Hadamard Parametrix and Short-Distance Structure

The Hadamard parametrix isolates the universal short-distance singularity of a free-field two-point function. Its singular coefficients are fixed locally by the metric and the Klein–Gordon operator; the smooth remainder carries the state dependence. This separation is why two admissible states can give different local expectation values while sharing the ultraviolet structure required for point-splitting.

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 local geometric form

Work in a geodesically convex neighborhood in four spacetime dimensions. With the global $(+---)$ metric, Synge’s world function $\sigma(x,x')$ is one half the signed squared geodesic distance and is positive for nearby timelike separation. It is convenient to put $s=-\sigma$. If $T$ is any local time function, define the boundary value

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

For $P_\xi=\Box+m^2+\xi R$, with the site’s signed $\xi$ convention, a Hadamard two-point function has the local form

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

The sign of the pole follows from using $s=-\sigma$; references that use spacelike-positive $\sigma$ display the same formula without this crosswalk. The time orientation in $s_\epsilon$ is fixed so that

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

The leading coefficient is $U=\Delta_{\mathrm{vV}}^{1/2}$, the square root of the van Vleck–Morette determinant. The smooth coefficient $V$ has a local covariant series whose transport recursion is fixed by $P_\xi$. The smooth $W_\omega$ is constrained by the bisolution and positivity conditions but is not fixed by geometry. In even dimensions the logarithm is essential; the power and logarithmic structures change with dimension. The geometric singular coefficients and their state-independent character are established in Fulling, Sweeny, and Wald 1978, pp. 257–264.

What the length scale changes

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

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

This is a smooth local shift and can be absorbed into $W_\omega$. It does not alter the wavefront set or the Hadamard class, but it matters when a local composite observable is assigned a renormalization prescription. Omitting $\ell$ is therefore dimensionally incomplete; changing it is not a change of state singularity.

First application: subtract two states

Let $\omega$ and $\omega'$ be scalar Hadamard states for the same operator and time orientation. In the same convex neighborhood their $U$ and $V$ terms agree, so

$$\omega_2(x,x')-\omega'_2(x,x') =\frac{1}{8\pi^2}\bigl(W_\omega(x,x')-W_{\omega'}(x,x')\bigr)$$

is smooth. Its coincidence limit is therefore well defined, and derivatives may be taken before coincidence. This is the basic state-comparison calculation behind differences of Wick squares and stress tensors. It does not by itself choose an absolute subtraction prescription.

Adversarial modifications

Two tempting alterations fail for different reasons.

Change the leading coefficient. Replacing $U$ by $U+F$ with generic smooth $F$ changes $F/s_\epsilon$, hence changes the singularity and generally prevents $P_\xi\omega_2=0$ modulo smooth terms. A smooth numerator is not a smooth change when divided by $s_\epsilon$.

Suppress the logarithmic data. For a massive or curvature-coupled four-dimensional field, setting $V=0$ generally violates the transport recursion. Writing a logarithm without $\ell$ also conceals a dimensionful ambiguity. Either way, the strongest surviving claim is merely a formal short-distance ansatz, not a Hadamard parametrix for the declared operator.

The local series is asymptotic and valid inside a convex normal neighborhood. Caustics and multiple geodesics require patching; the global condition is better stated microlocally through the equivalent wavefront characterization Radzikowski 1996, Theorem 5.1.

Exercise

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

Solution

Both parametrices acquire the same smooth shift $-2aV/(8\pi^2)$. It cancels in $\omega_2-\omega'_2$. Only the difference $W_\omega-W_{\omega'}$ remains, so state differences are independent of the common parametrix scale.

Domain and failure conditions

The parametrix occupies the central ultraviolet box in the construction map. The checkpoint $\omega_2-H$ smooth is meaningful only after positivity and the canonical antisymmetric part have made $\omega_2$ a state kernel; the smooth remainder then records state dependence without changing the universal singular terms.

Hadamard subtraction compares a state kernel with the geometric parametrix; smoothness of the difference licenses ultraviolet admissibility, not uniqueness. Schematic; not to scale.

Changing $U$, deleting a required logarithmic term, or hiding its length scale enters the insufficient-regularity branch of the failure map. The resulting expression may resemble the local form but is not a parametrix for the declared operator.

The transport equations and dimensionful logarithmic scale are part of the domain; omitting them removes the claimed Hadamard construction. Schematic; not to scale.

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.
• 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.