- #1
raintrek
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[SOLVED] The Dirac Equation
I'm trying to understand the following property of the Dirac equation:
[tex](i \gamma^{\mu}\partial_{\mu} - m)\Psi(x) = 0[/tex]
Acting twice with [tex](i \gamma^{\mu}\partial_{\mu} - m)[/tex]:
[tex](i \gamma^{\mu}\partial_{\mu} - m)^{2} \Psi(x) = 0[/tex]
[tex] = [ - \gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} - 2im\gamma^{\mu}\partial_{\mu} + m^{2}]\Psi = 0[/tex]
But then somehow the book jumps to this step:
[tex] = [ 1/2 \left{\{\gamma^{\mu}, \gamma^{\nu}\right}\} \partial_{\mu}\partial_{\nu} + m^{2}]\Psi = 0[/tex]
And I have no idea how it got there! I understand the { } denote an anticommutator, but I just can't see how the factor of 1/2 has appeared, where the minus has gone and where the middle term has gone. Help!
I'm trying to understand the following property of the Dirac equation:
[tex](i \gamma^{\mu}\partial_{\mu} - m)\Psi(x) = 0[/tex]
Acting twice with [tex](i \gamma^{\mu}\partial_{\mu} - m)[/tex]:
[tex](i \gamma^{\mu}\partial_{\mu} - m)^{2} \Psi(x) = 0[/tex]
[tex] = [ - \gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} - 2im\gamma^{\mu}\partial_{\mu} + m^{2}]\Psi = 0[/tex]
But then somehow the book jumps to this step:
[tex] = [ 1/2 \left{\{\gamma^{\mu}, \gamma^{\nu}\right}\} \partial_{\mu}\partial_{\nu} + m^{2}]\Psi = 0[/tex]
And I have no idea how it got there! I understand the { } denote an anticommutator, but I just can't see how the factor of 1/2 has appeared, where the minus has gone and where the middle term has gone. Help!