Formal scaling and renormalization cocycle

Assume Countable Choice.

Let $\mathbb{M}={\mathbb{R}}^4$ have Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$. A scalar local polynomial functional is a finite sum $F(\varphi )=c+{\sum }_{j=1}^N{\int }_{\mathbb{M}}{f}_j(x)P_j(j^{s_j}\varphi (x))d^{4}x,$ where $N$ is a finite nonnegative integer, $\varphi \in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$, $c\in \mathbb{C}$, and $f_j\in C_c^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$. For a finite integer $s_j\ge 0$, the $s_j$-jet is $j^{s_j}\varphi (x)=({\mathrm{\partial }}^{\alpha }\varphi (x){)}_{\alpha \in \{0,1,2,\dots {\}}^4,\mathrm{\mid }\alpha \mathrm{\mid }\le s_j}$; each $P_j$ is a complex polynomial in these finitely many jet coordinates. This is the local functional class used by the supplied scalar time-ordered system.

For every mass label $m\ge 0$, suppose a normalized scalar time-ordered system $T_n^{(m)}$ with the same local scalar contract is supplied. Define its coefficientwise formal S-map by ${\mathcal{S}}_m(V)={\sum }_{n=0}^{\mathrm{\infty }}\frac{1}{n!}{T}_n^{(m)}(V,\dots ,V),T_0^{(m)}=1,T_1^{(m)}=\mathrm{id}\mathrm{.}$ The series is formal in the interaction degree; no analytic convergence is asserted.

Use units $c=\mathrm{\hslash }=1$, so the mass label $m$ has mass dimension one. The scale parameter $\rho >0$ is dimensionless. On scalar field configurations put $(C_{\rho }\varphi )(x)={\rho }^{-1}\varphi ({\rho }^{-1}x),$ the engineering scaling of a four-dimensional scalar field. For a mass-indexed family of functionals $F=(F_m{)}_{m\ge 0}$, define $({\sigma }_{\rho }F{)}_m=C_{\rho }^{\ast }{F}_{{\rho }^{-1}m},(C_{\rho }^{\ast }F)(\varphi )=F(C_{{\rho }^{-1}}\varphi )\mathrm{.}$ Thus the component at mass $m$ is pulled back from the component at mass ${\rho }^{-1}m$; equivalently the mass label scales as $m\mapsto \rho m$. If $A=(A_m{)}_{m\ge 0}$ is instead a family of maps from local functionals (or their formal series) to local or microcausal functionals (or their formal series), define its induced scale conjugate, denoted ${\sigma }_{\rho }A{\sigma }_{\rho }^{-1}$, by $({\sigma }_{\rho }A{\sigma }_{\rho }^{-1}{)}_m=C_{\rho }^{\ast }\circ A_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1},(C_{\rho }^{\ast }{)}^{-1}=C_{{\rho }^{-1}}^{\ast }\mathrm{.}$ Here the rightmost pullback acts on the input, the leftmost pullback acts on every output coefficient, and formal indeterminates are fixed. In particular the mass-$m$ component of the scale-conjugated formal S-map is $({\sigma }_{\rho }\mathcal{S}{\sigma }_{\rho }^{-1}{)}_m=C_{\rho }^{\ast }\circ {\mathcal{S}}_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1}\mathrm{.}$ These functional and map actions are extended coefficientwise to formal series. They obey ${\sigma }_1=\mathrm{id}$ and ${\sigma }_{\rho \tau }={\sigma }_{\rho }{\sigma }_{\tau }$; the dilation sends compact supports to compact supports and normalizes the proper orthochronous Poincare action.

For $m\ge 0$, $\rho >0$ and each $n\ge 0$, define the conjugated sequence of multilinear maps componentwise by $T_{n,\rho }^{\mathrm{\prime }(m)}(F_1,\dots ,F_n):=C_{\rho }^{\ast }\text{\hspace{0.17em}⁣}\left[T_n^{({\rho }^{-1}m)}\right((C_{\rho }^{\ast }{)}^{-1}{F}_1,\dots ,(C_{\rho }^{\ast }{)}^{-1}{F}_n\left)\right]\mathrm{.}$ For $n=0$ this means $T_{0,\rho }^{\mathrm{\prime }(m)}=C_{\rho }^{\ast }1=1$; for $n=1$ it gives $T_{1,\rho }^{\mathrm{\prime }(m)}=\mathrm{id}$. Assume for every $m\ge 0$ and $\rho >0$ that this sequence, as well as the supplied sequence $(T_n^{(m)}{)}_{n\ge 0}$, satisfies the normalized local scalar time-ordered-system contract for the same free data at mass $m$, after transporting the free data at mass ${\rho }^{-1}m$ by the stated pullback. This is the scale-admissibility condition on the sequences. Their formal S-maps are distinct objects: by multilinearity and coefficientwise pullback, the S-map of $(T_{n,\rho }^{\mathrm{\prime }(m)}{)}_{n\ge 0}$ is exactly $({\sigma }_{\rho }\mathcal{S}{\sigma }_{\rho }^{-1}{)}_m$ displayed above. Define $Z_{\rho }=(Z_{\rho ,m}{)}_{m\ge 0}$ by the componentwise equations $C_{\rho }^{\ast }\circ {\mathcal{S}}_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1}={\mathcal{S}}_m\circ Z_{\rho ,m}(m\ge 0),$ where each $Z_{\rho ,m}$ is the unique formal local map with identity linear term, supplied by the finite-renormalization comparison proposition under these hypotheses. The notation $Z_{\rho ,m}(V)=V+O(V^2)$ means that all terms after the linear term have interaction degree at least two. This is a conditional definition: no existence of a time-ordered system for any particular model is asserted.

Local scalar time-ordered product contract

Renormalization maps form a formal composition group

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The time-ordered systems and scale-admissibility condition are supplied inputs. Their comparison map is determined uniquely by a triangular coefficient recursion under Countable Choice from the actual contraction contract. Constructing time-ordered systems is a separate theorem with its own Choice hypotheses; no existence of a system for a particular model is asserted here.