The Hadamard Parametrix

The Hadamard parametrix isolates the short-distance singularity that every physically admissible two-point function of a free scalar field must share. Its coefficients are fixed locally by the metric and the Klein–Gordon operator; the choice of state enters only through a smooth remainder. Deriving this split makes precise the statement that ultraviolet admissibility is local and geometric without pretending that geometry selects a unique state.

Required background. Partial Differential Equations and Green Functions supplies normally hyperbolic operators, distributional fundamental solutions, and transport along characteristics. The bounded local bridge used below is the eikonal identity for Synge’s world function and the action of the Klein–Gordon operator on a singular biscalar inside one convex normal neighborhood.

Helpful background. Flat-Spacetime Quantum Field Theory provides the Minkowski positive-frequency two-point function, Causality and Global Structure defines time orientation and global time functions, and Quantum Fields in Curved Spacetime provides the inherited sign and field-equation conventions.

World function and boundary-value prescription

Work in four spacetime dimensions with signature $(-,+,+,+)$ and

$$P=\Box-m^2-\xi R.$$

Let $\mathcal O$ be a geodesically convex normal neighborhood. For $x,x'\in\mathcal O$, Synge’s world function is one half of the signed squared geodesic interval,

$$\sigma(x,x')=\frac12 s(x,x')^2, \qquad \sigma_{;a}\sigma^{;a}=2\sigma.$$

Thus in Minkowski coordinates $\sigma=[-\Delta t^2+\lvert\Delta\boldsymbol x\rvert^2]/2$. Choose any smooth time function $T$ increasing toward the future and put $\tau=T(x)-T(x')$. With the positive-frequency convention $e^{-i\omega t}$, define

$$\sigma_\epsilon=\sigma+i\epsilon\tau+\frac12\epsilon^2, \qquad \epsilon>0,$$

and take the distributional boundary value as $\epsilon\downarrow0$. Replacing $T$ by another compatible time function does not change that boundary value. Reversing the sign of the $i\epsilon\tau$ term would reverse the frequency orientation, so the prescription is physical information rather than decoration.

For an arbitrary length scale $\ell$, the local singular kernel is written

$$H^{(N)}_{\ell,\epsilon}(x,x') =\frac{1}{8\pi^2} \left[ \frac{U(x,x')}{\sigma_\epsilon} +\sum_{n=0}^{N}v_n(x,x')\sigma^n \log\!\left(\frac{\sigma_\epsilon}{\ell^2}\right) \right].$$

The prefactor $1/(8\pi^2)$ follows from using the half-squared world function. Authors who call the full squared interval “$\sigma$” instead write $1/(4\pi^2)$. Declaring both conventions prevents a spurious factor-of-two disagreement.

The leading transport equation

Away from the diagonal, apply $P_x$ to $U/\sigma$. The identity $\sigma_{;a}\sigma^{;a}=2\sigma$ gives

$$P_x\!\left(\frac{U}{\sigma}\right) =\frac{P_xU}{\sigma} +\frac{-2\sigma^{;a}\nabla_aU+(4-\Box\sigma)U}{\sigma^2}.$$

Eliminating the unwanted $\sigma^{-2}$ term yields the first transport equation,

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

where square brackets denote coincidence $x\to x'$. Its unique solution in $\mathcal O$ is

$$U(x,x')=\Delta(x,x')^{1/2},$$

with $\Delta$ the van Vleck–Morette determinant,

$$\Delta(x,x')= \frac{-\det[-\sigma_{;ab'}(x,x')]} {\sqrt{-g(x)}\sqrt{-g(x')}}.$$

The sign makes $\Delta=1$ in Minkowski coordinates. This coefficient measures geodesic focusing: it compares the geodesic separation map with its flat-space value. It depends on local geometry but not on the state, boundary conditions outside $\mathcal O$, or a choice of modes. The restriction to a convex normal neighborhood avoids caustics and the ambiguity of choosing among several connecting geodesics.

Logarithmic coefficients and recursion

For a smooth biscalar $V$, direct differentiation gives

$$P_x(V\log\sigma) =(P_xV)\log\sigma +\frac{2\sigma^{;a}\nabla_aV+(\Box\sigma-2)V}{\sigma}.$$

Cancellation of the remaining $1/\sigma$ term determines $v_0$:

$$2\sigma^{;a}\nabla_av_0+(\Box\sigma-2)v_0=-P_xU.$$

At coincidence, $[\Box\sigma]=4$ and $[\Box\Delta^{1/2}]=R/6$, so

$$[v_0]=\frac12\left[m^2+\left(\xi-\frac16\right)R\right].$$

Expanding $V\sim\sum_{n\geq0}v_n\sigma^n$ and cancelling successive logarithmic terms gives, for $n\geq0$,

$$(n+1)\left[ 2\sigma^{;a}\nabla_a+\Box\sigma+2n \right]v_{n+1}=-P_xv_n.$$

Each equation is an ordinary transport problem along the unique geodesic joining $x'$ to $x$. The coincidence data are fixed by regularity, so $U$ and every $v_n$ are determined by $g$, $m$, and $\xi$. The derivation and normalization are equivalent to the local form displayed in Hollands and Wald 2015, arXiv v2, §2.1, pp. 14–15, PDF, after accounting for their full-squared-interval convention.

