Smeared free scalar field and its Poincare representation

Since $K_{\mu }f,K_{\mu }\bar{f}\in H_{\mu }$ by [F9], the field formula is defined on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. The first term $a(K_{\mu }\bar{f})$ is complex-linear in $f$ because complex conjugation is followed by the conjugate-linear dependence of $a$; the second is linear because both $a^{†}$ and $K_{\mu }$ are linear. Both terms preserve ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ by [F1,F2]. For real $f$, the shell vectors coincide, so ${\mathrm{\Phi }}_{\mu }(f)=a(K_{\mu }f)+a^{†}(K_{\mu }f)=\sqrt{2}{\mathrm{\Phi }}_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(K_{\mu }f){\mathrm{\mid }}_{{\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}}$. It is symmetric on the common finite-particle domain by [F2]. Substituting $h=\sqrt{2}{K}_{\mu }f$ and $k=\sqrt{2}{K}_{\mu }g$ into [F3]'s Weyl relation gives the stated scalar phase, and its vacuum Gaussian gives $⟨\mathrm{\Omega },W_{\mu }(f)\mathrm{\Omega }{⟩}_2=e^{-\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_2^2\mathrm{/}2}$.

For $g=(a,\mathrm{\Lambda })$, [F10] gives the one-particle representation and $U_{\mu }(g)K_{\mu }f=K_{\mu }{f}_g$. Since the affine action is real, $U_{\mu }(g)K_{\mu }\bar{f}=K_{\mu }\overline{f_g}$. The Fock lift [F7] intertwines creation and annihilation, so the field formula in step 1.1 gives ${\mathrm{\Gamma }}_s(U_{\mu }(g)){\mathrm{\Phi }}_{\mu }(f){\mathrm{\Gamma }}_s(U_{\mu }(g){)}^{-1}={\mathrm{\Phi }}_{\mu }(f_g)$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. The explicit Segal Weyl formula [F3] and the same lift give bounded Weyl covariance. Strong continuity of the Fock lifts follows from [F7] and [F10].

Since $a(K_{\mu }\bar{f})\mathrm{\Omega }=0$ and $⟨\mathrm{\Omega },a^{†}(h)\mathrm{\Psi }{⟩}_2=0$, the vacuum one-point function vanishes. Expanding the two fields, only $a(K_{\mu }\bar{f})a^{†}(K_{\mu }g)\mathrm{\Omega }$ contributes to the vacuum pairing; CCR [F2] makes that value $⟨K_{\mu }\bar{f},K_{\mu }g{⟩}_2$. Taking the first test to be $\bar{f}$ and the second $f$ gives $\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}^2\ge 0$; for real $f$ this is the ordinary diagonal. For an ordered product of $n$ real smeared fields, write the first field as creation plus annihilation. The creation part vanishes against the left vacuum; commute the annihilation part successively past each later field using [F2]. Each commutator is the corresponding two-point pairing, and the remaining annihilator kills the right vacuum. This yields the recursive sum over its possible partner and proves the Wick pairing rule by induction on $n$.

