Physics in discrete spaces (A): Space-Time organization

We put forward a model of discrete physical space that can account for the structure of space- time, give an interpretation to the postulates of quantum mechanics and provide a possible explanation to the organization of the standard model of particles.

We put forward a model of discrete physical space that can account for the structure of spacetime, give an interpretation to the postulates of quantum mechanics and provide a possible explanation to the organization of the standard model of particles. PACS numbers: 03.65.Ta ; 05.65.+b ; 12.91.-g

A MODEL OF SPACE-TIME
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 has usually an ontological status that is it requires no further explanations. However 4 the total number 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, spin-lattice or twister theories for example. We rather build the model on three statements that we cannot reject without jeopardizing physics itself. We consider the three statements and their mathematical formalizations in turn.

A. The universe does exist
The first statement is simply that the universe does exist that is some information can be obtained on the universe through experimental observations. Information is the key word. As a matter of fact since nothing else than information is available on the nature of the universe, at least for physicists, 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 not a signal, but the simplest 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 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 CB's states ( ) , called a Lagrangian, which is, at least approximately, minimized for the physically realizable states of the universe. The most general Lagrangian is written as an expansion over all possible clusters of CB's.  This eliminates the odd terms of ( ) Ψ Λ . In other ways all CB's must be treated on equal footing which compels the amplitudes of interactions of same order to be one another identical. That is for clusters implying ω CB's, a, b, …, c one has, for arbitrary a, b,…,c with ω an even number. For all pairs ab for example . It is assumed that the interaction amplitude ) (ω J decreases very rapidly with the number ω of CB's, in particular where the sign correlations are to be taken into account C. The universe is not frozen 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 CB's are subject to a degree of disorder whose amplitude is determined by a parameter b called "cosmic noise". Space-time is then treated as an ordinary thermodynamic system analogous to e.g. a ferromagnetic material. It can be studied by using the tools of statistical mechanics. This is not a trivial assertion because statistical mechanics rests upon two fundamental hypotheses. The first one is the ergodic hypothesis, according to which temporal averages may be replaced by ensemble averages. Since the concept of time is not yet defined only ensemble averages may be given a physical meaning. 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 state Ψ is given by the following Gibbs expression The second basic hypothesis of statistical mechanics is the existence of a reservoir. One may imagine that the total number of CB's is infinite and that CB N , the number of CB's belonging to our own universe, is just a finite part of this set. Then the reservoir is made of the set of CB's not belonging to our universe. To summarize we put forward in this essay a thermodynamic model of space-time. This model is basically discrete. It introduces three, and only three, sorts of free parameters b J J and , ) 4 ( ) 2 ( . In the model everything of our familiar physics is, a priori, lost, no more space or time, no more fields, no more particles. Everything has to be rebuilt. In section II we show how space-time emerges from the model, in section III we see how it can account for the basic postulates of quantum physics and in section IV we argue that it can explain the organization of the usual (Standard) model of elementary particles.

