nonintegrable aesthetic field theory

Ma& Compur. Modelling, Vol. IO, No. 8, pp. 571-581, 1988 Printed in Great Britain. All rights reserved

0895-7 I77/88 $3.00 + 0.00 Copyright 0 1988 Pergamon Press plc

NONINTEGRABLE AESTHETIC FIELD THEORY

M. MURASKIN

Physics Department, University of North Dakota, Grand Forks, ND 58201, U.S.A.

(Received May 1987; accepted for publication February 1988)

Communicated by X. J. R. Avula

Abstract-We review our research on a system of equations not satisfying integrability. We show how to formulate the no integrability theory within the “aesthetic” field framework. Although we emphasize this aesthetic field framework, many of our remarks concerning no integrability are not dependent on the aesthetic field concept. We show how to handle the no integrability theory by two different methods. We discuss some numerical work made on the no integrability system and show that the results lead to planar maxima and minima and are thereby suggestive of multiparticles. Although symmetry of mixed derivatives has been taken to be the norm in contemporary physics, it is not a compelling hypothesis, and we suggest such a restriction unduly limits our scope. In addition, the absence of symmetry for mixed derivatives may have considerable implications for physics. The way we introduce derivatives is consistent with the no integrability field equations and is the same as conventional derivatives when the integrability equations are satisfied.

1. EQUATIONS WITH ARBITRARY DATA AT A SINGLE POINT

Consider the equations

with a, b, c, d > 0. These oscillatory in behavior.

Consider the equations

dx - = ax - bxy,

dy dt dt-

- -cy +dxy, (1)

are the Volterra predator-prey relations [l]. Solutions for x and y are

dx dy dt-

- -x - 2xy, & = -y -x*+y2. (2)

These are the Henon-Heiles equations [2]. For certain choices of origin point data we get random type behavior.

These equations are samples of nonlinear equations in which the variables x and y are completely determined from data at a single point.

These equations are first-order coupled nonlinear equations with quadratic expressions appearing on the r.h.s. The solutions show varied and involved behavior. However, they are not suitable as field equations since they have no space derivatives appearing in them.

We have been working with a set of equations that have some important similarities with the equations discussed above. We have called these equations the “aesthetic field equations” as they have been obtained from a set of mathematical aesthetic principles. In particular, we assumed that all tensors, and the way these tensors change, be treated in a uniform way. We introduced a change function that determines the change of all tensors in a Cartesian space. Since the change function is a tensor, it therefore determines its own change as well. This leads to us a set of equations for ’ the change function itself:

Equation (3) is a nonlinear first derivative set of coupled field equations with quadratic expressions appearing on the r.h.s. The field at any point is determined in terms of data furnished at a single point.

A case can be made, in its own right, for studying field equations in which data is assigned at a single point.

571

572 M. MURASKIN

Consider hyperbolic equations. Here the field and its first time derivative are arbitrary on, say, a t = 0 hypersurface. This would mean that the particle structure at t = 0 is arbitrary. Also particle number at t = 0 would be arbitrary. How then, should we assign the number of particles at t = 0, especially in view of possible conservation laws? If the arbitrary field and its first derivative at t = 0 were assigned to be constant, how then, could we expect, in view of the symmetry assigned, that multiparticles develop? Arbitrariness at t = 0, thus, can be looked at as an incompleteness of hyperbolic theory (especially if one is concerned with particle structure).

This suggests that theories with data furnished at a single point, should be studied for reasons of their own. They have less arbitrariness than hyperbolic theories so that particle structure need not be considered arbitrary at some time. Furthermore, we have argued that equations with data furnished at a single point have varied and interesting solutions [I, 21.

2. INTEGRABILITY vs NO INTEGRABILITY

The aesthetic field equations are an example of a set of equations in which data is prescribed at a single point. Although we shall focus attention on this system, many of the remarks we make are not restricted to the particular set of equations studied here.

Let the origin point where the data is prescribed be denoted as P. We wish to calculate the field at some other point Q in terms of the data at P.

