Neutrino theory regarding rest masses

[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 differnt 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