observed when increasing the velocity of the electron. It can be shown that the relativistic propagator is similar in form to the non-relativistic one ensuring that the interference pattern remains the same and depends only on the magnetic ¯ux. There is no experimental evidence on these aspects. Another interesting quantum mechanical e€ect (and dual to the Aharonov±Bohm e€ect) arises through the so-called Aharonov±Casher e€ect in which the electron is replaced by uncharged magnetic dipoles (e.g., neutrons), and the whisker is replaced by an in®nite line charge with a charge per unit length of k0 [77]. There exists now an electric ®eld, Eˆ

k0 ^r; 2pr

but no magnetic ®eld. In a semiclassical approximation, the magnetic moment of a neutron, Z ln ˆ d3 xM;

…105†

…106†

is associated with a current density j ˆ $  M;

…107†

where M is the magnetisation density. If the particle (carrier) of the magnetic moment is moving with a velocity vn , special relativity demands the existence of an electrical charge density qˆ

c2

j
vn 1 p  2 j
vn ; 1 ÿ v2n =c2 c

…108†

and thus, the Lagrangian (98) corresponding to the Aharonov±Bohm e€ect must contain now the additional terms

32

The mathematical concept of homotopy is used in order to classify all possible trajectories between two points (P, Q) which a particle may cover. Two trajectories appertain to the same class of homotopy if they can be deformed in a continuous way (i.e. without the intervention of any cut) from one into an other. If there is no obstacle which imposes a circumnavigation of the particle around the obstacle, all the trajectories between P and Q appertain to the same (n ˆ 0) class of homotopy. If an obstacle is present, two di€erent trajectories appertain to the same class of homotopy if they circumnavigate around the obstacle in the same manner.

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

1 Lneutron ˆ mn v2n ÿ 2

Z

d3 xA0 q:

43

…109†

The total Lagrangian corresponding to a system composed by a point charge e of mass m0 moving with a velocity v and a neutron of mass mn moving with a velocity vn is given by Z 1 1 2 2 …110† L ˆ m0 v ‡ mn vn ‡ ev
A…r ÿ R† ÿ d3 xA0 q; 2 2 where A…r ÿ R† represents the vector potential generated by the magnetic dipole, and A0 ˆ

e 4pjr ÿ Rj

…111†

represents the scalar (Coulomb) potential generated by the charge e (see Fig. 19). Gauss' theorem indicates that the last term in the Lagrangian (110) can be written as Z Z 3 ÿ d xA0 q ˆ ÿ d3 xvn
…$  M†A0 ˆ ÿvn
E  ln ;

…112†

where the electric ®eld E is given by E ˆ ÿ$R A0 ˆ

e 4pjr ÿ Rj3

…R ÿ r†:

…113†

It is interesting to observe that the vector potential generated by the magnetic dipole can also be written as A…r ÿ R† :ˆ

1 …R ÿ r†  ln 1 ˆ E  ln ; 4p jr ÿ Rj3 e

…114†

and thus the Lagrangian (110) becomes 1 1 L ˆ m0 v2 ‡ mn v2n ‡ eA…r ÿ R†
v ÿ vn : 2 2

…115†

This last expression of the Lagrangian demonstrates the electric charge-magnetic moment duality characteristic of electromagnetic interactions. In the present case of the Aharonov±Casher e€ect v ˆ 0, and thus the phase di€erence (of the Aharonov±Bohm type) between paths 1 and 2 (see Fig. 17 adjusted to the Aharonov±Casher con®guration) is given by Z I k0 ^ R  ln ˆ k0 ln : …116† DhAC ˆ dtvn
E  ln ˆ dR
2pr The Aharonov±Casher e€ect has been con®rmed experimentally [78]. Another alternative to the Aharonov±Bohm e€ect may be realised with a rotating superconducting con®guration (see Fig. 20). In terms of the metric tensor and an inertial frame of reference, the Lagrangian de®ning the motion of an electron e in an electromagnetic ®eld Aa is given by q 1 …117† Linertial ˆ ÿm0 c2 gab x_ a x_ b ÿ eAa x_ a  ÿm0 c2 ‡ m0 v2 ÿ eA0 ‡ ev
A ‡


