- Email: [email protected]
A reexamination of unified field theory
ANNALS
OF PHYSICS
59,
27-41 (1970)
A Reexamination
of Unified
Field Theory
M. MURASKIN University of North Dakota, Grand Forks, North Dakota 58201 Received July 7, 1969
We study a nonlinear field theory with a nonsymmetric I’;, which has the property that all tensor functions of g,, , eSLi,r,!, , and 8, are treated in a uniform manner. All scalar functions of gij , eair r;, and a, are constants in such a theory. If the theory is to be meaningful, it is necessary for a set of consistency conditions to be satisfied. We, then, show that nontrivial solutions to these consistency equations do exist. We describe, in another paper, results of a finite difference approximation to the field equations which we have obtained using a computer. We show that the field equations are asymptotic to the conventional wave equation. We find particle-like behavior can be made to appear at an origin point. The theory does not involve any arbitrary functions. Thus, there is less arbitrariness in this theory as compared with other hyperbolic type theories.
1.
INTRODUCTION
What is characteristic of the universe is change. By universe, we mean a system that can be described by a set of numbers at each point. The way these numbers change from point to point is given by the field equations. This is the underlying hypothesis of any so-called Unijied Field Theory. Once we are given the field equations, then, in principle, we would know all that is possible about the universe. The search for such a basic set of field equations has had a long but unsucessful history. After the successesof General Relativity, Einstein sought to generalize the gravitational equations to include electromagnetism. Since then, many researchers have tried their hand at formulating a Unified Field Theory [l]. The results of their work can be summarized as follows. The end results of these investigations are sets of equations that are generalizations of Maxwell’s Equations and Einstein’s gravitational equations. Now, it is of course, possible to generalize Maxwell’s and Einstein’s equations in an infinite number of ways. There is, at present, no experimental confirmation of any of the equations proposed. Also, the equations are, invariably, so complicated that it is difficult to make clear-cut predictions of what the equations really imply. Thus, there is no reason to believe any of these possible generalizations, at this time. 27
28 However, above. The Lagrangian The field
MURASKIN
the difficulty with these generalizations lies deeper than our discussion conventional method for obtaining field equations is by means of a approach. equations emerge from the action principle,
where 9 is the Lagrangian density and g is the determinent of a symmetrical second-rank tensor. The problem, then, is to specify 9 and the field variables which are to be varied by the operation 6. Unfortunately, this problem suffers from a complete lack of uniqueness. What is usually done is to stipulate that the field equations should involve second derivatives of g,$ , at most. This limits the choice of 9 considerably. The only reason offered for this distinguished property of the second derivative is that Newton’s gravitational equations have such a structure. However, this can hardly be taken as a serious argument. Rainich [2] has proposed equations that involve fourth derivatives of gij . Still, there is no more reason to distinguish the fourth derivative than it was to distinguish the second. Incidently, Rainich’s equations have a structure that closely resembles Maxwell’s equations and Einstein’s equations. However, in view of the infinite number of generalizations possible, we should be leery of accepting this particular one. Thus, with the obvious arbitrariness of the basic equations thus far proposed, and with the difficult mathematics that one encounters in trying to learn what the proposed equations imply, the progress in Unified Field Theory is, we feel it is not unfair to say, unimpressive. One may feel somewhat uncomfortable with the whole approach based on an action principle. We may ask what is the meaning of the variational principle, (1.1). The answer is that it is a mathematical tool for obtaining field equations. We may then ask if there is a way to motivate the equations on physicuI rather than mathematical grounds. The difficulty is this: On a fundamental level, we have no rods, clocks, and particles to give us physical intuition. After all, we expect a particle to be a complicated manifestation of the field. Thus, we would agree with Einstein and Schrodinger, for example, that a mathematical approach is necessary. That is, in looking for the field equations, we must not use terms like particles, forces and other expressly physical terms. Thus, far, we have encountered some of the difficulties confronting Unified Field Theory. We might say that, in present approaches, the essential difficulty is that of mathematical inelegance. That is, the approach of Einstein and Schrodinger (a prototype of Unified Field Theory) distinguishes particular scalars and particular orders of derivatives. This leads to unconvincing theories.
UNIFIED FIELD THEORY
29
Let us take a different approach. We shall find it possible to motivate a theory which does not make the unreasonable restrictions of the preceding paragraph. There exists also a means to study what information is contained in our equations. Thus, hopefully, it may be eventually possible to see how reasonable the equations are. In the remainder of this article, we shall describe our approach to the problem.
2. FORMULATION
OF THE THEORY
We set up a coordinate system that is Cartesian in space and has a 4-th component x0 = ct with c constant. In fact, we may set up any coordinate system we like since coordinate systems are constructs of the mind. We choose the simplest coordinate system we can think of. We have introduced a time axis since we shall anticipate a wave character to our field equation (a propagation cone in a wave theory requires a hyperbolic g,J. We shall restrict ourselves to four dimensions. The basic equations are not dimension dependent in their derivation so we shall stay with four dimensions. We shall consider only linear coordinate transformations. We shall not make general coordinate transformations since they only complicate the state of affairs by bringing in coordinate fields. It is not that we couldn’t introduce general coordinates if we really insisted. It’s a question really of “who needs it.” In General Relativity, one has covariance under general coordinate transformations. This will not be the case in our approach. This principle of general covariance has come under strong attack in recent years. Fock [3] has argued that the empirical results of gravitational theory can be obtained without general covariance. In his work there is a subsidiary condition that is not covariant. That is, this condition takes on a more complicated form when one makes general coordinate transformations. Our belief is that the principle of general covariance mixes up field effects with coordinate effects and thus, should not be taken as a basic principle. We assume the existence of a universal change function that determines the change of vector functions according to dAi = r&Ajdxk.
(2-l)
We may ask why we have extracted an A, on the right side of (2.1). The reason is essentially that it leads to the desirable properties that we list later on. g,, is taken to act like a product of two vectors. Thus, its change is given by &ir = V’hti
+ r;k git) dxk.
The change function itself is taken to act like a product of three vectors.
(2.2)
30
MURASKIN
Thus, the change function determines its own change according to
drjk = (rAkr;t + q&r; - r;r;l) dxk.
(2.3)
A set of basis vectors is denoted by Pi (the invex 01tells which of the four basis vectors we are considering). Their change is given by de”, = Assuming continuity
fikeuj
dxk.
