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Jay R. Yablon
#14
Nov4-06, 03:39 PM
P: n/a
"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1158728994.691348.273450@m73g2000cwd.googlegroups.com...

>> I have been studying the Schwinger calculation carefully and closely.
>> If sigma_uv denotes the object constructed from antisymmetric
>> [gamma,
>> gamma] combinations, and p, p' designate the ingoing and outgoing
>> electron four-momenta, and alpha is the EM coupling, then it looks to
>> me
>> like the form of the Schwinger "correction" Lambda is:
>>
>> Lambda = (alpha/2pi)(i sigma (p'-p)/2m) (1)
>>
>> or, alternatively, decomposed:
>>
>> Lambda = (alpha/2pi)gamma - (alpha/2pi)((p'-p)/2m) (2)
>>
>> In both, the full vertex factor is:
>>
>> Gamma ^u = gamma^u + Lambda^u (3)
>>
>> Looking specifically at (2), when you say "anticommutators
>> proportional
>> to the Clifford identity?" do you mean that although the term
>> (alpha/2pi)gamma in (2) anticommutes proportionally to the Dirac
>> Clifford algebra generators gamma, the second term
>> (alpha/2pi)((p'-p)/2m)
>> does NOT? And, is this the source of the issue you have identified
>> here?

>
> First, I don't understand how you got your equations (1) and (2).
> Using
> Weinberg's QFT v.1 book as a reference, I'm looking at equation
> (11.3.29):
>
> Gamma^u = gamma^u F_1 + i/2 sigma^uv (p'-p)_v F_2,
>
> where F_1 and F_2 are scalar structure factors that depend only on the
> square of the momentum transfer. Also F_1 ~ 1 + O(alpha) and F_2 ~
> O(alpha). Some combination of these O(alpha) terms gives the
> correction
> to the magnetic moment. However, this equation looks neither like your
> (1) nor (2).


I checked Weinberg (11.3.29). My (1) and (2) above are special cases of
(11.3.29), where F_1 and F_2 are based on the EM Schwinger correction,
only. (See and contrast Ryder, eqs. (9.136) and (9.138).) I like
Weinberg (11.3.29) because it is perfectly general.

By the way, your term "i/2 sigma^uv (p'-p)_v" has an sic: it is either
"i/2 [gamma^u,gamma^v] (p'-p)_v" or "sigma^uv (p'-p)_v" but not "i/2
sigma^uv (p'-p)_v."

My main point is that Gamma^u = gamma^u F_1 + i/2 [gamma^u,gamma^v]
(p'-p)_v F_2, and that the anticommutation properties are determined by
gamma^u (which will commute with the identity) and sigma^uv (p'-p)_v =
i/2 [gamma^u,gamma^v] (p'-p)_v (which will not).

> And instead of wondering whether the anticommutators of Gamma^u are
> proportional to the identity or not, why don't you just check. Have
> you
> tried to compute the anticommutators?


Yes, I have, done a preliminary calculation. Using only the Schwinger
correction, the anticommutators {Gamma,Gamma}, to first order in alpha,
turn out to be equal to the Minkowski metric n_uv = {gamma,gamma}. The
terms containing sigma^uv (p'-p)_v which are not proportional to gamma^u
cancel one another precisely, again, to first order in alpha. I am
wondering what the meaning of this might be. Second order in alpha
*does* leave some residual terms not proportional to the gamma^u. Just
thinking out loud, I am wondering whether, perhaps, one might want to
impose a *condition* that these cancellations occur at all orders in
alpha as they do in first order using the Schwinger correction, which
might then drive what the terms look like at each order, and give us a
canonical way of generating higher order terms which will reproduce what
we get from calculating loop diagrams.

Jay.