There is no reason to expect the field at Q to be independent of the integration path from P. In order for the results at Q to be independent of path it is necessary to restrict the data at the origin point in aesthetic field theory. We see this as follows. The results are independent of path provided a set of integrability equations are satisfied. These equations are obtained by requiring that mixed derivatives be symmetric. From

a2r rk _ a2r;k -_ axfax’ ax/ax’ (4)

we obtain, with the use of the field equations (3),

with

R:,,,k = r$&,, - r#,,k + r{,,,i,rjk - r;&,,,. (6)

We note that if equation (5) is satisfied then mixed derivatives of any products of Ts (as well as contractions) will be symmetric.

The integrability equations are then 384 algebraic nonlinear conditions on the origin point data. There are only 64 pieces of data at the origin to begin with. Thus, the integrability equations severely restrict the origin point data.

One then pays a high price for requiring that the results at Q be independent of the integration path. One wonders if it is natural to have 384 restrictions on the origin point data or not.

In a simpler version of the theory the integrability equations take the form

RIk, = 0. (7)

In this case we have 96 conditions on the origin point data. This again constitutes a drastic restriction on the theory.

This line of thought suggests that we consider the case when the integrability equations are not required to be satisfied. This would mean that mixed derivatives of the field are not symmetric.

In current physics it is taken for granted that mixed derivatives are symmetric. The reason for this is understandable. Consider, as an example, the function f (x, y):

ww -Y’)

f(x,_Y)= x2+y2 k Y) + ((4 0) @a)

=o 6% Y> = (0, 0). (8b)

Nonintegrable aesthetic field theory 573

Then, by calculation of mixed derivatives, we see

everywhere except at (0,O). In general, so long as the function considered is well-behaved we can expect the mixed derivatives

to be symmetric. Thus, if we are to consider a field theory not satisfying the integrability equations we have to

face up to some pointed questions:

Can we formulate a field theory in an aesthetic way when integrability is not satisfied? How do we handle such a theory? How do we deal with the theorem that requires regular functions to have symmetric mixed derivatives? Provided that we can answer the above questions, what sort of things does the no integrability theory say? Is the no integrability theory consistent with multiparticle solutions?

These questions will be addressed in this paper. Before proceeding, we shall state some of our results. The no integrability ideas can be built into the aesthetic field theory in a natural way. Integrability unduly restricts the origin point data, making the no integrability situation a more attractive alternative. We shall obtain two different ways to handle the no integrability theory. We shall discuss solutions in a particular case to the no integrability system. We shall find, to the degree that our limited numerical work allows, that multiparticle solutions appear consistent with the equations.

In summary, the aims of our research project were:

(a) To study the consequences of mathematically aesthetic principles. (b) To study a system of field equations for which data is prescribed at a single point

rather than a hypersurface. (c) To study a system of equations for which integrability is not satisfied.

Some of our earlier results are listed below:

Field equations can be formulated according to mathematical aesthetics [3]. For certain choices of origin point data the field equations collapse so that sinusoidal solutions are exact solutions to the field equations [4]. For another choice of origin point data we found a particle solution in which the particle does not spread in time (soliton effect) [5]. For another choice of origin point data we found multiparticle solutions in which the particles are located in a symmetric way (lattice solutions) [6]. In this situation the integrability equations were not satisfied so an integration path was specified.

These results illustrate that equations having arbitrary data at a single point are capable of very interesting behavior, as was suggested in our discussion in Section 1.

The field equations (3) for the change function follow from the requirement that all tensors be treated in a uniform way, and is independent of whether integrability holds or not. The effect of the no integrability enters when we consider derivatives higher than the first, as we no longer expect these mixed derivatives to be symmetric.

The question arises whether the research project we have been involved in, which is of a basic mathematical character, has any relevance for physics.

The fundamental belief is that our tool for understanding lies in mathematical aesthetics, due to the limitations in empiricism. It is somewhat gratifying that aesthetic mathematical ideas can lead to the results in Refs [2-51 (briefly mentioned above).

In the latter part of this paper we shall look into some of our limited numerical results on no integrability. The results suggest that should it turn out that no integrability be required by nature, as we have suggested earlier, then we may expect rather deep implications for physics.

514 M. MURASKIN

3. HOW TO FORMULATE THE NO INTEGRABILITY THEORY IN AN AESTHETIC WAY

We first studied the no integrability system within aesthetic field theory in 1977 [7]. We found considerably more structure here compared to when integrability was satisfied.

