Quantum Theory — Hilbert Space, Spectra, Algebras

Job: State Hilbert-space QM axioms; deploy the spectral theorem; point to \(C^*\)-algebras/KMS for infinite systems; record scattering in one clean line.

Core

Hilbert space axioms (nonrelativistic QM)

  1. States: pure states \(=\) unit rays \([\psi]\subset\mathcal{H}\) (complex separable Hilbert space); general states \(=\) density operators \(\rho\ge 0\), \(\operatorname{Tr}\rho=1\).
  2. Observables: self-adjoint operators \(A\) (possibly unbounded) on \(\mathcal{H}\); outcomes \(=\) spectrum \(\sigma(A)\).
  3. Born rule: for pure \(\psi\) and spectral measure \(E_A\) of \(A\), \(\mathbb{P}(\Delta)=\|E_A(\Delta)\psi\|^2\); general \(\mathbb{P}(\Delta)=\operatorname{Tr}(\rho E_A(\Delta))\).
  4. Dynamics: strongly continuous unitary group \(U(t)=e^{-itH/\hbar}\) (Stone) with self-adjoint Hamiltonian \(H\); Schrödinger \(i\hbar\dot\psi=H\psi\); Heisenberg \(\dot A= (i/\hbar)[H,A]+\partial_t A\).
  5. Composition: \(\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B\); entanglement \(=\) nonproduct vectors.
  6. Measurement (idealized): Lüders/von Neumann update by spectral projections — foundationally subtle; operational POVM generalizations.

Symmetries: Wigner’s theorem — automorphisms of rays preserving transition probabilities \(\Leftrightarrow\) unitary or antiunitary operators. Continuous symmetry groups \(\Rightarrow\) projective unitary reps; lifts to true reps of covering groups (spin).

Spectral theorem — use in QM

Bounded self-adjoint \(A\): unique projection-valued measure (PVM) \(E\) on \(\sigma(A)\subset\mathbb{R}\) with $$ A=\int_{\sigma(A)}\lambda\,dE(\lambda),\qquad f(A)=\int f(\lambda)\,dE(\lambda) $$ for Borel \(f\) (bounded). Unbounded self-adjoint: same on domain; essential self-adjointness of symmetric operators decides uniqueness of dynamics/observables.

Application Mechanism
Position / momentum multiplication / differentiation on \(L^2(\mathbb{R}^d)\); Stone–von Neumann uniqueness of CCR (Schrödinger rep)
Functions of observables continuous / Borel functional calculus
Compatible observables commuting self-adjoints \(\Leftrightarrow\) joint PVM
Discrete spectra $A=\sum \lambda_n
Time evolution \(U(t)=\int e^{-it\lambda/\hbar}\,dE_H(\lambda)\)

Stone’s theorem. Strongly continuous one-parameter unitary groups \(\Leftrightarrow\) self-adjoint generators. Stone–von Neumann: irreducible reps of Weyl CCR on finite-dim phase space are unique up to unitary equivalence (fails in infinite dimensions / field theory — inequivalent representations).

Unbounded operators (hygiene)

Domain issues: \(A\) symmetric \(\not\Rightarrow\) self-adjoint. Deficiency indices \((n_+,n_-)\): self-adjoint extensions exist iff \(n_+=n_-\). Essential self-adjointness on a core \(\Rightarrow\) unique evolution. Physical Hamiltonians need essentially self-adjoint realizations (e.g. \(-\Delta+V\) under Kato conditions).

\(C^*\)-algebras and KMS (pointer)

\(C^*\)-algebra \(\mathfrak{A}\): Banach \(*\)-algebra with \(\|a^*a\|=\|a\|^2\). States: positive normalized linear functionals. GNS: state \(\omega\) \(\Rightarrow\) cyclic rep \((\mathcal{H}_\omega,\pi_\omega,\Omega_\omega)\) with \(\omega(a)=\langle\Omega,\pi(a)\Omega\rangle\).

