_{1}

^{*}

We see the whole universe as a collection of very simple binary physical systems. With this assumption, we put forward a detailed model of discrete spaces. Our own universe with its four dimensions, shared between one time-like dimension and three space-like dimensions, as well as the Minkowski metrics, are emerging properties of the model.

The natural phenomena are usually described in the framework of a four-dimensional space. This space has three equivalent space-like components, one time-like component and it is equipped with a Minkowski metrics. Space-time usually has an ontological status that it generally requires no further explanations.

However, the numbers of dimensions (3 the number of space-like dimensions, and 1 the number of time-like dimensions) are numerical experimental data. If one considers that the general purpose of physics is to build theories that account for numerical experimental data, the construction of a theory of space-time is a necessity. In this essay, we put forward such a model and we explore some of its consequences.

Any physical model rests upon a number of hypotheses and one can wonder what sort of hypotheses would form the basis of a relevant theory of space-time. We do not want to make any ad hoc hypothesis such as in string [^{ }for example. We even want the quantum or relativistic theories not to be prerequisites but to be consequences of the structure of space itself and to have no ontological status. The model that we propose here rests on three statements that we cannot reject without jeopardizing physics itself. We consider these three statements and their mathematical formalizations in turn.

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

The state

We write

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

where

with

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

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

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

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

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-ferro- magnetic, 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

The Lagrangian of a world point writes

In this expression, the Lagrangian

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

Here the binary interaction has been renormalized

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

To give an analytical expression to the components

To answer that question we must study more carefully the statistical mechanics of a world point made of d sub-world points. Let

The calculation, a classical calculation in statistical mechanics, is given in Appendix 1. The polarizations

The polarizations are obtained by solving the set of d equations given by:

the free energy per bit reduces to

The condition

When b is large enough,

for

The order parameter does not vanish and does not fluctuate if

Therefore the polarization components

d is called the dimensionality of internal space. Our space is 4-dimensional. This implies that

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

with orders 1, 1, 2, 3 and 3 respectively [

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

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

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

The possible states

where

. Table of characters of S_{4}

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 of characters of Γ_{4}

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

One defines the increment

or

for each component of

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

that is

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 parameters

The first order derivative of

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

or

for each component of

and the second order derivative of the vector field component

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

One introduces the coefficient

By using the diagonal expression of G and the two parameters

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

The metric tensor g is defined by

Since bJ > 1 its

and its unique element associated with

that is

The metrics is therefore Minkowskian. It would be Euclidian for

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

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.

The universe exists. The universe is globally ordered (it is not pure chaos). The universe shows some degree of disorder (it is not fully frozen). We present in this contribution a model of discrete universe fully and only based on these three very general statements. According to this model the universe is made of elementary physical systems called “cosmic bits”. The idea that the universe is made of bits is not new. Wheeler, for example, states that physics at large could be understood in terms of “It from bit” [

Besides the three statements there is, a priori, no other prerequisites, no landscape, no metrics, no fields, no particles. Everything has to be rebuilt. The 4-dimensional time-space continuum has been recovered in this contribution but it remains to prove that the postulates of quantum theory or the Lagrangian of general relativity for example can also be recovered. These topics are outside the scope of the present discussion. As a matter of fact the model does not bring any essentially new results but it allows many concepts that are introduced in physical theories without justifications to be given a physical interpretation. Let us finish this paper by a list of these concepts.

World point: this term has been introduced by Einstein to denote a point of the space-time continuum. In his context a world point is a mathematical point with zero dimension. Here a world point is a physical entity with a physical dimension l^{*}

Internal spaces: This notion is introduced in particles theory but is not given a physical interpretation. Here an internal space is the internal space of a world point that is the space spanned by all possible states

Generator G: the Lagrangian of a world point i in state

Gauge symmetry invariance: If G is invariant under the operations of a symmetry group the physical phenomena generated by G must be invariant under these operations, a property called. gauge symmetry invariance.

Space-time generation:

Minkowski metrics: the model generates a specific metrics with signature (−, +, +, +) that is the Minkowski metrics. It eliminates the ambiguity between the signatures (−, +, +, +) and (+, −, −, −) that are equivalent in special relativity.

Finally, the appearance of the Klein-Gordon equation and the equivalence principle (in Appendix 2) strongly suggests that the quantum theory is, so to speak, cosubstantial with our model.

I would like to thank Pr. Roger Maynard for his helpful remarks and comments and Dr. Ana Cabral for her careful reading of this text.