1. Introduction
A 2n-dimensional symplectic manifold (M, ω) describes the phase space of a finite dimensional classical Hamiltonian system in terms of the Poisson bracket {f, g} = ω(θf, θg) with respect to Hamiltonian vector fields θf, θg with f, g ∈ C∞(M). A Hamiltonian H ∈ C∞(M) introduces a ‘dynamical system’ on the basis of differential equations df = {H, f} such that trajectories can be determined in terms of n independent integrals of motion {f1 = H, f2, …, fn} with {fi, fj} = 0 for i ≠ j called an integrable system. In a general setting, the notion of integrability is on the basis of finding a set of independent integrals of motion which commute with each other with respect to a Poisson bracket [1]. Any subalgebra A ⫅ C∞(M) which obeys the equations {f, g} = 0 for f, g ∈ A is an integrable system [2]. There are alternative approaches to pass from classical to quantum setting where the quantum extension of integrability might have the challenge of understanding functional independence of integrals of motion [3, 4]. In one approach, deformation quantization of C∞(M) defines a noncommutative algebra C∞(M)[[ℏ]] where an integrable system A remains integrable after quantization if A[[ℏ]] is a commutative subalgebra of C∞(M)[[ℏ]] [2, 5]. In another approach, quantum inverse scattering method addresses the Yang–Baxter equation and its modifications as the master equations for the formulation of (quantum) integrable systems [6–10].
The quantum inverse scattering method has been introduced and developed for the calculation of correlation functions in completely quantum integrable systems. This method applies a certain associative algebra, which is neither finite dimensional nor Lie algebra, for the description of quantum integrable systems. This algebra is defined in terms of some generators, as functions of a continuous parameter, organized in a finite square matrix. Some quadratic relations on generators, encapsulated by the Yang–Baxter equation, recover the algebra. The trace of that finite square matrix forms a commutative family of operators which can be considered as integrals of motion for a quantum integrable system. The formulation of the quantum inverse scattering method has been developed on the basis of the theory of Hopf algebras where representations of quasitriangular Hopf algebras generate a certain class of matrices as solutions of the Yang–Baxter equation. In this regard, quantum algebras (groups) and their relation to braid groups and Hecke algebras are considered where a deformation of classical groups leads us to formulate quantum integrable systems in terms of solutions of the Yang–Baxter equation associated to the corresponding quantum groups [4, 7–13].
The present paper focuses on the interaction between the theory Hopf algebras and quantum integrable systems where thanks to Rota–Baxter algebras, topological Hopf algebra of renormalization and its core extension, a new class of quantum integrable systems will be determined. These new quantum integrable systems recover the evolution of solutions of quantum motions in gauge field theories.
1.1. Rota–Baxter algebras: from integrability to perturbative renormalization
The integrability problem of dynamical systems in classical or quantum cases can be algebraically formulated in terms of a matrix factorization program encoded by the Yang–Baxter equation r12r13r23 = r23r13r12 such that r: A ⨂ A → A ⨂ A is a linear operator where rij acts on ith and jth components of the tensor product A ⨂ A ⨂ A. Solutions of the Yang–Baxter equation determine quadratic algebras such that their representations generate quantum integrable systems [6, 9, 12, 14–16]. The quantum inverse scattering method provides a process to unify classical integrable models and their quantum version where the commutation relation of operators is encoded by the Yang–Baxter equation. This method has already been developed on the basis of a deformation theory of groups and Lie algebras where quantum groups and Hopf algebras are useful tools for the study of quantum integrable systems [1, 7, 11, 13].
Given a field ${\mathbb{K}}$ with characteristic zero, a unital ${\mathbb{K}}$-algebra A together with a ${\mathbb{K}}$-linear map R: A → A is called Rota–Baxter algebra of weight $\lambda \in {\mathbb{K}}$ if 1 ) can be extended to the Lie algebra ${{ \mathcal L }}_{A}$ with respect to the Lie bracket [x, y] = xy − yx, for all x, y ∈ A, to generate the Yang–Baxter equation
$\begin{eqnarray}R(x)R(y)+\lambda R(xy)=R(R(x)y+xR(y)),\end{eqnarray}$
for all x, y ∈ A. The Yang–Baxter equation and Rota–Baxter algebra are in a bijective correspondence. On the one hand, if the Rota–Baxter algebra (A, R) be an associative or pre-Lie ${\mathbb{K}}$-algebra, then the equation ( $\begin{eqnarray}[R(x),R(y)]+R([x,y])=R([R(x),y]+[x,R(y)]),\end{eqnarray}$
by applying a scale transformation R ↦ λ−1R for λ ≠ 0. On the other hand, for an associative ${\mathbb{K}}$-algebra A, a solution r = ∑is(i) ⨂ t(i) in the A-bimodule A ⨂ A of the Yang–Baxter equation $\begin{eqnarray}{r}_{13}{r}_{12}-{r}_{12}{r}_{23}+{r}_{23}{r}_{13}=\lambda {r}_{13},{r}_{13}:= \displaystyle \sum _{i}{s}_{(i)}\otimes 1\otimes {t}_{(i)},\end{eqnarray}$
defines a Rota–Baxter map R of weight λ on A given by R(x) = ∑is(i)xt(i) [12, 16–18].• A ${\mathbb{K}}$-linear map N: A → A on a unital ${\mathbb{K}}$-algebra A is called Nijenhuis operator if
$\begin{eqnarray}N(x)N(y)+{N}^{2}(xy)=N(N(x)y+xN(y)),\end{eqnarray}$
for all x, y ∈ A. If the Nijenhuis algebra (A, N) be an associative or pre-Lie ${\mathbb{K}}$-algebra, then the equation ( $\begin{eqnarray}[N(x),N(y)]+{N}^{2}([x,y])=N([N(x),y]+[x,N(y)]),\end{eqnarray}$
for x, y ∈ A. • Given Rota–Baxter algebra (A, R) with the idempotent map R, and $\lambda \in {\mathbb{K}}$, the map $N_{\lambda}:=R-\lambda \tilde{R}$ with $\tilde{R}=\mathrm{Id}_{A}-R$ is a Nijenhuis operator on A.
