Hadamard States and the Wavefront-Set Characterization

For a free scalar field in four spacetime dimensions, two descriptions of ultraviolet regularity are equivalent. One subtracts a locally constructed Hadamard parametrix and asks for a smooth remainder; the other allows exactly one time-oriented family of null covectors in the wavefront set of the two-point distribution. The equivalence is powerful because the first description supports point splitting while the second is stable under propagation and distributional operations. It characterizes an admissible singularity class—not a preferred vacuum, and not the existence of a positive state.

Required background. Green-hyperbolic operators and causal propagators supplies the field equation and commutator; propagation of singularities for hyperbolic fields supplies null Hamilton flow.

Helpful background. Higher-point microlocal spectrum conditions gives the hierarchy generated from a Hadamard two-point function; no-natural-state results and covariant state spaces explains why no universal preferred state is expected. The physical volume develops the local Hadamard parametrix, the two-point admissibility criterion, adiabatic states, and fermion and gauge-field variants.

The canonical chapter hypothesis–conclusion table separates the two-point equivalence from the additional hierarchy, positivity, and state-existence claims.

The Hadamard–Radzikowski equivalence

Let $(M,g)$ be a smooth, oriented and time-oriented, four-dimensional globally hyperbolic spacetime without boundary. Consider the real scalar operator

$$P=\Box_g+q, \qquad q\in C^\infty(M;\mathbb R),$$

which includes $q=m^2+\xi R$. It is normally hyperbolic and formally self-adjoint. Let $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ be its causal propagator. A state two-point distribution $\omega_2\in\mathcal D'(M\times M)$ is an exact bisolution and has the canonical antisymmetric part

$$P_x\omega_2=P_y\omega_2=0, \qquad \omega_2-\omega_2^{\mathrm{op}}=-iE, \qquad \omega_2^{\mathrm{op}}(x,y):=\omega_2(y,x).$$

It is also Hermitian and of positive type, $\omega_2(\bar f,f)\geq0$. Positivity is needed for $\omega_2$ to come from a state, but it is not what fixes the wavefront orientation.

Write $(x,k)\sim(y,k')$ when $x$ and $y$ lie on one null geodesic, $k$ is a nonzero covector whose metric dual $k^\sharp$ is tangent to that geodesic at $x$, and $k'$ is the parallel transport of $k$ to $y$. The coincident case includes every nonzero null $k$ with $k'=k$. If $k\triangleright0$ means future directed, define

$$\begin{aligned} \mathcal C^+ :=\bigl\{(x,k;y,-k'):\;&(x,k)\sim(y,k'),\\ &k\triangleright0\bigr\}, \qquad \mathcal C^-:=\operatorname{flip}(\mathcal C^+). \end{aligned}$$

Here the minus sign belongs to the second distribution slot; it does not reverse the parallel transport. The two-slot pairing diagram makes that distinction visible.

Now choose a geodesically convex neighborhood $C$. Let $\sigma(x,y)$ be Synge’s world function and, in the site’s $(+---)$ convention, put $s=-\sigma$. For a smooth local time function $T$ increasing toward the future, set

$$s_\epsilon(x,y) =s(x,y)+i\epsilon\bigl(T(x)-T(y)\bigr)+\frac12\epsilon^2.$$

The four-dimensional scalar Hadamard parametrix is the distributional boundary value

$$H_\ell(x,y) =\lim_{\epsilon\downarrow0}\frac{1}{8\pi^2} \left[ \frac{U(x,y)}{s_\epsilon(x,y)} +V(x,y)\log\!\frac{s_\epsilon(x,y)}{\ell^2} \right].$$

The transport equations determine $U$ and the complete Hadamard-coefficient jet of $V$ from $P$ and the local geometry; $\ell>0$ is an arbitrary length. For a merely smooth metric, $V$ here means an all-orders cutoff or Borel-summed representative of that jet. Different admissible summations change $H_\ell$ only by a smooth kernel. A two-point distribution is locally Hadamard when

$$\omega_2|_{C\times C}=H_\ell+W_C, \qquad W_C\in C^\infty(C\times C),$$

around every point. The parametrix solves the field equation only modulo a smooth kernel. The smooth remainder must complete it to the exact bisolution and encode the state-dependent information.

