Free Wick Products and Point Splitting

For a free scalar field, normal ordering and point splitting define selected composites by subtracting a declared reference two-point function before restricting to the diagonal. The result is an operator-valued distribution on a common dense domain, not an ordinary pointwise operator. It depends on the free reference state or subtraction kernel, and defining one Wick polynomial does not automatically define products of several such polynomials, interacting composites, or a renormalized stress tensor.

Required background. Coincident Products and Contact Terms supplies the diagonal extension problem and contact-term language. Wick’s Theorem and Free Gaussian Factorization supplies contractions, normal ordering, and the Gaussian pairing rule.

Helpful background. Products, Scaling Degree, and Extensions of Singular Distributions supplies the extension criterion used to diagnose products of the Wick polynomials defined here.

A reference two-point function defines the subtraction

Let $\phi$ be a free real scalar field and let $\omega$ be a selected centered quasifree Hadamard reference state: $\omega(\phi)=0$, its two-point distribution has the standard local free-field singular structure, and the difference between any two allowed references is smooth near the diagonal. This is what “admissible reference” means below. Write

$$W_\omega(x,y)=\omega\!\left(\phi(x)\phi(y)\right).$$

The smooth-difference property and its role in defining normal products are stated in Hollands and Wald 2001, § 2.1, PDF p. 9.

The reference-normal-ordered bilocal product is

$$N_\omega(x,y) =:\!\phi(x)\phi(y)\!:_\omega \equiv \phi(x)\phi(y)-W_\omega(x,y)\mathbf1.$$

This identity is distributional. Both terms are paired with a two-variable test function before subtraction. In the Minkowski vacuum, it agrees with the familiar creation–annihilation rule that moves creators to the left. Schwartz derives that operator normal ordering and the free Wick expansion in Schwartz 2014, §§ 7.A.1–7.A.2, pp. 100–103.

The ordering fixes the kernel. For the same reference state, define

$$D_{F,\omega}(x,y) =\omega\!\left(\mathrm T\{\phi(x)\phi(y)\}\right).$$

At separated points,

$$\mathrm T\{\phi(x)\phi(y)\} =N_\omega(x,y)+D_{F,\omega}(x,y)\mathbf1.$$

Thus an unordered product is paired with $W_\omega$, while a time-ordered product is paired with $D_{F,\omega}$. Copying one kernel into the other formula changes the distribution being defined.

Point splitting constructs the Wick square

Choose a real test function $f\in C_c^\infty(M)$ and a symmetric normalized profile $\rho\in C_c^\infty(\mathbb R^d)$ with

$$\rho(r)=\rho(-r), \qquad \int\mathrm d^d r\,\rho(r)=1.$$

In local Minkowski coordinates, set

$$F_\epsilon(x,y) =f\!\left(\frac{x+y}{2}\right) \epsilon^{-d}\rho\!\left(\frac{x-y}{\epsilon}\right).$$

The family $F_\epsilon$ converges formally to $f\,\delta_\Delta$, where $\delta_\Delta$ is supported on the diagonal. Concentration alone would not authorize pairing it with an arbitrary distribution. Here it is used as an admissible representative of the diagonal pullback: matrix elements on the microlocal domain of smoothness satisfy the wavefront condition, and the limit is taken in a topology in which that pullback is continuous. In the Minkowski vacuum, this domain may be taken as test-function-generated finite-particle vectors with smooth mode functions. The free Wick square relative to $\omega$ is therefore the weak operator-valued-distribution limit

$$:\!\phi^2\!:_\omega(f) =\lim_{\epsilon\downarrow0} \int\mathrm d^d x\,\mathrm d^d y\, F_\epsilon(x,y)N_\omega(x,y).$$

The limit is tested in matrix elements on that common invariant dense smooth domain. It is not an operator-norm limit. Within the admissible class it is independent of the approximate profile because it represents the unique diagonal pullback. The subtraction is chosen so that the remaining bilocal normal product admits that restriction in this free setting; the unsubtracted $\phi(x)\phi(y)$ generally does not.

