The correspondence between CM and QM

though the one point problem

We must notice the analogy

the “classical” problem in Hamiltonian dynamics	the quantum problem in Schrodinger/H picture
(physical) space is and time is	(physical) space is and time is
configucation space is set of all positions possible	the domain of the wave function is (= configuration space)
the phase space is set of all positions, momentum pairs possible whose “dimension” is	the “quantum phase space” is the set of all wave functions, set of all square integrable functions whose “dimension” is not finite!
Hamiltonian is a smooth function	the “Hamiltonian” is a Hermitian linear functions
Hamilton’s equations	Schrodinger equation
Observables are smooth functions (just like Hamiltonian)	Observables are Hermitian linear functions (just like quantum “Hamiltonian”)
The set of all observables is the set of all smooth functions, so where we can define the Poisson brackets	The set of all observables is the set of all Hermitian linear functions , this is a subspace of all linear functions and is closed in the commutator brackets
position
momentum
momentum “generates” translations - ( is the Hamiltonian vector field of , exponential of vector fields) - translation is a group action by onto the phase space - which is not a symmetry here! because if we have a non-constant potential	momentum “generates” translations - (exponential of linear maps) - translation is a group action by onto the phase space - which is not a symmetry here! because if we have a non-constant potential
angular momentum “generates” rotations - ( is the Hamiltonian vector field of , exponential of vector fields) - rotation is a group action by onto the phase space - which is a symmetry here! because if we have a non-constant potential	angular momentum “generates” rotations - (exponential of linear maps) - rotation is a group action by onto the quantum phase space - which is a symmetry here! because if we have a non-constant potential
the group action is symplectic, and more specifically Hamiltonian!	the above group actions act by unitary transformations because exponential of Hermitian maps are unitary
means we are one step closer to Liouville–Arnold integrability	means we can simultaneously diagonalize them one step closer to having an eigenbasis (is this actually correct?)

This can be said in brief by saying its a Lie algebra homomorphism from a Lie subalgebra of observables on (classical) phase space

This is precisely what quantization is!

This however, might not be unique, or even exist for any set .

This analogy can be generalised many folds (geometric quantization of a Hamiltonian group action).

plot twist: QM Hamiltonian dynamics

For finite dimensions, any anti-Hermitian matrix (so is Hermitian) can be written as a Hamiltonian vector field of some function

which is given by

This can be seen as the following statement

(I think this is correct) where the groups are Unitary group - Wikipedia and Symplectic group - Wikipedia.

Hence, what this means is

This just means we could easily work with Hamilton’s equations and the Hamiltonian rather than the Schrodinger equation with the operator .

Notice the Hamiltonian for a operator is

has a QM interpretation!

As a consequence of this chain of thought, I give the example of how two harmonic oscillators and the spin- particle share a phase space and their flows commute!

This can be generalized to infinite dimensions: Geometric formulation of quantum mechanics - arxiv.org/pdf/1503.00238.pdf.

What does this mean physically, or even (physics) philosophically?