Under the hypotheses above, the following two statements are equivalent:

1. $\omega_2$ has the local Hadamard form.

2. Its wavefront set is exactly

   $$\operatorname{WF}(\omega_2)=\mathcal C^+.$$

It is enough to establish Hadamard form on a suitable neighborhood of a complete Cauchy surface; the property then propagates globally. This is the Hadamard–Radzikowski equivalence Radzikowski 1996, Theorem 5.1, pp. 544–550. A modern state-level formulation, which does not require quasifreeness, is Khavkine and Moretti 2015, Theorem 9, pp. 47–49 (Open PDF).

Radzikowski’s original theorem also identifies these conditions with the statement that the associated Feynman distribution is a distinguished parametrix modulo a smooth kernel. That third formulation is the microlocal bridge used in the converse below; its precise sign depends on the convention used to define the time-ordered fundamental solution.

For a genuine Klein–Gordon two-point function, the apparently weaker inclusion $\operatorname{WF}(\omega_2)\subset\mathcal C^+$ already forces equality. Indeed,

$$\operatorname{WF}(E)=\mathcal C^+\cup\mathcal C^-,$$

while the transpose has wavefront set in $\mathcal C^-$. The commutator identity therefore leaves no allowed branch of $E$ missing from $\omega_2$. This argument uses the fixed antisymmetric part. For an arbitrary distribution the inclusion is not enough—the zero distribution satisfies it.

Why the local expansion gives the global cone

The proof has three layers, and each hypothesis has a separate job.

First, the boundary value $s+i0\,[T(x)-T(y)]$ is not decorative regularization. Its Fourier transform selects the future-frequency branch in the first slot, so the local parametrix has the oriented cone $\mathcal C^+$ rather than the union $\mathcal C^+\cup\mathcal C^-$. Changing $T$, the cutoff used to sum the Hadamard coefficients, or the length $\ell$ changes only the smooth remainder.

Second, applying $P$ confines every nonzero wavefront covector to the characteristic set. For a Klein–Gordon operator the principal symbol is $g^{-1}(k,k)$, so these covectors are null. Propagation of singularities then carries them along the null bicharacteristic flow, which is the cotangent lift of null geodesic flow. This turns the local pairing into the global relation $(x,k)\sim(y,k')$.

Third, global hyperbolicity supplies a complete Cauchy surface intersected by every inextendible null geodesic. One must still avoid a tempting but incomplete argument: a wavefront element on $M\times M$ may have one nonzero slot and one zero slot, so it is not enough simply to “propagate both slots to the Cauchy surface.” Propagation of Hadamard form to a second Cauchy neighborhood rules out these partial-zero cases and closes the proof. The repaired vector-bundle treatment spells out this step in Sahlmann and Verch 2001, Theorems 5.5 and 5.8 and the accompanying Remark (iii) (Open PDF). An explicit support-moving operator that transfers both test functions into a Cauchy neighborhood is given by Khavkine and Moretti 2015, Proposition 16(b), pp. 49–51 (Open PDF). The Cauchy-band propagation diagram shows the same geometry, while the original Cauchy-evolution theorem is Fulling, Sweeny, and Wald 1978, pp. 257–264.

Why the cone recovers the local expansion

Starting from $\operatorname{WF}(\omega_2)=\mathcal C^+$, the time orientation determines the distinguished Feynman prescription. Duistermaat–Hörmander uniqueness says that two distinguished parametrices with the same choice of characteristic relation differ smoothly. The local transport construction supplies one such parametrix, so the candidate $\omega_2$ differs from the local Hadamard boundary value by a smooth kernel. In this direction the wavefront set does not compute the state-dependent $W_C$; it proves that no other ultraviolet singularity is present.

This also clarifies the role of positivity. The equivalence compares two descriptions of singularity structure for a two-point function whose dynamics and commutator are already fixed. Positivity decides whether that distribution is a physical state covariance. It neither follows from the cone nor selects a unique member of the Hadamard class.

Smooth differences of Hadamard states

Let $\omega_2$ and $\widetilde\omega_2$ be two Hadamard two-point functions for the same operator and commutator, and set $d=\omega_2-\widetilde\omega_2$. Locally, both subtract the same geometric parametrix, so $d$ is smooth near the diagonal. There is also a short global microlocal proof that shows exactly where the common commutator enters.