THE ORGANIZATION OF SPACE
An introduction to this section is given in (1) A. World points We strive to construct a geometry starting from the three hypotheses that form the basis of our model of discrete space. First of all it is necessary to give a meaning to the concept of point since the point is the elementary object of any geometry. Because space is assumed to be discrete a physical point, here called a world point, cannot be infinitesimally small and cannot, therefore, be identified with a mathematical point. Let us consider a cluster W of n cosmic bits all each other connected through negative (ferromagnetic) binary interactions Then, according to the majority rule, one has  The condition therefore writes d bJ > . It yields a highest value for d ) ( Int bJ d = d is called the dimensionality of space. Our space is 4-dimensional. This implies that 5 4 < < bJ , that is 1 > bJ and the vacuum is asymmetric indeed. The free energy of a system of d sub world points is given by gives eq.(1). Expanding the logarithmic functions to fourth order yields a Landau type free energy is normalized. This gives rise to a local gauge invariance that states that physics must be left invariant under any rotation of φ in the internal space of a world point.
In other respects the labels of the polarization components are arbitrary and, therefore, relabelling the names of the components must not change the physical phenomena. This gives rise to another gauge invariance which states that physics must be left invariant under the operations of the group d S of permutations of d objects. Locality means that both criteria of invariance must be satisfied for each world point W individually.
A state Ψ of the universe as a whole can be reformulated in terms of polarizations. It is written as a column vector The Lagrangian (limited to second terms) is given by In mean field theories there are no polarization fluctuations. The free energy can then be expressed as Since F must be left unchanged under the permutations or rotations of polarization components the parameters G is a square, real, symmetric, d-dimensional matrix that operates in the internal space on the polarization components of a given cosmic point. The elements G µν describe the interaction between the µ and ν components of the polarization inside one and the same world point W. ∆ is a square, real, symmetric, matrix. An element ∆ ij describes the interaction that links a world point "i" to world point "j". ∆ ij is a sum of binary random variables n J / ± . Its offdiagonal coefficients are random variables of the order of To express the parameters J 0 and J 1 in terms of b, J and n, the three parameters of the model, it is necessary to identify the two expressions (2) and (4) of F . By expanding the logarithmic functions to second order and by using the definition of polarization components, one has The identification of eq. (4) with eq. (5) yields If a convenient form of G is (4), this form is not unique because any unitary transformation of this representation is also convenient, in particular the one that makes G diagonal. Since It is not degenerate. This subspace, of dimension 1 whatever d, will be called "time dimension". The other subspace corresponds to the eigenvalue and define the metric tensor g by . 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 r G 0 and t G 0 respectively. From r G 0 and t G 0 it is possible to define two constants, c and h , whose physical meaning will be discussed below . They are given by The usual meaning of space and time will be recovered below when we derive the Klein-Gordon equation. As a matter of fact, this organization of space is fully determined by the irreducible representations of groups of permutations of d objects. For example the permutation group 4 S of four objects has 24 ! 4 = elements. Since 4 S has 5 classes there are 5 irreducible representations that are with orders 1, 1, 2, 3 and 3 respectively. The table of characters of these representations is given in Table-I. ( ) which is a four dimensional representation of the permutation (1234)=>(2431). Let Γ 4 be this representation. Its characters are given in Table-II: a sum of two irreducible representations with dimensions 1 and 3 respectively. The properties of the group 4 S are important because they fully determine the main aspects of the standard model of particles (see Section 4).

THE POSTULATES OF QUANTUM THEORY
The physically realizable states Ψ of the universe are obtained by minimizing the Lagrangian . This amounts to finding the minima, with respect to T Ψ , of ( ) ( ) where λ is a Lagrange multiplier.The solution is an eigenvalue equation : It is unlikely, however, that Ψ really represents a physical state of the whole universe because it is unlikely for the coherence of Ψ to be preserved everywhere in the universe. Equation (6) must therefore be valid only for a (small) part of the universe wherein Ψ keeps its coherence. This part, comprized of N world points, is called a quantum system and ψ , the piece of Ψ that belongs to the quantum system, is called a quantum state. The model of discrete space imposes some properties to quantum states. These properties are usually expressed in the form of postulates which, therefore rather appear as consequences of the model itself. is also an eigenstates associated with κ . Therefore the set of eigenstates associated with κ makes a (Hilbert) vector space. This space is real-valued but it can be made complex-valued by introducing local phase factors and by letting This does not change the normalization condition 1 i exp an hermitian operator that has exactly the same properties as ∆ (in particular the same real eigenvalues) because the eigenvalue equation Transforming the real vector space into a complex vector space therefore has no effect whatsoever as far as physics is concerned.
The first postulate writes accordingly: The states of a quantum system constitute a Hilbert space, that is to say a complex vector space equipped with an inner product that is defined for all its vectors. Let ψ and χ be two

Remark
Linearity is the most central and striking property of quantum mechanics. In the present approach, linearity stems from the quadratic form of the Lagrangian Λ which is a result of limiting Λ to binary interactions. As far as quantum theory is concerned the influence of fourth order interactions is completely negligible.

B. Second postulate
Besides local symmetries that reflect the symmetry properties of world points, a quantum system may also display global symmetries. Global symmetries are operations carried out on a quantum system as a whole that leave the physics of the system unchanged. The set of such operations constitutes a finite group P that permute the n labels 'i,…,j' of world points. We recall that, according to Cayley theorem, any finite group can be considered as a sub-group of a permutation group. One may associate a linear operator P A to a global symmetry P by compelling the operator to remain unchanged under the operations P ω of P.

