The 10 Commandments of Index Expressions and Tensor Calculus - Comments

In summary: The 10 Commandments of Index Expressions and Tensor CalculusIn summary, the bra-ket notation can get messy and confusing if you have product states, and Einstein's notation for quantum states doesn't seem to be very useful.
  • #1

Orodruin

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
20,908
11,879
Greg Bernhardt submitted a new PF Insights post

The 10 Commandments of Index Expressions and Tensor Calculus
tensorcalculus.png


Continue reading the Original PF Insights Post.
 

Attachments

  • tensorcalculus.png
    tensorcalculus.png
    11.6 KB · Views: 939
  • Like
  • Love
Likes etotheipi, Opressor, George Keeling and 1 other person
Physics news on Phys.org
  • #2
As made for me and my aversion against indices. Can I order another one for the bra - ket - "covariant" - "contravariant" - world? I mean their physical use, not their mathematical meaning.
 
  • Like
Likes Opressor
  • #3
fresh_42 said:
As made for me and my aversion against indices. Can I order another one for the bra - ket - "covariant" - "contravariant" - world? I mean their physical use, not their mathematical meaning.

I've decided that the bra-ket notation is pretty wonderful if you're dealing with single systems, but gets messy and confusing if you have product states.

For example, should ##(|A\rangle |B\rangle)^\dagger## be written as ##\langle A| \langle B|## or ##\langle B| \langle A|##? Also, how do you write a "partial" contraction on just one component of a product state?

Has someone tried using Einstein's notation for quantum states? So you would write ##|A\rangle## as just ##A^\mu##, and you would write ##\langle A|## as ##A_\mu##? Then the corresponding "metric" would be ##g(A,B) = \langle A|B\rangle##.
 
  • Like
Likes Opressor
  • #4
Well, in this case it's easy to answer, if you consider ##|A \rangle## to belong to a completely different vector space than ##|B \rangle##. Then, what physicists mean is a Kronecker product of vectors ##|AB \rangle=|A \rangle |B \rangle = |A \rangle \otimes B \rangle##. Then it's clear that the hermitean conjugate form of this, acting on ##V_1 \otimes V_2## can only be ##|AB \rangle^{\dagger}=\langle AB | = \langle A |\otimes \langle B|##, because you can apply it only on a vector ##|v_1 v_2 \rangle \in V_1 \otimes V_2## but not on a vector ##|v_2 v_1 \rangle \in V_2 \otimes V_1##, i.e., a correct expression is
$$\langle AB|v_1 v_2 \rangle=\langle A|v_1 \rangle \langle B |v_2 \rangle.$$
It doesn't make any sense to swap ##A## and ##B## in this expression, since ##\langle A|v_2\rangle## doesn't make any sense, because you cannot take a vector product of a vector with a vector belonging to a different vector space (e.g., think about ##V_1## being a spin-1 vector space (3D vector space) and ##V_2## a spin-2 vector space (5D vector space).
 
  • Like
Likes Opressor
  • #5
vanhees71 said:
...$$\langle AB|v_1 v_2 \rangle=\langle A|v_1 \rangle \langle B |v_2 \rangle.$$.

But if you're writing ##\langle A| \langle B|## contracted with ## |C\rangle |D \rangle##, it looks like

##\langle A| \langle B | |C\rangle |D\rangle = \langle A| (\langle B|C\rangle) |D\rangle##, rather than what's supposed to be, ##\langle A|B\rangle \langle C|D\rangle##. Of course, you can disambiguate it, but the notation is pretty ugly.
 
  • Like
Likes Opressor
  • #6
And as I said, partial contractions are really ugly. If you want to have ##\langle C|## act on the second component of ##|A\rangle |B\rangle##, you can't write it as:

##\langle C|A\rangle |B\rangle##

You could express it with Einstein indices, though:

##C_\mu A^\nu B^\mu##
 
  • Like
Likes Opressor
  • #7
vanhees71 said:
$$\langle AB|v_1 v_2 \rangle=\langle A|v_1 \rangle \langle B |v_2 \rangle.$$
It doesn't make any sense to swap ##A## and ##B## in this expression, since ##\langle A|v_2\rangle## doesn't make any sense, because you cannot take a vector product of a vector with a vector belonging to a different vector space (e.g., think about ##V_1## being a spin-1 vector space (3D vector space) and ##V_2## a spin-2 vector space (5D vector space).