Since the results are dependent on path when integrability is not satisfied, the simplest way to handle the no integrability situation is to specify an integration path at the outset. We chose to integrate first along x0, then x3, then x2 and finally, x’. In this way we can reach any point we like in a unique manner.

Consider the following origin point data:

r:, = 1.0,

with the other r;, = 0. Using

with a general e;, which we have

r;, = -1.0, r;,= -1.0, r;,= 1.0

rjk = ei ebeYr* a J k BY

chosen to be

(10)

(11)

i

0.88 -0.42 -0.32 0.22

0.5 0.9 eY= o.2

-0.425 0.3

-0.55

0.44 -0.16 0.39 1.01 I

0.89 0.6 ’ (12)

we end up with a set of origin point data. Maps of all 64r;k showed three-dimensional maxima and three-dimensional minima located in a symmetric fashion. This sort of solution we call a lattice solution. The integration path mentioned above was used to obtain the maps. We also followed the trajectories of the lattice particle in time. We found that the lattice particles all underwent simple harmonic motion in the z-direction. In the x- and y-directions the lattice particles were stationary.

We can also obtain an infinite number of particles undergoing simple harmonic motion from a Lagrangian supplied with a few parameters (this, then enables us to introduce a Lagrangian into the aesthetic field theory). Thus, if we concern ourselves only with the location of the lattice particles it is not necessary to concern ourselves with the underlying no integrability field equations. A few parameters take into account the effect of the basic equations. However, the Lagrangian approach would deny us a deeper understanding.

Although it is simplest to deal with the no integrability system by specifying an integration path, the aesthetics of assigning one particular integration path, rather than treating all integration paths in a uniform way, are open to question.

For the time being let us assume that integrability is satisfied. From the notion of derivatives we write [8]:

rjk(x’ + u,dx’, x2 + azdx2, x3 + a3dx3, x0 + cr,dxO) = rjk(x’, x2, x3, x0) + a, 2 dx’

+a2!%dx2+ . . . a2l-I ‘kdx’dx2 + u’a2axlax2

. +cc @I- l> a2r;k 2 i

+.. I 2,

c”2- ‘) a rjk dX2dX2 --=dx’dx’ + N,~--- . a2ax2

. + ar20r,@1-i) a3rjk +.. ~ a3r;k 21 ax2ada.d

dx2dxldx, +a&l-Wl-2) 3! ax~adad

dx’dx’dx’ + . . . .

(13)

a, is the number of steps along x1, a2 is the number of steps along x2 etc. Since we have assumed integrability is satisfied for the present, all the mixed derivatives are

symmetric. Thus, we can write

a2r! ,k _ i a2rk ( a2rt

-3 axlad axlax' +ada':r > (14)


etc. Thus the r.h.s. of equation (13) can be written in terms of symmetrized derivatives without affecting the answer we get for rj, at any point.

We have written the r.h.s. in terms of symmetrized derivatives so as not to favor any particular ordering of derivatives. This is akin to not favoring one integration path over another.

From equations (3) all higher order derivatives of rjk can be expressed in terms of products of Fs. Then equation (13) enables us to calculate r& at any point in terms of the origin point data. Thus, equation (13) becomes

r&(x’ + cc,dx’, x2 + a,dx2, x3 + a3dx3, x0 + aodxO)

=rg.d, x*,x~,x~)+&{ -r;r$+rLr;+ rjmrg}pdX’ 1

+a,a,iB&,,dx’dx* + a,a3B&,dx’dx3 + ... , (15)

Equations used to calculate the field are not dependent on the values assigned to the data associated with the equation. This is true for equations (1) and (2). It is also true for hyperbolic equations. We shall make the same hypothesis here. That is, we assume that the manner in which we calculate the field at any particular point will not depend on the values assigned to the origin point data. Since the integrability equations are just conditions on the origin point data, we thus require that equation (15) holds whether integrability is satisfied or not. That is, (r$), need not obey any restrictions in equation (15).

Thus, in principle, we can handle the no integrability theory. However, the method is rather cumbersome. We can see that the second derivative contributions in equations (15) and (16) are already becoming somewhat unwieldy. Thus, in practice, we can only calculate rjk at a few points close to the origin.

