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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.

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