Existence and Deformation of Hadamard States

Hadamard states exist on globally hyperbolic spacetimes even when no timelike symmetry supplies a vacuum. The standard existence proof compares the target spacetime with a reference spacetime carrying a known Hadamard state, joins them through a causally controlled interpolating geometry, and transports the state by the time-slice property. Propagation of singularities then carries the Hadamard wavefront condition from one Cauchy neighborhood to the other.

Required background. Partial Differential Equations and Green Functions supplies well-posed Cauchy evolution, advanced and retarded propagators, and propagation of characteristic singularities. The bounded local bridge used here is that data in any neighborhood of a Cauchy surface determine a Klein–Gordon solution everywhere in its globally hyperbolic development.

Helpful background. Flat-Spacetime Quantum Field Theory provides explicit reference quasifree states, Causality and Global Structure defines Cauchy surfaces, temporal functions, and global hyperbolicity, and Quantum Fields in Curved Spacetime provides the algebraic and sign conventions used by the construction.

Statement with the necessary hypotheses

Consider two oriented, time-oriented, globally hyperbolic spacetimes $(M_0,g_0)$ and $(M_1,g_1)$ whose smooth Cauchy surfaces are orientation-preservingly diffeomorphic to the same manifold $\Sigma$. On each spacetime take the same scalar-field theory,

$$P_j=\Box_{g_j}-m^2-\xi R_{g_j},$$

or, more generally, compatible normally hyperbolic operators whose lower-order terms can be smoothly interpolated. Suppose a quasifree Hadamard state $\omega_0$ is known on the reference theory over $M_0$.

The deformation theorem says that one can construct a quasifree Hadamard state on the target theory over $M_1$, provided there is a globally hyperbolic interpolating spacetime with a Cauchy region isometric to a region of $M_0$ in the past and another isometric to a region of $M_1$ in the future. The same-Cauchy-topology and controlled-causality hypotheses are substantive. They ensure that both ends can occur in one product manifold $\mathbb R\times\Sigma$ with common Cauchy slices.

This is an existence theorem, not a uniqueness theorem. Different reference states, interpolating metrics, and identifications can produce different target states. The original deformation construction is given in Fulling, Narcowich, and Wald 1981, pp. 243–272, building on preservation of Hadamard form under Cauchy evolution in Fulling, Sweeny, and Wald 1978, pp. 257–264.

Building a causally controlled interpolation

Choose temporal coordinates near suitable Cauchy surfaces so that each end is represented on a portion of $\mathbb R\times\Sigma$. Construct a smooth Lorentzian metric $g_{\mathrm{int}}$ that agrees with $g_0$ for sufficiently early time and with $g_1$ for sufficiently late time. Between them, widen or narrow the light cones in a controlled order so that the coordinate slices remain spacelike Cauchy surfaces throughout.

A pointwise convex combination such as

$$(1-\chi)g_0+\chi g_1$$

is not a proof: the set of Lorentzian metrics with a prescribed time orientation is not convex in arbitrary components, and a careless interpolation can change signature or destroy the Cauchy property. The construction must instead choose lapse, shift, and spatial metric—or suitable comparison metrics—so that a common temporal covector remains timelike and causal diamonds stay controlled.

Select Cauchy surfaces $\Sigma_-\prec\Sigma_+$ inside the regions where the interpolating metric has already become exactly the reference and target metric, respectively. Let $N_-$ and $N_+$ be globally hyperbolic neighborhoods of these surfaces. Their inclusions into the corresponding full spacetimes and into the interpolating spacetime are Cauchy morphisms.

Transport by the time-slice property

Let $\mathcal A(M)$ denote the unital $*$-algebra of the Klein–Gordon field on $M$. The time-slice property says that an embedding whose image contains a Cauchy surface induces an algebra isomorphism. Applying it to the four Cauchy embeddings gives a chain

$$\mathcal A(M_0) \longleftarrow\mathcal A(N_-) \longrightarrow\mathcal A(M_{\mathrm{int}}) \longleftarrow\mathcal A(N_+) \longrightarrow\mathcal A(M_1).$$

Inverting the appropriate arrows produces a unital $*$-isomorphism $\alpha$ between the target algebra and the reference algebra. Define

$$\omega_1(A)=\omega_0\!\left(\alpha(A)\right), \qquad A\in\mathcal A(M_1).$$

Normalization and positivity are automatic:

$$\omega_1(\mathbf1)=1, \qquad \omega_1(A^*A) =\omega_0\!\left(\alpha(A)^*\alpha(A)\right)\geq0.$$

The canonical commutator and field equation are preserved because $\alpha$ is induced by causal Klein–Gordon evolution rather than by an arbitrary vector-space bijection. For quasifree $\omega_0$, the transported state remains quasifree because the induced map acts linearly on the smeared field generators.

Propagating the Hadamard property

In $N_-$, the transported two-point function agrees, under the isometry, with the known reference Hadamard two-point function. It therefore has the future-oriented paired-null wavefront set there. The Klein–Gordon operator has real principal symbol

$$p(x,k)=g^{ab}k_ak_b,$$

so the propagation-of-singularities theorem carries each wavefront covector along the null bicharacteristic flow. Because $N_-$ contains a Cauchy surface, every inextendible characteristic meets it. No new characteristic direction can appear between $\Sigma_-$ and $\Sigma_+$, and the prescribed orientation is retained under parallel transport.

