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Noether's Theorem

  1. Oct 20, 2009 #1
    1. Calculate the conserved charges and currents for a scalar theory whose action is invariant under infinitesimal spacetime translations and infinitesimal lorentz transformations

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

    ST: [tex] x^\alpha \to x^\alpha + \epsilon^\alpha [/tex]
    LT: [tex] x^\alpha \to x^\alpha + W^\alpha_\beta x^\beta \;\;;\;\; W_{\alpha \beta} = - W_{\beta \alpha} [/tex]

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

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

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

    [tex] (T_{\alpha \beta})^b_a = -i \delta^b_a x_\alpha \partial_{\beta} [/tex]

    Are these correct?

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