Scalar Epstein–Glaser induction at finite perturbative order

Set $T_0=1$ and $T_1=\mathrm{id}$, as required. Use the unique balanced representative [F11] in every positive field-degree entry; a constant entry is interpreted as the corresponding scalar times the lower multilinear map, consistently with its empty functional support. Perform the extension construction below first on balanced labels. Prescribe numerical coefficients at zero field on these labels and recover their total derivatives using [F11]. Reconstruct the full field dependence only by the single field-shift prescription below; do not choose separate corrections for its remaining field powers. The numerical derivative identities and that prescription will establish independence of the density presentation. Assume $T_k$ has been constructed with all contract properties for every $k<n$, where $n\ge 2$.

In ${\mathbb{M}}^n$ let ${\mathrm{\Delta }}_n=\{x_1=\cdots =x_n\}$. For every nonempty proper $I\subset \{1,\dots ,n\}$ define $C_I=\{(x_1,\dots ,x_n):x_i\mathrm{\notin }{J}^{-}(x_j)\text{ for all }i\in I,j\mathrm{\notin }I\}\mathrm{.}$ Each $C_I$ is open because the Minkowski causal relation is closed. To see these sets cover ${\mathbb{M}}^n\setminus {\mathrm{\Delta }}_n$, choose two distinct coordinates $p,q$ in a configuration. The open cone of past-timelike covectors cannot annihilate the nonzero displacement $p-q$, so choose a past-timelike covector $\tau$ with $\tau (p)\mathrm{\ne }\tau (q)$. Choose $s$ strictly between these values and different from the finitely many $\tau (x_j)$, and put $I=\{j:\tau (x_j)>s\}$. The hyperplane $\tau (x)=s$ is spacelike, and $\tau$ is strictly increasing on every future causal curve. Thus no $x_i$ with $i\in I$ can lie in $J^{-}(x_j)$ for $j\mathrm{\notin }I$, so the configuration lies in $C_I$.

Expand the finite local polynomial derivatives in the graph convention of [F1]. For each fixed $n$, finite jet order and finite polynomial degree allow only finitely many labeled multigraphs, and every edge derivative order is finite. By [F6], after the already constructed proper subgraph extensions, each numerical coefficient $t_{\mathrm{\Gamma }}$ on the punctured relative space ${\mathbb{R}}^{4(n-1)}\setminus \{0\}$ has a finite upper scaling bound and scaling degree ${\delta }_{\mathrm{\Gamma }}<+\mathrm{\infty }$.

On $C_I$, define the lower-order product $T_n^I=T_{\mathrm{\mid }I\mathrm{\mid }}(F_I){\star }_W{T}_{\mathrm{\mid }{I}^c\mathrm{\mid }}(F_{I^c})$. The factors are microcausal by [F2] and the inductive hypothesis, so [F3] makes the product well-defined; each contraction in its finite graph expansion uses only the pullbacks and distribution products supplied by [F4]. The graph-kernel restriction on this stratum is therefore a distribution, and no separate product at the total diagonal is taken here.

If $C_I$ and $C_J$ overlap, put $A=I\cap J$, $B=I\setminus J$, $C=J\setminus I$, and $D=(I\cup J{)}^c$. The defining inequalities give $A$ later than or spacelike to $B$, $C$ later than or spacelike to $D$, $A$ later than or spacelike to $C$, and $B$ later than or spacelike to $D$. They also imply that every point of $B$ is spacelike to every point of $C$: neither causal ordering is possible. Lower-order causal factorization rewrites both local expressions as $T_A{\star }_W{T}_B{\star }_W{T}_C{\star }_W{T}_D$; associativity follows by indexing each finite contraction term by its three pairwise contraction counts. Since $E$ is causally supported, $W=W^{\mathsf{T}}$ on the spacelike-separated supports of $B,C$, so locality gives $T_B{\star }_W{T}_C=T_C{\star }_W{T}_B$. Thus the two local expressions agree on the overlap, including empty blocks interpreted as units.

Apply [F5] to the open cover $\{C_I\}$ of ${\mathbb{M}}^n\setminus {\mathrm{\Delta }}_n$, and use [A1] to assign each partition function to one cover member containing its support. The sum of the corresponding partition functions times $T_n^I$ is locally finite and defines a distribution $T_n^0$ on the complement of ${\mathrm{\Delta }}_n$. Step 3.1 makes it independent of the partition choice locally; the cover is permutation-invariant, so a finite permutation average gives a symmetric extension on this complement. It satisfies causal factorization there by construction.

