# Do Feynman path integrals satisfy Bell locality assumption?

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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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Feynman diagrams deal with virtual particles, which are not constrained the same way as real particles.

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While QFT is also usually solved in symmetric way (Feynman diagram ensemble: paths + more complex scenarios like decays), I didn't want to go there.

The question of focus here is: what is really disproven by Bell theorem?

For example classical field theories like electromagnetism or general relativity seem local realistic with field as "hidden variable" - do Bell theorem disprove them? What does it mean?

My point is that maybe the misunderstanding comes from focusing on asymmetric ways like Euler-Lagrange or Schrodinger, which indeed satisfy assumptions of Bell theorem - should satisfy resultant inequalities, which are violated by physics - contradiction.

Physicists believe that physics is fundamentally CPT symmetric, so maybe we should focus on symmetric ways of solving these models like through the least action principle, Feynman path/diagram ensemble - what I think don't satisfy Bell's assumptions, resolving this problem (?).

Regarding virtual particles, to understand such abstract concepts I usually try to find analogues in topological solitons like sine-Gordon or 2D fluxons in superconductors.

For example here are simple 2D solitons of field of unitary vectors (e.g. with (|v|^2-1)^2 Higgs potential to regularize singularities to finite energy). In the bottom half there is negative-positive topological charge pair in various distances - tension of the field increases with their distance, leading their attraction (e.g. F ~ 1/distance).

What is crucial here is that such pair creation is a continuous process - we can invest much less energy than their mass to only start such process: fields deformation toward pair creation - what in Feynman diagrams language should correspond to virtual pair creation (?): ##### Share on other sites

The hidden variable conjecture that Bell type experiments test for has little to do with symmetric or assymmetry.

They are certainly involved but that isn't the issue. For example photons are symmetric while fermions are not. The Bell experiment can be performed using either fermions or bosons.

The hidden information involves whether or not an entangled particle contains the spin information of its particle pair in such a manner that when the superposition wavefunction collapses via measurement with a detector alignment whether or not there is information exchange between the entangled particles. This would necessarily require instantaneous superluminal signaling.

Bells test essentially shows us no hidden variables are involved so no superluminal signaling occurs. There is still contestation of this conclusion which quite frankly is good science.

Virtual particles in Feymann diagrams are represented by the internal lines in essence the propogators.

QFT doesn't particularly use the particle view such as in Bohm theory. In QFT all particles are field excitations. The pointlike attributes in wave particle duality can be explained by wavefunctions such as the DeBroglie and Compton wavelength.

Edited by Mordred
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Mordred, I have meant time/CPT symmetry - which is at heart of fundamental models like QFT, and seems an alternative way to dissatisfy the contradicted assumptions of Bell theorem.

Solving a model in symmetric way like Feynman path integrals (or the least action principle), the boundary conditions are symmetric: in both past and future, we would need two separate "hidden variables", what is a bit different than in Bell theorem.

The best argument would be having Bell violation construction from solving in symmetric way - here is such construction for Wick-rotated: Boltzmann path ensembles, e.g. using Ising realization: spatial instead of temporal symmetry:

Having a solution found with path ensembles, we can transform e.g. to Schrodinger, but its hidden state would be already chosen also accordingly to all future measurements - as in superdeterminism: https://en.wikipedia.org/wiki/Superdeterminism

14 hours ago, Mordred said:

Virtual particles in Feymann diagrams are represented by the internal lines in essence the propogators.

Sure, but asking about field configuration behind a Feynman diagram, e.g. with ~1/r^2 electric field around charges, what are virtual particles?

We think about pair creation as 0/1 process, but e.g. topological solitons show that it can be a continuous process - e.g. from a few keVs there could start field deformation similar as in electron-positron pair creation, which quickly returns to the original state of the field.