; 2 where gab ˆ d‡1; ÿ1; ÿ1; ÿ1; c is the Minkowski metric. In a non-inertial (accelerated) frame of reference or, equivalently an application of the Einstein principle of equivalence, in a non-permanent gravitational ®eld, the Lagrangian (117) takes the form q …118† Lnon-inertial ˆ ÿm0 c2 gab x_ a x_ b ÿ eAa x_ a : In the particular case of a rotating frame described by the transformations of coordinates (in what follows we consider that c ˆ 1)

44

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

2 3 2 t 1 6x7 60 6 7 6 6 7ˆ6 4y5 40 0 z

0 cos xt0 sin xt0 0

0 ÿ sin xt0 cos xt0 0

32 0 3 2 3 t t0 0 6 07 6 0 0 0 07 07 76 x 7 6 x cos xt ÿ y sin xt 7 76 0 7 ˆ 6 0 7; 0 54 y 5 4 x sin xt0 ‡ y 0 cos xt0 5 z0 z0 1

…119†

the metric tensor is given by 0 ˆ 1 ÿ …x  x0 †
…x  x0 †; g00

‰xa ˆ …t; x†Š;

0

…120†

g0i0

ˆ …x  x†i ;

…121†

gij0

ˆ ÿ dij ;

…122†

…i; j ˆ 1; 2; 3†:

Now the Lagrangian (118) becomes 1 Lnon-inertial  ÿm0 ‡ m0 v02 ÿ eA0;eff ‡ ev0
Aeff ‡


; 2

…123†

where m0 m0 0 …x  x0 †
…x  x0 † ˆ A0 ÿ …1 ÿ g00 †; 2e 2e m0 ˆ A ÿ …x0  x†; e m0 0 ˆ Ai ÿ g0i e

Aeff;0 ˆ A0 ÿ Aeff Aeff;i

…124† …125† …126†

represent e€ective scalar and vector potentials. With these quantities the Lagrange equation of motion, d oLnon-inertial oLnon-inertial ÿ ˆ0 ov ox dt

…127†

takes the form of an electromagnetic (Lorentz) equation of motion, 0 ˆ e…E eff ‡ x_ 0  Beff †; m0 x

…128†

where m0 x  …x  x0 †; e 2m0 x ˆB‡ e

E eff ˆ E ‡

…129†

B eff

…130†

are e€ective (or generalised) electric and magnetic ®elds, respectively. The special (inertial or gravitational) terms which emerge in these ®elds, noting that we have in this case a rotating system, correspond to the centrifugal and Coriolis forces. At this point we emphasise that we established a new physical signi®cance of the London moment [79]. Indeed, when a superconductor is subject to an angular velocity x relative to a local inertial frame, there emerges the London moment, i.e. a London magnetic ®eld B L given by BL ˆ

2m0 x e

…131†

inside the superconductor. We observe that the London magnetic ®eld represents precisely an extra magnetic ®eld due to a (inertial) non-permanent gravitational ®eld. London equation (131) expresses in fact the Meissner e€ect. Thus B eff ˆ 0 inside a (rotating) superconductor. Thus, the London moment is a consequence of a (general) relativistic invariance of the Meissner e€ect. The experimental setup shown in Fig. 20 is equivalent to that of a conventional Aharonov±Bohm e€ect. The e€ective vector potential is now given by

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

45

Fig. 19. A 3D coordinate system which describes the system consisting of a point charge e (an electron) and a point magnetic dipole (a neutron). Fig. 20. Aharonov±Bohm e€ect within a rotating superconductor (here, a rotating SQUID ± Superconducting quantum interference device). Here, as against the conventional Aharonov±Bohm con®guration (Fig. 17), the in®nite solenoid is replaced by an empty (i.e. nonsuperconducting) region with B eff 6ˆ 0 (for r < R). All other items of the experimental setup are embedded within a superconductor ^ The role of the two (di€raction) (where B eff ˆ 0 for r > R, the Meissner e€ect). The entire system is given an angular velocity x ˆ xk. slits used in the conventional Aharonov±Bohm con®guration is replaced here by two Josephson junctions. At a cosmic scale, this experimental arrangement may arise with superconducting cosmic strings and the presence of cosmic inhomogeneities can lead to cosmic Josephson junctions (weak links) and thus some intermittency routes to a cosmic chaos can be constructed.