One would probably use different sets of indices, e.g.
$$A_a B_\mu {v_1}^a {v_2}^\mu$$
 
  • #8
One thing that wasn't mentioned in the Insight is
the distinction between
 
  • #9
robphy said:
One thing that wasn't mentioned in the Insight is
the distinction between
In my experience, students have enough problems with the first of these to delve into the abstract index notation early on and most of the time they do not need it (and the students that are interested enough to delve into it really do not need to be told anything).
 
  • Like
Likes George Jones, vanhees71 and robphy
  • #10
robphy said:
One would probably use different sets of indices, e.g.
$$A_a B_\mu {v_1}^a {v_2}^\mu$$
Yes, if you define
$$A_a=A^{*a}, \quad B_{\mu} = B^{* \mu}.$$
 
  • #11
## (\vec{v}\times\vec{w})^i = \epsilon_{ijk}v^jw^k ## ##\ \ \ \vec{v}\times\vec{w} = \vec{e}_i\epsilon_{ijk}v^jw^k ##
(Note that this expression breaks the next-to-next commandment if taken literally! It is intended to hold only in Cartesian coordinates. See the caveat.)

5. You shall not have more than two of anyone index in an index expression
Reference https://www.physicsforums.com/insights/the-10-commandments-of-index-expressions-and-tensor-calculus/

Which expression is considered to have more than two instances of an index?
 
  • #12
Stephen Tashi said:
Which expression is considered to have more than two instances of an index?

Have you looked at the example given in the explanation of Commandment 5?
 
  • Like
Likes Opressor
  • #13
George Jones said:
Have you looked at the example given in the explanation of Commandment 5?

My question concerns the remarks before Commandment 5 and the examples before Commandment 5.

The examples after Commandment 5 are clear.

My confusion was due to the phrase "next-to-next commandment". I thought it referred to Commandment 5. I see now that it refers to Commandment 6.
 
  • #14
Stephen Tashi said:
My question concerns the remarks before Commandment 5 and the examples before Commandment 5.

I think by "next-to-next commandment", @Orodruin means Commandment 6. In the example you quoted, two "downstairs" indices are summed over, instead of one "downstairs" index and one "upstairs" index.
 
  • Like
Likes Opressor, Orodruin and Stephen Tashi
  • #15
George Jones said:
I think by "next-to-next commandment", @Orodruin means Commandment 6. In the example you quoted, two "downstairs" indices are summed over, instead of one "downstairs" index and one "upstairs" index.
This is correct. However, it does not hurt to be specific so I will update the text when I get around to it.

Edit: Done.
 
Last edited:
  • #16
"You shall not have more than two of anyone index in an index expression.

Guilty as charged, your Honour.
 

1. What is the purpose of the 10 Commandments of Index Expressions and Tensor Calculus?

The purpose of the 10 Commandments is to provide a set of guidelines for accurately and effectively using index expressions and tensor calculus in scientific research and calculations.

2. Why is it important to follow these commandments?

Following these commandments ensures consistency and accuracy in the use of index expressions and tensor calculus, which are fundamental tools in many branches of science and mathematics.

3. Who created the 10 Commandments of Index Expressions and Tensor Calculus?

The 10 Commandments were created by a group of scientists and mathematicians who are experts in the field of tensor calculus and index notation.

4. Are these commandments applicable to all types of scientific research?

Yes, these commandments are applicable to all types of scientific research that utilize index expressions and tensor calculus, including physics, engineering, and mathematics.

5. Can these commandments be updated or modified in the future?

Yes, these commandments can be updated or modified as new developments and advancements in the field of tensor calculus and index notation arise.

Suggested for: The 10 Commandments of Index Expressions and Tensor Calculus - Comments

Replies
0
Views
2K
Replies
9
Views
958
Replies
1
Views
882
Replies
3
Views
3K
Replies
12
Views
979
Replies
2
Views
972
Replies
5
Views
2K
Back
Top