# Dummy index and renaming if not a tensor

• binbagsss
In summary, the conversation discusses the rules for renaming dummy indices in expressions involving tensors and the Levi-Civita symbol. It is mentioned that there is no rule preventing the renaming of dummy indices in non-tensor objects. An example is given where the dummy index is renamed in one of the terms of an expression. Later, the conversation focuses on finding the variation with respect to a specific index in an expression involving the Levi-Civita symbol. Confusion arises when trying to factor out the variation due to the different indices in each term.
binbagsss
Homework Statement
see below
Relevant Equations
see below
Apologies it's been a little while since I've done this, but I believe the rule is, that if the object is not a tensor you can not rename the dummy index?

For example, i have the action

##\int d^3 x \epsilon^{uvp} A_u \partial v A_p ## and I want to write this in terms of the ##i## and ##0## components (so that I can take ##\frac{\delta S}{\delta A_0}## and so later on, I imagine, once I've intergreated by parts etc, it will become handy to rename the dummy indicies, but this can not be done for the levi-civita symbol...is this correct?

many thanks

binbagsss said:
[...] but I believe the rule is, that if the object is not a tensor you can not rename the dummy index?
I don't think there's any such rule. If an index is a (repeated) dummy summation index, you can rename it.

For example, i have the action
##\int d^3 x \epsilon^{uvp} A_u \partial v A_p ## and I want to write this in terms of the ##i## and ##0## components (so that I can take ##\frac{\delta S}{\delta A_0}## and so later on,[...]
You might not need to do that. E.g., $$\epsilon^{\mu\nu\rho} \frac{\delta A_\rho}{\delta A_0} ~=~ \epsilon^{\mu\nu\rho} \delta^0_\rho ~=~ \epsilon^{\mu\nu 0}$$

strangerep said:
I don't think there's any such rule. If an index is a (repeated) dummy summation index, you can rename it.

You might not need to do that. E.g., $$\epsilon^{\mu\nu\rho} \frac{\delta A_\rho}{\delta A_0} ~=~ \epsilon^{\mu\nu\rho} \delta^0_\rho ~=~ \epsilon^{\mu\nu 0}$$
Sorry , well I meant like if you have ##A^a B_a A_b B^b - A^aA_pB_aB^p## you can rename ##p## in the second term to ##b##say. (The reason I want to rename is to cancel the variation##\delta A_{\mu}##but in the two terms I have got they have different indices in this variation).. (but I will also read through what you said in a bit)

strangerep said:
I don't think there's any such rule. If an index is a (repeated) dummy summation index, you can rename it.

You might not need to do that. E.g., $$\epsilon^{\mu\nu\rho} \frac{\delta A_\rho}{\delta A_0} ~=~ \epsilon^{\mu\nu\rho} \delta^0_\rho ~=~ \epsilon^{\mu\nu 0}$$
okay that looks simple enough, so also using the chain rule on the variation ##\frac{\delta S}{\delta A_0}= \frac{\delta S}{\delta A_{\rho}}. \frac{\delta A_\rho}{\delta A_0}##?

Actually in this case, it was a variation wrt ##A_{i}## and the final answer is a expression involving ##\epsilon^{ij}## with only two-indices and not three, I'm really confused how one managed to go from an expression involving a three index levi-civta tensor to an expression involving only two.

Many thanks

binbagsss said:
Actually in this case, it was a variation wrt ##A_{i}## and the final answer is a expression involving ##\epsilon^{ij}## with only two-indices and not three, I'm really confused how one managed to go from an expression involving a three index levi-civta tensor to an expression involving only two.
Well, I can't be sure without seeing the full original context. Maybe because one of ##\epsilon## indices is ##0##?

(This is why we normally insist on a more complete "homework statement". The one in your opening post is not adequate for helping you properly.)

strangerep said:
Well, I can't be sure without seeing the full original context. Maybe because one of ##\epsilon## indices is ##0##?

(This is why we normally insist on a more complete "homework statement". The one in your opening post is not adequate for helping you properly.)
okay but in answer to the original question , say i have ##\epsilon_{abc}A^bB^{ac} ## can i rename say ## b \to d## as:##\epsilon_{adc}A^dB^{ac}##?, for the levi-civita symbol, as it's not a tensor, was my initial question

binbagsss said:
okay but in answer to the original question , say i have ##\epsilon_{abc}A^bB^{ac} ## can i rename say ## b \to d## as:##\epsilon_{adc}A^dB^{ac}##?, for the levi-civita symbol, as it's not a tensor, was my initial question
Yes. The summation convention doesn't care whether it's a tensor or a goose.

strangerep said:
Well, I can't be sure without seeing the full original context. Maybe because one of ##\epsilon## indices is ##0##?