• Nijenhuis and Rota–Baxter operators are the main sources of generating hierarchies of deformed (Lie) algebras [14–16].
The theory of Rota–Baxter algebras plays a fundamental role in relating renormalization of Feynman integrals and Dyson–Schwinger equations in quantum field theory to the Riemann–Hilbert problem. In this regard, the Connes–Kreimer theory formulates an interpretation of the BPHZ perturbative renormalization on the basis of the Hopf–Birkhoff factorization on complex Lie groups underlying the Riemann–Hilbert problem [17–26].
A gauge field theory Φ is defined in terms of a collection ${{ \mathcal A }}_{{\rm{\Phi }}}=\{{v}_{i},{e}_{j}\,:\,i,j\}$ of elementary particles ej and possible interactions vi among them. Feynman diagrams are main tools for a combinatorial interpretation of Green’s functions, their renormalization and solutions of their fixed point equations. A Feynman diagram Γ is a finite decorated ordered graph determined by a collection Γ[1] of edges, as the symbols of particles, and a collection Γ[0] of vertices, as the symbols of interactions, such that the total sum of input momenta is the same as the total sum of the output momenta. The collection Γ[1] contains a subset ${{\rm{\Gamma }}}_{{\rm{int}}}^{[1]}$ of internal edges which have both beginning and ending vertices and a subset ${{\rm{\Gamma }}}_{{\rm{ext}}}^{[1]}$ of external edges which have either beginning or ending vertex. A Feynman diagram is called 1PI, if it remains a connected graph after the removal of an arbitrary internal edge. A subgraph γ in Γ is a graph determined by subsets ${\gamma }_{{\rm{int}}}^{[1]}\subseteq {{\rm{\Gamma }}}_{{\rm{int}}}^{[1]}$, γ[0] ⫅ Γ[0] such that for each v ∈ γ[0], the graph ${{\rm{res}}}_{\gamma }(v)$ generated by shrinking all internal edges connected to v in γ is the same as the graph ${{\rm{res}}}_{{\rm{\Gamma }}}(v)$ generated by shrinking all internal edges connected to v in Γ [21, 27, 28].
Feynman rules provide a dictionary between Feynman diagrams and Feynman integrals such that the dimension of the space-time background and the domain of momentum parameter, which varies between zero and infinity, are the reason of the appearance of some sub-divergences in Feynman integrals. These sub-divergences are presented by nested loops in Feynman diagrams. The Zimmermann–Bogoliubov’s forest formula and $\bar{R}$-operation provides a recursive machinery to remove step by step these nested loops to generate some renormalized values extracted from these ill-defined integrals [21, 27, 28].
The Connes–Kreimer theory builds a graded connected commutative non-cocommutative Hopf ${\mathbb{K}}$-algebra ${H}_{{\rm{FG}}}({\rm{\Phi }})\,={\oplus }_{n\geqslant 0}{H}_{{\rm{FG}}}^{(n)}$ free generated by 1PI Feynman diagrams. This Hopf algebra is graded in terms of the loop number such that for each n, ${H}_{{\rm{FG}}}^{(n)}$ is the vector space of superficially divergent 1PI Feynman diagrams with the loop number n and products of 1PI Feynman diagrams with the overall loop number n. Its coproduct is given by
$\begin{eqnarray}{{\rm{\Delta }}}_{{\rm{ren}}}({\rm{\Gamma }})={\mathbb{I}}\otimes {\rm{\Gamma }}+{\rm{\Gamma }}\otimes {\mathbb{I}}+\displaystyle \sum _{\gamma }\gamma \otimes {\rm{\Gamma }}/\gamma ,\end{eqnarray}$
such that the sum is taken over all disjoint unions of 1PI subgraphs of Γ which have sub-divergences, called Feynman subdiagrams [19–21, 29]. The combinatorial version of this Hopf algebra, which is formulated on the polynomial algebra of non-planar rooted trees, is called Connes–Kreimer Hopf algebra and presented by HCK. If we decorate vertices of these trees by primitive (1PI) Feynman diagrams of the physical theory, then there exists an injective Hopf algebra homomorphism from HFG(Φ) to HCK(Φ) [17, 18, 28, 29].Let Adr be the ${\mathbb{K}}$-algebra of Laurent series with finite pole parts, called dimensional regularization algebra, equipped with an idempotent ${\mathbb{K}}$-linear map
$\begin{eqnarray}{R}_{{\rm{ms}}}:{A}_{{\rm{dr}}}\to {A}_{{\rm{dr}}},\,\displaystyle \sum _{n\geqslant -m}^{\infty }{a}_{n}{z}^{n}\mapsto \displaystyle \sum _{n\geqslant -m}^{-1}{a}_{n}{z}^{n},\end{eqnarray}$