To consider e.g. scattering of topological solitons, we also need to use ensemble of scenarios - Feynman diagrams exactly as in perturbative QFT. Such fluctuation toward pair creation would correspond to virtual particle creation there, like in vacuum polarization.

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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: ##### Share on other sites

Your missing way too many essential details and steps in the above. Mainly you need to work initially from the Langrangian and get the appropriate creation and annihilation operators.

Then use those operators in Fock space for particle number density for your Boltzmann distributions.

The amplitudes are essential to identify the meson nucleon  scattering etc.

As this is far too lengthy to cover in a forum post here is an article covering the above.

This should get you started into how the field propogators propogate the operators. AZ well as how the operators operate on the propogators.

With what you have above I wouldn't be able to identify which vertex legs would arise to answer the question mark in the above image.

Also make sure you have the necessary causality of the amplitudes in the x and y plane for the commutations. Detailed in the above article.

Edited by Mordred
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Just in case your not already aware ( the equations are primarily covariant in your image  which could be intentionally chosen)and also for other readers. You should identify your indices according to the Einstein summation convention for your indices. This will also apply to the Kronecker delta relations for symmetric antisymmetric and mixed terms.

More of an informative side note under gauge group theory.

The SO (3.1) Poincare group (spacetime under four vector etc) is a double cover $SU (2)\otimes SU(2)/\mathbb{Z}^2$. This will correspond to Fock space and the Hamilton. The $\mathbb{Z}$ is the helicity or parity operator. This will correspond to the right hand rules which the creation and annihilation operators follow as well.

How this applies to Bells test is somewhat covered in the next link here is a relevant quote (page 16)

" But it has been proved by Bell, that if one tries to attach random variables behind each observable and try to find (complicated) rules that explain the princi- ples of quantum mechanics, then this reaches an impossibility. Indeed, taking the spin of a particle in three well-choosen directions, one obtains three Bernoulli vari- ables, but their correlations cannot be obtained by any triplet of classical Bernoulli variables. "

The article specifically covers Fock space and the hermiteans.

Hope that helps.

Edited by Mordred
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Mordred, once again - this was supposed to be Bell theorem thread, e.g. if it also disproves Lagrangian formalism models like EM, GR, QFT - which are realistic and local.

But sure, I know especially perturbative QFT, and it is a very universal formalism - e.g. phonons are mechanical waves/distortions of atomic lattice, but QFT allows to treat them as real particles.
Soliton models are in classical field theories, there is also performed second quantization for them. From other perspective, considering scattering of solitons, we need to consider various possible scenarios (e.g. kink-antikink pair creation): ensemble of Feynman diagrams from perturbative QFT.

As QFT is generally built on classical field theory (e.g. EM -> QED), we should be able to understand fields corresponding to each Feynman diagram, like E ~1/r^2 around charged particles - I am only saying that we can also try to understand field for virtual particles, like in the bottom-left of soliton diagram above: just start of continuous process of pair creation.

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Yes I understood that. The soliton  is a quasi particle as opposed to a virtual particle. However it would still follow the rules for the field propogator in your Feymann diagrams it too must apply the conservation laws of the Eightfold wayen. As the mass and momentum would be off shell you wouldn't be able to determine any precise number density The off shell values would be indeterminimant. In this sense it won't necessarily preserve conservation of energy momentum except as the full propogator action. All particle production occurs in pairs regardless of quasi, virtual or real due to the numerous conservation laws such as charge, isospin, parity etc.

However in terms of hidden variables the quoted section on Bells hidden variables is still applicable. One thing you seldom see examined with hidden variables is that wavefunctions are 3 dimensional with longitude and transverse components. So this must applied in path integrals. As well as hidden variables. Those conservation laws also apply when the entangled particles are initially prepared.

This also applies to spin helicity as per photon helicity etc. You are involving a projection of spin according to the direction of motion.

