1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Deriving the continuity equation from the Dirac equation (Relativistic Quantum)

  1. Jun 14, 2008 #1
    So I am trying to derive the continuity equation:

    [tex]\frac{\partial}{\partial x^{\mu}}J^{\mu} = 0[/tex]

    From the Dirac equation:

    [tex]i\gamma^{\mu} \frac{\partial}{\partial x^{\mu}}\Psi - \mu\Psi = 0[/tex]

    And its Hermitian adjoint:

    [tex]i\frac{\partial}{\partial x^{\mu}}\overline{\Psi}\gamma^{\mu} - \mu\overline{\Psi} = 0[/tex]


    [tex]\overline{\Psi}=\Psi^{+}\gamma^{0}[/tex] (Dirac conjugate)

    3. The attempt at a solution
    By multiplying the Dirac equation on the right by [tex]\overline{\Psi}[/tex] and the adjoint on the right by [tex]\Psi[/tex] I get:

    [tex]i(\frac{\partial}{\partial x^{\mu}}(\gamma^{\mu}\Psi)\overline{\Psi} + \frac{\partial}{\partial x^{\mu}}(\overline{\Psi})\gamma^{\mu}\Psi) - \mu(\Psi\overline{\Psi} - \overline{\Psi}\Psi)=0[/tex]

    The first term is basically what I am after (except I am not 100% sure I can simply apply the product rule - what is the correct order?) which means I shoudl expect the second term to go to zero:

    [tex]\Psi\overline{\Psi} - \overline{\Psi}\Psi =0[/tex]

    But because [tex]\gamma^{0}[/tex] is a 4x4 matrix, [tex]\Psi[/tex] is a 4x1 and [tex]\overline{\Psi}[/tex] is a 1x4, I should also expect the second term to be multiplied by the 4x4 identity matrix (so that the subtraction makes sense). However the first term is NOT a constant multiplied by the identity so I don't see how this works.

    Any help would be greatly appreciated...
  2. jcsd
  3. Jun 15, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi toam! :smile:

    Don't you have to multiply one of them on the left? :confused:
  4. Jun 15, 2008 #3
    I tried that and got something else that didn't work. However I will try again because I was surprised that it didn't work so I may have made a mistake or missed something obvious...
  5. Jun 15, 2008 #4
    Ok so it turned out I had multiplied the wrong function on the left. It worked out quite simply when I fixed that. The lecture notes had erroneously shown both functions multiplied on the right.

    Thanks, tiny-tim.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook