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Deriving the dirac equation

  1. Jan 30, 2013 #1
    Hi i am trying to derive the Dirac equation of the form:
    [itex] [i\gamma^0 \partial_0 + i\frac{1}{a(t)}\gamma.\nabla +i\frac{3}{2}(\frac{\dot{a}}{a})\gamma^0 - (m+h\phi)]\psi [/itex] where a is the scale factor for expnasion of the universe.

    I understand that the matter action is [itex]S=\int d^{4}x e [\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - V(\phi) + i \bar{\psi}\bar{\gamma}^{\mu}\vec{D}_{\mu}\psi -(m+h\phi)\bar{\psi}\psi)] [/itex] but i don't understand firstly why there is a vierbein and not a [itex]\sqrt{-g}[/itex] term.

    I don't really understand why this is the case [itex]D_{\mu}=\frac{1}{4}\bar{\psi}\bar{\gamma}^{\mu} \gamma_{\alpha \beta}\omega^{\alpha \beta}_{\mu}[/itex] and why the arrow above the D is gone.

    And lastly I don't understand why [itex]\bar{\gamma}^{i}=\frac{1}{a(t)}\gamma^{i}[/itex]

    I understand that one needs to vary the action and i can do that bit but I don't understand some of these conversions, thx. I would appareciate any help that anyone can offer in tis challenge.
  2. jcsd
  3. Jan 31, 2013 #2


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    To deal with spinors in curved spacetimes (or even just curvilinear coordinates) you need to use a set of basis vectors. This is because the gamma matrices that obey {γμ, γν} = 2gμν aren't constant, so we use instead matrices referred to a basis, in which {γa, γb} = 2ηab.

    The covariant derivative is Dμ = ∂μ - (1/4)σabωabμ where σab is the usual Dirac matrix, and ωabμ are the Ricci rotation coefficients associated with the vierbein.

    I think the only reason there's an arrow over the D is to remind us that it acts on the spinor to its right.
  4. Jan 31, 2013 #3
    yeah thanks, i have a method to work on now.
    I know that one can relate the spin connection to the gamma matrices by: [itex] \Gamma_{\mu} [/itex] to [itex] \gamma[/itex] by [itex] [\Gamma_{\mu},\gamma^{\nu}][/itex] but is this simply a standard commutator relationship or is it something more because wouldn't [itex]\Gamma_{1}\gamma^{1} - \gamma^{1}\Gamma_{1} =0 [/itex] for example?
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