(2.4)
for the fields, we get (2.5)
This is our basic field equation. We also get agik -
a2
-
mmk
-
r:gim
=
gik:L
=
OF
and (2.7)
edli is the dual field defined by eOLiesi= PB . By writing gjj = e@jesjgorB
w-9
and (2.9) with g,, , r;,, constant (gUe is taken to be the Minkowski metric), we get from the conditions ag,&W = 0, W&/ax m = 0 that (2.5) and (2.6) are satisfied. This is consistent with our previous statement that I’jk acts like a product of three vectors and gij acts like a product of two vectors. Equation (2.5) says the covariant derivative of I’ik vanishes. We now state some easily verifiable predictions of the field equations. (a) The covariant derivatives of all tensor functions of gij , r;k and eq vanishes. (b) The covariant derivatives of all higher derivatives of these tensor functions vanish. (c) All scalars formed from g,j , I’& , Pi , and 8, are constants. Thus, we have a theory where all tensor functions are treated in a similar fashion since all have vanishing covariant derivatives. All higher derivatives of the field are treated in a similar fashion for the same reason. Also, no scalar function is preferred over any
UNIFIED
FIELD
THEORY
31
other scalar function. We may contrast this with the Lagrangian approach previously discussed. The hypothesis of continuity implies that there are no interior points of the universe for which all of the components of r,i, are zero. If there existed such an interior point where I’;k = 0, then by means of the field equation (2.9, we see that all the derivatives of I’ik at the point would be zero. Thus, it would be impossible to go from this point to a point where the field is not zero in a manner in which the field varies in a continuous fashion. Thus, the continuity assumption together with the field equations imply that the interior points have nonvanishing fields. 3. CONSISTENCY CONDITIONS
Thus, the system of equations we have proposed are simply motivated and have a certain aesthetic appeal. We may, next, point out that this is all well and good, but there may well be no nontrivial solutions to the equations. That is, rf,:, = 0 is 256 equations for 64 r$ which in turn are given in terms of 16 eni . Thus, all these nice aesthetic properties may be all spurious. We see from the field equations that the mixed partial derivatives of the fields are not, in general, symmetric. In order to obtain symmetries in the mixed derivatives, it is necessary for certain consistency conditions to be met. We find the requirement, (3.1) implies using (2.6) that we have &h&k
+ &&mk
= 0,
where
From (2.5), we get
The condition,
a2rik ---C axPax
implies
a2rjk axl'
32
MURASKIN
We must investigate whether the conditions (3.2) and (3.6) can be satisfied. It is necessary for them to be satisfied if (2.5) and (2.6) are to define a reasonable system. The symmetry of mixed derivatives of eai gives, using (2.7) R&
= 0.
(3.7)
The conditions (3.2) and (3.6) are, thus, satisfied from (3.7). It follows, from (3.1), (3.5), and Pea,/SxQxk = 62e”i/6xkaxj that the second mixed derivatives are symmetric for any set of products of eai , g,j and I’jk . Furthermore, since derivatives of eDLi, g,i, and I$ can be expressed in terms of eai, gig, and I$ by means of (2.5), (2.6), and (2.7), we find that all higher mixed derivatives of functions of the field are symmetric as well. From (3.4), (2.5), and (3.7) we get the consistency conditions, r;jr&,,
-
r,tjrAk
+
r&,rit,
-
r:kr;,,,
=
0.
(3.8)
From (2.7) and (2.9) we get for our field equation (3.9)
However, these equations involve the constants r;,, . The form (2.5) and (2.6) may be used if we do not want I’;,, to appear in the equations. We can obtain lYjk at a point R with coordinates x, y, z, ct in space time in terms of IY;k at the origin point P if we assume a Taylor expansion r;k
=
f
1
m,n,P,ac-Om! n! p! q!
axn ay azp
p xnymzqctp.
(3. IO)
From the field equations (2.5) we see that all the derivatives of rjk at P can be expressed in terms of I’:k(P). Therefore, we have that rik(R) is given in terms of r&(P). Thus, the problem of existence of solutions to our field equations will depend on whether there is a consistent choice of I’:k at the point P, i.e., whether there exists any solutions to (3.8). Equation (3.8) constitutes 96 equations (there is antisymmetry in the indices k and m) for 64 I’jilc . Thus, it is not obvious whether any solutions to (3.8) do exist. Note, we can explicity demonstrate that a/axP of (3.8) gives zero on using (2.5) and (3.8). We get zero also in a similar fashion when higher derivatives are applied to (3.8). Thus, the consistency equation (3.8) holds at all points if it holds at one point.
UNIFIED
FIELD
33
THEORY
We have found a simple solution of (3.8) that does not lead to constant rik and gij on use of (2.5) and (2.6). The set is
(3.11)
r& = r& = d. The other r;k are zero.,The difficulty with this set is that upon using (2.5), we find that all the r,e, that are zero at one point remain so everywhere. We can get around this problem as follows. We require that (2.9) holds at the point P. We identify the set (3.11) with r;,, . We need not require I’;,, to be constant (see Section 6). That is, we may consider a theory where (2.5) and (2.6) hold but not (2.7). By direct evaluation we see that (3.11) satisfies
rgAr," - r;Ar,:, + r;@r;,. - rp,a
= 0.
(3.12)
Then, from (2.9) we get that (3.8) is satisfied for an arbitrary choice of en, at the arbitrary point P. Once the parameters at a point are specified, the solution is unique. It is not clear what other simple sets of r& also satisfy (3.8). We have found many sets that do not. We have demonstrated by means of a computer that we can get solutions involving 64 I’& and that the I’jk vary from point to point in a nontrivial way. The solutions were generated using a finite difference approximation to the field equations. Even in the vicinity of the origin point (where the Pi were specified), we found interesting structure. We found field components that increase (decrease) for a while and the decrease (increase) as one moves away from the origin. The field theory has the property of not involving any arbitrary functions as is the case with conventional wave equations. All we have to do is prescribe a set of parameters at a point and the solution is then determined everywhere. It is still not clear how we should specify the parameters at a point. Since the universe is so large and the origin point where the parameters are specified are arbitrary, there is a tremendous range of choice for eUi at the origin point. On the other hand, it is not clear what sort of connection there may be between the choice of parameters and the boundary conditions for the universe. The whole question of boundary conditions is not understood. Thus, in summary, our equations have an aesthetic appeal; nontrivial solutions to the consistency equations do exist; and we have obtained finite difference approximation solutions to the field equations. We can not yet say whether solutions to the field equations are finite at all points in space.
34
MURASKIN
4. RELATIONSHIP OF FIELD EQUATIONS TO THE WAVE EQUATION We next ask what relationship do the field equations have with other equations known in mathematical physics. On the surface, these equations don’t seem to resemble any other equation in use today. Our study of the field equation has turned up some interesting results in this regard. We have found that the equation I’jikil = 0 is asymptotic to the conventional wave equation. In order to show this, we define a new eMi such that r;Y in Eq. (2.9) is a function of x. That is, we take rik = esjeykeaiF&(x).
(4-l)
Since r,&(x) is now a function of x, this means that we need a change function for a, ji indices. We find that (2.5) and (2.6) still can be made to hold. We shall require that F&(x) satisfy
2aP ax3 = Ar&, ar,8, -=a.9
--arfo ax0 - -BP 20,
Br310
3
(4.2) -=ar!l
Are
ax3
ar!2 -_-
12 3
_
a2
Are
11,
rf2 = cl , rfl = -rz2, with A, B constant. The remaining components of r;,,(x) are to be zero. Thus, we have that the nonvanishing components of r&(x) obey the conventional wave equation. If eai = aai, we have that the components of I’,:, have the same values as rgx), with rgx) h aving the property of obeying the wave equation. Instead of (2.4) we have -
aeai
a.9 = riktFt - I$t?“i
(4.3)
with %k
where J’ osk = g,Jik
(4.4).
