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
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 states that participate in the weak interaction are not mass-eigenstates
in general, but a superposition of mass eigenstates.
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
Such a transformation matrix is an operator in Hilbert space. It must be
unitary to preserve inner products between states in the Hilbert space.
I know the feeling.