This is the Community Post presenting the 2016 Wolfram Summer School Project of József Konczer, a Hungarian PhD student of theoretical physics, assisted by Todd Rowland.

The project notebook with the code is attached to this post.

Fundamental theories in Physics

The Holy Grail of theoretical physics would be a theory, which could describe all known phenomena in every situation. This would be called the Theory of Everything (ToE). In present times, all approaches to a ToE candidate—like String/M theory, Loop quantum gravity (and many others)—incorporate quantum mechanics from the beginning. This approach served surely very useful effective theories like the Standard Model itself, however this approach does not help to find an underlying deterministic theory from which quantum effects would emerge, like Einstein dreamed.

The standard argument for the unavoidability of quantum mechanics and uncertainty is the result of the Bohr–Einstein debates which was "won" by Bohr because the EPR paradox was tested by measurements (there is an ongoing test as well called the Big Bell test) and the result excludes local hidden variables. (The arguments can be found in details here and here). It has to be emphasized that in these arguments locality is a key assumption.

One can ask, what kind of nonlocal theory can be constructed, which still have predictive power, and is not based of conspiracy of Nature? There are only a few researchers who post this question openly, one is them is Gerard ’t Hooft who published recently a book based on collected papers. His approach is conservative (from main stream point of view), and mainly suggests, that if one quantize time, then in same basis the unitary time evolution considered in quantum mechanics become a permutation operator between special basis elements or "beable states". However not every time evolution has this property and typically the interacting theories fail to fulfill the requirements. A more bold, however much less understood theory (or framework) is what Stephen Wolfram described in NKS. The brief summary of his ideas can be found in this blog post. The main idea here, is to find a simple data structure, for instance a sparse graph, a simple discrete dynamics governed by a replacement rule, an interpretation for this cellular automaton (CA) and then investigate if we can observe similar phenomena what we see in our Universe.

Hints pointing toward the CA description

This is a speculative and highly subjective argumentation, however I think this blog post is an appropriate place to articulate my motives and do not stick to the objective style of research papers.

So first of all without going too deep into metaphysics I don't want to state things about Nature it self, I only talk about our description of it. The first successful and highly useful description of Nature was Newtons description, which heavily used the idea of continuity of space and time. This idea proved to be useful in description of solids, liquids and gases as well. However some ideas became so useful and popular, that we forgot that all of them is only our description and not Nature its self. Quantum effects and effects related to relativity reminded us, that under non standard circumstances old descriptions can fail. As I see, quantum effects have two message for us. The first is that quantities could be and should be described by discrete variables, and secondly that under a certain level systems can not be observed without disturbance. If we take into account that space and time even as we observe them are influenced by these quantized quantities, it is straightforward to deduce, that space and time should be quantized as well.

Before these findings in physics probability theory was developed. First it was used to analyze gambling situations where one don't know every information about the system. From this point of view it is clearly a strategy to manage our ignorance toward some details in a deterministic situation. However after some point physicists started to use probabilities as they were part of the phenomena, and not only our clever way to make inference from systems where we do not know every detail. Many physicists—including myself—where educated in the spirit of frequentist interpretation of probability theory, which is useful in some cases but as I think, prevents some questions to ask. I think this promotion of probability to an objective property contributed to the interpretation of quantum mechanics as well. As Jayns wrote in his book:

In current quantum theory, probabilities express our own ignorance due to our failure to search for the real causes of physical phenomena; and, worse, our failure even to think seriously about the problem. This ignorance may be unavoidable in practice, but in our present state of knowledge we do not know whether it is unavoidable in principle; the ‘central dogma’ simply asserts this, and draws the conclusion that belief in causes, and searching for them, is philosophically naive. If everybody accepted this and abided by it, no further advances in understanding of physical law would ever be made; indeed, no such advance has been made since the 1927 Solvay Congress in which this mentality became solidified into physics. But it seems to us that this attitude places a premium on stupidity; to lack the ingenuity to think of a rational physical explanation is to support the supernatural view.

However even if one thinks, that theories incorporating quantum mechanics are "only" effective theories, probably we can get intuitions from them. There is a recent result from AdS/CFT correspondence as an example for the EPR=ER conjecture. And a connecting paper of Leonard Susskind, concluding that:

What all of this suggests to me, and what I want to suggest to you, is that quantum mechanics and gravity are far more tightly related than we (or at least I) had ever imagined. The essential nonlocalities of quantum mechanics (the need for instantaneous communication in order to classically simulate entanglement) parallels the nonlocal potentialities of general relativity: ER=EPR.

