Marhl Comput. Modelling, Vol. 12, No. 6, pp. 721-128, 1989 0895-7177/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 0 1989 Pergamon Press plc

ALTERNATIVE APPROACH TO NO INTEGRABILITY FIELD THEORY

M . MURASKIN

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

(Receiwd May 1988; accepted for publication November 1988)

Communicated by X. J. R. Avula

Abstract-This is another paper in a series which illustrates that foregoing the rules of traditional calculus opens up a realm of possibilities. We study the “aesthetic field theory”, although the techniques are more general. We introduce a new integration procedure which is consistent with aesthetic principles. There is a single change function associated with each point rather than many change quantities at a point (there is a change quantity associated with each path going through any point in our previous work). The integration procedure is consistent with the rules of traditional calculus when the integrability equations are satisfied. We find that the new procedure, when applied to the data associated with the loop lattice, leads to multiple maxima and minima on a plane. Another lattice solution (point lattice) does not show evidence of being bounded in certain domains of the region studied. In the summation-over-paths method all the paths from the origin point are needed to calculate the field. Knowledge of the field on one hypersurface is not sufficient to obtain the field on the succeeding hypersurface. In the present approach to no integrability, we can calculate the field on one hypersurface from knowledge of the field on the proceeding hypersurface (without requiring knowledge of the past history). Thus, the new approach enables us to obtain this desirable feature of a hyperbolic theory without requiring the field to be arbitrary on a hypersurface.

1. INTRODUCTION

The calculus of Newton and Leibniz can be said to be the language of physics. The standard procedure is to obtain partial differential or integro-differential equations that describe a physical system and then seek solutions, whether exact or approximate. So far, it would appear that only quantum theory lies beyond such a program, although attempts have been made in this regard [l].

The notion of a field is basic to a framework such as the above. A field is a set of numbers obtained in a unique way at each point.

The computer has shown that a procedure such as the above is not essential in generating a set of numbers at each point [2-151. We obtain such a set of numbers whether the rules of calculus are obeyed or not.

We have been pursuing this line of thought for some time. For demonstration purposes we borrow some equations from aesthetic field theory [l&l 81:

aA aJ aL -=k,J, g=k,J, .u=klA, z=k,A, ax

aA aB aJ -=k,L, -=k,L, -=kkZA, $=k,A; k,=A-B, k,EJ-L. ay ay ay

(1)

Since A and B satisfy the same equations k, is constant. Since J and L satisfy the same equations k2 is constant.

From system (1) we see that

(a, - a,,)J = hk:. (2) The rules of the calculus demand k, = 0 or k2 = 0 for J regular. If k, = 0, A and B are constant and J and L are monotonic.

However, if we prescribe an integration path (for example, we may integrate first along y and then x) we obtain a lattice solution for the quantities A, B, J and L. An (x, y)-map for the quantity

721

122 M. MURASKIN

Fig. 1. (x,y)-Map of A in equations (1) when a path is specified by integrating first along y and then along x to reach any point. The map shows a lattice structure. The numbers on the map are 100 times

the actual values.

A is shown in Fig. 1. Thus, if we are willing to forego the rules of calculus, system (1) can prove interesting.

We have introduced derivatives in a way different from the norm [9]. These derivatives agree with conventional derivatives in the case where integrability is satisfied. Mixed derivatives are defined in a unique way so as to maintain consistency with the field equations, They are not intrinsically symmetric. With an eye to our numerical work, derivatives are introduced as finite differences.

We have introduced methods previously to handle no integrability equations such as system (1) [5,9, 10, 121. Since the results of integration depend on path, we have found, subject to simple and reasonable hypotheses, that the field at any point can be obtained from the following schematic equation:

F(U) = a c (contributions from a path). paths

Here N is the total number of paths. There is no backtracking and segments are taken along the Cartesian coordinate axes. F represents a component of Fj,.

Using the summation-over-paths technique, as well as other techniques, we have integrated the no integrability aesthetic field equations in various situations [2-151. We have obtained solutions which show multiple maxima and minima on a plane [5, 11, 131, and are thus suggestive of multiparticles.

Multiparticles in the form of loop particles were obtained in Ref. [14]. Not all lattice solutions appear to be consistent with multiparticles when we sum over paths (by a lattice solution we mean a system which gives a lattice when we specify an integration path).

System (1) leads to magnitudes which increase in size as we move away from the origin when we supplement the field equations by the superposition principle [equation (3)].

