Документ взят из кэша поисковой машины. Адрес оригинального документа : http://temporology.bio.msu.ru/EREPORTS/krugly_discrete.pdf Дата изменения: Fri Feb 28 02:00:24 2014 Дата индексирования: Fri Feb 28 02:36:11 2014 Кодировка:

Discrete mechanics: a kinematics for a particular case of causal sets
arXiv:1008.5169v1 [gr-qc] 30 Aug 2010
Alexey L. Krugly


Abstract The mo del is a particular case of causal set. This is a discrete mo del of spacetime in a microscopic level. In pap er the most general prop erties of the mo del are investigated without any reference to a dynamics. The dynamics of the mo del is intro duced in [arXiv: 1004.5077]. These two pap ers intro duce a consistent description of the mo del.

Quantum Information Laboratory, Institute of Physics and Physical Technologies, Moscow, Russia; akrugly@mail.ru.


1


Contents
1 INTRODUCTION 2 A MODEL 3 PROPERTIES OF D-GRAPHS 4 A SEQUENTIAL GROWTH 5 A N TI CH A I N S 6 A WIDTH OF D-GRAPH 7 AN IDENTIFICATION OF OBJECTS 8 A LIGHT CONE 9 CONCLUSION 3 4 7 10 12 16 18 20 21

2


1

INTRODUCTION

The idea of building simple mo dels to test the hypothesis of spacetime discreteness, or some aspect of it, is intriguing. One of such approach to quantum gravity is a causal set program (see e.g. [1, 2, 3, 4]. The central hypothesis is that on a microscopic scale, spacetime is a partially ordered set of purely discrete points. This idea is proposed by J. Myrheim [5] and G. 't Ho oft [6]. A causal set is a pair (C , ), where C is a set and is a binary relation on C satisfying the following properties (x, y , z are general points in C ): xx (irreflexivity), (acyclicity), (transitivity), (1 ) (2 ) (3 ) (4 )

{x | (x y ) (y x)} = (x y ) (y z ) (x z ) | A(x, y ) |<

(lo cal finiteness),

where A(x, y ) is an Alexandrov set of the elements x y . A(x, y ) = {z | x z y }. The lo cal finiteness means that the Alexandrov set of any elements is finite. Sets of elements are denoted by calligraphic capital Latin letters. One of the goals of the causal set programme is to investigate the emergence of continuous spacetime as an approximation of some kind of causal sets. The main idea is a faithful embedding of causal sets [7]. Such causal set must possesses specific properties. But spacetime can emerge from causal set only after some coarse graining. The primordial causal set can be unfaithfully embedded [8]. We can use this primordial causal set for the description of particles without any reference to spacetime. The simple particular mo del of the primordial causal set and its dynamics is intro duced in [9]. In this paper I investigate a kinematics of this mo del. It seems natural to follow the scheme that a theory is comprised of three components: kinematics, dynamics, and phenomenology. For causal sets, kinematics refers first of all to the kind of structure one has and its properties. The dynamics means what one might describe as the `equations of motion' of the causal set. The word `phenomenology' needs no definition. In the next section I intro duce a mo del. In section 3 the most general properties are investigated. In section 4 I present the result that is important for a dynamics. In section 5 the properties of slices are intro duced. The slice is the discrete analog of a spacelike hypersurface. In section 6 I prove the theorem that all slices have the same cardinality in this mo del. In section 7 I intro duce some ideas that concern the definition of physical ob jects. In section 8 one property of light cones is intro duced. In section 9 some open questions are discussed. 3


Figure 1: A chronon.

2

A MODEL

There are a set of primordial indivisible ob jects. They have not an internal structure. Consequently, they itself have not any internal properties except one. They exist. The property "existence" can adopt two values: "the primordial ob ject exists", and "the primordial ob ject do es not exist". There are two elementary pro cesses. The first is a creation of a primordial ob ject. This pro cess changes the value of the property "existence" from "the primordial ob ject do es not exist" to "the primordial ob ject exists". It is denoted by i . The second is an annihilation of a primordial ob ject. This pro cess changes the value of the property "existence" from "the primordial ob ject exists" to "the primordial ob ject do es not exist". It is denoted by j . In this mo del, any physical pro cess is a finite network of finite elementary pro cesses [10, 11, 12, 13, 14, 15]. Below a terminology of David Finkelstein is used. These two elementary pro cesses are called monads [13]. A propagation of the primordial ob ject is simply an ordered pair of creation and annihilation. This pro cess of propagation is called a chronon [10] (Fig. 1). The primordial ob ject can be destroyed only by the interaction with another primordial ob ject. The interaction of this second primordial ob ject means the change of its state. Only one kind of change is possible. This is the annihilation of the second primordial ob ject. Suppose the number of the primordial ob jects do es not change. This is a fundamental conservation law. We have a simplest interaction pro cess: two primordial ob jects are destroyed and two primordial ob jects are created. This pro cess is called a tetrad [14] or an x-structure [16] (Fig. 2). The x-structure consists of two monads of destruction and two monads of creation. Suppose any pro cess can be divided into x-structures. This symmetric dyadic kinematics is called X kinematics [15]. Suppose there is a universal causal order of monads. Consider only finite sets of x-structures. Such set forms a structure. This structure is called dgraph [9]. Consider simple examples. There are two connected d-graphs that consist of two x-structures (Fig. 3) and seven connected d-graphs that consist of three x-structures (Fig. 4). (The rigorous definition of a connected d-graph 4


