Monday, August 10, 2015

Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator?


In quantum field theory, with the field $\phi$ and the momentum $\pi$ being operators, their time evolution is governed (in the Heisenberg-picture) by the Heisenberg equation:


\begin{align} \dot{\phi} = \frac{i}{\hbar}[ \hat{H}, \phi] \\ \dot{\pi} = \frac{i}{\hbar}[ \hat{H}, \pi]. \\ \end{align}


Now, in case the Hamiltonian operator $\hat{H}=\int d^3x ~\hat{\cal H}$ can be written as an integral over the hamiltonian density $\hat{\cal H}$, and the fields and the momenta commute at non-equal positions, do the same equations hold as well with the Hamiltonian operator being replaced by it's density? What would the caveats be?



\begin{align} \dot{\phi} = \frac{i}{\hbar}[ \hat{\cal H}, \phi] \\ \dot{\pi} = \frac{i}{\hbar}[ \hat{\cal H}, \pi]. \\ \end{align}



Answer



You have $\hat{H} = \int d^3x \hat{\tilde{H}}(x)$. That implies that canonical Relations will be slightly altered.


For a Quantum field Operator $\hat{\phi}(x',t)$ distributed over space $x'$ and time $t$, you will have a relation like the following:


$[\hat{\tilde{H}}(x),\hat{\phi}(x',t)] = \frac {\partial}{\partial t} \hat{\phi(x',t)} \delta(x-x')$.


The Delta function factor ensures not only the commutation of Operators for nonequal space Points; also that after Integration over space, the ordinary commutation Relations are obtained


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