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Srednicki 43.10

by Lapidus
Tags: 4310, srednicki
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Jul1-11, 02:24 AM
P: 283
Trivial question...

How exactly does the minus sign arise in eq. 43.10? The sentence below states because the functional derivative goes through one spinor, but I can't see how that works...

book is online here

equation 43.10 is on pdf page 273

thank you
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Jul1-11, 03:07 AM
P: 185
i'll take a shot.

let me argue by analogy, maybe making the rule plausible. suppose i have anticommuting
numbers x,y,h. and i'm given the expression yx and i want to differentiate it with respect to x.

lacking any better choice i form the difference quotient
[tex]\frac{d}{dx}(yx) = \lim_{h \rightarrow 0} \frac{1}{h} ( y(x+h) - yx ) [/tex]
[tex] = \lim_{h \rightarrow 0} \frac{1}{h} yh [/tex]
now because the numbers are anticommuting i can't just cancel h. i have to first swap
yh or h^(-1) and y and then i can cancel.

[tex] = \lim_{h \rightarrow 0}\left( -y \frac{1}{h} h \right)= -y [/tex]
Jul2-11, 04:04 AM
P: 283
qbert, I thank you very much!

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