# Hopf Monoids in Perturbative Algebraic
Quantum Field Theory

William Norledge  Pennsylvania State University<EMAIL_ADDRESS>
###### Abstract.

We develop an algebraic formalism for perturbative quantum field theory (pQFT)
which is based on Joyal’s combinatorial species. We show that certain basic
structures of pQFT are correctly viewed as algebraic structures internal to
species, constructed with respect to the Cauchy monoidal product. Aspects of
this formalism have appeared in the physics literature, particularly in the
work of Bogoliubov-Shirkov, Steinmann, Ruelle, and Epstein-Glaser-Stora. In
this paper, we give a fully explicit account in terms of modern theory
developed by Aguiar-Mahajan. We describe the central construction of causal
perturbation theory as a homomorphism from the Hopf monoid of set
compositions, decorated with local observables, into the Wick algebra of
microcausal polynomial observables. The operator-valued distributions called
(generalized) time-ordered products and (generalized) retarded products are
obtained as images of fundamental elements of this Hopf monoid under the
curried homomorphism. The perturbative S-matrix scheme corresponds to the so-
called universal series, and the property of causal factorization is naturally
expressed in terms of the action of the Hopf monoid on itself by Hopf powers,
called the Tits product. Given a system of fully renormalized time-ordered
products, the perturbative construction of the corresponding interacting
products is via an up biderivation of the Hopf monoid, which recovers
Bogoliubov’s formula.

This paper is an abridged version of ‘Species-theoretic foundations of
perturbative quantum field theory’, arXiv:2009.09969

###### Contents

  1. I Hopf Monoids
    1. 1 The Algebras
    2. 2 $\boldsymbol{\Sigma}$ as a Hopf E-Algebra
    3. 3 Products and Series
    4. 4 Perturbation of T-Products
  2. II Perturbative Algebraic Quantum Field Theory
    1. 5 Spacetime and Field Configurations
    2. 6 Observables
    3. 7 Time-Ordered Products and S-Matrix Schemes
    4. 8 Interactions
    5. 9 Scattering Amplitudes

### Introduction

The theory of species is a richer, categorified version of analyzing
combinatorial structures in terms of generating functions, going back to André
Joyal [Joy81], [Joy86], [BLL98]. In this approach, one sees additional
structure by encoding processes of _relabeling_ combinatorial objects, that is
by modeling combinatorial objects as presheaves on the category S of finite
sets $I$ (the labels) and bijections $\sigma$ (relabelings). In this paper, we
are concerned with species p valued in complex vector spaces, i.e. functors of
the form

$\textbf{p}:\textsf{S}^{\operatorname{op}}\to\textsf{Vec},\qquad
I\mapsto\textbf{p}[I],\quad\sigma\mapsto\textbf{p}[\sigma]$

where Vec is the category of complex vector spaces. Explicitly, p consists of
a complex vector space $\textbf{p}[I]$ for each finite set $I$, and a
bijective linear map $\textbf{p}[\sigma]:\textbf{p}[I]\to\textbf{p}[J]$ for
each bijection $\sigma:J\to I$ such that composition of bijections is
preserved.

A highly structured theory of gebras111 meaning (co/bi/Hopf)algebras and Lie
(co)algebras internal to vector species has been developed by Aguiar-Mahajan
[AM10], [AM13], building on the work of Barratt [Bar78], Joyal [Joy86],
Schmitt [Sch93], Stover [Sto93b], and others. For the internalization, one
uses the Day convolution monoidal product
$\textbf{p}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{q}$
with respect to disjoint union and tensor product, given by

$\textbf{p}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{q}[I]=\textbf{p}\otimes_{\text{Day}}\textbf{q}[I]=\bigoplus_{S\sqcup
T=I}\textbf{p}[S]\otimes\textbf{q}[T].$

This may be viewed as a categorification of the Cauchy product of formal power
series.222 from the perspective of S-colored (co)operads, as defined in e.g.
[Pet13, Section 3], there is an equivalent description of these gebras as
(co)algebras over the left (co)action (co)monads of the (co)operads
$\textbf{Com}^{(\ast)}$, $\textbf{Ass}^{(\ast)}$, $\textbf{Lie}^{(\ast)}$
[AM10, Appendix B.5], which relates the gebras of this paper to structures
such as cyclic operads, which already appear in mathematical physics Various
decategorifications of Aguiar-Mahajan’s theory recovers the plethora of graded
combinatorial Hopf algebras which have been studied [AM10, Chapter 15].

On the other hand, quantum field theory (QFT) may be viewed as a kind of
modern infinite dimensional calculus. Perturbative quantum field theory (pQFT)
is the part of QFT which considers Taylor series approximations of smooth
functions. By an argument of Dyson [Dys52], Taylor series of realistic pQFTs
are expected to have vanishing radius of convergence. Nevertheless, if an
actual smooth function of a non-perturbative quantum field theory is being
approximated, then they are asymptotic series, and so one might expect their
truncations to agree to reasonable precision with experiment. This is indeed
the case.

There are two main synthetic approaches to (non-perturbative) QFT, which grew
out of the failure to make sense of the path integral analytically. There is
functorial quantum field theory (FQFT), which formalizes the Schrödinger
picture by assigning time evolution operators to cobordisms between
spacetimes. There is also algebraic quantum field theory (AQFT), going back to
[HK64], which formalizes the Heisenberg picture by assigning
$\text{C}^{\ast}$-algebras of observables to regions of spacetime. Low
dimension examples of AQFTs/Wightman field theories were rigorously
constructed in seminal work of Glimm-Jaffe and others [GJ68], [CJ70], [GJS74].

Perturbative algebraic quantum field theory (pAQFT) [Rej16], [Düt19], [Sch20,
nLab], due to Brunetti, Dütsch, Fredenhagen, Hollands, Rejzner, Wald, and
others, is (mathematically precise, realistic) pQFT based on causal
perturbation theory [Ste71], [EG73], [Sch95], due to Stückelberg, Bogoliubov,
Steinmann, Epstein, Glaser, Stora, and others. See [Düt19, Foreword] for an
account of the history. Following [IS78], [BF00], [DF01], in which one takes
the algebraic adiabatic limit to handle IR-divergences, pAQFT satisfies the
Haag-Kastler axioms of AQFT, but with $\text{C}^{\ast}$-algebras replaced by
formal power series $\ast$-algebras, reflecting the fact that pQFT deals with
Taylor series approximations. In this paper, we show that the construction and
structure of these formal power series algebras is naturally described in
terms of gebra theory internal to species.

For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on
Minkowski spacetime $\mathcal{X}\cong\mathbb{R}^{p,1}$, $p\in\mathbb{N}$
(pAQFT may be applied in more general settings, see e.g. [Hol08]). Therefore
for us, an off-shell field configuration $\Phi$ is a smooth function

$\Phi:\mathcal{X}\to\mathbb{R},\qquad x\mapsto\Phi(x).$

In particular, we do not impose conditions on the asymptotic behaviour of
$\Phi$ at infinite times. Let $\mathcal{F}_{\text{loc}}$ denote the space of
local observables $\emph{{A}}\in\mathcal{F}_{\text{loc}}$; these are
functionals of field configurations which are obtained by integrating
polynomials in $\Phi$ and its derivatives against bump functions on
$\mathcal{X}$. Let $\mathcal{F}$ denote the commutative $\ast$-algebra of
microcausal polynomial observables $\emph{{O}}\in\mathcal{F}$; these are
polynomial functionals of field configurations satisfying a microlocal-
theoretic condition known as microcausality, with multiplication the pointwise
multiplication of functionals, sometimes called the normal-ordered product.
Then $\mathcal{F}[[\hbar]]$ is a formal power series $\ast$-algebra in formal
Planck’s constant $\hbar$, called the (abstract, off-shell) Wick algebra, with
multiplication the Moyal star product for the Wightman propagator
$\Delta_{\text{H}}$ of the Klein-Gordan field

$\mathcal{F}[[\hbar]]\otimes\mathcal{F}[[\hbar]]\to\mathcal{F}[[\hbar]],\qquad\emph{{O}}_{1}\otimes\emph{{O}}_{2}\mapsto\emph{{O}}_{1}\star_{\text{H}}\\!\emph{{O}}_{2},$

sometimes called the operator product.

Perhaps the most fundamental Hopf monoid of Aguiar-Mahajan’s theory is the
cocommutative Hopf algebra333 we say ‘algebra’ and not ‘monoid’ since vector
species form a linear category of compositions $\boldsymbol{\Sigma}$, see
Section 1.2, which is a Hopf monoid internal to vector species defined with
respect to the Day convolution. (More familiar is perhaps a certain
decategorification of $\boldsymbol{\Sigma}$, which is the graded Hopf algebra
of noncommutative symmetric functions NSym, see [AM10, Section 17.3].) A
composition $F$ of $I$ is a surjective function of the form

$F:I\to\\{1,\dots,k\\},\qquad\text{for some}\quad k\in\mathbb{N}.$

The ordering $1>\dots>k$ is understood, so that $F$ models the $k^{\text{th}}$
ordinal with $I$-marked points. We let $S_{j}=F^{-1}(j)$, called the lumps of
$F$, and write $F=(S_{1},\dots,S_{k})$. Each component
$\boldsymbol{\Sigma}[I]$ is the space of formal linear combinations of
compositions $F$ of $I$,

$\boldsymbol{\Sigma}[I]=\Big{\\{}\mathtt{a}=\sum_{F}c_{F}\mathtt{H}_{F}\
\big{|}\ c_{F}\in\mathbb{C}\Big{\\}}.$

The multiplication

$\mu_{S,T}:\boldsymbol{\Sigma}[S]\otimes\boldsymbol{\Sigma}[T]\to\boldsymbol{\Sigma}[I],\qquad\mathtt{H}_{F}\otimes\mathtt{H}_{G}\mapsto\mathtt{H}_{FG}$

is the linearization of concatenating compositions (‘gluing’ via ordinal sum),
and the comultiplication

$\Delta_{S,T}:\boldsymbol{\Sigma}[I]\to\boldsymbol{\Sigma}[S]\otimes\boldsymbol{\Sigma}[T],\qquad\mathtt{H}_{F}\mapsto\mathtt{H}_{F|_{S}}\otimes\mathtt{H}_{F|_{T}}$

is the linearization of restricting compositions to subsets (‘forgetting
marked points’), where $S\sqcup T=I$.

Aspects of $\boldsymbol{\Sigma}$ have appeared in the physics literature as
follows. Firstly, Epstein-Glaser-Stora’s algebra of proper sequences [EGS75,
Section 4.1] is the action of $\boldsymbol{\Sigma}$ on itself by Hopf powers,
called the Tits product [AM13, Section 13], going back to Tits [Tit74].
Secondly, the primitive part
$\textbf{Zie}=\mathcal{P}(\boldsymbol{\Sigma})$444 the name ‘Zie’ comes from
[AM17], which is a Lie algebra internal to species, is essentially the
Steinmann algebra from e.g. [Rue61, Section 6], [BL75, Section III.1]. More
precisely, the Steinmann algebra is a graded Lie algebra based on the
structure map of the adjoint realization of Zie, see Section 1.7. Thirdly and
fourthly, and outside the scope of this paper, see below regarding work of
Losev-Manin and Feynman integrals.

The central idea of this paper is to formalize the construction of a system of
interacting time-ordered products in causal perturbation theory as the
construction of a homomorphism $\widetilde{\text{T}}$ of algebras internal to
species of the form

$\widetilde{\text{T}}:\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}.$

We describe this construction in a clean abstract setting in Section 3.1, and
then specialize to QFT in Section 7. Here, $\otimes$ is the Hadamard monoidal
product (=componentwise tensoring),
$\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}$ is the species given by
$I\mapsto(\mathcal{F}_{\text{loc}}[[\hbar]])^{\otimes I}$, and
$\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}$ is the algebra in species which
has the Wick algebra, with formal coupling constant g adjoined, in each
$I$-component,

$\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}[I]=(\mathcal{F}_{\text{loc}}[[\hbar]])^{\otimes
I},\qquad\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}[I]=\mathcal{F}[[\hbar,\emph{{g}}]].$

It follows that the data of a system of products $\widetilde{\text{T}}$ is
equivalently a homomorphism of $\mathbb{C}$-algebras

$\hat{\boldsymbol{\Sigma}}(\mathcal{F}_{\text{loc}}[[\hbar]])\to\mathcal{F}[[\hbar,\emph{{g}}]]$

where $\hat{\boldsymbol{\Sigma}}(-):\textsf{Vec}\to\textsf{Vec}$ is the
analytic endofunctor, or Schur functor, on vector spaces associated to
$\boldsymbol{\Sigma}$ [AM10, Section 19.1.2].555 the hat
$\hat{\boldsymbol{\Sigma}}$ is meant to suggest a kind of categorified Fourier
transform Decategorified versions of this formalization appear in graded Hopf
algebra approaches to pQFT [Bro09], [Bor11, p. 635]. In particular, there is
an interpretation of the Moyal deformation quantization in terms of Laplace
pairings (=coquasitriangular structures) [Fau01], [Bro09, Section 2.4].

Also related is the notion of a Losev-Manin cohomological field theory [LM00,
Theorem 3.3.1], [SZ11, Definition 1.3], where finite ordinals are replaced by
strings of Riemann spheres glued at the poles, giving a Hopf monoid structure
on the toric variety of the permutohedron, and $\boldsymbol{\Sigma}$ is
replaced by the ordinary homology of this toric variety. The Hopf monoid
structure of this toric variety is also central to modern approaches to
Feynman integrals [Bro17, p.6], [Sch18]. We shall study this Hopf monoid in
future work.

Explicitly, the homomorphism $\widetilde{\text{T}}$ consists of component
linear maps

$\widetilde{\text{T}}_{I}:\boldsymbol{\Sigma}[I]\otimes(\mathcal{F}_{\text{loc}}[[\hbar]])^{\otimes
I}\to\mathcal{F}[[\hbar,\emph{{g}}]],\qquad\mathtt{H}_{F}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}}\mapsto\widetilde{\text{T}}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}})$

for each finite set $I=\\{i_{1},\dots,i_{n}\\}$. This homomorphism should also
satisfy causal factorization, which says

$\widetilde{\text{T}}_{I}(\mathtt{a}\otimes\emph{{A}}_{i_{1}}\otimes\dots\emph{{A}}_{i_{n}})=\widetilde{\text{T}}_{I}(\\!\\!\\!\\!\underbrace{\mathtt{a}\triangleright\mathtt{H}_{G}}_{\text{Tits
product}}\\!\\!\\!\\!\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}})\qquad\text{for
all}\quad\mathtt{a}\in\boldsymbol{\Sigma}[I]$

whenever the local observables $\emph{{A}}_{i_{1}},\dots,\emph{{A}}_{i_{n}}$
respect the ordering of $I$ induced by the composition $G$, see 7.1.
Additional properties are often included, such as translation equivariance.

We can curry $\widetilde{\text{T}}$ with respect to the internal hom
$\mathcal{H}(-,-)$ for the Hadamard product, giving a homomorphism of algebras

$\boldsymbol{\Sigma}\to\mathcal{H}(\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]},\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}),\qquad\mathtt{H}_{F}=\mathtt{H}_{(S_{1},\dots,S_{k})}\mapsto\widetilde{\text{T}}(S_{1})\dots\widetilde{\text{T}}(S_{k}).$

The resulting linear maps

$\widetilde{\text{T}}(S_{1})\dots\widetilde{\text{T}}(S_{k}):(\mathcal{F}_{\text{loc}}[[\hbar]])^{\otimes
I}\to\mathcal{F}[[\hbar,\emph{{g}}]]$

are called interacting generalized time-ordered products. For each choice of a
field polynomial, the curried homomorphism is a ‘representation’ of
$\boldsymbol{\Sigma}$ as $\mathcal{F}[[\hbar,\emph{{g}}]]$-valued generalized
functions on $\mathcal{X}^{I}$, called operator-valued distributions since the
Wick algebra is often represented on a Hilbert space. The composition of the
time-ordered products $\widetilde{\text{T}}(I)$ with the Hadamard vacuum state

$\langle-\rangle_{0}:\mathcal{F}[[\hbar,\emph{{g}}]]\to\mathbb{C}[[\hbar,\emph{{g}}]],\qquad\emph{{O}}\mapsto\emph{{O}}(\Phi=0)$

are then translation invariant $\mathbb{C}[[\hbar,\emph{{g}}]]$-valued
generalized functions

$\text{G}_{I}:\mathcal{X}^{I}\to\mathbb{C}[[\hbar,\emph{{g}}]],\qquad(x_{i_{1}},\dots,x_{i_{n}})\mapsto\text{G}_{I}(x_{i_{1}},\dots,x_{i_{n}})$

called time-ordered $n$-point correlation functions. After taking the
adiabatic limit, and in the presence of vacuum stability, these functions may
be interpreted as the probabilistic predictions made by the pQFT of the
outcomes of scattering experiments, called scattering amplitudes, see Section
9. However, their values are formal power series in $\hbar$ and g, and so have
to be truncated.

Central to Aguiar-Mahajan’s work is the interpretation of
$\boldsymbol{\Sigma}$ (and other Hopf monoids) in terms of the geometry of the
type $A$ reflection hyperplane arrangement, called the (essentialized) braid
arrangement

$\text{Br}[I]=\big{\\{}\\{x_{i_{1}}-x_{i_{2}}=0\\}\subseteq\\!\\!\\!\\!\\!\\!\\!\underbrace{\mathbb{R}^{I}/\mathbb{R}\twoheadleftarrow\mathbb{R}^{I}}_{\text{quotient
by translations}}\\!\\!\\!\\!\\!\\!\\!:(i_{1},i_{2})\in I^{2},\ i_{1}\neq
i_{2}\big{\\}}.$

In causal perturbation theory, the braid arrangement appears as the space of
time components of configurations $\mathcal{X}^{I}$ modulo translational
symmetry [Rue61, Section 2], and the reflection hyperplanes are the coinciding
interaction points. Every real hyperplane arrangement A has a corresponding
adjoint hyperplane arrangement $\text{A}^{\vee}$ [AM17, Section 1.9.2]. The
free vector space $\mathbb{R}I$ on $I$ is naturally
$\operatorname{Hom}(\mathbb{R}^{I},\mathbb{R})$, and so the adjoint of the
braid arrangement is given by

$\text{Br}^{\vee}[I]=\bigg{\\{}\Big{\\{}\sum_{i\in S}x_{i}=\sum_{i\in
T}x_{i}=0\Big{\\}}\subseteq\underbrace{\operatorname{Hom}(\mathbb{R}^{I}/\mathbb{R},\mathbb{R})\hookrightarrow\mathbb{R}I}_{\text{sum-
zero subspace}}:(S,T)\in 2^{I},\ S,T\neq\emptyset\bigg{\\}}.$

In causal perturbation theory, the adjoint braid arrangement appears as the
space of energy components [Rue61, Section 2], and the hyperplanes correspond
to subsets going ‘on-shell’. The spherical representation of the adjoint braid
arrangement is called the Steinmann sphere, or Steinmann planet, e.g. [Eps16,
Figure A.4]. The chambers of the adjoint braid arrangement are indexed by
combinatorial gadgets called cells $\mathcal{S}$ [EGS75, Definition 6], also
known as maximal unbalanced families [BMM+12] and positive sum systems
[Bjo15].

