Srednicki 43.10: Minus Sign Explained

[Lapidus]

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 http://www.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

equation 43.10 is on pdf page 273

thank you

 

[qbert]

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
$$\frac{d}{dx}(yx) = \lim_{h \rightarrow 0} \frac{1}{h} ( y(x+h) - yx )$$
$$= \lim_{h \rightarrow 0} \frac{1}{h} yh$$
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.

$$= \lim_{h \rightarrow 0}\left( -y \frac{1}{h} h \right)= -y$$

 

[Lapidus]

qbert, I thank you very much!