Discrete space-time

Regge [regge61], Sorkin [sorkin87] and T.D.Lee [tdleeSSa], among others, have used a related idea to perform calculations on a discretize space-time see section 12.5.3. 

In this section wc explore three attempts at discretizing space-time with a radically unconveiitional view each assumes that it is the contiiiiium that is an approximation to a fundamentally discrete underlying substructure. The question is, What effect docs this assumption have on our understanding of fundamental physics ... 

An intriguing possibility thus presents itself. If some kind of a primordial information, and not higher-level constructs such as mass, energy, spin, and so forth, is indeed the real substance out of which all stuff is made - leaving aside for the moment, the question of form of that information - is it not natural to suppose that a discrete space-time structure, our heretofore pre-defined and static dynamical mediator, is itself built out of the same substance i.e. to suppose that space-time is not just a backdrop for information processing, there only to define what is local and what is not and where to and where from information is allowed to flow, but is itself a construct of primordial information This supposition is not entirely without precedent. 

Fig. 6.6 Scheme of discretized space-time domain using a uniform grid. 

In keeping with the current interest in tests of conservation laws, we collect together a Table of experimental limits on all weak and electromagnetic decays, mass differences, and moments, and on a few reactions, whose observation would violate conservation laws. The Table is given only in the full Review of Particle Physics not in the Particle Physics Booklet. For the benefit of Booklet readers, we include the best limits from the Table in the following text. Limits in this text are for CL=90% unless otherwise specified. The Table is in two parts Discrete Space-Time Symmetries, i.e., C, P, T, CP, and CPT and Number Conservation Laws, i.e., lepton, baryon, hadronic flavor, and charge conservation. The references for these data can be found in the the Particle Listings in the Review. A discussion of these tests follows. 

Discrete or continuous time. The link between the entropy fiow fg and the entropy S is the evolution operator. As already discussed in Chapter 10, this operator is not necessarily a differential one and discretization of time is easily admitted as an alternative to the classical theory. This opens some interesting perspectives on thermodynamics in discretized space-time. 

Fig. 3. Quantum solution of the test system of 3.3 for e = 1/100. computed numerically using Fourier pseudospectral methods in space and a syraplectic discretization in time. Reduced g -density f t)j dg versus t and qF Initial...

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

It is easy to see that this expression has two minima within the Brillouin zone. One minimum is at fc = 0 and gives the correct continuum limit. The other, however, is at k = 7t/a and carries an infinite momentum as the lattice spacing a 0. In other words, discretizing the fermion field leads to the unphysical problem of species doubling. (In fact, since there is a doubling for each space-time dimension, this scheme actually results in 2 = 16 times the expected number of fermions.)... 

To discretize this therrry, Lee assumes that in any n-dimensional volume O — L" V, there can be at most N measurements that determine the space-time position Xi of the, observation and the value of the field d>i at The ratio p = N/Q, = is therefore a fumlamental constant of the theory,... 

In this scheme, digital particles are still wandering localized clusters of informa-tionl but (conventional) variables such as space, time, velocity and so on become statistical quantities. Given that no experimental measurement to date has yet detected any statistical dispersion in the velocity of light, the sites of a hypothetical discrete underlying lattice can be no further apart than about 10 cm. 

Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. We call models of physics that assume Finite Nature Digital Mechanics. . ..we take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata. ... 

Fredkin s finite nature hypothesis makes no assumptions about the actual scale of space-time s discretization. It might be as large as current estimates of the... 

Later, Kuppermann and Belford (1962a, b) initiated computer-based numerical solution of (7.1), giving the space-time variation of the species concentrations from these, the survival probability at a given time may be obtained by numerical integration over space. Since then, this method has been vigorously followed by others. John (1952) has discussed the convergence requirement for the discretized form of (7.1), which must be used in computers this turns out to be AT/(Ap)2normalized forms of r and t. Often, Ar/(Ap)2 = 1/6 is used to ensure better convergence. Of course, any procedure requires a reaction scheme, values of diffusion and rate coefficients, and a statement about initial number of species and their distribution in space (vide infra). 

Quantum physics makes similar pronouncements when it states that the electron is not somewhere or sometime it is a cloud of probabilities and that is all one can say about it. A similar quality adheres to my idea of time and the comparison of time to an object. If time is an object, then the obvious question to be asked is what is the smallest duration relevant to physical processes The scientific approach would be to keep dividing time into still smaller increments in order to find out if a discrete unit exists. What one is looking for by doing this is a chronon, or a particle of time. I believe the chronon exists, but it is not distinct from the atom. Atomic systems are chronons atoms are simply far more complicated than had been suspected. I believe that atoms have undescribed properties that can account not only for the properties of matter, but for the behavior of space/time as well. 

We see that the size of the quantum of translation energy is around 1020 times smaller than kg T in this example. To a very good approximation we can treat translational energy levels as though they were continuous rather than discretely spaced. 

Figure8-1 Space-time grid for the one-dimensional diffusion equation, evidencing the explicit forward-difference, implicit backward-difference and C rank-Nicholson discretization schemes.

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

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