I know that quantum mechanics is sometimes called 0+1 dimensional quantum field theory. What is the meaning? How should we understand it?


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In field theory, a field can be thought of as a map from the spacetime $M$, usually a Lorentzian manifold---a particularly popular choice is $\Bbb R^{1,n-1}$ (Minkowski space)---to some other space. For instance, a scalar field $\phi$ can be viewed as a map $\phi:M\to \Bbb R$, or equivalently as a global section of the trivial (real) line bundle over $M$. The spacetime $M^n$ has one timelike direction and $n-1$ spacelike directions, and one can say that one studies $(n-1)+1$-dimensional field theory.

When doing mechanics, what do we use for a "field"? The position of the particle(s)! The position depends only on time, and hence we have maps $x_i:\Bbb R\to \Bbb R^n$ (in case the space in which the particles move is not simply $\Bbb R^n$, the target may be some other Riemannian manifold of dimension $n$) and fitting this into the general QFT picture we can identify $\Bbb R$ as our "spacetime", where we now have no spatial directions and only a timeline direction. Thus, we may call the theory a $0+1$-dimensional field theory. After quantization, one obtains a $0+1$-dimensional quantum field theory.

  • $\begingroup$ Excellent answer. Just a question: say we're talking about Galilean QM and QFT, both dealing with particles/fields of spin 0. If we take many-body QM, it's equivalent to an associated QFT (conversely, if we take a Galilean QFT, we can use the creation/annihilation operators to find the many-body Hamiltonian), or this is what many people are arguing (I can attach a link if you like). Following your arguments, this many-body QM is still a 0+1-dimensional QFT. But it's also equivalent to a QFT with scalar fields, so a 3+1-dimensional QFT. How do we reconcile the two? $\endgroup$ Commented Oct 20, 2021 at 23:13
  • $\begingroup$ I think when they say "Quantum Mechanics" in this context they are referring to QM with a fixed number of particles. If you are in many-body QM as studied in condensed matter physics, I believe that is indeed closer to a QFT than QM, and the equivalence pretty much breaks down. They still call it QM in that case because it is non-relativistic. $\endgroup$ Commented Feb 18, 2023 at 1:11

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