Cauchy–Schwarz gives $\mathrm{\parallel }{C}_h\mathrm{\parallel }\le \mathrm{\parallel }h\mathrm{\parallel }$ for $C_h(v)=⟨h,v{⟩}_2$. The tensor-extension bound and orthogonal symmetrizer in [F11] therefore give $\mathrm{\parallel }{a}_n(h)\mathrm{\parallel }\le \sqrt{n}\mathrm{\parallel }h\mathrm{\parallel }$ and $\mathrm{\parallel }{a}_n^{†}(h)\mathrm{\parallel }\le \sqrt{n+1}\mathrm{\parallel }h\mathrm{\parallel }$, the latter also using $\mathrm{\parallel }h\otimes v\mathrm{\parallel }=\mathrm{\parallel }h\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ from the product tensor pairing. Let $\mathrm{\Theta }$ have sectors only through $N$. These bounds and orthogonality of distinct output sectors imply $\mathrm{\parallel }{\mathrm{\Phi }}_{\mu }(f)\mathrm{\Theta }\mathrm{\parallel }\le (\sqrt{N}\mathrm{\parallel }{K}_{\mu }\bar{f}\mathrm{\parallel }+\sqrt{N+1}\mathrm{\parallel }{K}_{\mu }f\mathrm{\parallel })\mathrm{\parallel }\mathrm{\Theta }\mathrm{\parallel }$. Continuity of $K_{\mu }$ [F9] and conjugation [F12], followed by Hilbert Cauchy–Schwarz, makes every stated matrix coefficient continuous; complex linearity is step 1.1. Mutual adjoints [F2] give $⟨{\mathrm{\Phi }}_{\mu }(f)\mathrm{\Psi },\mathrm{\Theta }{⟩}_2=⟨\mathrm{\Psi },{\mathrm{\Phi }}_{\mu }(\bar{f})\mathrm{\Theta }{⟩}_2$ for finite-particle vectors. Thus every such $\mathrm{\Theta }$ lies in the adjoint domain with the asserted restricted adjoint action. Finite-particle invariance permits reversing every finite polynomial product by this identity, so for $\omega (B)=⟨\mathrm{\Omega },B\mathrm{\Omega }{⟩}_2$ one has $\omega (B^{\ast }B)=\mathrm{\parallel }B\mathrm{\Omega }{\mathrm{\parallel }}^2\ge 0$ and $\omega (I)=1$.

For a Schwartz test $\phi (z)$, define $G_{\phi }(p)=\int \phi (z)e^{-ip\circ z}{d}^{4}z$; for $F(x,y)\in \mathcal{S}({\mathbb{R}}^8)$ define $G_F(p)=\int F(x,y)e^{-ip\circ x}{e}^{ip\circ y}{d}^{4}x{d}^{4}y$. By [F12], these are continuous Schwartz transforms, the second followed by the injective linear frequency restriction $L$ determined by the displayed phases. Injectivity in finite dimensions gives $\mathrm{\mid }Lp\mathrm{\mid }\ge c\mathrm{\mid }p\mathrm{\mid }$ for some $c>0$; each derivative of a restricted transform is a fixed finite linear combination of its ambient derivatives at $Lp$. These bounds control every weighted restricted derivative by finitely many ambient Schwartz seminorms, proving the claimed restriction continuity and decay. In either case $\mathrm{\mid }G(p)\mathrm{\mid }\le q_4(G)(1+\mathrm{\mid }p\mathrm{\mid }{)}^{-4}$, with $q_4$ a continuous Schwartz seminorm. The shell integral of $(1+\mathrm{\mid }p\mathrm{\mid }{)}^{-4}$ is finite by [F4]: its radial behavior is $O(r)$ at the massless apex, $O(r^2)$ near zero when $\mu >0$, and $O(r^{-3})$ at infinity. Therefore the integrals of $G_{\phi }$ and $G_F$ against $d{\nu }_{\mu }$ define continuous complex-linear tempered distributions. On product tests $F(x,y)=f(x)g(y)$ the inner integral is $\overline{K_{\mu }\stackrel{‾}{f}(p)}{K}_{\mu }g(p)$; its shell integral is exactly the covariance from step 2.2. This proves the relative and bidistribution interfaces without assigning point operators.

The one-particle translation phase is $e^{i({\omega }_{\mu }(\mathbf{k})a^0-\mathbf{k}\cdot \mathbf{a})}$, so its real generator vector is $\varpi (\mathbf{k})=({\omega }_{\mu }(\mathbf{k}),-\mathbf{k})$. It is finite and measurable. The measure is sigma-finite by [F5], so [F8] applies and gives the joint PVM and vacuum at zero. On an $n$-particle sector the joint multiplier is ${\sum }_{r=1}^n({\omega }_{\mu }({\mathbf{k}}_r),-{\mathbf{k}}_r)$. Since $\mathrm{\mid }{\sum }_r{\mathbf{k}}_r\mathrm{\mid }\le {\sum }_r\mathrm{\mid }{\mathbf{k}}_r\mathrm{\mid }\le {\sum }_r{\omega }_{\mu }({\mathbf{k}}_r)$, every such value lies in the closed future cone; the vacuum value does too. Thus the joint spectral measure is supported there, as claimed.