Why Curved Spacetime Has No Preferred Vacuum

The absence of a natural preferred vacuum in curved spacetime is not the claim that useful vacua never exist. It is the claim that geometry and local covariance do not furnish one state, functorially and without extra choices, on every admissible spacetime. Distinguished stationary ground or KMS states remain possible on a fixed background when symmetry, spectral conditions, regularity, and boundary data support them.

Required background. Flat-Spacetime Quantum Field Theory supplies the Poincaré vacuum and spectral condition, while Variational and Symplectic Geometry defines the classical phase space whose positive-frequency splitting would select a Fock vacuum.

Helpful background. Partial Differential Equations and Green Functions provides the Cauchy evolution used to compare state choices, and Causality and Global Structure establishes the spacetime embeddings, causal complements, and global symmetries used in the no-go argument.

For immediate use, well-posed Cauchy evolution identifies data on different slices, while a compactly supported metric perturbation induces relative Cauchy evolution of the observable algebra. The no-go proof tests whether a proposed state responds nontrivially to that change.

Why positive frequency needs more geometry

In Minkowski spacetime, time translations are part of the Poincaré group. Requiring invariance, a positive energy-momentum spectrum, and the usual irreducibility and uniqueness conditions singles out the standard vacuum. Its positive-frequency modes transform as $e^{-i\omega t}$ with $\omega>0$. The coordinate $t$ is not doing the work by itself; the time-translation isometry and its positive generator are.

A generic globally hyperbolic metric has no timelike Killing vector. One may choose a Cauchy foliation and diagonalize an instantaneous quadratic Hamiltonian, but a different foliation produces a different splitting, and evolution need not preserve either one. The associated Bogoliubov coefficients may even fail the global Hilbert–Schmidt condition. Calling one such instantaneous vector “the vacuum” records a prescription, not a conclusion forced by the spacetime.

The simplest additional structure that can select a ground state is a complete timelike Killing flow together with a positive, well-defined generator in the field theory. A KMS state likewise refers to a chosen one-parameter automorphism group and an inverse temperature. Symmetry alone is not sufficient: zero modes, an operator unbounded below, boundaries, ergoregions, or superradiant sectors may obstruct existence or uniqueness. The fixed-background framework and its lack of generic preferred particles are reviewed in Hollands and Wald 2015, §1 and §2.1, PDF.

A precise natural-state question

The strongest version of “preferred” can be formulated without modes. A locally covariant theory assigns an algebra $\mathcal A(M)$ to each globally hyperbolic spacetime and an injective algebra map $\mathcal A(\psi)$ to each admissible causal embedding $\psi:M\to N$. A natural state would assign $\omega_M$ to every $M$ so that

$$\omega_M=\omega_N\circ\mathcal A(\psi)$$

for every such embedding. Thus restriction of the allegedly preferred state on $N$ must agree exactly with the allegedly preferred state selected on $M$. This is much stronger than choosing a state separately on one stationary spacetime.

The relevant hypotheses also include the timeslice property and dynamical locality, which identifies local physics defined by embeddings with local physics defined by invariance under metric perturbations outside a region. Extended locality says that algebras of causally disjoint regions intersect only in scalar multiples of the identity. These are substantive assumptions, not consequences of the word “quantum field.”

Fewster and Verch prove the following conditional result. Suppose a dynamically local theory has a natural state, and on some spacetime with noncompact Cauchy surfaces that state has a faithful GNS representation and the Reeh–Schlieder property for relatively compact regions. Then relative Cauchy evolution is trivial on that spacetime. If the theory also obeys extended locality, it is equivalent to the trivial theory. The exact hypotheses and conclusion are Fewster and Verch 2012, Definition 6.12 and Theorem 6.13.

This theorem does not say that every conceivable state-selection rule fails. Dropping dynamical locality, faithfulness, Reeh–Schlieder, noncompactness, or extended locality leaves the stated theorem inapplicable. Nor does it prohibit background-dependent families defined on a smaller class of spacetimes with additional structure.

How the no-go argument works

The proof begins with naturality. Relative Cauchy evolution is built from timeslice embeddings around a compact metric perturbation, so the natural state is invariant under it. In the GNS representation, that automorphism is implemented by a unitary $U[h]$ that fixes the cyclic vector.

Because the chosen spacetime has noncompact Cauchy surfaces, one can find a nonempty relatively compact region $O$ causally disjoint from the support of the perturbation $h$. Relative Cauchy evolution acts identically on $\mathcal A(O)$. Therefore $U[h]$ fixes every vector of the form $\pi(A)\Omega$ with $A\in\mathcal A(O)$. The Reeh–Schlieder hypothesis makes those vectors dense, so $U[h]$ is the identity. Faithfulness of the representation then implies that relative Cauchy evolution itself is trivial.

