Do Feynman path integrals satisfy Bell locality assumption?

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Do Feynman path integrals satisfy Bell locality assumption?

Postby Jarek » Sat Feb 15, 2020 5:07 am

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?
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby JohnDuffield » Sat Feb 15, 2020 9:43 am

Space is the "jello", Jarek. Not spacetime. Spacetime is something that models space at all times, and so is static. You can draw a worldline in it, but that worldline plots your motion through space over time. It doesn't plot your motion through spacetime. There is no motion in spacetime. Which means light doesn't curve because spacetime is curved. Instead it curves because space is "neither homogeneous nor isotropic". Light curves downwards in a gravitational field rather like sonar waves curve downwards in the sea:

Image

So local intrinsic curvature is nothing to do with gravity. However electromagnetism is. And that means local realistic "hidden variables" are not disproven.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Sat Feb 15, 2020 10:06 am

John, sure space is kind of "jello" from many perspectives e.g. EM field, materials ... including jello ;-)

But general relativity shows that time and space are quite similar, allowing to see the entire spacetime as kind of "4D jello" (sure: static) - satisfying local condition for tension: Einstein's equation determining intrinsic curvature from stress-energy tensor.

Local realistic theories are disproven for models solved in asymmetric way like Euler-Lagrange or Schrodinger, they have "hidden variable": state in the past.
But solving models in symmetric way like the least action principle, Feynman path/diagram ensembles, "hidden variables" are symmetric - in both past and future (nicely seen e.g. in Ising model). It has very different type of locality (as in static "4D jello") - does not have to satisfy such inequalities.
Nice animation about such symmetric 4D locality by Ojvind Bernander: https://www.youtube.com/watch?v=P-TgKIunUf0
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby minkwe » Sat Feb 22, 2020 10:58 am

Do Feynman path integrals satisfy Bell locality assumption?


This question is similar to asking: "Do wavefunctions smell like lemons"?

Bell's locality condition is an ontological condition that states that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past.

Feynman path integrals by themselves are mathematical constructions devoid of ontology. The two concepts do not mix. To ask such a question, you first have to state the ontology you ascribe to the path integrals and then the valid question would be if that ontology satisfies Bell's locality assumption. In some ontologies, the answer will be yes, and in others the answer will be no. But the main point is that you must relate the mathematical entities to a proper ontology before it even makes sense to start asking such questions.

https://arxiv.org/pdf/1805.08583.pdf
https://arxiv.org/pdf/1805.08583.pdf wrote:quantum theory usually start with a brief historical account of the experiments that were crucial for the development of the theory
to make plausible the postulates which define the mathematical framework [8, 12–15]. Subsequent efforts then go into mastering
linear algebra in Hilbert space, solving partial differential equations, and other abstract mathematical tools. The tendency to
focus on the elegant mathematical formalism [5], which, unfortunately, is far more detached from everyday experience than for
instance Newtonian mechanics or electrodynamics, promotes the “shut-up-and-calculate” approach [16]. Hermitian operators,
wave functions, and Hilbert spaces are conceptual, mental constructs which have no tangible counterpart in the world as we
experience it through our senses. The mathematical results that are derived from the postulates of a theoretical model are only
theorems within the axiomatic framework of that theoretical model. Theoretical physics uses axiomatic frameworks which have
a rich mathematical structure, allowing the proof of theorems. For instance, the Banach-Tarsky paradox [17] has no counterpart
in the world that humans experience. Taking the mathematical description for real is like opening bottles that contain very exotic
and sometimes magical substances. In other words, relating theorems derived within a mathematical axiomatic formalism to
observable reality is not a trivial matter
.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Sun Feb 23, 2020 12:52 am

minkwe,
This is not about human philosophy, but asking how nature objectively works - what in physics we decompose into mathematics.

There is commonly and successfully used Schrodinger equation to describe e.g. atoms, starting with properly predicting their energy levels.
It has alternative formulation by Feynman path integrals: https://en.wikipedia.org/wiki/Path_integral_formulation

While we can translate between their solutions, the solutions originally found with each of them have slightly different properties - the question is which formulation describes better what nature does?
Schrodinger usually has boundary conditions in the past and evolve them toward future ... but we could alternative use 'minus t' and evolve them toward past, unitary evolution allows for both.
Path integrals have boundary conditions in two past and future instead - we consider ensemble of all paths between them. It is symmetric way - not trying to enforce any time direction.

