Interpretation of Conjugate Momentum in Field Theory

Question

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)

Answer 1

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.

Answer 2

Conjugate momentum in field theory is really just an infinite particle generalization of conjugate momentum in classical mechanics. The reason we impose commutation relations using the conjugate momentum is due to Dirac's canonical quantization prescription. This is well explained by Qmechanic in this answer.

I'll briefly summarize the argument. To canonically quantize a classical theory you must first express it in Hamiltonian form. This involves defining the conjugate momentum, and imposing Hamilton's equations with Poisson brackets. You then promote Poisson brackets to commutators. It turns out that conjugate momentum is exactly the right variable that makes this approach work mathematically.

I'm not aware of a physical interpretation of conjugate momentum in field theory, but I'd be interested to hear other people's thoughts on this!

Answer 3

The conjugate momentum of the electromagnetic field $A_\mu$ is the electric field $\vec E$.