Although, in principle, equations (13H15) are quite simple, in practice they are far more difficult to work with than conventional theories since each point has to be treated in an individual way. We have developed another method for dealing with a nonintegrable system. This method [9] was applied to a simple set of no integrability equations. In this case we obtained, in principle, an analytic solution to the equations, although the solution is not in closed form.

The basis for our approach is the following schematic equation:

r(u)= 1

No.

( >

c (Contributions from integration along a path from P + U) (17)

of paths Paths starting with P and ending at U. Segments of path are along the coordinate axes and there is no “back-tracking”.

In our numerical studies of this equation calculations involving each path are made using a fourth-order Runge-Kutta approximation.

We can justify equation (17) on the grounds of symmetry and not allowing certain types of infinities. With a no integrability system the results at a point U depend on which path is employed in integrating from P to U. However, there is no reason to favor one path over another. Since no path is favored, we treat all paths in the same fashion. Furthermore, the results should agree with the special case when the values for each path contribution are the same (integrability satisfied). This can be achieved by dividing the r.h.s. of equation (17) by the total number of paths. Finally, we argue against including those paths in which there is “back-tracking” in the summation over paths. An example of such a path is given below:

576 M. MURASKIN

If we allow “back-tracking” then there will exist paths for a particular choice of grid that have infinite segments. Furthermore, the number of paths, again for a fixed grid, will also be infinite. The way to avoid these types of infinities, is to restrict the number of paths to those without “back-tracking”. We shall choose our segments of paths to be along the coordinate axes, as this is the way the equations are given [equations (3)].

Equation (17) can also be obtained directly from equation (13). We will demonstrate this for the simplest situation having the following four points (x’ = x, x2 = y):

The problem is to calculate the field at S in terms of the field at P when we do not have integrability.

We shall consider dx and dy as being finite with a view towards our numerical work. Derivatives are treated as finite differences. In this case equation (13) (expressed in terms of symmetrized derivatives) becomes

(18)

F represents a component of Fjk. For demonstration purposes we study the aesthetic field equations in a simple situation. We

choose rj, at P to be

r;,=A

r:,=A

r:,=A

r:,=A

r;,= -A

r:,= -A

r:,= -A

r;,= -A

r:,=J

r:,=J

r:,=J

r:,=J

r:,= -J

r;,= -J

r;,= -J

r:,= -J

r;2=A r13=A

ri2=A I-b=A

ri2=A r:,=A

rA2=A rh3=A

r:,= -A i-f,= -A

r:,= -A r:,= -A

r:,= -A r:,= -A

r;,= -A r;,= -A

ri2=L ly3=J

rG2=L ly3=J

r:2=~ r:3=J

ri2=L r&=J

r:,= -L ry,= -J

r;,= -L r;,= -J

rg= -L r;,= -J

r;,= -L r;,= -J

rlo=B

r:O=B

r:O=B

r:,=B

r:,= -B

r&=-B

r:o=-B

r&=-B

ly()=J

T&=J

lyo=J

r&=J

r$= -J

rg= -J

r&=-J

r&=-J. (19)

With this structure the field equations collapse into the following (as we will be concerned with maps in the x, y-plane we write down the equations in the two dimensions x and y):

aA aL aA aJ aL z=k~J, g=k,J. ;=Ak2, z=Ak2, &=k,L, $k,L, %=Ak,, 6=Ak,;

k,zA -B, k2=J-L.

(20)


The structure (19) is maintained at all points by the field equations. Thus, system (20) is also valid at all points. From the field equations we see that k, and k2 are constant.

When k, and k2 are both nonzero, system (20) leads to a lattice structure when we specify the integration path. We have taken an integration path that integrates y first and then x.

From system (20) we see

a,,J = k,k, L, a,J = k,k, J. (21)

Thus, the mixed derivatives are not symmetric and, therefore, we have a no integrability situation when k, and k, are nonzero.

The first question we may ask is how we define mixed derivatives in the no integrability theory in terms of finite differences. The answer is not unique anymore since results depend on path. There are two natural ways to define a mixed second derivative:

dxdy&J = J,(S) -J(R) - J(Q) + J(P) (22a)

or

dxdy&J = J,(S) - J(Q) - J(R) + J(P); Wb)

J,(S) is the value of J obtained by integrating from P -+ R+S, J*(S) is the value obtained by integrating from P + Q + S.