The primitive part Lie algebra $\textbf{Zie}=\mathcal{P}(\boldsymbol{\Sigma})$
(together with its dual Lie coalgebra $\textbf{Zie}^{\ast}$) has a natural
geometric realization over the adjoint braid arrangement [Rue61, Section 6],
[Ocn18, Lecture 33], [LNO19], [NO19], which results in cells $\mathcal{S}$
corresponding to certain special primitive elements
$\mathtt{D}_{\mathcal{S}}\in\textbf{Zie}[I]$, see Section 1.5. The special
elements were named Dynkin elements by Aguiar-Mahajan [AM17, Section 14.1 and
14.9.8]. It is shown in [NO19] that the Dynkin elements span Zie, but they are
not linearly independent. The relations which are satisfied by the Dynkin
elements are known as the Steinmann relations [Ste60b, Equation 44], see
Section 1.6, first studied by Steinmann in settings where
$\boldsymbol{\Sigma}$ is represented as operator-valued distributions. More
recently, they have been studied in the context scattering amplitudes, where
they appear to be related to cluster algebras [DFG18], [CHDD+19], [CHDD+20].

If we restrict a curried system of interacting generalized time-ordered
products to the primitive part Zie, then we obtain a Lie algebra homomorphism

$\textbf{Zie}\to\mathcal{H}(\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]},\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}),\qquad\mathtt{D}_{\mathcal{S}}\mapsto\widetilde{\text{R}}_{\mathcal{S}}.$

The operator-valued distributions $\widetilde{\text{R}}_{\mathcal{S}}$ which
are the images of the Dynkin elements $\mathtt{D}_{\mathcal{S}}$ are the
interacting generalized retarded products of the system, see e.g. [Ste60b],
[Ara61], [EG73, Equation 79]. In this paper, we give an exposition of the
Steinman algebra and Steinmann relations in Section 1.4, Section 1.5 and
Section 1.6.

Let $\textbf{L}\hookrightarrow\boldsymbol{\Sigma}$ be the Hopf subalgebra of
linear orders (=compositions with singleton lumps), and let
$\textbf{E}^{\ast}\hookrightarrow\boldsymbol{\Sigma}$ be the subcoalgebra of
compositions with one lump. Then we have the dictionary in Figure 1 between
products/vacuum expectation values and elements of $\boldsymbol{\Sigma}$. In
the commutative setting before Moyal deformation quantization, the species X
and E are similarly related to the smeared field and polynomial observables,
see Section 6.

|  | spanning set
---
| operator-valued distributions
---
| vacuum expectation values
---
| $\textbf{E}^{\ast}$
---
| universal series
---
$\mathtt{G}_{I}$
| time-ordered product
---
$\text{T}(I)$
| time-ordered $n$-point
---
function
| L
---
|  $\mathtt{H}$-basis linear orders
---
$\mathtt{H}_{\ell}$
| $\text{T}(i_{1})\dots\text{T}(i_{n})$
---
| Wightman $n$-point
---
functions
| $\boldsymbol{\Sigma}$
---
|  $\mathtt{H}$-basis set compositions
---
$\mathtt{H}_{F}$
| generalized time-ordered products
---
$\text{T}(S_{1})\dots\text{T}(S_{k})$
| generalized time-ordered
---
functions
| Zie
---
| Dynkin elements
---
$\mathtt{D}_{\mathcal{S}}$
| generalized retarded products
---
$\text{R}_{\mathcal{S}}$
| generalized retarded
---
functions
Figure 1. Dictionary between products/vacuum expectation values and elements
of the Hopf algebra $\boldsymbol{\Sigma}$.

In Section 4.1 and Section 8, we formalize the _perturbation_ of time-ordered
products in casual perturbation theory as follows. Our starting point is a
fully normalized system of generalized time-ordered products, that is a
homomorphism of algebras

$\text{T}:\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))}$

satisfying causal factorization, and such that the singleton components
$\text{T}_{\\{i\\}}$ are the natural inclusion

$\mathcal{F}_{\text{loc}}[[\hbar]]\hookrightarrow\mathcal{F}((\hbar)),\qquad\emph{{A}}\mapsto\,\,:\\!\emph{{A}}:.$

The corresponding operator-valued distributions are determined everywhere on
$\mathcal{X}^{I}$ by causal factorization, apart from on the fat diagonal
(=coinciding interaction points). In particular, off the fat diagonal, the
time-ordered products $\text{T}(I)$ are given by the Moyal star product
$\star_{\text{F}}$ with respect to the Feynman propagator $\Delta_{\text{F}}$
for the Klein-Gordon field. The terms of the product $\star_{\text{F}}$ may be
encoded in finite multigraphs, i.e. Feynman graphs. The remaining inherent
ambiguity means one has to make choices when extending the $\text{T}(I)$ to
the fat diagonal, and these choices form a torsor of the Stückelberg-Petermann
renormalization group. This is Stora’s elaboration [PS16], [Sto93a], [BF00] on
Stückelberg-Bogoliubov-Epstein-Glaser normalization [EG73], which constructs
the $\text{T}(I)$ inductively in $n=|I|$. We leave species-theoretic aspects
of renormalization, and possible connections to Connes-Kreimer theory [Pin00],
[GBL00], [BK05], [DFKR14], to future work.

In the original formulation by Tomonaga, Schwinger, Feynman and Dyson, would-
be time-ordered products are obtained by informally multiplying Wick algebra
products by step functions, which is in general ill-defined by Hörmander’s
criterion. This leads to the divergence of individual terms of the formal
power series, called UV-divergences. Then informal methods are used to obtain
finite values from these infinite terms [Sch95, Preface and Section 4.3].

The exponential species E, given by $\textbf{E}[I]=\mathbb{C}$ and
$1_{\mathbb{C}}\in\textbf{E}[I]$ denoted $\mathtt{H}_{I}$, has the structure
of an algebra in species by linearizing taking unions of sets,

$\mu_{S,T}:\textbf{E}[S]\otimes\textbf{E}[T]\to\textbf{E}[I],\qquad\mathtt{H}_{S}\otimes\mathtt{H}_{T}\mapsto\mathtt{H}_{I}.$

An E-module $\textbf{m}=(\textbf{m},\rho)$ is an associative and unital
morphism

$\rho:\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{m}\to\textbf{m}$

for m a species. Moreover, taking the inverse of $\mu_{S,T}$ as the
comultiplication turns E into a connected (co)commutative bialgebra, and so
the category of E-modules $\textsf{Rep}(\textbf{E})$ is a symmetric monoidal
category with monoidal product the Cauchy product of E-modules. In particular,
we may consider Hopf/Lie algebras internal to $\textsf{Rep}(\textbf{E})$,
which we call Hopf/Lie E-algebras.

The retarded $Y\downarrow(-)$ and advanced $Y\uparrow(-)$ Steinmann arrows are
(we formalize as) raising operators on $\boldsymbol{\Sigma}$, whose precise
definition is due to Epstein-Glaser-Stora [EGS75, p.82-83]. They define two
E-module structures on $\boldsymbol{\Sigma}$,

$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma},\quad\mathtt{H}_{Y}\otimes\mathtt{H}_{F}\,\mapsto\,Y\downarrow\mathtt{H}_{F}\qquad\text{and}\qquad\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma},\quad\mathtt{H}_{Y}\otimes\mathtt{H}_{F}\,\mapsto\,Y\uparrow\mathtt{H}_{F}.$

See Section 2.2. In particular, the retarded arrow is generated by putting
$\\{\ast\\}\downarrow\mathtt{H}_{(I)}=-\mathtt{H}_{(\ast,I)}+\mathtt{H}_{(\ast
I)}$.777 $(\ast I)$ denotes the composition of $\\{\ast\\}\sqcup I$ which has
a single lump Then

$Y\\!\downarrow\mathtt{H}_{(I)}=\underbrace{\sum_{Y_{1}\sqcup
Y_{2}=Y}\mu_{Y_{1},Y_{2}\sqcup
I}\big{(}\text{s}(\mathtt{H}_{(Y_{1})})\otimes\mathtt{H}_{(Y_{2}\sqcup
I)}\big{)}}_{\text{denoted $\mathtt{R}_{(Y;I)}$}}$

where $\text{s}:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ is the antipode of
$\boldsymbol{\Sigma}$. The Steinmann arrows were first studied by Steinmann
[Ste60b, Section 3], where $\boldsymbol{\Sigma}$ is represented as operator-
valued distributions. Here, the operator-valued distribution which corresponds
to $\mathtt{R}_{(Y;I)}\in\boldsymbol{\Sigma}[Y\sqcup I]$ is called the
retarded product $\text{R}(Y;I)$.888 note that some authors, e.g. [Düt19],
call $\text{R}(Y;i)$ the retarded product, and then call $\text{R}(Y;I)$ the
generalized retarded product

Since $\\{\ast\\}\downarrow(-)$ is a commutative biderivation of
$\boldsymbol{\Sigma}$ (2.1), the retarded Steinmann arrow gives
$\boldsymbol{\Sigma}$ the structure of a Hopf E-algebra, and Zie the structure
of a Lie E-algebra (similarly for the advanced arrow). There is an interesting
description of these Lie E-algebras in terms of the adjoint braid arrangement,
see Section 2.4. The Steinmann arrows are “two halves” of the restricted
adjoint representation
$\textbf{L}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$
of $\boldsymbol{\Sigma}$, which is reflected in [Ste60b, Equation 13]. This
directly corresponds to how the retarded $\Delta_{-}$ and advanced
$\Delta_{+}$ propagators are two halves of the causal propagator
$\Delta_{\text{S}}=\Delta_{+}-\Delta_{-}$.

Let
$\mathcal{H}^{\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}}(-,-)$
denote the internal hom for the Cauchy product of species, and let

$(-)^{\textbf{E}}=\mathcal{H}^{\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}}(\textbf{E},-).$

See Section 2.3 for a more explicit definition. See also [Nor20, Section 2]
for more details here regarding the differentiation between the j -colored
sets $I$ (physically, the source field) and the g-colored sets $Y$
(physically, the coupling constant). Then $(-)^{\textbf{E}}$ is an endofunctor
on species, which is lax monoidal with respect to the Cauchy product.
Therefore $\boldsymbol{\Sigma}^{\textbf{E}}$ is naturally an algebra, with
multiplication inherited from $\boldsymbol{\Sigma}$. Then, by currying the
retarded Steinmann action
$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$,
we obtain a homomorphism
$\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\textbf{E}}$. Similarly for the
setting with decorations, given a choice of adiabatically switched interaction
action functional
$\emph{{S}}_{\text{int}}\in\mathcal{F}_{\text{loc}}[[\hbar]]$, after acting
with the retarded Steinmann arrows and currying, we obtain the homomorphism

$\displaystyle\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}$
$\displaystyle\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]})^{\textbf{E}}$
$\displaystyle\mathtt{H}_{F}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}}$
$\displaystyle\mapsto\,\sum_{r=0}^{\infty}\underbrace{\downarrow\dots\downarrow}_{\text{$r$
times}}\mathtt{H}_{F}\otimes\underbrace{\emph{{S}}_{\text{int}}\otimes\dots\otimes\emph{{S}}_{\text{int}}}_{\text{$r$
times}}\,\otimes\,\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}}.$

Compare this with the formalism for creation-annihilation operators in [AM10,
Chapter 19]. Then, finally, the corresponding system of perturbed interacting
time-ordered products $\widetilde{\text{T}}$ is given by composing this
homomorphism with the image of T under the endofunctor $(-)^{\textbf{E}}$,

$\widetilde{\text{T}}:\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]})^{\textbf{E}}\xrightarrow{\text{T}^{\textbf{E}}}(\textbf{U}_{\mathcal{F}((\hbar))})^{\textbf{E}}\cong\textbf{U}_{\mathcal{F}((\hbar))[[\emph{{g}}]]}.$

See Section 4.1. It is a theorem of pAQFT that this does indeed land in
$\textbf{U}_{\mathcal{F}[[\hbar,\emph{{g}}]]}$.

Finally, in Section 3.3 and Section 7, we formalize S-matrices, or time-
ordered exponentials, as follows. Let $\operatorname{Hom}(-,-)$ denote the
external hom for species, which lands in vector spaces Vec. We let

$\mathscr{S}(-)=\operatorname{Hom}(\textbf{E},-).$

This is lax monoidal with respect to the Cauchy product. In the presence of a
generic system of products on an algebra a,

$\varphi:\textbf{a}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},$

series $\mathtt{s}\in\mathscr{S}(\textbf{a})$ of a

$\mathtt{s}:\textbf{E}\to\textbf{a},\qquad\mathtt{H}_{I}\mapsto\mathtt{s}_{I}$

induce
$\mathscr{S}(\textbf{U}_{\mathcal{A}})\cong\mathcal{A}[[\emph{{j}}\,]]$-valued
functions $\mathcal{S}_{\mathtt{s}}$ on $V$ as follows,

$\mathcal{S}_{\mathtt{s}}:V\to\mathcal{A}[[\emph{{j}}\,]],\qquad\emph{{A}}\mapsto\mathcal{S}_{\mathtt{s}}(\emph{{j}}\,\\!\emph{{A}}):=\sum_{n=0}^{\infty}\dfrac{\emph{{j}}\,^{n}}{n!}\varphi_{n}(\mathtt{s}_{n}\otimes\underbrace{\emph{{A}}\otimes\dots\otimes\emph{{A}}}_{\text{$n$
times}}).$

If $\varphi$ is a homomorphism of algebras, then

$\mathcal{S}_{(-)}:\mathscr{S}(\textbf{a})\to\text{Func}(V,\mathcal{A}[[\emph{{j}}\,]])$

is a homomorphism of $\mathbb{C}$-algebras. As a basic example, if we put
$\textbf{a}=\textbf{E}$, $\mathcal{A}=C^{\infty}(V^{\ast})$, and set
$\emph{{j}}\,=1$ at the end, then one can recover the classical exponential
function in this way.

For $c\in\mathbb{C}$, the so-called (scaled) universal series $\mathtt{G}(c)$
of $\boldsymbol{\Sigma}$ is given by sending each finite set to the (scaled)
composition with one lump,

$\mathtt{G}(c):\textbf{E}\to\boldsymbol{\Sigma},\qquad\mathtt{H}_{I}\mapsto\mathtt{G}(c)_{I}:=c^{n}\,\mathtt{H}_{(I)}.$

If we set $c=1/\text{i}\hbar$, then the function
$\mathcal{S}=\mathcal{S}_{\mathtt{G}(1/\text{i}\hbar)}$ above for a fully
normalized system of generalized time-ordered products
$\text{T}:\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))}$
recovers the usual perturbative S-matrix scheme of pAQFT,

$\mathcal{S}:\mathcal{F}_{\text{loc}}[[\hbar]]\to\mathcal{F}((\hbar))[[\emph{{j}}\,]],\qquad\emph{{A}}\mapsto\mathcal{S}(\emph{{j}}\,\\!\emph{{A}})=\sum_{n=0}^{\infty}\bigg{(}\dfrac{1}{\text{i}\hbar}\bigg{)}^{n}\dfrac{\emph{{j}}\,^{n}}{n!}\text{T}_{n}(\mathtt{H}_{(n)}\otimes\underbrace{\emph{{A}}\otimes\dots\otimes\emph{{A}}}_{\text{$n$
times}}).$

The image of $\mathcal{S}(\emph{{j}}\,\\!\emph{{A}})$ after applying
perturbation by the retarded Steinmann arrow and a choice of interaction
$\emph{{S}}_{\text{int}}\in\mathcal{F}_{\text{loc}}[[\hbar]]$ is

$\mathcal{Z}_{\emph{{g}}\emph{{S}}_{\text{int}}}(\emph{{j}}\,\\!\emph{{A}})=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\bigg{(}\dfrac{1}{\text{i}\hbar}\bigg{)}^{\\!r+n}\dfrac{\emph{{g}}^{r}\emph{{j}}\,^{n}}{r!\,n!}\,\text{R}_{r;n}(\underbrace{\emph{{S}}_{\text{int}}\otimes\dots\otimes\emph{{S}}_{\text{int}}}_{\text{$r$
times}}\,;\,\underbrace{\emph{{A}}\otimes\dots\otimes\emph{{A}}}_{\text{$n$
times}})$

where, by our previous expression for
$\mathtt{R}_{(Y;I)}=Y\downarrow\mathtt{H}_{(I)}$ (and letting
$\overline{\text{T}}$ denote the precomposition of T with the antipode of
$\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}$),
we have

$\text{R}_{Y;I}(\emph{{S}}_{\text{int}}^{\,Y};\emph{{A}}^{I})=\text{T}_{Y\sqcup
I}(Y\downarrow\mathtt{H}_{(I)}\otimes\emph{{S}}_{\text{int}}^{\,Y}\otimes\emph{{A}}^{I})=\sum_{Y_{1}\sqcup
Y_{2}=Y}\overline{\text{T}}_{Y_{1}}(\emph{{S}}_{\text{int}}^{\,Y_{1}})\star_{\text{H}}\text{T}_{Y_{2}\sqcup
I}(\emph{{S}}_{\text{int}}^{\,Y_{2}}\otimes\emph{{A}}^{I}).$

Then, since

$\mathcal{S}_{(-)}:\mathscr{S}(\boldsymbol{\Sigma})\to\text{Func}\big{(}\mathcal{F}_{\text{loc}}[[\hbar]],\mathcal{F}((\hbar))[[\emph{{g}}]]\big{)}$

is a homomorphism of $\mathbb{C}$-algebras, it follows that
$\mathcal{Z}_{\emph{{g}}\emph{{S}}_{\text{int}}}$ is given by

$\mathcal{Z}_{\emph{{g}}\emph{{S}}_{\text{int}}}(\emph{{j}}\,\\!\emph{{A}})=\mathcal{S}^{-1}(\emph{{g}}\emph{{S}}_{\text{int}})\star_{\text{H}}\mathcal{S}(\emph{{g}}\emph{{S}}_{\text{int}}+\emph{{j}}\,\\!\emph{{A}}).$

This is the generating function, or partition function, for time-ordered
products of interacting field observables, see e.g. [EG73, Section 8.1],
[DF01, Section 6.2], going back to Bogoliubov [BS59, Chapter 4]. In this
paper, we arrive at the generating function
$\mathcal{Z}_{\emph{{g}}\emph{{S}}_{\text{int}}}$ through purely Hopf-
theoretic considerations. However, it was originally motivated by attempts to
make sense of the path integral synthetically. For some recent developments,
see [Col16], [HR20].

#### Structure.

This paper is divided into two parts. In part one, we focus on developing
theory for the Hopf algebra of compositions $\boldsymbol{\Sigma}$ and its
primitive part Zie. In part two, we specialize to pAQFT for the case of a real
scalar field on Minkowski spacetime.

#### Acknowledgments.

We thank Adrian Ocneanu for his support and useful discussions. This paper
would not have been written without Nick Early’s discovery that certain
relations appearing in Ocneanu’s work were known in quantum field theory as
the Steinmann relations. We thank Yiannis Loizides and Maria Teresa Chiri for
helpful discussions during an early stage of this project. We thank Arthur
Jaffe for his support, useful suggestions, and encouragement to pursue this
topic. We thank Penn State maths department for their continued support.

##  Part I Hopf Monoids

###  1\. The Algebras

We recall the Hopf algebra of compositions $\boldsymbol{\Sigma}$, together
with its Lie algebra of primitive elements
$\textbf{Zie}\hookrightarrow\boldsymbol{\Sigma}$. We show that
$\boldsymbol{\Sigma}$ and Zie are naturally algebras over the exponential
species E. This will be a species-theoretic formalization of mathematical
structure discovered by Steinmann [Ste60b] and Epstein-Glaser-Stora [EGS75],
which, combined with a certain ‘perturbation of systems of products’
construction using the E-action, will recover the perturbative construction of
interacting fields in pAQFT, as in [EG73, Section 8.1], [DF01, Section 6.2],
going back to Bogoliubov [BS59, Chapter 4].