Once we recognize that restricting ourselves to the rules of calculus unnecessarily limits our hori- zons, we can introduce still other approaches that lead to a field at all points. We shall show that we may still proceed within the “aesthetic” program which we have been pursuing for some time.

2. ANOTHER APPROACH TO INTEGRATING THE FIELD EQUATIONS

When integrability is satisfied the field equations (1) define A (x, y), B(x, y), J(x, y) and L(x, y). When integrability is not satisfied we have used equations (1) to define the change of A, B, .I and L along an integration path. The full field at each point is then obtained by summing over paths, equation (3). Such a procedure elevates the notion of an integration path to a prominent position

No integrability field theory 123

in the theory. For each path there is a change quantity at each point. As we proceed farther away from the origin the number of paths becomes very great and, correspondingly, the number of change quantities becomes very great.

For example, consider the situation

T .

r’

U

R S

P l Q

The contributions along the path shown can be written (r represents a component of r$) as

[r,(U) - r,(s>l+ U-,(S) - J-W + P-(R) - r(p)1 + r(P).

Here T,(S) is the value of r obtained by integrating from P + R + S. T,(U) is the value of r obtained by integrating from P + R -+ S. T,(U) is the value of r obtained by integrating from P + R + S + U. The difference in r between two neighboring points is given by

rjk(w) - rj,(Y) = (rkkr; + r;,r; - r;rj,), dx’. (4)

In calculating r,(U) - T,(S) we use the quantity T,(S) on the r.h.s. of equation (4). T,(S) is thus a change quantity as it describes the change of all tensors (including itself) in going from one point along the path to a neighboring point.

The method described by equations (3) and (4) is a natural way that treats all paths in a uniform way.

However, this is not the only way to define a set of numbers at each point, when the results of integration depend upon path (integrability not satisfied).

In the new approach there is but one change function rg at each point (rather than a large number of change quantities at each point associated with the different paths going through each point). We shall demonstrate how the method works by showing how to calculate the field for a set of points close to the origin. This procedure can then be extended to all points. The method has been programmed for the computer and the results are given in Sections 3 and 4.

Consider the points in the x, y-plane

Ve W* Y*

S. To U.

We first integrate from P + S, then S --) V and from P + Q and Q + R using the aesthetic field equations:

dr;, = (r;,rj;l+ rg; - rjy:,) dx’. (5)

There is but one path going from P --* S + V as we do not allow for backtracking. Thus, by using the above procedure we now have r(V), r(S), r(P), r(Q) and T(R).

Using the results for T(S) and r(Q) we integrate from S + T and from Q + T, respectively. The results are different in general since the integrability restrictions are not required to be satisfied. The value of r at T is then obtained by averaging the two contributions since we treat the contributions to T(T) in a uniform way. We may write this as

r(T) = [contribution (S + T) + contribution (Q + T)] N (6)

Here N = 2 since there are two contributions. Next, this T(T) is used as a change function to integrate to U (and W later on). That is, we use equation (5) with T(T) on the r.h.s. We can get the full T(U) by combining the result with the integration from R + U, and finally dividing by 2.

To get r at W we use equation (5) to integrate from V + W and from T + W, again using an equation similar to equation (6) with N = 2. This gives the full field at W which is then used as the change function for the point W.

124 M. MURASKIN

It is clear that we can continue to obtain a unique r at all points in the x, y-plane by this method. We continually use equations similar to equation (6) with N = 2 (except on the axes where there is only one contribution to the field). We note that this manner of integration coincides with traditional calculus when the integrability conditions are satisfied by the origin point data (this is the case when the results are independent of integration path). This method employs the field equation (5) which implies that all tensors are treated in a uniform way [16, 171. We also have a unique change function at each point (in contrast to the path approach). The role of the field equations is now different from the path approach. The field equations determine changes to the neighboring point and no farther (except when integrating along the coordinate axes as there is but one path contribution here).

In four dimensions the change function determines the change to a neighboring point except now we go either in the X, y, z or x0 direction. Thus, we would use an equation like equation (6) with, at most, four contributions in which case we take N = 4. For points in the x, y-plane, off the axes, N = 2 is used. For points at x0 = 0, x + 0, y + 0 and z + 0, N = 3 would be used.

N represents the number of ways we can end up at a particular point. In the x, y-plane there are but two ways (off the axes). Consider some points in the + + quadrant below:

N L K .+. .