Bell theorem is for perspective of evolving state as in Schrodinger equation - assumed "hidden variables" would be hidden in some evolving state of the universe.

But what if (CPT symmetric) physics solves nature in symmetric way like path integrals - are local realistic "hidden variables" still forbidden in this case?
If they are, we have a big problem as e.g. geometry of spacetime in general relativity is kind of local realistic "hidden variable".

Please think about probability distribution of values inside Ising model (derived below) - basic condensed matter model, just Boltzmann distribution of sequences, analogous math as in Feynman path integrals.
Due to symmetry, we don't have probability distribution on Omega - leading to inequalities violated by physics, but we have two separate amplitudes on Omega - to be added then multiplied, exactly as in QM as it is the same mathematics.
There is not one "hidden variable" as in Bell, but two separate (psi) due to symmetry:

Image
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby gill1109 » Sun Feb 23, 2020 1:57 am

Jarek wrote:But what if (CPT symmetric) physics solves nature in symmetric way like path integrals - are local realistic "hidden variables" still forbidden in this case?
If they are, we have a big problem as e.g. geometry of spacetime in general relativity is kind of local realistic "hidden variable".

Bell's theorem says that QM is incompatible with locality+realism+no-conspiracy. If CPT symmetric physics reproduces the predictions of QM, and if it is local-realistic, then it violates no-conspiracy.

What's the problem? If it is helpful in solving mathematical physical problems, then use it.

I highly recommend the very intelligent and thoughtful discussion given by Boris Tsirelson in http://en.citizendium.org/wiki/Entanglement_(physics).

Let me also point out that there is also the possibility of admitting non-locality. See Ilja Schmelzer's detailed analysis. https://www.ilja-schmelzer.de/
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Sun Feb 23, 2020 2:53 am

Richard,
sure, we agree that solutions found in symmetric way do not satisfy Bell's assumptions - we can call it as dissatisfying "no conspiracy" assumption.
But this is a subjective human assumption - let's try to translate it to something more universal - such that "hypothetical aliens would agree".

One of the most fundamental questions of physics is does nature solve such models in symmetric way? - this is yes or no question, hypothetical aliens should have the same answer.
If no, all Lagrangian formalism leads to inequalities violated by physics - contradiction.
If yes, we agree that assumptions of Bell theorem are not satisfied - we don't get this contradiction.

So why can't we just answer "yes" and say that the Bell theorem issue is resolved?
Beside yes/no answers, is there a third option?
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby gill1109 » Sun Feb 23, 2020 3:06 am

Jarek wrote:Richard,
sure, we agree that solutions found in symmetric way do not satisfy Bell's assumptions - we can call it as dissatisfying "no conspiracy" assumption.
But this is a subjective human assumption - let's try to translate it to something more universal - such that "hypothetical aliens would agree".
One of the most fundamental questions of physics is does nature solve such models in symmetric way? - this is yes or no question, hypothetical aliens should have the same answer.
If no, all Lagrangian formalism leads to inequalities violated by physics - contradiction.
If yes, we agree that assumptions of Bell theorem are not satisfied - we don't get this contradiction.
So why can't we just answer "yes" and say that the Bell theorem issue is resolved?
Beside yes/no answers, is there a third option?

"No-conspiracy" is not a subjective human assumption. It is a mathematical property which aliens could also formulate. Bell's theorem, as I see it, is a true mathematical theorem about mathematical models.
Nature does not solve models. Nature just is. Mathematical models describe nature.
Non-locality is alive and well, even if not mainstream. I think we need to keep an open mind, and I think that experiments which actually test predictions of various (non-local) collapse theories are very interesting. I think that your approach is mathematically very interesting and I want to learn more about it. But life is short and I have many interests as well as many obligations. Discussing quantum foundations on internet fora like this is how I cope with difficult times in "real life".
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Sun Feb 23, 2020 3:48 am

Nature does not solve models. Nature just is. Mathematical models describe nature.

