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 ${\mathbb{R}}^3$ and time is $\mathbb{R}$	(physical) space is ${\mathbb{R}}^3$ and time is $\mathbb{R}$
configucation space is set of all positions possible ${\mathbb{R}}^3$	the domain of the wave function is ${\mathbb{R}}^3$ (= configuration space)
the phase space is set of all positions, momentum pairs possible ${\mathbb{R}}^3\times {\mathbb{R}}^3$ whose “dimension” is $6$	the ”quantum phase space” is the set of all wave functions, set of all square integrable functions ${\mathbb{R}}^3\to \text{C}$ $L^2({\mathbb{R}}^3,\text{C})$ whose “dimension” is not finite!
Hamiltonian is a smooth function ${\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$ $H(q,p)=\frac{p^2}{2m}+V(q)$	the “Hamiltonian” is a Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$ $\hat{H}=\frac{{\hat{p}}^2}{2m}+\hat{V}$
Hamilton’s equations	Schrodinger equation
Observables are smooth functions ${\mathbb{R}}^3\to \mathbb{R}$ (just like Hamiltonian)	Observables are Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$ (just like quantum “Hamiltonian”)
The set of all observables is the set of all smooth functions, so ${\mathcal{C}}^{\infty }({\mathbb{R}}^6)$ where we can define the Poisson brackets $\{,\}$	The set of all observables is the set of all Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$, this is a subspace of all linear functions and is closed in the commutator brackets $[,]$
position $q_i$	${\hat{q}}_i$
momentum $p_i$	${\hat{p}}_i=\text{pdvf}{q}_i$
momentum “generates” translations - $\exp (\alpha X_{p_i})$ ($X_{p_i}$ is the Hamiltonian vector field of $p_i$, exponential of vector fields) - translation is a group action by $({\mathbb{R}}^3,+)$ onto the phase space ${\mathbb{R}}^6$ - which is not a symmetry here! because $\{H,p_i\}\ne 0$ if we have a non-constant potential	momentum “generates” translations - $\exp (\alpha {\hat{p}}_i)$ (exponential of linear maps) - translation is a group action by $({\mathbb{R}}^3,+)$ onto the phase space $L^2({\mathbb{R}}^3)$ - which is not a symmetry here! because $[H,p_i]\ne 0$ if we have a non-constant potential
$L_i$	${\hat{L}}_i$
angular momentum “generates” rotations - $\exp (\theta X_{L_i})$ ($X_{L_i}$ is the Hamiltonian vector field of $L_i$, exponential of vector fields) - rotation is a group action by $SO(3)$ onto the phase space ${\mathbb{R}}^6$ - which is a symmetry here! because $\{H,L_i\}=0$ if we have a non-constant potential	angular momentum “generates” rotations - $\exp (\theta {\hat{L}}_i)$ (exponential of linear maps) - rotation is a group action by $SO(3)$ onto the quantum phase space $L^2({\mathbb{R}}^3)$ - which is a symmetry here! because $[H,L_i]=0$ 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
$\{f_1,f_2\}=0$ means we are one step closer to Liouville–Arnold integrability	$[{\hat{f}}_1,{\hat{f}}_2]=0$ means we can simultaneously diagonalize them $\to$ 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 $\mathcal{A}$ of observables on (classical) phase space $P$

$$\mathcal{A}\subseteq ({\mathcal{C}}^{\infty }(P),\{\})\to (\mathrm{End}(\mathcal{H}),[,])$$

This is precisely what quantization is!

This however, might not be unique, or even exist for any set $\mathcal{A}$.

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

plot twist: QM $\subset$ Hamiltonian dynamics §

For finite dimensions, any anti-Hermitian matrix $i\hat{H}$ (so $\hat{H}$ is Hermitian) can be written as a Hamiltonian vector field of some function

$$H:{\mathbb{C}}^N\to \mathbb{R}$$

which is given by

$$v\mapsto ⟨v\mid i\hat{H}v⟩$$

This can be seen as the following statement

$$U(N)\subseteq \mathrm{Sp}(2N,\mathbb{R})$$

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

Hence, what this means is

$$\text{finite dim QM}\subset \text{Hamiltonian dynamics}$$

This just means we could easily work with Hamilton’s equations and the Hamiltonian $H$ rather than the Schrodinger equation with the operator $\hat{H}$.

Notice the Hamiltonian for a operator $i\hat{H}$ is

$$v\mapsto ⟨v\mid i\hat{H}v⟩$$

has a QM interpretation!

As a consequence of this chain of thought, I give the example of how two harmonic oscillators and the spin-$\frac{1}{2}$ particle share a phase space ${\mathbb{R}}^4={\mathbb{C}}^2$ 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?