Bogoliubov Transformations and Unitary Implementability

A Bogoliubov transformation may preserve the canonical commutation relations without being implemented by any unitary operator on a chosen Fock space. For bosonic free fields, the decisive infinite-dimensional test is whether the antilinear coefficient $\beta$ is Hilbert–Schmidt. Failure means that the two global particle splittings define inequivalent Fock representations; it does not by itself imply inequivalent local physics.

Required background. Complex Structures and One-Particle Spaces defines the splittings being compared. Quasifree States and Two-Point Functions states the covariance conditions. Fock Space, Vacuum, and Particle Number fixes particle-number conventions.

Helpful background. Unbounded Operators, Domains, Closure, and Adjoints supplies operator-domain care. Haag’s Theorem gives a distinct source of representation inequivalence in interacting QFT.

Canonical mode mixing

Let $\{u_i\}$ and $\{v_j\}$ be complete positive-norm mode families for the same real field, normalized by the conserved Klein–Gordon product. Then

$$v_j=\sum_i\left(\alpha_{ji}u_i+\beta_{ji}u_i^*\right),$$

and the annihilation operators satisfy, with a corresponding convention,

$$b_j=\sum_i\left(\alpha_{ji}^*a_i-\beta_{ji}^*a_i^\dagger\right).$$

Preservation of the CCR gives

$$\alpha\alpha^\dagger-\beta\beta^\dagger=1, \qquad \alpha\beta^{\mathsf T}=\beta\alpha^{\mathsf T}.$$

These are canonical identities. They do not prove that a unitary $U$ exists with $b_j=Ua_jU^{-1}$.

The Hilbert–Schmidt criterion

The bosonic transformation is unitarily implementable on Fock space precisely when the antilinear part is Hilbert–Schmidt,

$$\lVert\beta\rVert_{\mathrm{HS}}^2 =\operatorname{Tr}(\beta\beta^\dagger)<\infty.$$

In a discrete mode basis this is $\sum_{ij}|\beta_{ij}|^2<\infty$; in a continuum it becomes the corresponding integral with the declared measure. The same quantity is the total expected $b$-particle number in the $a$-vacuum when that expression is meaningful:

$$\langle0_a|N_b|0_a\rangle =\operatorname{Tr}(\beta\beta^\dagger).$$

This physical form of the Shale criterion follows from the representation theory of the CCR Shale 1962, pp. 149–167.

Worked comparison

For a homogeneous scalar field, suppose the two splittings are diagonal in momentum:

$$v_{\mathbf k}=\alpha_k u_{\mathbf k}+\beta_k u_{-\mathbf k}^*, \qquad |\alpha_k|^2-|\beta_k|^2=1.$$

Then implementability per unit comoving volume is controlled by

$$\int\frac{\mathrm d^{d-1}k}{(2\pi)^{d-1}}\,|\beta_k|^2.$$

The ultraviolet and infrared ends must both be checked. A falloff $|\beta_k|\sim k^{-r}$ is ultraviolet integrable only if $2r>d-1$; a zero-mode divergence can independently spoil the infrared. This calculation is the required first application: canonical normalization is tested first, followed by the Hilbert–Schmidt integral.

Why finite truncations mislead

In a finite box with finitely many retained modes, every matrix $\beta$ is Hilbert–Schmidt. If the box size and ultraviolet cutoff are removed, the number of modes grows and the trace may diverge. Therefore a numerical unitary matrix at fixed truncation establishes only cutoff-level implementability.

The adversarial procedure is explicit: compute $\operatorname{Tr}\beta\beta^\dagger$ as both cutoffs are varied, state the order of limits, and require convergence. If it diverges, retain the canonical Bogoliubov relation but withdraw the global unitary-equivalence and finite-total-particle claims.

Local observables can remain comparable

Global Fock inequivalence is not the same as local disjointness. Quasifree Hadamard representations of the Klein–Gordon field satisfy strong local quasiequivalence results under stated hypotheses Verch 1994, Theorem 3.6. A divergent global number operator may thus coexist with mutually normal restrictions to bounded local algebras. Particle number is a global, representation-dependent diagnostic; local correlations require their own comparison.

Domain and failure conditions

Bogoliubov coefficients compare two one-particle descriptions within the state-and-representation portion of the construction map. Their canonical identities do not move a state automatically to the Hadamard, global-construction, or physical-selection boxes; Hilbert–Schmidt implementability is an additional global representation test.

Canonical mode mixing preserves the CCR, while a finite Hilbert–Schmidt norm is separately required for a global Fock-space unitary. Schematic; not to scale.

In the failure map, a finite box or cutoff is part of the approximation named in the first box. If $\operatorname{Tr}\beta\beta^\dagger$ diverges as the regulators are removed, the global unitary-equivalence claim stops, but the canonical transformation and possible local comparability remain.

Regulator removal sets the domain of the implementability claim; divergence removes global unitary equivalence without disproving the CCR relation. Schematic; not to scale.

For the distinction between this result and other state tests, see Domain and failure conditions.

Handoffs

GNS Representations, Local Normality, and Local Quasiequivalence develops that local comparison. Dynamical production rates belong to Particles, Detectors, and Nonadiabatic Production. The proof-level representation criterion continues in States, GNS Representations, and Folia.

References

• Shale, David. “Linear Symmetries of Free Boson Fields.” Transactions of the American Mathematical Society 103 (1962): 149–167. DOI.
• Verch, Rainer. “Local Definiteness, Primarity and Quasiequivalence of Quasifree Hadamard Quantum States in Curved Spacetime.” Communications in Mathematical Physics 160 (1994): 507–536. DOI.