T M

Contributions only come from directions where the change function is already known. That is, given the field at the origin and equation (5) the field at K (above) cannot be obtained without knowing the field at L.

Thus, the field equation (5) is augmented by an integration procedure, different from the procedure used previously [4-141. In equation (4) path-dependent quantities like T,(S) appear in the change equation. In equation (5) a unique change function for each point is used. Thus, the change equation is different in the two cases, even though equations (4) and (5) have a similar appearance.

The integration procedure is constructed according to reasonable tenants and is thus philosophi- cally consistent with the aesthetic field program. We have assumed the change equation (5), a unique change function at each point and that all contributions to the field at a point be treated in a uniform way. This procedure does not restrict the origin point data in any way.

We can define mixed second partial derivatives at the origin in the same manner as used in Ref. [9]. The mixed second derivatives are not symmetric. The introduction of higher mixed derivatives is not required in our procedure.

Should the change function be regular one can introduce derivatives in the usual way so that the mixed derivatives are symmetric. However, the fundamental field equations are not given in terms of these derivatives. Thus, working with the usual derivatives masks the underlying aesthetics which are given in terms of equation (5) and the integration procedure that requires a unique change function at each point.

Equation (3) is not useful here since we do not have equations to calculate the field along an extended path. Equation (5) does not have such a role.

3. RESULTS

We apply the method outlined in Section 2 to the data

r;z=1.0, l-:*=-1.0, r:,=-1.0, r&=1.0.

We use the same e: as Ref. [18].

(7)

This data has been studied extensively in the past. We first discussed it in Ref. [3] (note Fig. 2 therein). This figure was obtained by specifying an integration path. We studied the data in three spatial dimensions in Ref. [IO] and found that it led to a loop lattice when an integration path was specified. In Ref. [5] we showed that the summation-over-paths technique led to multiple maxima and minima on a plane with locations not as symmetric as a simple lattice. In Ref. [14], again with

No integrability field theory 725

Fig. 2. (x, y)-Map of r f, using the new method of integration. The grid used is 0.009375. The numbers on the map are 100 times the actual values.

this data, we showed that the summation-over-paths degree of freedom led to loop particles in three spatial dimensions. A hypothesis made was that the summation-over-paths degree of freedom is responsible for quantum behavior. However, the results of Ref. [14] were not suggestive of such a hypothesis.

System (7) has now been restudied using the new procedure in integrating equation (5). Figure 2 represents an (SC, y)-map for r I, using a 0.009375 grid. Although it is not as symmetric

in appearance as the loop lattice, some regularities are observed. In the + + and - - quadrants we observe minima of magnitude 0.64. In the + - and - + quadrants we observe maxima of magnitude 0.64. Note, the magnitude of both the maxima and minima is the same when we prescribe an integration path by integrating first in y and then in x. Also, this value appears for both maxima and minima when we use the summation-over-paths method. This suggests the possibility that all minima in the + + and - - quadrants have magnitude 0.64. The possibility exists as well that all maxima in the + - and - + quadrants have this same magnitude. On the other hand, as an example, we have two minima of magnitude 0.56 in the + + quadrant (one of these is not in the region shown in Fig. 2). Thus, we made a study of how the maxima and minima values were affected as we varied the grid size. Firstly, the -0.64 minima had the same magnitude for the differing grid sizes of 0.01875, 0.009375 and 0.0046875. In Table 1 we show how the other maxima and minima within the -t + quadrant were affected as we varied the grid size (values in the table are truncated after two places, so 0.18 and 0.17 are equivalent as far as the table is concerned). The last two entries in Table 1 are outside the range of Fig. 2.

We see that, except for the two minima located far from both the x- and y-axes, the magnitudes of the maxima and minima agree for all the grid sizes. We see that the region where errors are a factor is the diagonal area far from the origin, This is to be expected, considering the type of integration scheme under study.

Table 1 suggests that the 0.64 hypothesis mentioned earlier may be reasonable. The 0.0046875 computer run took 267 min of CPU time on the University of North Dakota 3090. Vectorization of the program is not expected to improve the situation in any significant way since the approach to the no integrability equations used here is intrinsically sequential. Due to the large amount of

726 M. MURASKIN

Table I. Values of maxima and minima in the + $ auadrant as a function of erid size

Grid size

0.01875 0.009375 0.0046875

Value of maxima and minima

0.60 -0.49 -0.48 0.17 0.60 -0.56 -0.56 0.17 0.61 -0.60 -0.60 0.18

time needed with a 0.0046875 grid we have been selective in using this grid and in using smaller grids.

