Do Feynman path integrals satisfy Bell locality assumption?
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.
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".
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?
Nature does not solve models. Nature just is. Mathematical models describe nature.
Non-locality is alive and well
Jarek wrote:minkwe,
This is not about human philosophy, but asking how nature objectively works - what in physics we decompose into mathematics.
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?
.
BTW, you didn't answer how the moving particle knows which path has the least action?
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.
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
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
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