- #1
latentcorpse
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If [itex]\{ \theta_i \}[/itex] are a set of Grassmann numbers then what is
[itex]\frac{\partial}{\partial \theta_i} ( \theta_j \theta_k \theta_l)[/itex]
I know that [itex]\frac{\partial}{\partial \theta_i} ( \theta_j \theta_k ) = \delta_{ij} \theta_k - \theta_j \delta_{ik}[/itex] - we need this to be the case becuse if we set [itex]j=k[/itex] then the LHS becomes the derivative of [itex]\theta_j^2=0[/itex] and so we need the RHS to vanish as well (hence the minus sign!)
However, now that there are three variables present, I am confused as to what should pick up a minus sign upon differentiation and what should not?
Thanks.
[itex]\frac{\partial}{\partial \theta_i} ( \theta_j \theta_k \theta_l)[/itex]
I know that [itex]\frac{\partial}{\partial \theta_i} ( \theta_j \theta_k ) = \delta_{ij} \theta_k - \theta_j \delta_{ik}[/itex] - we need this to be the case becuse if we set [itex]j=k[/itex] then the LHS becomes the derivative of [itex]\theta_j^2=0[/itex] and so we need the RHS to vanish as well (hence the minus sign!)
However, now that there are three variables present, I am confused as to what should pick up a minus sign upon differentiation and what should not?
Thanks.