- #1
spaghetti3451
- 1,344
- 33
Homework Statement
The motion of a complex field ##\psi(x)## is governed by the Lagrangian ##\mathcal{L} = \partial_{\mu}\psi^{*}\partial^{\mu}\psi-m^{2}\psi^{*}\psi-\frac{\lambda}{2}(\psi^{*}\psi)^{2}##.
Write down the Euler-Lagrange field equations for this system.
Verify that the Lagrangian is invariant under the infinitesimal transformation ##\delta \psi = i \alpha \psi, \delta \psi^{*} = - i \alpha \psi^{*}##
Derive the Noether current associated with this transformation and verify explicitly that it is conserved using the field equations satisfied by ##\psi##.
Homework Equations
The Attempt at a Solution
##\mathcal{L} = \partial_{\mu}\psi^{*}\partial^{\mu}\psi-m^{2}\psi^{*}\psi-\frac{\lambda}{2}(\psi^{*}\psi)^{2}##
Here's my derivation of the Euler-Lagrange field equations:
##\frac{\partial \mathcal{L}}{\partial \psi} = -m^{2}\psi^{*}-\lambda (\psi^{*}\psi)\psi^{*}##
and
##\frac{\partial \mathcal{L}}{\partial (\partial_{\rho}\psi)} = \frac{\partial}{\partial (\partial_{\rho}\psi)} \Big( \partial_{\mu}\psi^{*}\partial^{\mu}\psi \Big)= \frac{\partial}{\partial (\partial_{\rho}\psi)} \Big( \eta^{\mu\nu}\partial_{\mu}\psi^{*}\partial_{\nu}\psi \Big)=\eta^{\mu\nu}\frac{\partial}{\partial (\partial_{\rho}\psi)} \Big( \partial_{\mu}\psi^{*}\partial_{\nu}\psi \Big)=\eta^{\mu\nu}{\eta^{\rho}}_{\nu}\partial_{\mu}\psi^{*}={\eta^{\rho}}_{\nu}\eta^{\nu\mu}\partial_{\mu}\psi^{*}=\partial^{\rho}\psi^{*}##
so that
##\frac{\partial \mathcal{L}}{\partial \psi} - \partial_{\mu}\Big(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\psi)}\Big)=0##
##-m^{2}\psi^{*}-\lambda(\psi^{*}\psi)\psi^{*}-\partial^{2}\psi^{*}=0##
##(\partial^{2}+m^{2}+\lambda|\psi|^{2})\psi^{*}=0##
Furthermore,
##\frac{\partial \mathcal{L}}{\partial \psi^{*}} = -m^{2}\psi-\lambda (\psi^{*}\psi)\psi##
and
##\frac{\partial \mathcal{L}}{\partial (\partial_{\rho}\psi^{*})} = \frac{\partial}{\partial (\partial_{\rho}\psi^{*})} \Big( \partial_{\mu}\psi^{*}\partial^{\mu}\psi \Big)={\eta^{\rho}}_{\mu}\partial^{\mu}\psi=\partial^{\rho}\psi##
so that
##\frac{\partial \mathcal{L}}{\partial \psi^{*}} - \partial_{\mu}\Big(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\psi^{*})}\Big)=0##
##-m^{2}\psi-\lambda(\psi^{*}\psi)\psi-\partial^{2}\psi=0##
##(\partial^{2}+m^{2}+\lambda|\psi|^{2})\psi=0##
Am I correct so far?