Extended systems and generalized London equations

Volume 67B, number 3

PHYSICS LETTERS

EXTENDED

SYSTEMS AND GENERALIZED

11 April 1977

LONDON EQUATIONS

A. AURILIA lstituto Nazionale di bTsica Nucleare, Sezione di Trieste, Trieste, Italy

F. LEGOVINI Istituto di Fisica Teorica dell'Universita di Trieste, and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy

Received 25 January 1977 We propose a classical action for interacting extended objects in n spacetime dimensions. The theory of membranes closely resembles the classical Schwinger model. We explain this analogy and suggest a natural trapping mechanism for quarks according to Nambu's "electric" confinement scheme. Some time ago, Nambu suggested a scheme of quark confinement [ 1 ] which supports the view that relativistic London-type equations [ 2 - 4 ] may provide an appropriate phenomenological framework for describing the motion of extended objects. Nambu's approach originates from the classical interstring action considered by Kalb and Ramond [5]. More recently, the same string theory was reproposed by Lurid and Regge [6]. We wish to generalize this string theory to extended systems of any dimensionality. Our problem is that of describing the geometry and the dynamics of spatially closed timelike hypersurfaces embedded in an n-dimensional Minkowski space. The natural formalism for this appears to be the exterior calculus. Thus we consider the following system of defining equations for an ( n - r)-dimensional hypersurface (r = 1 ..... tl - 1 ) , parametrized by (n - r ) coordinates ~a:

xU=x~,(~a) ta=O,...,n-I a =O,...,n-r-

(1)

1 "

The invariant element of "area" is d Y. = dn-r ~X,ff~/],

r / - det (r/ab) ,

(2)

where rlab is the metric induced on the surface (3)

~x~ ~x u n~b = a~/--D- a~--g "

To this element of area we associate the (n - r)-form (4)

d x ui A ... A d x un-r =-dn-r~ 3 Ix ui .... ' xUn-r] ,

a [~0, ..., ~n-r- I] where the symbol ;) Ix ~ l .... , x un-r ]/a [~j0, ..., ~ n - r - 1 ] indicates the jacobian of the mapping ( 1). Then our general action for such (classical) system is

S=-M f[(-1)"-r-1 (dxU' ^'"^dxU"-r)2] I/2 ( -1)'-r-1 ~--r~. ] (n-r)]

e a{"d x U i A ' " A d x u n - r A " i ' " t a n - r

( _ l ) n - r - I fdnxFUi...Un_r+l 2 [ ~ r + i - ) . T] Ful'"Un-r*l " 299

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The "hypersurface tension" M has physical dimension (n - r) in units of mass, while the coupling constant e has physical dimensions (n/2) - r + 1. The action (5) is parametrization invariant and reflects a generalized gauge principle analogous to that of electrodynamics in ordinary Minkowski space. According to (5), in n spacetime dimensions the geometric degrees of freedom of the extended object contribute to the action in proportion to the invariant "area". Then, the minimal coupling of the surface elements of the time track of the extended object in Minkowski space leads to the introduction of a hierarchy of completely antisymmetric potentials A ut'''un-r(x). The corresponding antisymmetric field tensors FUl ...Un-r÷l (x) - ~ OUlAU2""Un-r÷l (x) , cycl.

(6)

are invariant under the generalized gauge transformation

A"' """"-~(x) + A"'"u"-~(x) + ~ a"~ a"2 ""~-~(x), cycl.

(7)

where AU2""Un-r(x) are antisymmetric but arbitrary tensor functions in n-dimensional spacetime. The Hamiltonian systems under consideration are constrained, thus reducing the effective number of dynamical degrees of freedom per space point. According to Dirac's formulation of constrained Hamiltonian systems [7], one finds that the ( n - r)-dimensional hypersurface effectively possesses r geometric degrees of freedom while the field tensor has (n- 5 independent components for r 1> 2 while for r = 1 the canonical variables are all constrained. It must be noted that these geometric degrees of freedom carry energy and momentum in contrast to those associated with a "region" of an n-dimensional spacetime (e.g., the MIT-bag in four dimensional Minkowski space [8]). By minimizing the action one derives the equations of motion in general form. This will be done in detail in a forthcoming article. Presently, we are mainly interested in the field equation OuI F m ...,n-r+l (x) = eG u~'''un-r+l ( x ) ,

(8)

where

Gm'"Un-r+' - f dYU'^'"^dYUn-~"~(n)(x-Y)= f dn-~ \(+[Y"+ a-~, ....~ ,Y"-++' l)+(.)[x_y(+0,...,+n-r-t)l

" (9)

Following Nambu [ 1], the London equation associated with a spatially closed hypersurface is the massive version of the field equation (8)

au F ul"''un-r÷~ (x) + m2A u2"''un-r+' (x) = eG uz'''un-r÷l (x),

(10)

with F m'''un-r+~ (x) explicitly given by eq. (6). Presumably, the mass term can arise, at the quantum level, via the spontaneous breakdown of our generalized gauge invariance [ 1,3]. In the static case, the source term in eq. (1 O) describes a 6-type singularity on the (oriented) hypersurface

X0= ~0;

xi=xi(~l,...,~n-r-l),

i = 1, ...,n -- 1.

