- #1

acipsen

- 3

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I am reading through Weinberg's "Quantum Theory of Fields" (vol. 1)

and I am somewhat confused about the signs in the cluster decomposition

of the S-matrix. Specifically, referring to eq. 4.3.2, let's say the term

coming from the partition

[itex]\alpha \to \alpha_1\alpha_2[/itex],[tex]\beta \to \beta_1\beta_2[/tex]

would be

[tex]+S^C_{\beta_1,\alpha_1}S^C_{\beta_2,\alpha_2}[/tex].

Lets now assume that permuting [tex]\alpha_1,\alpha_2[/tex] gives a sign, while permuting [tex]\beta_1,\beta_2[/tex] doesn't,

then the equivalent partition

[itex]\alpha \to \alpha_2\alpha_1[/itex],[tex]\beta \to \beta_2\beta_1[/tex]

would give rise to the term

[tex]-S^C_{\beta_2,\alpha_2}S^C_{\beta_1,\alpha_1}[/tex],

i.e. minus what I had before. This clearly cannot be right, but I'm not sure where

the flaw in my reasoning is. Does it have something to do with conservation

of [tex](-1)^F[/tex], where [tex]F[/tex] is the number of Fermions?

I would be grateful for any hints. And no, this is not homework , the course I'm

following is using Peskin & Schroeder.

Regards,

Asger