# Thread: Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?

1. Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk MERW.

But Boltzmann path ensemble has also spatial realization: 1D Ising model and its generalizations: Boltzmann distribution among spatial sequences of spins or some more complicated objects.

For E_uv energy of interaction between u and v neighboring spins or something more general, define M_uv = exp(-beta E_uv) as transition matrix and find its dominant eigenvalue/vector: M psi = lambda psi for maximal |lambda|. Now it is easy to find (e.g. derived here) that probability distribution of one and two neighboring values inside such sequence are:

Pr(u) = (psi_u)^2
Pr(u,v) = psi_u (M_uv / lambda) psi_v

The former resembles QM Born rule, the latter TSVF  the two ending psi come from propagators from both infinities as M^p ~ lambda^p psi psi^T for unique dominant eigenvalue thanks to Frobenius-Perron theorem. We nicely see this Born rule coming from symmetry here: spatial in Ising, time in MERW.

Having Ising-like models as spatial realization of Boltzmann path integrals getting Born rule from symmetry, maybe we could construct Bell violation example with it?

Here is MERW construction (page 9 here) for violation of Mermins Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1 inequality for 3 binary variables ABC, intuitively tossing 3 coins, at least 2 are equal (e.g. here is QM violation): From Ising perspective, we need 1D lattice of 3 spins with constraints  allowing neighbors only accordingly to blue edges in above diagram, or some other e.g. just forbidding |000> and |111>.

Measurement of AB spins is defect in this lattice as above  fixing only the measured values. Assuming uniform probability distribution among all possible sequences, the red boxes have correspondingly 1/10, 4/10, 4/10, 1/10 probabilities  leading to Pr(A=B) + Pr(A=C) + Pr(B=C) = 0.6 violation.

Could this kind of spin lattice construction be realized?

What types of constraints/interaction in spin lattices can be realized?

While Ising-like models provide spatial realization of Boltzmann path integrals, is there spatial realization of Feynman path integrals?

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Update: Born-like formulas from symmetry in Ising model (Boltzmann sequence ensemble): Pr(i)=(ψ_i)^2 where one amplitude ("hidden variable") comes from left, second from right:   2.

3. Another paper with Bell violation using Ising model: https://journals.aps.org/prl/abstrac...ett.123.170604  4. Construction of Bell violation can be extended further to quantum-like computers realized as Ising model: e.g. by somehow printing on a surface conditions for solving a given problem by Ising's: Boltzmann distribution among possible sequences.

Such Wick-rotated quantum gates seem a bit weaker computationally, but spatial realization allows to fix amplitudes from both directions: left and right, what seems(?) to allow to quickly solve NP-complete problems like 3-SAT (end of https://arxiv.org/pdf/1912.13300 ):   5. Let me ask a related question: do Feynman path integrals satisfy Bell locality assumption?

There are generally two basic ways to solve physics models:

1) Asymmetric, e.g. Euler-Lagrange equation in CM, Schrödinger equation in QM
2) Symmetric, e.g. the least action principle in CM, Feynman path integrals in QM, Feynman diagrams in QFT.

Having solution found with 1) or 2), we can transform it into the second, but generally solutions originally found using 1) or 2) seem to have a bit different properties - for example regarding "hidden variables" in Bell theorem.
The asymmetric ones 1) like Schrödinger equation usually satisfy assumptions used to derive Bell inequality, which is violated by physics - what is seen as contradiction of local realistic "hidden variables" models. Does it also concern the symmetric ones 2)?

We successfully use classical field theories like electromagnetism or general relativity, which assume existence of objective state of their field - how does this field differ from local realistic "hidden variables"?

Wanting to resolve this issue, there are e.g. trials to undermine the locality assumption by proposing faster-than-light communication, but these classical field theories don't allow for that.

So I would like to ask about another way to dissatisfy Bell's locality assumption: there is general belief that physics is CPT-symmetric, so maybe it solves its equations in symmetric ways 2) like through Feynman path integrals?

Good intuitions for solving in symmetric way provides Ising model, where asking about probability distribution inside such Boltzmann sequence ensemble, we mathematically get Pr(u)=(psi_u)^2, where one amplitude comes from left, second from right, such Born rule allows for Bell-violation construction. Instead of single "hidden variable", due to symmetry we have two: from both directions.

From perspective of e.g. general relativity, we usually solve it through Einstein's equation, which is symmetric - spacetime is kind of "4D jello" there, satisfying this this local condition for intrinsic curvature. It seems tough (?) to solve it in asymmetric way like through Euler-Lagrange, what would require to "unroll" spacetime.

Assuming physics solves its equations in symmetric way, e.g. QM with Feynman path integrals instead of Schrödinger equation, do Bell's assumptions hold - are local realistic "hidden variables" still disproven?  6. Originally Posted by Jarek Duda do Feynman path integrals satisfy Bell locality assumption?
Yes.

