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## Homework Statement

The Lagrangian density for a massive vector field ##C_{\mu}## is given by ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}## where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}##.

Derive the equations of motion and show that when ##m \neq 0## they imply ##\partial_{\mu}C^{\mu}=0##.

Further show that ##C_{0}## can be eliminated completely in terms of the other fields by ##\partial_{i}\partial^{i}C_{0}+m^{2}C_{0}=\partial^{i}\dot{C}_{i}##.

Construct the canonical momenta ##\Pi_{i}## conjugate to ##C_{i}, i=1,2,3## and show that the canonical momentum conjugate to ##C_{0}## is vanishing.

Construct the Hamiltonian density ##\mathcal{H}## in terms of ##C_{0},C_{i}## and ##\Pi_{i}##.

(Note: Do not be concerned that the canonical momentum for ##C_0## is vanishing. ##C_0## is non-dynamical - it is determined entirely in terms of the other fields using ##\partial_{i}\partial^{i}C_{0}+m^{2}C_{0}=\partial^{i}\dot{C}_{i}##.)

## Homework Equations

## The Attempt at a Solution

Given the Lagrangian density ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}##, where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}## and ##C_{\mu}## is a massive vector field,

##\frac{\partial \mathcal{L}}{\partial (\partial_{\rho}C_{\sigma})}=\frac{\partial}{\partial (\partial_{\rho}C_{\sigma})}\Big(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\Big)##

##\implies \frac{\partial \mathcal{L}}{\partial (\partial_{\rho}C_{\sigma})}=-\frac{1}{4}\frac{\partial}{\partial (\partial_{\rho}C_{\sigma})}[(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu})(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})]##

##\implies \frac{\partial \mathcal{L}}{\partial (\partial_{\rho}C_{\sigma})}=-\frac{1}{4}[({\eta^{\rho}}_{\mu}{\eta^{\sigma}}_{\nu}-{\eta^{\rho}}_{\nu}{\eta^{\sigma}}_{\mu})(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})+(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu})(\eta^{\rho\mu}\eta^{\sigma\nu}-\eta^{\rho\nu}\eta^{\sigma\mu})]##

##\implies \frac{\partial \mathcal{L}}{\partial (\partial_{\rho}C_{\sigma})}=-\frac{1}{4}[\partial^{\rho}C^{\sigma}-\partial^{\sigma}C^{\rho}-\partial^{\sigma}C^{\rho}+\partial^{\rho}C^{\sigma}+\partial^{\rho}C^{\sigma}-\partial^{\sigma}C^{\rho}-\partial^{\sigma}C^{\rho}+\partial^{\rho}C^{\sigma}]##

##=-(\partial^{\rho}C^{\sigma}-\partial^{\sigma}C^{\rho})##

##=-F^{\rho\sigma}##

and

##\frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{\partial}{\partial C_{\rho}}(\frac{1}{2}m^{2}C_{\mu}C^{\mu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{\partial}{\partial C_{\rho}}(\frac{1}{2}m^{2}\eta^{\mu\nu}C_{\mu}C_{\nu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{1}{2}m^{2}\eta^{\mu\nu}\frac{\partial}{\partial C_{\rho}}(C_{\mu}C_{\nu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{1}{2}m^{2}\eta^{\mu\nu}({\eta^{\rho}}_{\mu}C_{\nu}+{C_{\mu}\eta^{\rho}}_{\nu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{1}{2}m^{2}({\eta^{\rho}}_{\mu}\eta^{\mu\nu}C_{\nu}+{\eta^{\nu\mu}C_{\mu}\eta^{\rho}}_{\nu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{1}{2}m^{2}({\eta^{\rho}}_{\mu}C^{\mu}+{\eta^{\rho}}_{\nu}C^{\nu})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = \frac{1}{2}m^{2}(C^{\rho}+C^{\rho})##

##\implies \frac{\partial \mathcal{L}}{\partial C_{\rho}} = m^{2}C^{\rho}##

so that

##\frac{\partial \mathcal{L}}{\partial C_{\nu}}-\partial_{\mu}\Big(\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}C_{\nu})}\Big)=0 \implies m^{2}C^{\nu}+\partial_{\mu}F^{\mu\nu}=0##.

So, the equations of motion are ##m^{2}C^{\nu}+\partial_{\mu}F^{\mu\nu}=0##.

Am I correct so far?