=
-l%zk
9
(4.4)
. Thus, Eq. (2.6) is still satisfied using (2.8), (4.3), and
UNIFIED
FIELD
35
THEORY
We are considering the case where Fzti are constant. It is necessary to adopt (4.3), otherwise (4.1) will not be consistent with the basic equations (2.5) and (2.6). Equation (2.5) is satisfied using (4.1) and (4.3) if (4.5)
Now, we choose T;;, to be the following
with A, B constant. This choice satisfies (4.4). We, then, see by direct evaluation, using (4.6) and (4.2) that (4.5) is satisfied. Thus, we have shown in a theory based on (2.8), (4.1), (4.3) and (4.4) that (2.5) and (2.6) remain satisfied. In this theory, l-‘$ is a universal change function for tensor functions of gij, rii, , and a,. When we consider tensors with 01,p indices, we then need p,$ .l In (2.4), the LII,/3 indices were taken to be inert since, in this case, all 01,/3 tensors were constants. This is not the case, here, since it is useful to Iet F;,,(x) obey (4.2). We have found that it is, in fact, consistent with (2.5) and (2.6) to do this. We require that a%=, Peai (4.7)
axmaxk -aXkThen, from (4.3) and from the definition
of Rj,,
we get (4.8)
Using our choice of pPk given by (4.6), we get that the right side of (4.8) is zero. Also, making use of the nonzero FiY as given by (4.2), we find that the left side is zero as well. We here used rf2 = I’& , F,4 = --I’i2 in this regard (see Eq. 4.2). 1 We define the change function for oi, /I tensors to be p;,, = p& c$,~.F&, determines its own change according to (a/axs - SamLJ/axm)
This equation is identically satisfied for our choice of ii;;k given in (4.6). Thus, it is consistent to call i=iY a universal change function for functions formed from l’&,(x), I=&, a,, g-8. Since Sykis not a function of gii, I$, a,, we do not use r& as a change function when considering the k index in f& .
36
MURASKIN
In this section we have thus obtained another example of a set of r& that satisfies the consistency relations. We note there is no limiting process depending in a continuous way on a set of parameters such that l-‘i, goes into r&(x). The reason for this is that when l-‘jfi + r&(x), we end up with a/3 indices rather than i, j indices and, thus, we need f,& as a change function rather than I’$ . We may see this result explicitly as follows. We let eai --+ Sui in a continuous manner. From (4.1) we get (4.10) Then (2.5) goes into
This is not the conventional, wave equation and disagrees with (4.5). Thus, we have the situation that if eEi is strictly equal to aai then we get that I’& has the same components as r&(x) which obeys the wave equation. However, if we write eai = iPi + Pi , we see that if hai goes to zero in a continuous manner, (2.5) does not go into (4.5). We may, therefore, say that the conventional wave equation is asymptotic to the system (2.5), (2.6), and (3.7). A perturbation expansion using eOLi= ami + hai would then not be expected to lead to consistant results. We may describe the situation in the following way. eui # 6q is consistent with our theory. Also, eat = Sai is consistent with our theory. But, eai + @, , in a continuous manner, is not consistent with our theory. The choice of r;,,(x) is not unique. There are other components of r,&(x) for which (4.5) reduces to the wave equation. In a separate paper, Ref. (4), we show that Dirac plane waves in the z direction define a gij = g$” (where giy’ is the Minkowski metric) and a I’$ which has the same values as (4.6).2 When eai # Pi , then I’& is a complicated function involving eOLttimes a cosine or sine function. This is a hopeful sign, since it means that field components can have maxima and minima. This sort of structure would be a prerequisite for the existence of particles.
5. PARTICLE-LIKE
BEHAVIOR
Our next question is just what is it that we would like to learn from a basic field theory. The answer is that we would like to prove the existence of real particles. * Thus Dirac plane waves in the z direction are associated with a second derivative wave motion in the z direction (see (4.2)).
UNIFIED
FIELD
THEORY
37
Previous attempts in Unified Field Theory centered around generalizing Maxwell’s Equations and Einstein’s gravitational equations. We feel, on the other hand, that one should be cautious against any premature attempts of relating a basic field theory with the electrodynamic and gravitational fields. For discussion sake, we point out that Nelson [5] and Kolsrud [6] have suggested that Fik be related to the particle velocity vector by means of (5.1)
Thus, before we talk of Fik , we should discuss the problem of particles. vi in (5.1) refers to a particle that is not extended in space. Thus, various approximations and averaging techniques would be expected to be performed before we consider even the definition for Fik . Another way to look at the problem is as follows. From gij , IYjk , 8, , and edi, we can construct an infinite number of antisymmetric second rank tensors. The only way to say that a particular choice should be identified as the Maxwell Field is to examine the motion of a particle in the field. Thus, the primary objective of the field equations we would say is the demonstration of particle existence and elucidation of particle properties. The study of particle existence in nonlinear field theory is a science still in its infant stages. Let us summarize some examples of work by other investigators. Rosen [7] has shown that the nonlinear partial differential equation 8 - vq = 3qej
(5.2)
has the static solution,
e=
(z4g
-t” ; r2)1/2
(5.3)
2 and g are parameters. Another finite particle-like solution to a partial differential equation has been obtained by Born-Infeld [8]c They find (5.4)
with q, r. as parameters. It is reasonable to expect that we can generate more cases of this type of behavior. There are an infinite number of functions that are maximum about an origin. By differentiating such functions and by trial and error manipulation, we could hope to find some nonlinear partial differential equation which this function satisfies. Thus, there should be a whole class of equations that say things similar to (5.3) and (5.4). The equations of Rosen and Born-Infeld should, thus, be looked at as illustrative and not realistic. It may not be reasonable to expect symmetry about the origin in a general theory. Also, globally we would demand many particles rather than a single particle-like object.