The cited papers state, that spacetime structure can be understood as a net of entanglements, however maybe the statement can be reversed, and say that the phenomena of entanglement can be described by a nonlocal spacetime structure.

Among the mentioned hints, the existing theoretical constructions can help to find an appropriate interpretations as well. For example it can happen, that to describe our seemingly 3 dimensional space one has to describe space with higher effective dimensionality and interpret the entangled parts not just as connected regions, but as global structures in the extra dimensions.

After taking hints from existing theoretical constructions one can investigate, what kind of phenomena can appear in simple CA-s which mimic some parts of Nature.

Perhaps the most well known CA is Conway's Game of Life this is a 2D Cellular automaton where localized objects (called spaceships or gliders) can propagate, and can interact with each other. This behavior can remind us to particles, however the built in rectangular structure is reflected on the properties of spaceships, and there are no nonlocal connections between these object because of the locality of the rule. Both problems can be solved, if one tries to construct a CA without built in topology. (This construction will be described in detail.)

Another nice feature of special CA-s called substitution systems, is that for a structure living in the automaton, can not observe the absolute number of steps, or other structures beside him, only the causal net of implemented changes can be recognized from inside. This feature unites the relative space and time for observers or structures inside the system. It can remind us to causal network description of General Relativity.

A third hint from CA point of view is the typical appearance of complex behavior, which can lead to an effective probabilistic description of the system with a higher symmetry what the framework originally allowed. (For example CA description of flows) From disorder new effective order can emerge possible with higher symmetry.

The conjectured computational irreducibility of CA would replace the promised "free will" possibility of quantum mechanics with a different but in some sense similar concept. In this framework the faith of the Universe would be determined, but even an observer outside the system—God if one wishes—could not know the consequences only by letting the simulation run up to the desired point.

Furthermore a multiway CA dynamics is compatible with the many world interpretation of quantum mechanics, with the advantage, that the splitting points of histories are not observer dependent. In this framework the overall dynamics is deterministic, however structures living always on one branch of the evolution will witness an unavoidable true random behavior from the inside point of view.

Nature and our understanding of it

Of course it would be an arrogant attitude to force Nature to fulfill our philosophical expectations, however one can imagine how our description of it can change during time. There are several situations what one can imagine:

• There is a deterministic description which is valid in any situation (which can appear in our Universe)
  • This can be totally discreet
  • Or it can be continuous partially or in whole
• It can be, that after some point a truly random (or at least appearing to us) mechanism will appear, which can not be unfolded
• Or it can happen that construction of laws to describe Nature will never come to an end, and our understanding of reality will based on infinite set of possibly deterministic rules.
• And of course it can happen that something unexpected will turn out.

Without favoring any of the listed cases above, my main point is that the very first situation, namely that our universe can be described as a deterministic discrete system is not totally excluded. And the most natural way to understand it can be a CA description.

CA description candidate for our Universe

To have a CA description, one has to choose a data structure, a dynamics and an interpretation. (It has to be pointed out, that any CA can be simulated on another Turing complete CA with different interpretation of the states. Because of that any CA description is highly non unique. However one can try to choose a description which has the "simplest" interpretation.)

For a fundamental CA description one can choose simple graphs as data structure. This seems as a natural choice, because of its simplicity and because of its non fixed topology.

To have a chance to describe deterministic dynamics on this data structure, we even restrict the degree of nodes on the graph. One can try to find the threshold of complexity of the CA, and it seams that cubic graphs can already produce complicated enough structures. So one can set the data structure to a simple cubic graph.

The next step is to define an appropriate dynamics on this data structure. A natural approach is to introduce subgraph replacement rules, which means the following: If one finds a given subgraph pattern $H_1$ in the present graph $G$, then replace it with a compatible new graph $H_2$. It sounds simple, however there are many details which have to be fulfilled to get a dynamics with desired properties.

I mention here two properties of the patterns, which seems to be essential to get a substitution systems which generates a complex behavior and appears completely deterministic from the inside without the specification of the order of replacements in the system.

