Unitary

Definition

In representation theory, a unitary representation is a homomorphism from a group to the group of unitary operators on a Hilbert space. This means that the representation preserves the inner product structure of the space, which is crucial for ensuring that the concepts of orthogonality and norm are maintained. Unitary representations play an important role in quantum mechanics and provide insights into the symmetry properties of physical systems.

5 Must Know Facts For Your Next Test

1. Unitary representations are characterized by the property that for every group element, the corresponding operator is unitary, meaning it preserves inner products: \langle U(g)x, U(g)y \rangle = \langle x, y \rangle$ for all $x,y$ in the Hilbert space and group element $g$.
2. The characters of unitary representations are complex-valued functions that provide important information about the structure of the representation and are used in orthogonality relations.
3. The orthogonality relations for characters establish that different irreducible unitary representations of a finite group are orthogonal to each other when integrated over the group.
4. In the case of compact groups, all representations can be chosen to be unitary, highlighting the significance of unitary representations in understanding such groups.
5. Unitary representations have applications beyond mathematics, particularly in physics, where they describe symmetries and conservation laws in quantum mechanics.

Review Questions

• How does a unitary representation preserve the inner product structure in a Hilbert space?
  • A unitary representation preserves the inner product structure by ensuring that for any group element $g$ and any two vectors $x$ and $y$ in the Hilbert space, the inner product remains unchanged after applying the representation: \langle U(g)x, U(g)y \rangle = \langle x, y \rangle. This property is essential for maintaining concepts like orthogonality and distances between vectors, which are fundamental in quantum mechanics and functional analysis.
• Discuss the importance of orthogonality relations for characters in relation to unitary representations.
  • Orthogonality relations for characters are significant because they allow us to analyze how different irreducible unitary representations relate to each other. Specifically, these relations indicate that characters corresponding to distinct irreducible representations are orthogonal when integrated over the group. This leads to powerful tools in understanding representations, including decomposing representations into irreducible components and establishing connections between representation theory and number theory.
• Evaluate how unitary representations contribute to our understanding of symmetry in physical systems.
  • Unitary representations are crucial for understanding symmetries in physical systems because they mathematically encode how physical states transform under symmetry operations. In quantum mechanics, these transformations correspond to observable symmetries, such as rotations or translations, which conserve probabilities due to their unitary nature. The ability to utilize unitary representations enables physicists to apply representation theory techniques to analyze particles' behavior and interactions, thereby enhancing our grasp of fundamental principles like conservation laws and invariant properties within quantum systems.

Related terms

Hilbert Space: A complete vector space equipped with an inner product that allows for the generalization of geometric concepts such as distance and angle.

Group Representation: A way to represent the elements of a group as linear transformations of a vector space, allowing group operations to be expressed in terms of matrix multiplication.

Orthogonality: A property of vectors in a Hilbert space indicating that their inner product is zero, meaning they are perpendicular to each other.