The linear operator P
A reflects the physical (symmetry) properties of the system. It may be obtained by projecting the Lagrangian Λ on the trivial representation of P. The third postulate determines the dynamics of a quantum system: The dynamics of a quantum system is given by a Schrödinger equation where H is an operator called the Hamiltonian of the system. The quantum state i ψ at a world point 'i' is expressed in terms of its polarization components: where C is a 4-dimensional matrix that operates in the internal space of world points Let us now look more carefully at the expression of the Lagrange operator G ⊗ ∆ = Λ Any square matrix such as ∆ can be expressed, according to the LDU theorem of Banachiewicz, as a product of a lower triangular matrix L, a diagonal matrix A and an upper triangular matrix U. When the matrix is real and symmetric, as is the case for ∆, the two triangular matrices are each other transpose: where D is a triangular, here a random triangular matrix, that is D ij =0 for i<j. D T is the transpose matrix and A is diagonal. More precisely, since space is homogeneous, A is a spherical matrix (it is proportional to the unity matrix: A ij =aδ ij ) and one may take a=1. Hence One defines the increment of a polarization component µ ϕ i of world point "i" by The increment of the th ν component of the vector field i ψ along dimension µ is written as In appendix A2 we show that he operator D may be seen as a differential operator indeed. The partial derivatives of the quantum field components are given by where ∆ is here the usual three dimensional Laplacian. The parameters c and h that have been defined earlier, are introduced in that expression; c, called the velocity of light, is given by It is, in fact, a dimensionless, universal, parameter that determines the ratio between the standards of length and time.
In the non-relativistic limit, where the mass term 2 c m is much larger than the other terms, the equation may be approximated by The + sign corresponds to Schrödinger equation and we recover the third postulate. In conclusion the set of postulates of quantum theory may be seen as a consequence of the model of discrete space that we put forward in this essay.

THE ORGANIZATION OF ELEMENTARY PARTICLES
A. Bosons and fermions A quantum system, comprised of N world points, contains a particle P if the matrices G associated with the internal spaces of the world points of the system are all identical to 0 G the vacuum metrics except for one world point a where P G G = , a matrix characterized by the symmetry properties of P. The possible physical states of the quantum system are then solutions of the following eigenvalue equation is the set of all matrices of the quantum system. The symmetry of fundamental particles is traditionally associated with the irreducible representations of group S 2, the group of permutations of two particles. This group has two elements and two irreducible representations namely the even representation that is associated with the permutation properties of pairs of bosons and the odd representation that is associated with the permutation properties of pairs of fermions. This presentation is not satisfactory, however, for two reasons First, in this definition, the bosonic or fermionic character seems to be a property of pairs of particles and an isolated fermion, for example, is not defined. Secondly, it does not fit the idea that we support, that the nature of particles relates to their individual symmetry properties. In fact this problem has already been pointed out by theoreticians who believed that the symmetry of fundamental particles ought to be searched in the irreducible representations of a fundamental group and put forward the Poincaré's (or Lorentz) group. Our idea is close to this proposal with the difference that the relevant group would be the local group S 4 of permutations of 4 objects instead. The table of characters of 4 S has been given in Table I . P G must commute with any four dimensional representation of group S 4 and therefore it must transform according to direct sums of irreducible representations of S 4 . Let us first try to build the matrix P G by using 1 Γ , 2 Γ and 3 Γ .
The first possibility is to build P G as a direct sum of Γ 1 and Γ 3 ) 9 ( This corresponds to the solution 4=1+3. Cosmic points that are polarized along that symmetry display a bosonic polarization. Accordingly a bosonic polarization is described by a quadrivector. The other possibility is to build P G on the representation ( ) This corresponds to the solution 4=2+2. Cosmic points that are polarized along that symmetry display a fermionic polarization. The representation Γ 2 determines the fermionic, or spinor, character of the polarization. Since a fermionic polarization implies this representation Γ 2 twice, it is called a bispinor.