Edited by Mordred
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The soliton particle models I consider are based on Faber's model: repairing Maxwells' equations: to have built-in charge quantization (by using Gauss-Bonnet theorem as Gauss law - integrating field's curvature it gives topological charge inside: which has to be integer) and regularization of charge's electric field to finite energy (by using Higgs'-like potential allowing to deform electromagnetism into other interactions to prevent infinite energy in centers of particles).

Anyway, this is just Lagrangian mechanics, which becomes Maxwell's equations in vacuum (far from particles) - it has all conservation laws including Noether: of energy, momentum and angular momentum. Performing second quantization we should get ~QED which might not require renormalization as electric field of charge has already finite energy here. This regularization also has running coupling - deformations of Coulomb for very close particles.

Virtual particle is something different for topological solitons (like fluxons) - look at creation of minus-plus topological charge at the bottom of: Imagining it is electron-positron pair creation, the right hand side configuration uses fully formed particles: requires at least 2 x 511keV energy.

But generally this pair creation process is continuous - the left hand side configuration might require much less energy.

We can imagine e.g. vacuum polarization this way: brief fluctuations toward such left-most configuration ... in Feynman diagram representation it would be virtual pair creation, such brief fluctuation would not require entire 2 x 511keVs.

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Please don't use a drop box post the article on this site as a pdf. Why would the left hand configuration require less energy ?

When applying Higgs mechanism to Leptons you can apply the mass mixing angles ie via the PMNS mixing angles and CKMS mixing angles.

I am very conversant in how Higgs applies to mass terms of individual particles.

Do you have a good link on Faber's model. I'm not familiar with it.

Edited by Mordred
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Thanks now I know where you got the above image from so what your attempting makes more sense. I will study this more thoroughly later on.

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It starts much earlier, e.g. with 1D sine-Gordon model: https://en.wikipedia.org/wiki/Sine-Gordon_equation

Here is kink-antikink annihilation ( https://en.wikipedia.org/wiki/Topological_defect ) - their rest energy (mass) is released as massless excitations: Playing it backward we get pair creation - which is a continuous process: we can perform lower energy perturbation in this direction, what in perturbative QFT perspective would be virtual pair creation.

ps. nice "rubber band universe model" video: www.youtube.com/watch?v=nl5Qq5kUbEE

Edited by Duda Jarek
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Ok having looked at your soliton based theories a bit (though I have encountered certain applications for soliton before)

Let's get back to Bells hidden variables. You have already recognized the superluminal information exchange aspect (Ie spooky action at a distance)

Now solitons are from reading those articles Lorentz invariant, so really they behave much like particles under time dependent and independent treatments. You can readily define them to the four momentum of QFT. The Feymann path integrals also follow Lorentz invariance and employ the four momentum for all intensive purposes. This holds true regardless of assymetry or symmetric relations involved in both theories. So I really don't see how symmetry and assymetry will assist you in dealing with the causality violations with the spooky action aspect of hidden variables. Though quite frankly no action is needed to explain entanglement the theories such as Bohm are still around lol. Quite frankly I'm not going to look too deep into soliton based theories except as an assist in this thread. I prefer the standard QFT treatment and view of a particle. I would the hidden variable passage I quoted above is still applicable to the soliton treatment with regards to hidden variables.

Keep in mind my studies have all been cosmology applications as opposed to bothering with quantum interpretations etc. Obviously QFT, String theory, particle physics etc have applicability in Cosmology.

Superdeterminism I understand it as the lack of choice in detector settings by the experimentor. The freedom of choice loophole. So I don't see the connection to solitons or symmetry or assymmetry.

(Also keep in mind I don't bother with philosophy (metaphysics type arguments lol)

Edited by Mordred
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5 hours ago, Mordred said:

You have already recognized the superluminal information exchange aspect (Ie spooky action at a distance)

No, there is no superlaminal information exchange in EM, GR, or field theories I have considered.

