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I am currently having trouble with an exercise: writing the propagator of a particle coupled to a magnetic field.

So the lagrangian is [itex]L_A (\vec{x},\dot{\vec{x}}^2) = \frac{m}{2}\dot{\vec{x}} + e\vec{A}.\dot{\vec{x}}[/itex]

And it says that I should solve it in two different ways:

-by writing [itex]e\vec{A}.\dot{\vec{x}}[/itex] has a derivative relative to t

-by completing the square in the Lagrangian, ie something like [itex]L_A (\vec{x},\dot{\vec{x}}) = \frac{m}{2}(\dot{\vec{x}}+\frac{e}{m}\vec{A})^2 - \frac{e^2}{2m}\vec{A}^2[/itex] I guess.

But still, I develop a few lines more of calculus but I am stuck. I would be really happy if someone could explain how to proceed.

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# Path integral for a particle coupled to a magnetic field

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