I was wondering if the universal wave function can be thought of using a path integral approach? And if so, does each path correspond to a "branch" of the MWI of QM? Or have I misunderstood what is meant by a MWI branch?
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$\begingroup$ If you're asking about the MWI see physics.stackexchange.com/q/502211 $\endgroup$– alanfApr 1, 2022 at 9:30
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$\begingroup$ I am referring to the MWI of QM. But the question linked by alanf, while appreciated, certainly doesn't answer my question. $\endgroup$– SSDApr 1, 2022 at 14:25
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2 Answers
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Branches in the MWI are coarse-grained emergent things, and their number cannot be precisely quantified (see https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780199546961.001.0001/acprof-9780199546961-chapter-4 ). Depending on how finely you coarse-grain worlds and how the measurement is performed, more or less paths in the path integral may contribute significantly to each branch.
If you work only at the level of "individual" paths and define each path as a branch, like some sort of quantum Laplace's demon, then you must be very comfortable with some of your branches interfering with each other indefinitely rather than decohering into approximately separate worlds. By refusing to coarse-grain, you haven't really isolated any branching structure at all.
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$\begingroup$ Thanks for the reference. Your answer is incisive and appreciated. I suppose I'm not uncomfortable at all with paths (branches for the sake of this question) interfering indefinitely - as it seems to me only those branches which are very similar will contribute much. The dissimilar paths will tend to cancel with eachother. By course graining, I take it you mean considering branches as differing sequences of "measurements", right? Is there any problem in principle with considering us to live on one path, and measurements merely give info on which one? That's my sticking point. $\endgroup$– SSDApr 7, 2022 at 11:10
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1$\begingroup$ In other words, is there anything philosphically wrong with considering each course-grained history as simply incomplete information on what our "Laplace demon" knows as our "true" path? $\endgroup$– SSDApr 7, 2022 at 11:14
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$\begingroup$ The "The path integral in quantum-mechanical interpretation" section of this en.wikipedia.org/wiki/Path_integral_formulation article seems to suggest as much. Particularly this reference www2.perimeterinstitute.ca/personal/rsorkin/some.papers/… $\endgroup$– SSDApr 7, 2022 at 14:11
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$\begingroup$ Again, I want to emphasize how extraordinarily grateful I am for your clarifications on these points! I've been asking everywhere, and it seems there is not a wide-spread understanding of these concepts. $\endgroup$– SSDApr 7, 2022 at 14:16
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1$\begingroup$ Hm, interesting. Although I think you are now talking about an interpretation different enough from the regular MWI to deserve a new name. Especially as the MWI is fully deterministic (apart from our subjective self-locating uncertainty of which branch we are on), but probabilities seem to play a fundamental role in what you are describing. $\endgroup$ Apr 9, 2022 at 9:46
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Yes, you can use the path-integral approach in the many-worlds formulation of QM but no, the paths of the path integral do not correspond to the branches of the many-worlds formulation of QM. Branching is associated with measurements. On the other hand, the path integral makes no reference to entanglement or decoherence -- the two crucial elements of a measurement process. As an extreme example, consider the following: in a world of a single particle with nothing else, there is no branching. However, obviously, you can still use path integral to calculate the time-evolution of the wavefunction of this particle.
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1$\begingroup$ Hey, thanks for your answer. Unfortunately I'm still not satisfied. It seems to me that decoherence is trivially included in a universal path integral approach. All usual quantum mechanics should apply, and quantum uncertainty may be interpreted as self-locating uncertainty among all possible trajectories of the universe. The probabilities for divergent time evolution of previously identical universes is still given by the sum-over-histories approach... $\endgroup$– SSDApr 5, 2022 at 23:47
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$\begingroup$ In particular, were you to include the experimentalist, the measurement apparatus, and the system to be measured in the same (truly monstrous) path integral, you'd find the probability of measuring e.g. spin-up when the particle was actually spin-down as near zero, and vice versa, which of course is another perspective on decoherence. $\endgroup$– SSDApr 5, 2022 at 23:56
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$\begingroup$ This, according to my understanding, is because a whole lot of coincidences would have to happen to many trillions of particles (i.e. those making up the measurement apparatus, the experimenter, & the environment) in order to provide a faulty measurement, and the amplitudes of these coincidences end up cancelling out in the path integral. $\endgroup$– SSDApr 6, 2022 at 0:02