The wavefront-set sum rule gives

$$\operatorname{WF}(d)\subset\mathcal C^+.$$

The antisymmetric parts cancel, so $d=d^{\mathrm{op}}$. Transposition flips the two cotangent slots; hence

$$\operatorname{WF}(d) =\operatorname{flip}\!\bigl(\operatorname{WF}(d)\bigr) \subset\mathcal C^-.$$

Because $\mathcal C^+\cap\mathcal C^-=\varnothing$, one obtains $\operatorname{WF}(d)=\varnothing$ and therefore

$$\omega_2-\widetilde\omega_2\in C^\infty(M\times M).$$

Equality of two wavefront sets would not imply this for general distributions. Here smoothness follows from the one-sided Hadamard cone together with cancellation of the common commutator; the field equation is not needed in this short flip argument Sanders 2010, Lemma 2.9, preprint pp. 5–6 (Open PDF). This is why changing the Hadamard state changes Wick expectations and renormalized stress tensors by smooth, locally differentiable data rather than by new ultraviolet singularities.

Ultrastatic ground state from spectral calculus

Consider the ultrastatic spacetime

$$(M,g)=(\mathbb R\times\Sigma,\,dt^2-h),$$

where $(\Sigma,h)$ is compact and has no boundary. Write the field equation as

$$(\partial_t^2+A)\phi=0, \qquad A=-\Delta_h+q_\Sigma, \qquad q_\Sigma\in C^\infty(\Sigma;\mathbb R).$$

Assume that the self-adjoint elliptic operator $A$ is strictly positive: $A\geq\mu^2\mathbf1$ for some $\mu>0$. Choose an orthonormal eigenbasis

$$A\varphi_j=\omega_j^2\varphi_j, \qquad \omega_j\geq\mu.$$

Spectral calculus defines the ground-state two-point distribution by

$$\begin{aligned} \omega_{0,2}(t,x;t',y) &=\left\langle\delta_x, \frac{e^{-iA^{1/2}(t-t')}}{2A^{1/2}}\delta_y \right\rangle\\ &=\sum_j\frac{e^{-i\omega_j(t-t')}}{2\omega_j} \varphi_j(x)\overline{\varphi_j(y)}, \end{aligned}$$

where the series is understood distributionally. Each summand obeys the field equation in both arguments. For $f\in C_0^\infty(M)$, define

$$F_j:=\int_{\mathbb R\times\Sigma} e^{+i\omega_j t}\overline{\varphi_j(x)}f(t,x) \,dt\,d\operatorname{vol}_h(x).$$

Then

$$\omega_{0,2}(\bar f,f) =\sum_j\frac{|F_j|^2}{2\omega_j}\geq0,$$

which proves positive type instead of merely asserting it. Transposing the kernel changes $t-t'$ to its negative, so

$$\begin{aligned} \omega_{0,2}-\omega_{0,2}^{\mathrm{op}} &=-i\left\langle\delta_x, \frac{\sin\!\bigl(A^{1/2}(t-t')\bigr)}{A^{1/2}} \delta_y\right\rangle\\ &=-iE. \end{aligned}$$

Thus the normalization matches the site’s $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ convention.

The high-frequency step is microlocal. The square root $A^{1/2}$ is an elliptic pseudodifferential operator of order one with principal symbol $|\eta|_h$. Its half-wave propagator $e^{-itA^{1/2}}$ is a Fourier integral operator following the corresponding geodesic Hamilton flow. The phase $e^{-i\omega_j(t-t')}$ selects the future branch in the first spacetime slot, so

$$\operatorname{WF}(\omega_{0,2})\subset\mathcal C^+.$$

The commutator then forces equality, and the equivalence theorem converts that equality into local Hadamard form. The strict lower bound, static vacuum covariance, and extension of the pseudodifferential construction to compact $\Sigma$ are treated in Gérard and Wrochna 2014, §7.5 and §8.1, preprint pp. 29–30 (Open PDF). More generally, stationary ground and positive-temperature KMS states obey the microlocal spectrum condition under the hypotheses of Sahlmann and Verch 2000, Theorem 5.1 and the scalar-field applications (Open PDF).