Aeff

8 > <

m0 ^ xru e ˆ 2 > : m0 xR u ^ e r

9 if r < R; > =

…132†

> if r > R: ;

It emerges from this that the Aharonov±Bohm phase shift between paths 1 and 2 (see Fig. 20) can be written as Z I …133† Dhrot:super: ˆ q dtAeff
v ˆ 2e Aeff
dr ˆ 4pm0 xR2 ; where we have considered that, in a superconductor, a pair of electrons carries the electric charge q ˆ 2e. We note that the phase di€erence may be changed by variation of the angular velocity x. Generally, for a given junction area (usually a few square millimetres) the supercurrent satis®es the Josephson equation (see, for instance, [80]),   2eU t ‡ h0 ; …134† I ˆ Ic sin h ˆ Ic sin h  where Ic is the critical current. This equation reproduces directly the d.c. Josephson e€ect if U ˆ 0. and the a.c. Josephson e€ect (i.e. oscillations of a current with a frequency of mJ ˆ 2eU ) if a non-zero voltage (U 6ˆ 0) h is applied to the junction. In the Superconducting quantum interference device (SQUID) illustrated in Fig. 20, U ˆ 0, and thus the total (net) current ¯owing through this interferometer depends on the phase di€erence between the two paths 1 and 2. This phase shift can be written as Itotal ˆ Ic …sin h1 ‡ sin h2 † ˆ Imax cos

h1 ‡ h2 ; 2

…135†

46

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

where h1 ÿ h2 : …136† 2 However, since the SQUID is rotating and there emerges a London (``inertial'') magnetic ®eld, the maximum current must contain an extra phase di€erence of the Aharonov±Bohm type, i.e.     I I h1 ÿ h2 h1 ÿ h2 ‡ e Aeff
dr ˆ 2Ic sin ‡ e 2m0 ex dSempty Imax ˆ 2Ic sin 2 2 Imax ˆ 2Ic sin

ˆ 2m0 xSempty ;

…137†

where Sempty represents the area of the empty (non-superconducting) region between paths 1 and 2. We emphasise that the phase di€erence induced by a rotation may be changed by varying the angular velocity x. This e€ect has been con®rmed experimentally [81,82]. 16. Conclusion and open questions The main conclusions of the present work are as follows. (1) Astronomical observations of galaxy±galaxy and cluster±cluster correlations and, also of other largescale structures show that the space distribution of these astrophysical objects on scales less than 50 million light years is well approximated by a truncated fractal with dimension Dfractal  1:2. The observed limited fractal appears as it has grown from 2D sheet-like objects such as domain walls and cosmic string wakes. Such a (natural) tendency towards a 2D space distribution (which in condensed matter physics explains, for example, the fractional quantum Hall e€ect and a high Tc superconductivity) lead us to consider that, at a cosmic scale, there occurs a gravitational anyonisation and, on the other hand, the quantum Hall e€ect and superconductivity are caused by a quantum fractal structure. At very large scales, the galaxy distribution appears to shift from fractal power law to a random behaviour consistent rather with spatial homogeneity as predicted by Einstein's cosmological principle. Indeed, a pure (mono) fractal cannot explain the structure of the Universe because of the isotropy of microwave radiation. (2) However, the fractal structure of the Universe can still be maintained at any scale if we assume that the locally observed bounded galactic fractal represents in fact a fractallite which is merely a fractal element of a `poly-fractal' or a `quasi-fractal' Universe in which fractallites have di€erent orientations and hereby, at very large scales, the Universe seems homogeneous. Cosmic fractallites are localised in space independently and uniformly, at random, and can be considered as cosmic Levy ¯ights for which we can write a Schr odinger±Wheeler±DeWitt equation and de®ne a fractal wave function of the universe. The natural truncation of the cosmic fractals and their randomness can be explained from a crystallographic point of view. Cosmic fractallites are, ®rst, generated by primordial seeds or primeval ¯uctuations (density perturbations with a fractal string and sheet-like structure) and, then, by an aggregation of matter to these seeds resulting from a growth process governed by di€usion and gravity which lead to the observed distribution of galaxies in the Universe. (3) A fractal structure of spacetime, matter and ®elds can lead to deviations from the 1r Newtonian gravitational potential and also to a non-Einsteinian gravitation (Rossler conjecture). The various exotic `®fth', `sixth', and all such forces may have their origins in fractal±vacuum ¯uctuations of spacetime. (4) Not only spacetime and phase space can be endowed with a fractal structure, but also an internal symmetry space, which underlies for instance a Higgs ®eld, may have fractal (or crystal) features. Here, we have the situation that the topological defects occur in between domains of internal space with a di€erent direction of SSB (i.e. a di€erent direction of the Higgs ®eld). This is similar to the situation when a liquid crystallises and generates crystallites, crystal defects and a polycrystal. (5) If indeed (empty) space does have a fractal structure in the sense that, for example, particles are (self-) con®ned to move on pre-existing fractal trajectories, following the decay or evaporation of a material fractal structure there remains a fractal skeleton which represents an empty fractal space structure. Thus, a structured material formation can represent, in fact, a manifestation of a pre-existing fractal structure of