Figure 2: A x-structure.

Figure 3: Connected d-graphs that consist of two x-structures. will be given in section 4). A d-graph is not a graph. By definition, a graph is a set of vertexes and a binary relation (edges) over this set. We cannot describe external lines as in Feynman diagrams. A d-graph is like a Feynman diagram. There are internal lines or edges. These are chronons. They consist of two monads. Monads are halves of edges [14]. There are external monads. These external monads is useful for the description of dynamics [9]. Intro duce the axiomatic approach to this mo del. Consider the set G of monads and a binary relation (an immediate causal priority) over this set. By (i j ) denote an immediate causal priority relation of i and j . G satisfies the following axioms. i (!j (i j )) ( j (i j )), i ( j (i j )), j (!i (i j )) ( i (i j )), 5 (5 ) (6 ) (7 )


Figure 4: Connected d-graphs that consist of three x-structures. 6


j ( i (i j )).

(8 )

These axioms describe a chronon. There is no more than one monad j and do es not exist the monad j which immediately causally follows any i . There is no more than one monad i and do es not exist the monad i which immediately causally precedes any j . The pair (i j ) is called a chronon or an edge. The monad is called internal iff it is included in a chronon. Otherwise the monad is called external. The following axioms describe an x-structure. i !j (k (k i ) (k j )), i !j (k (i k ) (j k )). (9 ) (1 0 )

There is two and only two monads i and j which immediately causally precede any k . There is two and only two monads i and j which immediately causally follow any k . Each monad is included in a unique x-structure. By definition, two monads i and j are causally connected (i j ) iff there is the sequence (i k )(k l ) . . . (r j ). Such sequence of monads is called a saturated chain or a path. A causality is described by the following axiom. {i |(i k )(k l ) . . . (j i )} = . (1 1 ) We consider only finite sets of monads. |G | < . The set G of monads is d-graph if it satisfies these axioms. (1 2 )

3

PROPERTIES OF D-GRAPHS

Consider a main properties of d-graphs. Lemma 1 {i |(i k )(k l ) . . . (j i )} = . Pro of: Otherwise axiom (11) is not satisfies for k . The monad of any type is denoted by i . The monad i may By definition, two monads i and j are causally connected (i is the path (i k )(k l ) . . . (r j ). By definition, put A(i , j ) s i }. The set A(i , j ) is called an Alexandrov set of i ~ ~ definition, put A(i , j ) = {s |i s i }. The set A(i , j ) inclusive Alexandrov set of i and j . 7 (1 3 ) be i or i . j ) iff there = { s | i and j . By is called an


Lemma 2 The set {i |i G } is a causal set. The set {i |i G } is a causal set. The set {i |i G } is a causal set. Pro of: The monad i cannot immediately causally precede itself according axiom (6). Consequently axiom (1) follows from axioms (6) and (11). The union of two paths is a path iff the last monad of the first path coincides with the first monad of the second path. Consequently axioms (2) and (3) follows from this properties of paths and axiom (11). Axiom (4) follows from axioms (12). The pro of of the second and third sentences is the same. Consequently the considered mo del of d-graph is a particular case of a causal set. Theorem 1 If two monads are connected by an immediate causal priority theirs Alexandrov set is empty. Pro of: Two monads are connected by an immediate causal priority if they are included in the same edge or x-structure. The case of edge is trivial. Consider the second case. Denote these monads by i and j . They are included in Xj = {i , k , j , s }. If A(j , i ) is not empty it includes some monad m . We have j m and i m . Consequently k m . We have m i and m j . Consequently m s . k s by transitivity. But s k in x-structure. This contradiction proves the theorem. Two monads are related by the immediate causal priority if and only if they are causally connected and theirs Alexandrov set is empty. In this mo del the immediate causal priority or causal connection can be considered as a primordial property or as a consequence. ^ Lemma 3 Consider a d-graph G . Let T be the isomorphism from G to GT ^ ^ ^ such that T i = i , T j = j and T (i j ) = (j i ); then GT is a d-graph. Pro of: The pro of is an immediate checking of axioms (5) - (12). ^ The physical meaning of T is a time inversion. Lemma 4 The cardinality |{i |i G }| = |{i |i G }| is an even number. Pro of: Each monad is included in a unique x-structure. We can consider G as a set of x-structures. By N denote the number of x-structures in G . We have |{i |i G }| = |{i |i G }| = 2N . The past of the monad is the set of monads, which causally precede this monad. The past of i is denoted by P (i ) = {j |j i }. The future of the monad is the set of monads, which causally follow this monad. The future 8


