All comments

Repeat post with truncated matrices

We will consider the initial stage of reconciling the two approaches using the example of a spherically symmetric metric with a singularity on the three-dimensional unit sphere. Let be an orthogonal matrix () of order three such that , where . Then the Jacobian corresponding to the quadratic differential form ?? in a non-orthogonal coordinate system has the form

(1)  

where , and in the orthogonal coordinate system it is reduced to the form

(2)  

It only remains to check the differential form for harmonicity, which is almost obvious due to the fact that

(3)  

Ark, I continue searching into relations between pi and CFT. A very useful book is Gelfand, Graev, Vilenkinhttps://ikfia.ysn.ru/wp-content/uploads/2018/01/GelfandGraevVilenkin1962ru.pdf. It contains notions which have much in common with those considered in your recent posts.
I make some notes but fail to boil them into a smooth text… Still working with it.

Ark, I looked through the reasoning carefully, it is all clear, there is only one minor remark: when you write the sum for k=1, it is a bit misleading that k still stands there in the expression explicitly, in i_k and j_k. Wouldn’t it be better to omit these indices from the first sum?

The explanation is crystal clear, thank you very much. I was close to a solution, but I lacked persistence and confidence to work with deltas so freely.

“We can use Eq. (4) to get ({\bf How?})”
It is not evident for me how to use (4), I would rather use 16.5-5 from the Korn’s book,
which states that r-rank deltas are related to s-rank deltas with the factor (n-s)!/(n-r)!
In our case, s = r-1, we have coefficicient (n-r+1)!/(n-r)! = n-r+1, and that is exactly what is wanted.

As far as tensor densities are concerned, it is a new subject to me. So, the understanding is only intuitive but not rigorous at the moment.

Ark, there are a few places where bold and italic fonts are not working correctly, e.g.:
“The next question is: {\bf is it a tensor?}” and several cases below.

Now I am trying to grasp formulas (1) and (2) and how should we use them to get (3). Isn’t it easier just to take (3) as a definition?

As regards Why?s:

“to get a non-identically-zero symbol, we must have r n, then two or more of indices would have to coincide and we will get zero.

“Every index i_k must appear (and only once) among the indices j_1, j_r. Why?”

If no i_k appears among indices “on the other floor”, then, in view of (3), all (1,1)-deltas on the rhs would have different upper and lower indices, and hence no one of them could be nonzero. And if some i_k appear among j_l twice or more times, then we are lack of j_l=k, which was already used once to give delta^ik_jk and hence all other deltas have different upper and lower indices.

“If we apply the same permutation to upper and lower indices, the value of delta remains unchanged. Why?”

That is because each permultation (of upper or lower indices) only changes the sign of the result. If we make any permutation twice or any even number of times, we will get the same result.

Yeah the things Ark is hard on himself for are things I wouldn’t be able to notice anyways while the posts themselves are quite interesting. The idea that say CL(4) has 16 dimensions while the permutation of 4 things is 24 yet both have odd/even grade/parity is a little perplexing for me but apparently explaining the details for the Clifford algebra case does get into permutations and the grading is kind of just an end result that works. That Ark looks greatly at the details is great for me since things like number theory parallels are what draws me in; it certainly wasn’t my relatively basic math education.

I used the think I’d never understand even a little about things like EEQT and differential geometry beyond something like EEQT being GRW-like. Now EEQT being GRW-like seems like a very minor detail and I even see Ark’s EEQT and conformal structures work as related. I used to think of them as totally unrelated. It’s fun talking to AI about Ark’s overall work. It’s totally safe to bring up Ark to AI; Ark’s wife Laura and Ark’s friend Tony not so much. I now introduce Tony by introducing Ark first.

I’m probably interested in contravariant vs covariant indices related things for Born reciprocity position momentum phase space reasons but this may be more for future me at best. For Bradonjic unimodular volume form reasons, I like restricting to SL(n,R) though for Born reciprocity reasons I might like restricting even more to SP(n,R) and I’m not sure the pseudo tensors being treated as tensors still holds in this case. Course I’m not even sure what goes wrong parity-wise if you have pseudo tensors instead of tensors.

Ark, it seems to me that you’re being too hard on yourself. The post “It’s All About Permutations” was one of the most helpful for me: it reminded me how to deal with those scaring objects with multiple indices. As far as I remember, Penrose even introduced the notion of “arms and legs” in an attempt to make tensors easier to work with. But your explanations are more instructive; after thinking on the exercises, I truly felt more confident.

The very idea of ​​permutations is very profound, since it is not about quantities but rather about ordering, which is another face of a number.

And finally, in no case, the mysterious holographic images of the differently rearranged cats appearing in that post can be called a failure.

Moving to the present post, please take a look; there might be a typo here:
“But the matrix A^{i’}_i is the inverse of A^i_{j’}” –>
“But the matrix A^{i’}_i is the inverse of A^i_{i’}”,
with reference to the “Tensors on a picnic” post.

Ark, I am sorry for returning once again to the subject, but I am in doubt:

According to (16.5-7) epsilon with upper indices is related to epsilon with lower indices precisly like deltas with subscripts and superscripts being interchanged, right?

In compliance with the definition of delta-object in 16.5-2,
delta with i_r superscripts and k_r subscripts is equal to +1 or -1 if
“the ordered set of subscripts is obtained by an even or odd permutation of the set of superscripts”, and is zero for all other combinations.
This means that delta with i_r superscripts and k_r subscripts and delta with k_r superscripts and i_r subscripts take the same values simultaneously, don’t they?

So, epsilons with upper and lower indices appear to be the same anyway!

I feel that metrics should emerge somewhere, this is surely reasonable, but could you show how it happens rigorously?

Indeed, I didn’t like these two formulas the moment I saw them:
2) = 0 1
3) = 0 -1
I think the zeros should be removed.
Thus, according to these formulas, epsilon with subscripts is equal to epsilon with superscripts, and the metric is simply unity.
But I don’t know what this would look like if the metric were not Euclidean.