Neutrino theory regarding rest masses

[Orodruin]

Hi ChrisVer:

I made a correction at https://en.wikipedia.org/wiki/Hermitian_matrix . I hope it sticks.

Regards,
Buzz

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.

 

[Buzz Bloom]

Hi Orodruin:

The statement on Wikipedia was not wrong and should be reverted.

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 definition 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.

Thanks for your post,
Buzz

 

[Orodruin]

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.

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.

 

[Buzz Bloom]

Hi Orodruin:

You should not edit it while learning a subject just because you think it would be more pedagogical in a different way.

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.

I appreciate your sharing your thoughts about this with me.

Thanks for your post,
Buzz

 

[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.

 

[Orodruin]

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 realize that the change is a very very bad idea. This is the key part:

The distinction between real and something else, is with real vs imaginary, and not real vs complex.

 

[Buzz Bloom]

Hi ChrisVer and Oradruin:

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.

Thanks for your discussion,
Buzz

 

[Orodruin]

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.

 

[Vanadium 50]

Buzz, yesterday you didn't know what a Hermitian matrix was (and I am not certain you do even today). I don't understand why people feel compelled to edit Wikipedia on subjects that they are new to, but all you have done is left a mess on Wikipedia for someone else to clean up. You should revert it - the original definition was fine, and your new one is work.

 

[Orodruin]

In addition, you are quoting this very thread in the talk page as the reason for your edit. This could seriously damage the reputation of Physics Forums. Since you seem unwilling to revert the edit yourself, I am going to do it. Please stop doing things like this.

 

[vanhees71]

I don't understand the problem. A finite-dimensional complex matrix $\hat{M}$ is called Hermitean iff $\hat{M}^{\dagger}=\hat{M}$. In finite-dimensional unitary spaces a linear operator is self-adjoint iff it is Hermitean. That's it.

It's more complicated when it comes to hermitizity and self-adjointness in infinite-dimensional Hilbert spaces. There you have to distinguish between the weaker Hermitizity from the stronger self-adjointness.

 

[Orodruin]

I don't understand the problem.

The problem is that Buzz is equating "complex" with "not real" (i.e., having imaginary part not equal to zero).

 

[Buzz Bloom]

Hi @Vanadium 50 and @Orodruin:

I appologize for offending you. I am glad Orodruin restored the original text. You are probably not interested in my reasons for not doing it myself, and they are not particularly important anyway.

I very much enjoy, and learn from, the discussions in the PF. It never occurred to me that my referring on the Talk page about my being corrected in the PF might harm the PF. If you think it important to do so, I will edit that page and remove that reference.

Regretfully,
Buzz

 

[ChrisVer]

I didn't feel offended...(I hope nobody did- "speaking for others" )
What everyone has been trying to tell you (even after your 2nd correction) was that the original definition was better (and more correct) , and for that reason it should remain unchanged.
As for PF, as a result of dropping the quality of the definition (and for that referring the PF), you are also dropping the quality of PF...And nobody in here said that the original definition was wrong...so we didn't "correct" the definition, but we corrected your misinterpretation of it...

and as a final note: keep your thread on topic (this has gone a bit astray)...it's yours and you should protect it. The conversation about the hermitian conjugation and all that followed, is not helping in that...

 

[Buzz Bloom]

New Question

Quote from
https://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix#Parameterization .

The PMNS matrix is not necessarily unitary and additional parameters are necessary to describe all possible neutrino mixing parameters, in other models of neutrino oscillation and mass generation, such as the see-saw model, and in general, in the case of neutrinos that have Majorana mass rather than https://en.wikipedia.org/w/index.php?title=Irac_fermion&action=edit&redlink=1 .​

Does the quote mean that the PMNS mass mixing matrix includes values for probabilites that take into account both of two currently undecided possibilites regarding the nature of a neutrino's mass type: Majorana or Dirac?

 

[Orodruin]

Does the quote mean that the PMNS mass mixing matrix includes values for probabilites that take into account both of two currently undecided possibilites regarding the nature of a neutrino's mass type: Majorana or Dirac?

This makes no sense. I suggest you pick up a basic book in quantum mechanics before even attempting questions regarding neutrino oscillations. The PMNS matrix is not a matrix of probabilities, it is a mixing matrix. The PMNS matrix is equally applicable to Majorana and Dirac neutrinos, with some additional physical phases for Majorana neutrinos.

 

[Buzz Bloom]

Hi @Orobruin:

The PMNS matrix is not a matrix of probabilities, it is a mixing matrix. The PMNS matrix is equally applicable to Majorana and Dirac neutrinos, with some additional physical phases for Majorana neutrinos.

(underlining is mine)

Here is another quote from https://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix#Parameterization .

The PMNS matrix describes the amplitude that a neutrino of given flavor α will be found in mass eigenstate i. The probability that a neutrino of a given flavor α to be found in mass eigenstate i is proportional to |Uαi|² (underlining is mine)​

I am not certain how to reconcile the first underlined text from above quote from your post with the underlined text from Wikipedia quote. Are you simply making the technical distinction between an amplitude and it's absolute value square, that is, a probability. I am also not sure I understand exactly what you mean by "equally applicable" in the second underlined text from your post quote. Does this mean the following?

The form of the PMNS matrix can be applied equally well to both Dirac and Majorana neutrinos, but for the Majorana neutrinos some additional terms representng additional physical phases need to be included. Therefore the values of the matrix components would not be the same.​

The following concept is the reason I have made the interpretation in the above text. Dirac and Majorana neutrinos are not two different kinds of neutrinos, both of which existing in our real universe. Rather, they are two different theorectical possibilies regarding the nature of real neutrinos. Therefore, it would not make sense that a matrix of specific amplitude values would be applicable for both theories. Your post was helpful to me thinking about the question I asked in my previous post, and arriving at this concept. I believe my problem in understanding the Wikiedia quote in my previous post was a matter of what I see as ambiguous use of language, a common problem for me in learning from Wikikpedia articles.

I suggest you pick up a basic book in quantum mechanics before even attempting questions regarding neutrino oscillations.

I think this is an excellent suggesion. I have asked my town library to get a copy for me of Quantum mechanics : the theoretical minimum by Leonard Susskind and Art Friedman, and it is currently in transit from another library. I expect to start learning from it in a few days. This book was recommended by someone on the forum when I requested a suggestion, but I can't find that post now, so I can't tell you who made the recommendation.

Thanks for your post,
Buzz

 

[Orodruin]

I am not certain how to reconcile the first underlined text from above quote from your post with the underlined text from Wikipedia quote. Are you simply making the technical distinction between an amplitude and it's absolute value square, that is, a probability. I am also not sure I understand exactly what you mean by "equally applicable" in the second underlined text from your post quote. Does this mean the following?

It is simply not a matrix of probabilities. You can compute some pribabilities from it, but you need to take care due to oscillations.

 

[Buzz Bloom]

Hi Orodruin:

It is simply not a matrix of probabilities. You can compute some pribabilities from it, but you need to take care due to oscillations.

I get it. Calling it probabilities was an incorrect way of using the vocabulary.

BTW, the recommendation of Quantum mechanics : the theoretical minimum was made by bhobba in post #211 in the thread "is the cat alive, dead, both or unknown".

Thanks for the clarification re "probability",
Buzz