of i is denoted by F (i ) = {j |j i }. A monad is called maximal iff its future is an empty set. A monad is called minimal iff its past is an empty set. Lemma 5 is a m o n a d external mo o f m in im a l Any of a nad. m on maximal monad is a monad of a type . Any minimal monad type . The monad a maximal or a minimal monad iff it is The number of maximal monads in G is equal to the number ads.

Pro of: Each monad is included in a unique x-structure. In the x-structure, there are two monads of a type that are included in the past of both monads of a type . The monad of a type cannot be a minimal monad. The monad of a type is a maximal monad iff it is not included in a chronon. In the x-structure, there are two monads of a type that are included in the future of both monads of a type . The monad of a type cannot be a maximal monad. The monad of a type is a minimal monad iff it is not included in a chronon. Each chronon includes one monad of a type and one monad of a type . The number of internal monads of a type is equal to the number of internal monads of a type . By Lemma 4, |{i |i G }| = |{i |i G }|, so that the number of external monads of a type is equal to the number of external monads of a type . By definition, two chronons (i j ) and (r k ) are causally connected and (i j ) (r k ) iff j r . Lemma 6 The set of chronons {(i j )|(i j ) G } is a causal set. Pro of: The pro of is the trivial consequence of the partial order of monads. Consider two x-structures Xi = {i1 , i2 , i1 , i2 } and Xj = {j 1 , j 2, j 1 , j 2}. By definition, these two x-structures are causally connected and Xi Xj iff (i1 j 1 ) (i1 j 2 ) (i2 j 1 ) (i2 j 2 ). Lemma 7 The set of x-structures {Xi |Xi G } is a causal set. Pro of: The pro of is the trivial consequence of the partial order of monads. Lemma 8 There is the x-structure in the set {Xi |Xi G } such that this x-structure contains two maximal monads. There is the x-structure in the set {Xi |Xi G } such that this x-structure contains two minimal monads.

9


Figure 5: Three directed acyclic graphs. Pro of: The set {Xi |Xi G } is a finite partially ordered set. This set contains maximal and minimal elements. The maximal x-structure contains two maximal monads. The minimal x-structure contains two minimal monads. Define the isomorphism that takes each x-structure to the vertex of some graph and each chronon to the edge of this graph. We get the directed acyclic graph. All properties of this graph are the properties of the d-graph. We can use both mathematical languages: the d-graph and the directed acyclic graph. But the d-graph is more useful for dynamics [9]. For this reason the considered set of monads in this mo del is called a dynamical graph or d-graph. In the case of a d-graph, causal order of monads contain all information about the structure of this d-graph. This is not in the case of a directed acyclic graph. Consider three different graphs (Fig. 5). The causal order of the vertexes is the same for the graphs number 1 and 2: a b c. The causal order of the edges is the same for the graphs number 2 and 3: e1 e2 , e1 ||e3 , and e2 ||e3 .

4

A SEQUENTIAL GROWTH
We start from some given by one. This pro cedure is term `a classical sequential mo del of causal set.

The dynamics of this mo del is intro duced in [9]. d-graph G and add new x-structures to G one proposed in [17, 16]. Similar pro cedure and the growth dynamics' is proposed in [18] for another 10


