Canonical Commutation Relations and Their Representations

The formal identity $[Q,P]=i\hbar I$ hides unbounded-operator domains. The bounded Weyl relations provide a precise global alternative, but their representations still need continuity and irreducibility hypotheses before finite-mode uniqueness applies. This chapter connects these statements, shows why they are not interchangeable, and identifies the new representation problem that appears with infinitely many canonical modes.

Reading sequence

On-ramp. Begin with State Vectors, then complete Self-Adjoint Operators followed by The Unbounded Spectral Theorem, and also read Strongly Continuous Unitary Groups. Together these explain how to define the Weyl exponentials and recover their generators. Familiarity with translations, modulations, Fourier analysis, and finite-dimensional phase space is assumed.

Start with The Heisenberg Commutation Relation to see why finite matrices and everywhere-defined bounded operators cannot satisfy the exact relation. The interval example in Domain Subtleties of the Canonical Commutation Relations then shows why a valid commutator on a dense test space does not automatically integrate to a Weyl system. With that warning in hand, follow the main argument:

1. The Weyl Form of the Canonical Commutation Relations fixes

   $$T(a)=e^{-iaP/\hbar}, \qquad M(b)=e^{ibX/\hbar},$$

   and therefore

   $$T(a)M(b)=e^{-iab/\hbar}M(b)T(a).$$

2. The Schrödinger Representation realizes this regular irreducible Weyl system on $L^2(\mathbb R^n)$ and states the maximal domains of its position and momentum generators.

3. Before the uniqueness theorem, read Stone’s Theorem to recover the self-adjoint generators of strongly continuous unitary subgroups.

4. The Stone–von Neumann Theorem proves uniqueness up to unitary equivalence for finite $n$, fixed nonzero central character, regularity, and irreducibility.

Hypotheses that must remain separate

• Regularity is strong continuity and supplies self-adjoint generators.
• Irreducibility excludes invariant sectors and multiplicity spaces.
• Finite degrees of freedom excludes fields and thermodynamic limits.
• Fixed central character fixes the Weyl phase and the labeled value of $\hbar$.

Dropping one changes the representation problem. Irreducible Representations of CCR uses the bounded commutant to distinguish an irreducible system from a multiplicity space; having a cyclic vector alone is insufficient. Regularity Assumptions gives an exact but nonregular Weyl representation whose translations have no self-adjoint generator. It also explains why strong continuity, rather than norm continuity, is the relevant requirement.

Domains, observables, and infinite systems

Two companion pages connect the theorem sequence to calculations. Position and Momentum under Canonical Displacements tracks domains, spectral event probabilities, and moments under translations and modulations. Commutation Relations on Domains separates operator products from quadratic-form commutators, explaining why the uncertainty relation can remain meaningful when $QP\psi$ and $PQ\psi$ are not separately defined. Neither bounded Weyl identities nor a formal cancellation permits dropping those domain checks.

CCR Algebras packages the Weyl relations in an abstract observable algebra and separates that algebra from its states and Hilbert-space representations. Its norm topology does not itself impose regularity on the canonical labels. This distinction is essential for Inequivalent CCR Representations in Infinite Systems: the bosonic Shale criterion supplies an explicit family of transformations that are implementable at every finite cutoff but have no simultaneous unitary implementer in the chosen infinite-mode Fock representation.

The chapter therefore ends with two different limits on a familiar slogan. For finitely many modes, uniqueness has precise representation hypotheses. For infinitely many modes, even regular irreducible representations need not be equivalent, and the physical state becomes part of selecting a realization of the canonical algebra.

References

• J. Dereziński, “Introduction to Representations of the Canonical Commutation and Anticommutation Relations”, arXiv:math-ph/0511030, 2005 — finite and infinite canonical systems and implementability.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — operator domains, Weyl relations, and uniqueness.