called minimal subtraction, which projects each series onto its pole parts. The pair (Adr, Rms), which is a Rota–Baxter algebra, factorizes Adr = A− ⊕ A+, called Birkhoff factorization, such that ${A}_{-}={\mathbb{C}}[{z}^{-1}]$ and ${A}_{+}={\mathbb{C}}\{z\}$ is the ring of convergent power series or germs of holomorphic functions at z = 0. The Rota–Baxter map Rms extends to the ${\mathbb{K}}$-algebra (L(HFG(Φ), Adr), *ren) of linear maps from the renormalization Hopf algebra to Adr together with the convolution product $\begin{eqnarray}\begin{array}{l}{\psi {}_{1}\,\ast \,}_{{\rm{ren}}}{\psi }_{2}({\rm{\Gamma }})={\psi }_{1}\displaystyle \otimes {\psi }_{2}({{\rm{\Delta }}}_{{\rm{ren}}}({\rm{\Gamma }}))\\ \quad =\,\displaystyle \sum _{\gamma }{\psi }_{1}(\gamma ){\psi }_{2}({\rm{\Gamma }}/\gamma ).\end{array}\end{eqnarray}$
It gives the Rota–Baxter algebra $\begin{eqnarray}\begin{array}{l}{{\rm{RB}}}_{{\rm{\Phi }}}:= \left(L({H}_{{\rm{FG}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}){,\ast }_{{\rm{ren}}},{ \mathcal R }:L({H}_{{\rm{FG}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})\right.\\ \,\left.\to \,L({H}_{{\rm{FG}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}),\phi \mapsto {R}_{{\rm{ms}}}\circ \phi \right),\end{array}\end{eqnarray}$
such that its elements are factorized by the equation ψ = ψ−−1*renψ+ called Hopf–Birkhoff factorization [17, 18].A certain class of elements of the complex Lie group ${{\mathbb{G}}}_{{\rm{\Phi }}}({A}_{{\rm{dr}}})={\rm{Hom}}({H}_{{\rm{FG}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})\subset L({H}_{{\rm{FG}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})$ recover regularized Feynman rules of the physical theory. The Connes–Kreimer theory shows that the BPHZ perturbative renormalization can be recovered by the Hopf–Birkhoff factorization $({\phi }_{-}^{z},{\phi }_{+}^{z})$ of the regularized Feynman rules character φz such that
$\begin{eqnarray}\begin{array}{l}{\phi }_{-}^{z}:{H}_{{\rm{FG}}}({\rm{\Phi }})\to {A}_{-},\\ {\phi }_{-}^{z}({\rm{\Gamma }})=-{R}_{{\rm{ms}}}({\phi }^{z}({\rm{\Gamma }})+\displaystyle \sum _{\gamma }{\phi }_{-}^{z}(\gamma ){\phi }^{z}({\rm{\Gamma }}/\gamma )),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\phi }_{+}^{z}:{H}_{{\rm{FG}}}({\rm{\Phi }})\to {A}_{+},\\ {\phi }_{+}^{z}({\rm{\Gamma }})=\,{\phi }^{z}({\rm{\Gamma }})+{\phi }_{-}^{z}({\rm{\Gamma }})+\displaystyle \sum _{\gamma }{\phi }_{-}^{z}(\gamma ){\phi }^{z}({\rm{\Gamma }}/\gamma ).\end{array}\end{eqnarray}$
The Slavnov–Taylor and Ward–Takahashi elements, which present quantum gauge symmetries, generate Hopf ideals of the renormalization Hopf algebra. The compatibility of quantum gauge symmetries with this Hopf algebraic setting is recovered by the quotient Hopf algebras HFG(QCD)/IST and HFG(QED)/IWT [19, 20, 24, 30–32].The Hopf–Birkhoff factorization can be described in terms of the Baker–Campbell–Hausdorff formula and the Kontsevich’s bi-differential symplectic operator where a regularized Feynman rules character is interpreted as a deformation of the pointwise multiplication of some exponential functions via Kontsevich’s star product in the direction of the linear Poisson bracket [33].
The interconnection between Rota–Baxter structures and Hopf–Birkhoff factorization has been extended to min-plus semirings and their thermodynamic deformations. Rota–Baxter structures of weights ±1 on thermodynamic semirings are introduced to obtain components of the Hopf–Birkhoff factorization of min-plus characters of commutative Hopf algebras. In this setting, thermodynamic Rota–Baxter structures for the von Neumann entropy are formulated. The min-plus characters of the Manin’s renormalization Hopf algebra of the Halting problem [34, 35] encodes a fundamental bridge between the theory of computation and the theory of thermodynamic semirings underlying the BPHZ renormalization program and the theory of Rota–Baxter algebras [36–38].
1.2. Feynman graph limits
The fixed point equations of Green’s functions known as Dyson–Schwinger equations, which determine quantum motions in the physical theory, are reformulated by Hochschild cohomology of the renormalization Hopf algebra. Solutions of these equations are polynomials of powers of running coupling constants together with higher loop order Feynman diagrams as coefficients. For a family ${\{{\gamma }_{n}\}}_{n\geqslant 1}$ of primitive (1PI) Feynman diagrams in HFG(Φ), the recursive equation 12 ) is called combinatorial Dyson–Schwinger equation such that its solution X = ∑n≥0cnXn is given by the recursive relations 13 ), each Xn, which is built by grafting operator ${B}_{{\gamma }_{n}}^{+}$ on the lower order components Xj, for j < n, contributes to the order n of the equation DSE. The collection ${\{{X}_{n}\}}_{n\geqslant 0}$ generates a graded commutative Hopf subalgebra of HFG(Φ) [28, 30].