As a normalization check, the continuous-spectrum analogue on Minkowski space is

$$\omega_{0,2}(x,y) =\int_{\mathbb R^3} \frac{d^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}(t-t')+i\mathbf p\cdot(\mathbf x-\mathbf y)}, \qquad \omega_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}.$$

Its large-$|\mathbf p|$ covectors approach the future null cone, as the general principal-symbol argument predicts.

Strict positivity is an infrared hypothesis, not a cosmetic one. If $A$ has a zero mode, then $A^{-1/2}$ in the displayed formula is undefined. For a massless minimally coupled field on compact connected $\Sigma$, the constant spatial mode is the standard example. That finite-dimensional mode can instead be supplied with a positive covariance satisfying the CCR; its on-shell kernel is polynomial in $t,t'$ and hence smooth, so it does not alter the Hadamard wavefront set. However, the zero-frequency degree of freedom has no regular time-translation-invariant oscillator ground state. The naive spectral formula and its ground-state interpretation therefore fail even though the ultraviolet Hadamard question can still be repaired Fewster and Verch 2012, discussion after Corollary 5.4, preprint p. 24 (Open PDF).

This ultrastatic state is the reference object used in deformation and gluing constructions: transfer its Cauchy data through a controlled interpolating geometry, then use propagation to certify the target state. The classical construction is Fulling, Narcowich, and Wald 1981, §§IV–V, pp. 261–269.

A positive wrong-frequency state

The hypotheses can be stress-tested without inventing an informal “image singularity.” Let $\omega_{0,2}$ be any Hadamard two-point function and choose a constant $\alpha>0$. Define

$$w_\alpha =(1+\alpha)\omega_{0,2} +\alpha\omega_{0,2}^{\mathrm{op}}.$$

This candidate passes three strong tests.

• It is a bisolution because both summands are bisolutions.

• It is of positive type. The first summand is positive, and $\omega_{0,2}^{\mathrm{op}}(\bar f,f)=\omega_{0,2}(f,\bar f)\geq0$ after taking $g=\bar f$ in the positivity condition.

• It has the correct commutator:

  $$w_\alpha-w_\alpha^{\mathrm{op}} =\omega_{0,2}-\omega_{0,2}^{\mathrm{op}} =-iE.$$

Consequently $w_\alpha$ defines a quasifree state two-point function. Nevertheless,

$$\operatorname{WF}(w_\alpha) =\mathcal C^+\cup\mathcal C^-.$$

There is no cancellation: $\mathcal C^+$ and $\mathcal C^-$ are disjoint, and each summand is microlocally smooth near the other summand’s cone. Thus $w_\alpha$ is not Hadamard. In the ultrastatic mode picture it gives occupation number $\alpha$ to every frequency, including arbitrarily high ones; the occupation fails to decay in the ultraviolet and leaves a wrong-oriented singular branch. Subtracting the canonical $H_\ell$ cannot cancel the added positive- and negative-frequency singularities, so the remainder is not smooth.

This example separates the state axioms cleanly: dynamics, Hermiticity, positivity, and the canonical commutator do not imply the Hadamard condition. The missing input is ultraviolet frequency orientation.

Relation to the higher-point condition

For a centered quasifree state, Wick’s rule constructs every even $n$-point distribution from products of $\omega_2$, and all odd distributions vanish. The Hadamard cone then embeds each contraction into the directed graph cone, so the full hierarchy satisfies the smooth microlocal spectrum condition Brunetti, Fredenhagen, and Köhler 1996, Proposition 4.3 (Open PDF). Quasifreeness is sufficient but not necessary. For a generalized real free field with a c-number commutator, a Hadamard two-point function forces every truncated $n$-point distribution with $n\ne2$ to be smooth and yields the full microlocal spectrum condition Sanders 2010, Theorem 4.2 and Corollary 4.3, preprint p. 9 (Open PDF).

Thus a nonquasifree free-field state may contain independent smooth connected correlations, but not new connected wavefront singularities under these hypotheses. Genuinely independent singular connected functions become possible only after leaving the generalized-free, c-number-commutator setting, as in interacting theories. At $n=2$, the field equation and canonical commutator convert the one-sided cone into the Hadamard relation without any quasifree assumption. The higher-point page gives the exact graph statement and its scope.

