# Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $$\{ab,c\} = a\{b,c\} + \{a,c\}b$$ etc. But I am working with tensor densities, so a PB is defined as

$$\{A,B\} = \int\left[\frac{\delta A}{\delta \gamma_{ij}}\frac{\delta B}{\delta \pi^{ij}}-\frac{\delta B}{\delta \gamma_{ij}}\frac{\delta A}{\delta \pi^{ij}}\right]d^3x.$$

I have not been able to prove that the canonical PB properties are valid in this case. Any insights would be much appreciated.

• 1. What difficulty exactly do you have in showing the relations for your expression? The more detailed you are in your question, the likelier an answer will actually help you. 2. Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. – ACuriousMind Jun 20 at 10:42