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Four-vector differentiation

  1. Apr 16, 2014 #1
    Hi all, I'm working on some QFT and I've run into a stupid problem. I can't figure out why my two methods for evaluating

    [itex]
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x)
    [/itex]

    don't agree. I'm using the Minkowski metric [itex]g_{\mu\nu} = diag(+,-,-,-) [/itex] and I'm using [itex] \partial_\mu = (\frac{\partial}{\partial t},\vec{\nabla} ) [/itex]

    Method one (correct):

    [itex]
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x) = i\gamma^\mu \frac{\partial}{\partial x^\mu} \exp(-i p_\mu x^\mu) \\
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =i\gamma^\mu (-i p_\mu) \exp(-i p_\mu x^\mu) \\
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =\gamma^\mu p_\mu \exp(-i p_\mu x^\mu) \\
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =[\gamma^0 E - \gamma^1 p_x - \gamma^2 p_y - \gamma^3 p_z] \exp(-i p_\mu x^\mu)
    [/itex]

    Method 2 (incorrect):

    [itex]
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 \frac{\partial}{\partial t} - \gamma^1 \frac{\partial}{\partial x} - \gamma^2 \frac{\partial}{\partial y} - \gamma^3 \frac{\partial}{\partial z}] \exp(-iEt+ip_x x + i p_y y + i p_z z) \\
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 (-iE) - \gamma^1 (ip_x) - \gamma^2 (ip_y) - \gamma^3 (ip_z)] \exp(-iEt+ip_x x + i p_y y + i p_z z) \\
    i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=[\gamma^0 E + \gamma^1 p_x + \gamma^2 p_y + \gamma^3 p_z] \exp(-i p_\mu x^\mu)
    [/itex]

    What's going on? It feels like I'm going crazy.
     
  2. jcsd
  3. Apr 17, 2014 #2

    Bill_K

    User Avatar
    Science Advisor

    Should be
    [itex]i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 \frac{\partial}{\partial t} + \gamma^1 \frac{\partial}{\partial x} + \gamma^2 \frac{\partial}{\partial y} + \gamma^3 \frac{\partial}{\partial z}] \exp(-iEt+ip_x x + i p_y y + i p_z z)
    [/itex]

    You only need to insert minus signs when raising or lowering indices, or when the Minkowski metric is explicitly present. In this case, γμ is contravariant and ∂/∂xμ is covariant, so everything's fine, and the sum over μ is just a sum.
     
  4. Apr 17, 2014 #3
    Ohhh okay I get it. Thanks for your help!
     
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