For a fixed smooth real function $h$ define the field shift $({\tau }_{h}F)(\psi )=F(\psi +h)$ and the directional variation $D_{h}F(\psi )={\frac{d}{ds}F(\psi +sh)\mid }_{s=0}$. Both preserve the local polynomial class, and $\mathrm{supp}({\tau }_{h}F)\subseteq \mathrm{supp}F$. In addition to the contract, maintain inductively $D_h{T}_k(F_1,\dots ,F_k)={\sum }_i{T}_k(F_1,\dots ,D_h{F}_i,\dots ,F_k)$ for every smooth $h$; compact input supports make this meaningful without compact support of $h$. It holds for $k=0,1$. The finite contraction formula makes $D_h$ a derivation of ${\star }_W$, since $W$ is field independent and directional variations commute with every input variation. Hence the causal block products, and then their glued distribution $T_n^0$, obey the same identity off ${\mathrm{\Delta }}_n$. Applying it along $s\mapsto T_n^0({\tau }_{sh}{F}_1,\dots ,{\tau }_{sh}{F}_n)(\psi -sh)$ gives zero derivative, so $T_n^0({\tau }_h{F}_1,\dots ,{\tau }_h{F}_n)(\psi )=T_n^0(F_1,\dots ,F_n)(\psi +h)$. All expressions are finite polynomials in $s$, so this reasoning is algebraic. Put ${\nu }_n^0=T_n^0{\mathrm{\mid }}_{\psi =0}$, meaning the numerical density-label distributions with smearings away from ${\mathrm{\Delta }}_n$. They determine the full punctured product by the preceding shift identity, and satisfy the numerical total-derivative identities because evaluation at zero commutes with coordinate differentiation. During extension and Lorentz normalization make choices only for this numerical family; graphwise it consists of the fully contracted coefficients. Constant entries are fixed by the lower maps as in step 1.1.

For each $t_{\mathrm{\Gamma }}$ in relative dimension $d=4(n-1)$, proceed as follows. If ${\delta }_{\mathrm{\Gamma }}=-\mathrm{\infty }$, choose a compact cutoff $\chi =1$ near zero and use the no-subtraction dyadic series $T(\phi )=t((1-\chi )\phi )+{\sum }_{k\ge 0}t((\chi (2^k\cdot )-\chi (2^{k+1}\cdot ))\phi )$. Fix a compact annulus $K\subset {\mathbb{R}}^d\setminus \{0\}$ and a real $\omega <d$. On the complete fixed-support test space ${\mathcal{D}}_K$, put $L_{\rho }(\psi )={\rho }^{\omega }(t_{\mathrm{\Gamma }}{)}_{\rho }(\psi )$. Since ${\mathrm{sd}}_0(t_{\mathrm{\Gamma }})=-\mathrm{\infty }$, for every $\omega$ there is a decay exponent $\eta <\omega$, so $L_{\rho }(\psi )\to 0$ for each fixed $\psi$. On every compact scale interval $[ϵ,1]$ the scaled tests vary continuously in one fixed-support test space, so ${\sup }_{0<\rho \le 1}\mathrm{\mid }{L}_{\rho }(\psi )\mathrm{\mid }<\mathrm{\infty }$. The closed sets $E_N=\{\psi :{\sup }_{\rho }\mathrm{\mid }{L}_{\rho }(\psi )\mathrm{\mid }\le N\}$ cover ${\mathcal{D}}_K$. Completeness and Baire under DC, supplied by the already assumed AC, give ${\psi }_0\in {\mathcal{D}}_K$, an integer $m\ge 0$, and $a>0$ with ${\psi }_0+\{\zeta :p_m(\zeta )<a\}\subset E_N$ for some positive integer $N$. Subtraction gives $\mathrm{\mid }{L}_{\rho }(\zeta )\mathrm{\mid }\le 2N$ there; scaling $\zeta =a\psi \mathrm{/}(2p_m(\psi ))$ yields $\mathrm{\mid }{L}_{\rho }(\psi )\mathrm{\mid }\le (4N\mathrm{/}a)p_m(\psi )$ when $p_m(\psi )>0$, while $p_m(\psi )=0$ implies $\psi =0$. Hence $\mathrm{\mid }(t_{\mathrm{\Gamma }}{)}_{\rho }(\psi )\mathrm{\mid }\le C{\rho }^{-\omega }{p}_m(\psi )$ uniformly. This is the fixed-annulus Baire proof, with the arbitrarily negative decay exponent replacing the finite-real-degree hypothesis. Its $k$th shell is bounded by $C_{\phi }{2}^{-k(d-\omega )}$, proving convergence and continuity. For the scaled test ${\rho }^{-d}\phi (\cdot \mathrm{/}\rho )$ only shells of radius $\lambda \le C_{\phi }\rho$ occur, and their sum is $O({\rho }^{-\omega })$. The same series is independent of $\omega$, so arbitrarily low $\omega$ prove degree $-\mathrm{\infty }$; telescoping on punctured tests proves it extends $t$. Otherwise [F7] applies with its finite real degree. Choose the extension linearly on a basis of the balanced labels, and extend its total-derivative labels by [F11]. For any fixed finite jet order this uses a finite-dimensional polynomial space; compatible bases over all finite orders give a linear extension on their algebraic union. The graph and derivative bounds remain finite because total differentiation adds only its finite order. Descendant coefficients obey these engineering upper bounds, including any forced contact terms, rather than an incompatible requirement to preserve the exact degree of a vanished punctured derivative combination. The extension has degree ${\delta }_{\mathrm{\Gamma }}$ on each freely extended balanced graph coefficient. For the independent balanced coefficients of finite real degree [F8] gives the stated finite-jet ambiguity; for degree $-\mathrm{\infty }$, [F10] implies uniqueness because a nonzero difference supported at zero would have degree at least $d$. Lift the chosen relative extension back to ${\mathbb{M}}^n$ by translation: in coordinates $z_i=x_i-x_n$ and $y=x_n$, pair the relative distribution with $z\mapsto {\int }_{\mathbb{M}}\mathrm{\Psi }(y+z_1,\dots ,y+z_{n-1},y)d^{4}y$. The inner function is a compactly supported smooth test by differentiation under a compact integral. This defines a translation-covariant distribution extending the off-diagonal coefficient. Apply these extensions to the numerical family ${\nu }_n^0$ from step 5.1, obtaining a multilinear map ${\nu }_n^1$ from actual local functionals to $\mathbb{C}\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$. Its density-label kernels satisfy the numerical derivative identities by construction; thus [F11] makes it well-defined on the quotient by integration by parts.

