Graduate Fock spaces and inequivalent ccr representations

[Heidi]

TL;DR

Do physicist and mathematicians agree with the définition of inequivalent ccr representations?

Fock spaces use lists of integers (0 and 1 in the fermionic case) to describe set of particles.
a list 0 1 1 0 1 0 0 0 ... for 1 fermion in the second third and fifth state may be associated to the real number 0.0110100000...
so this set of list is not countable. a Fock space select a countable subset of such lists
the simpler is found if we choose the lists with a finite number or 1. it contains 0 0 0 0 0 0 ...
we can do the same thing in the bosonic case.
i read this in a book:
Two representations in different subsets are unitarily inequivalent to each other.
If two different subsets can be used as base of representations for the operators;
{a i , a i † } and {α i , α i † } in the set {|n 1 , n 2 , . . . , n i , . . . }, these two representations are
unitarily inequivalent to each other, which means that a vector of one representation
for the operators {a i , a i † } is not expressed by a superposition of base vectors of
another representation for the operators {α i , α i † }. In quantum mechanics, we can
express a vector of one representation by a superposition of base vectors of another
representation, because all possible representations are unitary equivalent.This
situation disappears in the case of quantum field theory.

this the definition of unitarily inequivalence for a physicist.
a mathematician would have spoken of commutativity and intertwiners.
Are they talking about the same thing?

 

[PeterDonis]

Moderator's note: Thread level changed to "A".

 

[Heidi]

It seems that physicists and mathematicians are talking about the same thing. But when a mathematician sees a bijective intertertwiner A between H1 and H2 with their own represention of a group or an algebra, the physicist sees only one hilbert space H (with its scalar product), A becomes the identity and there are two representations on H.
if all can be well computed they say that the representations are unitarily equivalent. But if the existence of the two representations leads to contradictions or ill defined things they say that there are unitatily inequivalent.

 

[WWGD]

Maybe @fresh_42 knows about this?

 

[PeterDonis]

i read this in a book

Which book?

the physicist sees only one hilbert space H (with its scalar product)

I'm not sure this is true; I think physicists consider unitarily inequivalent representations to lie in different Hilbert spaces. In the quote you give from a book, it says the basis vectors of one representation cannot be expressed as superpositions of the basis vectors of another representation, if the two representations are unitarily inequivalent. But if both sets of basis vectors belong to the same Hilbert space, each set must be expressible as superpositions of the other set, since it's a vector space axiom that any vector in the space can be expressed as a linear combination of a set of basis vectors.

 

[Heidi]

Which book?

the book is "liquid glass transition" written by Toyosuki Kitamura (chapter 3).