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The dynamics of binary alternatives for a discrete pregeometry
arXiv:1201.0005v1 [gr-qc] 28 Dec 2011
Alexey L. Krugly


Abstract A particular case of a causal set is considered that is a directed dyadic acyclic graph. This is a mo del of a discrete pregeometry on a microscopic scale. The dynamics is a sto chastic sequential growth of the graph. New vertexes of the graph are added one by one. The probability of each step dep ends on the structure of existed graph. The particular case of dynamics is based on binary alternatives. Each directed path is considered as a sequence of outcomes of binary alternatives. The probabilities of a sto chastic sequential growth are functions of these paths. The goal is to describ e physical ob jects as some selforganized structures of the graph. A problem to find self-organized structures is discussed. Keywords: causal set, random graph, self-organization. PACS: 04.60.Nc

1

Introduction

Consider a particular mo del of a discrete pregeometry. This is a directed dyadic acyclic graph. The edges are directed. Each vertex possesses two incident incoming edges and two incident outgoing edges. A vertex with incident edges is called an x-structure (Fig. 1). The mo del was intro duced by D. Finkelstein in 1988 [1]. The acyclic graph means that there is not a directed lo op. In this paper only such graphs are considered. Then they are called graphs for simplicity.
Scientific Research Institute for System Analysis of the Russian Academy of Science, 117218, Nahimovskiy pr., 36, k. 1, Moscow, Russia; akrugly@mail.ru.

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Figure 1: An x-structure. This mo del is the particular case of a causal set. 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 elements of 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 ) = {z | x z y }. The first three properties are irreflexivity, acyclicity, and transitivity. These are the same as for events in Minkowski spacetime. A(x, y ) is called an Alexandrov set of the elements x and y or a causal interval or an order interval. In Minkowski spacetime, an Alexandrov set of any pair of events is an empty set or a set of continuum. The lo cal finiteness means that an Alexandrov set of any elements is finite. The physical meaning of this binary relation is causal or chronological order. By assumption a causal set describes spacetime and matter on a microscopic level. In the considered mo del, the set of vertexes and the set of edges are causal sets. A causal set approach to quantum gravity has been intro duced by G. 't Ho oft [2] and J. Myrheim [3] in 1978. There are reviews of a causal set program [4, 5, 6, 7]. The goal of the considered mo del is to describe physical ob jects as some self-organized structures of the graph. This self-organization must be the consequence of dynamics.

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2

Sequential growth dynamics

The mo del of the universe is an infinite graph. But any observer can only know finite graph. In a graph theory, by definition, an edge is a relation of two vertexes. Consequently some vertexes of finite graph have less than four incident edges. These vertexes have free valences instead the absent edges. These free valences are called external edges as external lines in Feynman diagrams. They are figured as edges that are incident to only one vertex. There are two types of external edges: incoming external edges and outgoing external edges. An example of the graph with 2 vertexes (Fig. 2) possesses 1 edge, 3 incoming external edges, and 3 outgoing external edges. We can prove that the number of incoming external edges is equal to the number of outgoing external edges for any such graph [9]1 .

Figure 2: An example of the graph with 2 vertexes. A finite graph is a mo del of a part of some pro cess. The task is to predict the future of the pro cess or to reconstruct the past. We can reconstruct the graph step by step. The minimal part is a vertex. We start from some given graph and add new vertexes one by one. This pro cedure is proposed in papers of author [13, 14]. Similar pro cedure and the term `a classical sequential growth dynamics' is proposed by D. P. Rideout and R. D. Sorkin [15] for other mo del of causal set dynamics. We can add a new vertex to external edges. This pro cedure is called an elementary extension. There are four types of elementary extensions [10]. There are two types of elementary extensions to outgoing external edges (Fig. 3 and 4). This is a reconstruction of the future of the pro cess. In this and following figures the graph G is represented by a rectangle because it
It should be noted that a set of halves of edges is considered in papers [9, 10, 11]. The halves of edges as basic ob jects is intro duced by D. Finkelstein G. McCollum and in 1975 [12]. By some reasons, it is convenient to break the edge into two halves of which the edge is regarded as composed. The set of halves of edges in papers [9, 10, 11] is isomorphic to the considered graph.
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3


Figure 3: The first type of an elementary extension. can have an arbitrary structure. The edges that take part in the elementary extension are figured by bold arrows. First type is an elementary extension to two outgoing external edges (Fig. 3). The number n of incoming or outgoing external edges is not changed by this elementary extension. Second type is an elementary extension to one outgoing external edge (Fig. 4). The numbers n of incoming external edges and outgoing external edges have increased by 1. Similarly, there are two types of elementary extensions to incoming external edges (Fig. 5 and 6). These elementary extensions reconstruct the past evolution of the pro cess. Third type is an elementary extension to two incoming external edges (Fig. 5). The number n of incoming or outgoing external edges are not changed by this elementary extension. Fourth type is an elementary extension to one incoming external edge (Fig. 6). The numbers n of incoming external edges and outgoing external edges have increased by 1. We can prove that we can get every connected graph by a sequence of elementary extensions of these four types [9, Teorem 2]. By assumption, the dynamics of this mo del is a sto chastic dynamics. We can only calculate probabilities of different variants of elementary extensions.

