# Neutrino theory regarding rest masses

1. Jul 1, 2015

### Buzz Bloom

Hi fzero, ChrisVer, and Orodruin:

To summarize from the discussion: it is certain that:
(1) M must be Hermitian, that is, it's conjugate transpose is it's own inverse
(2) at least one of M's nine components is not real.​

Thanks for the discussion,
Buzz

Last edited: Jul 1, 2015
2. Jul 1, 2015

### Orodruin

Staff Emeritus
1) No. M is hermitian, but you have given the description of a unitary matrix. A hermitian matrix is equal to its own hermitian conjugate.
2) No. We do not know this. This is still to be determined experimentally.

3. Jul 1, 2015

### ChrisVer

You mean like determining δ=0?

4. Jul 1, 2015

### Orodruin

Staff Emeritus
Yes, if $\delta = 0$ or $\pi$, neutrino oscillations are not violating CP.

5. Jul 1, 2015

### Buzz Bloom

Hi Orodruin:

Sorry about the appearance of the special characters in the quote, I think there is somthing flaky in my computer.

Underlining in above quotes is mine.

Sorry about my confusion. The vocabulary for the variety of complex matrix types is not yet well re-embeded in my mind.

I also said:
at least one of M's nine componets is not real.​
You commented:
No. We do not know this. This is still to be determined experimentally.​

If my mind is now working OK, if M is Hermetian, then M's three diagonal components must all be real.
Also, Wikperida https://en.wikipedia.org/wiki/Hermitian_matrix defines a Hermetian matrix:
a square matrix with [at least some] complex entries that is equal to its own conjugate transpose (bracketed text my addition)​
Therefore M is Hermetian also implies that two or four or all six of it's non-diagonal element are complex.

Buzz

Last edited: Jul 1, 2015
6. Jul 1, 2015

### Orodruin

Staff Emeritus
No, your logic is failing here. The off diagonal terms can also be real, they do not need to be, but they may be. There may be zero non-real elements in the matrix.

Last edited: Jul 3, 2015
7. Jul 1, 2015

### ChrisVer

A real symmetric matrix is Hermitian.... Hermitianity is the relation that $A^\dagger = A$... A real symmetric matrix is satisfying the hermitianity condition.
$A = \begin{pmatrix} a & b \\ b &c \end{pmatrix}$ with $a,b,c \in \mathbb{R}$ has $A^\dagger = A^T =\begin{pmatrix} a& b \\ b & c\end{pmatrix}= A$.

In this case again, it's as I asked Orodruin too, if $\delta =0$ (or $\pi$) then the $e^{\pm i \delta}$ doesn't stand for a complex number...it's $\pm 1$...and there are no possible complex elements in $M$ (of course the real numbers are just a subset of the complex numbers)...except for a Majorana case(?).

8. Jul 1, 2015

### Buzz Bloom

Hi ChrisVer:

(I edited the garbled quote that my computer put above to try to make it look like the original. How did you enter the dagger? is it a TeX command? I think my computer flakiness is related to TeX.)

I will try to change the definition at Wikipedia to make it clear that a Hermitian matrix may have complex components.

Buzz

9. Jul 1, 2015

### ChrisVer

The dagger I use in latex is " ^\dagger ".. ^ is for the powering.
Well again I'm saying that the real numbers case is just a special case of the complex numbers [where the imaginary part vanishes] so there is nothing wrong in saying it is complex and it happening to be real...it's just that the extra operation of complex conjugation * is trivial.

10. Jul 1, 2015

### Buzz Bloom

11. Jul 1, 2015

### Orodruin

Staff Emeritus
The statement on Wikipedia was not wrong and should be reverted. Real numbers are a subset of complex numbers as Chris pointed out. What we argued against was your assertion that there had to be elements which were not real numbers. Do not edit Wikipedia unless you are 100% sure of what you are doing and have expertise in the field.

12. Jul 2, 2015

### Buzz Bloom

Hi Orodruin:

I agree with you completelty about the math. I found the original phrasing ambiguous and unnecessarily confusing, although correct mathematically. It seemed to suggest that the defintion of Hermitian implied at least one non-real component.

The discussion says clearly, "The diagonal elements must be real," and "Hence, a matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix."

However, I think that many who did not already know what a Hermitian matrix was, and who read only what appears to be the definition in the first sentence would make the wrong interpretation. I think my revision avoids this ambiguity. I think it is unreasonable to require someone reading about a mathematical term to read an entire article to understand the definition of the term when one sentence can be sufficient.

Buzz

Last edited: Jul 2, 2015
13. Jul 2, 2015

### Orodruin

Staff Emeritus
I strongly disagree. The definition was not the least bit unclear. A hermitian matrix is a matrix with complex entries which is its own hermitian conjugate. There is nothing ambiguous about that. The definition makes it perfectly clear that any matrix which satisfies this is hermitian. Real numbers are a subset of complex numbers and you should expect anyone who reads about hermitian matrices to know this. Therefore, a real and symmetric matrix is going to be hermitian. Saying that the elements "may be" complex is only adding confusion. In my opinion, you have destroyed a perfectly fine opening to a Wikipedia article.

Information on Wikipedia should be accurate and precise, which the original was. You should not edit it while learning a subject just because you think it would be more pedagogical in a different way. In general, people with significantly more experience in communicating the subject are going to have written the entries in the first place.

14. Jul 2, 2015

### Buzz Bloom

Hi Orodruin:

When I made the change I also added my reasons for the change to the talk page. If the more experienced people maintinaing Wikipedia articles agree with you, they will undo my change. I think they might possibly agree with me that pedagogical considerations are very important, and yet disgree with me that the article would benefit from the pedagolically oriented change I made, or that one was necessary in the original text -- or maybe not with respect to any combination of these possibilities.

Buzz

15. Jul 2, 2015

### ChrisVer

it is indeed a very bad mistake to use "may be" in a definition... it raises the ambiguity, when definitions should be fair and square...someone can say "then there might be the case that it is not be a complex number=>what is it?"... and strictly speaking "not a complex number" would also rule out the real numbers . The distinction between real and something else, is with real vs imaginary, and not real vs complex.

Last edited: Jul 2, 2015
16. Jul 2, 2015

### Orodruin

Staff Emeritus
Just because you misunderstood the definition does not mean that someone else will. In my mind, anyone who is well versed in the terminology of complex numbers should get the definition correctly. Add on top of that the reasons given by Chris and you should realise that the change is a very very bad idea. This is the key part:

17. Jul 2, 2015

### Buzz Bloom

You have convinced me that my pedagogical change can be impoved. I have added a word as follows:
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix that may have non-real complex entries, and that is equal to its own conjugate transpose
I do not see in what way this new definition is still ambiguous. I am hopeful that the Wikipedia people who look at my current change will agree that compared with the original text some pedagogical change would be helpful. If they see any ambiguity in my text, I hope they will improve it.

Buzz

18. Jul 2, 2015

### Orodruin

Staff Emeritus
Please just reverse it to what it was before. Even textbooks will give the definition that was there before. This new version of yours is even making it worse. It is completely unintuitive what "may have non-real complex components" means and the word "may have" has nothing to do in a definition as remarked by Chris. And next time you consider making a change, check the exact text with someone who is experienced on the subject before making the change.

19. Jul 2, 2015