Free-Field Quantization and Representations

Free-field quantization on curved spacetime is best understood as a sequence of logically separate constructions. Causal dynamics fixes an abstract CCR or CAR algebra. A positive-frequency or covariance choice selects a state. The state then produces a Hilbert-space representation, which may or may not be globally unitarily equivalent to another useful representation. Keeping those layers separate removes the need to assume a universal vacuum before any local observable can be defined.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical fields and Fock intuition; Partial Differential Equations and Green Functions provides advanced and retarded solution operators; Causality and Global Structure establishes causal support and Cauchy evolution; and Variational and Symplectic Geometry defines the reduced symplectic phase space.

From causal dynamics to quantum relations

For the real scalar field, the fixed convention is

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}, \qquad [\Phi(f),\Phi(g)]=iE(f,g)\mathbf1.$$

The compact Green-function bridge is as follows. Retarded and advanced operators solve the same inhomogeneous equation but have future and past support. Their difference solves the homogeneous equation, and formal self-adjointness makes its test-function pairing antisymmetric. On a globally hyperbolic spacetime without timelike boundary, quotienting compactly supported test functions by the field equation removes the degeneracy. The resulting symplectic space determines the Weyl phase

$$W([f])W([g]) =e^{-iE(f,g)/2}W([f]+[g]).$$

This construction, including the quotient, CCR functor, and Fock realization, is given in Bär, Ginoux, and Pfäffle 2007, §§4.2–4.7, pp. 116–153. Fermions begin instead with a positive Hermitian solution space and CAR. Their anticommutator form, adjoint, and positivity are not obtained by mechanically replacing the scalar commutator symbol.

The chapter stays within linear fields on prescribed, globally hyperbolic backgrounds. Gauge reduction, interacting QFT, renormalized composite observables, and metric backreaction require additional constructions. Boundaries or failures of global hyperbolicity also require boundary conditions or other input before the causal quotient above is available.

Three layers that must remain distinct

The algebra records the field equation, adjoint, and canonical relations. A state is a normalized positive functional on that algebra. Its GNS representation realizes the algebra on a Hilbert space with a cyclic vector. A Fock vacuum is therefore a vector associated with a chosen state and one-particle structure; it is not an element silently built into the abstract field algebra.

A compatible positive complex structure $J$ on an appropriate real phase-space completion supplies one such choice. Positivity means $\sigma(v,Jv)\gt0$ for every nonzero $v$, rather than positivity of every off-diagonal pairing. The corresponding one-particle inner product is normalized so that

$$2\operatorname{Im}\langle Kv,Kw\rangle=\sigma(v,w).$$

Symmetric or antisymmetric Fock completion then produces bosonic or fermionic creation and annihilation operators. The bosonic fields are unbounded and their products require a common dense domain; CAR generators are bounded. These analytic facts belong to a representation and must not be confused with the representation-independent causal relations.

Two complex structures can define the same algebra yet inequivalent Fock representations. For bosons, Shale’s theorem requires the antilinear Bogoliubov part to be Hilbert–Schmidt Shale 1962, Theorem 4.1, pp. 157–158. The fermionic Shale–Stinespring criterion instead tests the off-diagonal blocks relative to a CAR polarization. In either case, finite mixing of every individual mode does not imply global unitary implementability.

The six-step route

The pages are ordered so that no Hilbert-space choice is used before its assumptions are visible.

1. CCR and CAR Algebras derives the bosonic commutator from the causal propagator, constructs the Weyl phase with the matching sign, and contrasts the positive Hermitian data required by fermions.
2. Complex Structures and One-Particle Spaces states the compatibility, positivity, and continuity conditions that turn real classical data into a one-particle Hilbert space.
3. Fock Space and Field Operators builds symmetric and antisymmetric Fock spaces, identifies the finite-particle core, and checks field and number-operator normalizations.
4. Bogoliubov Implementability proves why the global Hilbert–Schmidt sum—not merely modewise finiteness—decides whether a bosonic symplectic map has a Fock-space unitary, and distinguishes the CAR theorem.
5. Quasifree States and n-Point Functions gives the field equation, canonical, reality, and positivity conditions on a two-point bidistribution and reconstructs higher moments by Wick pairing.
6. Why Curved Spacetime Has No Preferred Vacuum states the hypotheses of the natural-state no-go theorem and contrasts it with ground, KMS, conformal, and ultrastatic choices on particular backgrounds.

