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Vector and Axial vector currents

  1. Jul 27, 2009 #1
    1. The problem statement, all variables and given/known data
    I have to show that the vector current [tex] \vec{V}^\mu = \overline{\psi} \gamma^\mu \vec{T} \psi [/tex] and the vector axial current [tex] \vec{A}^\mu = \overline{\psi} \gamma^\mu \gamma_5\vec{T} \psi [/tex] satisfy this

    [tex]
    \partial_\mu \vec{V}^\mu = i\overline{\psi}[M,\vec{T}] \psi

    [/tex]


    [tex]
    \partial_\mu \vec{A}^\mu = i\overline{\psi}\{M,\vec{T}\}\gamma_5 \psi
    [/tex]

    The brackets are commutators and the braces are anticommutators
    2. Relevant equations
    Dirac's equation for [tex] \psi [/tex] and [tex]\overline{\psi} [/tex]. Also, M is a mass matrix because [tex] \psi [/tex] is extended to a vector in flavor space and M is a diagonal matrix (entries are the quark masses). T are the pauli matrices divided by 2 and we are working only with two flavors. The column vector [tex] \psi [/tex] contains two entries then, and each of these are a 4-component Dirac spinor.


    3. The attempt at a solution
    Using Dirac's equations:
    [tex]
    \overline{\psi}(i\gamma^\mu\stackrel{\leftarrow}{\partial_\mu} + M) &=& 0
    [/tex]
    [tex]
    \partial_\mu\overline{\psi} \gamma^\mu &=& i\overline{\psi}M
    [/tex]

    [tex]
    (i\gamma^\mu{\partial_\mu} + M)\psi &=& 0

    [/tex]
    [tex]
    \gamma^\mu\partial_\mu{\psi} &=& -iM{\psi}

    [/tex]
    when we replace it in the gradient [tex]\partial_\mu \vec{V}^\mu [/tex]

    [tex]
    \partial_\mu \vec{V}^\mu = \partial_\mu\overline{\psi} \gamma^\mu \vec{T} \psi + \overline{\psi}\gamma^\mu\vec{T}\partial_\mu \psi [/tex]


    there's no problem substituting [tex]\partial_\mu\overline{\psi} \gamma^\mu = i\overline{\psi}M [/tex] in the first term of this last gradient but for the second term i don't get [tex]\gamma^\mu\partial_\mu{\psi} [/tex] but rather [tex]\gamma^\mu\vec{T}\partial_\mu \psi [/tex] which differ just by commuting [tex]\vec{T} [/tex] and [tex]\gamma^\mu [/tex]

    Same applies to [tex]\partial_\mu\vec{A}^\mu [/tex]
     
  2. jcsd
  3. Jul 27, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    T is a matrix in flavor space. The gammas are matrices in spinor space. I.e. their indices are of different types. You can commute them.
     
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