####  1.1. Compositions

Let $I$ be a finite set of cardinality $n$. We think of $I$ as having ‘color’
j (physically, the source field). As a particular example of the set $I$, we
have the set of integers $[n]:=\\{1,\dots,n\\}$ (formally, we have picked a
section of the decategorification functor $I\mapsto n$). For $k\in\mathbb{N}$,
let

$(k):=\\{1,\dots,k\\}$

equipped with the ordering $1>\dots>k$. A _composition_ $F$ of $I$ of _length_
$l(F)=k$ is a surjective function $F:I\to(k)$. The set of all compositions of
$I$ is denoted $\Sigma[I]$,

$\Sigma[I]:=\bigsqcup_{k\in\mathbb{N}}\big{\\{}\text{surjective functions}\
F:I\to(k)\big{\\}}.$

We often denote compositions by $k$-tuples

$F=(S_{1},\dots,S_{k})$

where $S_{j}:=F^{-1}(j)$, $1\leq j\leq k$. The $S_{j}$ are called the _lumps_
of $F$. In particular, we have the length one composition $(I)$ for
$I\neq\emptyset$, and the length zero composition $(\,)$ which is the unique
composition of the empty set. The _opposite_ $\bar{F}$ of $F$ is defined by

$\bar{F}:=(S_{k},\dots,S_{1}),\qquad\text{i.e.}\quad\bar{F}^{-1}(j)=F^{-1}(k+1-j).$

Given a decomposition $I\\!=S\sqcup T$ of $I$ ($S,T$ can be empty), for
$F=(S_{1},\dots,S_{k})$ a composition of $S$ and $G=(T_{1},\dots,T_{l})$ a
composition of $T$, their _concatenation_ $FG$ is the composition of $I$ given
by

$FG:=(S_{1},\dots,S_{k},T_{1},\dots,T_{l}).$

For $S\subseteq I$ and $F=(S_{1},\dots,S_{k})\in\Sigma[I]$, the _restriction_
$F|_{S}$ of $F$ to $S$ is the composition of $S$ given by

$F|_{S}:=(S_{1}\cap S,\dots,S_{k}\cap S)_{+}$

where $(-)_{+}$ means we delete any sets from the list which are the empty
set.

For compositions $F,G\in\Sigma[I]$, we write $G\leq F$ if $G$ can be obtained
from $F$ by iteratively merging contiguous lumps. Given compositions $G\leq F$
with $G=(T_{1},\dots,T_{l})$, we let

$l(F/G):=\prod^{k}_{j=1}l(F|_{T_{j}})\qquad\text{and}\qquad(F/G)!:=\prod^{k}_{j=1}l(F|_{T_{j}})!\,.$

####  1.2. The Cocommutative Hopf Monoid of Compositions

Let

$\boldsymbol{\Sigma}[I]:=\big{\\{}\text{formal $\mathbb{C}$-linear
combinations of compositions of $I$}\big{\\}}.$

The vector space $\boldsymbol{\Sigma}[I]$ is naturally a right module over the
symmetric group on $I$, and these actions extend to a contravariant functor
from the category S of finite sets and bijections into the category Vec of
vector spaces over $\mathbb{C}$,

$\boldsymbol{\Sigma}:\textsf{S}^{\text{op}}\to\textsf{Vec},\qquad
I\mapsto\boldsymbol{\Sigma}[I].$

For $F$ a composition of $I$, let $\mathtt{H}_{F}\in\boldsymbol{\Sigma}[I]$
denote the basis element corresponding to $F$. The sets
$\\{\mathtt{H}_{F}:F\in\Sigma[I]\\}$ form the _$\mathtt{H}$ -basis_ of
$\boldsymbol{\Sigma}$.

In general, functors $\textbf{p}:\textsf{S}^{\text{op}}\to\textsf{Vec}$ are
called (complex) _vector species_ , going back to Joyal [Joy81], [Joy86].
Morphisms of vector species $\eta:\textbf{p}\to\textbf{q}$ are natural
transformations; they consist of a linear map
$\eta_{I}:\textbf{p}[I]\to\textbf{q}[I]$ for each finite set $I$ which
commutes with the action of the bijections. When $I=[n]:=\\{1,\dots,n\\}$, we
abbreviate $\eta_{n}:=\eta_{[n]}$.

We equip vector species with the tensor product
$\textbf{p}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{q}$
known as the _Cauchy product_ [AM10, Definition 8.5], given by

(1)
$\textbf{p}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{q}[I]:=\bigoplus_{I=S\sqcup
T}\textbf{p}[S]\otimes\textbf{q}[T].$

This is the Day convolution with respect to disjoint union of sets and tensor
product of vector spaces. In this paper, we consider algebraic structures on
species which are constructed using this tensor product. In particular, a
multiplication on a species p consists of linear maps

$\mu_{S,T}:\textbf{p}[S]\otimes\textbf{p}[T]\to\textbf{p}[I]$

and a comultiplication on p consists of linear maps

$\Delta_{S,T}:\textbf{p}[I]\to\textbf{p}[S]\otimes\textbf{p}[T],$

where we have a map for each choice of decomposition $I=S\sqcup T$ ($S,T$ can
be empty). We can then impose conditions like (co)associativity, see e.g.
[Nor20, Section 1.3].

Following [AM13, Section 11], $\boldsymbol{\Sigma}$ is a connected999 a
species p is _connected_ if $\textbf{p}[\emptyset]=\mathbb{C}$ bialgebra,
meaning it is naturally equipped with an associative, unital multiplication
and a coassociative, counital comultiplication, which are compatible in the
sense they satisfy the bimonoid axiom. See [AM10, Section 8.3.1] for details.
The multiplication and comultiplication are given in terms of the
$\mathtt{H}$-basis by

$\mu_{S,T}(\mathtt{H}_{F}\otimes\mathtt{H}_{G}):=\mathtt{H}_{FG}\qquad\text{and}\qquad\Delta_{S,T}(\mathtt{H}_{F}):=\mathtt{H}_{F|_{S}}\otimes\mathtt{H}_{F|_{T}}.$

We sometimes abbreviate
$\mathtt{H}_{F}\mathtt{H}_{G}:=\mu_{S,T}(\mathtt{H}_{F}\otimes\mathtt{H}_{G})$.
The unit and counit are given by

$\mathtt{1}_{\boldsymbol{\Sigma}}:=\mathtt{H}_{(\,)}\qquad\text{and}\qquad\epsilon_{\emptyset}(\mathtt{H}_{(\,)}):=1_{\mathbb{C}}.$

Let

(2)
$\overline{\mathtt{H}}_{F}:=\sum_{G\geq\bar{F}}(-1)^{l(G)}\,\mathtt{H}_{G}.$

Then [AM10, Theorem 11.38] (in the case $\textbf{q}=\textbf{E}^{\ast}_{+}$ and
$q=1$) shows that

(3) $\sum_{S\sqcup
T=I}\mathtt{H}_{F|_{S}}\overline{\mathtt{H}}_{F|_{T}}=0\qquad\text{and}\qquad\sum_{S\sqcup
T=I}\overline{\mathtt{H}}_{F|_{S}}\mathtt{H}_{F|_{T}}=0.$

In general, connected bialgebras are automatically Hopf algebras, and it
follows from (3) that the antipode
$s:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ is given by

$\text{s}_{I}(\mathtt{H}_{F})=\overline{\mathtt{H}}_{F}.$

The Hopf algebra $\boldsymbol{\Sigma}$ is the free cocommutative Hopf algebra
on the positive coalgebra $\textbf{E}^{\ast}_{+}$ [AM10, Section 11.2.5], and
so $\boldsymbol{\Sigma}\cong\textbf{L}\boldsymbol{\circ}\textbf{E}^{\ast}_{+}$
where ‘$\boldsymbol{\circ}$’ is plethysm of species and
$\textbf{L}\hookrightarrow\boldsymbol{\Sigma}$ is the subspecies of singleton
lump compositions ($=$linear orders).

There is a second important basis of $\boldsymbol{\Sigma}$, called the
_$\mathtt{Q}$ -basis_. The $\mathtt{Q}$-basis is also indexed by compositions,
and is given by

$\mathtt{Q}_{F}:=\sum_{G\geq
F}(-1)^{l(G)-l(F)}\dfrac{1}{l(G/F)}\mathtt{H}_{G}\qquad\text{or
equivalently}\qquad\mathtt{H}_{F}=:\sum_{G\geq
F}\dfrac{1}{(G/F)!}\mathtt{Q}_{G}.$

For $S\subseteq I$ and $F\in\Sigma[I]$, we have _deshuffling_

$F\parallel_{S}\,:=\begin{cases}F|_{S}&\quad\text{if $S$ is a union of lumps
of $F$}\\\ 0\in\boldsymbol{\Sigma}[S]&\quad\text{otherwise.}\end{cases}$
1010footnotetext:  not necessarily contiguous

The multiplication and comultiplication of $\boldsymbol{\Sigma}$ is given in
terms of the $\mathtt{Q}$-basis by

$\mu_{S,T}(\mathtt{Q}_{F}\otimes\mathtt{Q}_{G})=\mathtt{Q}_{FG}\qquad\text{and}\qquad\Delta_{S,T}(\mathtt{Q}_{F})=\mathtt{Q}_{F\parallel_{S}}\otimes\,\mathtt{Q}_{F\parallel_{T}}.$

####  1.3. Decorations

Given a complex vector space $V$, we can use $V$ to ‘decorate’
$\boldsymbol{\Sigma}$ in order to obtain an enlarged Hopf algebra
$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$. This goes as follows.

We have the species denoted $\textbf{E}_{V}$, given by

$\textbf{E}_{V}[I]:=V^{\otimes
I}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{V\otimes\dots\otimes V}_{\text{a
copy of $V$ for each $i\in I$}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$

The action of bijections is given by relabeling tensor factors.

######  Remark 1.1.

Notice species of the form $\textbf{E}_{V}$ are exactly the monoidal functors
$\textbf{E}_{V}:\textsf{S}^{\text{op}}\to\textsf{Vec}$.

We denote vectors by $\emph{{A}},\emph{{S}}\in V$, and we denote simple
tensors of $V^{\otimes I}$ by

$\emph{{A}}_{I}=\emph{{A}}_{i_{1}}\otimes\cdots\otimes\emph{{A}}_{i_{n}}\in
V^{\otimes I}$

where $I=\\{i_{1},\dots,i_{n}\\}$. If $\emph{{A}}_{i}=\emph{{A}}$ for all
$i\in I$, then we write

(4) $\emph{{A}}^{I}:=\emph{{A}}\otimes\cdots\otimes\emph{{A}}\in V^{\otimes
I}\qquad\text{and}\qquad\emph{{A}}^{n}:=\emph{{A}}^{[n]}\in V^{\otimes[n]}$

where $[n]=\\{1,\dots,n\\}$ as usual.

We let ‘$\otimes$’ denote the Hadamard product of species, which is given by
componentwise tensoring, see e.g. [Nor20, Section 1.2]. Then the species of
$V$-_decorated compositions_ $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$ is
given by

$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}[I]=\boldsymbol{\Sigma}[I]\otimes\textbf{E}_{V}[I]=\boldsymbol{\Sigma}[I]\otimes
V^{\otimes I}.$

Following [AM10, Section 8.13.4], $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$
is a connected bialgebra, with multiplication given by

$\mu_{S,T}\big{(}(\mathtt{H}_{F}\otimes\emph{{A}}_{S})\otimes(\mathtt{H}_{G}\otimes\emph{{A}}_{T})\big{)}:=\mathtt{H}_{F}\mathtt{H}_{G}\otimes\emph{{A}}_{S}\otimes\emph{{A}}_{T}$

and comultiplication given by

$\Delta_{S,T}(\mathtt{H}_{F}\otimes\emph{{A}}_{I}):=(\mathtt{H}_{F|_{S}}\otimes{\emph{{A}}_{I}}|_{S})\otimes(\mathtt{H}_{F|_{T}}\otimes{\emph{{A}}_{I}}|_{T}).$

The unit and counit are given by

$\mathtt{1}_{\boldsymbol{\Sigma}\otimes\textbf{E}_{V}}:=\mathtt{H}_{(\,)}\otimes
1_{\mathbb{C}}\qquad\text{and}\qquad\epsilon_{\emptyset}(\mathtt{H}_{(\,)}\otimes
1_{\mathbb{C}}):=1_{\mathbb{C}}.$

For
$\mathtt{H}_{F}\otimes\emph{{A}}_{I}\in\boldsymbol{\Sigma}\otimes\textbf{E}_{V}[I]$,
we have

$\sum_{S\sqcup
T=I}\mu_{S,T}\big{(}(\mathtt{H}_{F|_{S}}\otimes{\emph{{A}}_{I}}|_{S})\otimes(\overline{\mathtt{H}}_{F|_{T}}\otimes{\emph{{A}}_{I}}|_{T})\big{)}=\underbrace{\sum_{S\sqcup
T=I}\mathtt{H}_{F|_{S}}\overline{\mathtt{H}}_{F|_{T}}}_{\text{$=0$ by
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\ref{eq:inversion
relation for reverse time-ordered products})}}}\otimes\,\emph{{A}}_{I}=0$

and

$\sum_{S\sqcup
T=I}\mu_{S,T}\big{(}(\overline{\mathtt{H}}_{F|_{S}}\otimes{\emph{{A}}_{I}}|_{S})\otimes(\mathtt{H}_{F|_{T}}\otimes{\emph{{A}}_{I}}|_{T})\big{)}=\underbrace{\sum_{S\sqcup
T=I}\overline{\mathtt{H}}_{F|_{S}}\mathtt{H}_{F|_{T}}}_{\text{$=0$ by
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\ref{eq:inversion
relation for reverse time-ordered products})}}}\otimes\,\emph{{A}}_{I}=0.$

It follows that the antipode of $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$ is
given by

(5)
$\text{s}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I})=\overline{\mathtt{H}}_{F}\otimes\emph{{A}}_{I}.$

####  1.4. The Steinmann Algebra

The Hopf algebra $\boldsymbol{\Sigma}$ is connected and cocommutative, and so
the CMM Theorem applies, see [Nor20, Section 1.4]. We now describe the
positive111111 a species p is _positive_ if $\textbf{p}[\emptyset]=0$ Lie
algebra of primitive elements

$\mathcal{P}(\boldsymbol{\Sigma})\subset\boldsymbol{\Sigma}.$

For $I\in\textsf{S}$ a finite set, let a _tree_ $\mathcal{T}$ over $I$ be a
planar121212 i.e. a choice of left and right child is made at every node full
binary tree whose leaves are labeled bijectively with the blocks of a
partition of $I$ (a _partition_ $P$ of $I$ is a set of disjoint nonempty
subsets of $I$, called _blocks_ , whose union is $I$). The blocks of this
partition, called the _lumps_ of $\mathcal{T}$, form a composition called the
_debracketing_ $F_{\mathcal{T}}$ of $\mathcal{T}$, by listing them in order of
appearance from left to right. We denote trees by nested products
$[\,\cdot\,,\,\cdot\,]$ of subsets or trees, see Figure 2. We make the
convention that no trees exist over the empty set $\emptyset$.

Figure 2. Let $I$ be various subsets of $\\{1,2,3,4,5,6,7,8,9\\}$. The trees
$[4]$, $[1,23]$ ($\neq[23,1]$), $[[2,3],5]$, $[[24,[1,9]],678]$ are shown. The
debracketing of $[[24,[1,9]],678]$ is the composition $(24,1,9,678)$. If we
put $\mathcal{T}_{1}=[24,[1,9]]$ and $\mathcal{T}_{2}=[678]$, then
$[\mathcal{T}_{1},\mathcal{T}_{2}]$ would also denote this tree.

We define the positive species Zie by letting $\textbf{Zie}[I]$ denote the
vector space of formal $\mathbb{C}$-linear combinations of trees over $I$,
modulo the relations of antisymmetry and the Jacobi identity as interpreted on
trees in the usual way. Explicitly,

  1. (1)

(antisymmetry) for all trees of the form
$[\dots[\mathcal{T}_{1},\mathcal{T}_{2}]\dots]$ (writing a tree in this form
is equivalent to picking a node) we have

$[\dots[\mathcal{T}_{1},\mathcal{T}_{2}]\dots]+[\dots[\mathcal{T}_{2},\mathcal{T}_{1}]\dots]=0.$

  2. (2)

(Jacobi Identity) for all trees of the form
$[\dots[[\mathcal{T}_{1},\mathcal{T}_{2}],\mathcal{T}_{3}]\dots]$ we have

$[\dots[[\mathcal{T}_{1},\mathcal{T}_{2}],\mathcal{T}_{3}]\dots]+[\dots[[\mathcal{T}_{3},\mathcal{T}_{1}],\mathcal{T}_{2}]\dots]+[\dots[[\mathcal{T}_{2},\mathcal{T}_{3}],\mathcal{T}_{1}]\dots]=0.$

Then Zie is a positive Lie algebra in species, with Lie bracket
$\partial^{\ast}$ given by

$\partial_{S,T}^{\ast}(\mathcal{T}_{1}\otimes\mathcal{T}_{2}):=[\mathcal{T}_{1},\mathcal{T}_{2}].$

######  Remark 1.2.

We have that Zie is the free Lie algebra on the positive exponential species
$\textbf{E}^{\ast}_{+}$, and so the species Zie is also given by

$\textbf{Zie}[I]=\textbf{Lie}\boldsymbol{\circ}\textbf{E}^{\ast}_{+}[I]=\bigoplus_{P}\textbf{Lie}[P]$

where Lie is the species of the Lie operad, and the direct sum is over all
partitions $P$ of $I$.

The Lie algebra in species Zie is closely related to the Steinmann algebra
from the physics literature [BL75, Section III.1], [Rue61, Section 6].
Precisely, the Steinmann algebra is an ordinary graded Lie algebra based on
the structure map for the adjoint braid arrangement realization of Zie. The
adjoint braid arrangement realization of Zie is the topic of [LNO19], and the
fact that the Lie algebra there is indeed Zie was shown in [NO19].

Via the commutator bracket, $\boldsymbol{\Sigma}$ is a Lie algebra in species,
given by

$[\mathtt{H}_{F},\mathtt{H}_{G}]=\mathtt{H}_{F}\mathtt{H}_{G}-\mathtt{H}_{G}\mathtt{H}_{F}.$

Let

$[I;\text{2}]:=\big{\\{}\text{surjective functions}\ I\to\\{1,2\\}\big{\\}}$

denote the set of compositions of $I$ with two lumps. Since
$\boldsymbol{\Sigma}$ is connected, its positive Lie subalgebra of primitive
elements $\mathcal{P}(\boldsymbol{\Sigma})\subset\boldsymbol{\Sigma}$ is given
on nonempty $I$ by

$\mathcal{P}(\boldsymbol{\Sigma})[I]=\bigcap_{(S,T)\in[I;\text{2}]}\text{ker}\big{(}\Delta_{S,T}:\boldsymbol{\Sigma}[I]\to\boldsymbol{\Sigma}[S]\otimes\boldsymbol{\Sigma}[T]\big{)}.$

In particular, $\mathtt{Q}_{(I)}\in\mathcal{P}(\boldsymbol{\Sigma})[I]$ for
$I$ nonempty. Since Zie is freely generated by stick trees $[I]$, we can
define a homomorphism of Lie algebras by

$\textbf{Zie}\to\mathcal{P}(\boldsymbol{\Sigma}),\qquad[I]\mapsto\mathtt{Q}_{(I)}.$

To describe this explicitly, given a tree $\mathcal{T}$, let
$\text{antisym}(\mathcal{T})$ denote the set of $2^{l(F_{\mathcal{T}})-1}$
many trees which are obtained by switching left and right branches at nodes of
$\mathcal{T}$. For $\mathcal{T}^{\prime}\in\text{antisym}(\mathcal{T})$, let
$(\mathcal{T},\mathcal{T}^{\prime})\in\mathbb{Z}/2\mathbb{Z}$ denote the
parity of the number of node switches required to bring $\mathcal{T}$ to
$\mathcal{T}^{\prime}$. Then the homomorphism is given in full by

$\textbf{Zie}\to\mathcal{P}(\boldsymbol{\Sigma}),\qquad\mathcal{T}\mapsto\mathtt{Q}_{\mathcal{T}}:=\sum_{\mathcal{T}^{\prime}\in\text{antisym}(\mathcal{T})}(-1)^{(\mathcal{T},\mathcal{T}^{\prime})}\mathtt{Q}_{F_{\mathcal{T}^{\prime}}}.$

By [AM10, Corollary 11.46], this is an isomorphism. From now on, we make the
identification

$\textbf{Zie}=\mathcal{P}(\boldsymbol{\Sigma})$

and retire the notation $\mathcal{P}(\boldsymbol{\Sigma})$.