Consequently the interpolating two-point function is Hadamard throughout $M_{\mathrm{int}}$, including $N_+$. Transport through the future isometry and time-slice map then makes $\omega_1$ Hadamard on $M_1$. This step is the microlocal version of preservation under Cauchy evolution. Radzikowski’s proof makes the local-form/wavefront equivalence and propagation step precise in Radzikowski 1996, Theorem 5.1 and its proof, pp. 544–550; the paired-covector formulation used here is also recorded in Brunetti, Fredenhagen, and Köhler 1996, §3, pp. 639–641.

There are two independent checks. Algebraically, pullback along a unital $*$-isomorphism preserves positivity and the commutator exactly. Analytically, the wavefront set in the known end must propagate to precisely the future/past null pairing in the target end. Checking only one side would be insufficient: a bisolution can have the right singular directions without being positive, while a positive state can fail the Hadamard condition.

Choosing the reference end

A convenient reference is an ultrastatic spacetime

$$M_0=\mathbb R\times\Sigma, \qquad g_0=-dt^2+h,$$

for which separation in $t$ reduces the field equation to a spatial operator. When that operator has the required positive self-adjoint realization, its positive spectral square root defines a quasifree ground state, and standard pseudodifferential estimates show that the state is Hadamard. This supplies the known state needed by the deformation theorem.

The positivity qualification cannot be dropped. A zero mode or negative spectral sector can obstruct the simple ground-state formula. That does not prove nonexistence of Hadamard states; it means that this particular reference construction must be modified, for example by choosing a different reference geometry or a non-ground quasifree Hadamard covariance. Likewise, boundary conditions require a separately well-posed boundary theory rather than an unannounced use of the boundaryless theorem.

For example, on a compact ultrastatic Cauchy surface the massless spatially constant mode obeys $\ddot q=0$, not an oscillator equation with nonzero frequency. It has no normalizable ground vector, but one can choose Gaussian initial data with positive variances satisfying the uncertainty inequality and evolve $q(t)=q(0)+t p(0)$. This finite-dimensional modification contributes a smooth two-point kernel, so it does not spoil the Hadamard ultraviolet structure of the nonzero modes. A missing ground state is therefore not a missing admissible state. For more general reference operators, low spectral behavior requires its own construction rather than this single-mode example.

Failure modes and scope

If the proposed reference and target have nondiffeomorphic Cauchy surfaces, the displayed interpolation on one $\mathbb R\times\Sigma$ does not exist. One must not infer a state on the target by pretending that topology can change inside a globally hyperbolic cobordism of this form. The general existence argument instead chooses a reference with the target’s actual Cauchy topology.

A compactly generated chronology horizon is another adverse case: global hyperbolicity and the Cauchy time-slice chain fail at the horizon. The theorem therefore supplies no extension claim there. Similarly, an interpolating metric whose cones tip enough to create closed causal curves invalidates the construction before any state is transported.

The main limitation is conceptual as well as geometric. Deformation proves that the Hadamard class is nonempty; it provides no canonical member of that class. The choices deliberately disappear from the universal short-distance singularity, but they remain in the state-dependent smooth part of the two-point function.

Check your understanding

Assume the time-slice chain defines a unital $*$-isomorphism $\alpha:\mathcal A(M_1)\to\mathcal A(M_0)$ and $\omega_0$ is a Hadamard state. Separate the argument that $\omega_0\circ\alpha$ is a state from the argument that it is Hadamard, and identify the hypothesis used by each part.

Solution

The state argument is algebraic. Unitality gives normalization, while the star-homomorphism property writes the image of every positive element as $\alpha(A^*A)=\alpha(A)^*\alpha(A)$; positivity of the reference functional therefore gives a nonnegative result. The Hadamard argument is analytic and needs more than an abstract isomorphism. The map must arise from isometric identifications and Klein–Gordon Cauchy evolution, so the two-point distribution agrees with the reference distribution in a Cauchy neighborhood and remains a bisolution. Global hyperbolicity ensures that every null bicharacteristic meets that neighborhood, and propagation of singularities transports the paired future/past wavefront set to the target region. Thus positivity uses the unital star structure, whereas Hadamard regularity uses the geometric origin of the map, the field equation, and globally hyperbolic propagation.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Michael Köhler. “The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes.” Communications in Mathematical Physics 180, no. 3 (1996): 633–652. doi:10.1007/BF02099626. Open PDF.
• Fulling, Stephen A., Francis 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, no. 2 (1981): 243–272. doi:10.1016/0003-4916(81)90098-1.
• Fulling, Stephen A., Mark Sweeny, and Robert M. Wald. “Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime.” Communications in Mathematical Physics 63, no. 3 (1978): 257–264. doi:10.1007/BF01196934.
• Radzikowski, Marek J. “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time.” Communications in Mathematical Physics 179, no. 3 (1996): 529–553. doi:10.1007/BF02100096.