- #1
Dhyrim
- 1
- 0
The problem:
$$\mathcal{L} = F^{\mu \nu} F_{\mu \nu} + m^2 /2 \ A_{\mu} A^{\mu} $$
with: $$ F_{\mu \nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu} $$
1. Show that this lagrangian density is not gauge invariance
2.Derive the equations of motion, why is the Lorentzcondition still implied eventho there is no gauge invariance?
3.Construct the general solutions and discussI kinda know how to solve the first part, just by checking this transofrmation $$A_{\mu} \ \ to \ \ A_{\mu} + \partial_{\mu} f(x) $$
its clear that the lagrangian density is not invariant here.
I think the equations of motions should be:
$$ F^{\mu \nu} = A^{\nu} $$
and these are also, not gauge invariance. But why would the imply the condition:
$$\partial_{\mu} A^{\mu}$$
?
My guess it is because when you preform a gauge transform on the lagrangian density before deriving the equations of motion, you will get extra terms, these will make it so you get extra terms in the equations of motion, depending on this gauge transform, and they will vanish when this condition is in use.
I have no idea how to construct the general solution of this, can someone show me the answer to these?
Thx,
Dhyrim
$$\mathcal{L} = F^{\mu \nu} F_{\mu \nu} + m^2 /2 \ A_{\mu} A^{\mu} $$
with: $$ F_{\mu \nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu} $$
1. Show that this lagrangian density is not gauge invariance
2.Derive the equations of motion, why is the Lorentzcondition still implied eventho there is no gauge invariance?
3.Construct the general solutions and discussI kinda know how to solve the first part, just by checking this transofrmation $$A_{\mu} \ \ to \ \ A_{\mu} + \partial_{\mu} f(x) $$
its clear that the lagrangian density is not invariant here.
I think the equations of motions should be:
$$ F^{\mu \nu} = A^{\nu} $$
and these are also, not gauge invariance. But why would the imply the condition:
$$\partial_{\mu} A^{\mu}$$
?
My guess it is because when you preform a gauge transform on the lagrangian density before deriving the equations of motion, you will get extra terms, these will make it so you get extra terms in the equations of motion, depending on this gauge transform, and they will vanish when this condition is in use.
I have no idea how to construct the general solution of this, can someone show me the answer to these?
Thx,
Dhyrim