How to manipulate indices when Grassmannian numbers are present?

[LCSphysicist]

Homework Statement

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Relevant Equations

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Suppose i have a term like this one (repeated indices are being summed)

$$x = \psi^a C_{ab} \psi^b$$

Such that $C_{ab} = - C_{ba}$, and $\{\psi^a,\psi^b\}=0$. How do i evaluate the derivative of this term with respect to $\psi_r$?

I mean, my attempt g oes to here

$$\frac{\partial x}{\partial \psi_r} = C_{rb} \psi^b + \psi^r C_{ar}$$

But, i think this is zero!!, isnt? Ok, maybe we can't change a and b in $x$ because we have the anticommuting property of psi, but since in the term above we have one psi for each term, i can't see a problem in change a to b, using the C antisymmetry property and getting 0.

What is wrong?

 

[haushofer]

Why do you think it's zero?

Your calculation can't be right; look at the indices. Your second term should contain a free r-index, not a, and hitting the second psi with your derivative should give an addition minus sign.

 

[malawi_glenn]

What is the definition of derivative here? I.e. upper- vs lower case index

 

[vanhees71]

Also you must distinguish between "left" and "right" derivatives!

 

[LCSphysicist]

Why do you think it's zero?

Your calculation can't be right; look at the indices. Your second term should contain a free r-index, not a, and hitting the second psi with your derivative should give an addition minus sign.

Ups, just tiped it wrong.

So it should be

$$C_{rb} \psi^b + \psi^{a}C_{ar}$$

By the way

"Why do you think it's zero?"

Well,

$$C_{rb} \psi^b + \psi^a C_{ar} = C_{rb} \psi^b + \psi^b C_{br} = \psi^b ( C_{rb} - C_{rb} ) = 0$$

Where i have used that $C$ is anti-symmetric, and that since we are dealing only with indices, we "could" just say that $A_{bc}x^{c} = x^{c}A_{bc}$.

What is the definition of derivative here? I.e. upper- vs lower case index

Ok, let's be more specific, derivating it with respesct to $\psi^r$. So that the indices are fine.

Also you must distinguish between "left" and "right" derivatives!

What do you mean?

 

[malawi_glenn]

What do you mean?

http://ckw.phys.ncku.edu.tw/public/...s/1.5._DifferentiationInGrassmannAlgebras.pdf

 

[vela]

Ok, let's be more specific, derivating it with respect to $\psi^r$. So that the indices are fine.

The word you're looking for is differentiating.