Regular time-ordered contraction coefficients

Assume the Axiom of Countable Choice: for every sequence $(E_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a function $c$ with domain $\mathbb{N}$ such that $c(n)\in E_n$ for all $n$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). This premise is inherited from the construction of $H_{\mu }$ in Tempered time-ordered scalar two-point carrier; the finite contractions below require no additional choice assumption.

Set $X={\mathbb{R}}^4$. Fix $\mu \ge 0$ and the $(-+++)$, shell, Fourier, and coordinate conventions of Tempered time-ordered scalar two-point carrier. That current supplier states that its relative-coordinate tempered distribution $H_{\mu }$ is symmetric, agrees with the ordered Wightman kernels on strictly ordered time supports and on spacelike-separated supports, and has no prefactor $i$ in its definition. Its current carrier is a draft with a supplied conditional proof; the formula below uses that exact interface. Complex integrals of compactly supported regular kernels use the componentwise Riemann convention in Regular polynomial field functional algebra: write the integrand as its real part plus $i$ times its imaginary part and integrate each real component.

For open $\mathrm{\Omega }\subseteq {\mathbb{R}}^m$, $\mathcal{D}(\mathrm{\Omega })$ is the space of smooth compactly supported complex tests with the LF topology, and ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ is its continuous complex-linear dual, paired bilinearly with no conjugation (Test function space d of an open set, Test function topology, Distribution). Schwartz space consists of smooth functions with all polynomially weighted derivatives bounded; ${\mathcal{S}}^{\mathrm{\prime }}$ is its continuous complex-linear dual with bilinear pairing (Schwartz space and its seminorms, Schwartz topology and convergence, Tempered distribution).

For any relative-coordinate distribution $K\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^4)$, define its translation-invariant two-point bidistribution $\tilde{K}\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^4\times {\mathbb{R}}^4)$ by $⟨\tilde{K},\mathrm{\Psi }⟩=⟨K,z⟼{\int }_{{\mathbb{R}}^4}\mathrm{\Psi }(y+z,y)dy⟩,\mathrm{\Psi }\in \mathcal{D}({\mathbb{R}}^4\times {\mathbb{R}}^4)\mathrm{.}$ The inner function is a test function: its support lies in the difference of the two compact coordinate projections of $\mathrm{supp}\mathrm{\Psi }$; each derivative in $z$ is the Riemann integral of the corresponding first-variable derivative of $\mathrm{\Psi }$ over a compact rectangle containing the fixed compact second-variable projection; iterate the real single-coordinate differentiation rule over every derivative in $z$ and apply it to both real components, using the compact parameter-integral differentiation convention of Regular polynomial field functional algebra. On each fixed compact support space, every output derivative seminorm is bounded by the volume of a compact rectangle containing the second-variable projection times the corresponding source derivative seminorm. The support of the output lies in the compact difference of the two projections. Thus this map is continuous on each fixed-support test space, and hence on the LF test topology Test function topology. Thus the displayed formula defines a distribution without multiplying singular distributions. Write ${\mathsf{H}}_{\mu }={\tilde{H}}_{\mu }$.

For $F\in {\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{g}}({\mathbb{R}}^4)$, define its support as the finite union of the coordinate projections of the supports of its nonconstant symmetric coefficient kernels. This is a compact set and is presentation independent by Tensor-product tests are dense in smooth compact kernel topology. For compact $A,B\subset {\mathbb{R}}^4$, write $A{\succcurlyeq }_{\mathrm{o}\mathrm{u}\mathrm{t}}B$ when $x-y\mathrm{\notin }{C}_{-}:=\{(t,\mathbf{x})\in {\mathbb{R}}^4:t\le -\mathrm{\mid }\mathbf{x}\mathrm{\mid }\}\text{for every }x\in A,y\in B\mathrm{.}$ This means the first support has no point in the closed causal past of the second, using the supplier's fixed inertial chart.

