Wick Polynomials under Microlocal Conditions

Wick powers become operator-valued distributions when their coincident contractions are subtracted with Hadamard singular data and the remaining products satisfy the microlocal cone criterion. Normal ordering relative to a Hadamard state gives a valid representation-dependent construction; replacing its singular part by the local Hadamard parametrix exposes the state-independent local field.

Required background. Wavefront-set products, pullbacks, and pushforwards supplies the diagonal criterion; Hadamard states and the wavefront-set characterization supplies the universal singularity.

Helpful background. Local and microcausal functionals with Peierls brackets gives the functional domain; higher-point microlocal spectrum conditions controls products of the new fields.

Normal ordering and coincident limits

Fix a quasifree Hadamard state $\omega$ with two-point function $\omega_2$. The formal generating relation

$$:e^{i\phi(f)}:_\omega= \exp\!\left(\frac12\omega_2(f,f)\right)e^{i\phi(f)}$$

encodes normal-ordered powers after differentiation in $f$. At distinct points, Wick’s theorem subtracts every internal contraction by $\omega_2$. To obtain a field at one point, approximate the diagonal and prove convergence in a distribution space whose cone excludes the conormal directions that would obstruct pullback.

For the Wick square, a state-independent local formula is

$$:\!\phi^2\!:_H(f)= \int_M f(x)\lim_{y\to x} \big(\phi(x)\phi(y)-H(x,y)\mathbf1\big)d\mathrm{vol}_g(x),$$

where the limit denotes a microlocally justified point-split extension and $H$ is a local Hadamard parametrix. The difference $\omega_2-H$ is smooth near the diagonal, so in a Hadamard state

$$\omega(:\!\phi^2\!:_H(x))=[\omega_2-H](x,x)$$

is a smooth function. Higher Wick powers are obtained by the same contraction combinatorics. Brunetti, Fredenhagen, and Köhler prove existence of the diagonal limits for auxiliary Wick monomials and then that the resulting monomials are Wightman fields on a common dense invariant domain; Brunetti, Fredenhagen, and Köhler 1996, Proposition 5.3 and Theorem 5.7, pp. 15–18.

Comparing two Hadamard prescriptions

Let $\omega$ and $\omega'$ be Hadamard states. Their two-point difference

$$d(x,y)=\omega'_2(x,y)-\omega_2(x,y)$$

is a smooth symmetric bisolution. Wick’s combinatorial identity gives

$$:\!\phi^n\!:_{\omega'}= \sum_{j=0}^{\lfloor n/2\rfloor} \frac{n!}{(n-2j)!,j!,2^j} [-d(x,x)]^j:\!\phi^{n-2j}\!:_{\omega},$$

with the sign determined by which prescription is expressed in terms of the other. Every coefficient is smooth. For $n=2$, the expectation-value difference is simply

$$\omega(:\!\phi^2\!:_H)-\omega'(:\!\phi^2\!:_H) =[\omega_2-\omega'_2](x,x),$$

a smooth function. This is the concrete result used in Wick polynomials and Hadamard point splitting.

The construction also preserves the commutator rule

$$[ :\!\phi^n\!:(x),\phi(y) ]=inE(x,y):\!\phi^{n-1}\!:(x),$$

in the distributional sense. This provides an independent check on the factor $n$ and on the sign of the subtraction. In the flat vacuum it reduces to ordinary Fock normal ordering.

To see why the point-split notation denotes an operator-valued distribution rather than an illicit pointwise operator, smear first with a test function $f(x)$ and a family $\rho_\epsilon(x,y)$ approaching the diagonal. Form the separated-point, contraction-subtracted quadratic expression on the common finite-particle domain, and take $\epsilon\to0$ in matrix elements. The wavefront bounds give a distributional limit independent of the chosen approximating family. Repeating the argument for products of the resulting fields requires the enlarged microlocal domain: functional derivatives may not have covectors all in the closed future cone or all in the closed past cone. This “microcausal” exclusion is precisely what allows contractions with the causal propagator while retaining sequential continuity.

There are three different conclusions here. Existence says the smeared composite is well defined on a common invariant dense domain. The microlocal spectrum bound controls its matrix elements and products. Local covariance is stronger and is obtained only after replacing the state-dependent subtraction by geometrically specified Hadamard data and imposing compatibility across embeddings. None of these statements says that the unsmeared symbol $\phi(x)^2$ is an operator at a point in the Hilbert-space sense.

Adversarial non-Hadamard subtraction

Normal order against a two-point function with an additional wrong-oriented singularity. Its difference from $H$ is no longer smooth. Restricting the residual kernel to the diagonal can meet the diagonal conormal bundle, so the displayed coincident limit is undefined. Writing two divergent kernels with a formal minus sign does not prove their singular directions cancel.

The strongest surviving statement may be a normal-ordered expression at separated points or after additional smearing; it is not a local Wick field. Conversely, the Hadamard product criterion ensures existence of these free-field composites but does not classify their locally covariant finite curvature terms, construct time-ordered products, or select a physical state.

Exercises

1. Fourth power. Express $:\!\phi^4\!:_{\omega'}$ through $\omega$ and $d(x,x)$.

Solution

With the sign convention above, it is $:\!\phi^4\!:_\omega-6d(x,x):\!\phi^2\!:_\omega+3d(x,x)^2\mathbf1$. The coefficients count one and two pair contractions.

2. State difference. Why can a smooth $d(x,y)$ always be restricted to the diagonal?

Solution

Its wavefront set is empty, so it cannot intersect the diagonal conormal bundle. The ordinary pullback theorem therefore gives the smooth function $d(x,x)$.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Michael Köhler. “The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes.” Communications in Mathematical Physics 180 (1996): 633–652. DOI. Open PDF.
• 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.