Strongly continuous scalar one-particle Poincare action

The affine pullback $f_g(x)=f({\mathrm{\Lambda }}^{-1}(x-a))$ preserves Schwartz space. Repeated chain rule [F6] expresses each derivative as a finite linear combination of derivatives of $f$ at ${\mathrm{\Lambda }}^{-1}(x-a)$. Also $1+\mathrm{\mid }x\mathrm{\mid }\le C(1+\mathrm{\mid }{\mathrm{\Lambda }}^{-1}(x-a)\mathrm{\mid })$ for a finite constant $C$ depending on $a,\mathrm{\Lambda }$. Thus every polynomially weighted derivative remains bounded by the corresponding finite sum of Schwartz seminorms.

Multiplication by $e^{ip\circ a}$ preserves the $H_{\mu }$ norm. Composition with ${\mathrm{\Lambda }}^{-1}$ also preserves that norm by [F2]; it preserves null-equivalence classes for the same reason. Therefore $U_{\mu }(g)$ is a well-defined isometry. Its inverse is the corresponding formula for $g^{-1}$, so it is unitary. Direct substitution and Lorentz invariance of $p\circ x$ give $U_{\mu }(a,\mathrm{\Lambda })U_{\mu }(b,\mathrm{\Gamma })=U_{\mu }(a+\mathrm{\Lambda }b,\mathrm{\Lambda }\mathrm{\Gamma })$.

In the Fourier integral for $K_{\mu }{f}_g$, change variables $x=a+\mathrm{\Lambda }y$; [F5] gives $\mathrm{\mid }\det \mathrm{\Lambda }\mathrm{\mid }=1$ and [F7] justifies the change of variables. The integrand is absolutely integrable because $f$ is Schwartz; apply [F7] to the positive and negative parts of its real and imaginary components to obtain this change of variables for the complex integral. Lorentz invariance of the phase then gives $K_{\mu }{f}_g(p)=e^{ip\circ a}{K}_{\mu }f({\mathrm{\Lambda }}^{-1}p)\mathrm{.}$ Both sides belong to $H_{\mu }$ by [F4], so this equality is also equality of Hilbert-space vectors. This is the action formula used in step 1.2.

Let $h\in C_c^{\mathrm{\infty }}$ in the shell chart, with support away from the massless apex. If $g\to g_0$ in the Poincare group, $U_{\mu }(g)h$ converges pointwise to $U_{\mu }(g_0)h$. In a compact neighborhood of $g_0$, the supports lie in one compact chart subset $C$ away from the apex; ${\nu }_{\mu }(C)<\mathrm{\infty }$ by [F2]. The squared pointwise difference is bounded by $4\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}^2{\mathbf{1}}_C$, so [F8] gives norm continuity for this dense family. Density follows from [F3]. For arbitrary $v\in H_{\mu }$, choose such $h$ with $\mathrm{\parallel }v-h\mathrm{\parallel }<ϵ$; unitarity gives $\mathrm{\parallel }{U}_{\mu }(g)v-U_{\mu }(g_0)v\mathrm{\parallel }\le 2ϵ+\mathrm{\parallel }{U}_{\mu }(g)h-U_{\mu }(g_0)h\mathrm{\parallel }\mathrm{.}$ First take $g\to g_0$ and then $ϵ\downarrow 0$. This proves strong continuity on all of $H_{\mu }$.