In path integral formulation of QM Feynman show that for a photon every possible path should be taken.
I'm puzzled how this must be applied for entangled photons.
Let there is a pair starting at A. Photon 1 goes to B.
Photon 2 goes to BS C and then can go to B or to D.
Let it went ACD and is registered there. It's path is just ACD.
The photon 1 has path AB. But I don't know have I to include for it also the path ACB because its entangled twin can make it?
If you know and explain I would be very thankful. Please insert a textbook or article.
Ilian Peruhov
In the path integral formulation of quantum mechanics, as developed by Richard Feynman, the principle is that a particle like a photon does indeed explore every possible path from its start to its endpoint. Each path contributes to the overall probability amplitude of the particle's final position, with the classical path typically contributing the most because it has the stationary (extremum) action.
For entangled photons, the situation incorporates additional complexity due to their quantum correlation. Even though each photon individually explores all possible paths, the measurement outcomes are correlated. In your example, photon 1 and photon 2 start at point A, with photon 1 going to B and photon 2 potentially going to C, then either to B or D, finally being detected at D.
For photon 1, while it does take all paths from A to B, its state must be considered in the context of the state of photon 2 due to entanglement. When photon 2 is detected at D, the entanglement impacts the possible paths and the corresponding amplitudes of photon 1. However, this doesn’t mean that photon 1 needs to consider the path ACB physically. Instead, the entanglement influences the probability amplitudes associated with its paths, but each photon still explores its paths independently in terms of the physical space it covers.
The key here is that the entanglement affects the probability amplitudes and the resulting interference patterns, not that photon 1 would "know" or "choose" its path based on photon 2’s actions in a classical sense. The path ACB for photon 1 would be a valid path in the absence of any measurement, but once photon 2 is measured at D, the entanglement restricts the possible outcomes for photon 1 based on the quantum correlations.
For a more detailed understanding of these concepts, including how entanglement affects the calculations of path integrals, you might look into resources that delve into quantum mechanics and specifically the path integral approach to quantum field theory and quantum optics. A useful textbook on this subject is "Quantum Mechanics and Path Integrals" by Richard P. Feynman and Albert R. Hibbs. Additionally, a resource like "Principles of Quantum Mechanics" by R. Shankar can provide a solid foundation for understanding the broader implications of quantum mechanics on particle behavior.
hope this helps.
regard, H
Dear Harri
Thank you for considerations.
In fact I read Feynman and Hibbs long ago but I don't remember them to discuss entangled particles in the book. I will try to take a look.
About physical presence I don't discuss it. In CI e.g. orthodox QM interpretation the particle between measurement is not considered.
This is in fact a mathematical question. There are strict rules. One takes the allowed possible paths of least action.
So my question has only two answers:
1. We take paths ab and acb (and sum the integrals for them)for photon 1 and path acd for ph2
Or
2. Only ab (only this integral ab is taken) for ph1 and path acd for ph2
I didn't understand what do you support?
Or you think there is other alternatives?
Regards: Ilian
It's not that easy. The path integral defines the amplitude for a process, from which the transition probability can be computed. Now the path integral uses the classical action, that, for two non-interacting particles, is the sum over the action for each particle, so the path integral-and, thus, the probability distribution factorizes.
What's needed is the inclusion of an appropriate flux, since entanglement is a non-local property.
Dear Stan
Thank you very much.
I checked in Feynman Hibbs there is nothing about entangled particles.
Can you give a citation where the issue is explained? I searched the net too. There are some very recent papers from 2022 but not exactly on the question.
Do you consider twins as two particles in your approach?
What is that flux? Something like EM flux between charged particles?
I far as I see the issue is not resolved? Even not investigated?
Thank you in advance for reply.
Ilian
It's not considered in Feynman Hibbs at all. And, indeed, the issue doesn't seem to be resolved-indeed it hasn't been fully considered, cf. , however, here: https://edisciplinas.usp.br/pluginfile.php/5034806/mod_resource/content/1/Entanglement%20entropy%20in%20QFT.pdf
The link to the paper cited in this talk is here: https://arxiv.org/pdf/1801.10352
The flux is, indeed, similar to magnetic flux, though it doesn't have anything to do with electromagnetism.
Gnets - im also abit stcuk ehre bit I may have a solution or a direction.
I shall revert back.
Rgds
H
Thank you Stan.
I thought the answer is found a long time ago and is very easy. I didn't at all suspect one should go to QFT and holography to find answers for path integral. I am very curious why Feynman has said nothing on the matter?
I wonder of my example: are there not experimental results? I think that would easy to do! So far I don't find anything. Just a MZI with one photon on BS2 directly and the other one tru BS1 and registered not in BS2 but outside in a detector D1.
In fact I am on my way to make such experiment.
Regards: Ilian
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