Even though there are thoughts it should be extended to non-locality, the advantage of this formalism is that it can explain current double slit experiments without the need for non-locality. Originally Posted by Jarek Duda QM with Feynman path integrals instead of Schrödinger equation, do Bell's assumptions hold - are local realistic "hidden variables" still disproven?
Yes.

QM path integral formulation restores particle's trajectory and it is extended in the Sum-Over-Histories approach to Quantum Mechanics. Considering Bell's inequalities this sort of QM interpretation assumes locality, but the world is still probabilistically indeterministic like standard QM (for instance Copenhagen interpretation).

Unlike Copenhagen interpretation it restores reality of particle's trajectory but this restoration is still probabilistic in nature i.e. not predictable. There is no ignorance. This interpretation suffers from the time causality formalism violation because backward time paths are still allowed (see ref. https://www.perimeterinstitute.ca/pe...rs/63.eprb.pdf).

It should be noted that some interpretations of QM which even stronger are trying to restore classicality have failed. Consider Bohmian mechanics. Bohmian mechanics is deterministic, it also restores particle's trajectory but it is contextual. Bohmian mechanics does not restore real values of all observables before measurement. Bohm and Hiley wrote (1993):

"The context dependence of results of measurements is a further indication of how our interpretation does not imply a simple return to the basic principles of classical physics." Originally Posted by Jarek Duda are local realistic "hidden variables" still disproven?
Non-contextual/realistic hidden variables as wanted by EPR in quantum world are forbidden by KochenSpecker no-go theorem.

Sincerely,
Zlatan  7. Sorry, I have just noticed the response.

I think you meant solving Feynman path integral by considering paths up to the measurement moment?
No, solving physics in symmetric way does not emphasize any moment in time (e.g. of a given measurement).

In contrast, as for asking about situation inside a membrane, Ising model ... spacetime in GR, the current situation e.g. of measurement is between the boundary conditions.
Wanting to get its situation, you need to use propagators from both direction, have two separate "hidden variables", like in TSVF: https://en.wikipedia.org/wiki/Two-st...ctor_formalism

Please think about probability distribution inside Ising model - you will see that instead of single probability distribution on Omega (leading to inequalities violated by physics), you have two amplitudes on Omega governed by Born rule, Pr(i) =(psi_i)^2. Consequences of such ensembles operating on paths are well seen in looking trivial question: what stationary probability distribution in [0,1] should we expect?
Any diffusion/chaos say uniform rho=1 ... but QM says localized: rho~sin^2.
Assuming uniform distribution of paths up to a given moment e.g. of measurement, you would get rho~sin instead.
To get the proper rho~sin^2, you need to use ensemble of complete paths - measurement is inside them - diagram above.

ps. all the diagrams are from https://www.dropbox.com/s/m1m8uq0gygo2lzt/Ising.pdf  8. Originally Posted by Jarek Duda No, solving physics in symmetric way does not emphasize any moment in time (e.g. of a given measurement).
The extension of Feynman Path Integral as the interpretation of QM is not complete, so your stressing on symmetry does have justification. I have not read and not seen explanation (within the framework) of the Bohm version of EPR paradox which includes measurements on x and z spin as incompatible observables by uncertainty principle. I do not see how your emphasising on time symmetry has benefit. I would rather accept explanation that we have epistemic fault in our gene code to understand quantum world than to invoke any kind of spooky action in time's future. It is just my opinion.

Zlatan  9. Indeed, while Feynman path integrals are generally said to be equivalent with QM, there are some lacks regarding measurements here.
Above in #3 here, there is diagram for Shor's algorithm from perspective of Feynman path ensemble, but it uses paths which start in state preparation, end in measurements.

The question is how to define measurement inside such paths in ensemble.
It should have Born rule: add amplitudes over unmeasured variables, then multiply them to get probabilities.

I have thought about this question from perspective of analogous (Wick-rotated) but simpler: Boltzmann path ensemble, realized e.g. in Ising model.
A longer explanation is in page 9 of https://arxiv.org/pdf/0910.2724 ... looking at Stern-Gerlach experiment as idealization of measurement, its strong magnetic field does not allow to modify once chosen spin alignment.
We can use it as a general rule - during measurement: the measured variables are fixed, the unmeasured variables can freely change.

This rule can be taken to e.g. Ising's Boltzmann path ensemble - here is the 3 spins Bell violation diagram where we
- measure A and B spins: matrix I fixes their values,
- not measure C: matrix X allows it to freely change.
This way in path ensembles we get the Born rule: add amplitudes over unmeasured, the multiply to get probabilities. Jarek  Bookmarks
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