(This is why we normally insist on a more complete "homework statement". The one in your opening post is not adequate for helping you properly.)
Hi okay, apologies I am now stuck on this (this wasn't the origianl question nor intention of my post and hence why I did not state the full problem at the start, but I think it makes more sense to post it here since we went that way anyway..)So i have ## \epsilon^{uvp} a_u \partial_v a_p ## (as the integrand under a 3d integral) and I wantt o find the variation w.r.t ##a_i##. where ##i = 1,2 ## running only over the spatial indicies.

So, so far I have

## \epsilon^{nup}(a_u + \delta_u)\partial_v(a_p+\delta a_p) ##and so to order ##\delta a## after integrating the by parts the first term to order :

##\epsilon^{nup} (- \partial_v a_u \delta_p + \partial_v a_p \delta a_u ##.

And now this is where my confusion is as I can't factorise out ##\delta a ## to get ##\delta S / \delta a^c ## since both terms have different indicies on delta a .

So ## \delta S / \delta a^i= \delta S/ \delta a^ c . \delta c/ \delta a^i ##, however , where ##c## runs over all 3 indices, and ##i## only the spatial ones

but I'm stuck because i don't have ## \delta S/ \delta a^ c ## to even apply the chain rule since i can't factorise out ##a## as said above

thanks

binbagsss said:
## \epsilon^{nup}(a_u + \delta_u)\partial_v(a_p+\delta a_p) ##
Where did the "n" index on ##\epsilon## come from? A typo? I'm guessing what you meant was: $$\epsilon^{uvp}(a_u + \delta_u)\partial_v(a_p+\delta a_p) ~~~?$$

and so to order ##\delta a## after integrating the by parts the first term to order : ##\epsilon^{nup} (- \partial_v a_u \delta_p + \partial_v a_p \delta a_u ##.
And here there seem to be more typos? I guess you meant $$\epsilon^{uvp} (- \partial_v a_u \delta a_p + \partial_v a_p \delta a_u) ~~~~~?$$

And now this is where my confusion is as I can't factorise out ##\delta a ## to get ##\delta S / \delta a^c ## since both terms have different indicies on delta a .

So ## \delta S / \delta a^i= \delta S/ \delta a^ c . \delta c/ \delta a^i ##, however , where ##c## runs over all 3 indices, and ##i## only the spatial ones

but I'm stuck because i don't have ## \delta S/ \delta a^ c ## to even apply the chain rule since i can't factorise out ##a## as said above
OK. In baby steps, here's what to do. First split it into 2 pieces:
$$\epsilon^{uvp} \partial_v a_p \delta a_u ~-~ \epsilon^{uvp}\partial_v a_u \delta a_p$$
Now rename the indices in one of the terms, so that ##\delta a_p## appears in both terms. (I'll choose the 1st term and interchange ##u \leftrightarrow p##).
$$\epsilon^{pvu} \partial_v a_u \delta a_p ~-~ \epsilon^{uvp}\partial_v a_u \delta a_p$$
Now remember that you can interchange any 2 indices on the ##\epsilon## provided you also change the sign. So the above can become
$$-\epsilon^{uvp} \partial_v a_u \delta a_p ~-~ \epsilon^{uvp}\partial_v a_u \delta a_p$$
which is just $$-2 \epsilon^{uvp} \partial_v a_u \delta a_p ~~.$$ Can you continue from there?

binbagsss

## 1. What is a dummy index in scientific notation?

A dummy index is a placeholder variable used in mathematical expressions to represent a repeated summation or integration. It has no physical meaning and can be replaced by any other valid variable without changing the outcome of the equation.

## 2. How are dummy indices denoted?

Dummy indices are often denoted by Greek letters such as α (alpha), β (beta), γ (gamma), etc. However, they can also be denoted by any other letter or symbol, as long as it is not already being used as a variable in the equation.

## 3. What is the purpose of renaming a dummy index?

Renaming a dummy index is done to avoid confusion when working with multiple summations or integrals in the same equation. It allows for easier understanding and manipulation of the mathematical expression.

## 4. Can a dummy index be a tensor?

No, a dummy index cannot be a tensor. Tensors are physical quantities that have magnitude and direction, while dummy indices are merely placeholders in mathematical expressions. They serve different purposes and cannot be interchanged.

## 5. How do I know when to use a dummy index in my equations?

Dummy indices are used when there is a repeated summation or integration in an equation. They make it easier to represent and manipulate the expression, especially when dealing with multiple variables or indices. They are commonly used in physics, mathematics, and other scientific fields.

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