Registered: Jun 2006
Thank you for your reply. I understand what you are saying, and I may have a satisfactory answer for you.
The problem of anisotropy is sometimes stated that a continuum has no preferred directions, while a regular lattice surely does. Or that particles in a lattice should go farther (or faster) in certain preferred directions compared to others, which is not observed in nature.
I also want to state for the record that nature is whatever it is, and it is up to us to try to determine what it chose to be. If nature chose a space-time continuum, then I am a crackpot and doomed to failure. If nature chose a discrete space-time lattice, then it would be very cool to figure this out.
The short answer to the question of anisotropy is that the maximum speed in different directions within a discrete grid is always one unit of position per unit of time. After say, three ticks of time, a particle with constant motion will always be three cells away from its origin, so by definition it has traveled the same distance (no farther) at the same speed (no faster) in all directions, so I don't think the anisotropy question really has to do with distance or speed per se.
I think the question of an anisotropic space-time model is really about the shape of a light-sphere, or in a flat drawing, a light-circle.
This requires a visual aid (I love visual aids). The easiest piece of a discrete space-time lattice to draw here on the NKS Forum would be a 2-D slice of space-time showing two dimensions of space. A 2-D flatland can be tiled with triangles, squares, or hexagons, and it's easiest to draw squares, so let's discuss 2-D anisotropy in terms of a lattice of tiny squares, even though 2-D nature would probably have chosen a more symmetrical lattice of hexagons, rather than squares.
Figure 1. Deterministic light circle
If a unit particle originates at position (5,5), shown as an "o" in the lattice above, and if it were allowed to move in any direction, including diagonally, then after three deterministic moves it would be in one of the cells marked with an "x". This appears at first to be obviously not much of a circle, and hardly isotropic.
The paper on arXiv proposes that motion is probabilistic rather than deterministic. This results in a probability mapping of the final results, rather than a deterministic result as shown above. A probabilistic light circle (see Figures 2 and 3 below) will be a lot fuzzier than a deterministic light circle (see Figure 1 above).
If a unit particle originates at position (5,5), shown as an "o" in the lattice below, and if it is allowed to move probabilistically to the (R)ight, or (D)own, or (B)oth, with equal probability, i.e., in all directions with equal probability, then after three probabilistic moves it would have followed one of 27 possible paths:
Figure 2. Plotting these 27 positions gives the probability map of a quarter light circle.
This is very different from the deterministic light circle shown in Figure 1. Note that any particles that happen to go straight down will wind up at position (5,8) with 100% probability, but that any particles that happen to move along the diagonal (down and to the right) are likely to wind up at (7,7) about 86% of the time, and at (8,8) only 14% of the time. Note also that the vertical distance from (5,5) to (5,8) is 3.00 units, while the diagonal distances from (5,5) to (7,7) and (8,8) are about 2.83 and 4.24 respectively. This means that the average distance traveled in the diagonal direction will be 86% of 2.83 + 14% of 4.24 = about 3.03 units - very nearly the same as the 3.00 units in the vertical direction!
If we mark these 100% vertical and 86% diagonal cells for all four quadrants, the probabilistic results begin to look much more like a real circle:
Figure 3. Highlights of a probabilistic light circle.
If we keep in mind that the space-time lattice is likely to be at the Planck scale (as derived in the paper posted on arXiv), and that the lattice may be hexagons rather than squares, and that any measurement we can perform in the real world would involve a light circle many, many, many orders of magnitude larger than these grids, then it would become exceedingly difficult to detect the anisotropic deviation from a perfect circle.
Report this post to a moderator | IP: Logged