- #1
maverick280857
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Hi,
I'm working my way through Schwinger's paper (http://www.physics.princeton.edu/~mcdonald/examples/QED/schwinger_pr_82_664_51.pdf" ) and I came across the following identity
-(\gamma\pi)^2 = \pi_{\mu}^2 - \frac{1}{2}e\sigma_{\mu\nu}F^{\mu\nu}
where
\pi_{\mu} = p_{\mu} - eA_{\mu}
F^{\mu\nu} = \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}
\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]
(This is equation 2.33 of the paper, for those of you who refer to the pdf.)
I am trying to prove this identity, but I ran into some problems. First of all, since his equation 2.4 states
\frac{1}{2}\{\gamma_{\mu},\gamma_{\nu}\} = -\delta_{\mu\nu}
I'm guessing his sign convention for the metric is different. Also, shouldn't this be g_{\mu\nu} on the RHS instead of the Kronecker delta?
Returning to the identity, I know that
\gamma^{\mu}a_{\mu}\gamma^{\nu}b_{\nu} = a\cdot b - i a_{\mu}\sigma^{\mu\nu}b_{\nu}
(\slashed doesn't work)
In particular, setting a = b = \prod, this becomes
(\gamma \pi)^2 = \pi^2 - e\sigma^{\mu\nu}(\partial_{\mu}A_{\nu} + A_{\nu}\partial_{\mu})
Questions:
1. How does one proceed from here?
2. I seem to get no minus sign on the LHS. Is that because of Schwinger's metric?
Any suggestions and inputs would be greatly appreciated. I've been stuck on this step for a few hours now.
Thanks in advance.
I'm working my way through Schwinger's paper (http://www.physics.princeton.edu/~mcdonald/examples/QED/schwinger_pr_82_664_51.pdf" ) and I came across the following identity
-(\gamma\pi)^2 = \pi_{\mu}^2 - \frac{1}{2}e\sigma_{\mu\nu}F^{\mu\nu}
where
\pi_{\mu} = p_{\mu} - eA_{\mu}
F^{\mu\nu} = \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}
\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]
(This is equation 2.33 of the paper, for those of you who refer to the pdf.)
I am trying to prove this identity, but I ran into some problems. First of all, since his equation 2.4 states
\frac{1}{2}\{\gamma_{\mu},\gamma_{\nu}\} = -\delta_{\mu\nu}
I'm guessing his sign convention for the metric is different. Also, shouldn't this be g_{\mu\nu} on the RHS instead of the Kronecker delta?
Returning to the identity, I know that
\gamma^{\mu}a_{\mu}\gamma^{\nu}b_{\nu} = a\cdot b - i a_{\mu}\sigma^{\mu\nu}b_{\nu}
(\slashed doesn't work)
In particular, setting a = b = \prod, this becomes
(\gamma \pi)^2 = \pi^2 - e\sigma^{\mu\nu}(\partial_{\mu}A_{\nu} + A_{\nu}\partial_{\mu})
Questions:
1. How does one proceed from here?
2. I seem to get no minus sign on the LHS. Is that because of Schwinger's metric?
Any suggestions and inputs would be greatly appreciated. I've been stuck on this step for a few hours now.
Thanks in advance.
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