Local Covariant Wick Powers and Operator Products

A local Wick power must be the same construction in every spacetime, not merely a well-defined normal ordering in one Hilbert-space representation. Local covariance, almost homogeneous scaling, smooth and analytic background dependence, the commutator rule, and microlocal regularity reduce the ambiguity to finite local curvature polynomials.

Required background. Wick polynomials under microlocal conditions supplies the composites; locally covariant QFT as a functor supplies naturality.

Helpful background. Stress–energy response and background variation supplies metric dependence; equivalence, uniqueness, and comparison notions distinguishes prescription equality; free-field OPE preview, curved-space OPE and local short-distance expansions, Wick polynomials and Hadamard point splitting, and renormalized currents and charge density give applications.

Axioms for a prescription across spacetimes

For each globally hyperbolic background $(M,g,m^2,\xi)$, a Wick power $\Phi^k_M(f)$ belongs to the extended free-field algebra. If $\psi:M\to N$ is an admissible isometric embedding preserving orientation and time orientation, local covariance requires

$$\alpha_\psi(\Phi^k_M(f))=\Phi^k_N(\psi_*f).$$

The field is local because it depends only on the germ of the background near $\operatorname{supp}f$. It is Hermitian for real $f$, satisfies the expected commutator with the basic field, depends smoothly and analytically on admissible metric and parameter families, and scales with engineering dimension $k$ up to controlled logarithms under $g\mapsto\lambda^{-2}g$ and $m^2\mapsto\lambda^2m^2$. These conditions exclude arbitrary state functions and arbitrary coordinate functions.

Suppose $\Phi^k$ and $\widetilde\Phi^k$ are two prescriptions satisfying the axioms. Hollands and Wald prove

$$\widetilde\Phi^k=\Phi^k+ \sum_{i=0}^{k-2}\binom{k}{i}C_{k-i}\Phi^i,$$

where each $C_j$ is a local covariant scalar polynomial in the metric, curvature, their derivatives, $m^2$, and $\xi$, with the correct scaling dimension and analytic parameter dependence. This is Hollands and Wald 2001, Theorem 5.1, pp. 30–34. The theorem is a classification under all stated axioms; dropping locality or scaling enlarges the answer drastically.

The four-dimensional Wick square

In four dimensions $\phi$ has dimension one, so a scalar c-number correction to $\phi^2$ must have dimension two. Local covariance and polynomial dependence leave

$$\widetilde{\phi^2}(x)=\phi^2(x)+ \big(c_1m^2+c_2R(x)\big)\mathbf1,$$

with dimensionless real constants that may have the permitted analytic dependence on $\xi$. Terms such as $R^2$, $\Box R$, or $m^{-2}R^2$ have the wrong dimension or violate the polynomial/analytic hypotheses. The two allowed terms are precisely the mass and scalar-curvature mixing recorded in curvature counterterms and composite-operator mixing.

One way to construct a prescription is to subtract a local Hadamard parametrix. Its length scale produces an allowed curvature polynomial when changed. The uniqueness theorem then proves that every other prescription satisfying the same axioms differs only by the displayed finite terms; it does not say their numerical coefficients are fixed without renormalization conditions.

An independent check evaluates on flat spacetime. There $R=0$, so only $c_1m^2$ remains; in the massless flat theory no scalar correction of dimension two remains. A purported universal term that survives this check without another scale violates the scaling axiom.

Why must the difference be a curvature polynomial rather than an arbitrary covariant functional? The commutator axiom makes the highest field-dependent part of two prescriptions agree, so their difference is triangular in lower Wick powers. Locality and smooth dependence imply that each coefficient depends on only a finite jet of the metric and background parameters at the point. Covariance turns those jets into tensorial curvature combinations, while scaling and analyticity leave only finitely many monomials of the required engineering dimension. The proof therefore combines algebraic recursion with a local differential-invariant classification; dimensional analysis alone would miss the finite-jet and analytic restrictions.

For operator products, this distinction also separates an asymptotic local expansion from an equality of operators at separated points. Local coefficients can mix under a change of Wick prescription, with compensating changes of the composite basis. The allowed finite redefinition preserves local covariance but does not imply term-by-term invariance of a truncated operator-product expansion.

Adversarial coordinate subtraction

Choose a chart and define $\widetilde{\phi^2}(x)=\phi^2(x)+x^0\mathbf1$. Although this is a smooth local-looking field in that chart, an isometric embedding or a change of coordinates does not carry $x^0$ as a scalar built from the background. The naturality square fails. Replacing $x^0$ by an arbitrary scalar function does not help unless that function is part of the declared background category.

The converse boundary matters: covariance and dimensional analysis classify possible local shifts, but do not determine their constants, prove convergence of an OPE, or establish a physical renormalization condition. A state-dependent normal ordering can be perfectly meaningful in one representation while failing to be locally covariant.

Exercises

1. Dimension four. Why is $R$ allowed in the Wick square ambiguity but $R^2$ is not?

Solution

$R$ has engineering dimension two, matching $\phi^2$ in four dimensions. $R^2$ has dimension four and would require an inverse mass or another scale, excluded by the polynomial analytic scaling assumptions.

2. Massless flat limit. What ambiguity survives for $m=0$ and $R=0$?

Solution

Neither allowed scalar remains, so the Wick square is fixed within this class. This does not eliminate ambiguities in higher powers or time-ordered products.

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.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI. Open PDF.