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How to calculate path weights/complex amplitudes?

Posted 2 months ago
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According to the physical interpretation of the Wolfram model given by Wolfram and Gorard, the evolution of a multiway graph represents the evolution of a quantum system. The different paths that a system may take are to be summed together in some way similar to a Feynman path integral.

From Gorard's Some Quantum Mechanical Properties of the Wolfram Model:

> Such an interpretation brings forth strong connotations of the path integral formulation of quantum mechanics, in which the overall trajectory of a quantum system is taken to be described by a sum (or, more properly, a functional integral) over all possible trajectories, weighted by their respective amplitudes

or

[A branchlike hypersurface] may be considered to be a linear superposition of the basis eigenstates of the multiway system

My question is: how do I calculate the complex amplitude/path weight of a particular state in a multiway system?

The simplest answer that comes to mind is: just sum the number of ways you can get to a particular state. This is what Gorard has apparently been doing in his bulletin about the double-slit experiment. The problem with this approach is that the number of ways you can get to a particular state is a natural number. The amplitudes will be positive integer (or rational if you normalize the state) numbers, while in quantum mechanics

  1. amplitudes are in general complex numbers
  2. amplitudes contain often irrational or transcendental numbers
  3. amplitudes can also be negative. Destructive interference is not possible if the amplitudes will only add

In fact, the interference pattern that Gorard obtains in the bulletin is only due to the peculiar choice of a sorting algorithm for the strings, as explained by Matt Kelly in a comment.

The question remains. I couldn't find a satisfactory answer in Gorard's paper either:

these amplitudes are concretely specified by a sum of the incoming path weights for the associated vertex in themultiway graph, where these path weights are computed using the discrete multiway norm, as defined below

Therefore he seems to be using the discrete multiway norm, that he defines later at p.27. Basically he defines a multidimensional lattice (with a time axis) and embeds the multiway graph in the lattice. Then he calculates the Minkowski norm between the points in the lattice.

Unfortunately this norm is totally dependent on the particular choice of embedding (as discussed in this post), and therefore cannot be used to calculate amplitudes (which of course should not depend on the embedding)

Some months ago I was discussing the issue of path weights with Pavlo Bulanchuk and José Manuel Rodríguez Caballero in Quantum random walk, but we couldn't converge to a definite answer. It has been 5 months now. If anyone has found some insight on the issue it would be much appreciated.

I think that understanding how to calculate path weights would be a critical step in the comprehension of quantum mechanics in the context of the Wolfram model. I would also be satisfied is someone could tell me: "No-one has figured this out yet", but this would mean that our present understanding of the model is way behind Wolfram's claims in the One-Year Update.

Hello, there are different approaches to the Wolfram Physics Project. In my case, I am developing the idea that quantum states are just undecidable propositions and the measurement relative to a base is the addition of a new axiom to the observer's worldview. The inspiration of this version can be found in Wolfram's Consciousness Scheme. Why are there "negative probabilities" in this framework? I suppose the answer is given by the fact that the interference is like a contradiction in mathematics. An observer cannot have a contradiction in his worldview because he loses consciousness. Notice that we are conscious beings. Hence, we will see quantum mechanics as a consequence of the anthropic principle: in places where quantum mechanics is violated, some contradictions prevent the emergence of consciousness. I am not claiming that this is a scientific discovery; this is just an idea that may motivate some reformulation of quantum mechanics using a formalized version of consciousness. However, I need more time to do the math and derive the Schrödinger equation. I know that some people have used the connection between quantum mechanics and consciousness to justify pseudoscientific claims. Nevertheless, they do not have a formal definition of consciousness. After having a formal definition, this is no longer pseudoscience, but just mathematics and the claims are logical consequences of the axioms. Although Wolfram's Consciousness Scheme is not presented as an axiomatic system, it can likely be formalized in the framework of complexity theory. Furthermore, maybe the deductive method of the observer is not classical logic but the so-called linear logic or some variation. Conclusion: I do not know how to calculate path weights/complex amplitude, but I guess they are a consequence of the consistency of the axiomatic system of the observer. I hope to derive this result in the future. Maybe tomorrow, I will change my mind and follow another approach: this is science, not a dogma ;-)

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