# Search for Fundamental Rule of Physics

Search for Fundamental Rule of Physics

Project Proposal

Maksim Piskunov (Northeastern U) and Todd Rowland (Wolfram Research)

## Introduction

Introduction

One of the most important questions in Causal Set Theory is how to make a growth rule, which would generate a causal set resembling the real Universe [1]. In this proposal we suggest an approach for solving this problem.

Our approach is based on the observation that any sufficiently interesting computational system generates a causal set (See chapter 9 of [2]), which makes it interesting to study such systems from the fundamental physics point of view. We call such systems set substitution systems.

Specifically, consider a set , each element of which has some unique arbitrary internal structure (without loss of generality, you can think about it as an integer), and also every element contains a sequence of references to other elements, so that , where ={|B∈S} and is a reference from to . We define two elements as distinct if their internal structure is different: . Also, we say that ⋂=∅⇔∀A∈∀B∈:A≠B.

S={A}

N

A

A=(,)

N

A

R

A

R

A

R

AB

R

AB

A

B

A≠B⇔≠

N

A

N

B

S

1

S

2

S

1

S

2

Next, consider an evolution rule , which takes a subset of , satisfying a given condition : ⊂S|(S,), and replaces it with another set of new unique elements =(S,)=∅, replacing also references from compliment of to with references to : , where (A)=(,{()|∈}), where ()= if , and ()=,C∈ if .

S

in

S

C

S

in

C

S

in

S

out

replace

S

in

S

in

S

out

S

in

S

in

S

out

(S)={(A)|A∈S∖}⋃

relink

S

in

S

out

relink

N

A

R

R

AB

R

AB

R

A

R

R

AB

R

AB

B∈S∖

S

in

R

R

AB

R

AC

S

out

B∈

S

in

The sequence of consecutive applications of evolution rule starting with an initial condition produces an evolution history: We define =∖(⋂), =,, and by definition .

S

0

,=(),=(),...,=(),....

S

0

S

1

S

0

S

2

S

1

S

k

k

S

0

i

S

in

S

i

S

i+1

S

i

i

S

out

replace

S

i

i

S

in

∀i∈∀j∈j≥i:=∅

0

0

j

S

out

S

i

Using this evolution history, a causal network can be constructed, in which the set of vertices is the set of evolution events , where =(,), and the edge exists from to if and only if the evolution rule at step operates on some of the elements produced in step . In other words, the adjacency matrix of the causal network satisfies the condition: =1⇔≠∅.

(V,E)

V={|i∈}

v

i

0

v

i

S

i

S

i+1

v

i

v

j

j

i

a

ij

i

S

out

j

S

in

Then, the transitive closure of the causal network is a causal set. You can see that the set substitution system never produces cycles in a causal network because . Then, since ⊂, =∅. Therefore, , which implies that the causal network is cycle-free, therefore its transitive closure has no cycles as well.

∀j∈∀i∈j≥i:=∅

0

j

S

out

S

i

i

S

in

S

i

j

S

out

i

S

in

∀j∈∀i∈j≥i:=0

0

a

ji

Note, that in general may be multivalued because might be true for multiple subsets of simultaneously , and are always single valued), in which case multiple evolution histories are possible starting from the same initial condition. We are primarily interesting in causal invariant systems however, for which any such history produces the same causal network. Sufficient but not necessary condition for this is the requirement of (S,) to only be true simultaneously for non-overlapping subsets of : ≠∅¬S,∨¬S, (see page 504 of [2] for details). Closely related to causal invariance is the concept of confluence studied in computer science [9].

C

S

replace

relink

R

C

S

in

S

1

S

in

2

S

in

C

1

S

in

C

2

S

in

The set substitution system is general enough, so that it can describe any network-like system, in which evolution rule operates locally (i.e. it replaces only a small subset of the network at each step). For example, it can trivially describe mobile automata, Turing machines, or network rewrite systems. (See chapters 3 and 9 of [2] for details.)

Therefore, if we assume that the fundamental law of physics can be represented as a computational system, it is natural to assume that there is a causal set in the Universe. Also, it is natural to think that this causal set is not fundamental, but is emergent from a computational system described above.

Since this computational system is operating on a very small scale (most likely, close to Planck scale), it is hard to deduce the evolution rule from the known large-scale laws of physics. Since we don’t know how does the fundamental evolution rule look like, we want to try as many rules as possible in an unbiased way, for we don’t want to miss the correct one.

