Correspondence between Hamiltonian mechanics and QM

[guhan]

To which entitity (operators, wavefunctions etc) in quantum mechanics do the dynamical variables and the hamiltonian vector fields that they generate (through Symplectic structure of classical mechanics) correspond to?

 

[confinement]

The dynamical variables of position and momenta correspond to (linear) Hermitian operators. In particular, the commutator of two QM operators is given by the imaginary unit times the poisson bracket of the corresponding classical variables. The phase space of a quantum system is in general an infinite-dimensional Hilbert space, in contrast with the symplectic manifold of classical phase space.

 

[guhan]

Ok.

And to what (in QM) does the vector field, say $$\overline{df}$$, generated by a dynamical variable, say $$f$$ (in Classical mechanics) correspond to?

I think I can associate the propagator in QM to the vector field generated by the Hamiltonian in classical mech, but what about other vector fields?

 

[guhan]

I thought I would give this thread a proper burial...

The formulation of QM is through a homomorphism of Lie algebras from the poisson bracket in classical hamiltonian mech to commutation relationships in QM. This is to handle the correspondence principle used in physics, in a more general way. The requirements are more than just the existence of this homomorphism (the kernel of which are the constant functions, i.e. $$f=constant$$). Although we would be tempted to associate $$X_f$$ (the vector field generated by $$f$$ in classical Hamiltonian mechanics) with the operators in QM, it turns out that it does not satisfy some axioms of pre-quantization (specifically, the axiom that constant functions should be mapped to multiplication by that function). In the end, we can associate $$-iX_f + f + \theta (X_f)$$ (where $$d\theta = \Omega$$, the symplectic 2-form) to QM operators. Anyway, a full treatment of this is given in mathematical physics books under the topic of (geometric) 'pre-quantization'. As far as I understand, a closely related issue is proving to be a big hurdle in rigorously formulating topological QFT.

note: $$X_f = \overline{df}$$ = 1-vector field that is generated by $$f$$ and which corresponds to the 1-form $$df$$ (correspondence is through the isomorphism between the tangent and cotangent space that is induced by the symplectic 2-form $$\Omega$$ )