- #1

loops496

- 25

- 3

## Homework Statement

Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then

##[H,F]-i \partial_0 F##

## Homework Equations

For KG we have:

##H=\frac{1}{2} \int d^3\mathbf{x}(\Pi^2 + (\nabla \Phi)^2 + m^2 \Phi^2)##

##\Phi(x) = \int \frac{d^3 \mathbf{p}}{(2\pi)^3 \sqrt{2E_{\mathbf{p}}}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##

##\Pi(x) = \int \frac{-i d^3 \mathbf{p}}{(2\pi)^3 }\sqrt{\frac{2E_{\mathbf{p}}}{2}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##

##[\Phi(t,\mathbf{x}),\Pi(t,\mathbf{y})]=i\delta^{(3)}(\mathbf{x}-\mathbf{y})##

##\Phi(x) = \int \frac{d^3 \mathbf{p}}{(2\pi)^3 \sqrt{2E_{\mathbf{p}}}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##

##\Pi(x) = \int \frac{-i d^3 \mathbf{p}}{(2\pi)^3 }\sqrt{\frac{2E_{\mathbf{p}}}{2}}(a_{\mathbf{p}} e^{-ipx} + a_{\mathbf{p}}^\dagger e^{ipx} )##

##[\Phi(t,\mathbf{x}),\Pi(t,\mathbf{y})]=i\delta^{(3)}(\mathbf{x}-\mathbf{y})##

## The Attempt at a Solution

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I have written ##F## as a generic two variable polynomial in ##\Phi,\;\Pi## but I don't know how to tackle that commutator, any help or hint is appreciated.