Indeed nature just it, but the goal of physics is trying to systematize the rules according to which "nature works" - and generally this trial turns out surprisingly successful ...
... leading to models in various scales, which still leave freedom of solving them in asymmetric way (like Euler-Lagrange, Schrodinger) or symmetric (like least action, path/diagram ensembles).
So which are more appropriate to model what nature does?
Asymmetric leading to inequalities violated by nature, or maybe symmetric which don't lead to this contradiction?
Non-locality is alive and well

Sure, but Lagrangian formalism we successfully use has finite propagation speed - trying to conclude faster-than-light communication from Bell theorem we still have a problem.
This problem disappears if solving in symmetric way, like QFT by Feynman diagrams - by not satisfying "no conspiracy".

To summarize, having in mind Lagrangian formalism like general relativity, to solve the Bell theorem locality issue:
faster-than-light communication NO - Lagrangian formalism has finite propagation speed,
accepting symmetry YES - in the basic ways used to solve e.g. general relativity or QFT.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby minkwe » Sun Feb 23, 2020 6:47 am

Jarek wrote:minkwe,
This is not about human philosophy, but asking how nature objectively works - what in physics we decompose into mathematics.

That is exactly the domain of philosophy of science. You won't penetrate that question without dealing with the distinction between mathematical tricks and physical reality.

You keep reminding me about different mathematical tricks that work, but refuse to address the key fact that they are just mathematical tricks. Nobody is arguing that those tricks do not work.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby minkwe » Sun Feb 23, 2020 7:03 am

Jarek wrote:
Nature does not solve models. Nature just is. Mathematical models describe nature.

Indeed nature just it, but the goal of physics is trying to systematize the rules according to which "nature works" - and generally this trial turns out surprisingly successful ...
... leading to models in various scales, which still leave freedom of solving them in asymmetric way (like Euler-Lagrange, Schrodinger) or symmetric (like least action, path/diagram ensembles).
So which are more appropriate to model what nature does?
.

This is the Mind Projection Fallacy at work. The rules you come up with are for your understanding and predictions. They help you to organise human information in order to obtain correct answei. They are not "how nature works". If anything, they may be how nature appears to work. Making the link is nontrivial.

BTW, you didn't answer how the moving particle knows which path has the least action?
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Sun Feb 23, 2020 7:29 am

While the general view is that there has basically remained only QM+GR unification for complete description of nature, I agree that this is more complicated.

Anyway, still these working "mathematical tricks" like EM, GR, QFT, QM carry concrete pictures on nature ... and still require answering the fundamental question like if we should imagine that nature solves them in asymmetric (disproven by Bell) or symmetric way - so what is the answer?

BTW, you didn't answer how the moving particle knows which path has the least action?

I have responded in the second thread regarding path ensemble:
Great question - let's look at it from perspective of Ising model, in which mathematically we assume that physics "tries out all possible" sequences/configurations - weighting them in Botlzmann instead of Feynman way.
So this is statistical mechanics - in reality physics randomly perturbs the configuration space, leading to Boltzmann ensemble as the safest/statistically dominant for fixed energy - due to mathematically universal (also for aliens) Jaynes maximal uncertainty principle: https://en.wikipedia.org/wiki/Principle ... um_entropy

So Boltzmann sequence ensemble in space is effective statistical/mathematical description of some complex behavior.
QM is Feynman path ensemble in time - according to general relativity, we live in spacetime, could transform between space and time e.g. below black hole horizon ... so maybe this is again just analogous effective statistical/mathematical description of some complex behavior.

I am definitely not saying that path ensembles are fundamental description, only that Bell theorem has ruled out asymmetric ways of solving local realistic models like general relativity, leaving the symmetric ways, like: the least action principle, Einstein's equations, path/diagram ensembles, TSVF, meeting of propagators from both time directions.

We don't even try to solve general relativity in asymmetric way like Euler-Lagrange: "unrolling" evolving geometry of spacetime sounds like a nonsense.
Instead, we treat spacetime as static "4D jello" - satisfying local intrinsic curvature (tension) Einestein's equation - solving it in symmetric way.

Regarding the least action principle, we live in a spacetime - standard "3D jello" finds configuration minimizing energy/tension.
Particles are their trajectories in spacetime, together with surrounding they form kind of static "4D jello" - this time minimizing tension as action.

