Summation-over-Paths Degree of Freedom and the Loop Lattice

M. Muraskin

University of North Dakota

Department of Physics Grand Forks, North Dakota 58202

ABSTRACT

Previously we found different kinds of lattice solutions in the no-integrability aesthetic field theory when we specify an integration path. When we combine the lattice origin-point data and aesthetic field equations with the summation-over-paths degree of freedom, the lattice solution is altered in a dramatic way. In this paper we study in particular how a loop lattice system is affected by the summation over paths.

We find likely evidence for two loop particles. The other potential particle systems could be closed strings (loops), but increased computer capability would be necessary

to confirm this.

I. INTRODUCTION

We have embarked on a long-term project that explores the consequences of mathematically aesthetic principles. We may call such a program the calculus of aesthetics.

The underlying hypothesis we have made is that the foundation of physics lies in mathematical aesthetics. If not, how would we justify, in the end, one system of equations rather than another? One cannot expect empiricism to determine a basic system of equations by itself, as one can argue there are fundamental limits to experimental precision.

The set of aesthetic principles that we have been using is not meant to be rigid. In our work we have focused attention on certain ideas which by now have been shown to have considerable content. One considers the basic entities to be Cartesian tensors. Once one opens the door to tensors, there is no reason to promote a low-rank tensor conceptually over a high-rank tensor when it comes to studying the change of these tensors. The fact that low-rank

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198 M. MURASKIN

tensors are simple to work with is not adequate cause to favor them. The way the tensors change is determined by their derivatives. In a Cartesian space derivatives of tensors are themselves tensors, so the uniform treatment of tensors would also encompass the uniform treatment of derivatives of tensors.

We have shown that the treatment of all tensors, including derivatives of tensors, by means of a uniform procedure (with respect to the change of these tensors) leads to a set of nonlinear partial differential equations called the aesthetic field equations [ 11.

At the outset of the program we do not assume multiparticles, quantum mechanics, special relativity, curved space, Lagrangians, etc. On the other hand, the project has shown that the aesthetic program contains considerable structure. In a previous paper [2] we have obtained a solution with extremely complicated particle structure. Such solutions are too difficult at this point to analyze. Long-time averages would be useful, but we do not have the capability to calculate them at this time. For this reason, we have studied simple solutions-in particular the lattice solution. Note that the lattice solution is not put in by hand, as is done so often in the literature, but emerges from the aesthetic field equations. The lattice solution offers us the possibility of studying physical concepts which are not tied to the complexity of the solution.

In Reference [4] we study a lattice solution in a very crude approximation, called a naive approximation. This approximation is not reliable, but it nevertheless enables us to study the effect of a linear superposition (based on a summation over paths) on the motion of particles. We call a system in which particles have well-defined trajectories a C theory. Systems in which particles do not have well-defined trajectories are termed a Q theory. We have shown, in this case, that the linear superposition principle gives rise to a Q theory. A specification of integration path, on the other hand, leads to a C theory. Said in another way, the manner in which the field equations are integrated in the no-integrability aesthetic field theory can transform a C theory to a Q theory.

The equations are not of the hyperbolic variety. Data are arbitrary at a single point rather than on a hypersurface. Nevertheless, we have shown that it is possible to introduce a concept of time in which the intuitive nature of “flow” in time is nevertheless present [5].

In a simple situation [4] we were able to introduce the concept of Lagrangian, even though it was pointed out in Reference [I] that all scalars are constant in the theory. The way we introduced the Lagrangian involved following the motion of the particles.

Thus, by studying simple solutions to the aesthetic field equations we may hope to gain insight into physical concepts. This program is still in its rudimentary stages.

Loop Lattice Systems 199

In our initial paper in 1970 [6], all we were able to obtain as a conse- quence of mathematically aesthetic principles was an extremum in the field. Our aim in 1970 was to obtain what we call a cornputerscope. The analogy is made with the microscope: once we insert a slide, we have the capability of studying “hidden worlds.” With a computerscope one supplies origin-point data in conjunction with the aesthetic field equations. Then the computer generates pictures of model mathematical universes. Initially all we aimed for was a mathematical universe that had interesting particle content. The hope was that such a model universe would simulate our empirical universe at least in some domain. We now see that the program can be used to explore basic physical concepts.