Any d-graph consists of x-structures. Obviously, we can construct any d-graph by the sequence of the addition of x-structures one by one to the empty set in arbitrary order. In each step, we get the d-graph. It can be a disconnected d-graph. By definition, the d-graph G is called a connected d-graph iff for any monads i , j G there exists a sequence of immediately causally connected monads with the terminal monads i and j . The simple examples are given in (Fig. 3) and (Fig. 4). The particular case of the classical sequential growth dynamics [9] is based on the following Theorem. Theorem 2 Consider any connected d-graph GN that consists of N x-structures. There exists a sequence of the addition of N x-structures one by one to the empty set such that the fol lowing conditions are satisfied: · the result of this sequence is GN ; · in each step number N (i) N , the resulting d-graph GN nected d-graph;
(i)

is a con-

· in each step number N (i) N , the last added x-structure is a maximal or minimal x-structure in GN (i) . Pro of: Consider the sequence of destructions of GN . In each step we delete one x-structure. Consider any step number N (i) N . We must find the maximal or minimal x-structure in GN (i) such that we get the connected d-graph GN (i)-1 by deleting this x-structure. Consider an iterative pro cedure. Take a maximal x-structure X0 GN (i) and delete it. If the resulting d-graph GN (i)-1 is a connected d-graph consider the next step N (i) + 1. Otherwise we get two disconnected d-subgraphs Gm(1) and Gk(1) . Take a connected d-subgraph Gm(1) X0 . It consists of m(1) + 1 x-structures. Gm(1) X0 includes maximal and minimal x-structures. X0 is a maximal x-structure. Take a minimal x-structure X1 (Gm(1) X0 ) and delete it. If the resulting d-subgraph (Gm(1) X0 ) X1 is a connected d-graph; then GN (1) X1 is a connected d-graph. Otherwise we divide Gm(1) X0 into two disconnected d-subgraphs Gm(2) and Gk(2) . Assume X0 Gk(2) . Take a connected d-subgraph Gm(2) X1 . It includes maximal and minimal xstructures. X1 is a minimal x-structure. Take a maximal x-structure X2 (Gm(2) X1 ) and delete it. The result is the sequence of d-subgraphs Gm(l) · · · Gm(2) Gm(1) GN (1) . We have m(l) < · · · < m(2) < m(1) < N (i). Either we get the required x-structure in some step or in the last step, we get the d-subgraph Gm(l) that includes only one x-structure Xl . This x-structure is connected only with one x-structure Xl-1 . Consequently Xl is a maximal or minimal x-structure in GN (i) , and if we delete Xl we get the connected d-graph GN (i)-1 . 11


We have maximal or nected d-gra x-structures

the sequence of dest minimal x-structure ph by construction. is the reverse of the

ructions of GN . In each such that the residual The required sequence constructed sequence of

step we delete one d-graph is a conof the addition of destructions.

5

A NT I C HA I NS

A chain is a totally (or a linearly) ordered subset of monads. Every two monads of this subset are related by . A chain is a subset of a path. An antichain is a totally unordered subset of monads. Every two elements of this subset are not related by . The cardinality of an antichain is called a width of an antichain. A slice is a maximal antichain. Every monad in G is either in the slice or causal connected to one of its monads. The set of all maximal (or minimal) monads is a slice. A slice is an important subset of d-graph. The physical meaning is a discrete spacelike hypersurface. Consider the properties of a slice. Lemma 9 Any slice S G divides al l monads of G subsets. The first subset is the slice S . The second that for any i B1 there exists j S that i set B2 such that for any s B2 there exists r S in three non-overlapping subsets is a set B1 such j . The third subsets is a that r s .

Pro of: Any i S is causal connected with some j S by definition of / a slice. Suppose there exists 0 S such that there exist two monads 1 S / and 2 S that 1 0 and 0 2 . We get 1 2 . This contradiction proves the lemma. The subset B1 is called a past of the slice S and is denoted by P (S ). The subset B2 is called a future of the slice S and is denoted by F (S ). This property is common for all causal sets. Consider the following specific property of d-graphs. If in a sequence of immediately causally connected monads, each previous monad is a cause of the subsequent monad this sequence is called a directed path or a path. If in a sequence of immediately causally connected monads, each subsequent monad is a cause of the previous monad this sequence is called an opposite directed path. If in a sequence of immediately causally connected monads, some previous monad is a cause of the subsequent monad and some subsequent monad is a cause of the previous monad this sequence is called an undirected path. Theorem 3 Consider the slice S G , the past P (S ), the future F (S ) of this slice, any monad i P (S ) and, any monad j F (S ). If undirected path U P includes i and j , then there exists a monad 0 U P that 0 S . 12