For $n\ge 1$, let ${\mathcal{Q}}_n$ consist of finite coefficient families in $n$ independent real smooth fields ${\varphi }_i\in C^{\mathrm{\infty }}(X;\mathbb{R})$ of the form $Q({\varphi }_1,\dots ,{\varphi }_n)={\sum }_{\mathbf{r}}\int q_{\mathbf{r}}({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}){\prod }_{i=1}^n{\prod }_{a=1}^{r_i}{\varphi }_i(x_a^{(i)})d\mathbf{x},$ where $\mathbf{r}=(r_1,\dots ,r_n)\in {\mathbb{N}}_0^n$ ranges over a finite set, $q_{\mathbf{r}}\in C_c^{\mathrm{\infty }}(X^{r_1}\times \cdots \times X^{r_n};\mathbb{C})$, and each coefficient is symmetric within every block ${\mathbf{x}}^{(i)}=(x_1^{(i)},\dots ,x_{r_i}^{(i)})$. $X^0$ is a one-point space; an empty product and an integral over no variables leave the scalar coefficient unchanged. Here $d\mathbf{x}$ is the product of the $d^4{x}_a^{(i)}$ coordinate volume elements over all variables present. The product ${\prod }_i{F}_i({\varphi }_i)$ of regular polynomials is in ${\mathcal{Q}}_n$.

For $1\le i<j\le n$, define the one-pair contraction $D_{\mathsf{H}}^{ij}:{\mathcal{Q}}_n\to {\mathcal{Q}}_n$ linearly and termwise on each block coefficient $q_{\mathbf{r}}$. It is zero if $r_i=0$ or $r_j=0$; otherwise it has multidegree $\mathbf{r}-{\mathbf{e}}_i-{\mathbf{e}}_j$ and coefficient $q_{\mathbf{r}-{\mathbf{e}}_i-{\mathbf{e}}_j}^{\mathrm{\prime }}({\mathbf{x}}^{\mathrm{\prime }})=r_i{r}_j⟨{\mathsf{H}}_{\mu },(x,y)⟼q_{\mathbf{r}}(\dots ,x,\dots ,y,\dots )⟩,$ where $x$ occupies the first variable of block $i$, $y$ occupies the first variable of block $j$, and the uncontracted variables ${\mathbf{x}}^{\mathrm{\prime }}$ are held as parameters. The result remains symmetric within each remaining block. The parameter-pairing theorem Distribution pairing with smooth parameter families makes this coefficient smooth and compactly supported. The factor $r_i{r}_j$ counts the choices of one variable from each symmetric block, so this is exactly one functional derivative in each of the independent field slots followed by pairing against ${\mathsf{H}}_{\mu }$.

For $r\ge 1$, define full symmetrization by ${\mathrm{Sym}}_{r}h(x_1,\dots ,x_r)=\frac{1}{r!}{\sum }_{\pi \in S_r}h(x_{\pi (1)},\dots ,x_{\pi (r)}),$ with ${\mathrm{Sym}}_0$ the identity. For $Q\in {\mathcal{Q}}_n$, define $m_{n}Q$ by setting ${\varphi }_1=\cdots ={\varphi }_n=\varphi$ and replacing each remaining block coefficient by its full symmetrization across all variables. For a finite-degree $Q$, the formal exponential is defined coefficientwise by $\exp \text{\hspace{0.17em}⁣}\left(\eta {\sum }_{1\le i<j\le n}{D}_{\mathsf{H}}^{ij}\right)Q={\sum }_{r\ge 0}\frac{{\eta }^r}{r!}{\left({\sum }_{1\le i<j\le n}{D}_{\mathsf{H}}^{ij}\right)}^{r}Q,$ which terminates because each contraction lowers total degree by two. The parameter $\eta$ is formal and fixed by complex conjugation. Define $T_n^H(F_1,\dots ,F_n)(\varphi )=m_n\exp \text{\hspace{0.17em}⁣}\left(\eta {\sum }_{1\le i<j\le n}{D}_{\mathsf{H}}^{ij}\right){\prod }_{i=1}^n{F}_i({\varphi }_i),T_0^H=1,T_1^H(F)=F\mathrm{.}$ The tensor-product distribution theorem Tensor product distributions and iterated pairings and the parameter-pairing theorem above make every contraction well defined; no diagonal product occurs. Symmetry of ${\mathsf{H}}_{\mu }$ makes $T_n^H$ symmetric in its inputs.

For any $V\in {\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{g}}({\mathbb{R}}^4)$ and formal coupling $z$, define $S_z^H(V)={\sum }_{n=0}^{\mathrm{\infty }}\frac{(iz\mathrm{/}\eta {)}^n}{n!}{T}_n^H(V,\dots ,V)\text{with coefficients in }{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{g}}({\mathbb{R}}^4)[\eta ,{\eta }^{-1}]\mathrm{.}$ This is a coefficientwise formal series; the definition asserts no convergence in $z$, no time-ordered product for singular local functionals, and no continuum interacting field.