####  1.5. Type $A$ Dynkin Elements

Recall that the set of minuscule weights of (the root datum of)
$\text{SL}_{I}(\mathbb{C})$ is in natural bijection with $[I;\text{2}]$. We
denote the minuscule weight corresponding to $(S,T)$ by $\lambda_{ST}$. See
[NO19, Section 3.1] for more details.

A _cell_ 131313 also known as maximal unbalanced families [BMM+12] and
positive sum systems [Bjo15] [EGS75, Definition 6] over $I$ is (equivalent to)
a subset $\mathcal{S}\subseteq[I;\text{2}]$ such that for all
$(S,T)\in[I;\text{2}]$, exactly one of

$(S,T)\in\mathcal{S}\qquad\text{and}\qquad(T,S)\in\mathcal{S}$

is true, and whose corresponding set of minuscule weights is closed under
conical combinations, that is

$\lambda_{UV}\in\text{coni}\big{\langle}\lambda_{ST}:(S,T)\in\mathcal{S}\big{\rangle}\quad\implies\quad(U,V)\in\mathcal{S}.$

By dualizing conical spaces generated by minuscule weights, cells are in
natural bijection with chambers of the adjoint of the braid arrangement, see
[NO19, Section 3.3], [Eps16, Definition 2.5]. Their number is sequence A034997
in the OEIS. We denote the species of formal $\mathbb{C}$-linear combinations
of cells by $\textbf{L}^{\vee}$.

Associated to each composition $F$ of $I$ is the subset
$\mathcal{F}_{F}\subseteq[I;\text{2}]$ consisting of those compositions
$(S,T)$ which are obtained by merging contiguous lumps of $F$,

$\mathcal{F}_{F}:=\big{\\{}(S,T)\in[I;\text{2}]:(S,T)\leq F\big{\\}}.$

More geometrically, $\mathcal{F}_{F}$ is the subset corresponding to the set
of minuscule weights which are contained in the closed braid arrangement face
of $F$. Let us write $F\subseteq\mathcal{S}$ as abbreviation for
$\mathcal{F}_{F}\subseteq\mathcal{S}$.

Consider the morphism of species given by

(6)
$\textbf{L}^{\vee}\to\boldsymbol{\Sigma},\qquad\mathcal{S}\mapsto\mathtt{D}_{\mathcal{S}}:=-\sum_{\bar{F}\subseteq\mathcal{S}}(-1)^{l(F)}\mathtt{H}_{F}.$

The element $\mathtt{D}_{\mathcal{S}}$ is called the _Dynkin element_
associated to the cell $\mathcal{S}$. These special elements were defined by
Epstein-Glaser-Stora in [EGS75, Equation 1, p.26], and the name is due to
Aguiar-Mahajan [AM17, Equation 14.1] (see 1.3). In fact,
$\mathtt{D}_{\mathcal{S}}$ is a primitive element [AM17, Proposition 14.1],
and so we actually have a morphism $\textbf{L}^{\vee}\to\textbf{Zie}$.

For $i\in I$, let $\mathcal{S}_{i}$ denote the cell given by

$\mathcal{S}_{i}:=\big{\\{}(S,T)\in[I,\text{2}]:i\in S\big{\\}}.$

This is the cell corresponding to the adjoint braid arrangement chamber which
contains the projection of the basis element $e_{i}\in\mathbb{R}I$ onto the
sum-zero hyperplane. Let the _total retarded_ Dynkin element $\mathtt{D}_{i}$
associated to $i$ be given by

$\mathtt{D}_{i}:=\mathtt{D}_{\mathcal{S}_{i}}=-\sum_{\begin{subarray}{c}F\in\Sigma[I]\\\
i\in S_{k}\end{subarray}}(-1)^{l(F)}\mathtt{H}_{F}.$

These Dynkin elements are considered in [AM13, Section 14.5]. For $i\in I$,
let

$\bar{\mathcal{S}}_{i}:=\big{\\{}(S,T)\in[I,\text{2}]:i\in T\big{\\}}.$

This is the cell corresponding to the adjoint braid arrangement chamber which
is opposite to the chamber of $\mathcal{S}_{i}$. Let the _total advanced_
Dynkin element $\mathtt{D}_{\bar{i}}$ associated to $i$ be given by

$\mathtt{D}_{\bar{i}}:=\mathtt{D}_{\bar{\mathcal{S}}_{i}}=-\sum_{\begin{subarray}{c}F\in\Sigma[I]\\\
i\in S_{1}\end{subarray}}(-1)^{l(F)}\mathtt{H}_{F}.$

Figure 3. A cell $\mathcal{S}$ over $\\{1,2,3\\}$ (on the adjoint braid
arrangement) and its Dynkin element $\mathtt{D}_{\mathcal{S}}$ (on the
tropical geometric realization of $\boldsymbol{\Sigma}$, where the
multiplication embeds facets and the comultiplication projects onto facets,
see [NO19, Introduction])). In the presence of causal factorization, the time
component of the corresponding generalized retarded function $r_{\mathcal{S}}$
is a $\mathbb{C}[[\hbar,\emph{{g}}]]$-valued generalized function on the braid
arrangement with support the gray cone. The Dynkin element shown is
$\mathtt{D}_{\mathcal{S}}=\mathtt{D}_{3}=\mathtt{R}_{(12;3)}$. Its support
consists of those configurations such that the event labeled by $3$ can be
causally influenced by the events labeled by $1$ and $2$.

######  Remark 1.3.

More generally, Dynkin elements are certain Zie elements of generic real
hyperplane arrangements, which are indexed by chambers of the corresponding
adjoint arrangement. They were introduced by Aguiar-Mahajan in [AM17, Equation
14.1]. Specializing to the braid arrangement, one recovers the type $A$ Dynkin
elements $\mathtt{D}_{\mathcal{S}}$.

In [NO19], the following perspective on the Dynkin elements is given. The Hopf
algebra $\boldsymbol{\Sigma}^{\ast}$ which is dual to $\boldsymbol{\Sigma}$ is
realized as an algebra $\hat{\boldsymbol{\Sigma}}^{\ast}$ of piecewise-
constant functions on the braid arrangement. Then its dual, in the sense of
polyhedral algebras [BP99, Theorem 2.7], is an algebra
$\check{\boldsymbol{\Sigma}}^{\ast}$ of certain functionals of piecewise-
constant functions on the adjoint braid arrangement, i.e. those coming from
evaluating on permutohedral cones. We have the morphism of species

$\check{\boldsymbol{\Sigma}}^{\ast}\to(\textbf{L}^{\vee})^{\ast}$

defined by sending functionals to their restrictions to piecewise-constant
functions on the complement of the hyperplanes. Since the multiplication of
$\check{\boldsymbol{\Sigma}}^{\ast}$ corresponds to embedding hyperplanes,
this morphism is the indecomposable quotient of
$\check{\boldsymbol{\Sigma}}^{\ast}$ [NO19, Theorem 4.5]. Then, in [NO19,
Proposition 5.1], we see that taking the linear dual of this morphism recovers
the Dynkin elements map,

$\textbf{L}^{\vee}\to\boldsymbol{\Sigma},\qquad\mathcal{S}\mapsto\mathtt{D}_{\mathcal{S}}.$

(Here we have identified
$\boldsymbol{\Sigma}^{\ast}=\check{\boldsymbol{\Sigma}}^{\ast}$.) Therefore we
obtain the following.

######  Theorem 1.1 ([NO19]).

The morphism of species $\textbf{L}^{\vee}\to\textbf{Zie}$ is surjective.
Therefore the Dynkin elements $\\{\mathtt{D}_{\mathcal{S}}:\mathcal{S}\
\text{is a cell over $I$}\\}$ span Zie.

####  1.6. The Steinmann Relations

The Dynkin elements span Zie, but they are not linearly independent. The
relations which are satisfied by the Dynkin elements are generated by
relations known in physics as the Steinmann relations, introduced in [Ste60a],
[Ste60b].

Let a pair of _overlapping channels_ over $I$ be a pair
$(S,T),(U,V)\in[I;\text{2}]$ of two-lump compositions of $I$ such that

$S\cap U\neq\emptyset\qquad\text{and}\qquad T\cap U\neq\emptyset.$

Let $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, $\mathcal{S}_{3}$, $\mathcal{S}_{4}$
be four cells over $I$ with $(S,T),(U,V)\in\mathcal{S}_{1}$, and such that
$\mathcal{S}_{2}$, $\mathcal{S}_{3}$, $\mathcal{S}_{4}$ are obtained from
$\mathcal{S}_{1}$ by replacing, respectively,

$(S,T),(U,V)\mapsto(T,S),(U,V)$ $(S,T),(U,V)\mapsto(T,S),(V,U)$
$(S,T),(U,V)\mapsto(S,T),(V,U).$

Then, by inspecting the definition of the Dynkin elements (6), we see
that141414 we go through the argument for the basic $4$-point case in 1.1,
which is sufficient to exhibit the general phenomenon

$\mathtt{D}_{\mathcal{S}_{1}}-\mathtt{D}_{\mathcal{S}_{2}}+\mathtt{D}_{\mathcal{S}_{3}}-\mathtt{D}_{\mathcal{S}_{4}}=0.$

In general, a _Steinmann relation_ is any relation between Dynkin elements
obtained in this way, i.e. an alternating sum of four Dynkin elements which
are obtained from each other by switching overlapping channels only. This
definition of the Steinmann relations can be found in [EGS75, Seciton 4.3] (it
is given slightly more generally there for paracells).

An alternative characterization of the Steinmann relations in terms of the Lie
cobracket of the dual Lie coalgebra $\textbf{Zie}^{\ast}$ is [LNO19,
Definition 4.2]. Here, the Steinmann relations appear in the same way one can
arrive at generalized permutohedra, i.e. by insisting on type $A$
‘factorization’ in the sense of species-theoretic coalgebra structure. See
[NO19, Theorem 4.2 and Remark 4.2].

Thus, Dynkin elements satisfy the Steinmann relations. Moreover, they are
sufficient.

######  Theorem 1.2.

The relations which are satisfied by the Dynkin elements are generated by the
Steinmann relations. That is, if

$\textbf{Stein}[I]:=\big{\langle}\mathtt{D}_{\mathcal{S}_{1}}-\mathtt{D}_{\mathcal{S}_{2}}+\mathtt{D}_{\mathcal{S}_{3}}-\mathtt{D}_{\mathcal{S}_{4}}:\mathtt{D}_{\mathcal{S}_{1}}-\mathtt{D}_{\mathcal{S}_{2}}+\mathtt{D}_{\mathcal{S}_{3}}-\mathtt{D}_{\mathcal{S}_{4}}=0\text{
is a Steinmann relation}\big{\rangle}$

then

$\textbf{Zie}\cong{\raisebox{1.99997pt}{$\textbf{L}^{\vee}$}\left/\raisebox{-1.99997pt}{$\textbf{Stein}$}\right.}.$

###### Proof.

This follows by combining [LNO19, Theorem 4.3] with [NO19, Theorems 4.2 and
4.5]. ∎

######  Example 1.1.

Let us give the basic $4$-point example $I=\\{1,2,3,4\\}$, which takes place
on a square facet of the type $A$ coroot solid [LNO19, Figure 1]. Consider the
following four cells over $I$ (we have marked where they differ, the names
‘$s$-channel’ and ‘$u$-channel’ are from physics and refer to Mandelstam
variables),

$\mathcal{S}_{1}=\big{\\{}\underbrace{(23,14)}_{u\text{-channel}},(12,34),(1,234),(13,24),(13,24),(134,2),(3,124)\\}$
$\mathcal{S}_{2}=\big{\\{}(23,14),\underbrace{(34,12)}_{s\text{-channel}},(1,234),(13,24),(13,24),(134,2),(3,124)\big{\\}}$
$\mathcal{S}_{3}=\big{\\{}\underbrace{(14,23)}_{u\text{-channel}},(34,12),(1,234),(13,24),(13,24),(134,2),(3,124)\big{\\}}$
$\mathcal{S}_{4}=\big{\\{}(14,23),\underbrace{(12,34)}_{s\text{-channel}},(1,234),(13,24),(13,24),(134,2),(3,124)\big{\\}}.$

The $s$-channel and the $u$-channel overlap, and so we should now have

$\mathtt{D}_{\mathcal{S}_{1}}-\mathtt{D}_{\mathcal{S}_{2}}+\mathtt{D}_{\mathcal{S}_{3}}-\mathtt{D}_{\mathcal{S}_{4}}=0.$

To see this, let us assume throughout that $\mathtt{H}_{F}$ appears in the
$\mathtt{H}$-basis expansion (6) of $\mathtt{D}_{\mathcal{S}_{1}}$, i.e.
$\bar{F}\subseteq\mathcal{S}_{1}$. Then we have

($\spadesuit$)
$\bar{F}\subseteq\mathcal{S}_{1}\setminus\\{(12,34),(23,14)\\}\quad\implies\quad\bar{F}\subseteq\mathcal{S}_{1},\
\mathcal{S}_{2},\ \mathcal{S}_{3},\ \mathcal{S}_{4}.$

If $\bar{F}\nsubseteq\mathcal{S}_{1}\setminus\\{(12,34),(23,14)\\}$, then
either $(12,34)\in\bar{F}$ or $(23,14)\in\bar{F}$ but not both, since the
channels overlap. We then have

() $(12,34)\in\bar{F}\implies\bar{F}\subseteq\mathcal{S}_{1},\
\bar{F}\nsubseteq\mathcal{S}_{2},\ \bar{F}\nsubseteq\mathcal{S}_{3},\
\bar{F}\subseteq\mathcal{S}_{4}.$

We also have

() $(23,14)\in\bar{F}\implies\bar{F}\subseteq\mathcal{S}_{1},\
\bar{F}\subseteq\mathcal{S}_{2},\ \bar{F}\nsubseteq\mathcal{S}_{3},\
\bar{F}\nsubseteq\mathcal{S}_{4}.$

Notice that in all three cases ($\spadesuit$ ‣ 1.1), (() ‣ 1.1), (() ‣ 1.1),
the prefactors of $\mathtt{H}_{F}$ sum to zero in the four term alternating
sum of the Steinmann relation.

######  Remark 1.4.

In [NO19], the Steinmann condition is seen to be equivalent to the restriction
to generalized permutohedra in a certain local (or spherical) sense. Ocneanu
[Ocn18] and Early [Ear19] have studied an affine version of the Steinmann
condition, in the context of higher structures and matroid subdivisions. Here,
one observes that the (translated) hyperplanes of the adjoint braid
arrangement for the Mandelstam variables give three subdivisions of the
hypersimplex $\Delta(2,4)$ (octahedron).

See [BC19], [CGUZ19] for the closely related study of generalized Feynman
diagrams in generalized biadjoint $\Phi^{3}$-theory.

####  1.7. Ruelle’s Identity

Since the Dynkin elements span Zie, we can ask what is the description of the
Lie bracket of Zie in terms of the Dynkin elements. The answer is known in the
physics literature as Ruelle’s identity.

In order to state Ruelle’s identity, we need to notice the following. For
$S\sqcup T=I$, if $\mathcal{S}_{1}$ is a cell over $S$ and $\mathcal{S}_{2}$
is a cell over $T$, then $\mathcal{S}_{1}\sqcup\mathcal{S}_{2}$ describes a
collection of codimension one faces of the adjoint braid arrangement which are
supported by the hyperplane orthogonal to $\lambda_{ST}$ (in [LNO19], such
faces were called _Steinmann equivalent_). A cell $\mathcal{S}^{[S,T]}$ over
$I$ which satisfies

$\mathcal{S}^{[S,T]}\supseteq\mathcal{S}_{1}\sqcup\mathcal{S}_{2}\qquad\text{and}\qquad(S,T)\in\mathcal{S}^{[S,T]}$

corresponds to a chamber arrived at by moving (by an arbitrarily small amount)
from an interior point of a face of $\mathcal{S}_{1}\sqcup\mathcal{S}_{2}$ in
the $\lambda_{ST}$ direction. In particular, such cells always exist, but they
are not unique (the Steinmann relations exactly quotient out this ambiguity).
The chamber obtained by moving in the opposite direction corresponds to the
cell obtained by replacing $(S,T)$ with $(T,S)$ in $\mathcal{S}^{[S,T]}$.

######  Proposition 1.3 (Ruelle’s Identity [Rue61, Equation 6.6]).

For $S\sqcup T=I$, let $\mathcal{S}_{1}$ be a cell over $S$ and let
$\mathcal{S}_{2}$ be a cell over $T$. Let $\mathcal{S}^{[S,T]}$ be a cell over
$I$ which satisfies

$\mathcal{S}^{[S,T]}\supseteq\mathcal{S}_{1}\sqcup\mathcal{S}_{2}\qquad\text{and}\qquad(S,T)\in\mathcal{S}^{[S,T]}.$

Let $\mathcal{S}^{[T,S]}$ denote the cell obtained by replacing $(S,T)$ with
$(T,S)$ in $\mathcal{S}^{[S,T]}$. Then the Lie bracket of Zie is given by

(7)
$[\mathtt{D}_{\mathcal{S}_{1}},\mathtt{D}_{\mathcal{S}_{2}}]=\mathtt{D}_{\mathcal{S}^{[S,T]}}-\mathtt{D}_{\mathcal{S}^{[T,S]}}.$

###### Proof.

This result is clear from [LNO19, Section 5.2]; the Lie bracket which was
given to the adjoint braid arrangement realization of Zie (denoted there by
$\boldsymbol{\Gamma}$) coincides with (7). Alternatively, we can just
explicitly check, as in [EGS75, Section 4.3]. ∎

###  2\. $\boldsymbol{\Sigma}$ as a Hopf E-Algebra

We now recall the Steinmann arrows, which are (or we interpret as) actions of
the exponential species E on $\boldsymbol{\Sigma}$. We show that they give
$\boldsymbol{\Sigma}$ the structure of a Hopf E-algebra (=Hopf monoid internal
to E-modules) in two ways, and thus the primitive part
$\textbf{Zie}=\mathcal{P}(\boldsymbol{\Sigma})$ the structure of a Lie
E-algebra in two ways.

####  2.1. Derivations and Coderivations of $\boldsymbol{\Sigma}$

Let $Y=\\{y_{1},\dots,y_{r}\\}$ be a finite set with cardinality
$r\in\mathbb{N}$. We think of $Y$ as having ‘color’ g (physically, the
coupling constant). Given a species p, we have the $Y$-_derivative_
$\textbf{p}^{[Y]}$ of p, which is the species given by

$\textbf{p}^{[Y]}[I]:=\textbf{p}[Y\sqcup
I]\qquad\text{and}\qquad\textbf{p}^{[Y]}[\sigma]:=\textbf{p}[\text{id}_{Y}\sqcup\sigma].$

A _raising operator_ $u$ on p is a morphism of species of the form161616 for
raising operators, we often abbreviate $u(\mathtt{a}):=u_{I}(\mathtt{a})$

$u:\textbf{p}\to\textbf{p}^{[Y]},\qquad\mathtt{a}\mapsto u(\mathtt{a}).$

######  Remark 2.1.

Moreover, there is an endomorphism algebra of raising operators [Nor20,
Section 2.4], which features when considering modules internal to species, see
[Nor20, Section 5.1].