For any numerical multilinear family $\nu$ obtained here define its reconstruction by $\mathcal{R}(\nu )(F_1,\dots ,F_n)(\varphi )=\nu ({\tau }_{\varphi }{F}_1,\dots ,{\tau }_{\varphi }{F}_n)$. This is a finite expression: for $F=c+\int fP(j^s\psi )$, the shifted density is $c+\int fP(j^s\psi +j^s\varphi )$, whose finite Taylor expansion has smooth compact coefficient functions $f$ times products of jets of $\varphi$. Reduce each term by [F11] before applying $\nu$. Equality of actual functionals is preserved by ${\tau }_{\varphi }$, so numerical presentation independence proves full presentation independence. Also ${\tau }_{\varphi }{\tau }_{th}={\tau }_{\varphi +th}$; differentiating this finite expression gives $D_h\mathcal{R}(\nu )(F_1,\dots ,F_n)={\sum }_i\mathcal{R}(\nu )(F_1,\dots ,D_h{F}_i,\dots ,F_n)$. Thus every occurrence of a numerical sub-polynomial coefficient uses the same $\nu$, including across input field degrees. Conversely, the shift calculation in step 5.1 proves that any family satisfying this identity equals $\mathcal{R}$ of its zero-field value. Set $T_n^1=\mathcal{R}({\nu }_n^1)$; step 5.1 shows that it extends $T_n^0$. The Taylor factorials count the choices of uncontracted slots, leaving precisely the graph Wick factorials of [F1]; lower numerical field degrees supply all spectator-field terms, rather than new independent extensions. Total coordinate derivatives commute with field shifts at fixed $\varphi$ when the derivatives act on the full shifted density; the finite Leibniz rule therefore differentiates both the numerical coefficient and its spectator jets. This proves the full derivative identity and its engineering bounds, with the possible contact terms already allowed in step 6.1. The numerical kernels are translation invariant, so all their wavefront tuples have total covector sum zero, including at the extended diagonals. Differentiating the finitely many field jets under the distribution pairing defines the output derivative kernels continuously, by attaching the remaining field slots to their vertex positions. In their Fourier phases the frequency at each vertex is the sum of the attached slot frequencies; jet derivatives add only polynomial factors. Multiplication by the smooth compact vertex coefficients preserves the numerical wavefront bound. Thus a nonzero output tuple in an all-future or all-past cone has nonzero total frequency, and rapid center-coordinate Fourier decay of the localized numerical distribution excludes it; the proper projection wavefront estimate [F4] expresses the same exclusion for integrated vertex variables. Thus the output belongs to the stated polynomial microcausal class, with compact support and finite polynomial degree inherited from the finitely many vertex terms.