Maps with the different grid sizes show that the qualitative picture is the same in all three cases. The results of Fig. 2 show in a clearcut way multiple maxima and minima on a plane in the region

studied. Unlike the summation-over-paths results [12, 141, we do not see the density of planar maxima

and minima decreasing markedly as we move away from the origin. We note several examples of pairs of planar maxima (as well as pairs of planar minima) in close

proximity. In order to observe this doublet effect we need a sufficiently small grid. For example, in the - - quadrant in the diagonal direction in Fig. 2, we see a planar minimum with value -0.64. Close by it we have written a X . At this location, with a 0.0046875 grid we see a second planar minimum with value -0.62. With a 0.00234375 grid the doublet effect is confirmed with values of -0.64 and -0.63. This is consistent with the +0.64 hypothesis discussed earlier.

Other doublets can be seen in Fig. 2 even with the 0.009375 grid used therein. The contour lines around these doublets resemble those of Fig. 2 of Ref. [3]. Figure 2 of Ref.

[3] was obtained by using the same origin point data as was used in Section 3 of this paper. However, in Ref. [3] the map was obtained by specifying an integration path. When the results of Ref. [3] were extended to three spatial dimensions (as was done in Ref. [lo]) it was found that the minima (maxima) lie on closed loops. The similar patterns appearing here in Fig. 2 and in Fig. 2 of Ref. [3] may indicate that the present approach to no integrability may lead to loop particles as well. The loop particles of Ref. [3] are arranged in a symmetric lattice structure. Here the situation is more complicated.

4. ANOTHER SET OF DATA

We consider the system leading to equations (1). We took A = - 1, B = 0, J = 1 and L = - 1. This led to an unbounded region when we applied the summation-over-paths technique. Using the present integration technique we found no sign of bound in the + - quadrant for the quantity A.

This illustrates that the new integration procedure gives different results depending on the lattice system we are dealing with. System (1) gives a point lattice when we specify an integration path, while system (7) describes the loop lattice when an integration path is specified.

5. A FUNDAMENTAL DIFFERENCE BETWEEN THE SUMMATION-OVER-PATHS METHOD AND THE NEW APPROACH TO THE NO INTEGRABILITY

EQUATIONS

In this Section we note an important conceptual difference between the new approach to no integrability and the summation-over-paths procedure.

In both formulations we start with the field being arbitrary at a single point. The field is not arbitrary on a hypersurface as we have stressed on numerous occasions [2, 161. Arbitrariness on a hypersurface would imply that particle structure is arbitrary on a hypersurface. This could be looked at as incompleteness of the hyperbolic approach.

Nevertheless, we recognize from ordinary experience in wave theory that one can calculate the field on a hypersurface from knowledge of the field behavior on the preceding hypersurface. Knowledge of prior history before the previous hypersurface is not necessary. The question is whether such a situation is consistent with no integrability theory in which data is specified at a single point.

No integrability field theory

Fig. 3. Sketch showing how one calculates the field given values on the z = 0 plane in three-dimensional space-time. The arrows indicate where equation (5) has been used.

With the summation-over-paths method knowledge of the field on one hypersurface is not sufficient to determine the field on a subsequent hypersurface. One needs to know the contribution from all paths starting at the origin point in order to obtain the field at any point. Thus, past history extending back to the origin point is needed. The hope would be that there exists some approximation scheme such that the hypersurface information would be sufficient to determine how the field behaves on a succeeding hypersurface.

There is another approach to the summation-over-paths method which was discussed in Ref. [15]. Here the summation-over-paths concept was combined with the notion of “specification of path” to deal with no integrability equations. In Ref. [15] we use summation-over-paths to obtain the field throughout three-dimensional space. Then we specified an integration path to obtain the field at later instants. In the way we set things up, once we have the field at all points of space, there is but a single way to integrate the equations. We can only integrate in time. This approach introduces a fundamental distinction between time and space at the outset (one sums over paths for space and specifies an integration path for time).

In our present approach to no integrability, the information on any hypersurface is sufficient to determine the field on a succeeding hypersurface. From the succeeding hypersurface we can proceed to the next succeeding hypersurface etc. This feature does not rely on any approximations nor does it introduce a distinction between time and space at the outset.

