- #1
acipsen
- 3
- 0
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
[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
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
following is using Peskin & Schroeder.
Regards,
Asger