(11)

This hypersurface singularity is thus the generalization of the line singularity encountered in the description of vortices in the London approach to superconductivity. For n -- 4, r = 3, the action (5) describes the conventional electrodynamics of point charges. However, one can also consider the non trivial embedding in four dimensional Minkowski space of two and three dimensional hypersurfaces corresponding to spatially closed strings and membranes. The string theory derived from the action (5) is known [5, 6]. There are two geometric degrees of freedom while the interstring action is effectively mediated by a massless scalar field. For spatially closed membranes one finds only one geometric degree of freedom. A spherical membrane can only "breathe". A free closed membrane could be viewed as a classical realization of the single "bubble states" of the SLAC-bag model in the strong coupling limit [9]. In the Lorentz gauge auAUVO = 0, the 300

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11 April 1977

retarded potential is

AUVp(x ) = ~nnf d4y O(x 0 _ yO)6 [(x _ y ) 2 ] GUVP(y) ,

(12)

and GUVP(x) is defined by eq. (9). The corresponding field tensor is

FUV°O(x) = euv°°f(x),

e

f ( x ) = ~ e"at3~r~ f O ( x ° - y ° ) 8 ' [(x __y)2] (x - .v ) ~ d y ~ A d y . r A d y ~ .

(13, 14)

If the bubble is static,f(x) is proportional to the solid angle subtended by the bubble at the point of observation. In spite of the tensor character of the potential, the canonical variables aa'e completely constrained and the effective bubble interaction is purely of the "Coulomb-type". The potential of a spherical bubble increases with the radius of the bubble. Such behaviour is not accidental and is suggestive of a trapping mechanism for the hadron constituents in ordinary Minkowski space. Indeed, for r = 1 the action (5) will display the same qualitative features as above regardless of the number n of spacetime dimensions. Thus for n = 2 our action gives the classical version of the Schwinger model [ 10]. In two spacetime dimensions, spatially extended systems collapse into pointlike objects (as pointed out earlier, we have excluded "regions" from our considerations). A "bubble" reduces to a pair of point charges with opposite sign. For a static bubble, the field tensor FuL, is a step function and the "Coulomb potential" increases linearly with distance. In the quantized version of the model, this property results in the well known confinement mechanism for quarks [ 11 ]. The physical spectrum consists of a massive spinless field which, in the presence of an external source, obeys a field equation of London's type. Presumably, in the quantum field theory of bubbles, the Schwinger mechanism would set in precisely as it does in two dimensional quantum electrodynamics. This mechanism would lead to the following London-type equation for bubbles in equilibrium against the pressure of a surrounding superfluid,

([3 + mZ) AvO°(x) - Ou OVAO°U(x) + a u i)OA°UV(x ) - ~uO°AtaVO(x) = eGVO°(x) .

(15)

Eq. (15) is equivalent to the set of equations

(I--1 + m2)AUVO(x) = eGUVO(x),

auAUVO(x ) = 0 .

(16)

In the absence of a source term, the equations above describe the free field theory of a massive spinless field. This is precisely analogous to the situation encountered in the Schwinger model. The above considerations can be generalized to any extended object. A more detailed analysis of the general action will be given in a forthcoming article. We wish to thank Drs. A.P. Balachandran, D. Christodoulou and S. Hojman for many informative discussions. Also, we wish to thank Mr. P. Campbell and Dr. C.T.J. Dodson for comments on the manuscript and for drawing our attention to Tucker's lecture notes [12].

References

[1] [2] [3] [4] [5] [6]

Y. Nambu, Phys. Reports 23C (1976) 250. F. London and H. London, Proc. Roy. Soc. A149 (1935) 71. Y. Nambu, Phys. Rev. D10 (1974) 4262; A. Aurilia, Nucl. Phys. B92 (1975) 241. M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273. F. Lund and T. Regge, Unified Approach to Strings and Vortices with Soliton Solutions, Inst. for Adv. Study preprint (Princeton, May 1976). [7] P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University (New York, 1964); See also A.J. Hanson, T. Regge and C. Teitelboim, Constrained hamiltonian systems, to be published by Accademia dei Lincei, Rome. 301

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A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. I39 (1974) 3471. W.A. Bardeen et al., Phys. Rev. D11 (1974) 1094. J. Schwinger, Phys. Rev. 128 (1962) 2425. L. Susskind, Lectures at Bonn Summer School, 1974; L. Susskind and J. Kogut, Physics Reports 23C (1976) 348; C.A. Carlson, L.N. Chang, F. Mansouri and J.F. Willemsen, Phys. Rev. D10 (1974) 4218. [12] R.W. Tucker, Extended particles and the exterior calculus, Rutherford Laboratory preprint RL-76-022.

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