Scope beyond the scalar theorem

The displayed boundary-value formula is specific to a four-dimensional scalar normally hyperbolic operator. Other dimensions have different local Hadamard expansions. Vector-bundle wave operators and Dirac fields have analogous microlocal characterizations, but the fiber pairing, adjoint operation, and CCR or CAR must be stated explicitly; Sahlmann and Verch 2001, Theorem 5.8 (Open PDF) gives a dimension-$\geq3$ treatment and repairs a gap in the original scalar-style proof.

A timelike boundary may create reflected singular branches and requires a boundary-condition theorem. Gauge theories require constraints or subsidiary fields before positivity and the physical two-point function are identified. Finite-order adiabatic states generally provide Sobolev, rather than smooth, control. None of these cases is licensed by silently reusing the scalar equality.

Common pitfalls

Keeping only singular support. Singular support remembers that two points can be null related but forgets which frequency orientation occurs. Hadamard admissibility needs the full cotangent data.

Inferring a state from the cone. A distribution can have the correct wavefront set and still fail the field equation, Hermiticity, the commutator, or positive type. Test those conditions separately.

Claiming smoothness from equal wavefront sets. Equal wavefront sets alone do not make a difference smooth. For Hadamard two-point functions the common commutator makes the difference symmetric, forcing its wavefront set into two disjoint orientations.

Calling $A\geq0$ sufficient in the ground-state formula. The factor $A^{-1/2}$ requires a spectral gap away from zero. Zero modes need their own infrared treatment.

Propagating two nonzero slots without checking them. Product wavefront sets allow one slot covector to vanish. A complete Cauchy-neighborhood argument is what excludes that loophole.

Exercises

1. Change the Hadamard length. Let $H_\ell$ and $H_{\ell'}$ use the same $U,V,s_\epsilon$ but different positive length scales. Show that their difference is smooth and determine it.

Solution

Only the logarithm changes:

$$H_{\ell'}-H_\ell =\frac{1}{8\pi^2}V(x,y) \log\!\frac{\ell^2}{\ell'^2}.$$

The logarithm of the ratio of two constants is constant and $V$ is smooth. Thus the difference is smooth, can be absorbed into $W_C$, and cannot change the wavefront set. The scale remains relevant to the finite renormalization freedom of composite observables, but not to Hadamard admissibility.

2. Prove smoothness of state differences. Let $d=\omega_2-\widetilde\omega_2$ for two Hadamard two-point functions with the same commutator. Prove that $d$ is smooth without assuming the result.

Solution

The wavefront-set sum rule gives $\operatorname{WF}(d)\subset\mathcal C^+$. The commutators cancel, so $d-d^{\mathrm{op}}=0$. Hence

$$\operatorname{WF}(d) =\operatorname{WF}(d^{\mathrm{op}}) =\operatorname{flip}\!\bigl(\operatorname{WF}(d)\bigr) \subset\mathcal C^-.$$

The first covector is future directed in $\mathcal C^+$ and past directed in $\mathcal C^-$, so the cones are disjoint. Therefore $\operatorname{WF}(d)=\varnothing$, which is equivalent to $d\in C^\infty(M\times M)$.

3. Verify the ultrastatic covariance. Starting from the eigenfunction series, check the field equation, positivity, and commutator normalization.

Solution

For each mode,

$$(\partial_t^2+A_x) \left[e^{-i\omega_j(t-t')}\varphi_j(x)\right] =(-\omega_j^2+\omega_j^2) e^{-i\omega_j(t-t')}\varphi_j(x)=0,$$

and the same calculation works in the primed slot. Smearing gives the sum $\sum_j|F_j|^2/(2\omega_j)$, which is nonnegative. Finally,

$$\frac{e^{-i\omega_j\tau}-e^{+i\omega_j\tau}}{2\omega_j} =-i\frac{\sin(\omega_j\tau)}{\omega_j}.$$

Summing the modes produces the kernel of $-i\sin(A^{1/2}\tau)/A^{1/2}=-iE(\tau)$. Thus all three conditions hold with the stated sign convention.

4. Diagnose a zero mode. Suppose $A\varphi_0=0$ on compact $\Sigma$. Which part of the ground-state construction fails? Construct a positive CCR-compatible covariance for this mode and explain why it does not change the Hadamard wavefront set.

