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Are all quadratic terms in gauge fields necessarily mass terms?

  1. Feb 2, 2010 #1
    Can someone please help me out with mass terms in the general case for a lagrangian?

    It is known that for n scalar fields, any quadratic in these fields will be a mass term.
    For classical fields [tex]\varphi_{j}[/tex] with the most genral possible expression being [tex]M^{jk}\varphi_{j}\varphi_{k}[/tex] , the matrix [tex]M^{jk}[/tex] is guaranteed to be symmetric and so can be diagonalised with an orthogonal similarity transformation. So there is no argument there - we get a sum of squares after diagonalisation of the form [tex]\sum_{j} M^{jj}\varphi_{j}\varphi_{j}[/tex]

    For the case of gauge fields, however, it does not seem (??? help me out here!!) that just any quadratic at all will necessarily be a mass term. Here is the reasoning as I see it:

    No one would disagree that a term like [tex] M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu} [/tex] is definitely a mass term. Again, for classical fields the mass matrix [tex]M^{jk}[/tex] is guaranteed symmetric by the sum over symmetric terms and so is once more diagonalisable with an orthogonal similarity transformation.

    Things seem different for the most general case. For example with a sum like[tex] M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}[/tex] , one would expect that a Lorentz transformation can reduce the term[tex] M_{\mu\nu}^{jk}[/tex] to something of the form [tex]g_{\mu\nu}m^{jk}[/tex].

    In this way, the 'mass' term becomes [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] after the Lorentz transformation has been applied.

    Observe however, that for the Lorentz transformation [tex]L_{\alpha}^{\mu}[/tex] which achieves this change we have

    [tex]M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}[/tex] .

    Multiplying both sides of this equation by [tex]{(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}[/tex], we obtain the result that

    [tex]M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}[/tex].

    This follows from the orthogonality of the Lorentz transformations with respect to the metric [tex]g_{\alpha\beta}[/tex].

    So the upshot of this appears to be that unless terms are of the form [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] they cannot represent mass terms and are in fact, self interaction terms.

    Please help me out here, as this is the only way I can interpret quadratic terms in the gauge fields at present.
     
  2. jcsd
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