As mentioned before, the aesthetic field program has considerable con- tent. When we specify an integration path we have seen a multisoliton lattice system in a three-dimensional space-time [7]. The lattice particles undergo a loop motion in the third dimension. When we introduce the summation over-paths degree of freedom (this degree of freedom arises when we favor no integration path over any other path) we see what appears to be a more disorderly system, although the number of particles is considerably reduced from the case when an integration path is specified.

We have studied another solution when the maxima and minima lie on closed strings (loop lattice) [s]. Th’. 1 1 t’ 15 50 u ion is fully four-dimensional. When we studied this system with the added summation-over-paths degree of freedom, we found planar multiple maxima and minima arranged in a more disorderly way [9]. However, the map shown in Reference [9] was restricted to the region close to the origin, and furthermore we had to use a coarse grid in obtaining the map. Such a grid causes some concern so far as accuracy is concerned. We need the coarse grid because of the proliferation in the number of paths as one proceeds away from the origin. Since these results were initially obtained in Reference [9], we have developed the random-path approximation [lo]. This procedure enables us to integrate the field equations considerably farther from the origin and with a finer grid. The random-path approximation has been tested against the summation-over-paths techniques as well as the more accurate commutator method used in Reference [B]. In the commutator method, only one integration path is needed. There is a correction to the single-path results, which is calculated from the set of commutation relations for mixed derivatives. The commutator method is useful only when the field equations collapse into a simple set, so that all the commutators can be readily calculated. The commutator method is thus not practical for discussing the loop lattice appearing in the appendix of Refer- ence [S].

In this paper we shall study effects of the summation-over-paths degree of freedom on the loop lattice. The loop lattice was previously discussed in

200 M. MURASKIN

References [3], [a], and [9]. Use will be made of the random-path approxima- tion.

We have yet to establish within a four-dimensional framework whether the summation-over-paths degree of freedom can lead to three-dimensional maxima and minima. In Reference [S] we found no evidence for the existence of particles when the summation-over-paths degree of freedom was com- bined with a point lattice system. The new random-path approximation gives us the opportunity to make such a study for a loop lattice system.

II. THE CALCULUS OF NO INTEGRABILITY

By now we have written two review articles on aesthetic field theory [l, 111. The crucial additional feature of Reference [ll] is the introduction of the concept of no integrability.

We have emphasized previously that data are supplied at but a single point, rather than on a hypersurface as in hyperbolic theories. There is no reason to expect when integrating from the origin point to some particular point that the results should be independent of path. Independence of path can be looked upon as an undue restriction on the theory.

We can look at the situation in another way. Consider the two-dimensional

equations

C?A aB a1 aL z = k,J, z=k,J, z=k,A, z=k”A,

aA - = k,L, aY

aB - =k,L, aY

81 - =k,A, aY

aL - = k,A, aY (1)

k,=A- B,

k,=J-L.

These equations can be obtained from the aesthetic field theory 1111. From Equation (1) we see that k, and k, are constants.

By formally taking a/ay of the top line of (1) and a/&r of the second line

we see that

63, -a,,) J = k,k;. (2)

Since the mixed derivatives are not symmetric, we conclude that the results

Loop Lattice System 201

of integration depend on path. This fact can be easily confirmed by our numerical program.

We have studied the system (1) in some detail [ll]. The equations (1) have interesting content. When we specify an integration path by integrating along y first and then along X, the system (1) leads to a planar lattice structure when k , k o < 0.

There is an immense class of equations similar to (11, for which the mixed derivatives are not symmetric. It would be shortsighted to not consider such equations just because the mixed partial derivatives are not symmetric. One’s horizons are much too stringently limited by such an exclusion.

In a series of papers we have shown how to handle equations of the type (1) [2,4,5, g-131. In Reference [2] we show how to calculate the field at all points by evaluating products of the field at the origin. In References [9], [ 111, and [12] we discuss the summation-over-paths technique. An approximation to the summation-over-paths method (random-path-approximation) is devel- oped in Reference [lo]. In References [S] and [13] we establish the commuta- tor method as a technique for solving the no-integrability equations when the form of the equations is of a simple enough variety [for example, the system

WI. In the period before high-speed computers it is understandable why one

would exclusively consider cases where the mixed partial derivatives are symmetric. However, equations such as (1) can be readily handled using numerical techniques, and that has been done in the series of papers listed above. With the use of computers we have generated picutres of what equations such as (1) say. We have used the University of North Dakota IBM 3090 and the Cornell University supercomputer in the course of our studies.

