The first statement is simply that the universe does exist, that is, some information can be obtained on the uni- verse through experimental observations. Information is the key word. As a matter of fact, since nothing else be- sides information is available on the nature of the universe, at least for materialist philosophers, one can assume that information itself constitutes the fabrics of the physical world.

Information is measured in terms of an information unit or bit. A bit, here called a cosmic bit (CB), is the sim- plest physical object one can imagine. Accordingly, the first hypothesis of the model writes:

Our universe as a whole is entirely made of a finite, countable, set of cosmic bits. The state of a cosmic bit a, , is a binary variable analogous to an Ising (classical) spin.

The state of the universe is determined by a family of CBs states.

We write as a -dimensional vector whose norm is. In discrete spaces, if is finite as we assume it is, the states of the universe are necessarily normalized.

The second statement follows from the observation that the universe is not completely disordered and, therefore, that all possible states of the universe cannot be realized. As a consequence we must assume that there exists a functional of CBs states, called a Lagrangian, which is, at least approximately, minimized for the physi- cally realizable states of the universe. The most general Lagrangian is written as an expansion over all possible clusters of CBs:

where is an interaction parameter between the cosmic bits belonging to cluster. Nothing determines the overall orientation of and therefore one must have. This elimi- nates the odd terms of. In other ways, all CBs must be treated on equal footing which compels the am- plitudes of interactions of same order to be identical. That is, for clusters implying a number of CBs, one has, for arbitrary

with an even number. For example all pairs ab are such that. It is assumed that the interac- tion amplitude decreases very rapidly with the number of CBs, in particular and we shall limit the expansion of to clusters of 4 CBs. The signs of interactions remain to be determined. Since no knowledge exists, is taken as a random binary variable:. Likewise. Some correlations, however, possibly exist between the signs of second order interactions and those of fourth order interactions. In the present essay, it is assumed that the sign of obeys a majority rule, that is

Finally

where the sign correlations are to be taken into account.

The last statement follows from the observation that the states of the universe are never completely frozen, that is, order is not perfect. This implies that the CBs are subject to a degree of disorder whose amplitude is deter- mined by a parameter b called “cosmic noise”. Space-time is then treated as an ordinary thermodynamic system analogous to e.g. a magnetic material, more precisely to a special sort of spin glass. It can be studied by using the tools of statistical mechanics. This is not a trivial assertion because statistical mechanics rests upon two fun- damental hypotheses. The first one is the ergodic hypothesis, according to which temporal averages may be re- placed by ensemble averages. Since the concept of time is not yet defined, only ensemble averages may be given a physical meaning, at least for the time being. Ergodicity is then a natural hypothesis and this makes it possible to derive the statistical properties of space from usual statistical physics techniques. In particular, according to statistical physics, the probability for space to be in a state is given by the following Gibbs expression

where Z is the partition function

and b the cosmic noise parameter.

The second basic hypothesis of statistical mechanics is the existence of a reservoir that makes the noise b a well defined parameter. One may imagine that the total number of CBs is infinite and that, the number of CBs belonging to our own universe, is just a finite part of this set. Then the reservoir is made of the set of CBs not belonging to our universe.

To summarize, with these three statements, we put forward a thermodynamic model of space-time. This model is basically discrete. It introduces three, and only three, sorts of free parameters and. In the model, everything of our familiar physics is, a priori, lost, no more space, no more time, no more fields, and no more particles. Everything has to be rebuilt. In the present article, we start the process with the construction of space and time. A first issue on this subject has already been published in a previous contribution [4] but here we develop the model in more detail.

The interplay between second order and fourth order interactions gives rise to clusters of cosmic bits called world (or physical) points. Let us consider a cluster W of n cosmic bits all connected to each over through nega- tive (ferromagnetic) binary interactions

Then, according to the majority rule, one has

A world point is a cluster that minimizes its Lagrangian

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.

The Lagrangian of a world point writes

In this expression, the Lagrangian is limited to second order interactions because the fourth order in- teractions 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 com- prised 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 approxima- tion 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

Here the binary interaction has been renormalized so as to make the Lagrangian an ex- tensive 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.

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 parame- ters 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

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-consis- tent 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 polari- zation components, one has

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 eigen- values 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 ei- genvalue

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”.

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 in- variance 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 Table 1.

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 Table 2.

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 Lan- grangian of the system becomes

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.

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

Table 1Table 1

. Table of characters of S₄

Classes
	1	1	1	1	1
	1	−1	1	1	−1
	2	0	2	−1	0
	3	1	−1	0	−1
	3	−1	−1	0	1

Table 2Table 2

. Table of characters of Γ₄

Classes
	4	2	0	1	0

To introduce the notion of derivatives in the context of discrete spaces one must introduce a square N-dimen- sional matrix D, namely the square root of that is. Since the elements of are positive or nega- tive D is, in general, a complex matrix.

One defines the increment of a polarization of world point “i” by

or

for each component of. The first order derivative of along the axis is then defined by

where l^* is the smallest length that has a physically measurable meaning, that is the scale where the metrics is lost and also the scale where the distinction between the particles, be they fermions or bosons, disappears. Therefore l^* should be the scale where super symmetry theories (Susy) come into play. Accordingly, the metric scale l^* must be of the order of.

D may be seen as a differential operator because it is linear and it obeys the Leibniz formula.

Let us consider two states and. One has

that is and is linear indeed.

On the other hand

The second term vanishes because the elements of D are random

The third term is a second order term. It may also be ignored and one has

whence, the Leibniz formula.

is a scalar field but physics generally deals with vector fields that are vectors of the internal spaces of world points. The components of a vector field are given by

The parameters are the components of vector and the increment of the component is given by

The first order derivative of along the axis then writes

The connection between D and the usual classical first order derivatives is more carefully studied in Appendix 2.

Let us now consider second order derivatives.

The second order increment of a scalar function at world point “i” is

or

for each component of. The second order derivative of along the axis is given by

and the second order derivative of the vector field component along axis by

For trivial fibre bundles where the G matrix is the same whatever the world point, one entry of Equation (5) writes

One introduces the coefficient in both members of this equation and carries out the sum over index.

By using the diagonal expression of G and the two parameters and the final equation reads

recognized a set of four Klein-Gordon equations. Let us write

The metric tensor g is defined by

Since bJ > 1 its elements associated with are

and its unique element associated with is

that is

The metrics is therefore Minkowskian. It would be Euclidian for. It is worth pointing out that there is no more ambiguity on the sign of g (whereas relativistic mechanics does not distinguish between g and –g). The three dimensions of space and the unique dimension of time constitute a conformal space with dilatations factors given by and respectively. With and we recover the usual expression of the Klein-Gordon equation

The identification of Equation (6) with Equation (7) allows fundamental parameters to be expressed in terms of the basic parameters b, J and l^* of the discrete space model.

1) The speed of light c is a universal dimensionless constant given by

The speed of light diverges at the transition.

2) The constant of Planck writes

3) Finally the mass m of the particle associated with the field is given by (provided that).

The connection between the eigenvalue Equation (5) and the Klein-Gordon Equation (7) establishes the link between the discrete and the continuous descriptions of our universe.