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CKM Matrix and mass eigenstates

by da_willem
Tags: eigenstates, mass, matrix
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da_willem
#1
May21-07, 02:37 PM
P: 599
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...

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?!

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

Sorry, the more I learn the less I seem to know...
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strangerep
#2
May22-07, 02:21 AM
Sci Advisor
P: 1,927
Quote Quote by da_willem View Post
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...
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
da_willem
#3
May22-07, 01:47 PM
P: 599
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!

strangerep
#4
May23-07, 06:36 PM
Sci Advisor
P: 1,927
CKM Matrix and mass eigenstates

Quote Quote by da_willem View Post
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.


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