jdstokes
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[SOLVED] Mandl and Shaw 2.3
Show that the Lagrangian density
[itex]\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[/itex]
for the real vector field [itex]\varphi^\alpha[/itex] leads to the field equations
[itex][g_{\alpha\beta}(\square+ \mu^2)-\partial_\alpha\partial_\beta]\varphi^\beta=0[/itex]
and that the field satisfies the Lorentz condition [itex]\partial_\alpha \varphi^\alpha = 0[/itex].
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?
Show that the Lagrangian density
[itex]\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[/itex]
for the real vector field [itex]\varphi^\alpha[/itex] leads to the field equations
[itex][g_{\alpha\beta}(\square+ \mu^2)-\partial_\alpha\partial_\beta]\varphi^\beta=0[/itex]
and that the field satisfies the Lorentz condition [itex]\partial_\alpha \varphi^\alpha = 0[/itex].
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?