If directed path D P includes i and j , then there exists an unique monad 0 D P that 0 S . Pro of: Assume the converse. Then any monad of U P is included either in P (S ) or in F (S ). Consider two successive monads 1 and 2 such that 1 P (S ) and 2 F (S ). We have (1 2 ) or (2 1 ). There exist 3 S and 4 S such that 1 3 and 4 2 . If (2 1 ), then 4 3 . This is the contradiction. We have (1 2 ). If this is an edge, then 4 1 and 2 3 . We get 4 3 . This is the contradiction. If (1 2 ) is included in the x-structure this x-structure includes two monads 5 and 6 . We have (5 6 ), (1 6 ), and (5 2 ), then 2 3 or 6 3 , and 4 1 or 4 5 . In any case we get 4 3 . Consequently there exists a monad 0 U P that 0 S . In the path D P , all monads are causally connected, then the path can include only one monad of an antichain. This property possesses a clear physical meaning. Any slice includes a unique spacelike section of each physical pro cess that starts in the past, and ends in the future of this slice. This property is valid for the slice of edges and monads (halves of edges). This is not true for a slice of x-structures (or vertexes [19]). Two antichains B1 and B2 are called ordered antichains, and are denoted B1 B2 if there exists the pair of monads 1 B1 and 2 B2 such that 1 2 and do es not exist the pair of monads 3 B1 and 4 B2 such that 4 3 . The next Theorem is truth for more general case of a causal set than a d-graph. Theorem 4 Consider a causal set C . Suppose the cardinality of any slice is finite; then for any slice S0 there exists a set of slices {Si } such that this set satisfies the fol lowing properties: · S0 {Si }; · {Si } is a linearly ordered set; · {Si }=C . Pro of: Construct the linearly ordered sequence of slices {Si(f ) } such that {Si(f ) } = F (S0 ). Suppose F (S0 ) = . Otherwise S0 is a maximal element of {Si }. Take an element x1 F (S0 ) such that P (x1 ) F (S0 ) = . We can do this by a lo cal finiteness of C . If P (x1 ) F (S0 ) = we take any element of this set. We get the needed element by the finite number of iterative repetitions. Consider S01 = P (x1 ) S0 and S02 = S0 \ S01 . The set 13


x1 S02 is an antichain. By M0 denote the set of minimal elements of the set F (S0 ) \ (F (x1 ) F (S02 )). The set S1 = x1 S02 M0 is obviously a slice. We have F (S0 ) P (S1 ) = . Take an element xi F (S02 ). Similarly, construct a slice S2 if P (xi ) F (S1 ) = . If P (xi ) F (S1 ) = consider any element of this set. We get the needed element x2 by the finite number of iterative repetitions. We have P (x2 ) F (S1 ) = . Construct a slice S2 x2 . Consider the element xi again. We have |P (xi ) F (S1 )| > |P (xi ) F (S2 )| by a lo cal finiteness of C . We construct a slice Si-1 by the finite number of iterative repetitions such that P (xi ) F (Si-1 ) = . Construct a slice Si xi . Consider S0i = S0 Si . We have |S0i | < |S02 | by construction. Consider an element xj F (S0i ). The cardinality of any slice is finite, then we can construct a slice Sk F (S0 ) by the finite number of steps. If the set F (S0 ) is infinite the construction of the linearly ordered set of slices is an infinite pro cedure. Let prove that any element xs F (S0 ) belongs to some slice with finite number. The element xs cannot be between two slices Ss-1 and Ss because P (Ss ) F (Ss-1) = by construction. Let prove that xs cannot be in the future of infinite number of slices of {Si(f ) }. Consider P (xs ) F (S0 ). This is a finite set by a lo cal finiteness of C . We have |P (xs ) F (Sk )| < |P (xs ) F (S0 )| by construction. We get a slice Sm such that P (xs ) F (Sm ) = . Construct a slice St F (Sm ). We have xs F (St ) by construction. Either xs St or / xs Sr , where m < r < t. Similarly, construct the linearly ordered sequence of slices that precede S0 . The union of these two sequences and S0 is {Si } The set {Si } is called a full linearly ordered sequence of slices. We can describe the universe as a linearly ordered sequence of spacelike hypersurfaces by this Theorem. The following Lemma is a consequence of this Theorem. Lemma 10 Suppose the causal set C satisfy the conditions of Theorem 4; then for any two ordered slices (S1 S2 ) there exists an ordered set of slices {Si } such that S1 S2 (F (S1 ) P (S2 )) = {Si }. Pro of: The set S1 S2 (F (S1 ) P (S2 )) is a causal set and satisfies the conditions of Theorem 4. We can describe any pro cess as an evolution from an initial state to a final state by this Lemma. In general case, we cannot cho ose the linearly ordered sequence of slices {Si } as non-overlapping sets. For example, consider the simple d-graph that consists of two x-structures (Fig. 6). There exists a unique full linearly ordered sequence of slices. These slices are overlapping sets. In special relativity theory if two hyperplanes correspond to two relatively moving inertial observers the part of the first hyperplane is in the future of 14