As a particular example of the set $Y$, we have the set of formal symbols
$[r]:=\\{\ast_{1},\dots,\ast_{r}\\}$ (formally, we have picked a section of
the decategorification functor $Y\mapsto r$). We often abbreviate
$\ast=\ast_{1}$, also $\ast=\\{\ast\\}$ and $\ast I=\\{\ast\\}\sqcup I$. The
_derivative_ $\textbf{p}^{\prime}$ of p is the $Y$-derivative in the singleton
case $Y=\\{\ast\\}$, thus

$\textbf{p}^{\prime}[I]:=\textbf{p}^{[\ast]}[I]=\textbf{p}[\ast I].$

Following [AM10, Section 8.12.1], an _up operator_ $u$ on p is a raising
operator of the form $u:\textbf{p}\to\textbf{p}^{\prime}$. Writing
$u_{\ast}(\mathtt{a})=u(\mathtt{a})$ in order to specify the name of the
adjoined singleton, we call an up operator _commutative_ if

$u_{\ast_{2}}(u_{\ast_{1}}(\mathtt{a}))=u_{\ast_{1}}(u_{\ast_{2}}(\mathtt{a})).$

Raising operators can be obtained by iteratively applying commutative up
operators, see [Nor20, Section 5.4]. Following [AM10, Section 8.12.4], an up
operator on an algebra a is called an _up derivation_ if

(8) $u\big{(}\mu_{S,T}(\mathtt{a}\otimes\mathtt{b})\big{)}=\mu_{\ast
S,T}\big{(}u(\mathtt{a})\otimes\mathtt{b}\big{)}+\mu_{S,\ast
T}\big{(}\mathtt{a}\otimes u(\mathtt{b})\big{)}$

(it follows that $u(\mathtt{1}_{\mathtt{a}})=0$ if a is unital) and an up
operator on a coalgebra c is called an _up coderivation_ if

(9) $\big{(}u\otimes\text{id}+\text{id}\otimes
u\big{)}\circ\Delta_{S,T}(\mathtt{a})=\Delta_{\ast
S,T}\big{(}u(\mathtt{a})\big{)}+\Delta_{S,\ast T}\big{(}u(\mathtt{a})\big{)}.$

An _up biderivation_ on a bialgebra h is an up operator which is both an up
derivation and an up coderivation. The data of an up (co/bi)derivation on a
connected species h is equivalent to giving h the structure of an
L-(co/Hopf)algebra (= an (co/Hopf)monoid internal to L-modules). The data of a
commutative up (co/bi)derivation on h is equivalent to giving h the structure
of an E-(co/Hopf)algebra. See [Nor20, Section 5] for more details and proofs.

Thus, an up derivation $u$ of $\boldsymbol{\Sigma}$ is a morphism of species

$u:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad\mathtt{H}_{F}\mapsto
u(\mathtt{H}_{F})\qquad\quad\text{such that}\qquad
u(\mathtt{H}_{F}\mathtt{H}_{G})=u(\mathtt{H}_{F})\mathtt{H}_{G}+\mathtt{H}_{F}u(\mathtt{H}_{G}).$

An up derivation of $\boldsymbol{\Sigma}$ is determined by its values on the
elements $\mathtt{H}_{(I)}$, $I\in\textsf{S}$, since then

$u(\mathtt{H}_{F})=u(\mathtt{H}_{(S_{1})})\mathtt{H}_{(S_{2})}\dots\mathtt{H}_{(S_{k})}+\
\,\cdots\
\,+\mathtt{H}_{(S_{1})}\dots\mathtt{H}_{(S_{k-1})}u(\mathtt{H}_{(S_{k})}).$

An up derivation must have $u(\mathtt{H}_{(\,)})=0$, since
$\mathtt{1}_{\boldsymbol{\Sigma}}=\mathtt{H}_{(\,)}$. An up coderivation $u$
of $\boldsymbol{\Sigma}$ is a morphism of species

$u:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad\mathtt{H}_{F}\mapsto
u(\mathtt{H}_{F})\qquad\quad\text{such that}\qquad\Delta_{\ast
S,T}\big{(}u(\mathtt{H}_{F})\big{)}=u(\mathtt{H}_{F|_{S}})\otimes\mathtt{H}_{F|_{T}}.$

In particular, an up coderivation must have

$\Delta_{\ast
S,T}\big{(}u(\mathtt{H}_{(I)})\big{)}=u(\mathtt{H}_{(S)})\otimes\mathtt{H}_{(T)}.$

Therefore, an up biderivation $u$ of $\boldsymbol{\Sigma}$ must have

$u(\mathtt{H}_{(i)})=a_{1}\mathtt{H}_{(\ast,i)}+a_{2}\mathtt{H}_{(\ast
i)}+a_{3}\mathtt{H}_{(i,\ast)}\qquad\quad\text{where}\qquad
a_{1}+a_{2}+a_{3}=0\in\mathbb{C}.$

Motivated by this, given $a,b\in\mathbb{C}$, we define an up derivation
$u_{a,b}$ of $\boldsymbol{\Sigma}$ by

(10) $u_{a,b}:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad
u_{a,b}(\mathtt{H}_{(I)}):=-a\mathtt{H}_{(\ast,I)}+(a+b)\mathtt{H}_{(\ast
I)}-b\mathtt{H}_{(I,\ast)}.$

Towards an explicit description, consider the following example for
$I=\\{1,2,3\\}$,

$\displaystyle u_{a,b}(\mathtt{H}_{(12,3)})$
$\displaystyle=\,u_{a,b}(\mathtt{H}_{(12)})\mathtt{H}_{(3)}+\mathtt{H}_{(12)}u_{a,b}(\mathtt{H}_{(3)})$
$\displaystyle=(-a\mathtt{H}_{(\ast,12)}+(a+b)\mathtt{H}_{(\ast
12)}-b\mathtt{H}_{(12,\ast)})\mathtt{H}_{(3)}+\mathtt{H}_{(12)}(-a\mathtt{H}_{(\ast,3)}+(a+b)\mathtt{H}_{(\ast
3)}-b\mathtt{H}_{(3,\ast)})$
$\displaystyle=-a\mathtt{H}_{(\ast,12,3)}+(a+b)\mathtt{H}_{(\ast
12,3)}-b\mathtt{H}_{(12,\ast,3)})-a\mathtt{H}_{(12,\ast,3)}+(a+b)\mathtt{H}_{(12,\ast
3)}-b\mathtt{H}_{(12,3,\ast)}.$

From this, we see that in general

$u_{a,b}(\mathtt{H}_{F})=\sum_{1\leq m\leq
k}-a\mathtt{H}_{(S_{1},\dots,\ast,S_{m},\dots,S_{k})}+(a+b)\mathtt{H}_{(S_{1},\dots,\ast
S_{m},\dots,S_{k})}-b\mathtt{H}_{(S_{1},\dots,S_{m},\ast,\dots,S_{k})}.$

######  Theorem 2.1.

Given $a,b\in\mathbb{C}$, the morphism of species

$\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad\mathtt{H}_{F}\mapsto
u_{a,b}(\mathtt{H}_{F})$

is an up biderivation of $\boldsymbol{\Sigma}$ (it follows this gives
$\boldsymbol{\Sigma}$ the structure of a Hopf L-algebra).

###### Proof.

In the following, for $F=(S_{1},\dots,S_{k})$ a composition of $I$ and
$S\subseteq I$, we write

$(U_{1},\dots,U_{k}):=(S_{1}\cap S,\dots,S_{k}\cap S).$

In general, $(U_{1},\dots,U_{k})$ is a decomposition of $I$.

First, $u_{a,b}$ defines a derivation of $\boldsymbol{\Sigma}$ by
construction. To see that $u_{a,b}$ also defines a coderivation, we have

$\displaystyle\Delta_{\ast S,T}\big{(}u_{a,b}(\mathtt{H}_{F})\big{)}\ =\ $
$\displaystyle\ \ \ \ \Delta_{\ast S,T}\Bigg{(}\sum_{1\leq m\leq
k}-a\mathtt{H}_{(S_{1},\dots,\ast,S_{m},\dots,S_{k})}+(a+b)\mathtt{H}_{(S_{1},\dots,\ast
S_{m},\dots,S_{k})}-b\mathtt{H}_{(S_{1},\dots,S_{m},\ast,\dots,S_{k})}\Bigg{)}$
$\displaystyle=\ $ $\displaystyle\ \ \ \ \Bigg{(}\sum_{1\leq m\leq
k}-a\mathtt{H}_{(U_{1},\dots,\ast,U_{m},\dots,U_{k})_{+}}+(a+b)\mathtt{H}_{(U_{1},\dots,\ast
U_{m},\dots,U_{k})_{+}}-b\mathtt{H}_{(U_{1},\dots,U_{m},\ast,\dots,U_{k})_{+}}\Bigg{)}\otimes\mathtt{H}_{F|_{T}}$
$\displaystyle=\ $ $\displaystyle\ \ \ \
\Bigg{(}\sum_{\begin{subarray}{c}1\leq m\leq k\\\\[2.0pt]
U_{m}\neq\emptyset\end{subarray}}-a\mathtt{H}_{(U_{1},\dots,\ast,U_{m},\dots,U_{k})_{+}}+(a+b)\mathtt{H}_{(U_{1},\dots,\ast
U_{m},\dots,U_{k})_{+}}-b\mathtt{H}_{(U_{1},\dots,U_{m},\ast,\dots,U_{k})_{+}}\Bigg{)}\otimes\mathtt{H}_{F|_{T}}$
$\displaystyle+\ \underbrace{\Bigg{(}\sum_{\begin{subarray}{c}1\leq m\leq
k\\\\[2.0pt]
U_{m}=\emptyset\end{subarray}}\big{(}-a+(a+b)-b\big{)}\,\mathtt{H}_{(U_{1},\dots,U_{m-1},\ast,U_{m+1},\dots,U_{k})_{+}}\Bigg{)}}_{=0}\otimes\,\mathtt{H}_{F|_{T}}$
$\displaystyle=\ $ $\displaystyle\ \ \ \
u(\mathtt{H}_{F|_{S}})\otimes\mathtt{H}_{F|_{T}}.$

Therefore $u_{a,b}$ is a biderivation of $\boldsymbol{\Sigma}$. ∎

####  2.2. The Steinmann Arrows

We now recall the Steinmann arrows for $\boldsymbol{\Sigma}$, whose precise
definition is due to Epstein-Glaser-Stora [EGS75, p.82-83]. The Steinmann
arrows were first considered by Steinmann in settings where
$\boldsymbol{\Sigma}$ is represented as operator-valued distributions [Ste60b,
Section 3].

Let the _retarded Steinmann arrow_ be the up biderivation of
$\boldsymbol{\Sigma}$ given by

(11)
$\ast\downarrow(-):\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad\ast\downarrow\mathtt{H}_{F}:=u_{1,0}(\mathtt{H}_{F})=\sum_{1\leq
m\leq
k}-\mathtt{H}_{(S_{1},\dots,\ast,S_{m},\dots,S_{k})}+\mathtt{H}_{(S_{1},\dots,\ast
S_{m},\dots,S_{k})}.$

Let the _advanced Steinmann arrow_ be the up biderivation of
$\boldsymbol{\Sigma}$ given by

(12)
$\ast\uparrow(-):\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\prime},\qquad\ast\uparrow\mathtt{H}_{F}:=u_{0,1}(\mathtt{H}_{F})=\sum_{1\leq
m\leq k}\mathtt{H}_{(S_{1},\dots,\ast
S_{m},\dots,S_{k})}-\mathtt{H}_{(S_{1},\dots,S_{m},\ast,\dots,S_{k})}.$

We use this arrow notation from now on instead of ‘$u$’ in order to match the
physics literature. In particular

$\ast\downarrow\mathtt{H}_{(I)}=-\mathtt{H}_{(\ast,I)}+\mathtt{H}_{(\ast
I)}\qquad\text{and}\qquad\ast\uparrow\mathtt{H}_{(I)}=\mathtt{H}_{(\ast
I)}-\mathtt{H}_{(I,\ast)}.$

We have

$\ast\uparrow\mathtt{H}_{F}\,-\,\ast\downarrow\mathtt{H}_{F}=u_{-1,1}(\mathtt{H}_{F})=[\mathtt{H}_{(\ast)},\mathtt{H}_{F}].$

This identity appears often in the physics literature for operator-valued
distributions, e.g. [Ste60b, Equation 13], [EG73, Equation 83]. The
biderivation $u_{-1,1}$ gives $\boldsymbol{\Sigma}$ the structure of a Hopf
L-algebra. This L-action is the restriction of the adjoint representation of
$\boldsymbol{\Sigma}$. Notice the Steinmann arrows are commutative up
operators. By [Nor20, Proposition 5.4], we can restrict them to obtain up
derivations of Zie,

$\ast\downarrow(-):\textbf{Zie}\to\textbf{Zie}^{\prime},\qquad\mathtt{D}_{\mathcal{S}}\mapsto\ast\downarrow\mathtt{D}_{\mathcal{S}}\qquad\text{and}\qquad\ast\uparrow(-):\textbf{Zie}\to\textbf{Zie}^{\prime},\qquad\mathtt{D}_{\mathcal{S}}\mapsto\ast\uparrow\mathtt{D}_{\mathcal{S}}.$

Following [Nor20, Section 5], the Steinmann arrows equip $\boldsymbol{\Sigma}$
with the structure of a Hopf E-algebra (and Zie with the structure of a Lie
E-algebra) in two ways. The details are as follows. First, E is the
_exponential species_ , given by

$\textbf{E}[I]:=\mathbb{C}\qquad\text{for all}\quad I\in\textsf{S}.$

We denote $\mathtt{H}_{I}:=1_{\mathbb{C}}\in\textbf{E}[I]$. The exponential
species is an algebra in species when equipped with the trivial multiplication

$\mu_{S,T}:\textbf{E}[S]\otimes\textbf{E}[T]=\mathbb{C}\otimes\mathbb{C}\xrightarrow{\sim}\mathbb{C}=\textbf{E}[I],\qquad\mathtt{H}_{S}\otimes\mathtt{H}_{T}\mapsto\mathtt{H}_{I}.$

We have the following E-modules induced by the Steinmann arrows, as defined in
[Nor20, Equation 23],

$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma},\qquad\mathtt{H}_{Y}\otimes\mathtt{a}\,\mapsto\,Y\\!\downarrow\mathtt{a}:=\underbrace{y_{r}\downarrow\circ\cdots\circ
y_{1}\downarrow}_{\text{invariant of the order}}(\mathtt{a})$

and

$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\boldsymbol{\Sigma}\to\boldsymbol{\Sigma},\qquad\mathtt{H}_{Y}\otimes\mathtt{a}\,\mapsto\,Y\\!\uparrow\mathtt{a}:=\underbrace{y_{r}\uparrow\circ\cdots\circ
y_{1}\uparrow}_{\text{invariant of the order}}(\mathtt{a})$

where $Y=\\{y_{1},\dots,y_{r}\\}$ as usual. In particular, $Y\downarrow(-)$
and $Y\uparrow(-)$ are the Steinmann arrow raising operators obtained from
iterating the Steinmann arrow up operators $\ast\downarrow(-)$ and
$\ast\downarrow(-)$, as mentioned in Section 2.1. For example, the retarded
arrow $Y\downarrow(-)$ consists of a linear map of the form

$\boldsymbol{\Sigma}[I]\to\boldsymbol{\Sigma}[Y\sqcup I]$

for each choice of finite set $I$. For $Y=[r]:=\\{\ast_{1},\dots,\ast_{r}\\}$,
we abbreviate

$\downarrow(-):=\ast\downarrow(-),\qquad\downarrow\downarrow(-):=\\{\ast_{1},\ast_{2}\\}\downarrow(-),\quad\dots$

and similarly for the advanced arrow. Since the arrows are derivations, they
respect the multiplication of $\boldsymbol{\Sigma}$, and since the arrows are
coderivations, they respect the comultiplication of $\boldsymbol{\Sigma}$. It
follows that these E-actions give $\boldsymbol{\Sigma}$ the structure of a
Hopf monoid constructed internal to E-modules.

By inspecting the definitions, we see that

(13) $Y\\!\downarrow\mathtt{H}_{(I)}=\mathtt{R}_{(Y;I)}:=\\!\sum_{Y_{1}\sqcup
Y_{2}=Y}\overline{\mathtt{H}}_{(Y_{1})}\mathtt{H}_{(Y_{2}\sqcup
I)}\qquad\text{and}\qquad
Y\\!\uparrow\mathtt{H}_{(I)}=\mathtt{A}_{(Y;I)}:=\\!\sum_{Y_{1}\sqcup
Y_{2}=Y}\mathtt{H}_{(Y_{1}\sqcup I)}\overline{\mathtt{H}}_{(Y_{2})}.$

It follows that

$Y\\!\downarrow\mathtt{H}_{F}=\sum_{Y_{1}\sqcup\dots\sqcup
Y_{k}=Y}\mathtt{R}_{(Y_{1};S_{1})}\dots\mathtt{R}_{(Y_{k};S_{k})}\qquad\text{and}\qquad
Y\\!\uparrow\mathtt{H}_{F}=\sum_{Y_{1}\sqcup\dots\sqcup
Y_{k}=Y}\mathtt{A}_{(Y_{1};S_{1})}\dots\mathtt{A}_{(Y_{k};S_{k})}.$

The sums are over all decompositions $(Y_{1},\dots,Y_{k})$ of $Y$ of length
$l(F)$. We call
$\mathtt{R}_{(Y;I)},\mathtt{A}_{(Y;I)}\in\boldsymbol{\Sigma}[Y\sqcup I]$ the
_retarded_ and _advanced_ elements respectively. The _total retarded_ and
_total advanced_ elements are given by

$Y\\!\downarrow\mathtt{H}_{(i)}=\mathtt{R}_{(Y;i)}=\sum_{Y_{1}\sqcup
Y_{2}=Y}\overline{\mathtt{H}}_{(Y_{1})}\,\mathtt{H}_{(Y_{2}i)}\qquad\text{and}\qquad
Y\\!\uparrow\mathtt{H}_{(i)}=\mathtt{A}_{(Y;i)}=\sum_{Y_{1}\sqcup
Y_{2}=Y}\mathtt{H}_{(Y_{2}i)}\,\overline{\mathtt{H}}_{(Y_{1})}$

respectively.

######  Remark 2.2.

If we put $I=J\sqcup\\{i\\}$, then we have

$\mathtt{R}_{(J;i)}=\sum_{\begin{subarray}{c}S\sqcup T=I\\\ i\in
T\end{subarray}}\overline{\mathtt{H}}_{(S)}\,\mathtt{H}_{(T)}=-\sum_{\begin{subarray}{c}F\in\Sigma[I]\\\
i\in S_{k}\end{subarray}}(-1)^{l(F)}\mathtt{H}_{F}=\mathtt{D}_{i}$

and

$\mathtt{A}_{(J;i)}=\sum_{\begin{subarray}{c}S\sqcup T=I\\\ i\in
T\end{subarray}}\mathtt{H}_{(T)}\,\overline{\mathtt{H}}_{(S)}=-\sum_{\begin{subarray}{c}F\in\Sigma[I]\\\
i\in S_{1}\end{subarray}}(-1)^{l(F)}\mathtt{H}_{F}=\mathtt{D}_{\bar{i}}.$

####  2.3. Currying the Steinmann Arrows

Given a species p, we let $\textbf{p}^{\textbf{E}}$ denote the species given
by

$\textbf{p}^{\textbf{E}}[I]:=\prod_{r=0}^{\infty}\big{(}\textbf{p}^{[r]}[I]\big{)}^{\textsf{S}_{r}}.$

Here, $\textbf{p}^{[r]}$ is the $Y$-derivative of p for $Y=[r]$, and
$(-)^{\textsf{S}_{r}}$ denotes the subspace of $\textsf{S}_{r}$-invariants,
where $\textsf{S}_{r}$ is the symmetric group on $[r]$. We denote elements of
$\textbf{p}^{\textbf{E}}[I]$ using formal power series notation

$\sum_{r=0}^{\infty}\mathtt{x}_{r},\qquad\mathtt{x}_{r}\in\textbf{p}^{[r]}[I].$

Explicitly, $\mathtt{x}_{r}$ is an element of the vector space
$\textbf{p}[\\{\ast_{1},\dots,\ast_{r}\\}\sqcup I]$ which is invariant under
the action of permuting $\\{\ast_{1},\dots,\ast_{r}\\}$ and leaving $I$ fixed.

