Mandl and Shaw 2.3

  • Thread starter jdstokes
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  • #1
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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?
 

Answers and Replies

  • #2
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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 =)
 
  • #3
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
 
  • #4
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Ahh yes, thanks same.
 

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