Mandl and Shaw 2.3

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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?
 

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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
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Does it follow from some symmetry of the Lagrangian?

No, it is an algebraic property of the field equation. Just operate with [itex]\partial^{\alpha}[/itex].

sam
 
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Ahh yes, thanks same.
 

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