Proving Schwinger's Identity: A Challenge for Mathematicians

In summary, the conversation is about a mathematical identity in Schwinger's paper, where the participants are trying to prove it using given equations and notations. They discuss the possible differences in sign convention and metric used by Schwinger. One of the participants also mentions a particular equation that could be helpful in solving the identity. They end with asking for suggestions and inputs to proceed with the proof.
  • #1
maverick280857
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4
Hi,

I'm working my way through Schwinger's paper (http://www.physics.princeton.edu/~mcdonald/examples/QED/schwinger_pr_82_664_51.pdf" [Broken]) and I came across the following identity

[tex]-(\gamma\pi)^2 = \pi_{\mu}^2 - \frac{1}{2}e\sigma_{\mu\nu}F^{\mu\nu}[/tex]

where

[tex]\pi_{\mu} = p_{\mu} - eA_{\mu}[/tex]

[tex]F^{\mu\nu} = \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/tex]

[tex]\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu][/tex]

(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

[tex]\frac{1}{2}\{\gamma_{\mu},\gamma_{\nu}\} = -\delta_{\mu\nu}[/tex]

I'm guessing his sign convention for the metric is different. Also, shouldn't this be [itex]g_{\mu\nu}[/itex] on the RHS instead of the Kronecker delta?

Returning to the identity, I know that

[tex]\gamma^{\mu}a_{\mu}\gamma^{\nu}b_{\nu} = a\cdot b - i a_{\mu}\sigma^{\mu\nu}b_{\nu}[/tex]

(\slashed doesn't work)

In particular, setting [itex]a = b = \prod[/itex], this becomes

[tex](\gamma \pi)^2 = \pi^2 - e\sigma^{\mu\nu}(\partial_{\mu}A_{\nu} + A_{\nu}\partial_{\mu})[/tex]

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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  • #2
Ideas, suggestions, anyone?
 

1. What is the concept of identity in Schwinger's paper?

In Schwinger's paper, identity refers to the fundamental principle that all physical quantities are equal to their corresponding mathematical operators.

2. How does identity play a role in quantum mechanics?

In quantum mechanics, the concept of identity is crucial as it allows for the description of physical phenomena in terms of mathematical operators, which can then be used to calculate measurable quantities.

3. What is the significance of identity in Schwinger's paper?

The concept of identity in Schwinger's paper is significant as it provides a rigorous mathematical framework for understanding and solving problems in quantum mechanics.

4. Can identity be applied to other areas of physics besides quantum mechanics?

Yes, the concept of identity can also be applied to other areas of physics, such as classical mechanics, electromagnetism, and general relativity.

5. Are there any criticisms of the concept of identity in Schwinger's paper?

Some physicists have criticized the concept of identity in Schwinger's paper for being too abstract and not directly applicable to experimental observations. Others argue that it is a necessary tool for understanding the fundamental principles of quantum mechanics.

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