Irreducible Representations of the CCR

A Weyl representation is irreducible when no proper nonzero closed subspace is preserved by every represented canonical displacement. Equivalently, the only bounded operators commuting with all displacements are scalars. This hypothesis removes spectator multiplicities from the Stone–von Neumann theorem. It concerns the entire canonical representation, not the spectrum of one Hamiltonian or the purity of a chosen state.

Required background. The Weyl Form of the CCR defines the bounded relations. The Schrödinger Representation supplies a regular irreducible realization.

Invariant subspaces are reducing for the Weyl family

Let $W(z)$ be Weyl unitaries on $\mathcal H$. A closed subspace $\mathcal M$ is invariant if $W(z)\mathcal M\subseteq\mathcal M$ for every $z$. Since $W(z)^*=W(-z)$ is also in the family, invariance holds in both directions and for adjoints. Thus $\mathcal M^\perp$ is invariant as well.

Writing $E$ for the orthogonal projection onto $\mathcal M$, this is equivalent to

$$EW(z)=W(z)E\qquad\text{for every }z.$$

Such a subspace is called reducing. Irreducibility therefore means that the bounded commutant

$$W'=\{B\in\mathcal B(\mathcal H):BW(z)=W(z)B\text{ for all }z\}$$

contains no projections other than $0$ and $I$. The use of bounded operators avoids trying to define a commutant through formal unbounded products on unspecified domains.

Why the commutant must be scalar

If a bounded self-adjoint $B$ commutes with every Weyl unitary, its spectral projections do too. This follows from the bounded spectral calculus: commutation passes first through polynomials and then to the spectral projections by the relevant limiting construction. If the Weyl family is irreducible, every such projection is either zero or identity, forcing the spectral measure of $B$ to be concentrated at a single point. Thus $B=bI$.

For a general bounded $B\in W'$, both its real and imaginary self-adjoint parts belong to $W'$. Each is scalar, hence $B$ is scalar. Conversely, if $W'=\mathbb C I$, a reducing-subspace projection can only be zero or identity. This proves the complex Hilbert-space form of Schur’s lemma relevant here.

The Schrödinger example satisfies this criterion: commuting with all modulations forces a bounded operator to be a multiplication operator, and commuting also with all translations forces the multiplier to be constant almost everywhere. The proof belongs to The Schrödinger Representation; the present criterion explains how to use its conclusion.

Adding a spectator destroys irreducibility

Let $W_{\rm S}$ be the Schrödinger representation and take

$$\mathcal H=L^2(\mathbb R^d)\otimes\mathbb C^2, \qquad \widetilde W(z)=W_{\rm S}(z)\otimes I_2.$$

The Weyl phase and strong continuity are unchanged. But $I\otimes|0\rangle\langle0|$ is a nontrivial commuting projection, so this representation is reducible.

In fact its commutant is exactly $I\otimes M_2(\mathbb C)$. To see this, write a bounded operator as a $2\times2$ matrix of bounded operators on $L^2(\mathbb R^d)$. Commutation with $\widetilde W(z)$ requires every matrix entry to commute with $W_{\rm S}(z)$. Irreducibility of the latter makes every entry scalar, leaving an arbitrary matrix on the spectator factor.

This representation cannot be unitarily equivalent to one Schrödinger copy, because unitary equivalence carries commutants into commutants. The abstract Hilbert spaces can nevertheless be isomorphic as infinite-dimensional separable spaces. Representation equivalence must intertwine the labeled Weyl operators, not merely identify the vector spaces.

The general finite-degree regular multiplicity theorem is stated on the Stone–von Neumann Theorem page. Irreducibility is precisely the condition selecting a one-dimensional multiplicity space.

Cyclicity and dynamical symmetries answer different questions

A vector $\Omega$ is cyclic when the closed linear span of $W(z)\Omega$ is all of $\mathcal H$. In an irreducible representation, every nonzero vector is cyclic: the closed orbit span is nonzero and invariant under all Weyl operators, so it must be the whole space.

The converse implication from one cyclic vector to irreducibility is false for general star representations. For example, diagonal matrices on $\mathbb C^2$ have the cyclic vector $(1,1)$, but preserve both coordinate axes. A state-generated representation can therefore be cyclic by construction and still reducible; a GNS construction alone does not establish irreducibility.

Nor does a conserved quantity for one chosen Hamiltonian automatically lie in $W'$. Oscillator number commutes with the oscillator Hamiltonian, but not with arbitrary canonical displacements. Parity provides another instructive example: in the Schrödinger representation,

$$(\Pi\psi)(x)=\psi(-x), \qquad \Pi W(z)\Pi^*=W(-z).$$

Parity transforms the representation labels instead of commuting with each fixed labeled operator. Its even and odd subspaces need not be invariant under translations. Irreducibility of the CCR is therefore compatible with a Hamiltonian having symmetries and a nontrivial eigenbasis.

Exercise: equivalence preserves irreducibility

Suppose $VW_1(z)V^*=W_2(z)$ for a unitary $V$. Show that $B\mapsto VBV^*$ carries $W_1'$ onto $W_2'$, and use this to rule out equivalence between one and two identical irreducible copies.

Solution

Conjugating $BW_1(z)=W_1(z)B$ gives the required commutation with $W_2(z)$; conjugation by $V^*$ gives the inverse map. Scalars map to scalars and nontrivial projections to nontrivial projections. One irreducible copy has scalar commutant, whereas two copies possess the projection onto the first summand. Hence no intertwining unitary can exist.

References

• J. Dereziński, “Introduction to Representations of the Canonical Commutation and Anticommutation Relations”, arXiv:math-ph/0511030, 2005 — irreducibility, regular representations, and the multiplicity form of the finite-dimensional CCR theorem.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — the Schrödinger representation and Stone–von Neumann theorem.