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- Many people say that the two are equivalent. Is this true?

A lot of people say that Quantum Field theory (QFT) an Quantum Mechanics (QM) are equivalent. Yet, I've found others who dispute these claims. Among the counter-arguments (which I admittedly do not have the expertise to pick apart and check their validity in full) are the following:

1) While QFT can fully capture instantonic solutions, many-body QM cannot. I've seen instantons in the context of QM though.

2) While QFT can fully capture topological phenomena, many-body QM cannot. I know that QM can capture at least some topological phenomena, such as the integer quantum hall effect (although I do not know if the description of edge states is present in the context of QM, whereas in the context of QFT they are decribed by Chern-Simons theories).

3) I do not know if there is a many-body QM correspondence to topological field theories.

4) There are QFT systems in condensed matter physics which do not admit quasiparticle excitations, such as the strange metal phase in superconductors. I do not think (but I'm not sure) if such systems can be described by many-body QM.

5) Say we have a scalar field in QFT, which lives on a manifold, M. Then a scalar field is a function M->R. I know that when M is Euclidean with trivial topology, we're talking about Galilean QM which has a formulation in both QFT and many-body QM. If M is Minkowski, then I know QFT is the better framework to work in, although I do not know if you can also work in this space with many-body QM. Yet, if the manifol M is generally a curved, complex manifold with non-trivial topology, while QFT can handle it, I highly doubt that many-body QM can describe it.

6) Some say that the two are equivalent only in the case where the particle number in many-body QM is conserved. In elementary systems that I've studied, this turns out to be true. Alas, I do not know if it's true in general (i.e. when the particle number is not conserved).

On the other hand, I know that one-body QM can be rigorously considered as being a (0+1)-diimensional QFT (with fields being the particle's position, themselves only depending on time), I do not know if and how I would make such an analogy between many-body QM with a (some+some)-dimensional QFT. If such a mapping doesn't exist, then this would - in my humble, naive and maybe ignorant opinion - be a conclusive argument as to the two not being equivalent.

By not necessarily basing your arguments just on the points given above, what is your take on it?

1) While QFT can fully capture instantonic solutions, many-body QM cannot. I've seen instantons in the context of QM though.

2) While QFT can fully capture topological phenomena, many-body QM cannot. I know that QM can capture at least some topological phenomena, such as the integer quantum hall effect (although I do not know if the description of edge states is present in the context of QM, whereas in the context of QFT they are decribed by Chern-Simons theories).

3) I do not know if there is a many-body QM correspondence to topological field theories.

4) There are QFT systems in condensed matter physics which do not admit quasiparticle excitations, such as the strange metal phase in superconductors. I do not think (but I'm not sure) if such systems can be described by many-body QM.

5) Say we have a scalar field in QFT, which lives on a manifold, M. Then a scalar field is a function M->R. I know that when M is Euclidean with trivial topology, we're talking about Galilean QM which has a formulation in both QFT and many-body QM. If M is Minkowski, then I know QFT is the better framework to work in, although I do not know if you can also work in this space with many-body QM. Yet, if the manifol M is generally a curved, complex manifold with non-trivial topology, while QFT can handle it, I highly doubt that many-body QM can describe it.

6) Some say that the two are equivalent only in the case where the particle number in many-body QM is conserved. In elementary systems that I've studied, this turns out to be true. Alas, I do not know if it's true in general (i.e. when the particle number is not conserved).

On the other hand, I know that one-body QM can be rigorously considered as being a (0+1)-diimensional QFT (with fields being the particle's position, themselves only depending on time), I do not know if and how I would make such an analogy between many-body QM with a (some+some)-dimensional QFT. If such a mapping doesn't exist, then this would - in my humble, naive and maybe ignorant opinion - be a conclusive argument as to the two not being equivalent.

By not necessarily basing your arguments just on the points given above, what is your take on it?