The mapping $\textbf{p}\mapsto\textbf{p}^{\textbf{E}}$ extends to an
endofunctor on species. In particular, given a morphism of species
$\eta:\textbf{p}\to\textbf{q}$, we have the morphism $\eta^{\textbf{E}}$ given
by

(14)
$\eta^{\textbf{E}}:\textbf{p}^{\textbf{E}}\to\textbf{q}^{\textbf{E}},\qquad\sum_{r=0}^{\infty}\mathtt{x}_{r}\mapsto\sum_{r=0}^{\infty}\eta_{[r]\sqcup
I}(\mathtt{x}_{r}).$

A _series_ of a species p is a morphism of species of the form
$s:\textbf{E}\to\textbf{p}$. Notice the elements of
$\textbf{p}^{\textbf{E}}[I]$ are naturally series of the species
$Y\mapsto\textbf{p}^{[Y]}[I]$. See [Nor20, Section 3.2] for more details. For
the connection between $\textbf{p}^{\textbf{E}}$ and the internal hom for the
Cauchy product, see [Nor20, Section 2.3].

If a is an algebra in species, then so is $\textbf{a}^{\textbf{E}}$, see
[Nor20, Equation 12]. In particular, $\boldsymbol{\Sigma}^{\textbf{E}}$ is an
algebra, with multiplication given by

$\sum_{r=0}^{\infty}\mathtt{x}_{r}\otimes\sum_{r=0}^{\infty}\mathtt{y}_{r}\
\mapsto\
\sum_{r=0}^{\infty}\sum_{r_{1}+r_{2}=r}\dfrac{r!}{r_{1}!\,r_{2}!}\mu_{[r_{1}]\sqcup
S,[r_{2}]\sqcup T}(\mathtt{x}_{r_{1}}\otimes\mathtt{y}_{r_{2}}).$

######  Theorem 2.2.

We have the following homomorphisms of algebras in species,

$\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\textbf{E}},\qquad\mathtt{H}_{F}\mapsto\sum_{r=0}^{\infty}\,\sum_{Y_{1}\sqcup\dots\sqcup
Y_{k}=[r]}\mathtt{R}_{(Y_{1};S_{1})}\dots\mathtt{R}_{(Y_{k};S_{k})}$

and

$\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\textbf{E}},\qquad\mathtt{H}_{F}\mapsto\sum_{r=0}^{\infty}\,\sum_{Y_{1}\sqcup\dots\sqcup
Y_{k}=[r]}\mathtt{A}_{(Y_{1};S_{1})}\dots\mathtt{A}_{(Y_{k};S_{k})}.$

###### Proof.

The Steinmann arrows are commutative up biderivations of
$\boldsymbol{\Sigma}$, and so give $\boldsymbol{\Sigma}$ the structure of a
Hopf E-algebra. This result is then a special case of [Nor20, Theorem 5.1]. ∎

The homomorphisms of 2.2 are the unique extensions of the maps

$\mathtt{H}_{(I)}\mapsto\sum_{r=0}^{\infty}\mathtt{R}_{(r;I)}\qquad\text{and}\qquad\mathtt{H}_{(I)}\mapsto\sum_{r=0}^{\infty}\mathtt{A}_{(r;I)}$

to homomorphisms. In the application to causal perturbation theory, we shall
be interested in the decorated analog of these homomorphisms, see Section 4.1.

######  Remark 2.3.

These homomorphisms $\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}^{\textbf{E}}$
come from currying the E-actions of the Steinmann arrows. See [Nor20, Section
5.1] for details.

####  2.4. The Steinmann Arrows and Dynkin Elements

Figure 4. Schematic for the action of the retarded Steinmann arrow
$\ast\downarrow$ for $I=\\{1,2,3\\}$ on the Steinmann sphere (left) and the
tropical geometric realization of $\boldsymbol{\Sigma}$ (right, see [NO19,
Introduction]).

We now show that the restriction of the Steinmann arrows to Zie, which are
derivations for its Lie bracket, have an interesting description in terms of
cells, i.e. chambers of the adjoint braid arrangement.

Following [Eps16, Section 2], we define the commutative up operators

$\ast\downarrow(-):\textbf{L}^{\vee}\to{\textbf{L}^{\vee}}^{\prime},\qquad\ast\downarrow\mathcal{S}:=\big{\\{}(\ast
S,T),(S,\ast T),(I,\ast):(S,T)\in\mathcal{S}\big{\\}}$

and

$\ast\uparrow(-):\textbf{L}^{\vee}\to{\textbf{L}^{\vee}}^{\prime},\qquad\ast\uparrow\mathcal{S}:=\big{\\{}(\ast
S,T),(S,\ast T),(\ast,I):(S,T)\in\mathcal{S}\big{\\}}.$

These are indeed well-defined; $\ast\downarrow\mathcal{S}$ corresponds to the
adjoint braid arrangement chamber on the $I$ side of the hyperplane
$\lambda_{\ast,I}=0$ which has the face of $\mathcal{S}$ as a facet, and
$\ast\uparrow\mathcal{S}$ corresponds to the chamber on the $\ast$ side of the
hyperplane $\lambda_{\ast,I}=0$ which has the face of $\mathcal{S}$ as a
facet. See around [LNO19, Remark 2.2] for more details. Thus, it follows from
1.3 (Ruelle’s identity) that

$[\mathtt{H}_{(\ast)},\mathtt{D}_{\mathcal{S}}]=\mathtt{D}_{\ast\uparrow\mathcal{S}}-\mathtt{D}_{\ast\downarrow\mathcal{S}}.$

The induced E-modules are given by

$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{L}^{\vee}\to\textbf{L}^{\vee},\qquad\mathtt{H}_{Y}\otimes\mathcal{S}\mapsto
Y\downarrow\mathcal{S}:=\big{\\{}(Y_{1}\sqcup S,Y_{2}\sqcup T)\in[Y\sqcup
I;\text{2}]:(S,T)\in\mathcal{S}\ \text{or}\ S=I\big{\\}}$

and

$\textbf{E}\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}\textbf{L}^{\vee}\to\textbf{L}^{\vee},\qquad\mathtt{H}_{Y}\otimes\mathcal{S}\mapsto
Y\uparrow\mathcal{S}:=\big{\\{}(Y_{1}\sqcup S,Y_{2}\sqcup T)\in[Y\sqcup
I;\text{2}]:(S,T)\in\mathcal{S}\ \text{or}\ T=I\big{\\}}.$

######  Proposition 2.3.

Given a cell $\mathcal{S}$ over $I$, we have

$Y\downarrow\mathtt{D}_{\mathcal{S}}=\mathtt{D}_{Y\downarrow\mathcal{S}}\qquad\text{and}\qquad
Y\uparrow\mathtt{D}_{\mathcal{S}}=\mathtt{D}_{Y\uparrow\mathcal{S}}.$

###### Proof.

We consider the retarded case
$Y\downarrow\mathtt{D}_{\mathcal{S}}=\mathtt{D}_{Y\downarrow\mathcal{S}}$
only, since the advanced case then follows similarly. It is sufficient to
consider the case $Y=\\{\ast\\}$. We have

$\downarrow\mathtt{D}_{\mathcal{S}}=-\sum_{\bar{F}\subseteq\mathcal{S}}(-1)^{l(F)}\downarrow\mathtt{H}_{F}\qquad\text{and}\qquad\mathtt{D}_{\downarrow\mathcal{S}}=-\sum_{\bar{F}\subseteq\,\downarrow\mathcal{S}}(-1)^{l(F)}\,\mathtt{H}_{F}.$

So, the result follows if we have the following equality

$\sum_{\bar{F}\subseteq\mathcal{S}}(-1)^{l(F)}\sum_{1\leq m\leq
k}-\mathtt{H}_{(S_{1},\dots,\ast,S_{m},\dots,S_{k})}+\mathtt{H}_{(S_{1},\dots,\ast
S_{m},\dots,S_{k})}\overset{\mathrm{?}}{=}\sum_{\bar{G}\subseteq\,\downarrow\mathcal{S}}(-1)^{l(G)}\,\mathtt{H}_{G}.$

Indeed, notice that the $\mathtt{H}$-basis elements
$\mathtt{H}_{G}\in\boldsymbol{\Sigma}[\ast I]$ which appear on the LHS are
exactly those such that

$\bar{G}\subseteq\downarrow\mathcal{S}.$

Notice also that each $\mathtt{H}_{G}$ appears with total sign $(-1)^{l(G)}$,
since when $\ast$ is inserted as a singleton lump, thus increasing $l(G)$ by
one, it appears also with a negative sign. ∎

######  Remark 2.4.

This interpretation of the E-module structure of $\boldsymbol{\Sigma}$
restricted to the primitive part
$\textbf{Zie}=\mathcal{P}(\boldsymbol{\Sigma})$ in terms of the adjoint braid
arrangement suggests obvious generalizations of the Steinmann arrows in the
direction of [AM17], [AM20], since the generalization of Hopf monoids there is
via hyperplane arrangements.

######  Corollary 2.3.1.

We have the following homomorphisms of Lie algebras in species,

$\textbf{Zie}\to\textbf{Zie}^{\textbf{E}},\qquad\mathtt{D}_{\mathcal{S}}\mapsto\mathtt{D}_{(-)\downarrow\mathcal{S}}=\sum_{r=0}^{\infty}\mathtt{D}_{[r]\downarrow\mathcal{S}}=\mathtt{D}_{\mathcal{S}}+\mathtt{D}_{\downarrow\mathcal{S}}+\mathtt{D}_{\downarrow\downarrow\mathcal{S}}+\cdots$

and

$\textbf{Zie}\to\textbf{Zie}^{\textbf{E}},\qquad\mathtt{D}_{\mathcal{S}}\mapsto\mathtt{D}_{(-)\uparrow\mathcal{S}}=\sum_{r=0}^{\infty}\mathtt{D}_{[r]\uparrow\mathcal{S}}=\mathtt{D}_{\mathcal{S}}+\mathtt{D}_{\uparrow\mathcal{S}}+\mathtt{D}_{\uparrow\uparrow\mathcal{S}}+\cdots.$

###### Proof.

The Steinmann arrows are commutative up biderivations of Zie, and so give Zie
the structure of a Lie E-algebra. This result is then a special case of
[Nor20, Theorem 5.1]. ∎

###  3\. Products and Series

We now recall several basic constructions of casual perturbation theory in the
current, clean, abstract setting. We do this without yet imposing causal
factorization/causal additivity. We say e.g. ‘T-product’ and ‘R-product’ for
now, and then change to ‘time-ordered product’ and ‘retarded product’ in the
presence of causal factorization.

####  3.1. T-Products, Generalized T-Products, and Generalized R-Products

Let $V$ be a vector space over $\mathbb{C}$. Let $\mathcal{A}$ be a
$\mathbb{C}$-algebra with multiplication denoted by $\star$. Let
$\textbf{U}_{\mathcal{A}}$ be the algebra in species given by

$\textbf{U}_{\mathcal{A}}[I]:=\mathcal{A}.$

The action of bijections is trivial, and the multiplication is the
multiplication of $\mathcal{A}$.

The _positive exponential species_ $\textbf{E}^{\ast}_{+}$ is given by

$\textbf{E}^{\ast}_{+}[I]:=\mathbb{C}\qquad\text{if}\quad
I\neq\emptyset\qquad\text{and}\qquad\textbf{E}^{\ast}_{+}[\emptyset]=0.$

Let a _system of_ T _-products_ T be a system of products for the positive
exponential species $\textbf{E}^{\ast}_{+}$, as defined in [Nor20, Section
6.2]. This means T is a morphism of species of the form171717 recall the
definition and notation for $\textbf{E}_{V}$ from Section 1.3

$\text{T}:\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},\qquad\mathtt{H}_{(I)}\otimes\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{(I)}\otimes\emph{{A}}_{I})$

where recall $\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{V}$ is the Hadamard
product of species, given by

$\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{V}[I]:=\textbf{E}^{\ast}_{+}[I]\otimes\textbf{E}_{V}[I].$

Thus, if $I\neq 0$, we have

$\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{V}[I]\cong V^{\otimes I}.$

We abbreviate

(15)
$\text{T}_{I}(\emph{{A}}_{I}):=\text{T}_{I}(\mathtt{H}_{(I)}\otimes\emph{{A}}_{I}).$

Let $\mathcal{H}(\textbf{E}_{V},\textbf{U}_{\mathcal{A}})$ denote the species
of linear maps between components, given by

$\mathcal{H}(\textbf{E}_{V},\textbf{U}_{\mathcal{A}})[I]:=\operatorname{Hom}_{\textsf{Vec}}\\!\big{(}\textbf{E}_{V}[I],\textbf{U}_{\mathcal{A}}[I])=\operatorname{Hom}_{\textsf{Vec}}\\!\big{(}V^{\otimes
I},\mathcal{A}).$

We have that $\mathcal{H}(-,-)$ is the hom for the Hadamard product. Therefore
we can curry T to give the morphism of species

$\textbf{E}^{\ast}_{+}\to\mathcal{H}(\textbf{E}_{V},\textbf{U}_{\mathcal{A}}),\qquad\mathtt{H}_{(I)}\mapsto\text{T}(I)$

where $\text{T}(I)$ is the linear map

$\text{T}(I):V^{\otimes
I}\to\mathcal{A},\qquad\emph{{A}}_{I}\mapsto\text{T}_{I}(\emph{{A}}_{I}).$

The linear maps $\text{T}(I)$ are called T _-products_. Notice that T-products
are commutative in the sense that

$\text{T}_{I}\big{(}\textbf{E}_{V}[\sigma](\emph{{A}}_{I})\big{)}=\text{T}_{I}(\emph{{A}}_{I})\qquad\quad\text{for
all bijections}\quad\sigma:I\to I.$

This property holds because the system T is a morphism of species, and
bijections act trivially for $\textbf{U}_{\mathcal{A}}$. This commutativity
exists despite the fact that the algebra $\mathcal{A}$ is noncommutative in
general.

######  Remark 3.1.

In applications to QFT, we shall also have a causal structure on $V$. Then T
is meant to first order the vectors of $\emph{{A}}_{I}$ according to the
causal structure, and then multiply in $\mathcal{A}$, giving rise to this
commutativity.

Let the _system of generalized_ T _-products_ associated to a system of
T-products be the unique extension to a system of products for
$\boldsymbol{\Sigma}=\textbf{L}\boldsymbol{\circ}\textbf{E}_{+}^{\ast}$ which
is a homomorphism, as defined in [Nor20, Section 6.2]. Thus

$\text{T}:\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},\qquad\mathtt{H}_{F}\otimes\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I}):=\text{T}_{S_{1}}(\emph{{A}}_{S_{1}})\star\dots\star\text{T}_{S_{k}}(\emph{{A}}_{S_{k}}).$

The currying of T is denoted by

$\boldsymbol{\Sigma}\to\mathcal{H}(\textbf{E}_{V},\textbf{U}_{\mathcal{A}}),\qquad\mathtt{H}_{F}\mapsto\text{T}(S_{1})\dots\text{T}(S_{k}).$

The linear maps

$\text{T}(S_{1})\dots\text{T}(S_{k}):V^{\otimes
I}\to\mathcal{A},\qquad\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I})$

are called _generalized_ T-_products_. Let the _system of generalized_ R
_-products_ associated to a system of T-products be the restriction to the Lie
algebra of primitive elements Zie,

$\text{R}:\textbf{Zie}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},\qquad\mathtt{D}_{\mathcal{S}}\otimes\emph{{A}}_{I}\mapsto\text{R}_{I}(\mathtt{D}_{\mathcal{S}}\otimes\emph{{A}}_{I}):=\text{T}_{I}(\mathtt{D}_{\mathcal{S}}\otimes\emph{{A}}_{I}).$

This is a morphism of Lie algebras, where $\textbf{U}_{\mathcal{A}}$ is
equipped with the commutator bracket. The currying of R is denoted by

$\textbf{Zie}\to\mathcal{H}(\textbf{E}_{V},\textbf{U}_{\mathcal{A}}),\qquad\mathtt{D}_{\mathcal{S}}\mapsto\text{R}_{\mathcal{S}}.$

The linear maps

$\text{R}_{\mathcal{S}}:\textbf{E}_{V}[I]\to\mathcal{A},\qquad\emph{{A}}_{I}\mapsto\text{R}_{I}(\mathtt{D}_{\mathcal{S}}\otimes\emph{{A}}_{I})$

are called _generalized_ R-_products_. From the expansion (6) of Dynkin
elements $\mathtt{D}_{\mathcal{S}}$ in terms of the $\mathtt{H}$-basis, we
recover [EG73, Equation 79],

$\text{R}_{\mathcal{S}}=-\sum_{\mathcal{F}_{F}\subseteq\bar{\mathcal{S}}}(-1)^{k}\,\text{T}(S_{1})\dots\text{T}(S_{k}).$

######  Remark 3.2.

Consider a system of products of the form

$\text{Z}:\textbf{E}^{\ast}_{+}\otimes{\textbf{E}_{V}}\to\textbf{U}_{V},\qquad\mathtt{H}_{(I)}\otimes\emph{{A}}_{I}\mapsto\text{Z}_{I}(\emph{{A}}_{I}).$

Then we obtain a new T-product $\text{T}^{\prime}$, given by

$\text{T}^{\prime}:\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},\qquad\text{T}^{\prime}_{I}(\emph{{A}}_{I}):=\sum_{P}\text{T}_{P}\big{(}\text{Z}_{S_{1}}(\emph{{A}}_{S_{1}})\dots\text{Z}_{S_{k}}(\emph{{A}}_{S_{k}})\big{)}.$

The sum is over all partitions $P=\\{S_{1},\dots,S_{k}\\}$ of $I$. This
construction underlies renormalization in pAQFT [Düt19, Section 3.6.2], which
deals with the remaining ambiguity of T-products after imposing causal
factorization, and perhaps other renormalization conditions.

####  3.2. Reverse T-Products

The system of _reverse generalized_ T _-products_ $\overline{\text{T}}$ of a
system of generalized T-products is given by precomposing T with the antipode
(5) of $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$, thus

$\overline{\text{T}}:\boldsymbol{\Sigma}\otimes{\textbf{E}}_{V}\to\textbf{U}_{\mathcal{A}^{\operatorname{op}}},\qquad\overline{\text{T}}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I}):={\text{T}}_{I}\big{(}\overline{\mathtt{H}}_{F}\otimes\emph{{A}}_{I}\big{)}.$

Since the antipode is a homomorphism
$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\operatorname{op},\text{cop}}$
[AM10, Proposition 1.22 (iii)], this is a system of generalized T-products
into the opposite algebra $\textbf{U}_{\mathcal{A}^{\operatorname{op}}}$. The
image of $\mathtt{H}_{(I)}$ under the currying of $\overline{\text{T}}$ is
called the _reverse_ T _-product_

$\overline{\text{T}}(I):\textbf{E}_{V}[I]\to\mathcal{A}^{\text{op}}.$

From (2), we obtain

$\overline{\text{T}}(I)=\sum_{F\in\Sigma[I]}(-1)^{k}\,\text{T}(S_{1})\dots\text{T}(S_{k}).$

Note that reverse T-products in [EG73, Equation 11] are defined to be
$(-1)^{n}\,\overline{\text{T}}(I)$. Our definition agrees with [Sch20,
Definition 15.35].

