Mathematics

Functional Analysis

Functional analysis studies infinite-dimensional vector spaces and operators on them, providing the mathematical foundation for quantum mechanics, quantum field theory, and quantum gravity. It extends linear algebra to function spaces.

1. Normed Vector Spaces and Banach Spaces

Normed Vector Space

A normed vector space (V, ||·||) is a vector space V over ℝ or ℂ equipped with a norm||·||: V → [0,∞) satisfying:

  • $||v|| \geq 0$ with equality iff v = 0
  • $||\alpha v|| = |\alpha| ||v||$ for all scalars α
  • $||v + w|| \leq ||v|| + ||w||$ (triangle inequality)

Examples

ℝⁿ with Euclidean norm:

$$||x|| = \sqrt{\sum_{i=1}^n |x_i|^2}$$

L² space of square-integrable functions:

$$||f||_2 = \left(\int |f(x)|^2 dx\right)^{1/2}$$

L^p spaces for 1 ≤ p < ∞:

$$||f||_p = \left(\int |f(x)|^p dx\right)^{1/p}$$

Banach Space

A Banach space is a complete normed vector space—every Cauchy sequence converges to a limit in the space.

Examples: ℝⁿ, ℂⁿ, L^p spaces (1 ≤ p ≤ ∞), space of continuous functions C[a,b].

2. Inner Product Spaces and Hilbert Spaces

Inner Product

An inner product on a complex vector space V is a map $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{C}$ satisfying:

  • $\langle v, v \rangle \geq 0$ with equality iff v = 0
  • $\langle v, w \rangle = \overline{\langle w, v \rangle}$ (conjugate symmetry)
  • $\langle \alpha v + \beta w, u \rangle = \alpha\langle v, u \rangle + \beta\langle w, u \rangle$ (linearity in first argument)

The inner product induces a norm: $||v|| = \sqrt{\langle v, v \rangle}$

Hilbert Space

A Hilbert space ℋ is a complete inner product space—it's a Banach space with the additional structure of an inner product.

Examples:

  • ℂⁿ with $\langle x, y \rangle = \sum_i \overline{x_i} y_i$
  • L²(ℝ) with $\langle f, g \rangle = \int \overline{f(x)} g(x) dx$
  • Sequence space ℓ² with $\langle (a_n), (b_n) \rangle = \sum_n \overline{a_n} b_n$

Orthonormal Basis

A set $\{e_n\}$ is an orthonormal basis if:

$$\langle e_m, e_n \rangle = \delta_{mn}$$

and every v ∈ ℋ can be written as:

$$v = \sum_n \langle e_n, v \rangle e_n$$

Parseval's identity:

$$||v||^2 = \sum_n |\langle e_n, v \rangle|^2$$

Separable Hilbert Spaces

A Hilbert space is separable if it has a countable orthonormal basis. Quantum mechanics uses separable Hilbert spaces.

3. Linear Operators

Bounded Operators

A linear operator A: V → W between normed spaces is bounded if:

$$||A|| = \sup_{v \neq 0} \frac{||Av||}{||v||} < \infty$$

Bounded operators are continuous. The set of bounded operators B(V, W) forms a Banach space.

Adjoint Operator

For a bounded operator A: ℋ → ℋ on a Hilbert space, the adjoint A† satisfies:

$$\langle Av, w \rangle = \langle v, A^\dagger w \rangle \quad \text{for all } v, w \in \mathcal{H}$$

Special Classes of Operators

Self-adjoint (Hermitian): $A = A^\dagger$

$$\langle Av, w \rangle = \langle v, Aw \rangle$$

Physical observables in quantum mechanics are self-adjoint operators.

Unitary: $U^\dagger U = UU^\dagger = I$

$$\langle Uv, Uw \rangle = \langle v, w \rangle$$

Time evolution in quantum mechanics is unitary: $U(t) = e^{-iHt/\hbar}$

Projection: $P^2 = P, \, P^\dagger = P$

Projects onto a subspace. Measurement in quantum mechanics uses projection operators.

4. Spectral Theory

Spectrum

The spectrum σ(A) of an operator A consists of all λ ∈ ℂ for which (A - λI) is not invertible.

For finite dimensions, σ(A) = set of eigenvalues. In infinite dimensions, the spectrum may be continuous.

Point Spectrum (Eigenvalues)

λ is an eigenvalue if there exists v ≠ 0 with:

$$Av = \lambda v$$

The set of eigenvalues is the point spectrum σ_p(A).

Continuous Spectrum

λ is in the continuous spectrum σ_c(A) if (A - λI) is injective but not surjective, and its range is dense.

Example: Position operator $\hat{x}$ and momentum operator $\hat{p} = -i\hbar \frac{d}{dx}$ have purely continuous spectra ℝ.

Spectral Theorem

For a self-adjoint operator A on a Hilbert space:

  • Eigenvalues are real
  • Eigenvectors for distinct eigenvalues are orthogonal
  • A can be diagonalized via spectral decomposition

Spectral decomposition:

$$A = \int \lambda \, dE(\lambda)$$

where E(λ) is the spectral measure (projection-valued measure).

Physical Interpretation

In quantum mechanics:

  • Spectrum = possible measurement outcomes
  • Point spectrum = discrete energy levels (bound states)
  • Continuous spectrum = scattering states
  • Spectral decomposition = measurement postulate

5. Compact Operators

An operator K: ℋ → ℋ is compact if it maps bounded sets to relatively compact sets (closure is compact).