On a merely smooth spacetime the infinite series need not converge. The correct statement uses each finite truncation: increasing $N$ makes the error under $P_x$ progressively smoother. Treating the series as a convergent global Green function would add hypotheses that the local construction does not possess.

Universal singularity and state dependence

A Hadamard two-point distribution $W_{2,\omega}$ has, in every convex normal neighborhood, the form

$$W_{2,\omega} =\lim_{\epsilon\downarrow0}H^{(N)}_{\ell,\epsilon} +R_{N,\omega},$$

where $R_{N,\omega}$ has the differentiability required at the chosen truncation order. Equivalently, after comparing all orders, two Hadamard two-point functions differ by a smooth bisolution. This smooth difference carries state information such as occupation numbers, long-distance correlations, and boundary data.

Changing $\ell$ also changes the displayed remainder by a smooth, locally determined term. It is a subtraction-scale freedom, not a change in the wavefront singularity. The universal/state-dependent split is therefore: the singular equivalence class of $H$ is geometry-fixed, while the smooth completion is state-dependent. Radzikowski’s equivalence theorem identifies this local Hadamard form with the corresponding positive-frequency wavefront condition under the Klein–Gordon and commutator hypotheses; see Radzikowski 1996, Theorem 5.1, p. 544.

The construction is local. It does not prove that a positive global state with this singularity exists, and it does not choose among such states. Those questions require global hyperbolicity, propagation, positivity, and sometimes extra symmetry.

Minkowski normalization check

For $R=0$, $m=0$, and $\xi=0$ in Minkowski spacetime, $\Delta=1$ and all $v_n$ vanish. Therefore

$$H_\epsilon(x,x') =\frac{1}{8\pi^2\sigma_\epsilon} =\frac{1}{4\pi^2} \frac{1}{\lvert\Delta\boldsymbol x\rvert^2-(\Delta t-i\epsilon)^2}.$$

This is the standard positive-frequency massless two-point boundary value for the $e^{-i\omega t}$ convention. It checks the prefactor, the normalization of $\sigma$, and the sign of the $i\epsilon$ prescription simultaneously. For $m\neq0$, the coincidence value $[v_0]=m^2/2$ reproduces the leading logarithm in the small-separation expansion of the massive Minkowski two-point function.

For an explicit massive check, separate the points by a spacelike distance $r$. Then $W_m(r)=mK_1(mr)/(4\pi^2r)$, and the small-argument Bessel expansion gives

$$W_m(r)=\frac{1}{4\pi^2r^2} +\frac{m^2}{8\pi^2} \left[\log\frac{mr}{2}+\gamma_E-\frac12\right] +O(r^2\log r).$$

Since $\sigma=r^2/2$, the parametrix logarithm is $m^2\log(\sigma/\ell^2)/(16\pi^2)$, with exactly the same coefficient of $\log r$. The finite constant belongs to the smooth remainder and depends on $\ell$. This separates a normalization test from a choice of finite subtraction.

Check your understanding

Suppose an author uses the full signed squared interval $s^2=2\sigma$ and writes the leading singularity as $C/(s^2+i0\,\Delta t)$. Determine $C$ by comparison with the formula above, and explain what changes if the sign of the boundary-value term is reversed.

Solution

Since $1/(8\pi^2\sigma)=1/(4\pi^2s^2)$, the coefficient is $C=1/(4\pi^2)$, with the infinitesimal terms rescaled consistently when $s^2=2\sigma$ is used. This comparison must be made before taking the boundary-value limit; otherwise a factor hidden in the definition of the regulator can be mistaken for a normalization change. Reversing the sign of $i0\,\Delta t$ complex-conjugates the massless boundary value and swaps the future-directed and past-directed frequency orientations. It is not a harmless redefinition of $C$: it changes which argument of the two-point function carries the future covector and would conflict with the chosen $e^{-i\omega t}$ convention. The singular support stays on the null cone in either case, which is why checking only where the distribution is singular cannot detect the error; its cotangent orientation must also be checked.

References

• 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. 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, no. 3 (1996): 529–553. doi:10.1007/BF02100096.