$\begin{eqnarray}{\rm{DSE}}:X={\mathbb{I}}+\displaystyle \sum _{n\geqslant 1}{c}^{n}{\omega }_{n}{B}_{{\gamma }_{n}}^{+}({X}^{n+1}),\end{eqnarray}$
in the topological ring HFG(Φ)[[c]], with respect to the n − adic topology, encodes a class of Dyson–Schwinger equations underlying the running coupling c as a function of the bare coupling constant. For each n ≥ 1, ${B}_{{\gamma }_{n}}^{+}:{H}_{{\rm{FG}}}({\rm{\Phi }})\to {H}_{{\rm{FG}}}({\rm{\Phi }})$, called grafting operator, is a linear homogeneous operator which sends Γ to a linear expansion of Feynman diagrams built by all possible insertion places of Γ into γn in terms of types of external edges of Γ and types of vertices of γn. The equation ( $\begin{array}{l}{X}_{n}=\displaystyle \sum _{j=1}^{n}{\omega }_{j}\\ \qquad {\times B}_{{\gamma }_{j}}^{+}{\left(\displaystyle \sum _{{k}_{1}+...+{k}_{j+1}=n-j,\,{k}_{i}\geqslant 0}{X}_{{k}_{1}}...{X}_{{k}_{j+1}}\right)}\in {H}_{{\rm{FG}}}({\rm{\Phi }}),\,{X}_{0}={\mathbb{I}}.\end{array}$
In the formula (The solution X of the equation (12 ), as a perturbative series, is divergent for strong coupling constants c ≥ 1. The method of Feynman graphon models deals with these non-perturbative series as graph limits of sequences of a certain class of graph functions, called stretched Feynman graphons, with respect to the cut-distance topology [31, 39–41]. This setting has already provided a non-perturbative extension of the BPHZ renormalization [24, 25, 31, 42–45].
For a given σ-finite measure space (Ω ⫅ [0, ∞), μ), a stretched graphon W is a real valued bounded symmetric μ-measurable function on Ω × Ω. For the Lebesgue measure space ([0, 1], m), the phrase graphon is used instead of stretched graphon. An invertible μ-measure preserving transformation ρ on Ω defines a labeled stretched graphon Wρ such that Wρ(x, y) = W(ρ(x), ρ(y)). Stretched graphons W1, W2 are called weakly isomorphic (i.e. W1 ≈ W2) if there exist μ-measure preserving transformations ρ1, ρ2 on Ω and a stretched graphon W such that ${W}_{1}={W}^{{\rho }_{1}}$ and ${W}_{2}={W}^{{\rho }_{2}}$ almost everywhere. The relation ≈ is an equivalence relation on the set ${{ \mathcal W }}^{{\rm{\Omega }}}$ of all stretched graphons on Ω. There exists a complete Hausdorff metric structure on ${{ \mathcal W }}_{\approx }^{{\rm{\Omega }}}$ with respect to the cut-distance metric
$\begin{eqnarray}\begin{array}{l}{{\rm{d}}}_{{\rm{cut}}}({W}_{1},{W}_{2})={{\rm{\inf }}}_{{\rho }_{1},{\rho }_{2}}\,{{\rm{\sup }}}_{A,B\subset {\rm{\Omega }},\,\mu -{\rm{measurable}}}\quad \left|{\displaystyle \int }_{A\times B}({W}_{1}^{{\rho }_{1}}(x,y)-{W}_{2}^{{\rho }_{2}}(x,y)){\rm{d}}\mu (x){\rm{d}}\mu (y)\right|,\end{array}\end{eqnarray}$
such that inf is taken over μ-measure preserving transformations on Ω and W1 ≈ W2 ⇔ dcut(W1, W2) = 0. [46–48]
| (1.) Each Feynman diagram in HFG(Φ) has a unique graphon representation in ${{ \mathcal W }}_{\approx }^{[0,1]}$ for the Lebesgue measure space ([0, 1], m). | |
| (2.) Each Feynman graph limit has a unique graphon representation in ${{ \mathcal W }}_{\approx }^{[0,\infty )}$ for the Lebesgue measure space ([0, ∞), m). |
1. For Γ ∈ HFG(Φ) with ∣Γ∣ = n which has no overlapping loops, its rooted tree representation tΓ has n vertices such that each vertex vi ∈ tΓ is decorated by a primitive (1PI) Feynman subdiagram γi of Γ [28, 29]. Let αi > 0 be the weight of the vertex vi such as αi = 1/n for each 1 ≤ i ≤ n in a toy model. Define a partition σ ≔ {I1, …, In} of [0, 1] such that for each 1 ≤ i ≤ n, ${\rm{m}}({I}_{i})=\frac{{\alpha }_{i}}{{\sum }_{i=1}^{n}{\alpha }_{i}}$ and ${I}_{k}\cap {I}_{l}={\rm{\varnothing }}$ for k ≠ l. The boxes ${\{{I}_{i}\times {I}_{j}\}}_{1\,\leqslant \,i,j\,\leqslant \,n}$ determine a pixel picture presentation ${P}_{{\rm{\Gamma }}}^{\sigma }$ given by $(x,y)\in {I}_{k}\times {I}_{l}\mapsto {a}_{kl}\in {{\rm{Ad}}}_{{t}_{{\rm{\Gamma }}}}$ such that ${{\rm{Ad}}}_{{t}_{{\rm{\Gamma }}}}$ is the adjacency matrix of tΓ. The box Ik × Il is colored by black if there exists an edge between vk and vl in tΓ, otherwise it is colored by white. The equivalence class
$\begin{eqnarray}{[{P}_{{\rm{\Gamma }}}^{\sigma }]}_{\approx }=\{{W}^{\tau }\in {{ \mathcal W }}^{{\rm{\Omega }}}\,:\,{W}^{\tau }\approx {P}_{{\rm{\Gamma }}}^{\sigma }\},\end{eqnarray}$
