Positive chiral current and exponential test-coordinate model

Use $\mathcal{D}(\mathbb{R};\mathbb{C})=C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ with the fixed-support Fr'echet seminorms and locally convex inductive-limit topology defined in Positive smooth damping preserves the double-pole boundary. A distribution is a continuous complex-linear functional on this test space, paired bilinearly. Write ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ for the algebraic direct sum of the symmetric $n$-particle sectors in Schwartz/distribution spaces and Hilbert Fock conventions; it is dense and invariant under the creation and annihilation maps by Bosonic and fermionic Fock operators with exact domains. A linear map $T$ from compact smooth tests to operators on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ is an operator-valued distribution on this domain when, for every fixed $\mathrm{\Psi },\mathrm{\Xi }\in {\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, the coefficient $f\mapsto ⟨\mathrm{\Psi },T(f)\mathrm{\Xi }{⟩}_2$ is a distribution in the preceding sense. The same Fock definition gives ${\mathcal{F}}_s(\mathcal{H})=\{({\mathrm{\Psi }}_n{)}_{n\ge 0}:{\mathrm{\Psi }}_n\in {\mathcal{H}}_s^{(n)},{\sum }_{n\ge 0}\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}$ and vacuum $\mathrm{\Omega }=(1,0,0,\dots )$. The closed Segal field of Self-adjoint Segal fields and explicit Weyl unitaries is the closure of ${\varphi }_0(h)=2^{-1\mathrm{/}2}(a^{†}(h)+a(h))$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, and that supplier proves the Weyl group $t\mapsto W(th)$ has generator $i\mathrm{\Phi }(h)$.

For $f,g\in \mathcal{D}(\mathbb{R};\mathbb{C})$ set ${\phi }_{f,g}(s):={\int }_{\mathbb{R}}\overline{f(x+s)}g(x)dx\mathrm{.}$ A distribution $T$ is of positive type in this translation-invariant two-point convention when $⟨T,{\phi }_{f,f}⟩\in [0,\mathrm{\infty })$ for every such $f$. This is positivity of the tested covariance form, not positivity of an operator.

For the current, the vacuum two-point form means $W(f,g):=⟨\mathrm{\Omega },J(\bar{f})J(g)\mathrm{\Omega }{⟩}_2;$ a distribution $C$ is its covariance kernel in the convention above when $W(f,g)=⟨C,{\phi }_{f,g}⟩$ for all compact smooth $f,g$.

For $w\in \mathbb{C}$, define $\sinh w:=\frac{e^w-e^{-w}}{2},\cosh w:=\frac{e^w+e^{-w}}{2},$ using the complex exponential of The complex exponential by its power series, which extends the real exponential by $\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential, and define $\mathrm{csch}w:=1\mathrm{/}\sinh w$ where $\sinh w\mathrm{\ne }0$. Their derivatives are ${\sinh }^{\mathrm{\prime }}w=\cosh w$ and ${\cosh }^{\mathrm{\prime }}w=\sinh w$ by The complex exponential is entire and its complex derivative is itself and the chain rule. For real $x$, $\cosh x>0$ by positivity of the real exponential (The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$). In this item, for $b>0$ with units of inverse length and a length variable $s$, set $Y_b(s):=\frac{\sinh (bs)}{b}\mathrm{.}$ This is a smooth increasing diffeomorphism of $\mathbb{R}$: the real exponential series The real exponential function and the number $e$ by a power series gives $e^t\ge 1+t$ for $t\ge 0$, and reciprocal positivity The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$ gives $e^{-t}\le 1$ there. Thus $\sinh t\ge t\mathrm{/}2$ for $t\ge 0$, while $\sinh (-t)=-\sinh t$; in particular $\sinh t>0$ for $t>0$ and $\sinh t\to \pm \mathrm{\infty }$ as $t\to \pm \mathrm{\infty }$. Also, $\sinh v-\sinh u=2\cosh \text{\hspace{0.17em}⁣}\left(\frac{u+v}{2}\right)\sinh \text{\hspace{0.17em}⁣}\left(\frac{v-u}{2}\right)>0(v>u)\mathrm{.}$ Continuity and the intermediate-value property for real continuous functions (also used in the proof of The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\mathrm{\infty })$) give its range as $\mathbb{R}$. If $H_b=Y_b^{-1}$, monotonicity gives continuity of $H_b$, and the difference quotient at $y=Y_b(s)$ gives $H_b^{\mathrm{\prime }}(y)=\frac{1}{Y_b^{\mathrm{\prime }}(H_b(y))}=\frac{1}{\cosh (b{H}_b(y))}\mathrm{.}$ The right-hand side is smooth; induction with the chain rule shows $H_b$ is $C^{\mathrm{\infty }}$, which is the inverse regularity used in the test-coordinate pullback below.

