The conjugate momentum density, following as a conserved quantity with Noethers Theorem, from invariance under displacement of the field itself, i.e. $\Phi \rightarrow \Phi'=\Phi + \epsilon$, is given by $\pi=\frac{\partial L}{\partial ( \dot{\Phi})}$.
On the other hand the physical momentum density, following as a conserved quantity with Noethers Theorem, from invariance under translations, i.e. $\Phi(x) \rightarrow \Phi(x')=\Phi(x+\epsilon)$, is given by $\Pi = \frac{\partial L}{\partial ( \dot{\Phi})} \frac{\partial \Phi}{\partial x}$.
Does anyone know a enlightening interpretation of the conjugate momentum? Furthermore why do we, in quantum field theory, impose commutation relations with the conjugate momentum instead of the physical momentum?
(For brevity all possible indices are supressed)
Best Answer
The momentum you call $\Pi$ is the momentum in a certain spatial direction. The momentum you call $\pi$ is the momentum in a direction in field space.
For example, consider vertical waves on a horizontal string. The "field" $y(x, t)$ represents the vertical displacement of the string at point $x$. Then $\Pi$ is the ordinary horizontal momentum, since waves on the string move horizontally, while $\pi$ is the ordinary vertical momentum, since the string itself only moves up and down.