The answer to the question posed above is obtained by requiring consistency with equations (21). Writing system (20) as a set of total differential equations we obtain (see Appendix B)

We also obtain

J,(S)-J(R)-J(Q)+J(P)=k,k,Ldxdy. (234

J2(S) - J(R) -J(Q) + J(P) = k,k,Jdxdy. (23b)

These two expressions are not equal since k, and k, are nonzero. From equations (21) we see that the mixed derivatives should be defined according to definition

(22a). In this manner we can deduce a general rule. In evaluating a mixed derivative we first take finite

differences according to the index on the far left. Then we take finite differences according to the adjacent index etc. The mixed derivatives are not in general symmetric.

Evaluating equation (18) when we consider the adjacent points P, Q, R and S introduced previously, and rearranging terms, gives

r (S) = +{K, (S) - r WI + [r W - r @‘)I + r 0’))

+#X9 - r (Q)l + [r (Q) - r @‘)I + r @‘)I. (24)

For any two neighboring points along a path on the r.h.s. of equation (24) we assume

J’jk(H) - Tjk(G) = {rgr; + r&r; - r;r;,)odx’. (25)

Combining equations (24) and (25) we get the structure (17). Similar results occur when we go farther from the origin and have many more paths to deal with.

From structure (17) it is again clear that the field at each point must be calculated in an individual way. de la Pena [lo] has pointed out that the Feynman path integral method is suggestive of stochastic behavior. We see here that a summation-over-paths method is natural to a no integrability theory as well.

We wish to emphasize that equation (17), when expressed in terms of finite differences using the field equations (23, enables us to calculate the field at any point, regardless of whether integrability is satisfied or not. The hypothesis invoked is that the manner of calculating the field should not be affected by the choice for origin point data. Equations like (17) can be used to calculate the field when integrability is satisfied. Here the results from each path are the same.

We have constructed a no integrability system by not requiring that the origin point data satisfy the integrability conditions. We introduce mixed derivatives by means of finite differences. Elementary calculations employing a few points close to the origin, together with numerical studies, verify that the results depend on the choice of integration path, and that mixed derivatives are not

578 M. MURASKIN

symmetric [see expressions (22a,b) and (23a,b)]. Nevertheless, we have shown that the field can be calculated at any point in a unique way by means of equation (15) or (17).

We may at this point ask what is the role of field equations in a basic theory. Is it to give a function likef(x, y), as in equation (8) [(rjk[xm]) in our case]? We point out that this is not the purpose of the field equations in the no integrability theory. The role of the field equations (3) or (20) is to calculate rk along any integration path. This is illustrated in Ref. [9], where we showed how to solve (in principle) a simple no integrability system. We found sinusoidal behavior along any integration path. When we altered direction along the path, the amplitude, phase and wavelength of the sine function changed (also the quantity called c changed). We also noted that continuity was maintained along an integration path. Once we had calculated the contribution from the different paths the full field was obtained by summing over paths [equation (17)]. (See also Appendix C for a discussion of the analytic approach.) At this stage we can define for the points

P, Q, R and S

dx d&J = r (S) - r (Q) - r (R) + r (P). (26)

Note that here r represents a field component obtained from the summation-over-paths procedure. Suppose even that r is regular, so we would have a,r = 8,,r. Regardless of this, the field equations are not expressed in terms of these quantities. Working with these quantities would mask the underlying aesthetics, and the underlying lack of integrability, and would not enable us to avoid the restrictive character of the integrability equations.

In other words, the role of the field equations is to calculate quantities like r, (S) and T,(S) [see equation (24)]. Such quantities are related to the underlying aesthetics in our program and are not affected by the theorem of calculus requiring symmetric mixed derivatives for regular functions.

4. SOME NUMERICAL RESULTS IN THE NO INTEGRABILITY THEORY

From equation (17) we see that the numerical calculations will be restricted to regions near the origin. The number of paths needed when we have n, segments along x ‘, n2 along x2, n, along x3 and no along x0 is (n = n, + n2 + n3 + no)

n!

n,!n,!n,!n,!’ (27)

which becomes astronomical as we move away from the origin. We have chosen the origin point data to be

r:, = 1.0, r:,= 4.0, r:, = -1.0, r:, = 1.0; (28)

with ep given by equation (12). Results of the numerical work are given in Fig. 1. The numerical results are consistent with multiparticles. However, in order to observe a region with several maxima and minima we had to use a large grid and numerical errors are thus a factor that causes some concern.

