1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Constructing conserved current from lagrangian

  1. Aug 2, 2014 #1

    CAF123

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ where ##F_{\alpha \beta} = \partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}## which means $$\mathcal L = \frac{1}{4} (\partial^{\alpha}A^{\beta} - \partial^{\beta}A^{\alpha})(\partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}) + \frac{1}{2}m^2A^{\alpha}_{\alpha} \,\,\,\,(1)$$ The canonical energy-momentum tensor is, using the relation

    $$T^{\mu \nu}_c = -\eta^{\mu \nu} \mathcal L + \frac{\partial \mathcal L}{\partial (\partial_{\mu}\Phi)}\partial^{\nu}\Phi,\,\,\,\,(2)$$

    $$T^{\mu \nu}_c = F^{\mu \alpha}\partial^{\nu}A_{\alpha} - \eta^{\mu \nu}\mathcal L$$

    Then from ##T^{\mu \nu}_B = T^{\mu \nu}_c + \partial_{\rho}B^{\rho \mu \nu}##, it is found that $$B^{\alpha \mu \nu} = F^{\alpha \mu}A^{\nu}\,\,\,\,(3)$$ using the formula $$B^{\mu \rho \nu} = \frac{1}{2}i \left\{\frac{\partial \mathcal L}{\partial (\partial_{\mu}A_{\gamma})} S^{\nu \rho}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\rho}A_{\gamma})} S^{\mu \nu}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\nu}A_{\gamma})} S^{\mu \rho}A_{\gamma}\right\}$$ My question is how is this equation obtained and how did they obtain ##(3)##? Did they make use of the explicit form of the spin matrix for a vector field?
    2. Relevant Equations

    (My question is from Di Francesco et al 'Conformal Field Theory' P.46-47).

    3. The attempt at a solution
    $$B^{\alpha \mu \nu} = \frac{i}{2}\left\{F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} + F^{\mu \gamma}S^{\alpha \nu}A_{\gamma} + F^{\nu \gamma}S^{\alpha \mu}A_{\gamma}\right\}$$ from simplifying the above. Concentrate on the first term. Then $$F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} = F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b} (S_{ab})^c_dA^d$$ Inputting the form of $S$ for a vector field, I get $$F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b}(\delta^c_a \eta_{bd} - \delta^c_b \eta_{da})A^d$$ But simplifying this and writing the other terms does not yield the result. Did I make a mistake upon insertion of the spin matrix?
     
  2. jcsd
  3. Aug 11, 2014 #2

    CAF123

    User Avatar
    Gold Member

    Can anybody help with this at all? Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Constructing conserved current from lagrangian
Loading...