Tempered time-ordered scalar two-point carrier

For an integer $M\ge 2$, integration by parts in $\mathbf{x}$ gives $(1+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2{)}^M{\widehat{\varphi }}_t(\mathbf{k})=\widehat{(1-{\mathrm{\Delta }}_{\mathbf{x}}{)}^M\varphi (t,\cdot )}(\mathbf{k})\mathrm{.}$ Every boundary term vanishes because $\varphi$ is Schwartz. For any fixed $L>3$, Schwartz seminorms bound ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }(1-{\mathrm{\Delta }}_{\mathbf{x}}{)}^M\varphi (t,\mathbf{x})\mathrm{\mid }{d}^3\mathbf{x}\le C_M(\varphi )(1+\mathrm{\mid }t\mathrm{\mid }{)}^{-2}{\int }_{{\mathbb{R}}^3}(1+\mathrm{\mid }\mathbf{x}\mathrm{\mid }{)}^{-L}{d}^3\mathbf{x},$ so $\mathrm{\mid }{\widehat{\varphi }}_t(-\mathbf{k})\mathrm{\mid }\le C_M^{\mathrm{\prime }}(\varphi )(1+\mathrm{\mid }t\mathrm{\mid }{)}^{-2}(1+\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2{)}^{-M}$. The $t$ bound is integrable. In radial $\mathbf{k}$ coordinates the shell weight is at most a constant times $rdr$ because ${\omega }_{\mu }(r)\ge r$; hence its product with $(1+r^2{)}^{-M}$ is integrable at infinity and at $r=0$, including $\mu =0$. Thus the defining integral converges absolutely and is bounded by a finite Schwartz seminorm, proving that $H_{\mu }$ is tempered.

For each $\omega >0$, let $g_{\omega }(t)=e^{-i\omega \mathrm{\mid }t\mathrm{\mid }}\mathrm{/}(2\omega )$. It is continuous at $0$, with $g_{\omega }^{\mathrm{\prime }}(0+)=-i\mathrm{/}2$ and $g_{\omega }^{\mathrm{\prime }}(0-)=+i\mathrm{/}2$. For a compactly supported smooth test $\eta (t)$, integrating by parts separately on $(-\mathrm{\infty },0)$ and $(0,\mathrm{\infty })$ cancels the boundary terms involving ${\eta }^{\mathrm{\prime }}(0)$ and leaves the derivative jump times $\eta (0)$; since $g_{\omega }^{\mathrm{\prime }\mathrm{\prime }}=-{\omega }^2{g}_{\omega }$ off $0$, this proves $g_{\omega }^{\mathrm{\prime }\mathrm{\prime }}+{\omega }^2{g}_{\omega }=-i{\delta }_0$ as a one-dimensional distribution.

By [F4, F5] and Fubini's theorem [F7], the positive-frequency two-point kernel is $W_{\mu }(t,\mathbf{x})={\int }_{{\mathbb{R}}^3}\frac{d^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mu }(\mathbf{k})}{e}^{-i{\omega }_{\mu }(\mathbf{k})t+i\mathbf{k}\cdot \mathbf{x}}$ as a distribution. This is the slice of $H_{\mu }$ for $t>0$. For $t<0$, replacing $\mathbf{k}$ by $-\mathbf{k}$ identifies its slice with $W_{\mu }(-t,-\mathbf{x});$ the same substitution proves $H_{\mu }(t,\mathbf{x})=H_{\mu }(-t,-\mathbf{x})$. For real tests, the vacuum expectation of the field commutator is $W_{\mu }(f,g)-W_{\mu }(g,f)$; [F6] identifies it with a distribution supported in the causal cone. Thus the two Wightman orderings agree on spacelike-separated supports. For a compactly supported smooth spatial test supported away from $\mathbf{x}=0$, both slice pairings are continuous in $t$ by [F9] and the integrable bound in 1.1, so no additional term supported at $t=0$ occurs away from the coincidence point. The partial Fourier formula itself specifies the extension at coincidence.

The spatial Fourier multiplier of $\mathrm{\square }-{\mu }^2$ is $-{\mathrm{\partial }}_t^2-\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2-{\mu }^2=-({\mathrm{\partial }}_t^2+{\omega }_{\mu }(\mathbf{k}{)}^2)$. Applying it to the kernel in 2.1 and using 1.2 gives $i{\delta }_0(t)$ times $(2\pi {)}^{-3}\int e^{i\mathbf{k}\cdot \mathbf{x}}{d}^3\mathbf{k}$. Fourier inversion [F8] identifies the latter as ${\delta }_0(\mathbf{x})$. For $\mu =0$ the excluded apex has measure zero and the shell weight is locally integrable there, as established in 1.1; it contributes no additional distribution. Therefore $(\mathrm{\square }-{\mu }^2)H_{\mu }=i{\delta }_0$ in four dimensions.