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Representation of the updates in Wolfram Model by linear operators

Posted 6 months ago
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I was reading "Some Quantum Mechanical Properties of the Wolfram Model" by Gorard and technical introduction by Wolfram, and was trying to understand the construction of quantum evolution. I am not sure I am understanding it correctly at all.

At the end of the page 555-beginning of the page 556 Gorard writes:

Given our interpretation of the multiway system as a discrete analog of Some Quantum Mechanical Properties of the Wolfram Model https://doi.org/10.25088/ComplexSystems.29.2.537 (complex) projective Hilbert space, we may interpret rewrite relations as being linear operators $\hat{A}$ acting upon this space: $$ \hat{A}=\sumi{ei \hat{A}_i}$$ for basis vectors $e_i$ , and with each component operator $A_i$ yielding a corresponding eigenvalue $a_i$ when applied to the wavefunction ψ: $$ \hat{A}i\psi=ai \psi$$ where this wavefunction corresponds to a particular “branchlike hypersurface” in the multiway system, and the associated eigenvalue corresponds to the sum of path weights for a particular state or collection of states in some neighboring hypersurface (corresponding to the subsequent evolution step). Interpreting this rewrite relation as corresponding to some observable quantity $A$, it therefore follows that: $$ \hat{A} \psi = a \psi,$$ for some eigenvalue a, which itself corresponds to the measured value of $A$

I am not sure how exactly I should interpret this description. At first, I thought the description was trying to say something like this: branchlike hypersurfaces in the multiway graph correspond to the wave function, and each update corresponds to the application of the quantum operator $\hat{A}$, which multiplies the wavefunction by the number of paths from one hypersurface to the next. Then, when I saw that the commutator of two operators is imaginary, $i\hbar$, I concluded, that I must multiply the number of paths by additional $i \hbar$ (otherwise the commutator is integer). But then it stopped making sense to me again, because $\hbar$ has a specific units, and I would expect evolution operator to be unitless, which means, instead of $\hbar$ one must have a numerical constant. At this point, though, my thinking diverges with texts from Wolfram and Gorard, because none of them mentions any fundamental numerical constants.

I feel at this point I really need a help from someone, who understands realization of quantum evolution in Wolfram models. What is wrong and what is correct with my understanding?

As another example, we can also look at the Wolfram's description of quantum evolution (somewhat easier to interpret than Gorard's one):

But now we can make a potential identification with standard quantum formalism: we suppose that the Lagrangian density ℒ corresponds to the total flux in all directions (or, in other words, the divergence) of causal edges at each point in multiway space.

Unfortunately, there is no clear indication how one would exactly calculate an action from such description. It seems natural to think that the density ℒ is a property of nodes, and action corresponding to a path is a sum of ℒ over all nodes along the path. There is no direct confirmation of such interpretation in the texts, so I would like to get a confirmation that the interpretation is correct. And there is still a question of the arbitrary numerical constant that should be assigned to each path. Should I assume it to be equal to 1?

13 Replies

Regarding the presence of dimensional constants in the theory, you can find some ideas in the technical introduction: Units and Scales.

Unfortunately, I cannot help you with your other questions, because I am still trying to understand the other papers on general relativity (which I think that pedagogically precede the one you mentioned).

Posted 6 months ago

Thank you for the link. It was quite helpful in the sense that it clarified that Wolfram considers a single link in the multiway system to contribute about $10^{-116}$ to the phase of the path integral. We could assume this is the fundamental constant of "the" Wolfram model, which represents our universe.

Concerning the clarification of the paragraph:

But now we can make a potential identification with standard quantum formalism: we suppose that the Lagrangian density ℒ corresponds to the total flux in all directions (or, in other words, the divergence) of causal edges at each point in multiway space.

Complemented by:

By the way, it’s worth mentioning what a “flux of causal edges.” corresponds to. Each causal edge represents a causal connection between events that are, in a sense “carried” by some element in the underlying hypergraph (the “spatial hypergraph”). So a “flux of causal edges” is in effect the communication of activity (i.e. events), either in time (i.e. through spacelike hypersurfaces) or in space. (i.e. through timelike hypersurfaces). And at least in some approximation we can then say that energy is associated with activity in the hypergraph that propagates information through time, while momentum is associated with the activity that propagates information in space.