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48

47

space (which we call fractal skeleton). Di€raction and difusion experiments with a (empty space) skeleton and very low density matter could display evidence for a fractal structure of space. (6) A fractal structure of space can lead to a new interpretation of the Heisenberg uncertainty principle. (7) Possibly, the reader of the present paper will have wondered that we de®ne a (quasi) fractal structure of the Universe and, as well as, a (quasi) crystal structure. Leaving aside the fact that an anisotropic crystal may be considered as a particular kind of a fractal and that the process of crystallisation has some common features with PT (or symmetry breaking) in the early Universe, we note that some 3D galaxy-surveys demonstrate a periodicity: galaxies reside at discrete peaks separated by 130 hÿ1 0 Mpc in variuos directions. This may be also a sign of multi-connectedness of the Universe [53]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

El Naschie MS. Dimensional symmetry breaking, information and fractal gravity in Cantorian space. BioSystems 1998;46:41±6. Omohundro SM. Geometric perturbation theory in physics. Singapore: World Scienti®c; 1986. Grandy WT Jr., Foundation of statistical mechanics, vol. I: Equilibrium theory. Dordrecht: D. Reidel, 1987. Rueda A, Cavalleri G. Zitterbewegung in stochastic electrodynamics and implications on a zero-point ®eld acceleration mechanism. Nuovo Cimento 1983;6C(3):239±60. Peebles PJE. Principles of physical cosmology. Princeton, NJ: Princeton University Press; 1993. El Naschie MS. On the 26-dimensional Leech lattice and E1 . Chaos, Solitons & Fractals 1999;10(11):1813±9. DelaPe~ na L, Cetto AM. Quantum phenomena and the zeropoint radiation ®eld. Found Phys 1994;24(6):917±25. Felegara A. The relevance of the Algebraic approach to the quantum ®eld theory in curved backgrounds. Fortschr Phys 1989; 7(1-2):117±23. Semiz I. Black hole as the ultimate energy source. Am J Phys 1995;63(2):151±6. Argyris J, Ciubotariu C. Complexity in spacetime and gravitation I. From chaos to superchaos. Chaos, Solitons & Fractals 1998;9(10):1651±701. Leinaas J. Hawking radiation, the Unruh e€ect and the polarisation of electrons. Europhys News 1991;2(4):78±80. Matsas GEA. Do inertial electric charges radiate with respect to uniformly accelerated observers. Gen Rel Grav 1994;26(11): 1165±9. Leader E, Predazzi E. An introduction to gauge theories and the new physics. Cambridge: Cambridge University Press; 1982. Kibble T. Of particles, pencils, and uni®cation. Preprint 1993, Department of Physics, Imperial College, London, UK. Mahajan SM. Field, Particles and Universe. Lecture Notes, Spring College on Plasma Physics, 17 May ± 11 June 1993, H4-SMR 692/29, Preprint ICTP, Trieste. Hands S. Ripples at the heart of physics. Preprint 1993, Theory Division, CERN, Geneva, Switzerland. Copeland EJ. Cosmic strings. A particle physics solution to the large-scale structure of the Universe. CERN Courier 1991;31(6):17±9. Martin JL. General relativity: a guide to its consequences for gravity and cosmology. Chichester: Ellis Horwood Limited; 1988. Chang T. Maxwells equations in anisotropic space. Phys Lett 1979;70A(1):1±2. Piran T. On fravitational repulsion. Gen Rel Grav 1997;29:1363±70 (e-print http://xxx.lanl.gov/abs/gr-qc/9706049, Third prize in 1997 Gravity essay competition). Hagai, El-Ad, Piran T. Voids in the large-scale structure. Astrophys J 1997; 491: 421±39 (e-print archive, http://xxx.lanl.gov/abs/ astro-ph/9702135). Piran T, Lecar M, Goldwirth DS, NicolacidaCosta L, Blumenthal GR. Limits on the primordial ¯uctuation spectrum void sizes and CMBR anisotropy, e-print archive http://xxx.lanl.gov/abs/astro-ph/9305019, 17 May 1993: 23. Rosen G. Black holes associated with galaxies. Int J Theor Phys 1991;30(11):1517±20. Vilenkin A. Cosmic strings and other topological defects, In: Humitaka S, Takeo I, editors. Quantum gravity and cosmology. Singapore: World Scienti®c, 1985: 269±300. Witten E. Superconducting strings. Nucl Phys B 1985;249:557±69. Linet B. Electro-gravitational conversion in the spacetime of an in®nite straight superconducting cosmic string. Phys Lett 1990;146A(4):159±61. Carter B, Peter P, Gangui A, Avoidance of collapse by circular current-carrying cosmic string loops. Phys Rev D 1996;54:4647±62 (e-print archive, http://xxx.lanl.gov/abs/hep-ph/9609401, 18 September 1996). Brandenberger R, Carter B, Davis A-C, Trodden M. Cosmic vortons and particle physics constraints. Phys Rev D 1997;55:6059± 71 (e-print archive, http://xxx.lanl.gov/abs/hep-ph/9605382, 23 May 1996). Martinez VJ. Is the universe fractal? Science 1999;284:445±6. Peebles PJE. The fractal galaxy distribution. Physica D 1989;38:273±8. Luo X, Schramm DN. Fractals and cosmological large-scale structure. Science 1992;256:513±5. El Naschie MS. Time symmetry breaking, duality and Cantorian space-time. Chaos, Solitons & Fractals 1996;7(4):499±518. Argyris J, Ciubotariu C, Weingaertner WE. Progress in physical concepts of string and superstring theory ± Thirty years of string theory. Chaos, Solitons & Fractals 1999;10(2±3):225±56. DerSarkissian M. Possible evidence for gravitational Bohr orbits in double galaxies. Lettere al Nuovo Cimento 1985;44(8):629±36.

