Positive chiral current and exponential test-coordinate model

Throughout this item, $m\in {\mathbb{N}}_0$. For each integer $d\ge 1$ and compact $Q\subset {\mathbb{R}}^d$, let ${\mathcal{D}}_Q({\mathbb{R}}^d):=\{\phi \in C^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C}):\mathrm{supp}\phi \subseteq Q\},$ with the Fr'echet topology generated by $p_{Q,N}(\phi ):={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le N}{\sup }_{x\in Q}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid }(N\in {\mathbb{N}}_0),$ where $\alpha =({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{N}}_0^d$, $\mathrm{\mid }\alpha \mathrm{\mid }={\sum }_j{\alpha }_j$, and ${\mathrm{\partial }}^{\alpha }={\mathrm{\partial }}_{x_1}^{{\alpha }_1}\cdots {\mathrm{\partial }}_{x_d}^{{\alpha }_d}$. The compact-test space $\mathcal{D}({\mathbb{R}}^d)=C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$ has the locally convex inductive-limit topology over these fixed-support spaces (equivalently over any compact exhaustion of ${\mathbb{R}}^d$). A distribution on ${\mathbb{R}}^d$ is a continuous complex-linear functional on this topology, and its test pairing is bilinear; write ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ for the space of such distributions. For $d=0$, use ${\mathbb{R}}^0=\{\ast \}$ with measure one. In that case set $\mathcal{D}({\mathbb{R}}^0)=\mathbb{C}$ with its usual finite-dimensional topology.

Let ${\mathbf{1}}_m\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^m)$ denote integration against Lebesgue measure when $m\ge 1$, and set $⟨{\mathbf{1}}_0,a⟩=a$. The symbol $1$ in the Statement's tensor notation means this ${\mathbf{1}}_m$. Its continuity on fixed-support test spaces follows from finite Lebesgue measure of compact supports Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure; for $m=0$ it is evaluation on $\mathbb{C}$. For $B\in {\mathcal{D}}^{\mathrm{\prime }}(\mathbb{R})$ define the product distribution ${\mathbf{1}}_m\otimes B$ by $⟨{\mathbf{1}}_m\otimes B,\psi ⟩:=⟨B,s\mapsto {\int }_{{\mathbb{R}}^m}\psi (v,s)dv⟩,$ where the integral for $m=0$ is evaluation at $\ast$. The inner function is a compactly supported smooth test. For each fixed compact support, its seminorms are bounded by the corresponding $s$-derivative seminorms of $\psi$ times the finite measure of its $v$-projection Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure, so this definition is a continuous distributional pairing.

Assume Countable Choice and use Euclidean Lebesgue measure. Let $m\ge 0$. For $m=0$ take ${\mathbb{R}}^0$ to be a singleton with measure $1$. For each sufficiently small $ϵ>0$ let $d_{ϵ}:{\mathbb{R}}^m\times \mathbb{R}\to (0,\mathrm{\infty })$ be smooth and real-valued. Assume that for every compact $V\subseteq {\mathbb{R}}^m$ and every $R>0$ there are constants $c,C,M,{ϵ}_0>0$ such that, on $V\times [-R,R]$ and for $0<ϵ<{ϵ}_0$, $c\le d_{ϵ}(v,s)\le C,\mathrm{\mid }{\mathrm{\partial }}_s{d}_{ϵ}(v,s)\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}_s^2{d}_{ϵ}(v,s)\mathrm{\mid }\le M\mathrm{.}$ For $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ set $\widehat{\phi }(k)={\int }_{\mathbb{R}}\phi (s)e^{-iks}ds$ and define $B$ by $⟨B,\phi ⟩:={\int }_0^{\mathrm{\infty }}k\widehat{\phi }(k)dk\mathrm{.}$ For $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^m\times \mathbb{R};\mathbb{C})$ write $\widehat{\psi }(v,k)={\int }_{\mathbb{R}}\psi (v,s)e^{-iks}ds$ and define $K$ by $⟨K,\psi ⟩:={\int }_{{\mathbb{R}}^m}{\int }_0^{\mathrm{\infty }}k\widehat{\psi }(v,k)dkdv\mathrm{.}$ Then $B$ and $K$ are distributions, $B$ is the lower boundary denoted $-(s-i0{)}^{-2}$, and $K=1\otimes B$ in the explicitly defined pairing above. The smooth kernels $K_{ϵ}(v,s):=-(s-iϵd_{ϵ}(v,s){)}^{-2}$ converge to $K$ in the distributional sense: $\int K_{ϵ}(v,s)\psi (v,s)dvds\to ⟨K,\psi ⟩$ for every such $\psi$. When $s$ and $ϵ$ carry units of length, $d_{ϵ}$ is dimensionless.