Reference: Finally We May Have a Path to the Fundamental Theory of Physics

Notice that S. Wolfram is proposing a more fundamental framework than the usual framework of energy and momentum. The idea is not to compute energy, momentum and the Lagrangians from the Wolfram Model and then go back to the mainstream physics, but to use the Wolfram Model to explain physical phenomena without using the notions from the previous paradigm: energy, momentum, Lagrangian, etc. The association between energy and activity in the hypergraph that propagates information through time is a hint for substituting energy in the explanations by the role of the activity in the hypergraph that propagates information through time.

For example, imagine an explanation of some property of a black hole in terms of energy. This explanation should be easy to translate to an explanation in the Wolfram Model using the activity in the hypergraph that propagates information through time without any need to compute the energy for the explanation to be consistent. Of course, at some point, the activity in the hypergraph that propagates information through time should explain the results of the measurements of energy.

The obvious question is: What is the problem with energy, momentum, etc? Well, these are real numbers and maybe real numbers are not the best way to describe nature at its most fundamental level. Maybe a real number is too reductive. To use discrete relations rather than real numbers is one of the starting points of the Wolfram Model.

Here is a reflection about the role of explanations in physics due to David Deutsch.

Posted 6 months ago

The first quote you give is also by Wolfram (not by Gorard) from his technical introduction. Almost all of the phenomena we know comply with quantum mechanics, so it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree. I am talking here about the general structure of quantum mechanics, not some specific examples of particular fields or Schrodinger equation (which also would can be useful).

Here is a reference to the reproduction of a part of categorical quantum mechanics in the Wolfram Model: the ZX calculus.

I am not sure I see how the paper really helps with the questions in the original post. Could you elaborate, please (I am not particularly knowledgeable in ZX calculus or categorical quantum information theory, so would be nice if you could make your point more or less self-contained)?

The ZX paper is meant to answer the requirement

it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree

ZX calculus is an important part of quantum mechanics. The adjective categorical is not about the physics, but about the mathematical framework in which quantum mechanics is expressed. Of course, ZX calculus is not the end of the story, it is just the beginning.

From what I can see, ZX calculus is a particular way of doing quantum computing. I am not sure though, what are the conclusions of the ZX paper about the mapping between ZX calculus and Wolfram model. The paper says in the conclusion:

[We] Demonstrated that the diagrammatic rewriting formalism of the ZX-calculus can indeed be embedded and realized within the more general formalism of Wolfram model multiway operator systems, using a novel reformulation of the Wolfram model in terms of double-pushout rewriting systems and adhesive categories.

I do not understand this. ZX calculus is "a graphical language for reasoning about linear maps between qubits", while Wolfram Model is a class of algorithms for updating graphs/strings/etc (with some physical meaning attached to the procedure). What does it even mean, that a language (ZX calculus) is imbedded into the model? Does it mean that "some "words" of the language are realized sometimes in some Wolfram models"? Or does it mean that we can "translate" ZX-calculus language into something made of Wolfram notations? To me, saying "ZX-calculus can be embedded and realized within the more general formalism of Wolfram model", is like saying "calculus can be realized and embedded within a more general formalism of electromagnetism" (in short, nonsense).

ZX calculus is a particular way of doing quantum computing.

Yes, any quantum circuit can be decomposed as spiders in ZX calculus. I agree with the possibility that you proposed:

it mean that we can "translate" ZX-calculus language into something made of Wolfram notation [I would prefer to use the word "framework" rather than "notation"]

Concerning

"calculus can be realized and embedded within a more general formalism of electromagnetism" (in short, nonsense).

I agree that differential and integral calculus can be realized and embedded in the general formalism of electromagnetism, but this is circular rather than nonsense, because the formalism of electromagnetism, at least in mainstream physics, is based on calculus. If someone proposes a formalism of electromagnetism that is not based on calculus, it will be nontrivial whether calculus can be embedded in some way in this formalism.

