When (canonically) quantizing a classical system we promote the Poisson brackets to (anti-)commutators. Now I was wondering how much of Poisson bracket structure is preserved. For example for a classical (continuous) system we have(adsbygoogle = window.adsbygoogle || []).push({});

$$ \lbrace \phi(z), f(\Pi(y)) \rbrace = \frac{\delta f(\Pi(y))}{\delta \Pi(z)}, $$

where the derivative is the functional derivative and the bracket denotes the classical poisson bracket for fields

$$ \lbrace F, G \rbrace := \int \text{d} x \left[\frac{\delta F}{\delta \phi(x)} \frac{\delta G}{\delta \Pi(x)} - \frac{\delta G}{\delta \phi(x)} \frac{\delta F}{\delta \Pi(x)}\right]. $$

Does this mean that when we quantize using the rule

$$ \lbrace \phi, \Pi \rbrace \rightarrow -i \left[ \phi, \Pi \right]_{\pm}, $$

where + stands for commutator (bosons) and - stands for anti-commutator (fermions), we automatically obtain

$$ \left[ \phi(x), f(\Pi(y)) \right]_{\pm} = i\frac{\delta f(\Pi(y))}{\delta \Pi(z)},$$

and similar formulae for other structures of the Classical Poisson bracket, (say Hamilton equations of motion)? I was wondering this because I want to compute

$$ \left[ H_{12}, \frac{1}{E-H_{22}}\right], $$

where $$H_{12}$$ and $$H_{22}$$ are functions of several different (second-quantized) creation and annihilation operators. I was able to check the preservation of this particular rule for the first quantized situation

$$[x,p]= i,$$

here it is easy to check that this leads to

$$[x,f(p)] = i \partial_{p} f(p),$$

either by using a test function and the regular coordinate space representation of the operators x and p, or simply by plugging in a monomial function for f(p) and using the rules for commutation. I am not sure how to prove a generalized version of this though.

Any help would be appreciated.

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# Preservation of Poisson Bracket Structure upon quantization?

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