- #1

Heidi

- 418

- 40

- 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.

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.