Solution

The coefficient $1/(2\omega_0)$ diverges, so neither $A^{-1/2}$ nor the oscillator-vacuum covariance exists on the zero eigenspace. The zero mode solves $q_0''(t)=0$ and behaves as a free particle rather than an oscillator; write $q_0(t)=Q+tP$ with $[Q,P]=i$. Choose

$$\begin{gathered} \langle Q^2\rangle=a, \qquad \langle P^2\rangle=b,\\ \frac12\langle QP+PQ\rangle=c, \qquad c\in\mathbb R,\\ a>0, \qquad b>0, \qquad ab-c^2\geq\frac14. \end{gathered}$$

The uncertainty inequality makes this a positive covariance, while its antisymmetric part has the required CCR. Explicitly,

$$\begin{aligned} \langle q_0(t)q_0(t')\rangle &=a+c(t+t')+btt'\\ &\qquad+\frac{i}{2}(t'-t). \end{aligned}$$

Multiplication by $\varphi_0(x)\overline{\varphi_0(y)}$ gives a kernel polynomial in $t,t'$ and smooth in every spacetime variable, so it contributes no wavefront directions. The obstruction is infrared and invalidates the naive stationary ground-state formula, not the local ultraviolet Hadamard singularity.

5. Analyze the wrong-frequency admixture. For $w_\alpha=(1+\alpha)\omega_{0,2}+\alpha\omega_{0,2}^{\mathrm{op}}$, verify every state test and identify the one failed regularity test.

Solution

Linearity preserves the field equation and Hermiticity. Both terms are of positive type with positive coefficients. Transposition interchanges the two terms, so the factors proportional to $\alpha$ cancel from the antisymmetric part and leave $-iE$. These properties supply a quasifree state. However,

$$\operatorname{WF}(\omega_{0,2})=\mathcal C^+, \qquad \operatorname{WF}(\omega_{0,2}^{\mathrm{op}})=\mathcal C^-.$$

The cones are disjoint, so neither singularity can cancel the other and $\operatorname{WF}(w_\alpha)=\mathcal C^+\cup\mathcal C^-$. The state fails precisely the Hadamard orientation test.

6. Upgrade inclusion to equality. Assume a Klein–Gordon two-point function satisfies the canonical commutator and $\operatorname{WF}(\omega_2)\subset\mathcal C^+$. Prove that the inclusion is an equality.

Solution

Transposition gives $\operatorname{WF}(\omega_2^{\mathrm{op}})\subset\mathcal C^-$. If some covector tuple in $\mathcal C^+$ were absent from $\operatorname{WF}(\omega_2)$, then it would also be absent from the transposed kernel, whose wavefront set lies in the disjoint cone $\mathcal C^-$. It would consequently be absent from $\omega_2-\omega_2^{\mathrm{op}}=-iE$. This contradicts $\operatorname{WF}(E)=\mathcal C^+\cup\mathcal C^-$. Thus every point of $\mathcal C^+$ occurs in $\operatorname{WF}(\omega_2)$, proving equality.

Continue the microlocal chain

Next, Wick polynomials under microlocal conditions uses smooth Hadamard subtraction and Hörmander’s product criterion to define coincident composite fields. For physical state construction and propagation, continue with constructing Hadamard states and propagation of the Hadamard property.

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 (1996): 633–652. DOI. Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality of the Free Scalar Field.” Annales Henri Poincaré 13 (2012): 1675–1709. DOI. 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 (1981): 243–272. DOI.
• 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 (1978): 257–264. DOI.
• Gérard, Christian, and Michał Wrochna. “Construction of Hadamard States by Pseudodifferential Calculus.” Communications in Mathematical Physics 325 (2014): 713–755. DOI. Open PDF.
• Khavkine, Igor, and Valter Moretti. “Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction.” In Advances in Algebraic Quantum Field Theory, 191–251. Cham: Springer, 2015. 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.
• Sahlmann, Hanno, and Rainer Verch. “Passivity and Microlocal Spectrum Condition.” Communications in Mathematical Physics 214 (2000): 705–731. DOI. Open PDF.
• Sahlmann, Hanno, and Rainer Verch. “Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime.” Reviews in Mathematical Physics 13 (2001): 1203–1246. DOI. Open PDF.
• Sanders, Ko. “Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime.” Communications in Mathematical Physics 295 (2010): 485–501. DOI. Open PDF.