3

The dynamics of binary alternatives

Consider the dynamics that is based on binary alternatives. This alternative has two outcomes with probabilities 1/2. This pro cess has 1 bit of information. A binary alternative is considered as some primordial entities. This 4


Figure 4: The second type of an elementary extension.

Figure 5: The third type of an elementary extension.

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Figure 6: The fourth type of an elementary extension. idea is considered in a set of papers. For example I intro duce two citations of C. F. von Weizsacker. It is certainly possible to decide any large alterЕ native step by step in binary alternatives [16, p. 222]. . . . the decision of an elementary binary alternative is the elementary pro cess and hence the elementary interaction [17, p. 94]. Binary alternatives are discussed in the bo ok "Gravitation" [18, section 44.5] and other papers [12, 19, 20]). This list of references is by no means complete. There is a recent paper of M. Kober [21]. In the considered mo del a binary alternative is identified with an xstructure. Consequently the graph is a net of binary alternatives. Consider a directed path. Number outgoing external edges by Latin indices. Number incoming external edges by Greek indices. Latin and Greek indices range from 1 to n, where n is the number of outgoing or incoming external edges. If we cho ose a directed path from any incoming external edge number , we must cho ose one of two edges in each vertex (Fig. 7). Assume the equal probabilities for both outcomes independently of the structure of the graph. Then this probability is equal to 1/2. This is the binary alternative. Consequently if a directed path includes k vertexes, the choice of this path has the probability 2-k . We have the same choice for an opposite directed path. Intro duce an amplitude ai of causal connection of the outgoing external edge number i and the incoming external edge number . By definition, put
M

ai = ai =
m=1

2

-k (m)

,

(5 )

6


Figure 7: A choice of a directed path is a sequence of binary alternatives. where M is the number of directed paths from the incoming external edge number to the outgoing external edge number i and k (m) is the number of vertexes in the path number m. This definition has clear physical meaning. The causal connection of two edges is stronger if there are more directed paths between these edges and these paths are shorter. Consider a following algorithm to calculate the probabilities of elementary extensions [11]. There are three steps. The first step is the choice of the elementary extension to the future or to the past. By definition, the probability of this choice is 1/2 for both outcomes. A new vertex is added to one or two external edges. The second step is the equiprobable choice of one external edge that takes part in the elementary extension. This is an outgoing external edge if we have chosen the future evolution in the first step. Otherwise this is an incoming external edge. The probability of this choice is 1/n for each outcome. The third step is the choice of second external edge. Denote by pij the probability to cho ose the outgoing external edge number j if we have chosen the outgoing external edge number i in the second step. By definition, put
n

p ij =
=1

ai aj .

(6 )

Consider the meaning of this definition. The addition of a new vertex to two external edges forms a set of lo ops (Fig. 8). Each lo op is formed by two 7


Figure 8: A new lo op is generated by a new vertex. directed paths. We can describe a lo op as a pro duct of these paths. The probability of the elementary extension is directly proportional to the sum of new lo ops that are generated by this elementary extension. Similarly,
n

p



=
i= 1

ai ai ,

(7 )

where p is the probability to add a new vertex to two incoming external edges numbers and . The sum of all directed paths from any edge is equal to 1. We get the right normalization if we put the following definition for the probability to add a new vertex to one outgoing or incoming external edge, respectively.
n

p ii =
=1 n

ai ai ,

(8 )

p



=
i= 1

ai ai .

(9 )

We can calculate the probability of any elementary extension by using this algorithm. 8


Figure 9: The connection by the common past. In this mo del, causality is defined as the order of vertexes and edges. But the causality has a real physical meaning only if the dynamics agrees with causality. The dynamical causality can be formulated in the following form. The probability to add a new vertex to the future can only depend on the subgraph that precedes this vertex [15]. Similarly, the probability to add a new vertex to the past can only depend on the subgraph that follows this vertex. The considered algorithm agrees with causality and has clear physical meaning. The probability to add a new vertex to two outgoing external edges is greater if their common past is larger, and this common past has the stronger connection with these edges.

