# Momentum of a charged particle in a magnetic field

In the presence of a magnetic field, the canonical momentum of a charge particle changes from $p_{i}\equiv mv^i$ to $\pi_{i}\equiv p_{i}+eA_{i}$, where $e$ is the charge of the particle.

1. It is possible to define another kind of canonical momentum $\tilde{\pi}_{i}=p_{i}-eA_{i}$?

2. How can you show that this new definition of momentum is not gauge-invariant?

## 1 Answer

Just as $qV$ is potential energy, $q \vec{A}$ could easily be considered to be 'potential momentum', while ordinary momentum could be considered 'kinetic momentum'.
$${\vec p}_{total}={\vec p}_{kinetic}+{\vec p}_{potential}$$

While this 'total momentum' is not gauge invariant, conservation of total momentum is gauge invariant, and that is the critical item.

• Ah, I see! So, under a gauge transformation, $\tilde{\pi} \rightarrow \tilde{\pi}-e\nabla\chi$ for a gauge transformation $A\rightarrow A+\nabla\chi$, which is why $\tilde{\pi}$ is not gauge invariant? Nov 14, 2016 at 7:46
• That's right. The gauge choice is time invariant and so the time derivative will cause the extra gauge term to drop out when you look at conservation of momentum. Dec 6, 2016 at 9:14