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1 vote

### Partial derivatives of canonical momenta in Poisson brackets

It's just like in thermodynamics, you need to keep track which variables you are using as coordinates for your function. It's always a good idea to formulate things in a coordinate independent ...
1 vote

When evalauating partial derivatives you need to specify what is being kept fixed as well as what is varying. The Poisson bracket is $$\{F(p,q),G(p,q)\}=\sum_i \left(\frac{\partial F(q,p)}{\partial ... 5 votes ### Conceptual question Einstein-Hilbert action and QFT in curved spacetime I think that your question is about backreaction. One can perfectly have an action like$$S = \int d^4x\sqrt{-g}\left(R-2\Lambda -\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi - V(\phi)\right) ...
1. Relation between the actions The scalar action you wrote is the one for a minimally coupled field. One could use the more general action (I'm working with the $-+++$ convention and the $+++$ MTW ...
The Einstein-Hilbert action (1) describes empty spacetime. The action (2) describes matter. So if you want to describe a scalar in general relativity you need to consider $S_{EH} + S_M$. Similarly if ...