Existence, Deformation, and Gluing of Hadamard States

Every smooth globally hyperbolic Klein–Gordon spacetime admits Hadamard states, but the theorem is an existence statement about short-distance structure—not a recipe for a preferred vacuum. One robust proof deforms the metric to an ultrastatic one near an earlier Cauchy surface, imports a known ground state there, and transports it through time-slice isomorphisms. Propagation of singularities then carries the Hadamard wavefront condition to the entire target spacetime.

Required background. Hadamard states and their wavefront characterization supplies the microlocal spectrum condition. Algebraic free fields on curved spacetimes supplies the Klein–Gordon algebra and time-slice map.

Helpful background. No-natural-state results and covariant state spaces explains why existence does not select one state naturally. Propagation of singularities for hyperbolic fields supplies the global propagation step. Constructing Hadamard states develops the physical construction, while states of low energy gives a different, observer-dependent selection criterion.

The deformation existence theorem

Let $(M,g)$ be a smooth, oriented and time-oriented globally hyperbolic spacetime with no boundary, and let $P=\Box_g+m^2+\xi R$ be a real, normally hyperbolic, formally self-adjoint Klein–Gordon operator with smooth coefficients. A quasifree state is Hadamard when its two-point distribution $\omega_2$ is a bisolution, has antisymmetric part $iE$, is of positive type, and satisfies

$$\operatorname{WF}(\omega_2) =\{(x,k;x',-k'):(x,k)\sim(x',k'),\ k\ \text{future directed}\}.$$

The deformation theorem asserts existence of such a state. It does not assume stationarity, an asymptotic region, a preferred foliation, or a small curvature expansion. Its geometric input is global hyperbolicity, which gives a smooth Cauchy temporal function and a product representation $M\simeq\mathbb R\times\Sigma$. The classical construction of Fulling, Narcowich, and Wald is given in Fulling, Narcowich, and Wald 1981, §§ IV–V, pp. 261–269; the wavefront-set form and its propagation are sharpened in Radzikowski 1996, §§ 5–6, pp. 546–552.

There is a technical qualification at the static end. On an ultrastatic metric $g_0=\mathrm dt^2-h$ the spatial operator $A$ must admit a positive self-adjoint realization for the standard ground-state covariance $A^{-1/2}$ to be defined. A strictly positive mass is a simple sufficient condition in the usual boundaryless setting; zero modes need separate treatment. This endpoint hypothesis is part of the construction, not an extra assumption on the final metric.

How the Cauchy gluing works

Choose Cauchy surfaces $\Sigma_-\prec\Sigma_+$ in the target. Construct a second globally hyperbolic metric $g_d$ that equals an ultrastatic metric in the past of $\Sigma_-$ and equals $g$ in the future of $\Sigma_+$. The light cones in the interpolation are chosen so that $g_d$ remains globally hyperbolic. Let $M_-$ and $M_+$ be neighborhoods of the two Cauchy surfaces on which the appropriate metrics agree. Time-slice gives the chain of algebra isomorphisms

$$\mathcal A(M,g_0)\longleftarrow\mathcal A(M_-,g_0) \longrightarrow\mathcal A(M,g_d) \longleftarrow\mathcal A(M_+,g_d) \longrightarrow\mathcal A(M,g).$$

Pull the ultrastatic ground state through this chain. Positivity and normalization survive because each arrow is a $*$-isomorphism; the field equation and commutator survive because the maps are induced by causal propagators. Near $\Sigma_-$ the two-point function is the ultrastatic Hadamard two-point function. The propagation-of-singularities theorem carries its null covectors along the bicharacteristic flow, and the bisolution property propagates the Hadamard condition across a Cauchy surface. Thus the transported state is Hadamard on all of $(M,g)$.

This argument is sometimes called “gluing,” but it is not a convex interpolation of two-point functions by a spacetime cutoff. Multiplying separate bisolutions by a partition of unity generally violates the field equation, and cross terms may destroy positivity. The legitimate gluing takes place through a common Cauchy development and algebra isomorphisms. Pseudodifferential constructions provide another route by choosing Cauchy-surface covariances whose principal symbols split positive and negative frequency; see Gérard and Wrochna 2014, §§ 6–7, pp. 740–753. Agreement between these methods concerns the Hadamard singularity class, not equality of states.

The first curved-spacetime construction

The page on constructing Hadamard states carries out the deformation: an ultrastatic ground-state covariance is transported through a metric that agrees with the desired geometry in a future Cauchy neighborhood. The independent check is local. Subtract a Hadamard parametrix $H$ from the transported $\omega_2$ in one convex normal neighborhood; the difference must be smooth. Propagation then shows the same property in every normal neighborhood reached through the Cauchy development.

Existence is stable under adding a smooth symmetric bisolution small enough to preserve positivity. Consequently the theorem yields an infinite family, not uniqueness. It also says nothing by itself about thermal equilibrium, minimal energy, invariance under an isometry group, or backreaction regularity beyond what the Hadamard condition supports.

Failure boundary: symmetry is not transported

Suppose the future metric has no timelike Killing field. The transported state remains Hadamard, and if the starting state is pure then the algebra isomorphism preserves algebraic purity. But “stationary” has no target symmetry group with respect to which it could be defined. Even when the future admits a Killing flow, the deformation map need not intertwine that flow, so invariance does not follow. Demanding a future ground state therefore inserts a new spectral and symmetry problem. The strongest surviving conclusion is the existence of a quasifree Hadamard state.

A second adversarial check is to retain an ultrastatic zero mode while writing $A^{-1/2}$. The covariance then diverges. One must remove or control the zero mode, alter the endpoint state, or invoke a construction that does not rely on a strictly positive ground-state Hamiltonian.

Exercises

Why does an algebra isomorphism preserve positivity of the transported state but not stationarity?

Solution

If $\alpha:\mathcal A_1\to\mathcal A_2$ is a $*$-isomorphism and $\omega_1$ is positive, then $\omega_2(A)=\omega_1(\alpha^{-1}A)$ obeys $\omega_2(B^*B)=\omega_1((\alpha^{-1}B)^*\alpha^{-1}B)\geq0$. Stationarity would additionally require $\omega_2\circ\tau_t=\omega_2$ for a target time-translation automorphism $\tau_t$ and an intertwining relation between $\tau_t$ and the source dynamics. A generic deformation supplies neither.

References

• Fulling, Stephen A., Frank J. Narcowich, and Robert M. Wald. “Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime, II.” Annals of Physics 136 (1981): 243–272. DOI.
• Gérard, Christian, and Michał Wrochna. “Construction of Hadamard States by Pseudodifferential Calculus.” Communications in Mathematical Physics 325 (2014): 713–755. DOI; Open PDF.
• Radzikowski, Marek J. “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time.” Communications in Mathematical Physics 179 (1996): 529–553. DOI; Open PDF.