The cited notes develop the distribution framework and its calculus; they are not used as a substitute for the local estimate. The positive smooth damping result and its uniform error bound are proved in full above.

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.

Use $\sinh w=(e^w-e^{-w})\mathrm{/}2$, $\cosh w=(e^w+e^{-w})\mathrm{/}2$, and $\mathrm{csch}w=1\mathrm{/}\sinh w$ where defined, as fixed in Positive chiral current and exponential test-coordinate model. The same supplier states their derivatives and that real $\cosh$ is positive. A function on an open subset of $\mathbb{C}$ is holomorphic when it is complex differentiable at every point, as defined in Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions. For the real line, $\mathcal{S}(\mathbb{R})$ consists of all $\phi \in C^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ with $p_{m,n}(\phi )={\sup }_s\mathrm{\mid }{s}^m{\phi }^{(n)}(s)\mathrm{\mid }<\mathrm{\infty }$ for every $m,n\in {\mathbb{N}}_0$; its topology is generated by these seminorms. A tempered distribution is a continuous complex-linear functional on this topological space, with bilinear test pairing, as defined in Schwartz space and its seminorms, Schwartz topology and convergence, and Tempered distribution.

“Chiral current” refers to the current of Positive chiral current and exponential test-coordinate model, whose one-particle space has exactly the positive-frequency sector $k>0$. In the phrase “analytic kernel,” analytic means locally represented by a convergent complex power series on the set where $\sinh (bz)\mathrm{\ne }0$. Indeed, $\sinh (bz)$ is analytic by its exponential series; near any such $z_0$, put $a=\sinh (b{z}_0)$ and $q(z)=(\sinh (bz)-a)\mathrm{/}a$. On a sufficiently small disk $\mathrm{\mid }q(z)\mathrm{\mid }<1$, and $\frac{1}{{\sinh }^2(bz)}=a^{-2}{\sum }_{n=0}^{\mathrm{\infty }}(-1{)}^n(n+1)q(z{)}^n,$ which is a convergent local power series.

For the final scope sentence, a norm-continuous C*-algebra KMS system at $\theta >0$ means a complex Banach $\ast$-algebra $\mathcal{A}$ satisfying $\mathrm{\parallel }{a}^{\ast }a\mathrm{\parallel }=\mathrm{\parallel }a{\mathrm{\parallel }}^2$, a group $t\mapsto {\alpha }_t$ of bijective $\ast$-homomorphisms for which $t\mapsto {\alpha }_t(a)$ is norm-continuous for each $a\in \mathcal{A}$, and a state $\rho$ (a positive linear functional of norm one, with $\rho (1)=1$ when $\mathcal{A}$ is unital) such that for every $a,b\in \mathcal{A}$ there is a bounded function $F_{a,b}$, continuous on $0\le \mathrm{Im}z\le \theta$ and holomorphic on the interior, with $F_{a,b}(t)=\rho (a{\alpha }_t(b))$ and $F_{a,b}(t+i\theta )=\rho ({\alpha }_t(b)a)$. A detector transition probability would additionally require a specified detector, coupled initial state and measurement outcome. Neither structure is part of the present two-point Fourier calculation.