38
MURASKM
At this point, we need a criterion for particle-like. We would like our particle to exhibit some kind of extreme behavior as compared to its environment. In order to be general, at this stage, we shall suppose only that a component of the field has an extremum at some point. This is somewhat more general than requiring just a maximum as appears in Rosen and Born-Infeld. At this point, we do not wish to ignore the possibility of a saddle point particle which may perhaps be a representation of unstable particles. We shall show, here, that such extremum-type behavior can be made to appear at an arbitrary origin. In a later work, we shall study the structure of several such particle-like objects. Let us consider a particular component of the field, say, the determinent of g (this is particularly easy to handle). We get an extremum at P by first requiring that at P (from now on the index a goes from one to three)
-=. ag 0 axa
(5.5)
Notice, this is not a Lorentz invariant restriction. However, there is no experimental evidence or any other compelling reason that a particle should have a Lorentz invariant structure (9). Equation (5.5) gives three conditions on the field variables. It is an easy matter to satisfy (5.5) at the origin point P. From the definition of the determinant, we have
ag= swk-jgagik-axa
Using the field equations (2.6) we get (5.7) Thus, we satisfy (5.5) by requiring r& = 0 at the point P. A set of eDLiand I’;,, that lead to ria = 0 is [the set r&, satisfies the consistency conditions discussed previously in Eq. (3.8)]. rg : r,l, = rfz = 0.5, ril
= r& = 0.2,
ris = r& = 0.3, emi : el, = eB1= es, = eel =
-0.62, 0.8, 0.3, 0.6,
el, = e22 = e32 = eo2 =
ris = r& = 0.1, -0.76, es1 = -0.7, 0.4, e23 = 0.5, 0.7, e33 = 0.8, eo3 = 0.1, 0.9,
el, = 0.1, E:" 1 ;:' e”I 2 210:
(5.8)
UNIFIED
FIELD
THEORY
39
In fact, it is easy to see that there exists an infinite number of ways to choose l$ = 0 as we have 16 eai at our disposel. Since the conditions are satisfied so easily, it may not be unlikely that these conditions will be satisfied as well at space time points, other than our origin. In linear wave equations like Schrbdinger’s equation, we can, for example, have Gaussian wave packets. This is also a type of particle-like behavior. However, the particle-like structure is, here, effectively introduced by hand via arbitrary functions. In the Schriidinger’s equation, the wave function on a t = 0 hypersurface is arbitrary. We note, in our field theory we do not have arbitrary functions appearing. Thus, any particle-like structure in our case is a consequence of the equations and not of the arbitrary initial functions. We have pointed out several times that our equations, uniike conventional, hyperbolic equations, do not depend on arbitrary functions. We may ask next if this is a positive property or not. We would expect that a basic theory should have a minimum of arbitrariness. Thus, the field on an initial hypersurface should not be arbitrary but should be fixed by the theory. In our work, we still have arbitrariness in the parameters at a point (emi and r;,,). Although the specification of the parameters is somewhat unclear, we have considerably less arbitrariness in our theory than appears in other theories.
6.
DISCUSSION
At this point, we have argued that an extremum in a field component does exist and there is a potentiality for more such behavior. But how are we going to find out if real particles exist? That is, does a long time average of these particle-like objects lead to an object that has the properties of particles found in nature? Are we not essentially in the same difficulty as the previous Unified Field Theories we discussed. As we recall, these equations were so hopelessly nonlinear that it was hard to say just what information is contained in the equations. We shall comment on this problem momentarily. First, we would like to make some other unrelated comments. The basic field equations have a lot more information in them than is needed for most practical purposes. That is, every irregularity in a contour surface (surface on which a field component takes on a constant value) is not important so far as laboratory experiments are concerned. Thus, models of particles, say, that of the elastic rigid body particle of Hara and Goto [lo], may be in the right direction. Nevertheless, it still would be most gratifying from a conceptual point of view to verify the correctness of a Unified Field Theory. Another interesting feature of the equations that we have not discussed is the fact that our basic equations have no scale. In Schrodinger’s and other such
40
MURASKIN
equations having to do with matter, m, c, h, e enter in the equations. However, since the basic theory is to explain the existence of particles and their motion in the neighborhood of other particles and through the vacuum, it is not unrealistic to expect that such quantities as m, c, h, e do not appear in the basic equations. We would like to make some remarks about the 64 variable theory. We can adopt Eqs. (2.5) and (2.6) but not (2.7). We can think of a representation of this theory as being motivated by propagation concepts. That is, we would take as basic elements of the theory the propagation vectors. If we know everything about the way that propagation vectors behave, then, this defines the theory. This is not unlike the arguments used in electrodynamics where one talks about potentials. Any function affecting the motion of test particles is a field variable. Variables that can be changed without affecting the test particle are potentials. In a propagation theory, we would substitute the words propagation vectors for the words test particle. Now, gii can be taken to define a propagation cone and I’;k can be taken to define the change of propagation vectors. Howevers, emi is not directly related to the way that propagation vectors behave. Thus, we may seek to generalize equation (2.4) (so that we have a 64 variable theory rather than a 16 variable theory) since we may suppose that only propagation variables have I’;k as a change function. Let us now return to the problem of how to extract information from our field theory. Our equations are very easily treated by a computer using a finite difference approximation. We have already pointed out that field components can get larger and smaller as we procede away from the origin. A further study of computer solutions is underway. It is desirable to investigate larger regions of space-time than has been considered in this paper. We would have to be careful to minimize errors resulting from the approximate character of the finite difference method when we move far away from the origin. It is clear that since microscopes have a limitation to their resolving power, we will never be able to see an elementary particle in a microscope. The basic hope would be to find a correct set of field equations and then let a computer generate a picture of what is going on. This would have been an impossible dream when Unified Field Theories were first being contemplated. However, with a computerscope there may be some hope of progress. We would like to end with one additional comment. We may ask, is it really possible to know everything in principle about the universe given the basic field equations? Is man, therefore, capable of understanding everything? The answer to this is unambiguously, no. The trivial solution to the field equations is always possible and perfectly consistent. There can be no logical reason that disqualifies the trivial solution from existing. Thus, the fact we are here is beyond human comprehension.
UNIFIED FIELD THEORY
41
ACKNOWLEDGMENTS I would like to thank D. Schmitz and Professor T. Clark for their valuable assistance in programming the equations for the computer. 1 would like to also thank Professor Clark for his interest in the work and for his interesting comments. 1 am also grateful to Professor Max Dresden for discussing the details of the paper with me.
REFERENCES I. A. EINSTEIN, “Meaning of Relativity,” 5th ed., Princeton Univ. Press, Princeton, N. J., 1955; E. SCH~RDINGER,‘Space Time Structure,” Cambridge Univ. Press, London/New York, 1950; C. MOLLER, Mat. Fys. Skr., Danske Vid. Sefsk. 1(1961), 10; R. FINKELSTEIN, J. Math. Phys. 1 (1960), 441; D. LIPKIN, Nuovo Cimenro Suppl. 6 (1968), 1371; G. RAINICH, Trans. Amer. Math. Sot. 27 (1925), 106; M. SACHS, Nuovo Cimento 53B (1968), 398; D. SEN, “Fields and/or Particles,” Academic Press, Inc., New York, and Ryerson Press, 1968. Some more classical references may be found here. These are some examples of the theories being considered. This list is not meant to be complete. 2. G. RAINICH, Trans. Amer. Math. Sot. 27 (1965), 106. 3. V. FOCK, “Theory of Space Time and Gravitation,” Pergamon Press, London/New York, 1959.
M. MURASKIN, preprint. 5. E. NELSON, Phys. Rev. 150 (1966), 1079. 6. M. KOLSRUD, Nuovo Cimento 39 (1965), 504. 7. G. ROSEN, J. Math. Phys. 7 (1966), 2066. 8. M. BORN AND L. INFELD, Proc. Roy. Sot. A 144 (1934), 425. 9. L. LANDAU AND E. LIFSHITZ, “Classical Theory of Fields,” p. 47, Addison Wesley, Pub]. Inc., Reading, Mass., 1962; R. MORPURGO, Physics 2 (1965), 95; 0. HARA AND T. GOTO, Prog. Theoret. Phys. Suppl. 41 (1968), 56. 10. 0. HARA AND T. GOTO, Prog. Theoret. Phys. Suppl. 41 (1968), 56. 4.
