Hadamard Parametrix and Short-Distance Structure

The Hadamard parametrix isolates the universal short-distance singularity of a free scalar two-point function. Its singular coefficients are fixed locally by the metric and the Klein–Gordon operator, while a smooth remainder distinguishes states once the same parametrix convention is used. This separation makes state-to-state differences of point-split local expectation values finite without pretending that ultraviolet admissibility selects a preferred vacuum.

Required background. Green Functions and Causal Propagators fixes the Klein–Gordon and causal-kernel conventions. Products, Scaling Degree, and Distribution Extensions explains why coincidence limits of singular kernels need control.

Helpful background. Free-Field Wick Products and Point Splitting gives the flat-space model. Levi–Civita Connection, Geodesics, and Riemann Curvature supplies Synge’s world function and the van Vleck determinant. Quasifree States and Two-Point Functions supplies the state conditions imposed in addition to the local singular form.

The four-dimensional boundary value

Throughout this page, $(M,g)$ is a smooth, oriented and time-oriented, four-dimensional globally hyperbolic spacetime without boundary. The field is real, $m^2\geq0$ and $\xi\in\mathbb R$ are constants, and

$$P_\xi\phi=(\Box+m^2+\xi R)\phi=0$$

with the site’s $(+---)$ metric and curvature conventions. The construction below is local: choose a geodesically convex neighborhood $C$, so every pair $x,x'\in C$ is joined by a unique geodesic lying in $C$.

Synge’s world function $\sigma(x,x')$ is one half the signed squared geodesic distance and is positive for nearby timelike separation. It satisfies

$$\sigma_{;a}\sigma^{;a}=2\sigma.$$

Put $s=-\sigma$. Then $s$ is positive for spacelike separation and

$$s_{;a}s^{;a}=-2s, \qquad [s]=0, \qquad [\Box s]=-4,$$

where square brackets denote the coincidence limit $x'\to x$. Let $T$ be a smooth local time function increasing toward the future and define

$$s_\epsilon(x,x') =s(x,x')+i\epsilon\bigl(T(x)-T(x')\bigr)+\frac12\epsilon^2, \qquad \epsilon\downarrow0.$$

Use the branch of $\log z$ cut along the negative real axis. The limit is distributional: first smear in $x$ and $x'$, then take $\epsilon\downarrow0$. The local Hadamard singularity condition is that a candidate $\omega_2$ can be written

$$\omega_2(x,x') =\lim_{\epsilon\downarrow0}\frac{1}{8\pi^2} \left[ \frac{U(x,x')}{s_\epsilon(x,x')} +V(x,x')\log\!\left(\frac{s_\epsilon(x,x')}{\ell^2}\right) +W_{\omega,\ell}(x,x') \right],$$

with $W_{\omega,\ell}$ smooth on $C\times C$. Denote the boundary value containing the $U$ and $V$ terms, but not $W_{\omega,\ell}$, by $H_\ell$. Radzikowski uses the full spacelike-positive squared distance in the corresponding boundary prescription; converting that distance to $2s$ gives the pole normalization and $i\epsilon$ scaling above Radzikowski 1996, Eq. (3) and Definition 3.4.

The normalization has a quick flat-space check. In massless Minkowski space, $U=1$, $V=0$, $T=t$, and

$$s_\epsilon =\frac12\left(\lvert\mathbf x-\mathbf x'\rvert^2 -(t-t'-i\epsilon)^2\right).$$

The pole therefore becomes

$$\omega_{0,2}(x,x') =\frac{1}{4\pi^2} \lim_{\epsilon\downarrow0} \frac{1}{\lvert\mathbf x-\mathbf x'\rvert^2-(t-t'-i\epsilon)^2},$$

the standard positive-frequency Wightman boundary value. Transposing $x$ and $x'$ reverses the side of the light-cone boundary value, so the site’s causal convention is recovered:

$$\omega_2(f,g)-\omega_2(g,f)=-iE(f,g), \qquad E=G_{\mathrm{ret}}-G_{\mathrm{adv}}.$$

Transport fixes the singular coefficients

The pole coefficient is not an adjustable smooth numerator. Away from $s=0$, applying $P_{\xi,x}$ gives

$$P_{\xi,x}\!\left(\frac{U}{s}\right) =\frac{P_{\xi,x}U}{s} -\frac{2s^{;a}\nabla_aU+(\Box s+4)U}{s^2}.$$

