Unitary

Definition and Overview

In mathematics and physics, the term "unitary" refers to a property of certain matrices and operators that preserve the inner product in a complex vector space. A unitary matrix is a complex square matrix \( U \) that satisfies the condition \( U^\dagger U = UU^\dagger = I \), where \( U^\dagger \) is the conjugate transpose of \( U \) and \( I \) is the identity matrix. This property ensures that the matrix preserves the length of vectors and the angles between them, making it a fundamental concept in quantum mechanics and other fields.

Mathematical Definition

A matrix \( U \) is unitary if it satisfies the following condition: \[ U^\dagger U = UU^\dagger = I \] where: - \( U^\dagger \) is the conjugate transpose of \( U \). - \( I \) is the identity matrix.

For an operator \( \mathcal{U} \) on a complex Hilbert space \( \mathcal{H} \), \( \mathcal{U} \) is unitary if: \[ \langle \mathcal{U}x, \mathcal{U}y \rangle = \langle x, y \rangle \] for all \( x, y \in \mathcal{H} \), where \( \langle \cdot, \cdot \rangle \) denotes the inner product.

Properties of Unitary Matrices

Unitary matrices have several important properties: - **Norm Preservation**: For any vector \( v \) in a complex vector space, \( \|Uv\| = \|v\| \). - **Eigenvalues**: The eigenvalues of a unitary matrix lie on the complex unit circle, meaning they have an absolute value of 1. - **Orthogonality**: The rows and columns of a unitary matrix form an orthonormal basis. - **Invertibility**: A unitary matrix is always invertible, and its inverse is its conjugate transpose, \( U^{-1} = U^\dagger \).

Applications in Quantum Mechanics

In quantum mechanics, unitary operators are essential because they describe the evolution of quantum states. The time evolution of a quantum state \( |\psi(t)\rangle \) is governed by the Schrödinger equation, and the solution involves a unitary operator \( U(t) \): \[ |\psi(t)\rangle = U(t)|\psi(0)\rangle \] where \( U(t) \) is a unitary operator that depends on time. This ensures that the total probability is conserved over time.

Unitary Transformations

A unitary transformation is a linear transformation that preserves the inner product. If \( U \) is a unitary matrix, then the transformation \( v \mapsto Uv \) is a unitary transformation. These transformations are widely used in Fourier analysis, signal processing, and other areas of applied mathematics.

Unitary Groups

The set of all \( n \times n \) unitary matrices forms a group under matrix multiplication, known as the unitary group \( U(n) \). This group is a Lie group, meaning it has a smooth manifold structure and group operations that are smooth. The unitary group plays a significant role in various branches of mathematics and theoretical physics.

Unitary Representations

A unitary representation of a group \( G \) on a Hilbert space \( \mathcal{H} \) is a homomorphism \( \pi: G \to U(\mathcal{H}) \), where \( U(\mathcal{H}) \) is the group of unitary operators on \( \mathcal{H} \). Unitary representations are crucial in the study of harmonic analysis and the representation theory of groups.

Unitary Operators in Functional Analysis

In functional analysis, a unitary operator on a Hilbert space \( \mathcal{H} \) is a bounded linear operator \( U: \mathcal{H} \to \mathcal{H} \) that satisfies \( U^*U = UU^* = I \), where \( U^* \) is the adjoint of \( U \). These operators are essential in the spectral theory of operators and the study of operator algebras.

Examples and Special Cases

1. 1. Pauli Matrices

The Pauli matrices are a set of three \( 2 \times 2 \) complex matrices that are unitary and Hermitian. They are used in the study of spin in quantum mechanics: \[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]

1. 1. Fourier Transform

The discrete Fourier transform (DFT) matrix is an example of a unitary matrix. For an \( n \)-dimensional vector, the DFT matrix \( F \) is defined as: \[ F_{jk} = \frac{1}{\sqrt{n}} e^{-2\pi i jk / n} \] where \( j, k = 0, 1, \ldots, n-1 \).