with respect to Lebesgue measure preserving transformations τ on [0, 1] is the unique unlabeled stretched graphon associated to Γ. We use the notation ${W}_{{\rm{\Gamma }}}:= {[{P}_{{\rm{\Gamma }}}^{\sigma }]}_{\approx }$ and call it stretched Feynman graphon associated to Γ such that each ${W}_{{\rm{\Gamma }}}^{\tau }$ is called a labeled stretched Feynman graphon [31, 41]. 2. Thanks to (
$\begin{eqnarray}\parallel {P}_{{\rm{\Gamma }}}^{\sigma }{\parallel }_{{\rm{cut}}}={{\rm{\sup }}}_{A\times B\subset [0,\infty )\times [0,\infty )}{\left| {\int }_{A\times B}{P}_{{\rm{\Gamma }}}^{\sigma }(x,y){\rm{d}}{x}{\rm{d}}{y}\right|} ,\end{eqnarray}$
such that A, B are Lebesgue measurable subsets of [0, ∞) to define the distance function $\begin{eqnarray}{d}_{{\rm{cut}}}({W}_{{{\rm{\Gamma }}}_{1}},{W}_{{{\rm{\Gamma }}}_{2}}):= {{\rm{\inf }}}_{\tau ,\psi }\parallel {W}_{{{\rm{\Gamma }}}_{1}}^{\tau }-{W}_{{{\rm{\Gamma }}}_{2}}^{\psi }{\parallel }_{{\rm{cut}}}.\end{eqnarray}$
The subspace ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},[0,\infty )}\subset {{ \mathcal W }}_{\approx }^{{\rm{\Omega }}}$ of stretched Feynman graphons associated to Feynman diagrams in HFG(Φ) is a complete Hausdorff metric space. It topologically completes HFG(Φ) with respect to the distance function $\begin{eqnarray}{d}_{{\rm{cut}}}({{\rm{\Gamma }}}_{1},{{\rm{\Gamma }}}_{2}):= {d}_{{\rm{cut}}}({W}_{{{\rm{\Gamma }}}_{1}},{W}_{{{\rm{\Gamma }}}_{2}}).\end{eqnarray}$
Following measure theoretic tools in dealing with graph limits of sequences of sparse graphs [46], we call a sequence ${\{{{\rm{\Gamma }}}_{n}\}}_{n\geqslant 1}$ of Feynman diagrams is convergent if the sequence ${\left\{\frac{{W}_{{{\rm{\Gamma }}}_{n}}}{| | {W}_{{{\rm{\Gamma }}}_{n}}| {| }_{{\rm{cut}}}}\right\}}_{n\geqslant 1}$ converges to some $W\in {{ \mathcal W }}_{\approx }^{{\rm{\Omega }}}$ when n tends to infinity. Here, W is a graphon representation for an infinite tree (or forest) t∞ which is likely the graph limit of the sequence ${\{{t}_{n}\}}_{n\geqslant 1}$ when n → ∞ such that vertices of t∞ are decorated by some primitive (1PI) Feynman diagrams in HFG(Φ). Therefore t∞ generates a large Feynman diagram X such that WX ≈ W and ${W}_{X}={[{P}_{{t}_{\infty }}]}_{\approx }\in {{ \mathcal S }}_{\approx }^{{\rm{\Phi }},[0,\infty )}$ [41, 43, 45]. ☐In general, any stretched Feynman graphon can be defined in the way that it is identical to the original Feynman graphon except that the graph function ${W}_{X}:[0,1]\times [0,1]\to {\mathbb{R}}$ stretches to a new graph function ${W}_{X}^{{\rm{s}}}:{\rm{\Omega }}\subseteq [0,\infty )\,\times {\rm{\Omega }}\,\subseteq [0,\infty )\to {\mathbb{R}}$ such as
$\begin{eqnarray}{W}_{X}^{{\rm{s}}}(x,y)={\Space{0ex}{2.5ex}{0ex}\{}_{0,\,\,{\rm{otherwise}}}^{{W}_{X}(| | {W}_{X}| {| }_{1}^{1/2}x,| | {W}_{X}| {| }_{1}^{1/2}y),\,0\,\leqslant \,x,y\,\leqslant \,| | {W}_{X}| {| }_{1}^{-1/2}},\end{eqnarray}$
such that $| | {W}_{X}^{{\rm{s}}}| {| }_{1}=1$. The rescaling of the ground measure space together with the re-weighting techniques such as ${W}_{X}\mapsto \frac{1}{| | {W}_{X}| {| }_{{\rm{p}}}}{W}_{X}$, p≥1, are applied to associate non-trivial Feynman graph limits to sequences of Feynman diagrams with increasing loop numbers. figures 1, 2 and 3 present the procedure of associating Feynman graphon models to Feynman diagrams and their sequences. Further details about the structure of Feynman graphon models can be found in [31, 39–41, 43, 45].
Figure 1. Replacing Feynman diagrams with their rooted tree representations underlying the renormalization Hopf algebra to generate Feynman graphon models via lemma |
Figure 2. The rooted forest representation of some 1PI Green's function underlying the renormalization Hopf algebra. |
Figure 3. The domain of any Feynman graphon model which contributes to divergent perturbative series such as the one given in figure 2 is stretched to associate non-trivial Feynman graph limit (i.e. large Feynman diagram). Any Feynman graphon model defined on a stretched domain is called a stretched Feynman graphon. Further details are given in [31, 39, 40]. |
Let DSE be a combinatorial Dyson–Schwinger equation such that ${\{{Y}_{m}\}}_{m\geqslant 1}$ be the sequence of partial sums of its solution X with ${Y}_{m}={\sum }_{k=1}^{m}{c}^{k}{X}_{k}$. The solution X, as a large Feynman graph generated by a sequence of stretched Feynman graphons, is presented by a process of random graphs [31, 43, 45].
