Unitary representation

Definition

A unitary representation is a homomorphism from a group to the group of unitary operators on a Hilbert space, preserving the inner product structure. This means that the action of the group on the space is not only linear but also maintains the lengths and angles, making it a very structured way to understand symmetries in quantum mechanics and other areas of mathematics. Unitary representations play a key role in understanding linear representations and their properties, as well as in the application of Schur's Lemma and orthogonality relations, which provide essential tools for analyzing these representations.

5 Must Know Facts For Your Next Test

1. Unitary representations are essential for studying symmetries in both classical and quantum mechanics, as they allow for the exploration of how groups act on quantum states.
2. The preservation of inner products in unitary representations means that if two vectors are orthogonal, their images under the representation will also be orthogonal.
3. Every finite-dimensional unitary representation can be decomposed into irreducible representations, which are those that cannot be further simplified.
4. Schur's Lemma states that if two unitary representations are equivalent, any intertwining operator between them is either zero or an isomorphism.
5. Orthogonality relations among characters of unitary representations help in determining how different representations relate to each other and contribute to classification efforts.

Review Questions

• How does the property of preserving inner products in unitary representations influence their application in quantum mechanics?
  • The preservation of inner products in unitary representations is crucial for quantum mechanics because it ensures that physical states remain normalized and orthogonal under the action of symmetry transformations. This property guarantees that probabilities derived from quantum states, which depend on inner products, remain consistent. As such, unitary representations serve as the mathematical backbone for describing symmetry operations that govern physical systems.
• Discuss how Schur's Lemma relates to unitary representations and its significance in identifying irreducible components.
  • Schur's Lemma is fundamentally important in the study of unitary representations because it provides insights into when two representations are equivalent. Specifically, it states that if two equivalent unitary representations have an intertwining operator, then this operator must either be zero or an isomorphism. This lemma is significant for classifying irreducible components, as it helps identify distinct representations by ruling out non-equivalent cases based on the behavior of intertwining operators.
• Evaluate the role of orthogonality relations in understanding the structure of unitary representations and their implications in broader mathematical contexts.
  • Orthogonality relations serve as powerful tools for analyzing the relationships among characters of unitary representations, offering insights into their structure and classification. By revealing how characters interact with one another through inner products, these relations allow mathematicians to distinguish between different types of representations and understand their implications in various mathematical fields. In broader contexts, such as number theory and combinatorial designs, these relations contribute to a deeper understanding of symmetry and its applications across disciplines.

Related terms

Hilbert Space: A complete inner product space that provides a framework for studying quantum mechanics and various mathematical problems.

Group Homomorphism: A function between two groups that preserves the operation of the groups, meaning that it maps elements in a way that respects their algebraic structure.

Orthogonality Relations: These are equations that express how different representations of a group interact with one another, particularly concerning their inner products.