Journal of Modern Physics
Vol.5 No.8(2014), Article ID:46119,13 pages DOI:10.4236/jmp.2014.58067

Physics in Discrete Spaces: On Space-Time Organization

Figure 1. The model of space we put forward in this essay. Here 18 cosmic bits (small circles: black for, white for) are shared between 3 world points (large dotted circles) each comprised of cosmic bits. Heavy lines are for binary negative (ferromagnetic) interactions, dotted lines are for binary positive, anti-ferromagnetic, interactions (only a part of these interactions are represented in the graph). This graph has no geometrical signification. The cosmic bits are only but elements of a set. The world points are subsets of this set.

tions. It is therefore a random variable whose distribution is Gaussian and given by

.

may be seen as a degree of proximity, the larger the closer i and j, but this interpretation has only a local topological signification and for example cannot be seen as a distance since no global topology, no geometry and no metrics have been defined so far.

2.2. Statistical Properties

The Lagrangian of a world point writes

.

In this expression, the Lagrangian is limited to second order interactions because the fourth order interactions are completely negligible inside world points. Due to the interplay between the binary interactions and the cosmic noise b the world points may be polarized. The polarization of a world point i comprised of n cosmic bits is defined as the thermal average of the order parameter s:

.

The statistical properties of a world point W are determined by using the mean field theory which consists in replacing the dynamic variables by their statistical averages. In general, the mean field theory is an approximation but when the connectivity of the elements of the system is high enough the mean field is an exact theory. This is the case for four dimensional Ising or Heisenberg magnets. This is also the case for world points due to their complete connectivity. The polarization is then the solution of a self consistent equation given by

. (1)

Here the binary interaction has been renormalized so as to make the Lagrangian an extensive quantity. The polarization vanishes if. This situation is called symmetric vacuum. It does not vanish if, and then vacuum is asymmetric.

Another important property of mean field theories is the disappearance of fluctuations at least in the limit of infinitely large systems.

2.3. World Points Internal Spaces

We endow a world point with a (non-directly observable) organization by assuming that the polarization may be considered as the length of a vector in a d-dimensional abstract space called the internal space of W:

.

To give an analytical expression to the components of vector we pose the following question: can a world point be considered as a set of d subsets (sub-world points so to speak) such that the system obtained by putting these d sub-world points together, reproduces the polarization of the world point as a whole?

To answer that question we must study more carefully the statistical mechanics of a world point made of d sub-world points. Let (with), , be the number of cosmic bits associated with a sub world point of W. The polarization components are given by the statistical averages of the d order parameters with

.

The calculation, a classical calculation in statistical mechanics, is given in Appendix 1. The polarizations are obtained by minimizing the quantity, somehow similar to a free energy. In the framework of a mean field theory is given by

. (2)

The polarizations are obtained by solving the set of d equations given by: (the saddle point method). In the case where d = 1, that is to say if

the free energy per bit reduces to

.

The condition gives Equation (1).

When b is large enough, , the global polarization does not vanish and it does not fluctuate. This no fluctuation property is also desirable for the components to give to the properties of a vector, but this is not guaranteed. To illustrate this point we use a good approximation for the solution of the self consistent Equation (1)

for. Let us take for example. Then: Whereas a majority of CBs is oriented along, about 20% are oriented along. Therefore if the world point is divided into d sub world points the order parameters strongly depend on the way the sharing has been carried out. To cope with this difficulty we consider an isolated sub-world point. Its polarization is given by the following self-consistent equation:

.

The order parameter does not vanish and does not fluctuate if

Therefore the polarization components are well defined quantity if, the condition that we are looking for. This yields a highest value for d

.

d is called the dimensionality of internal space. Our space is 4-dimensional. This implies that , that is and the vacuum is asymmetric indeed.

By expanding the logarithmic functions to second order in Equation (2) and by using the definition of polarization components, one has

. (3)

The expression (3) is rewritten along

where G is a d-dimensional symmetric matrix whose elements are

.

G is called the space-time generator. A more convenient form of G is its diagonal representation. The eigenvalues of G are solutions of the following equation:

.

The diagonal representation identifies two and only two subspaces for G. The first one corresponds to the eigenvalue

.

It is not degenerate. This subspace, of dimension 1 whatever d, will be called “time type dimension”. The other subspace corresponds to the eigenvalue

.

This subspace, of dimension d − 1, will be called “space type dimensions”.

2.4. Gauge Symmetry Invariance

Nothing determines the orientation of the internal space of a world point. Therefore physics must be insensitive to any reorientation of the internal space or to any permutation of its axes. This generates two sorts of gauge invariance symmetry. Let us consider the permutation invariance. Then G must transform according to direct sums of irreducible representations of, the group of permutations of d objects. Let us for example consider four dimensional spaces. The permutation group of four objects has elements. Since has 5 classes there are 5 irreducible representations that are

with orders 1, 1, 2, 3 and 3 respectively [5] . The table of characters of these representations is given in Table1

The invariance of four dimensional matrices, such as G, under those transformations, requires the matrix to commute with the 24 matrices of permutations. An example of a permutation matrix is

which is a four dimensional representation of the permutation. Let be this representation. Its characters are given in Table2

From these tables it is deduced that

a sum of two irreducible representations with dimensions 1 (time type dimension) and 3 (space type dimension) respectively.

The state of the universe is now determined by a family of world point states and the Langrangian of the system becomes

, (4)

an expression that, via the Cartesian product, takes the global independence of internal spaces into account. The universe is now seen as a fibre bundle where forms the basis of the fibre bundle and G its fibres. If G is the same whatever the world point, the fibre bundle is trivial and we are dealing with flat spaces. If G is world point dependant, the fibre bundle is not trivial and we are dealing with general relativity.

3. Recovering the Space-Time Continuum

3.1. The Possible States of the Universe

The possible states of the universe are obtained by minimizing the Lagrangian (4) under the constraint (N is the number of world points) that is by minimizing the expression

where is a Lagrange multiplier. The solution is an eigenvalue equation

(5)

Table

Pierre Peretto