Mathematical Quantum Mechanics

Mathematical quantum mechanics begins where finite-matrix shorthand stops being reliable. An unbounded operator includes its domain; continuity must name a topology; an observable is represented through a spectral measure; and a theorem is only as strong as its hypotheses. This volume develops those points without separating them from their physical purpose.

The reading map has three connected branches. The operator route runs from self-adjointness through spectral measures and unitary generators to the canonical commutation relations. The projective side then splits: pure-state rays and transition-probability preservers lead to Wigner’s theorem, whereas countably additive projection probabilities lead from PVMs to Gleason’s theorem. A many-body branch starts from unitary dynamics and develops the volume-uniform locality estimate needed to control finite-to-infinite limits.

Enter the theorem sequence

Operators, spectra, and dynamics

1. Enter through Operators, Domains, and Spectra, whose on-ramp supplies the state-vector and functional-analysis capabilities needed for Self-Adjoint Operators.
2. From self-adjointness, complete both branches:
   • use Spectral Measures and Observables for its Projectors on-ramp, The Unbounded Spectral Theorem, and Projection-Valued Measures;
   • use Dynamics and Hamiltonians for Strongly Continuous Unitary Groups.
3. Rejoin the branches at Stone’s Theorem, which proves the two-way correspondence between unitary groups and self-adjoint generators.

Canonical representations

With the operator, spectral, and unitary-group branches complete, enter the Canonical Commutation Relations gateway:

1. The Weyl Form of the CCR replaces a domain-sensitive formal commutator by bounded unitary relations.
2. The Schrödinger Representation realizes those relations regularly and irreducibly on $L^2(\mathbb R^n)$.
3. Carry forward Stone’s Theorem from the operator route to recover the self-adjoint generators of the two unitary subgroups.
4. The Stone–von Neumann Theorem states exactly when that finite-degree representation is unique up to unitary equivalence.

Rays, symmetries, and probabilities

These are two branches, not one linear theorem sequence.

1. For ray symmetries, enter through Hilbert-Space Quantum Mechanics, whose on-ramp leads to Physical States as Rays. Then use the Symmetry, Representations, and Superselection gateway for Transition-Probability Preserving Maps → Wigner’s Theorem.
2. For projection probabilities, complete the Spectral Measures and Observables route through Projection-Valued Measures, then use the Foundational Theorems gateway to reach Gleason’s Theorem.

Many-body locality

After the unitary-dynamics route, enter Many-Body Statistical. Begin with Lieb–Robinson Bounds, which makes the metric, interaction decay norm, support prefactor, and finite-volume uniformity explicit. The article derives the exponential exterior cone through interaction-path iteration and then explains which additional steps are needed to construct an infinite-volume dynamics.

The theorem owner defines the lattice algebras and locality data it uses. Complete its unitary-dynamics prerequisite before entering this branch; no separate physical many-body chapter is required for that argument.

A reading map

The chapters can be read along three compact routes:

self-adjoint operators
    ├─ spectral theorem ─ PVMs ─ Gleason
    └─ strong unitary groups ─ Stone
                    └─ Weyl CCR ─ Schrödinger representation
                                      └─ Stone–von Neumann

physical rays ─ transition-probability preservers ─ Wigner

strong unitary dynamics ─ metric interactions ─ Lieb–Robinson bound
                                      └─ infinite-volume dynamics boundary

This diagram is a reading order, not the full prerequisite directed acyclic graph. For example, both Stone’s theorem and the Weyl relations also require the spectral-theorem capability, and Stone–von Neumann uses Stone’s theorem as well as the Weyl and Schrödinger pages. Each leaf page gives the authoritative hard-prerequisite edges as explicit capabilities rather than merely listing titles.

How to read a rigorous theorem page

Every theorem in this spine has four layers:

1. an exact statement and hypothesis ledger;
2. a worked consequence or model that can be checked directly;
3. a proof architecture naming imported nontrivial lemmas;
4. failure cases showing what changes when a hypothesis is removed.

The proof architecture is deliberate. The unbounded spectral theorem, Stone–von Neumann, Wigner, Gleason, and the modern Lieb–Robinson theorem rely on substantial constructions that should not be disguised as one-paragraph proofs. Their pages prove tractable domain calculations and examples, expose the decisive recursion or reduction steps, and locate imported construction theorems in authoritative references.

Conventions held fixed

Throughout these theorem treatments:

• inner products are conjugate-linear in the first slot and linear in the second;
• $D(A)$ is part of every unbounded operator;
• $E_A(\Delta)$ denotes the spectral projection of $A$ for a Borel set $\Delta$;
• $T(a)=e^{-iaP/\hbar}$ and $M(b)=e^{ibX/\hbar}$, so $T(a)M(b)=e^{-iab/\hbar}M(b)T(a)$;
• strong continuity means vector-norm continuity for each fixed vector;
• regularity, irreducibility, and surjectivity are distinct hypotheses.

These conventions are repeated only where a sign or domain is at risk.

Further topics

The routes above form the substantive theorem sequence. The wider volume also contains entries marked Planned whose articles are not yet written. These entries identify future coverage; they do not supply a theorem, proof, or prerequisite capability.

For compact theorem lookup after reading the owners, use the Formalism Theorem Cards or the Many-Body Results card. The cards retain statements and hypothesis checks; proofs and derivations remain in this volume.