CKM Matrix and mass eigenstates

In summary, the conversation discusses the concept of mass eigenstates and how they relate to operators, specifically in the context of the standard model and quantum mechanics. The speakers also touch on the role of unitary transformations and the importance of understanding operators in Hilbert space. They also discuss the complexities of understanding these concepts and the need for self-teaching.
  • #1
da_willem
599
1
First off, what is a mass eigenstate?! Is there a (hermitian) operator associated to mass? What should I picture when discussing a non-mass-eigenstate?! The same goes for a 'weak eigenstate' as the CKM matrix is supposed to be the basis transformation between these two... :rolleyes:

The I read that 'a linear transformation which diagonalizes the mass terms of the u-type quarks does not necessarily diagonalize those of the d-type quarks.'

What does this mean, and why not?! :grumpy:

Finally, I rwill be very much helped by any insight on the meaning of this matrix and its properties, e.g. why its unitary...? :blushing:

Sorry, the more I learn the less I seem to know...
 
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  • #2
da_willem said:
First off, what is a mass eigenstate?! Is there a (hermitian) operator associated to mass?
It's more convenient to use the mass^2 operator, i.e: E^2 - p^2. A mass
eigenstate is an eigenstate of this operator. Such a state has a determinate
mass value.

What should I picture when discussing a non-mass-eigenstate?!
I'm not sure what you mean by "picture". A "non-mass-eigenstate" is a
superposition of more than one mass eigenstate. It's mass is therefore
indeterminate (i.e: there's some probability of measuring any of the
mass values of the mass-eigenstates that have been superposed).

The same goes for a 'weak eigenstate' as the CKM matrix is
supposed to be the basis transformation between these two... :rolleyes:
The states that participate in the weak interaction are not mass-eigenstates
in general, but a superposition of mass eigenstates.

The I read that 'a linear transformation which diagonalizes the mass
terms of the u-type quarks does not necessarily diagonalize those of the d-type
quarks.' What does this mean, and why not?!
It means the operator (i.e: observable) corresponding to the flavor property
(u,d,etc) does not commute with the operator corresponding to the mass
property. So if you choose a Hilbert space basis corresponding to the
flavor eigenstates, they are in general a non-trivial superposition of mass
eigenstates.

Finally, I rwill be very much helped by any insight on the meaning
of this matrix and its properties, e.g. why its unitary...?
Such a transformation matrix is an operator in Hilbert space. It must be
unitary to preserve inner products between states in the Hilbert space.

Sorry, the more I learn the less I seem to know...
I know the feeling.

- strangerep
 
  • #3
Thanks very much for that! I guess that for any observable there is an operator, so also for (quark) mass and flavour. I just never came across (explicit representations of) such operators, so find them weird, rather than just an extension of what I know within the quantum formalism. Thanks very much again, for just showing that even in the standard model things are just like the hermitian operators and unitary transformations of ordinary quantum mechanics!
 
  • #4
da_willem said:
Thanks very much for that! I guess that for any observable there is an operator, so also for (quark) mass and flavour. I just never came across (explicit representations of) such operators, so find them weird, rather than just an extension of what I know within the quantum formalism. Thanks very much again, for just showing that even in the standard model things are just like the hermitian operators and unitary transformations of ordinary quantum mechanics!
I also needed much self-teaching before I finally realized that it's all about
Hilbert space operators. Or more precisely, it's all about relativistic QFT: unitary
operators in a multi-particle Hilbert space (aka Fock space) which has been
explicitly constructed so as to carry a (tensor product of) irreducible
representations of the Poincare group and the various internal symmetry
groups of the standard model.

- strangerep.
 

1. What is the CKM matrix?

The CKM matrix, also known as the Cabibbo-Kobayashi-Maskawa matrix, is a unitary matrix used to describe the mixing between different quark flavors in the Standard Model of particle physics. It encodes the relative strengths of the weak interactions between quarks of different flavors.

2. How does the CKM matrix relate to mass eigenstates?

The CKM matrix connects the mass eigenstates (states with definite masses) of quarks to the weak interaction eigenstates (states with definite weak interaction properties). The CKM matrix elements describe the probability of a quark of one flavor transforming into a quark of another flavor through a weak interaction.

3. What are the implications of the CKM matrix?

The CKM matrix provides a framework for understanding the observed pattern of quark mixing and the relative strengths of different weak interactions involving quarks. It also allows for predictions of rare decays and CP violation in particle interactions, which can provide insight into the nature of matter and antimatter in the universe.

4. How is the CKM matrix determined?

The CKM matrix elements are determined experimentally through measurements of particle interactions and decays involving quarks. The values of the matrix elements have been refined over the years as experimental techniques have improved, and the current values are consistent with the predictions of the Standard Model.

5. Are there any open questions or challenges related to the CKM matrix?

While the CKM matrix has been successful in describing quark mixing and weak interactions, there are still some open questions and challenges. For example, the precise values of some of the matrix elements are not well-constrained, and there is ongoing research into understanding the origin of the observed mixing pattern and the presence of CP violation in the CKM matrix.

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