The recurring finite oscillator fixes signs and factors. The recurring infinite tower exposes what finite-dimensional intuition misses: every finite truncation may have a unitary even when the full transformation does not. Quasifree covariance provides a third check, because the antisymmetric part can be correct while positivity still fails. The thermal oscillator then shows how a mixed state changes the symmetric covariance while preserving the commutator and Wick pairing. For fermions the upper covariance bound $C\leq\mathbf1$ expresses Pauli exclusion; it is not the bosonic uncertainty condition.

Reading routes

For a fast route to local algebraic QFT, follow steps 1, 5, and 6. This route answers what the equations determine, how states are encoded, and why local admissibility replaces a universal particle vacuum. It is the shortest preparation for Hadamard states and locally covariant observables.

For particle creation or horizon applications, follow all six steps. The middle three pages explain how positive frequency becomes a Fock representation and when two mode choices are globally comparable by a unitary. Skipping the implementability criterion can turn a well-defined modewise calculation into an unjustified assertion about one global Hilbert space.

For fermionic fields, read steps 1 and 3 with particular attention to Hermitian positivity, bounded CAR operators, and polarization. Then use steps 4 and 5 for the fermionic off-diagonal Hilbert–Schmidt condition and signed Wick expansion. The bosonic symplectic formulas provide an analogy, not a substitute.

Symmetry, admissibility, and limitations

On Minkowski spacetime, Poincaré symmetry and the spectrum condition distinguish the standard vacuum. A complete timelike Killing flow with a suitable positive generator can distinguish a ground state on a fixed stationary background; the same chosen flow defines a KMS condition. These statements have hypotheses: zero modes, boundaries, ergoregions, or superradiance can obstruct existence or uniqueness.

They also do not contradict the no-natural-state result. Under dynamical locality, a faithful Reeh–Schlieder GNS representation on a spacetime with noncompact Cauchy surfaces, and extended locality, an exactly covariant natural state assignment would force the theory to be trivial Fewster and Verch 2012, Definition 6.12 and Theorem 6.13. Dropping a hypothesis makes that theorem inapplicable; restricting to a symmetric class of backgrounds is a different question.

Quasifree is also not synonymous with physically admissible. Positivity and Wick factorization do not impose the Hadamard singularity structure. Nor does unitary inequivalence prove that local predictions disagree: global inequivalence can coexist with local quasiequivalence for suitable state classes. The durable hierarchy is therefore algebra first, admissible states second, and representation-dependent particles only when the geometry and observable justify them.

Integrative diagnostic

Suppose two asymptotic positive-frequency choices for a scalar field are related by diagonal coefficients

$$\beta_n=(n+1)^{-1/2}, \qquad \alpha_n=\sqrt{1+\lvert\beta_n\rvert^2}.$$

Does this change the CCR algebra? Is the transformation unitarily implementable on the first Fock space? Can the second choice still define a state, and does the calculation identify a natural vacuum on arbitrary curved spacetimes?

Solution

The CCR algebra is unchanged because the symplectic form and causal commutator are unchanged. Each mode satisfies the bosonic normalization, but the global sum of squared antilinear coefficients is harmonic and diverges, so Shale’s Hilbert–Schmidt condition fails and no unitary implements the full transformation on the first Fock space. A positive second complex structure can nevertheless define a quasifree state and its own GNS representation; failure of one unitary identification is not failure of the state. Finally, neither asymptotic prescription is a natural assignment on every spacetime. A symmetry or boundary condition may distinguish one on a particular background, but the mode calculation alone cannot produce a universal preferred vacuum.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. EMS Press, 2007. DOI 10.4171/037.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality and Covariance: What Makes a Physical Theory the Same in All Spacetimes?” Annales Henri Poincaré 13 (2012): 1613–1674. DOI 10.1007/s00023-012-0165-0.
• Shale, David. “Linear Symmetries of Free Boson Fields.” Transactions of the American Mathematical Society 103, no. 1 (1962): 149–167. DOI 10.1090/S0002-9947-1962-0137504-6.
• Wald, Robert M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, 1994. Publisher record.