- #1
spookyfish
- 53
- 0
I am trying to derive the conserved charge from the symmetry of the action under Lorentz transformations, but I am doing something wrong.
Noether's theorem states that the current is
[tex]
J^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)} \delta\phi - T^{\mu \nu}\delta x_\nu
[/tex]
For an infinitesimal Lorentz transformations
[tex]
\Lambda^\mu _\nu = \delta^\mu_\nu +\omega ^\mu_\nu
[/tex]
I get
[tex]\delta x_\nu = \omega_{\nu \sigma} x^\sigma, \quad \delta \phi = -\omega^\mu _\nu x^\nu \partial_\mu \phi [/tex]
This gives
[tex]
J^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)}(-\omega^\nu _\sigma x^\sigma \partial_\nu \phi) -T^{\mu \nu}\omega_{\nu \sigma} x^\sigma
[/tex]
where
[tex]
T^{\mu \nu} = \frac{\partial \cal L}{\partial(\partial_\mu \phi)} \partial^\nu \phi -\cal L \eta^{\mu \nu}
[/tex]
so
[tex]
J^\mu = -\omega_{\nu \sigma} x^\sigma \left(2\frac{\partial \cal L}{\partial(\partial_\mu \phi)}\partial^\nu \phi - \cal L \eta^{\mu \nu} \right)
[/tex]
this is "almost" the correct form but I have this factor of 2 which is wrong. I will be happy to know where is my mistake
Noether's theorem states that the current is
[tex]
J^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)} \delta\phi - T^{\mu \nu}\delta x_\nu
[/tex]
For an infinitesimal Lorentz transformations
[tex]
\Lambda^\mu _\nu = \delta^\mu_\nu +\omega ^\mu_\nu
[/tex]
I get
[tex]\delta x_\nu = \omega_{\nu \sigma} x^\sigma, \quad \delta \phi = -\omega^\mu _\nu x^\nu \partial_\mu \phi [/tex]
This gives
[tex]
J^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)}(-\omega^\nu _\sigma x^\sigma \partial_\nu \phi) -T^{\mu \nu}\omega_{\nu \sigma} x^\sigma
[/tex]
where
[tex]
T^{\mu \nu} = \frac{\partial \cal L}{\partial(\partial_\mu \phi)} \partial^\nu \phi -\cal L \eta^{\mu \nu}
[/tex]
so
[tex]
J^\mu = -\omega_{\nu \sigma} x^\sigma \left(2\frac{\partial \cal L}{\partial(\partial_\mu \phi)}\partial^\nu \phi - \cal L \eta^{\mu \nu} \right)
[/tex]
this is "almost" the correct form but I have this factor of 2 which is wrong. I will be happy to know where is my mistake