Canonical Commutation Relations

The formal identity $[X,P]=i\hbar I$ hides unbounded-operator domains. This chapter begins instead with bounded unitary relations, then asks when their representation is regular, irreducible, and unique.

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 supply the three hard capabilities needed to define the Weyl exponentials and recover their generators. Familiarity with translations, modulations, Fourier analysis, and finite-dimensional phase space is assumed.

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 does not merely weaken a proof; it changes the representation problem. Nonregular representations, reducible multiples, and inequivalent infinite-system representations all lie outside the irreducible uniqueness conclusion.

Scope of the argument

The three theorem treatments above, together with Stone’s theorem as their cross-chapter prerequisite, form one complete argument from bounded relations to their canonical representative and its theorem-level uniqueness. Other CCR entries marked Planned do not yet contain articles and are not required for this argument.