A relativistic particle of rest mass $m$ and electric charge $q$ follows a worldline $x^{\mu}(\lambda)$ in Minkowski spacetime where $\lambda=\lambda(\tau)$ is an arbitrary parameter which increases monotonically with the proper time $\tau$. We consider the motion of the particle in a background electromagnetic field with four-vector potential $A^{\mu}(x)$ between initial and final values of the proper time denoted $\tau_{i}$ and $\tau_{f}$ respectively.

(i) Write down an action for the particle's motion. Explain what is meant by a gauge transformation of the electromagnetic field. How does the action change under a gauge transformation?

(ii) Derive an equation of motion for the particle by considering the variation of the action with respect to the worldline $x^{\mu}(\lambda)$. Setting $\lambda=\tau$ show that your equation of motion reduces to the Lorentz force law,

$$m \frac{d u^{\mu}}{d \tau}=q F^{\mu \nu} u_{\nu}$$

where $u^{\mu}=d x^{\mu} / d \tau$ is the particle's four-velocity and $F^{\mu \nu}=\partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu}$ is the Maxwell field-strength tensor.

(iii) Working in an inertial frame with spacetime coordinates $x^{\mu}=(c t, x, y, z)$, consider the case of a constant, homogeneous magnetic field of magnitude $B$, pointing in the $z$-direction, and vanishing electric field. In a gauge where $A^{\mu}=(0,0, B x, 0)$, show that the equation of motion $(*)$ is solved by circular motion in the $x-y$ plane with proper angular frequency $\omega=q B / m$.

(iv) Let $v$ denote the speed of the particle in this inertial frame with Lorentz factor $\gamma(v)=1 / \sqrt{1-v^{2} / c^{2}}$. Find the radius $R=R(v)$ of the circle as a function of $v$. Setting $\tau_{f}=\tau_{i}+2 \pi / \omega$, evaluate the action $S=S(v)$ for a single period of the particle's motion.