III. THE LOOP LATTICE

Consider the following origin-point data for I&:

l-i2 = 1.0, r,: = - 1.0,

(3) rp,= -1.0, ril=i.o.

I$ are obtained from

(4)

202 M. MURASKIN

where eui is chosen to be

e”i = I 0.88 -0.42 0.22

0.5 0.9 0.3 0.44 -0.16 1.01

(5)

0.2 -0.55 0.6

This set of data (with the 3 and 0 indices interchanged) was first introduced in Reference [3]. The results are summarized by Figure 2 of Reference [3]. This figure was obtained by specifying an integration path. For these data we integrate along 1~’ z t, then Z, then y, and finally X. These same data were further studied in the appendix of Reference [8]. Here the calculation was extended to three space dimensions. It was found that three-dimensional maxima and minima lie on closed loops. The loops are located in a symmetric way throughout the three-dimensional space. We call such a system a loop lattice.

The addition of the summation-over-paths degree of freedom to this system was done in Reference [9] (see Figure 1 there). The results indicate planar multiple maxima and minima in a more disorderly-looking arrange- ment than in the lattice. In Figure 1 of Reference [9] we had no way to deal with the path proliferation problem, so we were unable to integrate very far from the origin. We also chose to confine ourselves to the X, y plane. As a result of this we were unable to establish whether the summation-over-paths degree of freedom is consistent with multiparticles in four-dimensional space-time. We may add that the existence of particles in four-dimensional space-time using the summation-over-paths degree of freedom has not been established for any set of data. Should such particles be found, their structure would be of interest.

We are now in a position to study the loop lattice, when we sum over paths, in a more accurate way, using the random-path approximation.

An x, y map at t = 0 is given in Figure 1 for the representative component Iii,. The grid used was 0.0375 except for the region around certain maxima and minima, where a finer grid was used. The number of random paths was taken to be 500, although we get the same picture as Figure 1 if it is taken to be 100. The results show that the O.3-grid results of Reference [9] are qualitatively correct. The maxima and minima in Figure 1 all have magnitude 0.62kO.02. This is the same magnitude as when we specify an integration path. Assuming that this pattern continues throughout the plane, we have a way to monitor the accuracy of the calculation. If a maximum or minimum magnitude falls outside the range 0.62 f 0.02, we refine the grid until the magnitude of the maximum or minimum falls within this range. This was done in constructing Figure 1.

Loop Lattice Systems 203

FIG 1. Data (3), (4), and (5) used in conjunction with the random-path approximation to

the summation-over-paths method. The map is for a representative component, r:,. Map is at

z = 0, t = 0. The grid is 0.0375, and the number of random paths is 500. Numbers in the figures

are 100 times actual numbers.

Unfortunately, in this study we have not had unlimited computer time available to us. Thus, we have had to make our computer runs selective.

We first focused on the - 63 planar minimum in Figure 1 (-63 is 100 times the actual number). We studied the location of this planar minimum as a function of .a. The results are given in Table 1. Note that x, y, .z are all given in units of 0.3.

For a large region - 11 I .z I 14 this planar minimum lies within the range - 0.62 f 0.02. Outside of this z range we still can get the minimum to lie within -0.62 +0.02 if we refine the grid. However, if we increase the magnitude of z further, we find that finer grids do not give rise to a planar minimum lying within the -0.62*0.02 range.

Consider the planar minimum at z = 22 with value -0.314 (using a 0.0375 grid). When we arrived at this point with a 0.01875 grid, we got a value of -0.337. With a 0.009375 grid we get -0.363, and with a 0.0046875 grid we got -0.377. In the last case we took the number of paths to be 300. In the other cases it was taken to be less. Furthermore, we covered the region of the minimum with a 0.009375 grid and never found a value greater than - 0.373.

