Is there a way of characterizing entanglement between states in a path integral formalism? If so, does this shed some light on the apparently non-local effects of quantum mechanics?
$\begingroup$
$\endgroup$
6
-
$\begingroup$ Why path integrals? Path integrals are used in quantum field theory which is used for studying fundamental dynamics, while entanglement is a concept usually associated with states. Shouldn't one rather consider a formalism that more geared toward represented states then dynamics? There are functional integral approaches for representing states in terms of there most general form. $\endgroup$– flippiefanusCommented Feb 13 at 13:05
-
1$\begingroup$ @flippiefanus I would put it the other way round. Entanglement is a fundamental concept in quantum mechanics. It is therefore an extremely reasonable question to ask how does it appear and how can it be calculated in the path-integral formulation, which is after all, fully equivalent to any other formulation of QM. $\endgroup$– By SymmetryCommented Feb 13 at 15:54
-
$\begingroup$ @BySymmetry. Strictly speaking entanglement is not a fundamental concept. It is a consequence of the quantized nature of interactions together with conservation principles. $\endgroup$– flippiefanusCommented Feb 14 at 3:54
-
$\begingroup$ @BySymmetry. As for the "fully equivalent to any other formulation..." The states in QFT are not as sophisticated as those in other formalisms of QM. The reason is that QFT is focussed of fundamental dynamics. $\endgroup$– flippiefanusCommented Feb 14 at 3:58
-
$\begingroup$ @flippiefanus that might be how things are viewed in your field. Where I work in Condensed matter and Non-equillibrium many-body systems we spend a lot of time finding ways to apply tools from field theory to more complex states and asking precisely the type of question posted here $\endgroup$– By SymmetryCommented Feb 14 at 10:12
|
Show 1 more comment
1 Answer
$\begingroup$
$\endgroup$
Yes, I've worked out exactly how one can analyze entanglement using the path integral in a pair of published papers; first, second.
Note that here we're using the discrete sort of path integral more commonly known as "sum over histories", avoiding the actual integrals by assuming we know what path each particle is on.
To summarize, these papers lay our recipes for translating any entangled state into a path-integral-friendly setup, and then showing how to calculate the probabilities by adding up the complex amplitudes and squaring the result. The results match quantum predictions, but each amplitude is calculated in spacetime rather than in Hilbert space or configuration space.
The trick for the second paper was figuring out how to measure a pair of qubits in an arbitrary entangled basis. The resolution isn't very practical (one would never actually use the experimental geometry in the laboratory), but it works on paper, again generating the correct outcomes. Most of the foundational discussion about insights from this alternate viewpoint can be found in the second paper.
That ties into your final question, about exactly what this says about the "nonlocality" of entanglement itself. In our view, it says that entanglement is not discontinuously-nonlocal, there aren't mysterious connections directly from one region of space to another. Instead, it pushes much more for an "all at once" viewpoint, where the whole history in spacetime is viewed as one entity, so that the links between distant regions are restored via the past or the future particle trajectories. That in turn supports future-input-dependent (aka retrocausal) accounts of entanglement, as summarized in this third Rev Mod Phys paper.
