Cluster decomposition of the S-matrix

[acipsen]

Hi,

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
$\alpha \to \alpha_1\alpha_2$,$$\beta \to \beta_1\beta_2$$
would be
$$+S^C_{\beta_1,\alpha_1}S^C_{\beta_2,\alpha_2}$$.
Lets now assume that permuting $$\alpha_1,\alpha_2$$ gives a sign, while permuting $$\beta_1,\beta_2$$ doesn't,
then the equivalent partition
$\alpha \to \alpha_2\alpha_1$,$$\beta \to \beta_2\beta_1$$
would give rise to the term
$$-S^C_{\beta_2,\alpha_2}S^C_{\beta_1,\alpha_1}$$,
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 $$(-1)^F$$, where $$F$$ 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

 

[tiny-tim]

Welcome to PF!

Hi Asger! Welcome to PF!

(have an alpha: α and a beta: β and try using the X² tag just above the Reply box )

Since α_i and β_i together form a separate experiment, for each i, the number of fermions in α_i will be odd only if the number in β_i is.

So swapping two experiments will give the same number of fermion interchanges for α as for β.

 

[acipsen]

Hi Tim!

Thanks for the answer. I'm sure Weinberg thought that it too obvious to mention.

Regards,
Asger

 

[ansgar]

Hi Tim!

Thanks for the answer. I'm sure Weinberg thought that it too obvious to mention.

Regards,
Asger

That is Weinberg's style :) If it is too simple, he does not mention it ;)