We imagine that space and time are very different, but GR shows that they are nearly the same - spatial construct like jello, Ising model should have temporal analogs, and they have: like action optimization, Feynman path integrals.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby gill1109 » Mon Feb 24, 2020 9:50 pm

Jarek wrote:So this is statistical mechanics - in reality physics randomly perturbs the configuration space, leading to Boltzmann ensemble as the safest/statistically dominant for fixed energy - due to mathematically universal (also for aliens) Jaynes maximal uncertainty principle: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy

The Wikipedia page on Jaynes' maximal uncertainty principle makes it clear that the "principle" is not universally agreed by human scientists. About alien scientists, I wouldn't know. There is no agreement at all as to its formulation. There are attempts to derive it from combinatorial arguments which seem to me to be circular. https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Tue Feb 25, 2020 1:05 am

Richard,
to understand the maximum entropy principle, its universality, let's look at the simplest case/question: there is length n sequence of 0/1, what its percentage of '1' is the safest assumption?
The number of sequences with 'p' percentage of '1' is
binomial(n,pn) ~ exp(n * h(p))
for h(p) = -p ln p - (1-p) ln(1-p) Shannon entropy, this is asymptotic behavior (heart of information theory) derived e.g. using https://en.wikipedia.org/wiki/Stirling% ... roximation

So assuming uniform probability distribution among sequences, the safest assumption is maximizing entropy p=1/2 due to being in exponent - asymptotically subset described by 'p' completely dominates the entire population.

Generally, we use Boltzmann entropy H=log(|Omega|), e.g. for combinations above, now we split Omega into subsets corresponding to various statistical parameters like 'p' above - entropy maximization means focusing focusing on subset described by parameters which allow it to asymptotically dominate entire Omega, like for https://en.wikipedia.org/wiki/Typical_set
Not knowing anything more, the safest assumption are parameters maximizing entropy.

I believe hypothetical advanced aliens would also notice this universal mathematics, which is at heart of statistical physics, e.g. in properly predicting spectrum of stars: https://en.wikipedia.org/wiki/Planck%27s_law
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby gill1109 » Thu Feb 27, 2020 4:40 am

Jarek wrote:Richard,
to understand the maximum entropy principle, its universality, let's look at the simplest case/question: there is length n sequence of 0/1, what its percentage of '1' is the safest assumption?
The number of sequences with 'p' percentage of '1' is
binomial(n,pn) ~ exp(n * h(p))
for h(p) = -p ln p - (1-p) ln(1-p) Shannon entropy, this is asymptotic behavior (heart of information theory) derived e.g. using https://en.wikipedia.org/wiki/Stirling% ... roximation

So assuming uniform probability distribution among sequences, the safest assumption is maximizing entropy p=1/2 due to being in exponent - asymptotically subset described by 'p' completely dominates the entire population.

Generally, we use Boltzmann entropy H=log(|Omega|), e.g. for combinations above, now we split Omega into subsets corresponding to various statistical parameters like 'p' above - entropy maximization means focusing focusing on subset described by parameters which allow it to asymptotically dominate entire Omega, like for https://en.wikipedia.org/wiki/Typical_set
Not knowing anything more, the safest assumption are parameters maximizing entropy.

I believe hypothetical advanced aliens would also notice this universal mathematics, which is at heart of statistical physics, e.g. in properly predicting spectrum of stars: https://en.wikipedia.org/wiki/Planck%27s_law

The "safest assumption" is the assumption which costs you least harm. That has got nothing at all to do with physics! It is about protecting an agent from making poor predictions, in a minimax sense. You want to minimise the worst damage which could occur. Are the laws of physics set up so that agents like us are least exposed to risk when predicting the future? I doubt it.
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Re: Do Feynman path integrals satisfy Bell locality assumpti

Postby Jarek » Thu Feb 27, 2020 5:00 am

Richard, you are supposed to be statistician.
Imagine there is a sequence of n white and black balls - that's all you know. Having to assume some percentage of white balls, what would be the safest assumption? 10%? 90%? 50%?

Image

There are 2^n such sequences, but asymptotically they are completely dominated by p=1/2 percentage subset, so this is the safest assumption while not knowing anything more, due to entropy maximiziation.
Boltzmann distribution (e.g. in Ising model) is its generalization by weighting with energy, e.g. to focus only on sequences having a chosen energy ... and it is at heart of statistical physics, which leads to universal mathematical results agreeing with nature, like energy spectrum of stars.

This is pure mathematics, we can observe in nature consequences of its universality .. there are more of them, like https://en.wikipedia.org/wiki/Zipf%27s_law
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