Figure 6: Four ordered slices {1 , 2 , 4 }, {1 , 2 , 4 }, {1 , 3 , 4 }, and { 1 , 3 , 4 } . the second hyperplane and the part of the second hyperplane is in the future of the first hyperplane. There are such slices in the d-graph. But in the d-graph such slices can be non-overlapping sets. For example, consider the simple d-graph that consists of three x-structures (Fig. 7). Lemma 11 Suppose G is a d-graph, S is a slice, G0 is a d-subgraph, M1 is a set of minimal monads of G0 , and M2 is a set of maximal monads of G0 ; then the set S1 = S0 M01 M02 is a slice in G0 where S0 = S G0 , M01 = M1 F (S ), and M02 = M2 P (S ). Pro of: Let prove that S1 is an antichain. Take a minimal monad 1 M01 . There exists a monad 2 S that 2 1 by assumption. If there exists a monad 3 S1 that 1 3 we have 2 1 3 in G . This is the contradiction. Take a maximal monad 4 M02 . There exists a monad 5 S that 4 5 by assumption. If there exists a monad 6 S1 that 6 4 we have 6 4 5 in G . This is the contradiction. Suppose 1 4 , then 2 1 4 5 . This is the contradiction. Consequently S1 is an antichain. Let prove that S0 is a slice. Assume the converse. Then there exists a monad 7 G0 that has not causal connection with the monads of S0 , M01 , and M02 in G0 , and do es not belong to these sets. We have 7 S because / 7 G0 and 7 S0 . In G , there exists a path D P between 7 and some / 15


Figure 7: Two non-overlapping slices {3 , 4 , 5 , 6 } and {3 , 4 , 5 , 6 }. monads of S because S is a slice. Assume this is the path from 7 . The case of the path to 7 is similar. There is not the path from 7 to S0 in G0 by assumption. Consequently some monads of D P belong to G0 and some monads do not belong. Start from 7 and go to the first monad of D P that do es not belong to G0 . The previous monad 8 of D P is a maximal monad of G0 . This monad belong to P (S ). Consequently 8 M02 . We have the causal connection between 7 and M02 in G0 . This contradiction proves the theorem. This Lemma is an algorithm of a construction of slices in d-subgraphs. It is truth for any causal set.

6

A WIDTH OF D-GRAPH

The cardinality of a slice is called a width of this slice. Theorem 5 Al l slices of a d-graph G have the same width. This width is cal led a width of G . Pro of: Denote by N the number of x-structures in GN . The pro of is by induction on N . For N = 1, there is nothing to prove. There are two slices with two monads in each slice. By the inductive assumption, if GN consists of N x-structures all slices of GN have the same width n. We must prove that 16


Figure 8: The addition of a new x-structure to one maximal monad.

Figure 9: The addition of a new x-structure to two maximal monads. if GN +1 consists of N + 1 x-structures all slices of GN +1 have the same width. By Theorem 2, we can get any GN +1 by addition of a maximal or minimal x-structure to some GN . Consider the case of a maximal x-structure. The case of a minimal x-structure is similar. There are two cases: the addition of a new x-structure {1 , 2 , 1 , 2 } to one maximal monad 0 (Fig. 8) and to two maximal monads 01 and 02 (Fig. 9). Consider the first case. The set {Si } of all slices in GN can be divided in two subsets: {Si1 |0 Si1 } and {Si2 |0 Si1 }. Consider Si1 . We get / three slices from Si1 in GN +1 . This is Si1 2 , (Si1 \ 0 ) 1 2 , and (Si1 \ 0 ) 1 2 . The width of these new slices is n + 1. Consider Si2 . The monad 0 has causal connection with some monads of Si2 . Consequently the monads 1 , 1 , and 2 have causal connections with some monads of Si2 . We 17


get one slice from Si2 in GN +1 . This is Si2 2 . The width of this new slice is n + 1. In GN +1 , the width of all slices is n + 1. Consider the second case. The set {Si } of all slices in GN can be divided in four subsets: {Si1 |01 Si1 , 02 Si1 }, {Si2 |01 Si2 , 02 Si2 }, {Si3 |01 / / / / Si3 , 02 Si3 }, and {Si4 |01 Si4 , 02 Si4 }. We get five slices from Si1 in GN +1 . This is Si1 , (Si1 \ 01 ) 1 , (Si1 \ 02 ) 2 , (Si1 \ (01 01 )) 1 2 , and (Si1 \ (01 01 )) 1 2 . We get two slices from Si2 in GN +1 . This is Si2 and (Si2 \ 01 ) 1 . The case of Si3 is the same. We get one slice from Si4 in GN +1 . This is Si4 . In GN +1 , the width of all slices is n. This Theorem has a clear physical meaning. This is the conservation law of the number of primordial indivisible ob jects. If there is the conservation law of the number of primordial ob jects in each elementary interaction (xstructure) there is the conservation law of the number of primordial ob jects in each pro cess. This Theorem is the base of the classification of d-graphs and d-subgraphs. We can classify all physical pro cesses in microscopic level by using the number of primordial ob jects that simultaneously take part in these pro cesses.