Dynamical locality propagates this trivial response into the local net: every nonempty local algebra becomes the whole algebra. Choose two causally disjoint regions. Extended locality says their intersection contains only scalars, while the preceding step says each is the whole algebra. Hence the whole algebra consists only of scalars—the trivial theory. The contradiction shows that a nontrivial theory satisfying all the hypotheses cannot possess such a natural state.

The key step is not “curvature creates particles.” It is that exact functorial state selection would make the state invariant under every compactly supported change of background geometry, while local cyclicity and dynamical response prevent that invariance in a nontrivial theory.

Distinguished states on symmetric backgrounds

On an ultrastatic spacetime $ds^2=-dt^2+h_{ij}dx^i dx^j$, the Klein–Gordon equation is $\partial_t^2\phi=-A\phi$, where $A=-\Delta_h+m^2+\xi R$ acts on $L^2(\Sigma,d\mathrm{vol}_h)$ with its specified self-adjoint domain. A strictly positive lower bound $A\geq\epsilon\mathbf1$, $\epsilon\gt0$, is a convenient sufficient spectral assumption for the usual ground-state construction from $A^{1/2}$. More general spectra require an infrared check on $A^{-1/2}$ acting on the chosen data. Nonnegativity alone does not settle that check.

For example, a massless minimally coupled scalar on a compact flat spatial torus has a spatially constant mode. Its equation is $\ddot q_0=0$, and its Hamiltonian is the free-particle Hamiltonian $p_0^2/2$, rather than an oscillator of positive frequency. On the usual $L^2(\mathbb R,dq_0)$ representation there is no normalizable lowest-energy vector. This explicit zero mode shows why time-translation symmetry alone does not supply a Fock ground state for the full field. Removing it or quantizing only a derivative field changes the observable theory.

On a stationary spacetime, a complete timelike Killing flow can support a ground state, and the same flow defines the automorphisms entering a KMS condition. On spacetimes with a bifurcate Killing horizon, invariance, quasifree structure, and nonsingularity near the horizon lead to strong uniqueness and thermality results under carefully stated assumptions Kay and Wald 1991, main theorems, pp. 93–136. These results do not assert existence for every stationary black hole; rotating cases and superradiance require special care.

Conformal symmetry can also distinguish a state for a conformally invariant field when a specified conformal map transports a known state. In an expanding FLRW spacetime, by contrast, there is generally no global timelike Killing symmetry. Adiabatic prescriptions and states fixed by asymptotic or initial conditions are useful families, but they depend on order, time choice, or boundary data. Their multiplicity illustrates the absence of a universal choice without making any one of them illegitimate.

Thus the answer to the chapter’s guiding question has two levels. A timelike Killing symmetry with a suitable positive generator is the standard extra structure for a ground state, and a chosen stationary flow supports KMS states. Stronger horizon or conformal structures can distinguish other states. None of these fixed-background selections defines a natural vacuum on the entire category of curved spacetimes.

What replaces a universal vacuum

The algebraic field relations are state independent. Physical calculations then use a class of admissible states rather than one universal vector. For free fields, the Hadamard condition controls the local ultraviolet singularity, while ground, KMS, adiabatic, symmetry, or initial-data conditions address different global questions. A detector response is tied to a worldline and switching function; it need not be interpreted through a representation-independent particle count.

These replacements also preserve the boundary between fixed-background QFT and semiclassical gravity. A state may have a renormalized stress–energy expectation value, but solving for the metric sourced by that expectation value is a further backreaction problem. Nothing in the no-natural-state theorem evolves the geometry or selects a self-consistent semiclassical spacetime.

Exercise: fixed-background distinction

Compare two proposals. Proposal A chooses the lowest instantaneous Hamiltonian state on every Cauchy slice of every spacetime. Proposal B chooses the invariant ground state on a fixed ultrastatic spacetime whose spatial Klein–Gordon operator is strictly positive. Which proposal is geometrically distinguished, and does either evade the natural-state no-go theorem?

Solution

Proposal A depends on a foliation and generally changes under a different time coordinate; states selected on overlapping spacetimes need not restrict compatibly. It therefore does not define a natural state and is not geometrically preferred. Proposal B uses an actual time-translation isometry and a positive generator, so it can define a distinguished ground state on that fixed ultrastatic background. It does not evade or contradict the no-go theorem, because it is not an assignment on every spacetime and does not claim covariance under all admissible embeddings. The contrast is between a background-dependent symmetric choice and a universal natural choice.

References

• 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.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI 10.1016/j.physrep.2015.02.001.
• Kay, Bernard S., and Robert M. Wald. “Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon.” Physics Reports 207, no. 2 (1991): 49–136. DOI 10.1016/0370-1573(91)90015-E.