The real logarithm is also $C^{\mathrm{\infty }}$ on $(0,\mathrm{\infty })$: its supplied derivative is $1\mathrm{/}x$ The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, and direct algebra gives ${\lim }_{h\to 0}\frac{(x+h{)}^{-n}-x^{-n}}{h}=-n{x}^{-n-1}(x>0,n\ge 1)\mathrm{.}$ Induction therefore differentiates ${\log }^{\mathrm{\prime }}(x)=x^{-1}$ to every order. Hence $U_{\kappa }$ and $u_{\kappa }$ in the Statement are smooth inverse coordinates.

The one-dimensional chiral model here is exactly the single positive-frequency sector $k\in (0,\mathrm{\infty })$ in $\mathcal{H}=L^2((0,\mathrm{\infty }),dk;\mathbb{C})$, with no negative-frequency sector; $k$ is Fourier-dual to the length coordinate $U$. With dimensionless compact tests, $j_{\kappa }$ is a model density per unit length, $C_{\kappa }$ has units of inverse length squared, and its Fourier spectrum has units of inverse length; these are mathematical model units, not empirical assignments.

Assume Countable Choice. Let $\mathcal{H}=L^2((0,\mathrm{\infty }),dk;\mathbb{C}),⟨h,q{⟩}_2={\int }_0^{\mathrm{\infty }}\overline{h(k)}q(k)dk,$ using the second-variable-linear convention. For $f\in \mathcal{D}(\mathbb{R};\mathbb{C})=C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ define $h_f(k)=\sqrt{\frac{k}{4\pi }}{\int }_{\mathbb{R}}f(U)e^{ikU}dU(k>0)\mathrm{.}$ Then $h_f,h_{\bar{f}}\in \mathcal{H}$, and $f\mapsto h_f$ is continuous on each fixed-compact-support test space. With the bosonic Fock conventions of Schwartz/distribution spaces and Hilbert Fock conventions, define on its finite-particle domain $J(f)=a(h_{\bar{f}})+a^{†}(h_f)\mathrm{.}$ This is a complex-linear operator-valued distribution on that invariant domain. For real $f$, its closure is $\overline{J(f)}=\sqrt{2}\mathrm{\Phi }(h_f)$, where $\mathrm{\Phi }(h_f)$ is the closed Segal field of Self-adjoint Segal fields and explicit Weyl unitaries; the corresponding Weyl one-parameter group has generator $i\mathrm{\Phi }(h_f)$. The vacuum two-point form is positive type and has boundary kernel $C(s)=-\frac{1}{4\pi (s-i0{)}^2},$ where $s-i0$ means the distributional limit from $s-iϵ$, $ϵ\downarrow 0$.

For $\kappa >0$ with units of inverse length, let $u,U$ have units of length and set $U_{\kappa }(u)=-{\kappa }^{-1}{e}^{-\kappa u},u_{\kappa }(U)=-{\kappa }^{-1}\log (-\kappa U)(U<0)\mathrm{.}$ For $g\in \mathcal{D}(\mathbb{R};\mathbb{C})$ put $f_g(U)=g(u_{\kappa }(U))$ for $U<0$ and $f_g(U)=0$ for $U\ge 0$, and define $j_{\kappa }(g)=J(f_g)$. The zero extension is smooth and compactly supported. Its vacuum covariance is the positive-type distribution $C_{\kappa }(s)=-\frac{{\kappa }^2}{16\pi }{\mathrm{csch}}^2\text{\hspace{0.17em}⁣}(\frac{\kappa }{2}(s-i0))\mathrm{.}$ Here the notation is fixed as follows: for $b=\kappa \mathrm{/}2$, set $Y_b(s)=\sinh (bs)\mathrm{/}b$. Since $Y_b^{\mathrm{\prime }}(s)=\cosh (bs)>0$ and $Y_b(s)\to \pm \mathrm{\infty }$ as $s\to \pm \mathrm{\infty }$, it is a smooth increasing diffeomorphism. Define ${\mathrm{csch}}^2(b(s-i0)):=b^{-2}{Y}_b^{\ast }((y-i0{)}^{-2})$, where $⟨Y_b^{\ast }T,\phi ⟩:=⟨T,\phi (Y_b^{-1}(y))(Y_b^{-1}{)}^{\mathrm{\prime }}(y)⟩\mathrm{.}$ This is a one-dimensional chiral test-coordinate model. It asserts no restriction of a four-dimensional field to a null curve.

“Positive” refers to positive type of the vacuum covariance, as proved by $⟨h_f,h_f{⟩}_2\ge 0$; it does not assert that $J(f)$ is a positive operator. The cited curved-spacetime literature motivates the exponential coordinate model but is not a supplier for this Fock construction or its local distributional pullback. No collapse geometry, detector probability, experiment, or physical postulate is part of the lemma.