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Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such pair forming an irreducible representation of the Weyl relations,

$U_sV_t = e^{is\cdot t}V_tU_s$

is unitarily equivalent to the Schrödinger representation, and hence that all such representations are unitarily equivalent. (Note: the Weyl relations in this context are equivalent to the canonical commutation relations (CCRs) $[Q,P]\psi=i\psi$ for all $\psi$ in the common dense domain of $Q$ and $P$, where $Q$ and $P$ are the generators of $V$ and $U$.)

Question: Is there a known analogue of this result in the context of classical Hamiltonian mechanics?

I don't know of a classical analogue of the Weyl relations. But there is a classical analogue of the CCRs, which is the Poisson bracket $\{q,p\}=1$. So, here's how I imagine a classical analogue of the Stone-von Neumann theorem might look (just a rough attempt, really!).

Let $\mathcal{M}$ be a smooth $2n$-dimensional manifold and $\omega$ a symplectic form on $\mathcal{M}$. Let $\xi = (q,p)$ be any global coordinate system on $\mathcal{M}$, and let $Q:\mathcal{M}\rightarrow\mathbb{R}$ and $P:\mathcal{M}\rightarrow\mathbb{R}$ be the projections onto $q$ and $p$, respectively. Then (conjecture): all such pairs ($Q$, $P)$ satisfying,


where $\{\cdot,\cdot\}$ is the Poisson bracket associated with $(\mathcal{M}, \omega)$, are related by a single canonical transformation.

Does this seem like a reasonable way to formulate the classical analogue? Is the status of this conjecture obvious? Your thoughts are appreciated!

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Let n=1. What you get is the famous Jacobian conjecture... In classical limit representation correspond to symplectic leaves. R^2n is symplectic - so only 1 symplectic leave with accordance to Stone-vonNeuman... –  Alexander Chervov Oct 19 '11 at 13:25

1 Answer 1

Expanding on Chervov's comment: the Jacobian conjecture for two variables conjectures that if a polynomial map $(x,y) \to (X,Y)$ has for its Jacobian $\partial(X,Y)/\partial(x,y)$ a nonzero constant, then this polynomial map has a polynomial inverse. (The chain rule, plus the fact that over ${\mathbb C}$ polynomials have zeros, yields the truth of the converse: if the map has polynomial inverse, then its Jacobian is a nonzero constant.) If $(x,y) = (q,p), (X,Y) = (Q,P)$ then $\{Q, P \}$ is the Jacobian of this transformation. So: an affirmative answer for the Jacobian conjecture would precisely yield a 'yes' to your 'reasonable formulation' in the polynomial category when $n =1$.

Going back to Stone-von-Neumann and the Weyl relations you wrote down suggests adding the hypothesis that the flows of the Hamiltonian vector fields for both $Q$ and $P$ are complete. With that addition, you have a chance of a 'yes' answer when $n=2$, perhaps in the smooth category.

Note: your hypothesis imply that your manifold is ${\mathbb R}^{2n}$.

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