The stretched Feynman graphon ${W}_{{Y}_{m}}\in {{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$ corresponding to Ym is a direct sum of stretched Feynman graphons ${W}_{{X}_{k}}$, 1 ≤ k ≤ m of weights m(Ck) = ck such that for k ≠ l, ${C}_{k}\cap {C}_{l}={\rm{\varnothing }}$ and ${\cup }_{k=1}^{m}{C}_{k}\subseteq {\rm{\Omega }}$. Thanks to lemma
$\begin{eqnarray}{{\rm{lim}}}_{{m}_{i}\to \infty }\,{W}_{{Y}_{{m}_{i}}}={W}_{X}\iff {{\rm{lim}}}_{m\to \infty }{Y}_{m}=X,\end{eqnarray}$
with respect to the metric (For each m, let Rm be a random graph built by uniformly choosing m points a1, …, am from ${\cup }_{k=1}^{m}{C}_{k}$ such that $\frac{1}{| | {W}_{{Y}_{m}}| {| }_{1}}{W}_{{Y}_{m}}$ determines the probability value for the existence of an edge between ai and aj in Rm. The sequence ${\{{R}_{m}\}}_{m\geqslant 1}$, which converges to an infinite random graph R∞ defined on ${\cup }_{k=1}^{\infty }{C}_{k}$, provides a process of random graphs for the description of the large Feynman diagram X. ☐
1.3. Achievements
The present paper addresses a fundamentally new approach for the construction of quantum integrable systems in quantum theories with infinite degrees of freedom on the basis of the topological Hopf algebraic non-perturbative renormalization and the theory of Rota–Baxter algebras. This new setting relates the problem of quantum integrability in quantum field theory to deformation theory of a certain class of noncommutative associative algebras where the intersection of symplectic geometry and noncommutative geometry is capable of providing required equations for the determination of motion integrals.
Generally speaking, the algebraic approach for the formulation of quantum integrable systems is on the basis of the Yang–Baxter equations [4, 6–13, 49]. A certain class of Rota–Baxter algebras, which contribute to renormalization of Feynman diagrams in quantum field theories, generate the Yang–Baxter equations [17, 18]. The method of Feynman graphon models provided an extension of the BPHZ program in dealing with Feynman graph limits which contribute to solutions of quantum motions in quantum field theories [24, 25, 31, 40]. Thanks to this background, the present paper provides some new progress in the direction of Drinfeld’s Hopf algebraic framework. We explain the structure of the topological Hopf algebra of renormalization ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ on the space of stretched Feynman graphons and its core extension ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$ on the subspace of real valued stretched graphons corresponding to finite graphs without self-loops. These topological Hopf algebras, which encode a non-perturbative extension of the BPHZ renormalization, are applied to formulate some new noncommutative associative algebras (i.e. Lemma 2.1 and theorem 2.2 ). A deformation theory for these noncommutative algebras is formulated on the basis of Dubois–Violette’s differential graded algebras and the theory Nijenhuis algebras. This setting leads us to achieve a symplectic geometry background for the space of quantum motions of a gauge field theory (i.e. Theorems 3.1 and 3.3 ). This symplectic geometry background determines some new quantum integrable systems which contribute to the evolution of solutions of quantum motions in gauge field theories (i.e. Corollaries 3.2 , 3.4 and 3.5 ).
2. Renormalization of Feynman graph limits
Given a gauge field theory Φ, consider the graded free commutative ${\mathbb{K}}$-algebra ${{ \mathcal H }}_{{\rm{gr}}}({\rm{\Phi }})={\oplus }_{n=0}^{\infty }{{ \mathcal H }}_{{\rm{gr}}}^{(n)}$ generated by stretched Feynman graphons in ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$ associated to 1PI Feynman diagrams in HFG(Φ). For each n≥1, ${{ \mathcal H }}_{{\rm{gr}}}^{(n)}$ is the ${\mathbb{K}}$-vector space generated by WΓ with ${\rm{\Gamma }}\in {H}_{{\rm{FG}}}^{(n)}({\rm{\Phi }})$ such that ${{ \mathcal H }}_{{\rm{gr}}}^{(0)}={\mathbb{K}}\lt {W}_{{\mathbb{I}}}\gt =\,{\mathbb{K}}$. The zero graphon ${W}_{{\mathbb{I}}}$ is considered for the class of Feynman diagrams with zero loop number. For ${W}_{{{\rm{\Gamma }}}_{1}}\in {{ \mathcal H }}_{{\rm{gr}}}^{({n}_{1})}$ and ${W}_{{{\rm{\Gamma }}}_{2}}\in {{ \mathcal H }}_{{\rm{gr}}}^{({n}_{2})}$, if Feynman diagrams Γ1, Γ2 are weakly isomorphic, then n1 = n2.
Sequences of Feynman diagrams can be characterized in terms of the asymptotic densities of their corresponding stretched Feynman graphons such that Feynman diagrams / large Feynman graphs with different densities might have a similar structure. A real valued stretched Feynman graphon on a σ-finite measure space (Ω ⫅ [0, ∞), μ) can be projected to its canonical version defined on the Lebesgue measure space ([0, 1), m) by the application of rescaling and re-weighting techniques encoded by measure preserving or affine transformations. In other words, the Banach space ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},[0,1)}$ has a central role in the method of Feynman graphon models [31, 40–43].