OF PHYSICS
59,
27-41 (1970)
A Reexamination
of Unified
Field Theory
M. MURASKIN University of North Dakota, Grand Forks, North Dakota 58201 Received July 7, 1969
We study a nonlinear field theory with a nonsymmetric I’;, which has the property that all tensor functions of g,, , eSLi,r,!, , and 8, are treated in a uniform manner. All scalar functions of gij , eair r;, and a, are constants in such a theory. If the theory is to be meaningful, it is necessary for a set of consistency conditions to be satisfied. We, then, show that nontrivial solutions to these consistency equations do exist. We describe, in another paper, results of a finite difference approximation to the field equations which we have obtained using a computer. We show that the field equations are asymptotic to the conventional wave equation. We find particle-like behavior can be made to appear at an origin point. The theory does not involve any arbitrary functions. Thus, there is less arbitrariness in this theory as compared with other hyperbolic type theories.
1.
INTRODUCTION
What is characteristic of the universe is change. By universe, we mean a system that can be described by a set of numbers at each point. The way these numbers change from point to point is given by the field equations. This is the underlying hypothesis of any so-called Unijied Field Theory. Once we are given the field equations, then, in principle, we would know all that is possible about the universe. The search for such a basic set of field equations has had a long but unsucessful history. After the successesof General Relativity, Einstein sought to generalize the gravitational equations to include electromagnetism. Since then, many researchers have tried their hand at formulating a Unified Field Theory [l]. The results of their work can be summarized as follows. The end results of these investigations are sets of equations that are generalizations of Maxwell’s Equations and Einstein’s gravitational equations. Now, it is of course, possible to generalize Maxwell’s and Einstein’s equations in an infinite number of ways. There is, at present, no experimental confirmation of any of the equations proposed. Also, the equations are, invariably, so complicated that it is difficult to make clear-cut predictions of what the equations really imply. Thus, there is no reason to believe any of these possible generalizations, at this time. 27
28 However, above. The Lagrangian The field
MURASKIN
the difficulty with these generalizations lies deeper than our discussion conventional method for obtaining field equations is by means of a approach. equations emerge from the action principle,
where 9 is the Lagrangian density and g is the determinent of a symmetrical second-rank tensor. The problem, then, is to specify 9 and the field variables which are to be varied by the operation 6. Unfortunately, this problem suffers from a complete lack of uniqueness. What is usually done is to stipulate that the field equations should involve second derivatives of g,$ , at most. This limits the choice of 9 considerably. The only reason offered for this distinguished property of the second derivative is that Newton’s gravitational equations have such a structure. However, this can hardly be taken as a serious argument. Rainich [2] has proposed equations that involve fourth derivatives of gij . Still, there is no more reason to distinguish the fourth derivative than it was to distinguish the second. Incidently, Rainich’s equations have a structure that closely resembles Maxwell’s equations and Einstein’s equations. However, in view of the infinite number of generalizations possible, we should be leery of accepting this particular one. Thus, with the obvious arbitrariness of the basic equations thus far proposed, and with the difficult mathematics that one encounters in trying to learn what the proposed equations imply, the progress in Unified Field Theory is, we feel it is not unfair to say, unimpressive. One may feel somewhat uncomfortable with the whole approach based on an action principle. We may ask what is the meaning of the variational principle, (1.1). The answer is that it is a mathematical tool for obtaining field equations. We may then ask if there is a way to motivate the equations on physicuI rather than mathematical grounds. The difficulty is this: On a fundamental level, we have no rods, clocks, and particles to give us physical intuition. After all, we expect a particle to be a complicated manifestation of the field. Thus, we would agree with Einstein and Schrodinger, for example, that a mathematical approach is necessary. That is, in looking for the field equations, we must not use terms like particles, forces and other expressly physical terms. Thus, far, we have encountered some of the difficulties confronting Unified Field Theory. We might say that, in present approaches, the essential difficulty is that of mathematical inelegance. That is, the approach of Einstein and Schrodinger (a prototype of Unified Field Theory) distinguishes particular scalars and particular orders of derivatives. This leads to unconvincing theories.
UNIFIED FIELD THEORY
29
Let us take a different approach. We shall find it possible to motivate a theory which does not make the unreasonable restrictions of the preceding paragraph. There exists also a means to study what information is contained in our equations. Thus, hopefully, it may be eventually possible to see how reasonable the equations are. In the remainder of this article, we shall describe our approach to the problem.
2. FORMULATION
OF THE THEORY
We set up a coordinate system that is Cartesian in space and has a 4-th component x0 = ct with c constant. In fact, we may set up any coordinate system we like since coordinate systems are constructs of the mind. We choose the simplest coordinate system we can think of. We have introduced a time axis since we shall anticipate a wave character to our field equation (a propagation cone in a wave theory requires a hyperbolic g,J. We shall restrict ourselves to four dimensions. The basic equations are not dimension dependent in their derivation so we shall stay with four dimensions. We shall consider only linear coordinate transformations. We shall not make general coordinate transformations since they only complicate the state of affairs by bringing in coordinate fields. It is not that we couldn’t introduce general coordinates if we really insisted. It’s a question really of “who needs it.” In General Relativity, one has covariance under general coordinate transformations. This will not be the case in our approach. This principle of general covariance has come under strong attack in recent years. Fock [3] has argued that the empirical results of gravitational theory can be obtained without general covariance. In his work there is a subsidiary condition that is not covariant. That is, this condition takes on a more complicated form when one makes general coordinate transformations. Our belief is that the principle of general covariance mixes up field effects with coordinate effects and thus, should not be taken as a basic principle. We assume the existence of a universal change function that determines the change of vector functions according to dAi = r&Ajdxk.
(2-l)
We may ask why we have extracted an A, on the right side of (2.1). The reason is essentially that it leads to the desirable properties that we list later on. g,, is taken to act like a product of two vectors. Thus, its change is given by &ir = V’hti
+ r;k git) dxk.
The change function itself is taken to act like a product of three vectors.
(2.2)
30
MURASKIN
Thus, the change function determines its own change according to
drjk = (rAkr;t + q&r; - r;r;l) dxk.
(2.3)
A set of basis vectors is denoted by Pi (the invex 01tells which of the four basis vectors we are considering). Their change is given by de”, = Assuming continuity
fikeuj
dxk.
