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A Classical Yang-Mills theory

  1. Apr 9, 2017 #1
    Let us consider a classical field theory with gauge fields ##A_{\mu}^{a}## and a scalar ##\phi^{a}## such that the Lagrangian is gauge-invariant under the transformation of

    1. the gauge fields ##A_{\mu}^{a}## in the adjoint representation, with dimension ##D_{\bf R}##, of the gauge group ##SU(N)##, and
    2. the scalar ##\phi^{a}## in the fundamental representation, with dimension ##N##, of the gauge group ##SU(N).##

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    1. Why can we represent ##\phi## as a traceless Hermitian ##N \times N## matrix, so that ##\phi = \phi^{a}T^{a}## where the ##T^a## are the representation matrices in the fundamental representation?

    2. Why can we write down the variation of ##\phi## under a gauge transformation with gauge parameters ##\theta^{a}## as

    $$\delta\phi = ig[\theta^{a}T^{a},\phi]$$

    and the gauge covariant derivative as

    $$D_{\mu}\phi = \partial_{\mu}\phi - igA_{\mu}^{a}[T^{a},\phi]?$$
     
  2. jcsd
  3. Apr 10, 2017 #2

    dextercioby

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    The only explanation that I know of is the one by the Craiova school: C. Bizdadea, E. M. Cioroianu, M. T. Miauta , I. Negru, and S. O. Saliu.
    Lagrangian cohomological couplings among vector fields
    and matter fields, Ann. Phys. (Leipzig) 10 (2001) 11––12, 921––934
     
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