There exists a graded commutative non-cocommutative Hopf algebra structure on ${{ \mathcal H }}_{{\rm{gr}}}({\rm{\Phi }})$, topologically completed by (
The renormalization coproduct (
$\begin{eqnarray}\begin{array}{l}{{\rm{\Delta }}}_{{\rm{ren}},{\rm{gr}}}({W}_{{\rm{\Gamma }}})={W}_{{\mathbb{I}}}\displaystyle \otimes {W}_{{\rm{\Gamma }}}+{W}_{{\rm{\Gamma }}}\displaystyle \otimes {W}_{{\mathbb{I}}}+\displaystyle \sum _{\gamma }{W}_{\gamma }\displaystyle \otimes {W}_{{\rm{\Gamma }}/\gamma },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{S}_{{\rm{ren}},{\rm{gr}}}({W}_{{\rm{\Gamma }}})=-{W}_{{\rm{\Gamma }}}-\displaystyle \sum _{\gamma }{S}_{{\rm{ren}},{\rm{gr}}}({W}_{\gamma }){W}_{{\rm{\Gamma }}/\gamma },\\ \,{S}_{{\rm{ren}},{\rm{gr}}}({W}_{{\mathbb{I}}})={W}_{{\mathbb{I}}},\end{array}\end{eqnarray}$
such that Wγ is the stretched Feynman graphon associated with the non-trivial disjoint union γ of 1PI Feynman subdiagrams of Γ. Cut-distance topologically complete this Hopf algebra to obtain the graphon version of topological Hopf algebra of renormalization ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$. The linear and bounded property of (Thanks to Milnor–Moore theorem [21], the graded dual of ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ is the complex infinite dimensional Lie group ${{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({A}_{{\rm{dr}}})={\rm{Hom}}({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})$. Suppose Δ* is the infinitesimal punctured disk around z = 0 in the complex plane which restores the regularization parameter. The space of loops ${{\boldsymbol{\Delta }}}^{* }\to {{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({A}_{{\rm{dr}}})$, $z\mapsto {\tilde{\phi }}^{z}$ determines regularized unrenormalized Feynman rules characters on the space of stretched Feynman graphons such that
$\begin{eqnarray}{W}_{{\rm{\Gamma }}}\in {{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})\mapsto {\tilde{\phi }}^{z}({W}_{{\rm{\Gamma }}})={\phi }^{z}({\rm{\Gamma }}),{\phi }^{z}\in {{\mathbb{G}}}_{{\rm{\Phi }}}({A}_{{\rm{dr}}}).\end{eqnarray}$
For a sequence ${\{{W}_{{{\rm{\Gamma }}}_{n}}\}}_{n\geqslant 1}$ in ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ which converges to $W\in {{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ with respect to the metric ( $\begin{eqnarray}\begin{array}{l}{{\rm{RB}}}_{{\rm{gr}},{\rm{\Phi }}}:= (L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}){,* }_{{\rm{ren}},{\rm{gr}}}\,{,}\\ \quad { \mathcal R }:L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})\to L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})),\end{array}\end{eqnarray}$
recovers the renormalization of stretched Feynman graphons where $L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})$ is the noncommutative unital associative algebra of linear maps from ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ to Adr with respect to the convolution product *ren,gr defined by the coproduct ( $\begin{eqnarray*}{S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}(W)={{\rm{lim}}}_{n\to \infty }{S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}({W}_{{{\rm{\Gamma }}}_{n}})\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl} & = & {{\rm{lim}}}_{n\to \infty }-{R}_{{\rm{ms}}}\\ & & \times ({\tilde{\phi }}^{z}({W}_{{{\rm{\Gamma }}}_{n}})+\displaystyle \sum _{{\gamma }_{n}}{S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}({W}_{{\gamma }_{n}}){\tilde{\phi }}^{z}({W}_{{{\rm{\Gamma }}}_{n}/{\gamma }_{n}})),\\ & & {S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}({W}_{{\mathbb{I}}})=1,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}{* }_{{\rm{ren}},{\rm{gr}}}{\tilde{\phi }}^{z}(W)\\ \quad =\,{{\rm{lim}}}_{n\to \infty }{S}_{{R}_{{\rm{ms}}}}^{{\tilde{\phi }}^{z}}{* }_{{\rm{ren}},{\rm{gr}}}{\tilde{\phi }}^{z}({W}_{{{\rm{\Gamma }}}_{n}}),\end{array}\end{eqnarray}$
with respect to ${{\rm{\Delta }}}_{{\rm{ren}},{\rm{gr}}}({W}_{{{\rm{\Gamma }}}_{n}})$. ☐On the one hand, for each $r\in {{ \mathcal A }}_{{\rm{\Phi }}}$, the 1PI Green’s function Gr and its partial sums [30] are given by combinatorial formal series 18 ) to obtain the topological Hopf algebra ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }})$. It means that solutions of combinatorial Dyson–Schwinger equations are recovered by ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }})$ [31, 39, 40].
$\begin{eqnarray}\begin{array}{l}{G}^{r}={\mathbb{I}}\pm \displaystyle \sum _{{\rm{\Gamma }},\,{\rm{res}}({\rm{\Gamma }})=r}{c}^{| {\rm{\Gamma }}| }\frac{{\rm{\Gamma }}}{{\rm{Sym}}({\rm{\Gamma }})},\,\\ {G}_{\leqslant m}^{r}={\mathbb{I}}\pm \displaystyle \sum _{{\rm{\Gamma }},\,{\rm{res}}({\rm{\Gamma }})=r,\,| {\rm{\Gamma }}| \,\leqslant \,m}{c}^{| {\rm{\Gamma }}| }\frac{{\rm{\Gamma }}}{{\rm{Sym}}({\rm{\Gamma }})}.\end{array}\end{eqnarray}$
The collection $\{{W}_{{G}_{\leqslant m}^{r}}\,:\,r\in {{ \mathcal A }}_{{\rm{\Phi }}},\,m\geqslant 1\}$, as a dense subset in ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$, makes ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$ a separable Banach space. On the other hand, each sequence ${\{{{\rm{\Gamma }}}_{n}\}}_{n\geqslant 1}$ of Feynman diagrams has a Cauchy subsequence ${\{{{\rm{\Gamma }}}_{{n}_{i}}\}}_{i\geqslant 1}$ with respect to the cut-distance topology. The Feynman graph limit of this subsequence, which exists in ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }})$, has a graphon representation in ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$. Therefore ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$ topologically completes the renormalization Hopf algebra with respect to the metric (• There exists an injective homomorphism of Hopf algebras Γ ↦ WΓ from ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }})$ to ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$.