Therefore, we propose for this project to systematically look at many simplest evolution rules (it was demonstrated in [2] that even simple rules can produce complex, including computationally universal, behavior), observe their behavior, and possibly connect our observations to known laws of physics.

## Strategy

Strategy

It is not possible to run simulations with the general set substitution system described above. Since is arbitrary, the problem of computing the result of an evolution step might be undecidable. Hence, we are going to consider many special cases of the set substitution system. Since we know that space-time in the Universe doesn’t have a rigid structure (compare to cellular automata for example), it seems to be the most reasonable to consider systems operating on graphs, or generalizations of graphs. It is worth mentioning though, that cellular automata are also studied in the context of physics [8].

### Enumeration of special cases of set substitution system

Enumeration of special cases of set substitution system

Therefore, the first step of the project is to implement code for graph replacement, which will work for generalized graphs. The preliminary list of supported graph types includes:

◼

Multigraphs with arbitrary vertex degrees.

◼

Graphs might be directed, undirected, or combinations of the two.

◼

There might be integers or boolean variables attached to graph vertices and / or edges.

◼

Edges outgoing from vertices might be indistinguishable, distinguishable, or more generally, might have any discrete symmetry group attached to them. For example, in ordinary graphs, edges are indistinguishable. Another example one may consider, is 3-valent graphs in which every vertex has edges labeled , and the edge is distinct from the edge . Yet another example is to consider planar graphs, in which the edges have cyclic symmetry, so that sequence of edges is the same as , but different from .

1,2and3

(,1)(,1)

v

1

v

2

(,2)(,1)

v

1

v

2

(1,2,3)

(2,3,1)

(3,2,1)

◼

Finally, we might also consider hypergraphs.

Todd Rowland will help mentor Maksim Piskunov, and in return, the graph replacement code may be used in Mathematica. Todd Rowland could also be part of the thesis committee.

### Computer Experiments

Computer Experiments

After the code is written, the next step is to run as many computer experiments as possible. The computer experiments will be systematic, and will be made as unbiased as possible. Discovery Cluster might be helpful here, for the experimental phase is parallelizable, and computationally intensive.

### Analysis

Analysis

The final step is to analyze the data produced by computer experiments. The main goal of this analysis is to find candidate universes, and there will be certain technical challenges along the way.

The evolution rule given some initial conditions is considered a candidate universe if it was not proven that it cannot possibly represent the real Universe.

For example, some rules will terminate, that is, yield evolution histories, for which the state of the system is constant after a certain number of steps. Such histories are obviously not candidate universes. Other rules may produce trees, or uniform grids, which are also not plausible for the real Universe.

However, there will undoubtedly be systems, for which the evolution history will look random, and it will not be clear whether it represents the real Universe or not. The goal is to analyze such systems as deeply as possible, and either exclude them as implausible, or reproduce some known laws of physics (most importantly, Quantum Field Theory and General Relativity) with them.

The main challenge of this approach is that the computational system is most likely operating on Planck scale, while the known laws of physics only operate at a much larger scale.

Therefore, it will not be possible to generate an evolution history large enough to directly compare it to known laws of physics. The preliminary idea for mitigating this problem is to construct effective approximate models for the evolution on larger scales, and build a ladder of such models, which will eventually on large enough scale be equivalent to known laws of physics.

### Publishing

Publishing

The partial results will be published during the analysis phase once physically significant findings are obtained. One journal, which will likely accept papers about evolution of set substitution systems is Complex Systems. If some known physics will be reproduced, the results may be published in traditional physics journals, such as Physical Review Letters. This project will also be a significant part of Maksim Piskunov’s doctoral dissertation.

### Rough Schedule

Rough Schedule

For this proposal we can make a rough schedule of the project phases. We expect to have clean working code for at least one special case of set substitution system within a few months (by January 1, 2016). Most considered cases will be implemented within a year from now (that is by November 2016). The computer experiments for each special case of set substitution system will have to wait for the code, but once some code is done, the experiments can be done concurrently with further code development. The main phase of the computer experiments should be done within months after the code is done. The full analysis might be an undecidable problem, so it is hard to predict how much time it will take, but we expect to have partial results within months after the experiments are finished. Some basic results might be obtained immediately after the experiments.