48 [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]

J. Argyris et al. / Chaos, Solitons and Fractals 12 (2001) 1±48 Isham C. Quantum gravity. In: Davies P, editor. The new physics, Cambridge: Cambridge University Press, 1992. Ball RC, Witten TA. Causality bound on the density of aggregates. Phys Rev A 1984;29(5):2966±7. Ord GN. Fractal space-time and the statistical mechanics of random walks. Chaos, Solitons & Fractals 1996;7(6):821±43. Mandelbrot BB. The fractal geometry of nature. New York: Freeman; 1983. Kolb E, Turner MS. The early universe. San Francisco: Addison-Wesley; 1989. Argyris J, R ossler OE. Distance-proportional redshift in 1+e dimensional matter. Preprint, University of T ubingen, DTC, June 1999. DeSabbata V, Sivaram C. Fifth force as a manifestation of torsion. Int J Theor Phys 1990;29(1):1±6. Corinaldesi E, Strocchi F. Relativistic wave mechanics. Amsterdam: North-Holland; 1963. Nelson E. Derivation of the Schr odinger equation from Newtonian mechanics. Phys Rev 1966;150:1079±85. Reuter M. Gravitational anyonisation. Preprint DESY 91-007, ITP-UH 1/91, Institute of Theoretical Physics, University of Hannover, February 1991. Gasperini M. Broken Lorentz symmetry and the dimension of space-time. Phys Lett B 1986;180(3):221±4. El Naschie MS. Dimensional symmetry breaking and gravity in Cantorian space. Chaos, Solitons & Fractals 1997;8(5):753±9. Long DR. Vacuum polarisation and non-Newtonian gravitation. Nuovo Cimento 1980;55B(2):252±6. El Naschie MS. A note on quantum gravity and Cantorian spacetime. Chaos Solitons & Fractals 1997;8(1):131±3. Vilenkin A. Cosmic strings and domain walls. Phys Rep 1985;121(5):263±315. Sornborger A. The structure of cosmic string wakes. PhD Thesis, Department of Physics, Brown University, 1996. Eddington AS. The Mathematical Theory of Relativity. Cambridge: Cambridge University Press; 1924. Luminet J-P, Starkman GD, Weeks JR. Is space ®nite? Scienti®c American 1999;280(4):68±75. Lachieze-Rey M, Luminet J-P. Cosmic topology. Phys Rep 1995;254(3):135±214 (xxx.lanl.gov/abs/gr-qc/9605010). Bak D, Kim SP, Kim SK, Soh K-S, Yee JH. Fractal di€raction grating, Preprint available at http://xxx.lanl.gov/abs/physics/ 9802007 on the World Wide Web. Nottale L. Scale relativity, fractal space-time and quantum mechanics. Chaos, Solitons & Fractals 1994;4:361±88. Reprinted in: El Naschie MS, R ossler OE, Prigogine I. Quantum mechanics, di€usion and chaotic fractals. Oxford: Elsevier, 1995: 51±78. Uozumi J, Peiponen K-E, Savolainen M, Silvennoinen R, Asakura T. Demonstration of di€raction by fractals. Am J Phys 1994;62(3):283±5. Nelson E. Quantum ¯uctuations. Princeton, NJ: Princeton University Press; 1985. Ord GN. Fractal space-time: a geometric analogue