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

• Heidi
In summary, the paper discusses the proof of non-equivalent representations from Schur's lemma. It states that in the thermodynamical limit, the Hilbert spaces constructed over the respective vacuum states are orthogonal. From the second Schur's lemma, it is then concluded that the two representations of CCR (Weyl-Heisenberg algebra) cannot be connected by a unitary transformation. The paper also mentions that there are multiple formulations of Schur's lemma, and the one cited in reference 33 is used. The lemma states that if there are two irreducible representations of a group on two different vector spaces, and there exists a linear map that intertwines the representations, then either the map is 0 or the representations are
Heidi
Homework Statement
proof of non equivalent representation from schur s lemma
Relevant Equations
non equivalence
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?

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

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?

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.

## 1. How does the second Schur's lemma relate to representations in science?

The second Schur's lemma is a fundamental concept in representation theory, which is a mathematical framework used to study the symmetries and structure of physical systems. It states that for a given irreducible representation, any linear transformation that commutes with all elements of the group must be a scalar multiple of the identity. This has important implications for identifying and classifying different representations in science, as well as understanding the relationships between them.

## 2. Can you give an example of how the second Schur's lemma is applied in science?

One example where the second Schur's lemma is applied is in the study of molecular vibrations. The vibrational modes of a molecule can be represented by a set of linear transformations, which form a representation of the molecular symmetry group. The second Schur's lemma can then be used to identify and classify different vibrational modes and their symmetries, providing valuable insights into the molecular structure and behavior.

## 3. What is the significance of inequivalent representations in science?

Inequivalent representations are representations that cannot be transformed into each other by any unitary transformation. In science, this means that they describe fundamentally different symmetries and structures of physical systems. Understanding and identifying inequivalent representations is crucial in fields such as quantum mechanics, where different representations can correspond to different physical observables and have important consequences for the predictions and interpretations of experiments.

## 4. How is the concept of equivalence related to the second Schur's lemma?

Equivalence is a fundamental concept in representation theory that refers to the relationship between different representations of a group. The second Schur's lemma plays a key role in determining whether two representations are equivalent or not. If two representations are not equivalent, then they are inequivalent and describe fundamentally different symmetries and structures.

## 5. Are there any limitations or exceptions to the second Schur's lemma?

While the second Schur's lemma is a powerful tool for understanding representations, there are some limitations and exceptions to its applicability. One limitation is that it only applies to finite-dimensional representations. Additionally, there are certain types of groups, such as non-compact groups, for which the second Schur's lemma does not hold. In these cases, alternative methods must be used to analyze and classify representations.

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