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First some basic notation. I am referencing from Weinberg's "The Quantum Theory of Fields" volume II, pg 29. I'm hoping that the notation is relatively standard.
The question refers to the nilpotency of the s operator defined as follows:
The BRST symmetry transformation is parametrized by an infinitesimal constant [tex] \theta [/tex] that anticommutes with all ghost ( [tex] \omega [/tex] ) and fermionic ( [tex] \psi [/tex] ) matter fields.
A fermionic matter field transforms as [tex] \delta _{ \theta } \psi = i t_ {\alpha} \theta \omega _{ \alpha } \psi [/tex] and we define the operator s by [tex] \delta _{ \theta } F \equiv \theta s F [/tex] where F is a smooth functional of fermionic matter, gauge, ghost, and Nakanishi-Lautrup fields. (In this post I am only going to address the case of F as a single fermionic matter field [tex] \psi [/tex]. ) Also, the [tex]t_{ \alpha } [/tex] form a representation of the Lie algebra of the associated gauge group.
The problem is to show that [tex]ss \psi = 0[/tex] . Weinberg starts with the following equation:
[tex] \delta _{ \theta } s \psi = i t_{ \alpha } \delta _{ \theta } ( \omega _{ \alpha } \psi [/tex] )
If this equation is taken as a given I can go on to complete the proof. However I have two issues with this equation:
1) I am presuming a typo here. The operator [tex] \delta _{ \theta } s [/tex] makes no sense to me. I can only assume that Weinberg meant to write [tex] \delta _{ \theta } (s \psi ) [/tex]. But Weinberg's books have very few typos, so I am not certain about this being an error.
2) By its definition s is clearly a kind of derivative operator. So s acting on a matter field implies that [tex]s \psi [/tex] is also a matter field and thus transforms as one under a BRST transformation. Thus I say:
[tex] \delta _{ \theta } ( s \psi ) = i t_{ \alpha } \theta \omega _{ \alpha } ( s \psi ) [/tex]
[tex]= i t_{ \alpha } \theta \omega _{ \alpha } ( \theta ^{-1} \delta _{ \theta } \psi )[/tex]
[tex]= i t_{ \alpha } \omega _{ \alpha } \delta _{ \theta } \psi [/tex]
Now
[tex]\omega _{ \alpha } \delta _{ \theta } \psi = \delta _{ \theta} ( \omega _{ \alpha } \psi ) - ( \delta _{ \theta } \omega _{ \alpha } ) \psi [/tex]
From the BRST transformation of the ghost field [tex] \omega _{ \alpha } [/tex] we obtain:
[tex]\omega _{ \alpha } \delta _{ \theta } \psi = \delta _{ \theta } ( \omega _{ \alpha } \psi ) - \left ( - \frac{1}{2} \theta C_{ \alpha \beta \gamma } \omega _{ \beta } \omega _{ \gamma } \right ) \psi[/tex]
where the [tex]C_{\alpha \beta \gamma }[/tex] are the structure constants for the Lie algebra.
Putting it all together gives, finally:
[tex]\delta _{ \theta } ( s \psi ) = i t_{ \alpha } \delta _{ \theta } ( \omega _{ \alpha } \psi ) + \frac{i}{2} t_{ \alpha } \theta C_{ \alpha \beta \gamma } \omega _{ \beta } \omega _{ \gamma } \psi [/tex]
The first term on the right is what Weinberg states. The second one is a problem. Am I right in saying that the last term is zero due to a combination of the anti-commutivity of the ghost fields and the antisymmetry of the [tex]C_{ \alpha \beta \gamma }[/tex] in the [tex]\beta[/tex] and [tex]\gamma[/tex]?
Thanks in advance for the help!
-Dan Boyce
The question refers to the nilpotency of the s operator defined as follows:
The BRST symmetry transformation is parametrized by an infinitesimal constant [tex] \theta [/tex] that anticommutes with all ghost ( [tex] \omega [/tex] ) and fermionic ( [tex] \psi [/tex] ) matter fields.
A fermionic matter field transforms as [tex] \delta _{ \theta } \psi = i t_ {\alpha} \theta \omega _{ \alpha } \psi [/tex] and we define the operator s by [tex] \delta _{ \theta } F \equiv \theta s F [/tex] where F is a smooth functional of fermionic matter, gauge, ghost, and Nakanishi-Lautrup fields. (In this post I am only going to address the case of F as a single fermionic matter field [tex] \psi [/tex]. ) Also, the [tex]t_{ \alpha } [/tex] form a representation of the Lie algebra of the associated gauge group.
The problem is to show that [tex]ss \psi = 0[/tex] . Weinberg starts with the following equation:
[tex] \delta _{ \theta } s \psi = i t_{ \alpha } \delta _{ \theta } ( \omega _{ \alpha } \psi [/tex] )
If this equation is taken as a given I can go on to complete the proof. However I have two issues with this equation:
1) I am presuming a typo here. The operator [tex] \delta _{ \theta } s [/tex] makes no sense to me. I can only assume that Weinberg meant to write [tex] \delta _{ \theta } (s \psi ) [/tex]. But Weinberg's books have very few typos, so I am not certain about this being an error.
2) By its definition s is clearly a kind of derivative operator. So s acting on a matter field implies that [tex]s \psi [/tex] is also a matter field and thus transforms as one under a BRST transformation. Thus I say:
[tex] \delta _{ \theta } ( s \psi ) = i t_{ \alpha } \theta \omega _{ \alpha } ( s \psi ) [/tex]
[tex]= i t_{ \alpha } \theta \omega _{ \alpha } ( \theta ^{-1} \delta _{ \theta } \psi )[/tex]
[tex]= i t_{ \alpha } \omega _{ \alpha } \delta _{ \theta } \psi [/tex]
Now
[tex]\omega _{ \alpha } \delta _{ \theta } \psi = \delta _{ \theta} ( \omega _{ \alpha } \psi ) - ( \delta _{ \theta } \omega _{ \alpha } ) \psi [/tex]
From the BRST transformation of the ghost field [tex] \omega _{ \alpha } [/tex] we obtain:
[tex]\omega _{ \alpha } \delta _{ \theta } \psi = \delta _{ \theta } ( \omega _{ \alpha } \psi ) - \left ( - \frac{1}{2} \theta C_{ \alpha \beta \gamma } \omega _{ \beta } \omega _{ \gamma } \right ) \psi[/tex]
where the [tex]C_{\alpha \beta \gamma }[/tex] are the structure constants for the Lie algebra.
Putting it all together gives, finally:
[tex]\delta _{ \theta } ( s \psi ) = i t_{ \alpha } \delta _{ \theta } ( \omega _{ \alpha } \psi ) + \frac{i}{2} t_{ \alpha } \theta C_{ \alpha \beta \gamma } \omega _{ \beta } \omega _{ \gamma } \psi [/tex]
The first term on the right is what Weinberg states. The second one is a problem. Am I right in saying that the last term is zero due to a combination of the anti-commutivity of the ghost fields and the antisymmetry of the [tex]C_{ \alpha \beta \gamma }[/tex] in the [tex]\beta[/tex] and [tex]\gamma[/tex]?
Thanks in advance for the help!
-Dan Boyce