(2.4)
for the fields, we get (2.5)
This is our basic field equation. We also get agik -
a2
-
mmk
-
r:gim
=
gik:L
=
OF
and (2.7)
edli is the dual field defined by eOLiesi= PB . By writing gjj = e@jesjgorB
w-9
and (2.9) with g,, , r;,, constant (gUe is taken to be the Minkowski metric), we get from the conditions ag,&W = 0, W&/ax m = 0 that (2.5) and (2.6) are satisfied. This is consistent with our previous statement that I’jk acts like a product of three vectors and gij acts like a product of two vectors. Equation (2.5) says the covariant derivative of I’ik vanishes. We now state some easily verifiable predictions of the field equations. (a) The covariant derivatives of all tensor functions of gij , r;k and eq vanishes. (b) The covariant derivatives of all higher derivatives of these tensor functions vanish. (c) All scalars formed from g,j , I’& , Pi , and 8, are constants. Thus, we have a theory where all tensor functions are treated in a similar fashion since all have vanishing covariant derivatives. All higher derivatives of the field are treated in a similar fashion for the same reason. Also, no scalar function is preferred over any
UNIFIED
FIELD
THEORY
31
other scalar function. We may contrast this with the Lagrangian approach previously discussed. The hypothesis of continuity implies that there are no interior points of the universe for which all of the components of r,i, are zero. If there existed such an interior point where I’;k = 0, then by means of the field equation (2.9, we see that all the derivatives of I’ik at the point would be zero. Thus, it would be impossible to go from this point to a point where the field is not zero in a manner in which the field varies in a continuous fashion. Thus, the continuity assumption together with the field equations imply that the interior points have nonvanishing fields. 3. CONSISTENCY CONDITIONS
Thus, the system of equations we have proposed are simply motivated and have a certain aesthetic appeal. We may, next, point out that this is all well and good, but there may well be no nontrivial solutions to the equations. That is, rf,:, = 0 is 256 equations for 64 r$ which in turn are given in terms of 16 eni . Thus, all these nice aesthetic properties may be all spurious. We see from the field equations that the mixed partial derivatives of the fields are not, in general, symmetric. In order to obtain symmetries in the mixed derivatives, it is necessary for certain consistency conditions to be met. We find the requirement, (3.1) implies using (2.6) that we have &h&k
+ &&mk
= 0,
where
From (2.5), we get
The condition,
a2rik ---C axPax
implies
a2rjk axl'
32
MURASKIN
We must investigate whether the conditions (3.2) and (3.6) can be satisfied. It is necessary for them to be satisfied if (2.5) and (2.6) are to define a reasonable system. The symmetry of mixed derivatives of eai gives, using (2.7) R&
= 0.
(3.7)
The conditions (3.2) and (3.6) are, thus, satisfied from (3.7). It follows, from (3.1), (3.5), and Pea,/SxQxk = 62e”i/6xkaxj that the second mixed derivatives are symmetric for any set of products of eai , g,j and I’jk . Furthermore, since derivatives of eDLi, g,i, and I$ can be expressed in terms of eai, gig, and I$ by means of (2.5), (2.6), and (2.7), we find that all higher mixed derivatives of functions of the field are symmetric as well. From (3.4), (2.5), and (3.7) we get the consistency conditions, r;jr&,,
-
r,tjrAk
+
r&,rit,
-
r:kr;,,,
=
0.
(3.8)
From (2.7) and (2.9) we get for our field equation (3.9)
However, these equations involve the constants r;,, . The form (2.5) and (2.6) may be used if we do not want I’;,, to appear in the equations. We can obtain lYjk at a point R with coordinates x, y, z, ct in space time in terms of IY;k at the origin point P if we assume a Taylor expansion r;k
=
f
1
m,n,P,ac-Om! n! p! q!
axn ay azp
p xnymzqctp.
(3. IO)
From the field equations (2.5) we see that all the derivatives of rjk at P can be expressed in terms of I’:k(P). Therefore, we have that rik(R) is given in terms of r&(P). Thus, the problem of existence of solutions to our field equations will depend on whether there is a consistent choice of I’:k at the point P, i.e., whether there exists any solutions to (3.8). Equation (3.8) constitutes 96 equations (there is antisymmetry in the indices k and m) for 64 I’jilc . Thus, it is not obvious whether any solutions to (3.8) do exist. Note, we can explicity demonstrate that a/axP of (3.8) gives zero on using (2.5) and (3.8). We get zero also in a similar fashion when higher derivatives are applied to (3.8). Thus, the consistency equation (3.8) holds at all points if it holds at one point.
UNIFIED
FIELD
33
THEORY
We have found a simple solution of (3.8) that does not lead to constant rik and gij on use of (2.5) and (2.6). The set is
(3.11)
r& = r& = d. The other r;k are zero.,The difficulty with this set is that upon using (2.5), we find that all the r,e, that are zero at one point remain so everywhere. We can get around this problem as follows. We require that (2.9) holds at the point P. We identify the set (3.11) with r;,, . We need not require I’;,, to be constant (see Section 6). That is, we may consider a theory where (2.5) and (2.6) hold but not (2.7). By direct evaluation we see that (3.11) satisfies
rgAr," - r;Ar,:, + r;@r;,. - rp,a
= 0.
(3.12)
Then, from (2.9) we get that (3.8) is satisfied for an arbitrary choice of en, at the arbitrary point P. Once the parameters at a point are specified, the solution is unique. It is not clear what other simple sets of r& also satisfy (3.8). We have found many sets that do not. We have demonstrated by means of a computer that we can get solutions involving 64 I’& and that the I’jk vary from point to point in a nontrivial way. The solutions were generated using a finite difference approximation to the field equations. Even in the vicinity of the origin point (where the Pi were specified), we found interesting structure. We found field components that increase (decrease) for a while and the decrease (increase) as one moves away from the origin. The field theory has the property of not involving any arbitrary functions as is the case with conventional wave equations. All we have to do is prescribe a set of parameters at a point and the solution is then determined everywhere. It is still not clear how we should specify the parameters at a point. Since the universe is so large and the origin point where the parameters are specified are arbitrary, there is a tremendous range of choice for eUi at the origin point. On the other hand, it is not clear what sort of connection there may be between the choice of parameters and the boundary conditions for the universe. The whole question of boundary conditions is not understood. Thus, in summary, our equations have an aesthetic appeal; nontrivial solutions to the consistency equations do exist; and we have obtained finite difference approximation solutions to the field equations. We can not yet say whether solutions to the field equations are finite at all points in space.
34
MURASKIN
4. RELATIONSHIP OF FIELD EQUATIONS TO THE WAVE EQUATION We next ask what relationship do the field equations have with other equations known in mathematical physics. On the surface, these equations don’t seem to resemble any other equation in use today. Our study of the field equation has turned up some interesting results in this regard. We have found that the equation I’jikil = 0 is asymptotic to the conventional wave equation. In order to show this, we define a new eMi such that r;Y in Eq. (2.9) is a function of x. That is, we take rik = esjeykeaiF&(x).
(4-l)
Since r,&(x) is now a function of x, this means that we need a change function for a, ji indices. We find that (2.5) and (2.6) still can be made to hold. We shall require that F&(x) satisfy
2aP ax3 = Ar&, ar,8, -=a.9
--arfo ax0 - -BP 20,
Br310
3
(4.2) -=ar!l
Are
ax3
ar!2 -_-
12 3
_
a2
Are
11,
rf2 = cl , rfl = -rz2, with A, B constant. The remaining components of r;,,(x) are to be zero. Thus, we have that the nonvanishing components of r&(x) obey the conventional wave equation. If eai = aai, we have that the components of I’,:, have the same values as rgx), with rgx) h aving the property of obeying the wave equation. Instead of (2.4) we have -
aeai
a.9 = riktFt - I$t?“i
(4.3)
with %k
where J’ osk = g,Jik
(4.4).