Setting Role
Finite QM \(\mathfrak{A}=\mathcal{K}(\mathcal{H})\) or \(B(\mathcal{H})\); normal states \(=\) density matrices
Infinite systems local algebras; thermodynamic limit
AQFT nets \(\mathcal{O}\mapsto\mathfrak{A}(\mathcal{O})\); vacuum / charged sectors as reps
KMS equilibrium at inverse temperature \(\beta\)

KMS condition. Dynamics \(\alpha_t\in\mathrm{Aut}(\mathfrak{A})\); state \(\omega\) is \(\beta\)-KMS if for all \(a,b\) there is \(F_{a,b}\) analytic in the strip \(0<\mathrm{Im}\,z<\beta\) with boundary values $$ F(t)=\omega\bigl(a\,\alpha_t(b)\bigr),\qquad F(t+i\beta)=\omega\bigl(\alpha_t(b)\,a\bigr). $$ Equivalent (when applicable) to Gibbs \(\omega(a)=\operatorname{Tr}(e^{-\beta H}a)/\operatorname{Tr}e^{-\beta H}\) in finite systems; defines equilibrium without a global density matrix in the infinite volume limit. Tomita–Takesaki theory links modular automorphism group to KMS.

Scattering (one-liner)

Scattering theory: compare full dynamics \(e^{-itH}\) to free \(e^{-itH_0}\) via wave operators \(\Omega^\pm=s\text{-}\lim_{t\to\pm\infty}e^{itH}e^{-itH_0}P_{\mathrm{ac}}(H_0)\) (when limits exist); scattering operator \(S=(\Omega^+)^*\Omega^-\) maps free in-states to free out-states and encodes cross sections. Cook / Kato–Rosenblum / Mourre estimates give existence and asymptotic completeness in \(N\)-body QM under regularity/decay assumptions.

Free quantum fields (bridge)

One-particle Hilbert space \(\mathfrak{h}\); Fock \(\mathcal{F}(\mathfrak{h})=\bigoplus_n \mathfrak{h}^{\otimes_{\pm} n}\); field operators as distributions with CCR/CAR. Relativistic invariance and locality constrain two-point functions (Wightman). Interacting theories: see foundations axioms; constructive existence is the hard analytic problem.

See it

Hydrogen atom. \(H=-\Delta/(2m)-Ze^2/|x|\) on \(L^2(\mathbb{R}^3)\) essentially self-adjoint on \(C_c^\infty\); discrete spectrum \(E_n\sim -1/n^2\) from ODE / SO(4) symmetry; continuous spectrum \([0,\infty)\) for scattering states. Spectral theorem decomposes \(\mathcal{H}=\mathcal{H}_{\mathrm{pp}}\oplus\mathcal{H}_{\mathrm{ac}}\) (no singular continuous for this potential under standard analysis).

Inequivalent representations. Infinite spin chain / free field on infinite volume: ground states at different temperatures or mass parameters can yield unitarily inequivalent GNS representations — cannot implement all dynamics inside a single fixed Fock space. Forces algebraic / representation-independent formulations.

Check

Why must observables be self-adjoint, not merely symmetric?

Spectral theorem and real spectrum / functional calculus require self-adjointness. Symmetric operators may lack enough spectral projections; time evolution \(e^{-itA}\) needs a self-adjoint generator (Stone).

Show that pure vector states are extreme points of the convex set of density operators (idea).

If \(|\psi\rangle\langle\psi|=\lambda\rho_1+(1-\lambda)\rho_2\) with \(0<\lambda<1\), taking matrix elements / support forces \(\rho_i=|\psi\rangle\langle\psi|\) (rank-one projections are extreme in the state space).

Pitfalls

  • Domain of products: \([A,B]\) may be subtle for unbounded \(A,B\); formal CCR need Weyl form for rigorous uniqueness theorems.
  • Superselection: not all unit rays are physically coherent; charge sectors restrict superposition.
  • “Collapse” is not part of unitary dynamics; measurement axioms are additional structure (or emergent in decoherence programs).
  • Thermodynamic limit: \(\operatorname{Tr}e^{-\beta H}\) meaningless for infinite systems — use KMS / \(C^*\) states.

Refs

Reed–Simon I–IV; Hall, Quantum Theory for Mathematicians; Takesaki, Theory of Operator Algebras; Bratteli–Robinson; Haag, Local Quantum Physics; Simon, Functional Integration and Quantum Physics.

Next

Foundations · Gauge theory · Index · Functional analysis · Representation theory · SPDE · Stochastic processes · Spectral theory — self-adjoint extensions, spectral theorem for quantum observables · AQFT / scattering — the rigorous S-matrix / LSZ / Haag–Ruelle