# Do the canonical commutation relations have any connection to geometry?

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?

• This is an interesting subject, it would be very difficult for me to properly answer here, however, I think You can take a look at the paper "from equations of motions to canonical commutation relations". It contains a lot of discussion on the theme, as well as a very good bibliography on the subject. – Ittiolo Feb 7 '17 at 15:11

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.
$^1$ 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.