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?