Point Splitting and Wick Squares

The formal product $\Phi(x)^2$ is not an observable: the two-point function diverges as its arguments coincide. Point splitting repairs the product by keeping the points distinct, subtracting the universal Hadamard singularity, and only then taking coincidence. The result is finite for every Hadamard state and local in the background geometry, but it is not unique because equally admissible subtractions may differ by finite curvature terms.

Required background. Stress–Energy and Conservation provides the metric-variation definition and conservation logic used for local observables. Curvature supplies the Riemann and Ricci tensors entering the Hadamard coefficients.

Helpful background. Flat-Spacetime Quantum Field Theory defines ordinary normal ordering. Partial Differential Equations and Green Functions supplies parametrices and causal fundamental solutions. Causality and Global Structure develops convex normal neighborhoods. Quantum Fields in Curved Spacetime provides the volume-level progression from Hadamard states to renormalized observables.

The singularity before coincidence

Consider a real scalar field obeying

$$P\Phi=(\Box-m^2-\xi R)\Phi=0.$$

For a state $\omega$, its two-point distribution is $W_\omega(x,x')=\omega(\Phi(x)\Phi(x'))$. We take Synge’s world function $\sigma(x,x')$ to be one half of the signed squared geodesic distance and use the future-directed $i0$ prescription inherited from the Hadamard condition. In a convex normal neighborhood in four dimensions, a local parametrix may be written schematically as

$$H_\ell(x,x')=\frac{1}{8\pi^2} \left[ \frac{U(x,x')}{\sigma_+(x,x')} +V(x,x')\ln\!\left(\frac{\sigma_+(x,x')}{\ell^2}\right) \right].$$

Here $U$ and the asymptotic coefficients of $V$ are fixed recursively by the geometry, $P$, and the normalization $U(x,x)=1$; $\ell$ is an arbitrary reference length. Smooth additions to $H_\ell$ are deliberately not hidden in this notation. The all-orders Hadamard singularity has a smooth remainder; a finite truncation instead leaves a remainder of finite differentiability, and its order must suffice for the coincidence limit being taken. A Wick square needs fewer derivatives than a stress tensor. The singular coefficients are state independent even though the remainder contains the state dependence; see The Hadamard Parametrix for the finite-order construction.

Subtracting after setting $x'=x$ would be meaningless: both terms would already be divergent. The order “split, subtract, then coincide” is the essential renormalization step.

Deriving the Wick square

The expectation value of a provisional Wick square is

$$\omega\!\left(:\!\Phi^2\!:_\ell(x)\right) =\lim_{x'\to x}\left[W_\omega(x,x')-H_\ell(x,x')\right].$$

The limit is path independent because the bracketed biscalar is smooth. At the algebraic level the same subtraction defines a locally constructed field, not merely an expectation value in one representation. Smearing with a compactly supported test function gives a genuine algebra element.

This procedure differs from normal ordering relative to a chosen reference state $\omega_0$,

$$W_\omega(x,x')-W_{\omega_0}(x,x').$$

Reference-state normal ordering is finite when both states are Hadamard, but a preferred $\omega_0$ is generally unavailable and need not transform naturally under spacetime embeddings. The geometric parametrix avoids that dependence. Hollands and Wald 2001, §§2–4, PDF formulate locality and covariance precisely and show how the microlocal domain supports products of the resulting fields.

Two independent checks are immediate. In Minkowski spacetime, for the standard massless vacuum and the conventional flat parametrix, the smooth remainder vanishes and the Wick-square expectation is zero. More generally, for two Hadamard states $\omega$ and $\omega'$, subtraction cancels from their difference:

$$\omega(:\!\Phi^2\!:_\ell) -\omega'(:\!\Phi^2\!:_\ell) =\left[W_\omega-W_{\omega'}\right],$$

where brackets denote coincidence. The right-hand side is finite and independent of the chosen common subtraction scheme.

Why the finite answer is not unique

Locality and covariance do not select one finite part. In four dimensions, two admissible Wick squares for the same scalar field may differ by

$$:\!\Phi^2\!:'(x)-:\!\Phi^2\!:(x) =c_1m^2+c_2R(x),$$

times the identity, with dimensionless constants $c_1,c_2$ subject to the stated scaling convention. Changing $\ell$ shifts the smooth coincidence limit by a local expression derived from $V(x,x)$. This is renormalization freedom, not residual state dependence.

The distinction matters experimentally and conceptually. Differences between states at fixed background and scheme are unambiguous, whereas an absolute quoted value must state the length scale and finite curvature prescription. One may impose an additional normalization, such as vanishing in the Minkowski vacuum, but that condition alone cannot fix a curvature term that vanishes in flat spacetime.

The ambiguity is tightly constrained: an arbitrary function of position would violate locality, covariance, scaling, or smooth dependence on the metric. The full uniqueness theorem for higher Wick powers is developed on the next page; here the dimensional form already explains why “finite” does not mean “unique.”

A thermal value that can be calculated directly

In four-dimensional Minkowski spacetime, compare the massless thermal state at temperature $T=1/\beta$ with the vacuum. With both evaluated using the same subtraction, the Bose occupation factor $n_k=(e^{\beta k}-1)^{-1}$ gives

$$\begin{aligned} \langle\Phi^2\rangle_\beta-\langle\Phi^2\rangle_0 &=\int\frac{d^3\boldsymbol k}{(2\pi)^3}\frac{n_k}{k}\\ &=\frac{1}{2\pi^2\beta^2} \int_0^\infty\frac{x\,dx}{e^x-1} =\frac{T^2}{12}. \end{aligned}$$

Expanding $(e^x-1)^{-1}=\sum_{n\ge1}e^{-nx}$ evaluates the integral as $\sum_{n\ge1}n^{-2}=\pi^2/6$. This finite positive answer is a state difference, obtained without separately subtracting two divergent coincidence limits. It vanishes as $T\to0$ and has the mass dimension two required of a field square. With the massless Minkowski vacuum normalized to zero, it is also the absolute thermal Wick-square value.

Checks and limitations

Point splitting is local and therefore remains meaningful without a global mode decomposition. It does require a Hadamard state class: subtracting the Hadamard parametrix from a more singular state need not leave a smooth remainder. It also assumes a smooth globally hyperbolic spacetime without an untreated boundary. Boundary-induced singularities and image terms require a boundary-compatible state and subtraction analysis.

For derivatives of fields, one must differentiate the biscalar before coincidence and parallel-transport tensor indices to a common point. Those operations introduce further local terms. In particular, applying the classical stress-tensor bidifferential operator does not automatically produce a conserved quantum stress tensor; the conservation-restoring finite term is treated in Renormalized Stress–Energy Tensor.

Exercise

Let $\omega_1$ and $\omega_2$ be Hadamard states for the same scalar field and background. Show why their Wick-square difference is independent of both the singular parametrix and an allowed common shift $c_1m^2+c_2R$.

Solution

Write each expectation as $[W_i-H_\ell]+c_1m^2+c_2R$. Subtracting the two expressions cancels $H_\ell$ because the field equation and background are the same, and it cancels the finite geometric shift because the same renormalization prescription is used. The remainder is $[W_1-W_2]$, which exists because the difference of two Hadamard two-point functions is smooth. A comparison using different prescriptions would retain their finite local difference and would not be scheme independent.

References

• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI. Open PDF.
• Wald, Robert M. “The Back Reaction Effect in Particle Creation in Curved Spacetime.” Communications in Mathematical Physics 54 (1977): 1–19. DOI.