7

AN IDENTIFICATION OF OBJECTS

In relativity theory, any ob ject is a set of unconnected points at one time instant. Similarly, in discrete mechanics, any ob ject is an antichain. An antichain has one property. This is cardinality. If d-graph describes k ob jects the slice is the union of k non-intersecting antichains. The interaction of ob jects is a monad exchange. Consider a simple example (Fig. 10). Two ob jects interact. Before interaction each ob ject consists of 3 monads. After interaction the first ob ject consists of 2 monads and the second ob ject consists of 4 monads. We can describe this interaction using state vectors and creation and destruction operators. Denote these ob jects before and after interaction by |3 , |3 , |2 , and |4 , respectively. We have |4 = a |3 , ^ ^ |2 = a |3 (1 4 )

for the interaction. The creation and destruction operators was used for creation and destruction of the edges of the graph in [20]. But this mo del is undirected graph. In this simple example the ob jects are clearly visible. But we must have the formal algorithm to divide the slice in antichains. We cannot get such 18


Figure 10: Two interacting ob jects.

19


algorithm if we consider only the slice. We must consider the past and the future of the slice. The simplest criterion is based on a d-graph distance. This is the number of monads in the shortest undirected path between two monads. We can define the antichain as an ob ject if the d-graph distance between any two monads of this antichain is less than the d-graph distance between any monad of this antichain and any other monad of the slice. In our example, such ob jects are the antichains {1 , 2 , 3 } and {4 , 5 , 6 }. In general case, an ob ject can include some monads such that the d-graph distance between these monads and some monads of this ob ject is not less than the d-graph distance between these monads and some other monads of the slice. Such monads can be included in several ob jects, and these ob jects have fuzzy boundaries. A slice can form a hierarchy of ob jects. The d-graph distance between any two monads of any ob ject of the k -level is less than the d-graph distance between any monads that are included in the same ob ject of the (k + 1)-level but are not included in the same ob ject of the k -level. Consider the evolution of ob jects. There are two ordered slices S1 S2 . These slices are divided in the ob jects. The question is: what ob jects of S2 are future states of the ob jects of S1 ? The weakest criterion is causal connections of monads. Two ob jects are two stage of the same ob ject if some monads of the first ob ject are causally connected or coincide with some monads of the second ob ject. In our example, the antichain {7 , 8 , 9 } is the future stage of the antichain {1 , 2 , 3 }, and the antichain {10 , 11 , 12 } is not. But in this case, both antichains {13 , 14 } and {15 , 16 , 17 , 18 } are the future stages of the antichain {1 , 2 , 3 }. We can use the average d-graph distance to distinguish these antichains. But in this example we have the same distance. We can use some intensity of causal connections. For example, this can be the average number of paths between monads of the previous stage and monads of the future stage. Two ob jects are two stages of the same ob ject if their monads have the greatest average intensity of causal connections.

8

A LIGHT CONE

The physical meaning of the sets P (i ) = {j |j i } and F (i ) = {j |j i } is a past light cone and a future light cone of i , respectively. The cardinality of the antichain P (i ) S (or F (i ) S ) is called the width of the past (or future) light cone in S . Lemma 12 Consider the past light cone P (0 ), the future light cone F (0) and two slices S1 and S2 in the d-graph G . If Bp1 Bp2 , where the antichain 20


Bp1 = S1 P (0 ) and the antichain Bp2 = S2 P (0 ); then the width of P (0 ) in S1 is not less than the width of P (0 ) in S2 . If Bf 1 Bf 2 , where the antichain Bf 1 = S1 F (0 ) and the antichain Bf 2 = S2 F (0 ); then the width of F (0 ) in S1 is not greater than the width of F (0 ) in S2 . Pro of: Let prove this lemma for a past light cone. In this case, the pro of for a future light cone is a consequence of Lemma 3. Consider a d-subgraph G0 of G . It consist of all x-structures Xi such that some monads of Xi belong to P (0 ). Obviously, G0 = P (0 ) M, where M is the set of some maximal monads in G0 . Consider slices S01 Bp1 and S02 Bp2 in G0 . By Lemma 11, S01 = Bp1 M01 and S02 = Bp2 M02 , where M01 M and M02 M. Some monads of M02 can belong to the future light cone of some monads of Bp1 . But some monads of M01 cannot belong to the future light cone of some monads of Bp2 . Otherwise, these monads of M01 belong to the future light cone of some monads of Bp1 because Bp1 Bp2 . Consequently M01 M02 . By Theorem 5, all slices has the same width. Consequently Bp1 Bp2 . In this mo del, the horizon of any observer cannot shrinks.

