Re: Gauge Transformations in Momentum Space?

Eugene Stefanovich wrote:

> [...]
> OK, let's skip superconductivity and talk about neutrinos.
> I looked at Blasone-Vitiello paper. This is a good example
> of what seems confusing about UIR for me.
>
> They find a unitary transformation which makes flavor
> eigenstates (or creation-annihilation operators) from
> mass eigenstates (or creation-annihilation operators).
> This transformation also changes the vacuum vector.
> In particular, it makes the new vacuum |0'> orthogonal
> to the old vacuum |0>.
>
> I have two questions:
>
> 1. In my opinion this construction does not mean that the
> new vacuum lies in a different Fock state.

I presume you meant Fock "space". :-)

> This wouldn't be the case even if all components of |0'>
> in the old basis were "zero" in the limit of infinite volume.

Actually, it would. I.e: it means that the new vacuum
cannot be expressed as a mathematically meaningful
superposition of vectors in the old Fock space.

Unfortunately, I have to go away for a couple of days, so
I can't respond more thoroughly right now. But I promise
to do so when I return. For now, I'll just make a few general
remarks:

It is probably true that *all* the vectors in the new Fock space
are orthogonal to *every* vector in the old Fock space,
(though I haven't explicitly checked this for the
Blasone-Vitiello - I should remedy that). However, Umezawa
contains the essence of such a calculation, though he
leaves it to the reader to fill in a fair bit of detail. Do you
have access to a copy of Umezawa? It explains a lot of
the theory of UIR much better than I can. The basic idea
is that every new vector is orthogonal to every old vector.
Therefore, none of the new vectors can be expressed as
superpositions of the old vectors. That's essentially what
defines a UIR.

I'll follow up on this, and your 2nd question, more fully
in a few days time.

.