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Homework Help: Trying to Make a (p[slash])^2 operator - is this right?

  1. Jul 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Find out what p[slash]p[slash] is (Feynman slash-notation), because Maple doesn't like it when you feed it p[slash]p[slash], and let it uber-"FOIL" out the (four non-commuting terms) x (four non-commuting terms), where the "x" denotes plain Jane matrix-multiplication.


    2. Relevant equations
    [tex]\begin{array}{c}
    [{\partial _\mu },{\gamma _\nu }] \equiv 0 \\
    {\gamma _0}^2 = - {\gamma _i}^2 \equiv {\bf{I}} \\
    \end{array}[/tex]


    3. The attempt at a solution

    [tex]\begin{array}{c}
    {p_{{\rm{slash}}}}{p_{{\rm{slash}}}} = - ({\gamma _\mu }{\partial ^\mu })({\gamma _\nu }{\partial ^\nu }) \\
    = - \left( {{\gamma _0}{\partial ^0} - \vec \gamma \bullet \vec \partial } \right)\left( {{\gamma _0}{\partial ^0} - \vec \gamma \bullet \vec \partial } \right) \\
    = - \left( {{\gamma _0}^2{{({\partial ^0})}^2} + (\vec \gamma \bullet \vec \gamma ){\nabla ^2} - 2({\gamma _0}{\partial ^0})(\vec \gamma \bullet \vec \partial )} \right) \\
    {p_{{\rm{slash}}}}{p_{{\rm{slash}}}} = - \left( {({\bf{I}}){{({\partial ^0})}^2} + ( - 3{\bf{I}}){\nabla ^2} - 2({\gamma _0}{\partial ^0})(\vec \gamma \bullet \vec \partial )} \right) \\
    \end{array}[/tex]
     
    Last edited: Jul 6, 2010
  2. jcsd
  3. Jul 6, 2010 #2
    sorry! LaTeX is bad.
     
  4. Jul 6, 2010 #3
    Um...writing text below the thread-start actually fixed the problem :-p
     
  5. Jul 7, 2010 #4
    No, it is not right. To get the right answer, begin by noting that pslash^2 is symmetric in d^mu and d^nu.

    There is one more relevant equation that the gammas satisfy that you need to solve this, as well.
     
  6. Jul 8, 2010 #5
    Hi chrispb, thank you for stopping to help us answer this question.

    May I ask: does "symmetric" mean pslash^2 is equal to its own transpose?
     
  7. Jul 8, 2010 #6
    No, I just mean [tex]{\gamma _\mu }{\partial ^\mu }{\gamma _\nu }{\partial ^\nu }={\gamma _\mu }{\gamma _\nu }{\partial ^\mu }{\partial ^\nu }={\gamma _\mu }{\gamma _\nu }{\partial ^\nu }{\partial ^\mu }[/tex].

    What this means is you can write [tex]{\partial ^\mu }{\partial ^\nu }=\frac{1}{2}({\partial ^\mu }{\partial ^\nu }+{\partial ^\nu }{\partial ^\mu })[/tex] free of charge.
     
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