B. A family of elementary particles
Super-symmetry theory (SUSY) puts forward that fermions and bosons may be considered as two aspects of the very same objects. In a spirit close to that of SUSY it is suggested here that the fundamental particles are objects that associate world points with different symmetries. A particle would be made of a pair of coupled world points, one undergoing a fermionic polarization (10) and the other a bosonic polarization (9). According to this model the state of a particle is represented in a 16-dimensional vector space that is obtained by the direct product of the two 4-dimensional spaces associated with the members of the pair. The states must therefore transform as ( ) ( ) Every bracket is related to a given type of particle P. As a result a family of particles is comprised of four types of particles. This fits the Standard Model. Moreover all these particles have a fermionic character since they all transform according to Γ 2 . One also observes that there are two classes of particles.
-a)-The first class is associated with the two representations that mix Γ 1 with Γ 2 . It may characterize the lepton particles. The particle associated with one of the two representations is a charged lepton, that associated with the other representation is the associated neutrino .
-b)-The other class associates Γ 3 with Γ 2 : It may characterize the quark particles. The particles associated with one of the two representations are then the three colour versions of one quark, the particles associated with the other representation are the three colour versions of the other quark.
The physical size of a particle would be 2l* where l* is the size ( cm 10 ) of a world point. This is the smallest segment that may be given a physical meaning. It is possibly related to the minimal segment that arises from the non commutative geometry theory of the Standard Model of Connes. If, instead of using the mirror insensitive representations 1 Γ and 3 Γ one appeals to the mirror If P ψ is the quantum state of a system that contains a particle, P ψ the quantum state of the associated antiparticle is simply given by C is the charge conjugation operator. The four antiparticles have the same physical properties as the particles except for their electric charges. For a discussion on the CPT theorem see Appendix A3.
We have so far obtained a convenient description of the organization of one family of elementary particles but one knows that there exist three families of particles with identical properties except for the masses. Since the symmetry properties are the same for the three families one must admit that the matrices P G are the same for the three families. The particles of the first family, however, are stable whereas the particles belonging to the other families are unstable. Therefore the existence of these families cannot be looked for in some new type of symmetry similar to SU(3) for example. At this point the question remains open.

DISCUSSIONS AND CONCLUSIONS
In this first contribution we put forward a model of space-time where we assume that the universe as a whole is made of elementary physical objets, called cosmic bits, whose states are entirely determined by one, and only one, binary variable.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' (2). There is however a fundamental difference between his approach and ours. In the Wheeler approach the bits are to be understood as bits of information that is signals transmitted from an emitter to a receiver. The physical laws are the results of computations carried out on those bits by a huge sort of universal computer, a Turing machine for example, according to convenient programs. The physical world would be the result of these computations and the physicists would be the receivers. In our approach there is no program and no programmer behind the stage. The bits are physical objects, not signals, that constitutes a system somehow similar to a ferromagnetic powder. The process that moves the bits is purely physical and determined by statistical physics. Moreover in our approach time and space are treated on equal footing, in the spirit of relativity theory, so avoiding the philosophical problems arising from the necessary existence of a clock driving the computer. Obviouly the difficulty is to show that the general laws of physics can be recovered from so simple a model. It seems that the difficulty can be overcome in most cases. In this contribution the structure of space-time has been recovered, the main postulates of quantum theory have been reestablished and the organization of elementary particles shown to fit that of the Standard Model of particles. More remains to be done, for example to describe the APPENDIX A1: The partition function of world points W.   For D to be a differential operator it is necessary to show that it is linear and that it obeys the Leibniz formula.
Let us consider two quantum states φ and η . One has and D 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. Both may be ignored and one has represents the state of a particle. The state φ of the associated anti-particle is obtained from φ through a mirror transformation that is a sign change of one of its components. Nothing, however, determines which component has to be modified and, therefore, all signs have to be changed. We note that if the polarization ϕ of a world point is a solution of the equation Finally the operation of reversing the direction of one axis, say the z direction, amounts to a reflection of space (a mirror symmetry operation) in any number of space dimensions, and in three space dimensions it is equivalent to reflecting all space coordinates, because one can add an additional rotation of 180 degrees in the x-y plane to complete the transformation. Let P be the operator that reflects all space coordinates. P amounts to changing the signs of the three spatial components 2 ϕ , 3 ϕ and 4 ϕ .