6

6

### Criteria of Success

Criteria of Success

The project will be unambiguously successful if the fundamental theory of physics is discovered. However, since that might be an impossible task for 4 years, we also define some intermediate goals:

◼

Implementation of high quality code for special cases of set substitution system, which professional researchers can use.

◼

Enumeration of a variety of special cases of set substitution system.

◼

Systematic covering of the simplest rules.

◼

Finding an evolution generating a causal set, which can be approximated by flat or curved spacetime.

◼

Finding an evolution of the set substitution system, which behavior is not trivial, but at the same time does not look completely random, that is, a class 4 system (see chapter 3 of [2]).

◼

Implementation of an elaborate set of rules, which can filter out evolutions which cannot represent the real Universe. The more complex filter is developed, the more systems will be filtered out.

## Notes on plausibility

Notes on plausibility

One challenge which has to be overcome in order for this project to be successful, is the identification of spacetime-like networks. This problem might be separated into two distinct parts: the analysis of local geometry (that is, metric tensor), and global geometry (that is, topology).

While both problems are not solved yet of now, their solution does not seem impossible. For example, measuring how the ball volumes around a graph vertex or geodesic change with radius might help determining the Ricci scalar at that vertex (see page 1052 of [2]). Also note, that while not enough to determine the metric structure, the information about ball sizes includes information about vertex degrees (balls of radius ) and local clustering coefficient (balls of radius ).

1

2

Moving to topology, while an algorithm for determining it is not known to us, there were previous attempts to construct it [7], and we think it is plausible that it can be done despite significant challenges.

An important thing to mention that while it is important to have a rigorous formulation of whether a particular network might be approximated by a continuous manifold, it is not required for the success of this project since the goal here is to only find plausible rules representing the Universe. Also note, that since we are going to study concrete networks, the question of approximating the network with a metric space can be approached in a very direct way, which gives us a significant advantage comparing to other pregeometric approaches.

Another challenge is to demonstrate that our approach will be able to produce Lorentz invariant systems. In order to understand why it might be possible, let us consider a set substitution system in which is multivalued, and therefore the evolution history is not unique. It is demonstrated in chapter 9 of [2] that such, so called causal invariant, systems can still produce unique causal sets, no matter in which order is applied. Then, since the state of the system at every step represents a space at a particular time, the different order of application would produce a space in a different frame of reference. And since does not depend on evolution history, but only on a current state of the system, it means that the evolution rule does not depend on the choice of a reference frame. If on top of that we have a finite speed of information propagation through the system (that is, finite speed of light), we will have two main postulates of special relativity.

Finally, one of the most serious objections to these deterministic causal networks comes from quantum mechanics. We can appeal to what Stephen Wolfram has said in chapter 9 of [2], that the apparent randomness aspect is not a problem due to intrinsic randomness, but that to overcome the Bell inequality the networks have to have long range connections. From the definitions we are using, the existence of long range connections is natural because the whole universe comes from a finite (and small) set. This has implications to the study of geometry. We can also appeal to Gerhard t’Hooft who, more recently, has been investigating quantum arising from deterministic systems [6]. We would like to be able to give plausible explanations for the observance of quantum phenomenon in actual causal sets.

## References

References

Introduced the idea of causal set. Emphasized the importance of dynamics.

Demonstrated how to go from a substitution system to a causal network and gave plausible explanations for how a fundamental theory could be found.

Explained how simple rules can generate complex behavior, and how to identify candidate rules in a computational space.

Studied special cases of network substitution systems using approaches similar to ours. Found some nontrivial substitution rules.

Studied character string substitution systems. Found interesting rules, including a computationally universal one.

Explored another type of network rules found in [2]. Searched for nontrivial behaviors. This rule type is lower on our list because it involves a global update, but it is possible to consider a local version.

Suggests that quantum theory might arise from a deterministic system.

Studied the networks arising from network substitutions in order to identify their global geometry and topology (unpublished). An initial study.

Famous for a digital physics theory involving cellular automaton. One of the many differences is that the causal sets from substitution systems are not necessarily rigid like those from cellular automaton.

A well-studied concept in rewriting theory from Computer Science. Many references exist, but we’re still looking for a good one. This is related to the causal invariance as described in [2] on page 504.