14. The digit-transition

Suppose that physics, or rather nature, is considered anal­ogous to a great chess game with millions of pieces in it, and we are trying to discover the laws by which the pieces move.

—Richard Feynman

The most novel dimensional unit is the digit­transition. We are assuming that the information in each cell of the RUCA is 2-state or 3-state; one bit (for 2-state) or one trit (for 3-state) per cell. We will use the word "digit" to mean a bit or a trit. Of course such systems are possible with any number of states per cell. Each cell contains just one digit, therefore a digit-transition must take place over space, time or space-time. Let us as­sume that the digit is a bit. The digit-transition would then be either a 1 to 0 transition or a 0 to 1 transition. It could be temporal or spatial. If a given cell changes from a 1 to a 0 in a par­ticular time step then we have a temporal tran­sition. Whether or not it changes during an odd or even time step is related to other derived units such as charge. The frequency of temporal digit-transitions is related to energy. If a cell is a 1 while at the same time its neighbor is a 0, then we have a spatial digit-transition. Spatial digit-transitions also have phase, and the frequency of spatial digit-transitions is related to momentum.

All the other units of the physics of DM are derived from D, L and T. In DM, the dimensions of the RUCA's digit-transition is the same as for Planck's constant. A series of digit-transitions can have angular momentum and a digit transition's numerical value is very simply related to Planck's constant.

The principle of least action takes on new signif­icance in DM. Since action and the digit-transition are the same unit, least action seems related to the least amount of information in a process.

As we have begun to understand DM, we have realized that there must be certain universal laws that govern the behavior of such systems. For ex­ample, "There can be no information without a means of its representation." This means that if a particle is moving with a particular velocity, it must be true that there is an interpretation of a conglomeration of bits of information in the sys­tem that represent the information that describes the particle's velocity. This information may be spread out over a considerable volume of space-time, superimposed onto a great deal of other un­related information. "For a process to evolve, there must be a means of interpreting and transforming the representational information." For example, if a particle is to move, then the information that represents its velocity must interact with the co­ordinate system (be, in effect, processed by the RUCA) to produce the change in position over a series of time steps that is in accord with the velocity. If a field is accelerating a particle, this means that the information that represents the the state of the field must interact with the infor­mation that represents the velocity of the particle so as to change the information that represents the velocity of the particle. Of course we may find that the information that represents the state of the field is carried by a digital version of a boson.

Certain quantities must he conserved. There are some things loosely called "information" and in DM the thing called "information" is conserved.

Depending on both the rule and the initial condi­tions, there will be other conserved quantities that would correspond to energy, momentum, angular momentum, charge, etc.

Every RUCA transits from one state to another along a closed trajectory. The trajectory consists of a finite sequence (ring) of states of the total system. Each state has one successor state (be­cause the RUCA is deterministic) and one pre­decessor state (because the RUCA is reversible) The length of the trajectory is finite (because the RUCA is finite). We will show a physically im­practical but mathematically correct method of determining a unique quantity for each trajectory. Since this quantity can be determined for each tra­jectory and because it is different for each different trajectory, it is a unique conserved quantity. This means that there is a one-to-one mapping between trajectories and distinct sets of conserved quan­tities. In other words for a given RUCA that is in a particular state, there is a value of a con­served quantity for the trajectory associated with the state that is different than that value for any state that is on a different trajectory. This can be proven as follows.

Assume that the RUCA is on a trajectory of length T. Assume that the number of cells in the x direction is X, and similarly Y for y and Z for z. For each of the XYZ values of the set [x, y, z, t] we map the state of every cell, C_x,y,z into a number which we will store in a two-dimensional array D_t,s by scanning in a regular way. The digits of all of the cells become the digits of an ordinary integer. Given that we first scan through the entire space with the t subscript equal to 0, then with it equal to 1, continuing until t equals the length of the trajectory, there will be 48XYZ different ways to generate the one-dimensional array. The subscript s in D_t,s ranges over the 48XYZ ways of scanning. This is be­cause we can start at any of the XYZ cells and we can scan through the cells in any of 48 orders, +x,+y,+z;+x,+y,-z;+x,+z,+y,..., 8 ways of distributing the signs and 6 different orders for x, y and z. We will do them all. We then consider each of the D_t,s as an integer. Every different D is unique to the particular trajectory. Of all of the D's produced, we choose the smallest value V (it is allright if there are many instances of that value).

V is a unique quantity of the trajectory, so we may say that it is a conserved quantity. This approach generates the same value for the con­served quantity V for distinct trajectories that dif­fer from each other only by displacements, rota­tions (through angles relating the coordinate axes) or reflections. Every different trajectory has a dif­ferent value for V. The ergodic hypothesis for DM is different because of V. A DM system will pass through every state in the trajectory it is on, but there will most likely be many other distinct tra­jectories that are identical with respect to all high level descriptions, such as the values of other con­served physical quantities, but which differ only in the value of V.