If we prescribe an integration path in the manner described previously, we obtain a lattice solution. Inspection of Fig. 1 shows that the sum over paths has led to a less symmetric situation.

5. SOME REMARKS CONCERNING A ROLE

FOR NO INTEGRABILITY IN PHYSICS

We have shown that a no integrability theory is natural and “aesthetic” in the sense that integrability leads to undue restrictions on the origin point data within the aesthetic field theory. We have shown that the no integrability theory can be handled easily from a conceptual point of view by two different methods. In a practical sense we expect that the no integrability theory will test present computer capabilities.

Although symmetry of mixed derivatives is taken for granted in physics we have argued that this viewpoint is unecessarily restrictive. Nonsymmetry of mixed derivatives does not prevent us from constructing a well-defined and meaningful theory.


Fig. 1. Map for Fi, using summation-of-paths technique. Numbers are 100 times actual values.

The question then, is whether there is a useful role for no integrability. It has been suggested by Godfrey [l 11, Pandres [12] and the present author [13, 141, that no integrability may be a key to understanding quantum theory.

We have already noted that a summation-over-paths technique appears in no integrability theory as well as in quantum theory.

Classical derivations of the continuity equation when we deal with energy as well as charge make use of symmetry of mixed derivatives. On the other hand, energy is not conserved in quantum theory [15] due to “quantum fluctuations”.

The field is calculated on a point-by-point basis in no integrability theory. In Ref. [13] attention was focused on equation (17). This equation represents a superposition

principle that is absent when integrability is satisfied. We have investigated the effect of this superposition principle on the motion on particles, albeit in the simplest approximation to equation (17). When we evaluated the field from a single path we found a continuous trajectory for the particle, which we could follow as long as we desired. When we considered equation (17) in the simplest approximation we found that the trajectory was no longer something we could follow in time.

The simplest approximation to equation (17) hardly carries the full information of the no integrability theory, but already the character of the solution is profoundly altered.

Hopefully more information about the no integrability theory, and its possible linkage to quantum theory will be uncovered by future research.

REFERENCES

I. G. F. Simmons, Differential Equalions with Applications in Historical Noles, p. 214. McGraw-Hill, New York (1972). 2. M. Henon and C. Heiles, The applicability of the third integral of motion: some numerical experiments. ASP. J. 69,

73 (1964). 3. M. Muraskin, Particle behavior in aesthetic field theory. Int. J. theor. Phys. 13, 303 (1975). 4. M. Muraskin, Sinusoidal solutions to the aesthetic field equation. Fdns Phys. 10, 237 (1980). 5. M. Muraskin, Further studies in aesthetic field theory. Hadronic J. 7, 540 (1984). 6. M. Muraskin, Aesthetic field theory: a lattice of particles. Hadronic J. 7, 296 (1984). 7. M. Muraskin and B. Ring, Increased complexity in aesthetic field theory. Fdns Phys. 7, 451 (1977).

580 M. MURASKIN

8. J. Mathews and R. Walker, Mathematical Methods of Physics, p. 329. Benjamin, Reading, Mass. (1964). 9. M. Muraskin, Sinusoidal decomposition of lattice solutions. Hadronic J. Suppl. 2, 600 (1986).

10. de la Pena, Proceedings of the Latin School of Physics. World Science (1982). 11. J. Godfrey, Quantum mechanics in the formulations of London & Weyl. PRL 1365 (1984). 12. D. Pandres Jr, Quantum geometry from coordinate transformation relating quantum observers. Inf. J. theor. Phys. 23,

839 (1984). 13. M. Muraskin, Trajectories of lattice particles. Hadronic J. Suppl. 2, 620 (1986). 14. M. Muraskin, Aesthetic fields without integrability. Hadronic J. 8, 279 (1985). 15. L. Landau and E. Lifschitz, Quantum Mechanics-Nonrelativistic Theory, 3rd edn, p. 158. Pergamon Press, New York

11977). 16. Jy Wheeler, Geomerrodynamics, p, 225. Academic Press, New York (1960).