Keep together all components that mix under Lorentz transformations. At fixed input field degrees and homogeneous derivative degrees, take the finite-dimensional space $L$ of balanced input-label tensors and put $O=\mathbb{C}$, the trivial zero-field output representation. Enlarge $L$ to contain its full Lorentz orbit. The substitution $p_a\mapsto {\mathrm{\Lambda }}^{\mathsf{T}}{p}_a$ of [F11] defines $R_L$, and set $R_O(\mathrm{\Lambda })=1$; linear substitutions preserve field degree and homogeneous derivative degree, so these enlargements remain finite. In relative dimension $d=4(n-1)$ put $A_{\mathrm{\Lambda }}(z_1,\dots ,z_{n-1})=(\mathrm{\Lambda }{z}_1,\dots ,\mathrm{\Lambda }{z}_{n-1})$ and $U_{\mathrm{\Lambda }}=(A_{{\mathrm{\Lambda }}^{-1}}{)}^{\ast }\otimes R_O(\mathrm{\Lambda })$ on ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)\otimes O$. Regard the chosen zero-field numerical coefficients of ${\nu }_n^1$ collectively as $t^1\in {\mathrm{Hom}}_{\mathbb{C}}(L,{\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)\otimes O)$, with combined action $(\mathrm{\Lambda }\cdot t)(b)=U_{\mathrm{\Lambda }}t(R_L(\mathrm{\Lambda }{)}^{-1}b)$. These are representations by the pullback and substitution laws. On the punctured relative space the collective coefficient $t^0$ is fixed by this combined action: the free kernels are invariant, their differentiated components transform by the chain rule, and the already covariant lower-order products agree on the causal-cover overlaps. A fixed coordinate derivative component need not be scalar invariant.

For the finite block in step 7.2 choose a common finite engineering upper bound $B$ compatible with its homogeneous label degrees and the finitely many lower-order counterterms. Linear coordinate changes commute with positive dilations and preserve these bounds. Put $J_N={\mathrm{span}}_{\mathbb{C}}\{{\mathrm{\partial }}^{\alpha }{\delta }_0:\mathrm{\mid }\alpha \mathrm{\mid }\le N\}$, where $N=\lfloor B-d\rfloor$ when $B\ge d$, and put $J_N=0$ when $B<d$. The permitted local corrections form the invariant subspace $V$ of ${\mathrm{Hom}}_{\mathbb{C}}(L,J_N\otimes O)$ satisfying the labelwise engineering bounds and the linear identities of the balanced coefficients. This space is finite-dimensional. Lorentz substitutions preserve each homogeneous degree and delta-derivative order, while [F11] makes the derivative identities equivariant, proving invariance of this subspace. Differentiate the combined action to obtain $\rho (X)$ and put $c(X)=\rho (X)t^1$. Restriction commutes with the action, and step 7.2 gives $\rho (X)t^0=0$, so $c(X)$ is supported at the relative origin. It obeys the same engineering bounds and identities; [F8] and [F10] therefore place it in $V$. Smooth differentiation is justified by differentiating the transformed compact test densities in their fixed-support smooth topology; the finite-order distribution bound then permits pairing. The representation law gives $c([X,Y])=\rho (X)c(Y)-\rho (Y)c(X)$, so $c:\mathfrak{s}\mathfrak{o}(1,3)\to V$ is exactly a real-linear cocycle in the finite-dimensional complex module required by [F9].

