Do the canonical commutation relations have any connection to geometry?

Question

I was wondering if the canonical commutation relations have any connection to geometry? If so, could you explain the connection in fairly simple and intuitive terms?

Answer 1

Well, this is a fairly broad topic.

1. Here is one way CCRs arise from a rather large class of geometries: Given a Fedosov manifold $(M,\omega, \nabla)$ [i.e. a manifold $M$ endowed with a symplectic $2$-form $\omega$ with a compatible torsion-free connection$^1$ $\nabla$], Fedosov proved the existence of an associative star product $$f\star g~=~fg +{\cal O}(\hbar)\tag{1}$$ of functions $f,g\in C^{\infty}(M)$ in the context of deformation quantization$^2$. The star commutator $$[f\stackrel{\star}{,}g]~:=~f\star g-g\star f~=~i\hbar\{f,g\}_{PB}+{\cal O}(\hbar^2) \tag{2}$$ corresponds to a commutator of operators via a symbol-operator correspondence map. The Darboux theorem ensures the local existence of canonical coordinates $(q^1, \ldots, q^n,p_1, \ldots, p_n)$. CCRs therefore make sense.

2. For an elementary discussion of the connection between commutators & Poisson brackets, see e.g. this Phys.SE post.

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$^1$ A symplectic manifold $(M,\omega)$ always has such connection $\nabla$; it is however not unique. See also e.g. this related Phys.SE post.

$^2$ Kontsevich extended the construction of an associative star product (1) to an arbitrary Poisson manifold.

Answer 2

Roughly speaking, the gauge potential is identical to connection where the gauge potential is related to amplitude (field variable).

The commutation relation in quantum theory can be written as the commutation relation of field variables (amplitude).

So it is natural to have the commutation relation expressed in connection.