- #1
latentcorpse
- 1,444
- 0
So we have an action
[itex]I[x,p;e]=\int d \lambda \left( \dot{x}^\mu p_\mu - e (p^2+m^2) \right)[/itex]
where e is the einbein and [itex]p^2=g^{\mu \nu} p_\mu p_\nu[/itex]
We're asked to find [itex]\delta I[/itex] given the variations
[itex]\delta x^\mu=\epsilon(\lambda) \xi^\mu(x)[/itex] and [itex]\delta p_\mu = - \epsilon(\lambda) \partial_\mu \epsilon^\nu p_\nu[/itex]
So I find that [itex]\delta \dot{x}^\mu = \dot{\epsilon}(\lambda) \xi^\mu(x)[/itex]
and we have that
[itex]\delta I = \int d \lambda \left( \delta ( \dot{x}^\mu p_\mu ) - \delta ( ep^2+em^2) \right)[/itex]
[itex]=\int d \lambda \left( \delta \dot{x}^\mu p_\mu + \dot{x}^\mu \delta p_\mu -2e g^{\mu \nu} p_\mu \delta p_\nu \right)[/itex] since [itex]\delta m=0[/itex]
Then, this gives by substitution
[itex]= \int d \lambda \left( \dot{\epsilon} \xi^\mu p_\mu - \epsilon \dot{x}^\mu \partial_\mu \xi^\nu p_\nu + 2eg^{\mu \nu} p_\mu \epsilon \partial_\nu \xi^\rho p_\rho \right)[/itex]
But this appears to be going "off course" so I reckon I've messed up but I can't see anything wrong with it!
[itex]I[x,p;e]=\int d \lambda \left( \dot{x}^\mu p_\mu - e (p^2+m^2) \right)[/itex]
where e is the einbein and [itex]p^2=g^{\mu \nu} p_\mu p_\nu[/itex]
We're asked to find [itex]\delta I[/itex] given the variations
[itex]\delta x^\mu=\epsilon(\lambda) \xi^\mu(x)[/itex] and [itex]\delta p_\mu = - \epsilon(\lambda) \partial_\mu \epsilon^\nu p_\nu[/itex]
So I find that [itex]\delta \dot{x}^\mu = \dot{\epsilon}(\lambda) \xi^\mu(x)[/itex]
and we have that
[itex]\delta I = \int d \lambda \left( \delta ( \dot{x}^\mu p_\mu ) - \delta ( ep^2+em^2) \right)[/itex]
[itex]=\int d \lambda \left( \delta \dot{x}^\mu p_\mu + \dot{x}^\mu \delta p_\mu -2e g^{\mu \nu} p_\mu \delta p_\nu \right)[/itex] since [itex]\delta m=0[/itex]
Then, this gives by substitution
[itex]= \int d \lambda \left( \dot{\epsilon} \xi^\mu p_\mu - \epsilon \dot{x}^\mu \partial_\mu \xi^\nu p_\nu + 2eg^{\mu \nu} p_\mu \epsilon \partial_\nu \xi^\rho p_\rho \right)[/itex]
But this appears to be going "off course" so I reckon I've messed up but I can't see anything wrong with it!