Eliminating the $s^{-2}$ term and matching the flat tangent-space singularity require

$$2s^{;a}\nabla_aU+(\Box s+4)U=0, \qquad [U]=1.$$

This is an ordinary differential equation along each radial geodesic. The van Vleck–Morette determinant

$$\Delta_{\mathrm{vV}}(x,x') =-\frac{\det[-\nabla_a\nabla_{b'}\sigma(x,x')]} {\sqrt{-g(x)}\sqrt{-g(x')}}$$

where $g(x)=\det g_{ab}(x)$, obeys the same transport equation and has coincidence value one. Uniqueness along the geodesic therefore gives

$$U=\Delta_{\mathrm{vV}}^{1/2}.$$

Now write the logarithmic coefficient as a formal covariant expansion

$$V\sim\sum_{n=0}^{\infty}V_n s^n.$$

The remaining $s^{-1}$ term fixes $V_0$:

$$2s^{;a}\nabla_aV_0+(\Box s+2)V_0=-P_{\xi,x}U.$$

The logarithmic terms then give, for $n\geq0$,

$$2s^{;a}\nabla_aV_{n+1}+(\Box s-2n)V_{n+1} =-\frac{1}{n+1}P_{\xi,x}V_n.$$

These transport equations determine every $V_n$ from the local geometry and $P_\xi$; Moretti records the general smooth-potential recursion in full-squared-distance and opposite-signature conventions Moretti 2003, Appendix A, Eqs. (55)–(62), Open PDF. At coincidence,

$$[V_0]=\frac12[P_{\xi,x}U] =\frac12\left[m^2+\left(\xi+\frac16\right)R\right],$$

where $[\Box\Delta_{\mathrm{vV}}^{1/2}]=R/6$ in the site’s curvature convention. This is a useful sign check: the site’s four-dimensional conformal value is $\xi=-1/6$.

For a merely smooth metric the formal series for $V$ need not converge. On this page, $H_\ell$ means an all-orders smooth representative obtained by a cutoff or Borel-type summation having the complete Hadamard coefficient jet at $s=0$; two admissible all-orders choices differ smoothly. If only finite regularity is needed, one may instead truncate deeply enough to define $H_\ell^{[N]}$ with $\omega_2-H_\ell^{[N]}\in C^N$. Such a finite-order statement must not be substituted for the $C^\infty$ remainder used in the Hadamard-state definition below. The singular data are therefore unambiguous even though a chosen smooth representative is not. The original Fulling–Sweeny–Wald result establishes propagation of Hadamard singularity structure for the real massless scalar anticommutator in its stated conventions Fulling, Sweeny, and Wald 1978, pp. 258–264; it is not being used here as the source of the general-$P_\xi$ Wightman normalization.

The logarithmic sector is present in the generic four-dimensional construction, but its coefficient can vanish. For the massless Minkowski field, $P_\xi U=0$ and the transport equations give $V_n=0$ for every $n$. By contrast, nonzero mass, nonconformal curvature coupling, or curvature tails generally produce $V\neq0$. Thus one may verify $V=0$ in a special model, but may not delete it from the general ansatz.

What the three terms know

Term	What fixes it	What to check
$U/s_\epsilon$	Local geometry; $U=\Delta_{\mathrm{vV}}^{1/2}$	Universal leading light-cone pole and $[U]=1$
$V\log(s_\epsilon/\ell^2)$	Local geometry and $P_\xi$ through transport	Subleading singularity; $V$ may vanish in special models, while $\ell$ records a smooth local convention
$W_{\omega,\ell}$	The state, after fixing the same parametrix and scale convention	Smoothness and scale shift; field equation and positivity belong to the completed $\omega_2$

Calling $W_{\omega,\ell}$ “the state-dependent part” always presumes a fixed choice of $V$, cutoff summation, and $\ell$. The invariant comparison is the smooth difference of two state two-point functions, not either smooth remainder in isolation.

