# Mandl and Shaw 2.3

## Main Question or Discussion Point

[SOLVED] Mandl and Shaw 2.3

Show that the Lagrangian density

$\mathcal{L} = -\frac{1}{2}\partial_\alpha \varphi_\beta \partial^\alpha \varphi^\beta + \frac{1}{2} \partial_\alpha \varphi^\alpha \partial_\beta \varphi^\beta + \frac{\mu^2}{2}\varphi_\alpha \varphi^\alpha$

for the real vector field $\varphi^\alpha$ leads to the field equations

$[g_{\alpha\beta}(\square+ \mu^2)-\partial_\alpha\partial_\beta]\varphi^\beta=0$

and that the field satisfies the Lorentz condition $\partial_\alpha \varphi^\alpha = 0$.

The first part is a simple matter of using the Lagrange equation for the fields. I'm not sure where this Lorentz condition comes from. Does it follow from some symmetry of the Lagrangian?

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This is the Proca field? As I'm not sure whether this is a homework problem of yours or not... I'll just assume you've consulted wikipedia already =)

samalkhaiat
No, it is an algebraic property of the field equation. Just operate with $\partial^{\alpha}$.