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Homework Help: Gauge symmetries of a theory

  1. Mar 13, 2010 #1
    1. The problem statement, all variables and given/known data

    I want to derive Gauge symmetries of the following gauge theory:

    [tex]S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}^{\;\;IJ} [/tex]

    Where [tex]B[/tex] is an antisymmetric tensor of rank two and [tex]F[/tex] is the curvature of a connection [tex]A[/tex] i.e: [tex]F=dA+A\wedge A[/tex], [tex]\mu,\nu...[/tex] are space-time indices and [tex]I,J...[/tex] are Lie Algebra indices (internal indices) I would like to find its symmetries.

    2. Relevant equations

    I rewrite the Lagrangian by splitting time and space indices [tex]\{\mu,\nu...=0..3\}\equiv \{0; i,j,...=1..3\}[/tex] I find:

    [tex]L = \int d^3 x\;(P^i_{\;IJ}\dot{A}_i+B_i^{\,IJ}\Pi^i_{\,IJ}+A_0^{\;IJ}\Pi_{IJ})[/tex]

    Where [tex]\dot{A}_i = \partial_0 A_i[/tex], [tex]P^i_{\;IJ} = 2\epsilon^{ijk}B_{jk\,IJ}[/tex] is hence the conjugate momentum of [tex]A_i^{\,IJ}[/tex]

    [tex]B_i^{\,IJ}[/tex] and [tex]A_0^{\;IJ}[/tex] being Lagrange multipliers we obtain respectively two primary and two secondary constraints:

    [tex]\Phi_{IJ} = P^0_{\;IJ} \approx0[/tex]

    [tex]\Phi_{\;\;IJ}^{\mu\nu} = P^{\mu\nu}_{\;\;IJ} \approx0[/tex]

    [tex]\Pi^i_{\,IJ} = 2\epsilon^{ijk}F_{jk\,IJ} \approx0[/tex]

    [tex]\Pi_{IJ}=(D_i P^i)_{IJ} \approx0[/tex]

    Where [tex]P^0_{\;IJ}[/tex] are the conjugate momentums of [tex]A_0^{\,IJ}[/tex] and [tex]P^{\mu\nu}_{\;\;IJ}[/tex] those of [tex]B_{\mu\nu}^{\;\;IJ}[/tex]. Making these constraints constant in time produces no further constraints.

    Whiche gives us a general constraint:

    [tex]\Phi = \int d^3 x \;(\epsilon^{IJ}P^0_{\,IJ}+\epsilon_{\mu\nu}^{IJ}\;P^{\mu\nu}_{\;\;IJ}+\eta^{IJ}\Pi_{IJ}+\eta_i^{IJ}\Pi^i_{\;IJ})[/tex]

    3. The attempt at a solution

    Each quantity [tex]F[/tex] have thus a Gauge transformation [tex]\delta F = \{F,\Phi\}[/tex] where [tex]\{...\}[/tex] denotes the Poisson bracket.

    Knowing that this theory have the following Gauge symmetry:

    [tex]\delta A = D\omega[/tex]

    [tex]\delta B = [B,\omega][/tex]

    Where [tex]\omega[/tex] is a 0-form, I would like to retrieve these transformations using the relation below. (where [tex]\Phi[/tex] is considered as the generator of the Gauge symmetry) but my problem is that I don't know how to proceed, I already did this with a Yang-Mills theory and it worked... but for this theory it seems to le intractable! Someone to guide me?
     
  2. jcsd
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