What the length scale changes

The reference length $\ell>0$ makes the logarithm dimensionless. Replacing $\ell$ by $e^a\ell$ gives

$$V\log\!\left(\frac{s_\epsilon}{\ell^2}\right) \longmapsto V\log\!\left(\frac{s_\epsilon}{\ell^2}\right)-2aV.$$

For the same $\omega_2$, the smooth remainder must consequently transform as

$$W_{\omega,e^a\ell}=W_{\omega,\ell}+2aV.$$

This shift changes neither the wavefront set nor the Hadamard class. It does matter when an absolute local composite observable is assigned a renormalization prescription. Writing $\log s_\epsilon$ without displaying a scale is dimensionally incomplete and hides the convention, but it is not a new singularity or a failure of Hadamard regularity: an implicit choice of units is an implicit choice of $\ell$.

Smooth remainder versus Hadamard state

For any bidistribution $K$, the statement $K-H_\ell\in C^\infty$ is mathematically meaningful. It says that $K$ has the same local singularity as the chosen parametrix. It does not by itself say that $K$ is a physical two-point function. For a real scalar state, $\omega_2$ must also be

• a distributional bisolution: $P_{\xi,x}\omega_2=P_{\xi,x'}\omega_2=0$;
• Hermitian: $\overline{\omega_2(f,g)}=\omega_2(\bar g,\bar f)$;
• of positive type: $\omega_2(\bar f,f)\geq0$;
• normalized to the canonical antisymmetric part: $\omega_2(f,g)-\omega_2(g,f)=-iE(f,g)$.

In the quasifree sector these covariance data determine the state; for a general state they are necessary properties of its two-point function. A Hadamard state is a state satisfying those algebraic and field-equation conditions whose two-point function differs locally from $H_\ell$ by a smooth kernel. Radzikowski proves the local/global and positive-frequency wavefront-set equivalence in the original scalar setting and permits the mass term to be replaced by a smooth potential Radzikowski 1996, § 2, pp. 531–532; Definition 3.4, pp. 534–535; Theorem 5.1, pp. 544–550. The two-point characterization for a state that need not be quasifree is stated explicitly in Khavkine and Moretti 2015, Theorem 9, pp. 47–49 (Open PDF).

First application: a positive smooth rank-one excitation

Take the Minkowski vacuum $\omega_0$ for $P=\Box+m^2$. Let $u$ be a real smooth Klein–Gordon solution obtained from smooth compactly supported Cauchy data, and let $\lambda\geq0$. Shifting the field by the classical solution $\sqrt{\lambda}\,u$ produces a coherent state with

$$\omega_{\lambda,2}(x,x') =\omega_{0,2}(x,x')+\lambda u(x)u(x').$$

Every required check is explicit. The added kernel is a bisolution because $Pu=0$, it is symmetric so it does not change the commutator, and it is smooth. It is also positive:

$$\bigl(\omega_{\lambda,2}-\omega_{0,2}\bigr)(\bar f,f) =\lambda\left\lvert\int_M u(x)f(x)\,\mathrm d\mathrm{vol}_g(x)\right\rvert^2 \geq0.$$

Hence $\omega_\lambda$ and $\omega_0$ have identical $U$ and $V$ and are both Hadamard. With the same point-splitting prescription their Wick-square difference is immediately reproducible:

$$\langle\phi^2(x)\rangle_{\omega_\lambda,\mathrm{ren}} -\langle\phi^2(x)\rangle_{\omega_0,\mathrm{ren}} =\lambda u(x)^2.$$

Applying the same stress-tensor bidifferential operator before coincidence similarly gives $\lambda T^{\mathrm{cl}}_{\mu\nu}[u]$. All local geometric counterterms cancel in this state difference; an absolute value still requires a declared renormalization prescription.

Common pitfalls

Treating a smooth numerator as a smooth correction. Replacing $U$ by $U+F$ adds $F/s_\epsilon$. Unless $F$ vanishes to the required order and preserves the transport equation, this changes the singularity and prevents the parametrix equation from holding modulo smooth terms.

Deleting the logarithm by inspection. The correct test is the transport recursion. The massless Minkowski theory passes with $V=0$; a generic massive or curvature-coupled theory does not.

Confusing a hidden scale with insufficient regularity. An unstated $\ell$ is a dimensional and reproducibility defect. A wrong $U$, a wrong nonzero $V$, or a failed transport equation is a singularity defect.