The first one is a non overlapping property of the pattern graph(s) $ H_1 $. This means, that $H_1$ has a special structure such as there is no cubic graph $G$, where two subgraphs can be found which are isomorphic to $H_1$ and have nonzero intersection. The following rule does not fulfill this requirement because there is a cubic graph, where two intersecting copies of $H_1$ can be found

The second requirement is non triviality, which gives a constraint for $H_2$. In this case we wish to have $H_2$, that there exists a cubic graph $G$, which contains a subgraph $H_1$, where after the replacement $H_1 \rightarrow H_2$ there can be found a new pattern $H_1$, which intersects with $H_2$ but has parts outside $H_2$ as well. (Without this property only self similar or frozen (where there is no more pattern which can be changed) graphs can be generated from finite initial graphs.) The pictures show visually the requirement:

After setting some rule, which fulfill these requirements, we have to find an initial graph, apply the rule many times and find an interpretation for the result. It has to be pointed out, that the actual graph structure at a given step can not be observed from an inside point of view. What an inner observer, or a structure can explore is the causal structure which is generated by the replacements. (For details see NKS chapter 9, section 13 ) This is similar to the causal set program.

So a natural way of interpretation of the emerging causal net is that it is a discretization of some kind of spacetime. And local propagating disturbances relative to the overall average structure are particle like excitations, which can have nonlocal connections relative to the average large scale structure. However from AdS/CFT insights it can happen, that we have to interpret particles for example as global structures in a higher dimensional bulk spacetime, which have ends on a boundary-like smaller dimensional surface.

My contribution to the project

During the 3 weeks of 2016 Wolfram summer school I set a framework where the steps of a substitution are precisely defined, and in which the substitutions can be effectively performed even for relatively big graphs. Furthermore I tested a numerical approach to measure the effective dimensionality of the emergent graph structure after sufficiently many steps.

Unfortunately I could not test this framework with rules which could give complex, deterministic behavior, so I could benchmark this machinery on a simple, point to triangle rule, which gives a fractal-like structure. If we interpret this graph as space, then this simple dynamics results a $D=\log(3)/\log(2)=1.58$ dimensional fractal space. Here is a graph of the generated fractal Universe after 100 steps, started from a tetrahedron:

And the neighborhood structure of this space:

Further directions

This project to find deterministic CA description for our Universe is in its infant stage. The framework is more or less set, but it needs tremendous work to investigate possible dynamics and analyze the results of simulations. An outline of a huge project would be the following:

• List the possible rules, which fulfill the non overlapping and non trivial conditions
• Investigate their long term behavior starting from simple initial graphs
• Find quantities and a method of their measurement which can be determined from generated causal graphs
• Find fixed points of the dynamics which preserve long scale dimensionality and possibly other quantities
• List and investigate local disturbances near these fixed points
• After setting an interpretation analyze the particle-like structures (gliders of this dynamics)
• Develop an effective field theory which can describe an effective behavior of the system near to the fixed points
• Match these field theories with the Standard Model of particle physics
• Find out new predictions of the derived effective field theories, which can be tested by measurements

Conclusion

In my project I could set a framework and show a trivial example for a deterministic graph evolution model. During the summer school I was not fortunate enough to find dynamics which produce complex behavior, however to find an appropriate rule seems reachable in the near future. Hopefully a dynamics producing complex topology would be interesting enough to inspire much more people and after some point a serious investigation of the field could be started.

I think personally, that proving or even disproving that this framework to describe Nature can be worked out is an extremely interesting challenge and deserves further theoretical research.

In the end I would like to thank my mentor Todd Rowland, and the whole Wolfram summer school team for the organization and I really hope that there will be a continuation of this project.

Last but not least I thank for all the summer school participants for great discussions and a lifelong experience!

Further comments

I try to collect here some useful comments of my friends and collegues, who kindly read my post, and responded in person:

There is a concept named Digital physics which has a much longer history what I suggested, and probably the earliest pioneer of the field was Konrad Zuse. Fortunatelly his thesis—Calculating Space or “Rechnender Raum”—is now translated into English and has a modern, LaTeX typesetting.

Beside NKS there is another relevant book, which can serve as an extended list of references and valuable material in its own, written by Andrew Ilachinski with the title Cellular Automata A Discrete Universe.

There is an ongoing "mini revolution" in the description of AdS/CFT based on Tensor Networks. The original paper on the topic can be found here.

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