How do I formulate a quantum version of Hamiltonian flow/symplectomorphisms in phase space to have a "geometric", quantum version of Noether's theorem

Question

I'm currently exploring how Noether's theorem is formulated in the Hamiltonian formalism. I've found that canonical transformations which conserve volumes in phase space, these isometric deformations in phase space, correspond to Hamiltonian vector fields. I believe that a quantity constant along the flow of the vector fields correspond to conserved quantities? I am unsure.

What I'm the most curious about is how to go about a quantum version and a field theoretic version of Hamiltonian flow/hamiltonian formulation of Noether's theorem?

Answer 1

I am not sure to understand your issue.

The condition of dynamical conservation of $f=f(q,p)$, as you correctly wrote, is $X_H(f)=0$, where $H=H(q,p)$ is the Hamiltonian function of the system. On the other hand, just in view of the definition of Hamiltonian field, $$X_H(f) = \{H,f\} =-\{f,H\}= -X_f(H).$$ That means that:

$f$ is conserved if and only if $H$ itself is invariant under the group of (canonical) transformations induced by $f$.

Finally, taking advantage of the Lie bracket of vector fields since $$[X_f,X_g]= X_{\{f,g\}},$$ $X_H(f)=0$ is equivalent to $$[X_H, X_f]=0.$$ This latter says that:

every motion of the system (an integral line of $X_H$) is transformed to a motion under the action of the transformation group generated by $f$ (and this is equivalent to the dynamical conservation of $f$).

This is a perfect beautiful picture which illustrates how the Hamiltonian version of Noether’s theorem is much more advanced and comprehensive than the Lagrangian one.

The quantum version is not very different from this. Just replace one parameter groups of canonical transformations for strongly continuous one parameter groups of unitary transformations. I stress that I am not referring here to some “quantization procedure” used to quantize a classical Hamiltonian system such that it preserves the structures and the relations written above. That is a very difficult (and hopeless) issue. I am just saying that, the Noether theorem in quantum theory has the same structure as in classical Hamiltonian physics.