204 M. MUBASKIN

TABLE 1

LOCATION OF A PLANAR MINIMUM AS A FUNCTION OF Z

Magnitude of planar minimum 2 x Y when 0.0375 grid is used

22 -15 1

20 - 14 3

18 - 12 4

16 -9 5

14 -7 5

12 -4 5

10 -2 5

8 -1 5

6 1 5

4 2,3 4,5 2 4 4

0 5 4

-1 6 3

-3 7 3

-5 8 2

-7 9 1

-9 10 1

-11 11 0

- 13 12 -1

- 15 12 -2

- 17 12 -4

-19 12 -5

-23 9 -8

-26 10 -7

-0.32

- 0.47

- 0.56

- 0.59

- 0.60

- 0.61

- 0.62

- 0.63

- 0.64

-0.61

- 0.64

- 0.62

- 0.63

- 0.63

- 0.63

- 0.62

- 0.61

- 0.60

- 0.59

- 0.59

- 0.57

- 0.56

- 0.49

- 0.31

At z = -26 the value shown in Table 1 for Iii, is -0.31. With a 0.0046875 grid the value was - 0.35, well outside the range - 0.62 _+ 0.02.

The evidence tells us that the string particle is bounded in .a (that is, the 0.62f0.02 magnitude region is bounded in z).

Can we understand better the boundedness of the 0.62 f 0.02-magnitude string? In order to see what is going on we need larger maps. Taking into account our limitation of computer time, we mapped regions of three-dimen- sional space using a 0.075 grid with the number of paths chosen between 100 and 250. This led to planar maxima and minima lying in roughly the same positions, but magnitudes not lying in the range 0.62 kO.02 (the distance between the points in the figures is now 0.6 rather than the 0.3 appearing in Figure 1). Previously we have seen that using a grid coarser than we would like still leads to qualitatively correct results. Furthermore, in sensitive regions we have refined the grid to 0.0375.

Loop Lattice Systems 205

FE. 2. Map at 2 = 0, 1= 0 for rj,. Grid is 0.075, which is a coarse grid. Contour lines are + 0.40. Number of paths has values between 100 and 250. This plot is similar to Figure 1 except

the domain is larger and the mesh size is doubled.

-6

-A-

FIG. 3. The regions labeled A in Figure 1 are studied for - 8 5 z I 8 (z is in units of 0.6). Contour lines are -0.40. We see that the A regions approach and/or join one another for z = ~-8, giving evidence for a closed string particle (loop particle). A number in the Figure

adjacent to a contour line represents the value of z for which the contour line has the shape indicated.

206 M. MLJRASKIN

FIG. 4. The regions labeled B in Figure 2 are studied for -8 _< ; I 8 (z in units of 0.6).

Contour lines are 0.40. .@in we have evidence for a closed string particle.

Figure 2 represents a z = 0, t = 0 map similar to Figure 1 except that it involves a larger region done with a coarser grid (0.075 rather than 0.0375). In Figure 2 we have marked the maximum and minimum regions A, B, C, and D. Contour lines have the magnitude+O.40.

Figure 3 shows (using a 0.075 grid) that the regions marked A approach one another and join at z = 8. At z = - 8 they practically join. Figure 3 superimposes the locations of the A regions as z varies from - 8 to +8 in steps of 2 (the values of .a are in units of 0.6). The contour lines in Figure 3 are -0.40.

In Figure 4 we do the same thing for the regions marked B in Figure 2. Again we see that the B regions approach one another. The contour lines in the figure are kO.40.

The evidence of Figure 3 and Figure 4 points towards A and B as loop particles, and C, D could also represent loop particles. However, greater computer capabilities would be needed to draw conclusions here.

This work represents the first evidence for three-dimensional particle systems arising within the summation-over-paths approach.

IV. CONCLUSION

The conclusion we reach from the data is that the summation-over-paths degree of freedom when applied to the loop lattice alters the picture we get

hop Lattice Systems 207

in a drastic way (note Figure 1 as compared to Figure 2 of Reference [3]. The evidence, as indicated by Figures 3 and 4, points towards A and B represent- ing three-dimensional particles in the shape of a loop. Particles C and D could also be closed strings, although further computer capability. would be needed to make a determination.

Assuming that we have a system of loop particles, there is a difference from the loop lattice. For the loop lattice the centers of the loops are situated in a lattice configuration. The data here suggest that the strings may “loop” around the origin region. We have no information whether this is the case for maxima and minima further from the origin.

The norm in physics is to take mixed partial derivatives as symmetric. This paper is another in a series that aims at understanding what effects are implied by not requiring symmetric mixed partial derivatives so that results depend on integration path. With the advent of high-speed computers we have the capability of obtaining pictures of such systems.

A totally different approach to nonintegrable systems is p!rsued in Refer- ences [15] and [16].

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