I don't think the discussion here is focused on non-relativistic QM. The question is if the two approaches are equivalent in general, not in specific cases such as Euclidean spaces.
Yes, this does seem to work for general topologies. But, do you happen to know if the quantum mechanical path...
Very interesting!
Can many-body QM also treat more complicated manifolds, such as complex ones (Kahler, Calabi-Yau) with arbitrary topologies (curled-up dimensions, holes, etc)?
I always found QFT to be much more convenient when dealing with anything else than a flat Euclidean space (maybe with some non trivial topologies too, such as the ones arising in solid state physics where we impose periodic boundary conditions).
I guess, the question is more precisely put as "in...
I didn't insist on anything. I just wrote what I thought was true and admitted to the fact that it might be a naive thought. But, ok, sure, what's your point then?
I don't see anything productive from this. The conversation has shifted to the meaning of charge conjugation in gravity, which is...
Understood.
So, can QM actually treat any underlying manifold space(time) with any topology, like QFT does? I've seen QM being able to be formulated for Euclidean (of course) and Lorentzian spacetimes, but what about general curved spacetimes with non-trivial topologies? QFT can handle it, but...
So, in principle you can employ a second quantization approach in which you inlcude the fundamental particles involved in the interactions of a such a condensed matter system, but in practice this is intractable and this is why researchers use different methods in such cases?
If so, it indeed...
I see. But what about cases where we know for a fact that there are no quasiparticle excitations? Doesn't that mean that a "particle approach" is invalid in those cases?
I'm talking about both QFT and many-body QM as general frameworks rather than models. You can construct a QFT no matter what the underlying manifold is. On the other hand, while many-body QM is used for some manifolds with non-trivial topologies (such as when you impose periodic boundary...
I can't seem to tell if you're being sarcastic or just using good old fashioned humor!
In any case, me not knowing if charge conjugation has a meaning in GR might actually be the essence of my question at the end of the day. I do suspect -maybe naively- that there should be some analogue to it...
So, essentially, if I understood correctly, the many-body QM (which includes the second quantization and the effective field theories that you mentioned) is a subset of QFT? Based on your answer, I can't see how many-body QM can properly treat particles/fields existing in a generic manifold of...
This is true. I ultimately agree. My initial problem was that a lot of what I said were things that are commonly found in books so I thought (maybe wrongly) that references wouldn't be needed. The rest was my speculation, which I believe I wrote them in a way to reflect that.
In any case, I hope...