I'm going back over some of my old notes for a current project, and I'm trying to figure out if I made an error or if I once knew something that I've now forgotten.
Consider a local field theory containing a set of fields $\phi^{(a)}$, for which the Lagrangian density is $$ \mathcal{L}(\phi^{(a)}, \dot{\phi}^{(a)}, \partial_i \phi^{(a)} ). $$ Here, $\partial_i$ stands for spatial derivatives only, i.e., we've already done a decomposition into a preferred foliation of spacetime (if necessary.) We can define a set of conjugate field momenta via the usual relation: $$ \pi^{(a)} \equiv \frac{\delta \mathcal{L}}{\delta \dot{\phi}^{(a)} }. $$ The Hamiltonian density will then be $$ \mathcal{H} = \sum_a \pi^{(a)}\dot{\phi}^{(a)} - \mathcal{L}. $$ We can then define a Poisson bracket for field quantities, of the form $$ \{ f, g \} \equiv \sum_a \left[ \frac{ \delta f}{\delta \phi^{(a)}} \frac{ \delta g}{\delta \pi^{(a)}} - \frac{ \delta g}{\delta \phi^{(a)}} \frac{ \delta f}{\delta \pi^{(a)}}\right], $$ where $f$ and $g$ are in principle any two quantities that depend on the fields and the momenta.
Here are my questions:
- Is it always the case under these definitions that $$ \{ \partial_i f, g \} \overset{?}{=} \partial_i \{f, g\} $$ for any two quantities $f$ and $g$?
- Is it always the case under these definitions that $$ \{ \partial_i f, \mathcal{H} \} \overset{?}{=} \partial_i \{f, \mathcal{H} \} $$ for any quantity $f$?