To illustrate how this works we shall first assume a three-dimensional space-time. From the origin point data we can determine a planar (x, y)-map in the same manner as in Fig. 2. Then from the origin point P we can integrate to Q (see Fig. 3). From the known field at S and Q we can calculate the field at T. From the field at M and T we can calculate the field at K. In a similar way we can calculate the field at R and W. Then from the field at T, R and F (Fig. 4) we can

K U C

T H

t

L 30 I/” R ,

/ i!

F /

-- - 7 --I-+- - ,’ / / /

/

Fig. 4. Sketch showing how we use the field at the points shown in Fig. 3 to calculate the field on a subsequent plane in z.

128 M. MURASKIN

calculate the field at H using N = 3. From the field at L, H and K we can calculate the field at U etc. In this manner we can calculate the field at z + AZ using the information at z. Using the field at z + AZ we can calculate the field at z + AZ etc.

In four dimensions, we obtain the field everywhere in three-dimensional space from the origin point data, as above. In a straightforward extension of the three-dimensional procedure we can then calculate the field at x0 + Ax0 etc. The entire past history is not required to calculate the field on a succeeding hypersurface.

Thus, the new approach enables us to obtain this desirable feature found in hyperbolic theories without requiring the field to be arbitrary on a hypersurface.

6. SUMMARY

We have introduced a new method to integrate the aesthetic field equations. The method employs a single change function defined at all points rather than a different change quantity at each point for each path going through the point. In this sense the new method can be thought of as an improvement over our previous approach to no integrability aesthetic field theory. The method enables us to compute the change function at all points in space. We found that the loop lattice data gives rise to multiple maxima and minima on a plane when supplemented by the new integration scheme. Not all lattice solutions behave like system (7). System (1) gives rise to what appears to be an unbounded situation (at least for one of the domains studied).

One of the desirable features of hyperbolic theory, whereby the field can be calculated from information on a hypersurface, also appears in the new version of no integrability theory, without requiring the field to be arbitrary on a hypersurface.

The new method gives identical results to the traditional calculus when the integrability equations are satisfied. We have argued that the traditional calculus is severely restrictive [note discussion concerning equations (l)]. Going beyond the calculus opens up a new realm of possibilities.

REFERENCES

1. E. Nelson, Derivation of the Schrodinger equations from Newtonian mechanics. Phys. Rev. 150, 1079 (1966). 2. M. Muraskin and B. Ring, Increased complexity in aesthetic held theory. F&IS Phys. 7, 451 (1977). 3. M. Muraskin, Aesthetics fields: a lattice of particles. Hadronic J. 7, 296 (1984). 4. M. Muraskin, More aesthetic field theory. Hadronic J. 8, 287 (1985). 5. M. Muraskin, Aesthetic fields without integrability. Hudronic J. 8, 279 (1985). 6. M. Muraskin, Sinusoidal decompositions of lattice solutions. Hadronic J. (Suppl.) 2, 600 (1986). 7. M. Muraskin, Trajectories of lattice particles. Hadronic J. (Suppl.) 2, 620 (1986). 8. M. Muraskin, Introduction of derivatives which are not necessarily symmetric. Preprint. 9. M. Muraskin, Nonintegrable aesthetic field theory. Math1 Comput. Modelling 10, 571 (1988).

10. M. Muraskin, Use of commutation relations in no integrability aesthetic field theory. Appl. Math. Compur. 29, 271 (1989).

11. M. Muraskin, Study of different lattice solutions in aesthetic field theory. Appl. Math. Comput. 30, 73 (1989). 12. M. Muraskin, Study of a three-component lattice system. Preprint. 13. M. Muraskin and R. Molmen, Approximation to summation over paths in aesthetic field theory. Preprint. 14. M. Muraskin, Summation over paths, degree of freedom and the loop lattice. Preprint. 15. M. Muraskin, On the nature of-time. Intr J. Math. math. Sci. (in press). 16. M. Muraskin. Particle behavior in aesthetic field theorv. Int. J. theor. Phvs. 13, 303 (1975) 17. M. Muraskin; Sinusoidal solutions to the aesthetic field equations. Fdns khys. i0, 23; (1980). 18. M. Muraskin and B. Ring, A two particle collision in aesthetic field theory. Fdns Phys. 5, 513 (1975).