What is relevant of the embedding of ZX calculus into the Wolfram Model is that it corresponds to the interpretation of quantum mechanics in this model, i.e., it is a confirmation that the interpretation of quantum mechanics in the Wolfram Model is not just an isolated idea, but it is related to existing parts of quantum mechanics, namely, the ZX calculus. Here are two lectures of Jonathan concerning this embedding of ZX calculus into the Wolfram Model. The audience are experts in categorical quantum mechanics: Cambridge, Oxford.

I would prefer to use the word "framework" rather than "notation"

What is "framework"? By "translating into Wolfram's notation", I meant just enriching words and notations of Wolfram Models to include definitions from ZX calculus.

I agree that differential and integral calculus can be realized and embedded in the general formalism of electromagnetism

What do you mean by this? Like, the whole phrase is just a non-statement for me... basically, just a collection of words (no intend to insult here, the phrase really doesn't make any sense to me).

What is "framework"? By "translating into Wolfram's notation", I meant just enriching words and notations of Wolfram Models to include definitions from ZX calculus.

I prefer to use the word framework, rather than notation, because a framework is conceptual (interaction among several concepts), whereas a notation is just a way to write something.

What do you mean by this? (Me: I agree that differential and integral calculus can be realized and embedded in the general formalism of electromagnetism)

Imagine an intelligent extraterrestrial civilization where the formalism of electromagnetism was discovered before than differential and integral calculus because these people are more related to electromagnetism than to mechanics for some historical reason. Then they visit Earth and they find a textbook of differential and integral calculus. In order to understand this textbook, they will express calculus using their formalism of electromagnetism. For example, in order to understand analytic functions of a complex variable they may say: "people from Earth call analytic function what we call static electric fields in a region of the plane containing no electric charge".

In this sense, calculus can be developed from the formalism of electromagnetism, but for historical reasons, it is done the other way around: electromagnetism is developed using calculus. This is the reason why Bob Coecke used the phrase When worlds collide...in a good way! (his world of categorical quantum mechanics and the world of the Wolfram Model). In the example above, the worlds are the formalism of electromagnetism and the world of calculus.

"people from Earth call analytic function what we call static electric fields in a region of the plane containing no electric charge"

This sounds like they are just using different words for the same thing. They would still need to write partial differential equation for the field. Fields, functions and differential equations are more general things than electric fields, and Maxwell equations. You can explain second through the first, but not the other way around.

Anyway, I don't know what we are arguing about (and I don't want to argue about it). Have you read the paper? Can you explain, what Gorard does there? Does it help with the questions in this post? You are sending references with 100 pages in each, without explaining the context or how the references are related to the questions in this thread, or where I should even look. It doesn't help. I asked specific questions and I need specific directions. If you don't know the answer to the questions, just say it (or don't say anything at all). If you know there are no answers yet, just say the questions were not worked out yet. If you know the information is contained in a specific source, and it answers my questions, then tell where I can find it. If you have a source, which you suspect may be helpful, but you have no idea if it actually is, then say directly that you have no idea if the source is of help because you haven't read it yet. Please, don't waste my and your time. Just give the useful information, suggestions, clarifications. I asked questions to clarify technical details of Wolfram model. Do you have anything specific to say about the questions?

Just to summarize, I answered your question:

it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree

with the reference of the Wolfram Model reproducing ZX calculus (a part of quantum mechanics). The other parts are work-in-progress. Concerning your other question

Representation of the updates in Wolfram Model by linear operators

I think that it is the other way around: the linear operators from quantum mechanics correspond to something in the Wolfram Model, but it is not that something in the Wolfram Model should be expressed as linear operators. I disagree with Jonathan's point of view (I am not saying that he is wrong, I just say that my conceptualization is different from his conceptualization)

we may interpret rewrite relations as being linear operators \hat{A} acting upon this space:

I prefer the other way around: the linear operator should be interpreted in the framework of the Wolfram model. The interpretation of the updates in the Wolfram Model as a linear operator is -in my opinion- a return to the previous paradigm.

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