4

Physical ob jects

Usually in physical theory, physical ob jects are considered as primary entities in instants of time. Then a pro cess is secondary, is a mapping of the ob jects or of their initial to their final states. An ob ject has some state if it has some structure. But in relativity theory, structures cannot exist in instants of time. Consider a simple example (Fig. 9). This is a 2-dimensional Minkowski space. Points a and b are simultaneous in the considered frame of reference. This instant of time is denoted by t0 . By definition, these points cannot have any connections in t0 . They can be connected only by the common past (the shaded triangle). The nearest point of this common past is the point c in the instant of time t1 . The points a and b form a structure only together 9


with c. This structure has a duration t0 - t1 . In the instant of time we have unconnected points. Any structure has finite duration. Consequently any structure is a pro cess. According to relativity, the world is a collection of pro cesses (events) with an unexpectedly unified causal or chronological structure. Then an ob ject is secondary; is a long causal sequence of pro cesses, a world line. [22, p. 2923]. An antichain is a totally unordered subset of edges. Every two edges of this subset are not related by causal connection. A slice is an inextendible antichain. Every edge in the graph is either in the slice or causal connected to one of its edges. The set of all outgoing (or incoming) external edges is a slice. In the considered mo del, a slice of edges is a discrete analog of spacelike hypersurface. We can define ob jects (structures) in the slice of outgoing external edges by their common past. The connection of a pair of outgoing external edges numbers i and j (i = j ) is defined by the probabilities pij . There is a scale hierarchy of the matter in the universe. Elements of the more deep level have stronger couplings. In the considered mo del, the strong coupling of the pair of the outgoing external edges numbers i and j is the high value of pij . This mo del is useful for numerical simulation. Now this investigation is started [23]. We start from 1 vertex and calculate 500 steps. There are many variants of the growth for a big graph. But usually there are very few variants with high probability. These are the couplings of outgoing external edges in the deepest level of ob jects. We can define a threshold value p0 for the deepest level of ob jects. If pij p0 , the pair of the outgoing external edges numbers i and j belong to the same ob ject of the deepest level. Let pij , where i = j be elements of square matrix p(out) with zero main diagonal. Transform p(out). Replace pij by 1 if pij p0 . Replace pij by 0 if pij < p0 . We get the matrix s(out). Consider s(out) as an adjacency matrix of some undirected graph S (out). We have the isomorphism that takes each outgoing external edge to the vertex of S (out). The vertexes numbers i and j of S (out) are connected by an edge iff pij p0 . By definition, an ob ject of the deepest level is a connected subgraph of S (out). A connected subgraph includes nonextendible cliques. By definition, a clique is a subgraph such that each pair of its vertexes is connected by an edge. In general case, a connected subgraph includes a set of overlapping and non-overlapping nonextendible cliques. These cliques form a frame of the ob ject. The cliques have the following property for the considered algorithm of sequential growth. Consider the addition of a new vertex to the outgoing external edges numbers i and j . These two outgoing external edges become internal edges, and two new outgoing external edges appear. We must delete 10


the vertexes numbers i and j of S (out) and add two new vertexes. We get new graph S1 (out). If the vertexes numbers i and j belong to the clique C of S (out), the two new vertexes belong to the clique C1 of S1 (out). We get the clique C1 by deletion the vertexes numbers i and j and addition two new vertexes. Consequently the interior interactions in the frame of the ob ject cannot destroy this frame. Similarly, we can define ob jects in the slice of incoming external edges by their common future. We can define ob jects for arbitrary slice of edges. Consider two edges numbers a and b of this slice. Define an amplitude aia of causal connection of the outgoing external edge number i and the edge number a. Define an amplitude aa of causal connection of the edge number a and the incoming external edge number . Consider an amplitude pab of coupling of the edges number a and b. By definition, put
n n

p

ab

=
=1

aa ab +
i= 1

aai aib .

(1 0 )

If the edges numbers a and b are outgoing external edges, definition (10) coincides with (6). If the edges numbers a and b are incoming external edges, definition (10) coincides with (7).

5

Conclusion

I hope that the existence of the small quantity of preferable variants of the growth is a symptom of self-organization. It is necessary to develop the metho ds to detect and analyze repetitive symmetrical self-organized structures during the numerical simulation of the sequential growth. One metho d is to consider the evolution of the graph S (out) during the sequential growth. We can search the connected subgraphs of S (out) by the algorithm of depthfirst search (DFS). We can search the cliques of these connected subgraphs by the Bron-Kerbosh algorithm [24]. This is the task for further investigation. I am grateful to Alexandr V. Kaganov and Vladimir V. Kassandrov for discussions and Ivan V. Stepanian for collaboration in numerical simulations.

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