9

CONCLUSION

This paper and the paper [9] intro duce a consistent description of the mo del: a kinematics and a dynamics. However this is an abstract mo del without any connections with physical phenomena. The goal of this mo del is to get the theory of particles. In this approach, particles must be identified with some structures of a d-graph in a like manner as in [13, 21] Also, some properties of structures must be identified with physical quantities such as a mass, an energy, a momentum, a charge etc. The properties of particles are considered now as manifestations of symmetry. Consequently the first open question is the investigation of symmetry of structures in a d-graph.

ACKNOWLEDGEMENTS
I am grateful to several colleagues for extensive discussions on this sub ject, especially Alexandr V. Kaganov and Vladimir V. Kassandrov.

References
[1] R. D. Sorkin, Causal sets: Discrete gravity, (notes for the Valdivia summer scho ol), in Lectures on Quantum Gravity, Proceedings of the 21


Valdivia Summer School, Valdivia, Chile, January 2002, edited by Andrґs Gomberoff and Don Marolf. (Springer, New York, 2005), 305e 327, (arXiv:gr-qc/0309009). [2] F. Dowker, Causal sets as discrete spacetime, Contemp. Phys. 47, 1-9 (2 0 0 6 ). [3] J. Henson, The causal set approach to quantum gravity, in Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter, chapter 21, edited by D. Oriti (Cambridge University Press, Cambridge, 2009), (arXiv:gr-qc/0601121). [4] P. Wallden, Causal Sets: Quantum gravity from a fundamentally discrete spacetime, arXiv:1001.4041 [gr-qc]. [5] J. Myrheim, Statistical Geometry, CERN preprint TH-2538, 1978. [6] G. 't Ho oft, Quantum gravity: a fundamental problem and some radical ideas, in Recent Development in Gravitation, Proceedings of the 1978 Cargese Summer Institute, edited by M. Levy and S. Deser (Plenum, New York/London 1979), 323-345. [7] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Space-time as a causal set, Phys. Rev. Let. 59, 521-524 (1987). [8] A. L. Krugly, Unfaithful embedding of causal sets, arXiv: 1006.2320 [physics.gen-ph]. [9] A. L. Krugly, Discrete mechanics: a sequential growth dynamics for causal sets, and a self-organization of particles, arXiv: 1004.5077 [grqc]. [10] D. Finkelstein, Space-Time Co de, Phys. Rev. 184, 4, 1969-1971 (1969). [11] D. Finkelstein, Space-Time Co de. II, Phys. Rev. D 5, 2, 320-328 (1972). [12] D. Finkelstein, Space-Time Co de. III, Phys. Rev. D 5, 12, 2923-2931 (1 9 7 2 ). [13] D. Finkelstein, Space-Time Co de. IV, Phys. Rev. D 9, 8, 2219-2231 (1 9 7 4 ). [14] D. Finkelstein and G. McCollum, Unified quantum theory in Quantum theory and the structures of time and space V.1, edited by L. Castell, M. Drieschner, and C. F. von Weizsacker (Munhen, Wienna, Hauser, 1975), Ё Ё 15- 54. 22


[15] D. Finkelstein, "Superconducting" causal net, Int. J. Theor. Phys. 27, 4 7 3 - 5 1 9 (1 9 8 8 ). [16] A. L. Krugly, Causal set dynamics and elementary particles, Int. J. Theor. Phys. 41(1), 1-37 (2002). [17] A. L. Krugly, The model of discrete spacetime (Moscow, Monolog, 1998) [in Russian], http://www.chronos.msu.ru/lab-kaf/Krugly/ekrpublic.html [18] D. P. Rideout and R. D. Sorkin, A classical sequential growth dynamics for causal sets, Phys. Rev. D 61, 024002-1 - 024002-16 (2000), (arXiv: gr-qc/9904062). [19] S. Ma jor, D. Rideout, and S. Surya, Spatial Hypersurfaces in Causal Set Cosmology, Class. Quant. Grav., 23, 4743-4752 (2006), (arXiv: gr-qc/0506133). [20] I. Dadiґ, and K. Pisk, Dynamics of Discrete-Space Structure, Int. J. c Theor. Phys., 18, 345-358 (1979). [21] S. O. Bilson-Thompson, A Topological Mo del of Composite Preons, arXiv: hep-ph/0503213.

23