Wightman Functions and Spectral Support

Vacuum expectation values turn the operator-valued Wightman axioms into a hierarchy of scalar tempered distributions. Their translation invariance, Lorentz covariance, positive type, spectral support, adjoint relation, and local permutation identities retain exactly the information needed for reconstruction. The spectrum condition appears most sharply as support of Fourier transforms in products of forward cones.

Required background. Wightman fields, domains, and axioms supplies the operator formulation; positivity, spectrum, covariance, and locality distinguishes the hypotheses; and tempered distributions and Fourier calculus fixes the distributional transform rules.

Helpful background. Spectral decomposition of two-point functions gives the physical reading of the two-point measure, and Poincaré covariance and the spectrum condition gives the geometric interpretation.

Vacuum distributions

For a scalar Wightman field, define

$$W_n(f_1\otimes\cdots\otimes f_n) =\langle\Omega,\phi(f_1)\cdots\phi(f_n)\Omega\rangle, \qquad W_0=1.$$

Because every smeared field preserves the common domain and its matrix elements are tempered, the Schwartz kernel theorem gives $W_n\in\mathcal S'(M^n)$. These are distributions, not functions required to have pointwise values. Translation invariance makes them depend on $n-1$ relative coordinates. With

$$\xi_j=x_j-x_{j+1},\qquad W_n(x_1,\ldots,x_n)=w_n(\xi_1,\ldots,\xi_{n-1}),$$

and the Fourier convention $\widehat w(q)=\int e^{iq\cdot\xi}w(\xi)\,\mathrm d^{4(n-1)}\xi$, the spectrum condition implies

$$\operatorname{supp}\widehat w_n \subseteq(\overline V_+)^{n-1}.$$

This relative-momentum statement is equivalent to a condition on partial sums of the $n$ full momenta, together with total momentum conservation. It follows by inserting the joint spectral measure of $P^\mu$ between adjacent fields. The sign of the cone reverses if the Fourier sign or the definition of $\xi_j$ is reversed; a cone statement without conventions is incomplete.

To see the equivalence explicitly, Fourier-transform all $x_i$ with the same $e^{+ip_i\cdot x_i}$ convention. Translation invariance gives $\sum_{i=1}^n p_i=0$. Rewriting

$$\sum_{i=1}^n p_i\cdot x_i =\sum_{j=1}^{n-1}q_j\cdot\xi_j, \qquad q_j=\sum_{i=1}^{j}p_i,$$

shows that the relative variables are precisely the partial momentum sums. The support condition is therefore $q_j\in\overline V_+$ for every cut between the $j$th and $(j+1)$st field. This “energy flowing through every cut” picture is a useful check on signs and on proposed momentum-space correlators.

Positivity, covariance, and locality

Let $\underline f=(f_0,f_1,\ldots,f_N)$ with $f_n\in\mathcal S(M^n)$, and let the involution reverse the arguments and complex-conjugate:

$$f_n^*(x_1,\ldots,x_n)=\overline{f_n(x_n,\ldots,x_1)}.$$

Hilbert-space positivity becomes the Wightman positivity inequality

$$\sum_{j,k=0}^{N}W_{j+k}(f_j^*\otimes f_k)\geq0.$$

It is much stronger than $W_2(\bar f\otimes f)\geq0$: it constrains the entire hierarchy and all polynomial states. Hermiticity gives $W_n(x_1,\ldots,x_n)^*=W_n(x_n,\ldots,x_1)$ in distributional form. Poincaré covariance acts simultaneously on all arguments. Locality says that adjacent arguments may be exchanged, with the relevant statistics sign, after smearing in a region where their difference is spacelike. These properties and the reconstruction conditions are stated together in Streater and Wightman 2016, §§ 3-3–3-4, pp. 106–131.

The free scalar two-point distribution

This calculation is the rigorous counterpart of the spectral decomposition of two-point functions.

For the massive free scalar,

$$W_2(x-y)=\Delta_+(x-y;m^2) =\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}} e^{-ip\cdot(x-y)}, \qquad p^0=E_{\mathbf p}>0.$$

Consequently,

$$\widehat W_2(p)=2\pi\,\theta(p^0)\delta(p^2-m^2),$$

up to the displayed Fourier-normalization convention. Its support is the positive-energy mass shell, which is Lorentz invariant under the proper orthochronous group and lies in $\overline V_+$. Positivity follows directly:

$$W_2(\bar f\otimes f) =\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}} \left|\widetilde f(E_{\mathbf p},\mathbf p)\right|^2\geq0.$$

The antisymmetric part is $i\Delta_m(x-y)$, whose support lies in the causal cone, so the commutator vanishes at spacelike separation. This single example displays three different notions that should not be conflated: mass-shell support, positivity of a quadratic form, and causal support of the commutator.

For higher free-field functions, Wick’s theorem builds every term from such positive-frequency two-point distributions. The partial-sum support condition survives even though a single Wick term may route momenta differently: the operator derivation with spectral projections proves the support of the complete distribution without relying on a diagrammatic routing choice.

Replace $\theta(p^0)\delta(p^2-m^2)$ by the symmetric measure $\delta(p^2-m^2)$. Lorentz invariance under the full Lorentz group remains, but negative-energy support appears. The resulting distribution cannot be a vacuum two-point function of a positive-energy translation representation. This adversarial modification shows why Lorentz invariance does not imply the spectrum condition.

What the hierarchy does and does not determine

If the vacuum is cyclic, all inner products of polynomial field states are values of the $W_n$. Hence the complete hierarchy determines the cyclic field representation up to unitary equivalence. A two-point function alone does not determine an interacting theory: distinct higher functions can share it, and an arbitrary choice of $W_4$ may violate positivity or locality. Gaussianity is an additional condition, expressed by Wick pairings of all even functions and vanishing odd functions.

As an independent support check, insert a spectral projection $E(dp)$ between two fields. Since $E$ is supported in $\overline V_+$, each relative Fourier variable is future directed. Conversely, the commutator contains the difference between positive- and negative-frequency boundary values, so its causal support is not the same statement as one-sided spectral support.

Exercises

Verify directly that the free two-point distribution is of positive type.

Solution

Smear the first argument with $\bar f$ and the second with $f$. Fourier transformation restricts both factors to the positive mass shell and produces the modulus square shown above. The measure $\mathrm d^3\mathbf p/((2\pi)^3 2E_{\mathbf p})$ is positive, so the integral is nonnegative. No pointwise interpretation of $W_2$ is used.

References

• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.
• Wightman, Arthur S. 1956. “Quantum Field Theory in Terms of Vacuum Expectation Values.” Physical Review 101: 860–866. DOI.