Inequivalent CCR Representations in Infinite Systems

Infinitely many canonical modes admit regular irreducible CCR representations that are not related by a unitary intertwiner. This is outside the finite-phase-space hypothesis of Stone–von Neumann uniqueness. A useful concrete test is bosonic squeezing: each finite set of mode squeezes is unitarily implementable, but an infinite collection is implementable in the original Fock representation only when its creation-operator mixing satisfies a square-summability condition.

Required background. The Stone–von Neumann Theorem identifies the finite-mode uniqueness hypotheses. CCR Algebras explains what it means to represent the same labeled algebra. Bosonic Fock space, creation and annihilation operators, and the vacuum are assumed for the example below.

Infinite modes change the representation problem

One canonical pair already acts on an infinite-dimensional Hilbert space such as $L^2(\mathbb R)$. This is compatible with Stone–von Neumann uniqueness because its phase space has finite dimension. A countably infinite collection of canonical pairs has an infinite-dimensional phase space, even when its Fock Hilbert space remains separable.

Two representations $\pi_1,\pi_2$ of the same Weyl algebra are unitarily equivalent when there is a unitary $U$ satisfying

$$U\pi_1(W(z))U^*=\pi_2(W(z)) \qquad\text{for every label }z.$$

An abstract isomorphism of their Hilbert spaces is insufficient. It must preserve the specified canonical observables. Nor does inequivalence mean that their restrictions to every finite collection of modes differ: global representations can agree in the finite-mode sense while failing to admit one simultaneous intertwiner for the whole algebra.

A bounded symplectic change of infinitely many modes

Let the one-particle space be $\ell^2(\mathbb N)$ and use its bosonic Fock representation, with $[a_n,a_m^*]=\delta_{nm}I$ on the finite-particle domain. For a bounded real sequence $(r_n)$, define modewise

$$b_n=\cosh(r_n)a_n+\sinh(r_n)a_n^*.$$

The hyperbolic identity $\cosh^2r_n-\sinh^2r_n=1$ preserves each canonical commutator, and different modes still commute. In real quadratures the transformation is $q_n\mapsto e^{r_n}q_n$, $p_n\mapsto e^{-r_n}p_n$. Boundedness of $(r_n)$ makes this a bounded invertible symplectic transformation of the underlying real one-particle Hilbert space.

The displayed unbounded mode equations are notation for a well-defined transformation of bounded Weyl operators. Composing the Fock representation with that symplectic automorphism gives another regular irreducible representation. The remaining question is whether a unitary on the original Fock space implements the automorphism.

The bosonic Shale criterion

For a bounded invertible symplectic transformation in a bosonic Fock representation, the Shale theorem states that a unitary implementer exists exactly when the antilinear part is Hilbert–Schmidt. In the diagonal example the condition is

$$\sum_{n=1}^{\infty}\sinh^2(r_n)<\infty.$$

The theorem includes both necessity and sufficiency. Its proof requires the structure of Fock representations and is imported here; Dereziński states it as Theorem 24, with the equivalent Hilbert–Schmidt conditions preceding it.

If $r_n=r\ne0$ for every mode, the sum diverges, so the transformed representation is not unitarily equivalent to the original one with the fixed canonical labels. If $r_n=1/n$, the sum converges; if $r_n=1/\sqrt n$, it diverges. For the last two examples, $\sinh r_n\sim r_n$ reduces the question to familiar series tests.

Every finite truncation of all three sequences is implementable. Thus existence of a unitary at each cutoff is weaker than existence of a unitary for the infinite transformation. There is no contradiction with the finite-degree uniqueness theorem.

What the squeezed-vacuum overlap diagnoses

For one mode, a normalized vector annihilated by $b=\cosh r\,a+\sinh r\,a^*$ has vacuum overlap $(\cosh r)^{-1/2}$. For finitely many modes, the transformed vacuum is the product of these squeezed vacua, with unsqueezed vacuum in all remaining modes. Denote it by $\Omega_N$ after squeezing the first $N$ modes.

For $M>N$, phases can be chosen so that

$$\langle\Omega_N,\Omega_M\rangle =\prod_{n=N+1}^{M}(\cosh r_n)^{-1/2}.$$

With a fixed nonzero $r$, these overlaps become small when $M-N$ is large, so the unit vectors $\Omega_N$ are not Cauchy in the original Fock norm. The finite-mode vacuum construction therefore fails to produce a vector limit there.

This calculation explains the mechanism but is not a replacement for Shale’s theorem. In particular, zero overlap with one reference vacuum does not establish representation inequivalence: ordinary orthogonal states can live in the same representation. Likewise, divergence of a number expectation alone is not sufficient, since some legitimate Fock vectors have an infinite expected particle number. The implementability theorem supplies the decisive conclusion.

Representation choice carries physical information

In an infinite system, a vacuum, equilibrium state, or asymptotic background can select a representation through its observable expectations. The same abstract canonical algebra can then have inequivalent realizations appropriate to different physical settings. This does not mean every representation is equally appropriate for a given experiment; dynamics, energy conditions, locality, and the chosen state still constrain the model.

The present result concerns the stable representation theory of canonical bosonic systems. Thermal limits, phases, and quasi-equivalence involve additional distinctions and belong to Representations in Thermodynamic Limits. It also does not prove a claim about arbitrary interacting relativistic field theories: the example is a precisely specified linear canonical transformation in Fock space.

Exercise: a summability threshold

Take $r_n=n^{-\alpha}$ with $\alpha>0$. Determine the range for which the diagonal transformation is implementable. What happens at the boundary?

Solution

The sequence is bounded, so the bounded symplectic setup applies. Since $r_n\to0$, $\sinh^2r_n\sim n^{-2\alpha}$. Shale’s criterion is therefore satisfied exactly for $\alpha>1/2$. At $\alpha=1/2$ the series diverges logarithmically, and no implementer exists on the original bosonic Fock space despite the slow divergence and the validity of every finite truncation.

References

• J. Dereziński, “Introduction to Representations of the Canonical Commutation and Anticommutation Relations”, arXiv:math-ph/0511030, 2005, Sections 6.3–6.4 — bounded symplectic maps and the bosonic Shale theorem.
• D. Shale, “Linear Symmetries of Free Boson Fields”, Transactions of the American Mathematical Society 103, 149–167, 1962 — the original bosonic implementability theorem.