=
-l%zk
9
(4.4)
. Thus, Eq. (2.6) is still satisfied using (2.8), (4.3), and
UNIFIED
FIELD
35
THEORY
We are considering the case where Fzti are constant. It is necessary to adopt (4.3), otherwise (4.1) will not be consistent with the basic equations (2.5) and (2.6). Equation (2.5) is satisfied using (4.1) and (4.3) if (4.5)
Now, we choose T;;, to be the following
with A, B constant. This choice satisfies (4.4). We, then, see by direct evaluation, using (4.6) and (4.2) that (4.5) is satisfied. Thus, we have shown in a theory based on (2.8), (4.1), (4.3) and (4.4) that (2.5) and (2.6) remain satisfied. In this theory, l-‘$ is a universal change function for tensor functions of gij, rii, , and a,. When we consider tensors with 01,p indices, we then need p,$ .l In (2.4), the LII,/3 indices were taken to be inert since, in this case, all 01,/3 tensors were constants. This is not the case, here, since it is useful to Iet F;,,(x) obey (4.2). We have found that it is, in fact, consistent with (2.5) and (2.6) to do this. We require that a%=, Peai (4.7)
axmaxk -aXkThen, from (4.3) and from the definition
of Rj,,
we get (4.8)
Using our choice of pPk given by (4.6), we get that the right side of (4.8) is zero. Also, making use of the nonzero FiY as given by (4.2), we find that the left side is zero as well. We here used rf2 = I’& , F,4 = --I’i2 in this regard (see Eq. 4.2). 1 We define the change function for oi, /I tensors to be p;,, = p& c$,~.F&, determines its own change according to (a/axs - SamLJ/axm)
This equation is identically satisfied for our choice of ii;;k given in (4.6). Thus, it is consistent to call i=iY a universal change function for functions formed from l’&,(x), I=&, a,, g-8. Since Sykis not a function of gii, I$, a,, we do not use r& as a change function when considering the k index in f& .
36
MURASKIN
In this section we have thus obtained another example of a set of r& that satisfies the consistency relations. We note there is no limiting process depending in a continuous way on a set of parameters such that l-‘i, goes into r&(x). The reason for this is that when l-‘jfi + r&(x), we end up with a/3 indices rather than i, j indices and, thus, we need f,& as a change function rather than I’$ . We may see this result explicitly as follows. We let eai --+ Sui in a continuous manner. From (4.1) we get (4.10) Then (2.5) goes into
This is not the conventional, wave equation and disagrees with (4.5). Thus, we have the situation that if eEi is strictly equal to aai then we get that I’& has the same components as r&(x) which obeys the wave equation. However, if we write eai = iPi + Pi , we see that if hai goes to zero in a continuous manner, (2.5) does not go into (4.5). We may, therefore, say that the conventional wave equation is asymptotic to the system (2.5), (2.6), and (3.7). A perturbation expansion using eOLi= ami + hai would then not be expected to lead to consistant results. We may describe the situation in the following way. eui # 6q is consistent with our theory. Also, eat = Sai is consistent with our theory. But, eai + @, , in a continuous manner, is not consistent with our theory. The choice of r;,,(x) is not unique. There are other components of r,&(x) for which (4.5) reduces to the wave equation. In a separate paper, Ref. (4), we show that Dirac plane waves in the z direction define a gij = g$” (where giy’ is the Minkowski metric) and a I’$ which has the same values as (4.6).2 When eai # Pi , then I’& is a complicated function involving eOLttimes a cosine or sine function. This is a hopeful sign, since it means that field components can have maxima and minima. This sort of structure would be a prerequisite for the existence of particles.
5. PARTICLE-LIKE
BEHAVIOR
Our next question is just what is it that we would like to learn from a basic field theory. The answer is that we would like to prove the existence of real particles. * Thus Dirac plane waves in the z direction are associated with a second derivative wave motion in the z direction (see (4.2)).
UNIFIED
FIELD
THEORY
37
Previous attempts in Unified Field Theory centered around generalizing Maxwell’s Equations and Einstein’s gravitational equations. We feel, on the other hand, that one should be cautious against any premature attempts of relating a basic field theory with the electrodynamic and gravitational fields. For discussion sake, we point out that Nelson [5] and Kolsrud [6] have suggested that Fik be related to the particle velocity vector by means of (5.1)
Thus, before we talk of Fik , we should discuss the problem of particles. vi in (5.1) refers to a particle that is not extended in space. Thus, various approximations and averaging techniques would be expected to be performed before we consider even the definition for Fik . Another way to look at the problem is as follows. From gij , IYjk , 8, , and edi, we can construct an infinite number of antisymmetric second rank tensors. The only way to say that a particular choice should be identified as the Maxwell Field is to examine the motion of a particle in the field. Thus, the primary objective of the field equations we would say is the demonstration of particle existence and elucidation of particle properties. The study of particle existence in nonlinear field theory is a science still in its infant stages. Let us summarize some examples of work by other investigators. Rosen [7] has shown that the nonlinear partial differential equation 8 - vq = 3qej
(5.2)
has the static solution,
e=
(z4g
-t” ; r2)1/2
(5.3)
2 and g are parameters. Another finite particle-like solution to a partial differential equation has been obtained by Born-Infeld [8]c They find (5.4)
with q, r. as parameters. It is reasonable to expect that we can generate more cases of this type of behavior. There are an infinite number of functions that are maximum about an origin. By differentiating such functions and by trial and error manipulation, we could hope to find some nonlinear partial differential equation which this function satisfies. Thus, there should be a whole class of equations that say things similar to (5.3) and (5.4). The equations of Rosen and Born-Infeld should, thus, be looked at as illustrative and not realistic. It may not be reasonable to expect symmetry about the origin in a general theory. Also, globally we would demand many particles rather than a single particle-like object.