of relativistic quantum mechanics. J Phys A: Math Gen 1983;16:1869±84. Collins PDB, Martin AD, Squires EJ. Particle physics and cosmology. New York: Wiley; 1989. Sterman G. Quantum ®eld theory. Cambridge: Cambridge University Press; 1993. Davies P. Superforce. London: Unwin Hyman Ltd, 1987. El Naschie MS. Stress, stability and chaos in structural engineering: an energy approach. London: McGraw-Hill; 1990. Nicolis G. Physics of far-from-equilibrium systems and self-organisation. In: Davies P, editor. The new physics, Cambridge: Cambridge University Press, 1992. Peat FD. Superstrings and the search for the theory of everything. contemporary Books, 1988. Rae IM. Quantum mechanics. 3rd ed. Bristol: Institute of Physics Publishing, 1993. Lawrie ID. A uni®ed grand tour of theoretical physics. 3rd ed. Bristol: Institute of Physics Publishing, 1990. Argyris J, Ciubotariu C. A metric for an Evans±Vigier ®eld. Found Phys Lett 1998;11(2):141±63. Bohm D, Aharonov Y. Signi®cance of electromagnetic potentials in the quantum theory. Phys Rev 1959;115:485±91. Bohm D, Aharonov Y. Further considerations on electromagnetic potentials in the quantum theory. Phys Rev 1961;123:1511±24. Peshkin M, Tonomura A. The Aharonov±Bohm e€ect. New York: Springer; 1989. Holstein BR. Variations on the Aharonov±Bohm e€ect. Am J Phys 1991;59(12):1080±5. Felsager B. Geometry, Particles and ®elds. Odense: Odense Uniersity Press; 1981. Feynman RP, Hibbs AR. Quantum mechanics and path integrals. New York: McGraw-Hill; 1965. M ollenstedt G, Bayh W. The continuous variation of the phase of electron waves in ®eld free space by means of the magnetic vector potential of a solenoid. Phys Bl att 1962;18:299±310. Tonomura A, Matsuda T, Suzuki R, Fukuhara A, Osakabe N, Umezaki H, Endo J, Shinagawa K, Sugita Y, Fujiwara H. Observation of Aharonov±Bohm e€ect by electron holography. Phys Rev Lett 1982;48:1443±6. Gamboa J, Rivelles VO. Quantum mechanics of relativistic particles in multiply connected spaces and the Aharonov±Bohm e€ect. J Phys A 1991;24:L659±66. Aharonov Y, Casher A. Topological quantum e€ects for neutral particles. Phys Rev Lett 1984;53:319±21. Cimmins A, Cimmino A, Opat GI, Klein AG, Kaiser H, Werner SA, Arif M, Clothier R. Observation on the topological Aharonov±Casher phase shift by neutron interferometry. Phys Rev Lett 1989;63:380±3. London F. Super¯uids. vol. I. New York: Wiley, 1950. Argyris J, Ciubotariu C. A nonlinear chaotic gravitational-wave detector. Chaos, Solitons & Fractals 1999;10(1):51±76. Semon MD. Experimental veri®cation of an Aharonov±Bohm e€ect in rotating reference frames. Found Phys 1982;12:49±57. Zimmerman JE, Mercereau JE. Compton wavelength of superconducting electrons. Phys Rev Lett 1965;14:887±8.