Apply [F9] to this module and cocycle to obtain $v\in V$ with $c(X)=\rho (X)v$. Thus $t^1-v$ has zero infinitesimal combined defect, retains the engineering bounds, and agrees with $t^0$ off the diagonal. Its combined orbit is constant along each one-parameter subgroup by the preceding smooth test-pairing argument. Along a smooth group path $g(s)$ the orbit derivative is the transported infinitesimal action of $g(s{)}^{-1}{g}^{\mathrm{\prime }}(s)$ and hence is zero. In the identity component every point is reachable by a piecewise smooth path: the set reachable from the identity and its complement are open, using straight paths in coordinate balls. Connectedness therefore gives full $S{O}^{+}(1,3)$ covariance. Work separately in homogeneous balanced-label degrees, whose finite spaces are Lorentz invariant, and recover total-derivative descendants by [F11]; these specify one corrected numerical map ${\nu }_n^2$ on their algebraic direct sum. Use $T_n^2=\mathcal{R}({\nu }_n^2)$ for every field degree. Reconstruction commutes with Poincare transformations, since ${\tau }_{\gamma \cdot \varphi }(\gamma \cdot F)=\gamma \cdot ({\tau }_{\varphi }F)$ and the numerical family is equivariant; step 7.1 proves its field independence and representation independence. In particular a correction chosen on a numerical label is reused in all occurrences of that label as a sub-polynomial, never chosen again for each spectator-field degree. The choices of these finite-block corrections are permitted by [A1]. Translation covariance is retained because every correction is a relative delta jet lifted by the same center-coordinate integral as in step 6.1.

Symmetrize the corrected numerical family over the finite permutation group and reconstruct, denoting the result again by $T_n^2$; this preserves covariance, field independence and the finite scaling bounds. The following symmetric diagonal correction and its polarization impose unitarity. Let $D_n(V)$ be the coefficient of $g^n$ in ${\mathcal{S}}_g(V{)}^{\ast }{\star }_W{\mathcal{S}}_g(V)-1$ after the lower orders have been made unitary. Associativity and $(F{\star }_{W}G{)}^{\ast }=G^{\ast }{\star }_W{F}^{\ast }$ follow from the finite contraction formula and $W^{\ast }=W^{\mathsf{T}}$. On each causal cover set $C_I$, factorization reduces the defect to products of lower-order unitary blocks, so $D_n$ vanishes away from ${\mathrm{\Delta }}_n$. It is self-adjoint because the full product ${\mathcal{S}}_g(V{)}^{\ast }{\star }_W{\mathcal{S}}_g(V)$ is self-adjoint. The graphwise scaling bounds and [F8] place each diagonal part of $D_n$ in a finite local jet space with the permitted derivative order. Define the correction by $\frac{i^n}{{\mathrm{\hslash }}^{n}n!}\mathrm{\Delta }{T}_n(V,\dots ,V)=-\frac{1}{2}{D}_n(V)\mathrm{.}$ This is an allowed formal correction: every order-$n$ defect term has $\mathrm{\hslash }$-degree at least $-n$, so multiplication by ${\mathrm{\hslash }}^n$ inverts the displayed prefactor without introducing negative powers of $\mathrm{\hslash }$; polarization gives a symmetric diagonal $n$-linear map. To check compatibility with field independence, polarize $D_n$ on real local inputs and extend complex multilinearly in its $n$ inputs. Each term is a product of lower maps and the current $T_n^2$, with every input occurring once, so the derivation identity of step 5.1 and step 7.1 imply $D_h{D}_n(F_1,\dots ,F_n)={\sum }_i{D}_n(F_1,\dots ,D_h{F}_i,\dots ,F_n)$. Involution also commutes with real field shifts. By the converse in step 7.1, $D_n=\mathcal{R}(d_n)$ for $d_n=D_n{\mathrm{\mid }}_{\varphi =0}$. Consequently the displayed correction is exactly the reconstruction of the numerical correction $-{\mathrm{\hslash }}^{n}n!d_n\mathrm{/}(2i^n)$, with no independent field-degree choices. The numerical correction inherits the total-derivative identities, covariance and permutation symmetry from the defect; [F11] and step 7.1 transfer them to every functional presentation. In the order-$n$ defect, this changes only the two terms paired with the unit coefficient, adding $-D_n\mathrm{/}2+(-D_n\mathrm{/}2{)}^{\ast }=-D_n$. Thus it cancels the defect, preserves the graph engineering upper bounds and covariance, and leaves all off-diagonal causal products unchanged. Formal unitarity holds through order $n$.

The construction at order $n$ uses only finite graph sums, finite-dimensional jet corrections and the already constructed orders below $n$. The Axiom of Choice supplies dependent choice for the recursive order-by-order selections and Countable Choice for the microlocal contractions in [F3]. Induction therefore constructs the sequence for every finite order and verifies every clause of the system contract in [F1]. No limit in $g$ is taken.