The goal of this thread was not solitons or virtual particles (separate topics), but to discuss a different possibility not to satisfy type of locality used in derivation of Bell inequality: by solving these models in symmetric way: through the least action principle, Feynman path/diagram ensembles.

I was told by Richard Gill that it does not satisfy "no-conspiracy" hidden assumption - what agrees with https://en.wikipedia.org/wiki/Counterfactual_definiteness  : " no conspiracy (called also "asymmetry of time")".

While we could transform such solution to asymmetric picture: of Euler-Lagrange or Schrodinger, symmetric ways lead to solutions with different properties - due to using symmetric boundary conditions: in past and future.

It is nicely seen in Ising model which is mathematically nearly the same (Feynman -> Boltzmann path ensemble), using more intuitive: spatial instead of temporal symmetry to get Born rule. For example probability distribution of value inside Ising sequence is Pr(i) = (psi_i)^2 due to symmetry: one amplitude comes from left, second from right. Sketch of derivation: Once again, by symmetry I mean time/CPT symmetry here: solving using the least action principle, path/diagram ensembles - please just think about finding probability distribution inside Ising model, e.g. in sketch of derivation above.

I also don't like philosophy talk. To reduce the number of philosophical ambiguous assumptions in Bell theorem, I prefer to focus on simpler Mermin's - for 3 binary variables ABC:
Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1
which is nearly "tossing 3 coins, at least 2 are equal".
Its derivation doesn't need any ambiguous "locality", "realism", just "there exists Pr(ABC) probability distribution" assumption:
Pr(A=B) = P(000) + P(001) + P(110) + P(111)
Pr(A=B) + Pr(A=C) + Pr(B=C) = 2P(000) + 2P(111) +sum_ABC P(ABC) = 2P(000) + 2P(111) + 1 >= 1

But this inequality is violated in QM formalism ( https://arxiv.org/pdf/1212.5214  ) .

It has only one trivial assumption: "there exists Pr(ABC) probability distribution" leads to Bell inequalities, which are violated by physics - hence somehow this assumption is non-physical.

And Ising model is a nice toymodel to understand how this assumption might not be satisfied: states are defined there with amplitudes on Omega instead of probabilities - to get probabilities we need to add over unmeasured variables, then multiply (Born rule).

5 hours ago, Mordred said:

Now solitons are from reading those articles Lorentz invariant, so really they behave much like particles under time dependent and independent treatments.

Exactly, all these considered field theories are Lorentz invariant, what already for sine-Gordon gives all special relativity effects, like scaling of mass/momentum, Lorentz invariance, and even time dilation for oscillating breathers ( https://en.wikipedia.org/wiki/Breather ) - their number of 'ticks' is reduced with velocity: If someone wants to understand special relativity, the best way is studying sine-Gordon.

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• 4 weeks later...
On 2/16/2020 at 2:48 PM, Duda Jarek said:

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.

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"?

The Schrödinger equation allows violations of the Bell inequalities.  Classical theories (EM, GR) don't allow them, indepedent of any CPT or whatever symmetry.  QFT allows violations of the BI.  The formulations used for QFT, CPT symmetry and so on, are irrelevant for this question.

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Schrödinger equation assumes existence of wavefunction (realism), is defined by using values and derivatives - how does it differ from locality?

So does it satisfy assumptions of Bell theorem - leading to inequalities which are violated by physics?

If you solve it with Feynman path integrals instead, this issues vanishes as we have kind of two hidden variables (amplitudes): https://en.wikipedia.org/wiki/Two-state_vector_formalism

It is visualized in the last slide here: https://www.dropbox.com/s/m1m8uq0gygo2lzt/Ising.pdf : The bottom right is looking simple but thought provoking question: what stationary probability distribution on [0,1] should we expect?

Any diffusion, chaos says uniform rho=1 ... QM says localized rho ~ sin^2.

Uni-directional uniform path ensemble gives rho ~ sin ... finally symmetric ensemble of complete paths gives required sin^2.

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