The diagonal smearing, dense domain, and reference-dependent normal products are constructed in Hollands and Wald 2001, § 2.2, PDF pp. 10–15.

In the usual shorthand,

$$:\!\phi^2\!:_\omega(x) =\lim_{y\to x} \left[ \phi(x)\phi(y)-W_\omega(x,y)\mathbf1 \right],$$

but the smeared formula above is the definition. The arrow $y\to x$ does not specify an ordinary pathwise limit.

Three checks fix the construction

First, the reference expectation vanishes:

$$\omega\!\left(:\!\phi^2\!:_\omega(f)\right)=0.$$

Second, in any other admissible free state $\omega'$ for which $W_{\omega'}-W_\omega$ is smooth near the diagonal,

$$\omega'\!\left(:\!\phi^2\!:_\omega(x)\right) =\left[W_{\omega'}(x,y)-W_\omega(x,y)\right]_{y=x}.$$

Third, Wick factorization gives, at separated points,

$$\omega\!\left( :\!\phi^2\!:_\omega(x) :\!\phi^2\!:_\omega(y) \right) =2W_\omega(x,y)^2,$$

and in time order

$$\omega\!\left( \mathrm T\{:\!\phi^2\!:_\omega(x) :\!\phi^2\!:_\omega(y)\}\right) =2D_{F,\omega}(x,y)^2.$$

The factor $2$ counts the two cross-pairings. These formulas expose two different limitations. For a Hadamard $W_\omega$, the one-sided wavefront set makes $W_\omega(x,y)^2$ a canonical bidistribution, but its further pullback to the diagonal $x=y$ generally fails. The time-ordered square $D_{F,\omega}^2$ is instead defined off the diagonal and requires an extension through it, with local contact ambiguity. Normal ordering each factor has solved neither same-center operation.

Changing the reference changes the local composite

Let $\omega$ and $\omega'$ be admissible free quasifree references whose two-point difference is smooth, and define

$$d_{\omega\omega'}(x) =\left[W_\omega(x,y)-W_{\omega'}(x,y)\right]_{y=x}.$$

Subtracting the two definitions gives the exact change-of-reference law

$$:\!\phi^2\!:_{\omega'}(x) =:\!\phi^2\!:_\omega(x) +d_{\omega\omega'}(x)\mathbf1.$$

The sign follows directly: the primed definition subtracts $W_{\omega'}$, so relative to the unprimed definition it adds $W_\omega-W_{\omega'}$. For higher powers the Gaussian combinatorics gives

$$:\!\phi^n\!:_{\omega'} =\sum_{k=0}^{\lfloor n/2\rfloor} \frac{n!}{2^k k!(n-2k)!} d_{\omega\omega'}^k :\!\phi^{n-2k}\!:_\omega.$$

For example,

$$:\!\phi^4\!:_{\omega'} =:\!\phi^4\!:_\omega +6d_{\omega\omega'}:\!\phi^2\!:_\omega +3d_{\omega\omega'}^2\mathbf1.$$

Reference dependence is therefore local and controlled, but it is real. Vacuum normal ordering is not a state-independent definition.

The reference-change isomorphism and its Gaussian coefficients are given in Hollands and Wald 2001, § 2.2, PDF pp. 10–15.

Wick products are distributions with domains

The notation $:\!\phi^n\!:(x)$ suppresses four pieces of data:

Suppressed datum	Why it matters
reference two-point function	determines which contractions are subtracted
ordering	selects Wightman, Feynman, Euclidean, or contour kernels
test function and topology	gives meaning to the diagonal limit
operator domain	controls products and matrix elements of unbounded operators

On the declared test-function-generated smooth domain, a smeared free Wick polynomial maps smooth finite-particle vectors to finite-particle vectors, though it need not be bounded or self-adjoint on that domain without further analysis. A real test function and formal Hermiticity are not by themselves a complete self-adjointness theorem.