####  3.3. T-Exponentials

For details on series in species, see [AM10, Section 12]. The (scaled)
_universal series_ $\mathtt{G}(c)$ is the group-like series of
$\boldsymbol{\Sigma}$ given by

$\mathtt{G}(c):\textbf{E}\to\boldsymbol{\Sigma},\qquad\mathtt{H}_{I}\mapsto\mathtt{G}(c)_{I}:=c^{n}\,\mathtt{H}_{(I)}\qquad\quad\text{for}\quad
c\in\mathbb{C}.$

The fundamental nature of this series is described in [AM13, Section 13.6].
The series $\text{s}\circ\mathtt{G}(c)$ which is the composition of
$\mathtt{G}(c)$ with the antipode s of $\boldsymbol{\Sigma}$ is given by

(16)
$\text{s}\circ\mathtt{G}(c):\textbf{E}\to\boldsymbol{\Sigma},\qquad\mathtt{H}_{I}\mapsto\big{(}\text{s}\circ\mathtt{G}(c)\big{)}_{I}=c^{n}\,\overline{\mathtt{H}}_{(I)}.$

Let $\mathcal{A}[[\emph{{j}}\,]]$ denote the $\mathbb{C}$-algebra of formal
power series in the formal symbol j with coefficients in $\mathcal{A}$. Given
a system of generalized T-products

$\text{T}:\boldsymbol{\Sigma}\otimes{\textbf{E}}_{V}\to\textbf{U}_{\mathcal{A}}$

let the T-_exponential_ $\mathcal{S}:=\mathcal{S}_{\mathtt{G}(c)}$ of this
system be the $\mathcal{A}[[\emph{{j}}\,]]$-valued function on the vector
space $V$ associated to the series $\mathtt{G}(c)$, as constructed in [Nor20,
Section 6.3]. Thus, we have181818 we use the abbreviations (4) and (15), and
also $\text{T}_{n}:=\text{T}_{[n]}$

(17)
$\mathcal{S}:V\to\mathcal{A}[[\emph{{j}}\,]],\qquad\emph{{A}}\mapsto\mathcal{S}(\emph{{j}}\,\\!\emph{{A}})=\sum^{\infty}_{n=0}\dfrac{c^{n}}{n!}\text{T}_{n}\underbrace{\big{(}\emph{{j}}\,\\!\emph{{A}}\otimes\cdots\otimes\emph{{j}}\,\\!\emph{{A}}\big{)}}_{\text{
$n$ times
}}:=\sum^{\infty}_{n=0}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\text{T}_{n}(\emph{{A}}^{n}).$

By [Nor20, Equation 34] and (16), the T-exponential for the system of reverse
T-products is the inverse of $\mathcal{S}$ as an element of the
$\mathbb{C}$-algebra of functions
$\text{Func}(V,\mathcal{A}[[\emph{{j}}\,]])$, given by

$\mathcal{S}^{-1}:V\to\mathcal{A}[[\emph{{j}}\,]],\qquad\emph{{A}}\mapsto\mathcal{S}^{-1}(\emph{{j}}\,\\!\emph{{A}}):=\sum^{\infty}_{n=0}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\overline{\text{T}}_{n}(\emph{{A}}^{n})=\sum^{\infty}_{n=0}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\text{T}_{n}(\overline{\mathtt{H}}_{(n)}\otimes\emph{{A}}^{n}).$

Therefore

$\mathcal{S}(\emph{{j}}\,\\!\emph{{A}})\star\mathcal{S}^{-1}(\emph{{j}}\,\\!\emph{{A}})=\mathcal{S}^{-1}(\emph{{j}}\,\\!\emph{{A}})\star\mathcal{S}(\emph{{j}}\,\\!\emph{{A}})=1_{\mathcal{A}}$

for all $\emph{{A}}\in V$. This appears in e.g. [EG73, Equation 2].

###  4\. Perturbation of T-Products

For the perturbation of T-products by a certain up coderivation of E which
gives the S-matrix scheme
$\mathcal{S}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=\mathcal{S}(\emph{{g}}\emph{{S}}+\emph{{j}}\,\\!\emph{{A}})$,
see [Nor20, Section 10.1].

####  4.1. Perturbation of T-Products by Steinmann Arrows

Suppose we have a system of generalized T-products

$\text{T}:\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}},\qquad\mathtt{H}_{F}\otimes\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I}).$

Following [Nor20, Section 6.4], given a choice of decorations vector
$\emph{{S}}\in V$, we can use the retarded Steinmann arrow (11) to perturb T
as follows.

Recall the decorated Hopf algebra $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$
from Section 1.3. Recall also the derivative
$(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\prime}$ of
$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$ from Section 2.1, given by

$(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\prime}[I]=\boldsymbol{\Sigma}[\ast
I]\otimes V\otimes V^{\otimes I}.$

We have the up derivation of $\boldsymbol{\Sigma}\otimes\textbf{E}_{V}$ which
is the decorated analog of the retarded Steinmann arrow, given by

$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\prime},\qquad\mathtt{H}_{F}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\otimes\emph{{A}}_{i_{n}}\
\mapsto\
\ast\downarrow\mathtt{H}_{F}\otimes\emph{{S}}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\otimes\emph{{A}}_{i_{n}}.$

This is indeed still an up derivation by [Nor20, Proposition 6.4]. Analogous
to the setting without decorations, we have the induced raising operators and
associated E-action by iterating, which, after currying, give us the
homomorphism

$\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\textbf{E}}$
$\mathtt{H}_{F}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\otimes\emph{{A}}_{i_{n}}\
\mapsto\ \sum^{\infty}_{r=0}\underbrace{\downarrow\dots\downarrow}_{\text{$r$
times}}\mathtt{H}_{F}\otimes\underbrace{\emph{{S}}\otimes\dots\otimes\emph{{S}}}_{\text{$r$
times}}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\otimes\emph{{A}}_{i_{n}}.$

This is a homomorphism by [Nor20, Theorem 5.1]. Then, a new ‘perturbed’ system
of generalized T-products is given by composing this homomorphism with
$\text{T}^{\textbf{E}}$ (defined in (14)),

$\widetilde{\text{T}}:\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to(\boldsymbol{\Sigma}\otimes\textbf{E}_{V})^{\textbf{E}}\xrightarrow{\text{T}^{\textbf{E}}}(\textbf{U}_{\mathcal{A}})^{\textbf{E}}\cong\textbf{U}_{\mathcal{A}[[\emph{{g}}]]}.$

For the result that
$(\textbf{U}_{\mathcal{A}})^{\textbf{E}}\cong\textbf{U}_{\mathcal{A}[[\emph{{g}}]]}$,
see [Nor20, Section 4].

######  Remark 4.1.

The fact $\widetilde{\text{T}}$ is still a homomorphism, and is thus still a
generalized system of products, depends crucially on the fact the Steinmann
arrow is a derivation [Nor20, Theorem 5.1], and that $(-)^{\textbf{E}}$ is a
monoidal functor [Nor20, Section 2.5]. We can similarly perturb a system of
generalized R-products, which uses the fact the Steinmann arrow is a
biderivation.

We now unpack all this formalism to give a fully explicit description of the
new perturbed system of products. Let us abbreviate

$\emph{{S}}_{Y}\emph{{A}}_{I}=\emph{{S}}_{y_{1}}\otimes\dots\otimes\emph{{S}}_{y_{r}}\otimes\emph{{A}}_{i_{1}}\otimes\dots\otimes\emph{{A}}_{i_{n}}\in\textbf{E}_{V}[Y\sqcup
I].$

Let

(18)
$\text{R}_{Y;I}(\emph{{S}}_{Y};\emph{{A}}_{I}):=\underbrace{\text{T}_{Y\sqcup
I}(\mathtt{R}_{(Y;I)}\otimes\emph{{S}}_{Y}\emph{{A}}_{I})=\sum_{Y_{1}\sqcup
Y_{2}=Y}\overline{\text{T}}_{Y_{1}\sqcup\emptyset}(\emph{{S}}_{Y_{1}})\star\text{T}_{Y_{2}\sqcup
I}(\emph{{S}}_{Y_{2}}\emph{{A}}_{I})}_{\text{by
}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\ref{eq:retardadvan}})}$

Then the new perturbed system is given by191919 we abbreviate
$\text{R}_{r;I}(\emph{{S}}^{\,r};\emph{{A}}_{I}):=\text{R}_{[r];I}(\emph{{S}}^{\,[r]};\emph{{A}}_{I})=\text{R}_{[r];I}(\underbrace{\emph{{S}}\otimes\dots\otimes\emph{{S}}}_{\text{$r$
times}}\,;\emph{{A}}_{I})$

$\widetilde{\text{T}}:\boldsymbol{\Sigma}\otimes\textbf{E}_{V}\to\textbf{U}_{\mathcal{A}[[\emph{{g}}]]},\quad\mathtt{H}_{F}\otimes\emph{{A}}_{I}\mapsto\sum_{r=0}^{\infty}\
\sum_{r_{1}+\,\cdots\,+r_{k}=r}\dfrac{\emph{{g}}^{r}}{r!}\text{R}_{r_{1};S_{1}}(\emph{{S}}^{\,r_{1}};\emph{{A}}_{S_{1}})\star\cdots\star\text{R}_{r_{k};S_{k}}(\emph{{S}}^{\,r_{k}};\emph{{A}}_{S_{k}}).$

In particular, the restriction to
$\textbf{E}_{+}^{\ast}\otimes\textbf{E}_{V}$, i.e. the new perturbed
T-product, is given by

$\displaystyle\widetilde{\text{T}}_{I}(\emph{{A}}_{I})$
$\displaystyle=\sum_{r=0}^{\infty}\dfrac{\emph{{g}}^{r}}{r!}\text{R}_{r;I}(\emph{{S}}^{\,r};\emph{{A}}_{I})$
$\displaystyle=\text{T}_{I}(\emph{{A}}_{I})+\underbrace{\emph{{g}}\,\text{T}_{\ast_{1}I}(\downarrow\mathtt{H}_{(I)}\otimes\emph{{S}}\emph{{A}}_{I})+\dfrac{\emph{{g}}^{2}}{2!}\text{T}_{\ast_{2}\ast_{1}I}(\downarrow\downarrow\mathtt{H}_{(I)}\otimes\emph{{S}}\emph{{S}}\emph{{A}}_{I})+\cdots}_{\text{perturbation}}\
.$

Similar, we can perturb a system of generalized T-products using the advanced
Steinmann arrow.

We let $\mathcal{V}_{\\!\emph{{g}}\emph{{S}}}$, respectively
$\mathcal{W}_{\\!\emph{{g}}\emph{{S}}}$, denote the T-exponential (as defined
in (17)) for the new perturbed system of generalized T-products using the
retarded, respectively advanced, Steinmann arrows. Thus

$\mathcal{V}_{\\!\emph{{g}}\emph{{S}}}:V\to\mathcal{A}[[\emph{{g}},\\!\emph{{j}}\,]],\qquad\mathcal{V}_{\\!\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}}):=\sum_{n=0}^{\infty}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\,\widetilde{\text{T}}_{n}(\emph{{A}}^{n})=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\dfrac{\emph{{g}}^{r}\emph{{j}}\,^{n}c^{r+n}}{r!\,n!}\,\text{R}_{r;n}(\emph{{S}}^{\,r};\emph{{A}}^{n})$

and

$\mathcal{W}_{\\!\emph{{g}}\emph{{S}}}:V\to\mathcal{A}[[\emph{{g}},\\!\emph{{j}}\,]],\qquad\mathcal{W}_{\\!\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}}):=\sum_{n=0}^{\infty}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\,\widetilde{\text{T}}_{n}(\emph{{A}}^{n})=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\dfrac{\emph{{g}}^{r}\emph{{j}}\,^{n}c^{r+n}}{r!\,n!}\,\text{A}_{r;n}(\emph{{S}}^{\,r};\emph{{A}}^{n})$

where

$\text{A}_{Y;I}(\emph{{S}}_{Y};\emph{{A}}_{I}):=\underbrace{\text{T}_{Y\sqcup
I}(\mathtt{A}_{(Y;I)}\otimes\emph{{S}}_{Y}\emph{{A}}_{I})=\sum_{Y_{1}\sqcup
Y_{2}=Y}\text{T}_{Y_{1}\sqcup
I}(\emph{{S}}_{Y_{1}}\emph{{A}}_{I})\star\overline{\text{T}}_{Y_{2}\sqcup\emptyset}(\emph{{S}}_{Y_{2}})}_{\text{by
}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\ref{eq:retardadvan}})}.$

######  Theorem 4.1.

We have

$\mathcal{V}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=\mathcal{S}^{-1}(\emph{{g}}\emph{{S}})\star\mathcal{S}(\emph{{g}}\emph{{S}}+\emph{{j}}\,\\!\emph{{A}})\qquad\text{and}\qquad\mathcal{W}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=\mathcal{S}(\emph{{g}}\emph{{S}}+\emph{{j}}\,\\!\emph{{A}})\star\mathcal{S}^{-1}(\emph{{g}}\emph{{S}}).$

###### Proof.

We have

$\text{R}_{r;I}(\emph{{S}}^{\,r};\emph{{A}}_{I})=\sum_{Y_{1}\sqcup
Y_{2}=[r]}\overline{\text{T}}_{Y_{1}\sqcup\emptyset}(\emph{{S}}^{Y_{1}})\star\text{T}_{Y_{2}\sqcup
I}(\emph{{S}}^{Y_{2}}\emph{{A}}_{I}).$

Then

$\displaystyle\mathcal{V}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=$
$\displaystyle\
\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\dfrac{\emph{{g}}^{r}\emph{{j}}\,^{n}c^{r+n}}{r!\,n!}\,\text{R}_{r;n}(\emph{{S}}^{\,r};\emph{{A}}^{n})$
$\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\dfrac{\emph{{g}}^{r}\emph{{j}}\,^{n}c^{r+n}}{r!\,n!}\sum_{Y_{1}\sqcup
Y_{2}=[r]}\overline{\text{T}}_{Y_{1}\sqcup\emptyset}(\emph{{S}}^{Y_{1}})\star\text{T}_{Y_{2}\sqcup[n]}(\emph{{S}}^{Y_{2}}\emph{{A}}^{n})$
$\displaystyle=$ $\displaystyle\
\sum^{\infty}_{r=0}\dfrac{\emph{{g}}^{r}c^{r}}{r!}\overline{\text{T}}_{r+0}(\emph{{S}}^{\,r})\star\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\dfrac{c^{n}}{n!}\text{T}_{r+n}(\emph{{S}}^{\,r}\emph{{A}}^{n})$
$\displaystyle=$ $\displaystyle\
\mathcal{S}^{-1}(\emph{{g}}\emph{{S}})\star\mathcal{S}(\emph{{g}}\emph{{S}}+\emph{{j}}\,\\!\emph{{A}})$

The proof for $\mathcal{W}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})$
is similar. ∎

######  Corollary 4.1.1 (Bogoliubov’s Formula [BS59, Chapter 4]).

We have

(19)
$\widetilde{\text{T}}_{i}(\emph{{A}})=\dfrac{1}{c}\,\dfrac{d}{d\emph{{j}}\,}\Bigr{|}_{\emph{{j}}\,=0}\mathcal{V}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}}).$

###### Proof.

We have

$\dfrac{d}{d\emph{{j}}\,}\mathcal{V}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=\dfrac{d}{d\emph{{j}}\,}\sum_{n=0}^{\infty}\dfrac{\emph{{j}}\,^{n}c^{n}}{n!}\,\widetilde{\text{T}}_{n}(\emph{{A}}^{n})=\sum_{n=1}^{\infty}\dfrac{\emph{{j}}\,^{n-1}c^{n}}{(n-1)!}\,\widetilde{\text{T}}_{n}(\emph{{A}}^{n}).$

Then, putting $\emph{{j}}\,=0$, we obtain

$\dfrac{d}{d\emph{{j}}\,}\Bigr{|}_{\emph{{j}}\,=0}\mathcal{V}_{\emph{{g}}\emph{{S}}}(\emph{{j}}\,\\!\emph{{A}})=c\,\widetilde{\text{T}}_{1}(\emph{{A}}).\qed$

This formula was originally motivated by the path integral heuristic, see e.g.
[Sch20, Remark 15.16].

####  4.2. R-Products and A-Products

The linear maps $\text{R}(Y;I)$ which are given by

$\text{R}(Y;I):\textbf{E}_{V}^{[Y]}[I]\to\mathcal{A},\qquad\emph{{S}}_{Y}\emph{{A}}_{I}\mapsto\text{R}_{Y;I}(\emph{{S}}_{Y}\emph{{A}}_{I})$

are called R _-products_. In the case of singletons $I=\\{i\\}$, the maps
$\text{R}(Y;i)$ are called _total_ R _-products_. By (13), R-products are
given in terms of T-products and reverse T-products by

$\text{R}(Y;I)=\sum_{Y_{1}\sqcup
Y_{2}=Y}\overline{\text{T}}(Y_{1})\star\text{T}(Y_{2}\sqcup I).$

Then

$\widetilde{\text{T}}(I)=\sum_{r=0}^{\infty}\dfrac{c^{r}}{r!}\text{R}(r;I).$

In a similar way, we can define the A-_products_ $\text{A}(Y;I)$, so that

$\text{A}(Y;I)=\sum_{Y_{1}\sqcup Y_{2}=Y}\text{T}(Y_{1}\sqcup
I)\star\overline{\text{T}}(Y_{2}).$

The total R-products are both R-products and generalized R-products, which is
due to the double description appearing in 2.2. A related result is [AM13,
Proposition 109].

######  Remark 4.2.

In the literature, the total retarded products in our sense are sometimes
called retarded products, and the retarded products in our sense are then
called generalized retarded products, e.g. [Pol58], [Düt19, Exercise 3.3.16].

##  Part II Perturbative Algebraic Quantum Field Theory

We now apply the theory we have developed to the case of a real scalar quantum
field on a Minkowski spacetime, as described by pAQFT.202020 although pAQFT
deals more generally with perturbative Yang-Mills gauge theory on curved
spacetimes Mathematically, the important extra property is a causal structure
on the vector space of decorations $V$, which allows one to impose causal
factorization. Connections between QFT and species have been previously
studied in [Abd04], [Far11], [GK18].

Our references for pAQFT are [DF01], [Rej16], [Düt19], [Sch20]. We mainly
adopt the notation and presentation of [Sch20]. Key features of pAQFT are its
local, i.e. sheaf-theoretic, approach, the (closely related) use of adiabatic
switching of interaction terms to avoid IR-divergences, and the interpretation
of renormalization as the extension of distributions to the fat diagonal to
avoid UV-divergences. The Wilsonian cutoff, sometimes called heuristic quantum
field theory, may be rigorously formulated within pAQFT [BDF09], [Düt12],
[Düt19, Section 3.8], [Sch20, Section 16].

###  5\. Spacetime and Field Configurations

Let $\mathcal{X}\cong\mathbb{R}^{1,p}$ denote a $(p+1)$-dimensional Minkowski
spacetime, for $p\in\mathbb{N}$. Thus, $\mathcal{X}$ is a real vector space
equipped with a metric tensor which is a symmetric nondegenerate bilinear form
$\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ with signature $(1,p)$. The
bilinear form gives rise to a volume form on $\mathcal{X}$, which we denote by
$\text{dvol}_{\mathcal{X}}\in\Omega^{p+1}(\mathcal{X})$. For regions of
spacetime $X_{1},X_{2}\subset\mathcal{X}$, we write

$X_{1}\\!\vee\\!\\!\wedge X_{2}$

if one cannot travel from $X_{1}$ to $X_{2}$ on a future-directed timelike or
lightlike curve. We have the set valued species $\mathcal{X}^{(-)}$ given by

$I\ \mapsto\ \mathcal{X}^{I}:=\big{\\{}\text{functions}\
I\to\mathcal{X}\big{\\}}.$

For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on
$\mathcal{X}$. Therefore, let $E\to\mathcal{X}$ be a smooth real vector bundle
over $\mathcal{X}$ with one-dimensional fibers. An (off-shell) _field
configuration_ $\Phi$ is a smooth section of the bundle $E\to\mathcal{X}$,

$\Phi:\mathcal{X}\hookrightarrow E,\qquad x\mapsto\Phi(x).$

The space of all field configurations, denoted $\Gamma(E)$, has the structure
of a Fréchet topological (real) vector space.

