How to manipulate indices when Grassmannian numbers are present?

In summary, the conversation discusses evaluating the derivative of a term involving repeated indices and Grassmann variables. The calculation provided in the conversation appears to have a mistake in the second term, which should contain a free r-index instead of a. The correct calculation is given as C_{rb} \psi^b + \psi^{a}C_{ar}. The conversation also mentions the importance of distinguishing between left and right derivatives, and provides a resource for further reading on differentiation in Grassmann algebras.
  • #1
LCSphysicist
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Homework Statement
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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?
 
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  • #2
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.
 
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  • #3
What is the definition of derivative here? I.e. upper- vs lower case index
 
  • #4
Also you must distinguish between "left" and "right" derivatives!
 
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  • #5
haushofer said:
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}##.

malawi_glenn said:
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.

vanhees71 said:
Also you must distinguish between "left" and "right" derivatives!
What do you mean?
 
  • #7
LCSphysicist said:
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. :smile:
 
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FAQ: How to manipulate indices when Grassmannian numbers are present?

How do Grassmannian numbers affect the manipulation of indices in tensor operations?

Grassmannian numbers, or Grassmann variables, are anticommuting numbers used in supersymmetry and quantum field theory. When manipulating indices with these numbers, one must account for their anticommutative property. Specifically, swapping two Grassmannian numbers introduces a negative sign, which affects tensor operations involving these variables. For example, if θ and φ are Grassmann numbers, then θφ = -φθ.

What is the correct way to handle indices when differentiating with respect to Grassmannian numbers?

When differentiating with respect to Grassmannian numbers, the differentiation rules differ from those of ordinary numbers. The derivative of a Grassmann number with respect to itself is 1, and the derivative with respect to a different Grassmann number is 0. Moreover, due to their anticommutative nature, the order of differentiation matters. For instance, if θ and φ are Grassmann numbers, then ∂/∂θ (θφ) = φ and ∂/∂φ (θφ) = -θ.

How do you handle summation indices in expressions involving Grassmannian numbers?

When summing over indices involving Grassmannian numbers, it is crucial to maintain the correct order to respect their anticommutative properties. If an expression involves a sum over Grassmannian numbers, each term in the series must preserve the order of the variables. Any reordering of indices within the sum will introduce a sign change. For example, Σ_i θ_i φ_i should be handled carefully to avoid incorrect sign changes.

What are the implications of Grassmannian numbers on the symmetrization and antisymmetrization of indices?

Grassmannian numbers inherently require antisymmetrization due to their anticommutative nature. When dealing with indices that involve Grassmannian numbers, one must antisymmetrize the indices. For example, if θ and φ are Grassmannian numbers, then an antisymmetric combination like θφ - φθ = 0 must be considered. This property is crucial in constructing antisymmetric tensors and spinors in supersymmetry theories.

How do Grassmannian numbers influence the handling of matrix indices?

When dealing with matrices whose elements are Grassmannian numbers, the anticommutative property must be taken into account during matrix multiplication. Specifically, the product of two matrices A and B, where A and B have Grassmannian elements, will involve anticommuting variables. Therefore, the order of multiplication and the associated indices must be handled carefully to avoid sign errors. For instance, if A_ij and B_jk are Grassmannian elements

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