All-order scalar causal extension and finite renormalization — Examples

Define the positive-frequency massless distribution by $w(t,\mathbf{x})={\int }_{{\mathbb{R}}^3}\frac{e^{-i\mathrm{\mid }\mathbf{k}\mathrm{\mid }t+i\mathbf{k}\cdot \mathbf{x}}}{2\mathrm{\mid }\mathbf{k}\mathrm{\mid }(2\pi {)}^3}{d}^3\mathbf{k}\mathrm{.}$ This is a distribution: pairing a compact smooth spacetime test gives its rapidly decreasing Fourier transform on the null mass shell, and $\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^{-1}{d}^3\mathbf{k}$ is integrable at zero and has polynomial growth at infinity. The mass-shell relation gives $Pw=0$. The measure is the restriction of $\theta (k^0)\delta ((k^0{)}^2-\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2)d^{4}k$, hence is invariant under the proper orthochronous Lorentz group; its reflection gives $w^{\ast }=w(-t,-\mathbf{x})$. Scaling $\mathbf{k}$ shows that $w$ is nonzero homogeneous of degree $-2$. With a damping factor $e^{-ϵ\mathrm{\mid }\mathbf{k}\mathrm{\mid }}$, angular integration and the elementary Laplace integral give $w_{ϵ}(t,\mathbf{x})=\frac{1}{4{\pi }^2}\frac{1}{\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2-(t-iϵ{)}^2}\mathrm{.}$ The damped spectral integrals converge to $w$ on tests by the preceding absolute Fourier bound.

This datum has precisely the required wavefront condition. At any nonzero point of the null cone, the preceding formula is, in a normal coordinate to that cone, a nonvanishing smooth multiple of a one-sided boundary value $(s\pm i0{)}^{-1}$. Its Laplace representation has Fourier support in one half-line and nonzero constant magnitude there; convolution with a compact cutoff therefore gives precisely that one-sided conormal direction. The sign selected by $t-iϵ$ makes the covector future null under the contract's metric-dual convention. Away from the cone the formula is smooth. At the origin, convolution of the null-shell measure with the rapidly decreasing transform of a test cutoff decreases arbitrarily fast in every cone separated from the future null cone: on that cone the frequency difference is bounded below by a fixed positive multiple of the sum of the two frequency magnitudes, and the mass-shell measure has polynomial growth. Thus no other origin directions occur. Every future null origin direction does occur, by closedness of the wavefront set and approach along nonzero null points with that conormal direction. In the two-point lift $W(x,y)=w(x-y)$ the covectors are $(\xi ,-\xi )$ by [F4], giving exactly the specified relation. The boundary-value difference $w-w(-t,-\mathbf{x})$ is supported on the null cone, so $E=(W-W^{\mathsf{T}})\mathrm{/}i$ has causal support. This verifies the free $W$ conditions in [F1].

Define the time-ordered distribution by its spatial Fourier representation $h_F(t,\mathbf{x})={\int }_{{\mathbb{R}}^3}\frac{e^{-i\mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{\mid }t\mathrm{\mid }+i\mathbf{k}\cdot \mathbf{x}}}{2\mathrm{\mid }\mathbf{k}\mathrm{\mid }(2\pi {)}^3}{d}^3\mathbf{k}\mathrm{.}$ It acts continuously on tests by first taking their spatial Fourier transform, which decreases rapidly uniformly on compact time intervals; the same polynomial measure bound as in step 1.1 is integrable after this decay. It is symmetric under $(t,\mathbf{x})\mapsto (-t,-\mathbf{x})$, equals the required ordering of $w$, and is homogeneous of degree $-2$, hence has scaling degree $2$. Off zero, time ordering and step 2.1 give exactly the permitted Feynman wavefront directions. At zero every relative covector is allowed by [F1]. Lorentz transformations preserve its off-zero ordering, since time order is preserved on causal pairs and the two orderings agree on spacelike pairs; the transformed distribution is another degree-$2$ extension of the same punctured kernel. Uniqueness from [F3] gives Lorentz invariance globally. Translation invariance of $H_F(x,y)=h_F(x-y)$ is immediate. Thus $(W,H_F)$ is an admissible free datum, with all its stated hypotheses checked.

For a fixed spatial frequency $\mathbf{k}\mathrm{\ne }0$, $e^{-i\mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{\mid }t\mathrm{\mid }}\mathrm{/}(2\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ is continuous at $t=0$, and its first derivative has jump $-i$. Twice differentiating it as a one-dimensional distribution therefore gives $({\mathrm{\partial }}_t^2+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2)\frac{e^{-i\mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{\mid }t\mathrm{\mid }}}{2\mathrm{\mid }\mathbf{k}\mathrm{\mid }}=-i\delta (t)\mathrm{.}$ The integrable small-frequency bound and the rapidly decreasing spatial test transform justify this differentiation inside the pairing; the value at the single frequency zero is immaterial. Fourier inversion [F4] yields $P{h}_F=-i{\delta }_0$ in four spacetime dimensions. Its punctured restriction is zero, of degree $-\mathrm{\infty }$, while the forced contact term has degree $4$. Two undifferentiated linear vertices have a contraction of degree $2<4$, whose unique degree-preserving extension is $h_F$. Replacing the first vertex $\int (Pf)\varphi$ by the equal functional $\int fP\varphi$ in [F2] forces its differentiated contraction to be $P{h}_F=-i{\delta }_0$. Requiring instead that the zero punctured differentiated coefficient extend with degree $-\mathrm{\infty }$ forces zero, giving different values for the same input functional. This refutes the Statement. The engineering bound is $2+2=4$, which permits precisely this contact term and is compatible with the corrected contract.

The graph in the Example has two distinct vertices and two parallel internal edges, so its numerical relative kernel is the product of the two Euclidean edge kernels, $C_0(z{)}^2=u(z)$. By [F2], this punctured distribution has scaling degree $4$; the causal extension theorem [F1] supplies the separately stated Lorentzian graph construction for admissible scalar free data at the same formal order. The Euclidean calculation uses [F2] and makes no identification of the Euclidean kernel with that Lorentzian graph.

Here the relative dimension is $4$ and the scaling degree is also $4$. Thus $\lfloor 4-4\rfloor =0$, so [F4] says every two same-degree extensions differ by $c{\delta }_0$ for a scalar $c\in \mathbb{C}$; no derivative of ${\delta }_0$ is allowed. This proves the local ambiguity claim without identifying the Euclidean kernel with $H_F$.

Applying the exact Euclidean calculation in [F2] to the chosen scale parameters gives $T_{\mu }-T_{\nu }=(8{\pi }^2{)}^{-1}\log (\mu \mathrm{/}\nu ){\delta }_0$. Since the difference is supported at $z=0$ and is proportional to ${\delta }_0$, it is local. The calculation concerns only the stated Euclidean extension family, so it supplies no Lorentzian amplitude or measured probability.