Flow of observables

Suppose a classical Hamiltonian system $(M,\omega)$ with canonical coordinates. A function $f\in \mathcal{C}^{\infty}(M)$ has associated a Hamiltonian vector field $X_f$ with a flow $\phi_s$. Now, we can define a family of transformations in the algebra of smooth functions (i.e., observables)

$$ \tilde{\phi}_s: \mathcal{C}^{\infty}(M) \longmapsto \mathcal{C}^{\infty}(M) $$

by means of

$$ g \longmapsto \tilde{\phi}_s(g)=g \circ \phi_s $$

This is completely related to the Heisenberg vs Schrodinger picture discussion. The flow $\phi_s$ in $M$ is some kind of evolution of states, like in the Schrodinger picture. And the flow $\tilde{\phi}_s$ corresponds to Heisenberg picture.

Observe that

$$ \frac{d}{ds} \tilde{\phi}_s(g)=\frac{d}{ds}(g\circ \phi_s)=dg_{\phi_{s}(-)} \cdot X_f=X_f(g\circ \phi_s)=\{\tilde{\phi}_s(g),f\} $$

and

$$ \tilde{\phi}_0(g)=g. $$

And also, $\tilde{\phi}_s$ respect the Poisson algebra structure.

Reciprocally, if we start with a function $f$ and we are able to associate a family of transformations $\{\tilde{\phi}^f_s\}$ from $\mathcal{C}^{\infty}(M)$ to $\mathcal{C}^{\infty}(M)$ that preserve the structure of Poisson algebra and such that

$\frac{d}{d s} \tilde{\phi}^f_s(g)=\left\{\tilde{\phi}^f_s(g),f \right\}$

$\tilde{\phi}^f_0(g)=g$

we can define a flow $\phi_s$ in $M$ such that

$$ \tilde{\phi}^f_s(g)=\tilde{\phi}_s(g) $$

We proceed by taking the vector field $\{-,f\}$ and the associated flow $\phi_s$, and using the two required properties, above. Since

$$ \frac{d}{ds} \tilde{\phi}_s(g)=\frac{d}{d s} \tilde{\phi}^f_s(g) $$

and

$$ \tilde{\phi}_0(g)=\tilde{\phi}^f_0(g) $$

we can show that $\tilde{\phi}_s(g)=\tilde{\phi}^f_s(g)$ by fixing a point $P\in M$.

We indeed have a duality, between transformations in $M$ and transformation in $\mathcal{C}^{\infty}(M)$. Why do we pass from the flow in the phase space $M$ to the "flow of observables"? Because this way, I guess, we can formulate everything in terms of algebras, and we get a deeper insight in the connection with Quantum Mechanics. Compare this with the discussion of derivations and 1-parameter group of automorphisms of the matrix algebra.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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