CCR Algebras

The Weyl C*-algebra packages canonical commutation relations as identities among abstract bounded unitaries. It separates the canonical algebra from a chosen Hilbert-space realization and state. Strong continuity is then imposed on representations or states; it is not built into the C*-norm dependence of the abstract Weyl generators. This page concerns this bounded Weyl algebra, rather than the polynomial algebra generated by formal unbounded position and momentum symbols.

Required background. The Weyl Form of the CCR fixes the symplectic sign. Operator Algebra Minimum distinguishes abstract algebras, states, and representations.

Abstract Weyl generators and their product

Let $S$ be a real vector space with a nondegenerate antisymmetric bilinear form $\sigma$. Fix $\hbar\ne0$ and introduce unitary symbols $W(z)$, $z\in S$, satisfying

$$W(0)=I, \qquad W(z)^*=W(-z),$$ $$W(z)W(z') =e^{i\sigma(z,z')/(2\hbar)}W(z+z').$$

For finite canonical phase space we use $z=(q,p)$ and $\sigma(z,z')=p\cdot q'-q\cdot p'$, following the chapter’s Weyl convention. The same algebraic construction can use an infinite-dimensional symplectic space. No Hilbert-space dimension has yet been chosen.

Finite linear combinations $F=\sum_jc_jW(z_j)$ form a unital star algebra with this multiplication. Associativity is ensured by the cocycle identity

$$\sigma(z,z')+\sigma(z+z',z'') =\sigma(z',z'')+\sigma(z,z'+z''),$$

which follows from bilinearity. Thus the phase is constrained data, not an optional convention that can be discarded while multiplying displacements.

The universal norm separates algebra from representation

In every unitary realization, the norm of $F$ is at most $\sum_j|c_j|$. One can therefore take the supremum of its operator norms over representations, quotient any null elements, and complete. The resulting universal C*-algebra is the Weyl algebra $\mathcal W(S,\sigma)$.

Its universal property means that any family of Hilbert-space unitaries satisfying the same relations determines a representation of this algebra. The completion supplies a normed algebra of bounded observables; it does not create unbounded $Q$ and $P$ as elements of that algebra. In a regular representation they arise instead as generators of represented one-parameter subgroups.

The existence and structural properties of this construction are C*-algebra theorems; Bratteli and Robinson give the detailed treatment. The important separation is that a relation in $\mathcal W$ can be represented in several Hilbert spaces, and a state can select a representation through GNS construction. Those steps are not part of choosing the symplectic multiplication law alone.

The Weyl generators are discontinuous in algebra norm

For $z\ne0$, nondegeneracy permits the scalar $\sigma(y,z)$ to range over every real value as $y$ varies. The relations give

$$W(y)W(z)W(y)^* =e^{i\sigma(y,z)/\hbar}W(z).$$

Unitary conjugation preserves spectrum. Therefore the spectrum of $W(z)$ is invariant under every rotation of the unit circle. Since the spectrum of a unitary is nonempty and contained in that circle, it must be the whole circle. The C*-norm then gives

$$\|W(z)-I\|=2\qquad(z\ne0).$$

In particular, $t\mapsto W(tz)$ is not norm continuous at zero. This does not obstruct a regular representation, in which $t\mapsto\pi(W(tz))$ is strongly continuous on every fixed vector. Algebra norm and strong operator topology ask different questions. The regularity page supplies explicit regular and nonregular realizations.

The distinction also explains a terminology trap. A polynomial Weyl algebra, an algebra obtained by integrating a continuous Weyl representation, and this universal unital Weyl C*-algebra are related constructions but should not be identified merely because each is called a CCR algebra.

State characteristic functions and positivity

A state $\omega$ defines a characteristic function $\chi(z)=\omega(W(z))$. It satisfies $\chi(0)=1$ and $\chi(-z)=\overline{\chi(z)}$. Positivity imposes the stronger finite-matrix condition

$$\sum_{j,k}\overline{c_j}c_k e^{-i\sigma(z_j,z_k)/(2\hbar)} \chi(z_k-z_j)\geq0$$

for every finite choice of coefficients and phase-space points. This is obtained by expanding $\omega(F^*F)$. Omitting the symplectic phase would test a different, commutative characteristic-function problem.

Conversely, a normalized function obeying this twisted positivity condition defines a state on the Weyl algebra. One proves this by constructing its positive functional on finite Weyl sums and the associated bounded unitary GNS representation. This is the algebraic version of reconstructing a quantum state from canonical displacement expectations.

Regularity can be tested at the state level. In the GNS representation with cyclic vector $\Omega$,

$$\|(\pi(W(tz))-I)\Omega\|^2 =2-2\operatorname{Re}\chi(tz).$$

Continuity of $\chi(tz)$ at zero for every $z$ gives continuity on the cyclic vector. The Weyl relations transfer it to every $\pi(W(y))\Omega$, and density plus the unitary norm bound extend it to the whole representation. Thus the state condition supplies regularity, but not irreducibility or finite-dimensional phase space.

Exercise: inspect the two-point positivity test

Apply the state-positivity condition to the points $0,z$. What elementary bound on $\chi(z)$ follows, and why is it not sufficient by itself to define a state?

Solution

The matrix is $\begin{pmatrix}1&\chi(z)\\\overline{\chi(z)}&1\end{pmatrix}$. Positivity requires determinant $1-|\chi(z)|^2\geq0$, so $|\chi(z)|\leq1$. A state must also pass every larger finite matrix test, including the nontrivial Weyl phases between different points. Pointwise boundedness alone contains none of those consistency conditions.

References

• O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics, 2nd ed., Springer, 1997 — the Weyl algebra, its states, and regularity in the CCR setting.
• J. Dereziński, “Introduction to Representations of the Canonical Commutation and Anticommutation Relations”, arXiv:math-ph/0511030, 2005 — canonical representations and their relation to bounded Weyl operators.