######  Remark 5.1.

We can always pick an isomorphism
$(E\to\mathcal{X})\cong(\mathcal{X}\times\mathbb{R}\to\mathcal{X})$, which
induces an isomorphism $\Gamma(E)\cong C^{\infty}(\mathcal{X},\mathbb{R})$, so
that field configurations are modeled as smooth functions
$\mathcal{X}\to\mathbb{R}$.

Let $E^{\ast}\to\mathcal{X}$ denote the dual vector bundle of $E$, and let the
canonical pairing be denoted by

$\langle-,-\rangle:E^{\ast}\otimes E\to\mathbb{R}.$

Let a _compactly supported distributional section_ $\alpha$ be a distribution
of field configurations

$\alpha:\Gamma(E)\to\mathbb{R},$

i.e. an element of the topological dual vector space of $\Gamma(E)$, which is
modeled as a sequence $(\alpha_{j})_{j\in\mathbb{N}}$ of smooth compactly
supported sections of the dual bundle $E^{\ast}\to\mathcal{X}$,

$\alpha_{j}:\mathcal{X}\hookrightarrow E^{\ast},\qquad j\in\mathbb{N},$

where the modeled distribution is recovered as the following limit of
integrals,

$\Gamma(E)\to\mathbb{R},\qquad\Phi\mapsto\underbrace{\int_{x\in\mathcal{X}}\big{\langle}\alpha(x),\Phi(x)\big{\rangle}\text{dvol}_{\mathcal{X}}:=\lim_{j\to\infty}\int_{x\in\mathcal{X}}\langle\alpha_{j}(x),\Phi(x)\rangle\text{dvol}_{\mathcal{X}}}_{\text{sometimes
called \emph{generalized function} notation}}.$

The space of all compactly supported distributional sections is denoted
$\Gamma_{\text{cp}}^{\prime}(E^{\ast})$. By e.g. [Bär15, Lemma 2.15], _all_
distributions $\Gamma(E)\to\mathbb{R}$ may be obtained as compactly supported
distributional sections in this way.

We can pullback the vector bundle $E^{\ast}$ to $\mathcal{X}^{I}$ along each
canonical projection

$\mathcal{X}^{I}\to\mathcal{X}^{\\{i\\}}\cong\mathcal{X},\qquad i\in I.$

The tensor product of these $n$ many pullback bundles is the exterior tensor
product bundle $(E^{\ast})^{\boxtimes I}$. This defines a presheaf of smooth
vector bundles on S,

$\textsf{S}^{\operatorname{op}}\to\textsf{Diff}_{/\mathcal{X}},\qquad
I\mapsto(E^{\ast})^{\boxtimes I}.$

By taking compexified compactly supported distributional sections
${{\Gamma^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(-):=\Gamma_{\text{cp}}^{\prime}(-)\otimes_{\mathbb{R}}\mathbb{C}$,
we obtain the complex vector species
${{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})$,
given by

${{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})[I]:={{\Gamma^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}\Big{(}(E^{\ast})^{\boxtimes
I}\Big{)}.$

Of course,
${{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})$
does not ‘factorize’ in the sense that it is not a monoidal functor,

(20)
${{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})[I]\ncong{{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})[i_{1}]\otimes\dots\otimes{{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})[i_{n}]$

where $I=\\{i_{1},\dots,i_{n}\\}$. There are more distributional sections then
just those coming from the tensor product.

###  6\. Observables

An off-shell _observable_ O is a smooth functional of field configurations
into the complex numbers,

$\emph{{O}}:\Gamma(E)\to\mathbb{C},\qquad\Phi\mapsto\emph{{O}}(\Phi).$

The space of all observables is denoted Obs. We can pointwise multiply
observables, sometimes called the _normal ordered product_ , so that
observables form a commutative $\mathbb{C}$-algebra,

$\text{Obs}\otimes\text{Obs}\to\text{Obs},\qquad\emph{{O}}_{1}\otimes\emph{{O}}_{2}\mapsto\emph{{O}}_{1}\cdot\emph{{O}}_{2}$

where

$\emph{{O}}_{1}\cdot\emph{{O}}_{2}(\Phi):=\\!\\!\\!\underbrace{\emph{{O}}_{1}(\Phi)\emph{{O}}_{2}(\Phi)}_{\text{multiplication
in $\mathbb{C}$}}\\!\\!\\!.$

Thus, we may form the commutative algebra in species
$\textbf{U}_{\text{Obs}}$, given by $\textbf{U}_{\text{Obs}}[I]=\text{Obs}$.

A _linear observable_ $\emph{{O}}\in\text{Obs}$ is an observable which is
additionally a linear functional, that is

$\emph{{O}}(\Phi_{1}+\Phi_{2})=\emph{{O}}(\Phi_{1})+\emph{{O}}(\Phi_{2})\qquad\text{and}\qquad\emph{{O}}(c\Phi)=c\emph{{O}}(\Phi)\qquad\text{for}\quad
c\in\mathbb{C}.$

The space of linear observables is denoted $\text{LinObs}\subset\text{Obs}$.
In particular, for each spacetime event $x\in\mathcal{X}$, we have the _field
observable_ $\boldsymbol{\Phi}(x)\in\text{LinObs}$, given by

$\boldsymbol{\Phi}(x):\Gamma(E)\to\mathbb{C},\qquad\Phi\mapsto\Phi(x).$

We now show how linear observables and so-called polynomial observables arise
species-theoretically, via (generalized) systems of products for the species E
and $\textbf{X}=\mathcal{P}(\textbf{E})$.

Let X denote the species given by
$\textbf{X}\big{[}\\{i\\}\big{]}:=\mathbb{C}$ for singletons and
$\textbf{X}[I]:=0$ otherwise. We denote
$\mathtt{H}_{i}:=1\in\textbf{X}\big{[}\\{i\\}\big{]}$. We have the following
morphism of species,

$\textbf{X}\otimes{{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})\to\textbf{U}_{\text{Obs}},\qquad\mathtt{H}_{i}\otimes\alpha\mapsto\bigg{(}\Phi\mapsto\int_{x\in\mathcal{X}}\big{\langle}\alpha(x),\Phi(x)\big{\rangle}\text{dvol}_{\mathcal{X}}\bigg{)}.$

This is like a system of products for X, however
${{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})$
does not factorize (20), and so cannot be written in the form
$\textbf{E}_{V}$. It follows from [Bär15, Lemma 2.15] that the colimit (as
defined in [AM10, Remark 15.7]) of the species which is the image of this
morphism is the space of linear observables LinObs. The currying of this map
is given by

$\textbf{X}\to\mathcal{H}\big{(}{{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast}),\textbf{U}_{\text{Obs}}\big{)},\qquad\mathtt{H}_{i}\mapsto\boldsymbol{\Phi}_{i}=\boldsymbol{\Phi}$

where

$\boldsymbol{\Phi}(\alpha):=\bigg{(}\Phi\mapsto\int_{x\in\mathcal{X}}\big{\langle}\alpha(x),\Phi(x)\big{\rangle}\text{dvol}_{\mathcal{X}}\bigg{)}.$

If we restrict $\boldsymbol{\Phi}$ to bump functions
$b\in\Gamma_{\text{cp}}(E^{\ast})\otimes_{\mathbb{R}}\mathbb{C}$, also called
‘smearing functions’, then one might call the linear map

$\boldsymbol{\Phi}:\Gamma_{\text{cp}}(E^{\ast})\otimes_{\mathbb{R}}\mathbb{C}\to\text{Obs},\qquad
b\mapsto\boldsymbol{\Phi}(b)$

an ‘observable-valued distribution’, and this is sometimes referred to as ‘the
(smeared) field’. The field observable $\boldsymbol{\Phi}(x)$ is recovered by
evaluating $\boldsymbol{\Phi}$ on the Dirac delta function $\delta_{x}$
localized at $x$. One views $b$ as the smearing of a Dirac delta function,
hence smearing functions and smeared field.

We extend the smeared field by replacing X with E to define the following
morphism of species,

$\textbf{E}\otimes{{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast})\to\textbf{U}_{\text{Obs}},\qquad\mathtt{H}_{I}\otimes\alpha_{I}\mapsto\bigg{(}\Phi\mapsto\int_{\mathcal{X}^{I}}\big{\langle}\alpha_{I}(x_{i_{1}},\dots,x_{i_{n}}),\Phi(x_{i_{1}})\dots\Phi(x_{i_{n}})\big{\rangle}\text{dvol}_{\mathcal{X}^{I}}\bigg{)}.$

This is like a system of products for E, but again without factorization. The
colimit of the species which is the image of this morphism is the vector space
of _polynomial observables_ , as defined in e.g. [Sch20, Definition 7.13],
denoted

$\text{PolyObs}\subset\text{Obs}.$

(Alternatively, if we restrict the limit of this map
$\mathscr{S}({{\boldsymbol{\Gamma}^{\prime}}^{\mathbb{C}}}_{\\!\\!\\!\\!\\!\\!\text{cp}}(E^{\ast}))\to\text{Obs}[[\emph{{j}}\,]]$
to finite series and set $\emph{{j}}\,=1$, then we recover [Düt19, Definition
1.2.1].) The space of _microcausal polynomial observables_ $\mathcal{F}$ is
the subspace

$\mathcal{F}\subset\text{PolyObs}$

consisting of those polynomial observables which satisfy a certain microlocal-
theoretic condition called _microcausality_ , see [Düt19, Definition 1.2.1
(ii)]. Following [Düt19, Definition 1.3.4], the space of _local observables_

$\mathcal{F}_{\text{loc}}\subset\text{Obs}$

consists of those observables obtained by integrating a polynomial with real
coefficients in the field and its derivatives (‘field polynomials’) against a
bump function
$b\in\Gamma_{\text{cp}}(E^{\ast})\otimes_{\mathbb{R}}\mathbb{C}$. Importantly,
we have a natural inclusion

$\mathcal{F}_{\text{loc}}\hookrightarrow\mathcal{F},\qquad\emph{{A}}\mapsto\
:\emph{{A}}:.$

Let $\mathcal{F}_{\text{loc}}[[\hbar]]$ and $\mathcal{F}[[\hbar]]$ denote the
spaces of formal power series in $\hbar$ with coefficients in
$\mathcal{F}_{\text{loc}}$ and $\mathcal{F}$ respectively, and let
$\mathcal{F}((\hbar))$ denote the space of Laurent series in $\hbar$ with
coefficients in $\mathcal{F}$.

Applying Moyal deformation quantization with formal Planck’s constant $\hbar$,
$\mathcal{F}[[\hbar]]$ is a formal power series $\ast$-algebra, called the
(abstract, off-shell) _Wick algebra_ , with multiplication the Moyal star
product [Düt19, Definition 2.1.1] defined with respect to the Wightman
propagator $\Delta_{\text{H}}$ for the Klein-Gordan field [Düt19, Section
2.2],

$\mathcal{F}[[\hbar]]\otimes\mathcal{F}[[\hbar]]\to\mathcal{F}[[\hbar]],\qquad\emph{{O}}_{1}\otimes\emph{{O}}_{2}\\\
\mapsto\emph{{O}}_{1}\star_{\text{H}}\emph{{O}}_{2}.$

We may form the algebra in species $\textbf{U}_{\mathcal{F}[[\hbar]]}$, or,
allowing negative powers of $\hbar$, $\textbf{U}_{\mathcal{F}((\hbar))}$.

###  7\. Time-Ordered Products and S-Matrix Schemes

For $\emph{{A}}\in\mathcal{F}_{\text{loc}}[[\hbar]]$, let
$\text{supp}(\emph{{A}})$ denote the spacetime support of A. Given a
composition $G$ of $I$, we say that
$\emph{{A}}_{I}\in\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}[I]$
_respects_ $G$ if

$\text{supp}({\emph{{A}}_{i_{1}}})\vee\\!\\!\wedge\
\text{supp}(\emph{{A}}_{i_{2}})\qquad\quad\text{for
all}\quad(i_{1},i_{2})\quad\text{such that}\quad
G|_{\\{i_{1},i_{2}\\}}=(i_{1},i_{2}).$

Consider a system of T-products (as defined in Section 3.1) of the form

$\text{T}:\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))},\qquad\mathtt{H}_{(I)}\otimes\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{(I)}\otimes\emph{{A}}_{I})=\text{T}_{I}(\emph{{A}}_{I}).$

Since $\boldsymbol{\Sigma}$ is the free algebra on $\textbf{E}^{\ast}_{+}$, we
have the unique extension to a system of generalized T-products

$\text{T}:\boldsymbol{\Sigma}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))},\qquad\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I}):=\text{T}_{S_{1}}(\emph{{A}}_{S_{1}})\star_{\text{H}}\dots\star_{\text{H}}\text{T}_{S_{k}}(\emph{{A}}_{S_{k}}).$

Then:

  1. 1.

(perturbation) we say that T satisfies _perturbation_ if the singleton
components $\text{T}_{i}$ are isomorphic to the inclusion
$\mathcal{F}_{\text{loc}}[[\hbar]]\hookrightarrow\mathcal{F}((\hbar))$, that
is

${\text{T}}_{i}(\emph{{A}})=\,:\\!\emph{{A}}\\!:$

  2. 2.

(causal factorization) we say that T satisfies _causal factorization_ if for
all compositions $(S,T)$ of $I$ with two lumps, if
$\emph{{A}}_{I}\in\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}[I]$ respects
$(S,T)$222222 explicitly, $\text{supp}({\emph{{A}}_{i_{1}}})\vee\\!\\!\wedge\
\text{supp}(\emph{{A}}_{i_{2}})$ for all $i_{1}\in S$ and $i_{2}\in T$ then

(21)
$\text{T}_{I}(\mathtt{H}_{(I)}\otimes\emph{{A}}_{I})=\text{T}_{I}(\mathtt{H}_{(S,T)}\otimes\emph{{A}}_{I}).$

Let a (fully normalized) _system of time-ordered products_ be a system of
T-products which satisfies perturbation and causal factorization. The
corresponding unique extension of T to $\boldsymbol{\Sigma}$ is called the
associated _system of generalized time-ordered products_. After currying

$\boldsymbol{\Sigma}\to\mathcal{H}(\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]},\textbf{U}_{\mathcal{F}((\hbar))}),\qquad\mathtt{H}_{F}\mapsto\text{T}(S_{1})\dots\text{T}(S_{k})$

the linear maps

$\text{T}(S_{1})\dots\text{T}(S_{k}):\mathcal{F}_{\text{loc}}[[\hbar]]^{\otimes
I}\to\mathcal{F}((\hbar)),\qquad\emph{{A}}_{I}\mapsto\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I})$

are called _generalized time-ordered products_. The linear maps $\text{T}(I)$
are called _time-ordered products_. After fixing a field polynomial, so that
each $\emph{{A}}_{i_{j}}$ of $\emph{{A}}_{I}$ is determined by a bump function
$b_{i_{j}}$, they are usually presented in generalized function notation as
follows,

$\text{T}_{I}(\emph{{A}}_{i_{1}}\otimes\cdots\otimes\emph{{A}}_{i_{n}})=\int_{\mathcal{X}^{I}}\text{T}(x_{i_{1}},\dots,x_{i_{n}})b_{i_{1}}(x_{i_{1}})\dots
b_{i_{n}}(x_{i_{1}})dx_{i_{1}}\dots dx_{i_{n}}$

where $(x_{i_{1}},\dots,x_{i_{n}})\mapsto\text{T}(x_{i_{1}},\dots,x_{i_{n}})$
is an ‘operator-valued’ generalized function. See e.g. [EG73, Section 1.2].

Given compositions $F=(S_{1},\dots,S_{k})$ and $G=(U_{1},\dots,U_{l})$ of $I$,
let

$\mathtt{H}_{F}\triangleright\mathtt{H}_{G}:=\mathtt{H}_{(U_{1}\cap
S_{1},\dots,U_{l}\cap S_{1},\dots\dots,U_{1}\cap S_{k},\dots,U_{l}\cap
S_{k})_{+}}.$

This is called the _Tits product_ , going back to Tits [Tit74]. See [AM13,
Section 13] for more on the structure of the Tits product, where it is shown
it is given by the action of $\boldsymbol{\Sigma}$ on itself by Hopf powers.
See also [AB08, Section 1.4.6] for the context of other Coxeter systems and
Dynkin types.

######  Proposition 7.1.

Let

$\text{T}:\textbf{E}^{\ast}_{+}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))}$

be a system of T-products which satisfies causal factorization. Given a
composition $G=(U_{1},\dots,U_{k})$ of $I$, and
$\emph{{A}}_{I}\in\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}[I]$ which
respects $G$, then

$\text{T}_{I}(\mathtt{a}\otimes\emph{{A}}_{I})=\text{T}_{I}(\mathtt{a}\triangleright\mathtt{H}_{G}\otimes\emph{{A}}_{I})\qquad\text{for
all}\quad\mathtt{a}\in\boldsymbol{\Sigma}[I].$

###### Proof.

We have

$\text{T}_{I}(\mathtt{H}_{G}\otimes\emph{{A}}_{I})=\underbrace{\text{T}_{U_{1}}(\emph{{A}}_{U_{1}})\star_{\text{H}}\dots\star_{\text{H}}\text{T}_{U_{k}}(\emph{{A}}_{U_{k}})=\text{T}_{I}(\emph{{A}}_{I})}_{\text{by
repeated applications of causal factorization}}.$

Observe that the action
$\mathtt{H}_{F}\mapsto\mathtt{H}_{F}\triangleright\mathtt{H}_{G}$, for
$F\in\Sigma[I]$, replaces the lumps of $F$ with their intersections with $G$.
But we just saw that
$\text{T}_{I}(\emph{{A}}_{I})=\text{T}_{I}(\mathtt{H}_{G}\otimes\emph{{A}}_{I})$,
and so it follows that

$\text{T}_{I}(\mathtt{H}_{F}\otimes\emph{{A}}_{I})=\text{T}_{I}(\mathtt{H}_{F}\triangleright\mathtt{H}_{G}\otimes\emph{{A}}_{I}).$

Since the claim is true for the $\mathtt{H}$-basis, it is true for all
$\mathtt{a}\in\boldsymbol{\Sigma}[I]$. ∎

######  Corollary 7.1.1.

If $\mathtt{a}\triangleright\mathtt{H}_{G}=0$, then

$\text{T}_{I}(\mathtt{a}\otimes\emph{{A}}_{I})=0$

for all $\emph{{A}}_{I}\in\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}[I]$
which respect $G$.

The restriction of T to the primitive part Lie algebra is called the
associated _system of generalized retarded products_ ,

$\text{R}:\textbf{Zie}\otimes\textbf{E}_{\mathcal{F}_{\text{loc}}[[\hbar]]}\to\textbf{U}_{\mathcal{F}((\hbar))}.$

The image of the Dynkin elements $\mathtt{D}_{\mathcal{S}}$ under the currying
of R are the _generalized retarded products_ $\text{R}_{\mathcal{S}}$, see
e.g. [EG73, Equation 79]. It follows from 7.1.1 and the structure of Dynkin
elements under the Tits product that generalized retarded products have nice
support properties. This is described in [EGS75].

Given a system of generalized time-ordered products