# Noether's Theorem

1. Oct 20, 2009

### div curl F= 0

1. Calculate the conserved charges and currents for a scalar theory whose action is invariant under infinitesimal spacetime translations and infinitesimal lorentz transformations

2. $$L = (\partial_\alpha \phi^\dagger)(\partial^\alpha \phi) - V$$
$$j^{\alpha, \beta} = i \frac{\partial \mathcal{L}}{\partial (\partial_{\beta} \phi_a)} (T^{\alpha})^b_a \phi_b$$

ST: $$x^\alpha \to x^\alpha + \epsilon^\alpha$$
LT: $$x^\alpha \to x^\alpha + W^\alpha_\beta x^\beta \;\;;\;\; W_{\alpha \beta} = - W_{\beta \alpha}$$

3. For the spacetime translation part, I've calculated the generators of the transformation to be: $$(T_\alpha)^b_a = - i \delta^b_a \partial_\alpha$$, the conserved currents and charges to be:
$$j^{\alpha,\beta} = (\partial^\alpha \phi^\dagger)(\partial^\beta \phi) + (\partial^\beta \phi^\dagger)(\partial^\alpha \phi)$$

$$Q^\alpha = \int d^3 x \left[(\partial^0 \phi^\dagger)(\partial^\alpha \phi) + (\partial^\alpha \phi^\dagger)(\partial^0 \phi)$$

For the lorentz transformation I've calculated the generators of the transformation to be:

$$(T_{\alpha \beta})^b_a = -i \delta^b_a x_\alpha \partial_{\beta}$$

Are these correct?

Can they be simplified further into things like the Hamiltonian/Total Energy etc?