38
MURASKM
At this point, we need a criterion for particle-like. We would like our particle to exhibit some kind of extreme behavior as compared to its environment. In order to be general, at this stage, we shall suppose only that a component of the field has an extremum at some point. This is somewhat more general than requiring just a maximum as appears in Rosen and Born-Infeld. At this point, we do not wish to ignore the possibility of a saddle point particle which may perhaps be a representation of unstable particles. We shall show, here, that such extremum-type behavior can be made to appear at an arbitrary origin. In a later work, we shall study the structure of several such particle-like objects. Let us consider a particular component of the field, say, the determinent of g (this is particularly easy to handle). We get an extremum at P by first requiring that at P (from now on the index a goes from one to three)
-=. ag 0 axa
(5.5)
Notice, this is not a Lorentz invariant restriction. However, there is no experimental evidence or any other compelling reason that a particle should have a Lorentz invariant structure (9). Equation (5.5) gives three conditions on the field variables. It is an easy matter to satisfy (5.5) at the origin point P. From the definition of the determinant, we have
ag= swk-jgagik-axa
Using the field equations (2.6) we get (5.7) Thus, we satisfy (5.5) by requiring r& = 0 at the point P. A set of eDLiand I’;,, that lead to ria = 0 is [the set r&, satisfies the consistency conditions discussed previously in Eq. (3.8)]. rg : r,l, = rfz = 0.5, ril
= r& = 0.2,
ris = r& = 0.3, emi : el, = eB1= es, = eel =
-0.62, 0.8, 0.3, 0.6,
el, = e22 = e32 = eo2 =
ris = r& = 0.1, -0.76, es1 = -0.7, 0.4, e23 = 0.5, 0.7, e33 = 0.8, eo3 = 0.1, 0.9,
el, = 0.1, E:" 1 ;:' e”I 2 210:
(5.8)
UNIFIED
FIELD
THEORY
39
In fact, it is easy to see that there exists an infinite number of ways to choose l$ = 0 as we have 16 eai at our disposel. Since the conditions are satisfied so easily, it may not be unlikely that these conditions will be satisfied as well at space time points, other than our origin. In linear wave equations like Schrbdinger’s equation, we can, for example, have Gaussian wave packets. This is also a type of particle-like behavior. However, the particle-like structure is, here, effectively introduced by hand via arbitrary functions. In the Schriidinger’s equation, the wave function on a t = 0 hypersurface is arbitrary. We note, in our field theory we do not have arbitrary functions appearing. Thus, any particle-like structure in our case is a consequence of the equations and not of the arbitrary initial functions. We have pointed out several times that our equations, uniike conventional, hyperbolic equations, do not depend on arbitrary functions. We may ask next if this is a positive property or not. We would expect that a basic theory should have a minimum of arbitrariness. Thus, the field on an initial hypersurface should not be arbitrary but should be fixed by the theory. In our work, we still have arbitrariness in the parameters at a point (emi and r;,,). Although the specification of the parameters is somewhat unclear, we have considerably less arbitrariness in our theory than appears in other theories.
6.
DISCUSSION
At this point, we have argued that an extremum in a field component does exist and there is a potentiality for more such behavior. But how are we going to find out if real particles exist? That is, does a long time average of these particle-like objects lead to an object that has the properties of particles found in nature? Are we not essentially in the same difficulty as the previous Unified Field Theories we discussed. As we recall, these equations were so hopelessly nonlinear that it was hard to say just what information is contained in the equations. We shall comment on this problem momentarily. First, we would like to make some other unrelated comments. The basic field equations have a lot more information in them than is needed for most practical purposes. That is, every irregularity in a contour surface (surface on which a field component takes on a constant value) is not important so far as laboratory experiments are concerned. Thus, models of particles, say, that of the elastic rigid body particle of Hara and Goto [lo], may be in the right direction. Nevertheless, it still would be most gratifying from a conceptual point of view to verify the correctness of a Unified Field Theory. Another interesting feature of the equations that we have not discussed is the fact that our basic equations have no scale. In Schrodinger’s and other such
40
MURASKIN
equations having to do with matter, m, c, h, e enter in the equations. However, since the basic theory is to explain the existence of particles and their motion in the neighborhood of other particles and through the vacuum, it is not unrealistic to expect that such quantities as m, c, h, e do not appear in the basic equations. We would like to make some remarks about the 64 variable theory. We can adopt Eqs. (2.5) and (2.6) but not (2.7). We can think of a representation of this theory as being motivated by propagation concepts. That is, we would take as basic elements of the theory the propagation vectors. If we know everything about the way that propagation vectors behave, then, this defines the theory. This is not unlike the arguments used in electrodynamics where one talks about potentials. Any function affecting the motion of test particles is a field variable. Variables that can be changed without affecting the test particle are potentials. In a propagation theory, we would substitute the words propagation vectors for the words test particle. Now, gii can be taken to define a propagation cone and I’;k can be taken to define the change of propagation vectors. Howevers, emi is not directly related to the way that propagation vectors behave. Thus, we may seek to generalize equation (2.4) (so that we have a 64 variable theory rather than a 16 variable theory) since we may suppose that only propagation variables have I’;k as a change function. Let us now return to the problem of how to extract information from our field theory. Our equations are very easily treated by a computer using a finite difference approximation. We have already pointed out that field components can get larger and smaller as we procede away from the origin. A further study of computer solutions is underway. It is desirable to investigate larger regions of space-time than has been considered in this paper. We would have to be careful to minimize errors resulting from the approximate character of the finite difference method when we move far away from the origin. It is clear that since microscopes have a limitation to their resolving power, we will never be able to see an elementary particle in a microscope. The basic hope would be to find a correct set of field equations and then let a computer generate a picture of what is going on. This would have been an impossible dream when Unified Field Theories were first being contemplated. However, with a computerscope there may be some hope of progress. We would like to end with one additional comment. We may ask, is it really possible to know everything in principle about the universe given the basic field equations? Is man, therefore, capable of understanding everything? The answer to this is unambiguously, no. The trivial solution to the field equations is always possible and perfectly consistent. There can be no logical reason that disqualifies the trivial solution from existing. Thus, the fact we are here is beyond human comprehension.
UNIFIED FIELD THEORY
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ACKNOWLEDGMENTS I would like to thank D. Schmitz and Professor T. Clark for their valuable assistance in programming the equations for the computer. 1 would like to also thank Professor Clark for his interest in the work and for his interesting comments. 1 am also grateful to Professor Max Dresden for discussing the details of the paper with me.
REFERENCES I. A. EINSTEIN, “Meaning of Relativity,” 5th ed., Princeton Univ. Press, Princeton, N. J., 1955; E. SCH~RDINGER,‘Space Time Structure,” Cambridge Univ. Press, London/New York, 1950; C. MOLLER, Mat. Fys. Skr., Danske Vid. Sefsk. 1(1961), 10; R. FINKELSTEIN, J. Math. Phys. 1 (1960), 441; D. LIPKIN, Nuovo Cimenro Suppl. 6 (1968), 1371; G. RAINICH, Trans. Amer. Math. Sot. 27 (1925), 106; M. SACHS, Nuovo Cimento 53B (1968), 398; D. SEN, “Fields and/or Particles,” Academic Press, Inc., New York, and Ryerson Press, 1968. Some more classical references may be found here. These are some examples of the theories being considered. This list is not meant to be complete. 2. G. RAINICH, Trans. Amer. Math. Sot. 27 (1965), 106. 3. V. FOCK, “Theory of Space Time and Gravitation,” Pergamon Press, London/New York, 1959.
M. MURASKIN, preprint. 5. E. NELSON, Phys. Rev. 150 (1966), 1079. 6. M. KOLSRUD, Nuovo Cimento 39 (1965), 504. 7. G. ROSEN, J. Math. Phys. 7 (1966), 2066. 8. M. BORN AND L. INFELD, Proc. Roy. Sot. A 144 (1934), 425. 9. L. LANDAU AND E. LIFSHITZ, “Classical Theory of Fields,” p. 47, Addison Wesley, Pub]. Inc., Reading, Mass., 1962; R. MORPURGO, Physics 2 (1965), 95; 0. HARA AND T. GOTO, Prog. Theoret. Phys. Suppl. 41 (1968), 56. 10. 0. HARA AND T. GOTO, Prog. Theoret. Phys. Suppl. 41 (1968), 56. 4.