Extending one geodesic formula through caustics. The displayed construction is valid in a convex normal neighborhood. Multiple geodesics and caustics require compatible local patches; the global statement is the wavefront-set criterion.

Exercises

1. Flat-space normalization and sign

Set $m=0$, $R=0$, $U=1$, $V=0$, and $T=t$ in Minkowski space. Recover the displayed vacuum Wightman function and explain why transposition gives the site’s $-iE$ antisymmetric part.

Solution

Here $s=\tfrac12(\lvert\Delta\mathbf x\rvert^2-\Delta t^2)$, so

$$s_\epsilon=\frac12\left(\lvert\Delta\mathbf x\rvert^2-(\Delta t-i\epsilon)^2\right).$$

Multiplying $1/s_\epsilon$ by $1/(8\pi^2)$ gives $1/(4\pi^2)$ times the standard positive-frequency denominator. Interchanging the arguments sends $\Delta t-i0$ to $-(\Delta t+i0)$ and reverses the light-cone boundary value. Its discontinuity is the Pauli–Jordan distribution; with $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ this is $-iE$.

2. The first transport checksum

Use the $V_0$ transport equation to compute $[V_0]$. What happens at the site’s massless conformal value $\xi=-1/6$?

Solution

At coincidence, $s^{;a}=0$ and $[\Box s]=-4$. Therefore

$$-2[V_0]=-[P_{\xi,x}U], \qquad [V_0]=\frac12[P_{\xi,x}U].$$

Using $[U]=1$ and $[\Box U]=R/6$ gives

$$[V_0]=\frac12\left[m^2+\left(\xi+\frac16\right)R\right].$$

For $m=0$ and $\xi=-1/6$, this coincidence value vanishes. That check alone does not prove that the full off-diagonal $V$ vanishes on a generic curved spacetime; the remaining transport equations must also be solved.

3. Verify the rank-one state comparison

For $K(x,x')=\lambda u(x)u(x')$, verify smoothness, the bisolution property, preservation of the antisymmetric part, and positivity.

Solution

Smoothness follows from $u\in C^\infty(M)$. Since $Pu=0$, both $P_xK$ and $P_{x'}K$ vanish. The kernel is symmetric, so $K(f,g)-K(g,f)=0$ and the original commutator is unchanged. Finally,

$$K(\bar f,f)=\lambda\left\lvert\int_M u f\,\mathrm d\mathrm{vol}_g\right\rvert^2\geq0.$$

Thus adding $K$ preserves every two-point state condition and, because $K$ is smooth, preserves the Hadamard class.

4. Change the logarithmic scale

Show that changing $\ell$ cannot change the difference of two Hadamard two-point functions for the same field operator and the same parametrix convention.

Solution

Under $\ell\mapsto e^a\ell$, the logarithmic term changes by $-2aV$ and each smooth remainder changes by $+2aV$. For two states $\omega$ and $\omega'$ the common shift cancels:

$$W_{\omega,e^a\ell}-W_{\omega',e^a\ell} =W_{\omega,\ell}-W_{\omega',\ell}.$$

Therefore $\omega_2-\omega'_2$ and every local observable difference computed with the same prescription are independent of the common parametrix scale.

Domain and failure conditions

The checkpoint $\omega_2-H_\ell$ smooth is a well-defined comparison for any bidistribution; it licenses a Hadamard state only when the field equation, Hermiticity, positivity, and canonical antisymmetric part also hold. Changing $U$, deleting a nonzero $V$ required by transport, or using coefficients for the wrong operator is a singularity defect. Failing to display $\ell$ instead leaves the singular class unchanged but makes the smooth convention and any absolute composite-observable prescription incomplete. The transport equations and the $U/V/W$ table above provide the direct checks for these distinct failures.

Other state tests are compared in Domain and failure conditions.

Handoffs

Hadamard Admissibility and the Two-Point Wavefront Criterion gives the global directional formulation. Absolute definitions of renormalized local observables belong to Renormalized Stress Tensor: Axioms and Curvature Ambiguities. The equivalence theorem and higher-dimensional variants continue in Hadamard States and Wavefront Characterization.

References

• 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.
• 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.
• Moretti, Valter. “Comments on the Stress-Energy Tensor Operator in Curved Spacetime.” Communications in Mathematical Physics 232 (2003): 189–221. 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.