APPENDIX A

In this section we would like to elaborate on certain features discussed in Ref. [3]. We quote from Wheeler [16]:

“Two views of the nature of physics in sharp contrast: (1) The space-time continuum serves only as an arena for the struggles of fields and particles. These

entities are foreign to geometry. They must be added to geometry to produce any physics. (2) There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and

other fields are only manifestations of the bending of space. Physics is geometry.”

We have adopted alternative (1) as a cornerstone of our approach. A,, g,,, e:, r;k etc. are a set of tensor fields that change from point to point in a space that we take to be Cartesian at the outset. Tensors are defined with respect to constant coordinate transformations. Transformations that do not change from point to point cannot simulate any dynamics, as dynamics implies a change of field from point to point.

We then hypothesize a change relationship for the various tensor fields. For a vector field we require

dA, = T$A,dx’. (A.1)

Since all tensor fields are treated in a uniform way with respect to change we obtained in Ref. [3],

87-q’. mnp /ax’+ TVk” m,,,,l-:_, + T” mp .I-&, + = 0; (‘4.2)

T” mnp, is a tensor constructed from the fields A,, g,, f Jk, er , ak etc. Equations (A.2) hold for any Tuk...,,,, and, thus, the set (A.2) constitutes an infinite number of equatrons. We can abbreviate set (A.2) as

T”k. mv ,,=o. (A.3)

We emphasize that ; I is not a covariant derivative since our coordinate system does not change from point to point. All fields change according to set (A.2). dx’ is a tensor by our definition of the notion of a tensor. However, it is not

a field, as we have adopted Wheeler’s alternative (1). Thus, we shall require d(dx’) = 0. That is, the role of space-time is to be that of an “arena” in which the fields A,, g,,, r;, change from point to point.

APPENDIX B

We furnish here some additional steps used in obtaining equation (23a). From system (20) we have for changes along x and y, respectively:

dJ = Ak,dx and dA = Lk,dy. (B.1)

These equations hold for neighboring points. Thus, for the points P, Q, R and S appearing in Section 3, we have

J,(S)-J(R)=A(R)k,dx and -J(Q)+J(P)= -A(P)k,dx. (B.2)

Thus, we have

J,(S)-J(R)-J(Q)+J(P)=A(R)k,dx-A(P)k,dx.

From equations (B.1) we make use of

A (R) = A (P) + k, L (P)dy.

Therefore,

(B.3)

(B.4)

This is equation (23a).

J,(S)-J(R)-J(Q)+J(P)=k,k,L(P)dxdy. (W

We obtain higher mixed derivatives, so as to have consistency with the field equations, by the rule outlined in the text. We give another example of the rule below:

Consider the points To .U

Re OS

PO l Q

Then, the quantity (a,,,J) dy dy dx is evaluated to be

@,,,J)dy dy dx = J,(U) - J(T)- 2[J,(S) - J(R)]+ J(Q) - J(P). (B.6)

Here J,(U) is the value obtained by integrating J from P --t R + T + U and J, (S) is the value of J obtained by integrating from P -+ R -+ S.


APPENDIX C

In this section we discuss an analytic approach to the simple no integrability system (20). This approach was first discussed in Ref. 191.

Along the x-axis a second derivative is defined in the usual way since the effect of nonsymmetry of mixed derivatives does not appear here.

Thus, for the points

P Q ti

we get a2A ax2= lim

AW)-2A(Q)+A@‘) Ax-0 AxAx ’

(C.1)

From the system of equations (20) we get

iY,A = k, k,A. (C.2)

For k, k, < 0 we have sinusoidal solutions to the equations. The parameters involved in the sinusoidal solution can be obtained from the field equations [9]. Thus, the structure of A (as well as B, J, L) is determined in an analytic way along the x-axis for as great a distance as we wish. At any particular point along the x-axis we can use the results obtained in the above fashion as origin point data. We can then integrate along y. From system (20) we get

a,,,,A = k,k,A. C.3)

Thus A is sinusoidal along y as well. Again, the parameters can be fixed from the equations. In this manner we can calculate the field along any path. The full field is then obtained from summing over paths according to equation (17).

The analytic approach is hampered by the large number of paths needed as we proceed away from the origin. The lattice solution to system (20) is characterized by sinusoidal behavior along integration paths.

nonintegrable aesthetic field theory

Documents