Equivalent Definitions

K is compact iff:

  • K maps bounded sequences to sequences with convergent subsequences
  • K is the norm limit of finite-rank operators

Properties

  • Compact operators form a closed two-sided ideal in B(ℋ)
  • If K is compact and self-adjoint, it has a countable spectrum of eigenvalues converging to 0
  • Eigenvectors form an orthonormal basis (spectral theorem for compact operators)

Example: Integral Operators

The operator:

$$(Kf)(x) = \int K(x, y) f(y) dy$$

is compact if K(x,y) is continuous and the domain is bounded.

Relevance to Physics

Compact operators arise in quantum mechanics for perturbations, scattering theory, and Green's functions. They have discrete spectra, unlike differential operators which typically have continuous spectra.

6. Distributions (Generalized Functions)

Distributions (generalized functions) extend the concept of functions to include objects like the Dirac delta "function".

Test Functions

Define the space of test functions $\mathcal{D}(\mathbb{R}^n)$ = smooth functions with compact support (C^∞_c).

Distribution

A distribution T is a continuous linear functional on $\mathcal{D}$:

$$T: \mathcal{D} \to \mathbb{C}, \quad \phi \mapsto T[\phi] = \langle T, \phi \rangle$$

Dirac Delta Distribution

The Dirac delta δ is defined by:

$$\langle \delta, \phi \rangle = \phi(0)$$

Informally: $\int \delta(x) \phi(x) dx = \phi(0)$

Shifted delta:

$$\langle \delta_a, \phi \rangle = \phi(a)$$

Derivatives of Distributions

The derivative of a distribution T is defined by:

$$\langle T', \phi \rangle = -\langle T, \phi' \rangle$$

Example: Derivative of Heaviside step function θ(x):

$$\theta'(x) = \delta(x)$$

Applications to Physics

  • Quantum mechanics: Position eigenstates |x⟩, momentum eigenstates |p⟩
  • Green's functions: $(\Box + m^2)G(x-y) = \delta^4(x-y)$
  • Point charges: $\rho(x) = q\delta^3(x - x_0)$
  • Initial conditions: Distributions specify boundary/initial data

7. Sobolev Spaces

Sobolev spaces H^k combine integrability conditions on functions and their weak derivatives, essential for PDEs and quantum field theory.

Definition

The Sobolev space H^k(ℝⁿ) (or W^{k,2}) consists of functions f ∈ L² whose weak derivatives up to order k are also in L²:

$$H^k(\mathbb{R}^n) = \left\{f \in L^2 : \partial^\alpha f \in L^2 \text{ for } |\alpha| \leq k\right\}$$

with norm:

$$||f||_{H^k}^2 = \sum_{|\alpha| \leq k} ||\partial^\alpha f||_{L^2}^2$$

Weak Derivatives

A function g is the weak derivative ∂f if:

$$\int g(x) \phi(x) dx = -\int f(x) \partial\phi(x) dx$$

for all test functions φ. Weak derivatives may exist even when classical derivatives don't.

H¹ Space

The space H¹ consists of functions f ∈ L² with first weak derivatives in L². This is the natural setting for energy functionals:

$$E[f] = \int (|\nabla f|^2 + |f|^2) dx$$

Applications

  • PDEs: Weak solutions to elliptic, parabolic, hyperbolic equations
  • Quantum field theory: Regularization of quantum fields
  • General relativity: Weak formulations of Einstein equations
  • Gauge theory: Configuration spaces for gauge fields

Sobolev Embedding Theorem

Under appropriate conditions, H^k embeds (continuously or compactly) into L^p or C^m spaces, relating integrability to smoothness.

8. Applications to Quantum Mechanics and Quantum Gravity

Quantum Mechanics

State space: Separable Hilbert space ℋ (e.g., L²(ℝ³) for a particle)

Observables: Self-adjoint operators (position $\hat{x}$, momentum $\hat{p}$, Hamiltonian $\hat{H}$)

Time evolution: Schrödinger equation

$$i\hbar \frac{\partial}{\partial t}|\psi\rangle = \hat{H}|\psi\rangle$$

Solution: $|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle$, where $U(t) = e^{-i\hat{H}t/\hbar}$ is a unitary operator (Stone's theorem)

Quantum Field Theory

Fock space: Direct sum of n-particle Hilbert spaces

$$\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{H}_n$$

Creation/annihilation operators: Unbounded operators on Fock space satisfying commutation/anticommutation relations

Renormalization: Functional analysis techniques handle infinities (distributions, operator-valued distributions)

Loop Quantum Gravity

Kinematical Hilbert space: L²(𝒜̄/𝒢) where 𝒜̄ is the space of generalized SU(2) connections

Spin network basis: Countable orthonormal basis labeled by graphs with SU(2) representations

Geometric operators: Area and volume operators are self-adjoint with discrete spectra

$$\hat{A}_S = 8\pi\gamma \ell_P^2 \sum_{I \in S} \sqrt{\hat{J}_I(\hat{J}_I + 1)}$$

where J_I are SU(2) Casimirs (spins) at surface intersections

String Theory

Worldsheet theory: 2D conformal field theory with Virasoro algebra (infinite-dimensional Lie algebra)

String Hilbert space: Fock space built from oscillator modes

BRST cohomology: Uses functional analysis to define physical states and gauge-fix string theory

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