How can second Schur's lemma lead to inequivalent representations?

[Heidi]

Homework Statement

proof of non equivalent representation from schur s lemma

Relevant Equations

non equivalence

i am reading this paper.
after equation 16 the author (Blasone) writes that
In the thermodynamical limit this goes to zero, i.e. the Hilbert spaces con-
structed over the respective vacuum states are orthogonal. From the second
Schur’s lemma [33] it then follows that the two representations of CCR (Weyl-
Heisenberg algebra) cannot be connected by a unitary transformation.
As there are many formulations of Schurs lemma i read the one in the cited referent 33.
let U(g) and U'(g) be 2 irreductible reps of G on V and V' and let A be a linear map from V' to V such that AU'(g) = U(g) A for all g in G then A = 0 or U and U' are equivalent.

here the paper says that inequivalence can be deduced from a 0 limit. Do you see why?
thanks in advance.

 

[Heidi]

Homework Statement: proof of non equivalent representation from schur s lemma
Homework Equations: non equivalence

i am reading this paper.
after equation 16 the author (Blasone) writes that
In the thermodynamical limit this goes to zero, i.e. the Hilbert spaces con-
structed over the respective vacuum states are orthogonal. From the second
Schur’s lemma [33] it then follows that the two representations of CCR (Weyl-
Heisenberg algebra) cannot be connected by a unitary transformation.
As there are many formulations of Schurs lemma i read the one in the cited referent 33.
let U(g) and U'(g) be 2 irreductible reps of G on V and V' and let A be a linear map from V' to V such that AU'(g) = U(g) A for all g in G then A = 0 or U and U' are equivalent.

here the paper says that inequivalence can be deduced from a 0 limit. Do you see why?
thanks in advance.

 

[Heidi]

I think that i found how inequivalence can be deduced from the lemma.
we have here a map A from U to U' which intertwins the representations. the lemma tells that we have
two exclusive possibilities : A = 0 or equivalent representation.
here it may be proved that A maps all the vectors of U to the 0 vector of U', so the representations art inequivalent.