Derivatives can be treated by differentiating the bilocal field and subtraction kernel before taking the diagonal restriction. That procedure can define selected free composites such as $:\!\partial_\mu\phi\partial_\nu\phi\!:_\omega$. It must retain derivative contacts and the same domain qualifications; it is not permission to manipulate singular point fields algebraically.

Why this is not the interacting prescription

The free construction uses a c-number two-point function and Gaussian contractions. In an interacting theory:

• a composite can mix with every local operator allowed by symmetries and dimension;
• the subtraction generally depends on a regulator, scheme, and scale;
• products of renormalized insertions need additional collision counterterms;
• anomalous dimensions and renormalization-group evolution enter;
• normal ordering with respect to a free vacuum is not invariant under a change of interacting description.

Curved spacetime adds a separate issue: there may be no preferred vacuum. A locally covariant Wick polynomial is built from geometric short-distance data and admits curvature-dependent finite ambiguities; subtracting one state’s complete two-point function is generally not the local-covariant prescription. Stress tensors add conservation, improvement, anomaly, and curvature constraints beyond the scalar Wick square.

The failure of state normal ordering to supply the local-covariant stress-tensor prescription, and the replacement by local Hadamard-parametrix Wick powers, are developed in Hollands and Wald 2001, § 3, PDF pp. 18–20; § 5.2, PDF pp. 34–36.

These limitations do not make free Wick products useless. They provide exact free composites, organize perturbative calculations, and expose which parts of a later renormalized definition are genuinely new.

Common mistakes with point splitting

“Subtract the divergence.” A subtraction must name a distribution, ordering, state or local parametrix, and finite part. The phrase alone does not define an operator.

“Take the points together along any path.” The invariant object is a diagonal restriction or a smeared limiting family. A coordinate path can hide direction dependence and does not specify the distribution topology.

“Normal ordering makes every later product finite.” It removes contractions internal to the declared normal product. Collisions between two normal products can still require new extensions.

“Zero reference expectation means zero operator.” The Wick square has zero expectation in its reference state but nonzero fluctuations and nonzero matrix elements.

“All reference changes are harmless constants.” For $:\!\phi^2\!:$ the change is a local c-number; higher powers mix with lower Wick powers, and derivatives can produce nonconstant local terms.

Check your understanding

Check 1: recover the change-of-reference sign

Write both definitions as $\phi^2-W\mathbf1$. Subtracting $W_{\omega'}$ instead of $W_\omega$ adds $W_\omega-W_{\omega'}$, so $:\!\phi^2\!:_{\omega'}=:\!\phi^2\!:_\omega+d_{\omega\omega'}\mathbf1$.

Check 2: count the Wick-square pairings

Each field at $x$ must contract with one field at $y$. There are $2!$ assignments, giving $2W_\omega(x,y)^2$ for the ordered Wightman product and $2D_{F,\omega}(x,y)^2$ after time ordering.

Check 3: distinguish two collision problems

Defining one Wick square removes the internal $x_1\to x_2$ singularity inside that composite. For two Wightman-ordered Wick squares, the cross-contraction square is a bidistribution but its same-center pullback can fail. For their time-ordered product, extending the off-diagonal Feynman square carries a local ambiguity. The first definition supplies neither second operation.

Check 4: identify the kernel from the ordering

For the unordered bilocal product subtract $W_\omega(x,y)$. For the time-ordered product subtract $D_{F,\omega}(x,y)$. Both leave the same bilocal normal product at separated points, but the contractions encode different orderings.

Continue beyond the free subtraction

• Free-Field OPE Preview uses these Wick composites in a bounded free short-distance expansion.
• Renormalized Composite-Operator Insertions develops interacting definitions, mixing, scheme and scale dependence, and anomalous dimensions.
• Wick Polynomials and Hadamard Point Splitting treats Hadamard subtraction and locally covariant products in curved spacetime.
• Local Covariant Wick Powers and Operator Products develops theorem-level constructions and finite ambiguities.

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, arXiv:gr-qc/0103074.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.