• Each 1PI Green’s function Gr given by (
$\begin{eqnarray}{{\rm{lim}}}_{m\to \infty }\,{G}_{\leqslant m}^{r}={G}^{r}\,\iff \,{{\rm{lim}}}_{m\to \infty }\,{W}_{{G}_{\leqslant m}^{r}}={W}_{{G}^{r}}.\end{eqnarray}$
• The sequences ${\{{W}_{{G}_{\leqslant m}^{r}}\}}_{m\geqslant 1}$ and ${\{{W}_{{G}_{m}^{r}}\}}_{m\geqslant 1}$ are weakly isomorphic which means that both converge to ${W}_{{G}^{r}}$. • The collection $\{{W}_{{G}_{\leqslant m}^{r}}\in {{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}\,:\,r\in {{ \mathcal A }}_{{\rm{\Phi }}},\,m\geqslant 1\}$ recovers a basis for the vector space structure of ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$.
The core Hopf algebra is an extension of the renormalization Hopf algebra where the condition on superficial degree of divergence on subgraphs in the renormalization coproduct is canceled. In other words,
$\begin{eqnarray}{{\rm{\Delta }}}_{{\rm{cor}}}({\rm{\Gamma }})=\displaystyle \sum _{\gamma }\gamma \otimes {\rm{\Gamma }}/\gamma ,\end{eqnarray}$
such that the sum is taken over all subgraphs in Γ independent of the fact that they might have subdivergence or not [50]. The pair renormalization–core coproducts are applied to formulate a bi-Heyting algebra structure on the space of Feynman diagrams and Feynman graph limits of a physical theory. The corresponding Heyting space provides some new topological tools for the study of solutions of quantum motions in the context of subsystems [39, 40].Let ${{ \mathcal W }}_{\approx }^{{\rm{\Omega }},{\rm{FGrp}}}\subset {{ \mathcal W }}_{\approx }^{{\rm{\Omega }}}$ be the complete Hausdorff topological subspace of those stretched graphons corresponding to finite graphs without self-loops and graph limits of their sequences with respect to the cut-distance topology. There exists a topological core Hopf algebra structure generated by ${{ \mathcal W }}_{\approx }^{{\rm{\Omega }},{\rm{FGrp}}}$ which recovers the renormalization of Feynman graph limits of physical theories.
Let ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}$ be the free commutative algebra generated by stretched graphons in ${{ \mathcal W }}_{\approx }^{{\rm{\Omega }},{\rm{FGrp}}}$. For each n≥1, ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(n)}$ is the ${\mathbb{K}}$-vector space generated by WG with ∣G∣ = n such that ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(0)}={\mathbb{K}}\lt {W}_{{\mathbb{I}}}\gt ={\mathbb{K}}$. For ${W}_{{G}_{1}}\in {{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{({n}_{1})}$ and ${W}_{{G}_{2}}\in {{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{({n}_{2})}$, if G1 and G2 are weakly isomorphic, then n1 =n2. For the cases $G\in {{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(0)}$ or $G\in {{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(1)}$, WG = 0 almost everywhere which means that ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(0)}\,={{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{(1)}={\mathbb{K}}$. The core coproduct (
$\begin{eqnarray}{{\rm{\Delta }}}_{{\rm{core}},{\rm{gr}}}({W}_{G})=\displaystyle \sum _{H}{W}_{H}\otimes {W}_{G/H}\end{eqnarray}$
such that the sum is taken over all subgraphs H in G where G/H is a new finite graph as the result of deleting all vertices and edges of H in G. Cut-distance topologically complete this new Hopf algebra to obtain the topological core Hopf algebra ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$. Apply (
$\begin{eqnarray}\begin{array}{l}{{\rm{RB}}}_{{\rm{core}},{\rm{gr}}}:= (L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}}),{\ast }_{{\rm{core}},{\rm{gr}}},\\ \,{ \mathcal R }:L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}})\to L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}})),\end{array}\end{eqnarray}$
which recovers a renormalization program in terms of the Hopf–Birkhoff factorization of characters in the complex Lie group $\begin{eqnarray}{{\mathbb{G}}}_{{\rm{core}},{\rm{gr}}}({A}_{{\rm{dr}}})={\rm{Hom}}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}})\subset L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}}).\end{eqnarray}$
On the one hand, the renormalization Hopf algebra is a quotient of the core Hopf algebra [50]. On the other hand, given a gauge field theory Φ, the renormalization Hopf algebra HFG(Φ) is embedded in a decorated version of the Connes–Kreimer Hopf algebra of non-planar rooted trees HCK(Φ) [28]. Non-planar rooted trees are simple finite graphs without self-loops. Therefore there exists an injective homomorphism of Hopf algebras t ↦ Wt from HCK(Φ) to ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}$. The combination of these facts together with lemma
3. Poisson brackets of quantum integrability
The main idea of this section is to relate the problem of quantum integrability in gauge field theories to the deformation theory of noncommutative associative algebras generated by Rota–Baxter structures. Table (33 ) summarizes algebraic structures, which will be introduced in this section, for the construction of quantum integrable systems from symplectic geometry models. 
The Rota–Baxter algebras ${{\rm{RB}}}_{{\rm{core}},{\rm{gr}}},{{\rm{RB}}}_{{\rm{gr}},{\rm{\Phi }}}$, given by (31 ) and (24 ), are central tools of the formulation of a deformation theory for noncommutative associative unital algebras $(L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}}),{* }_{{\rm{core}},{\rm{gr}}})$ and $(L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}),{\ast }_{{\rm{ren}},{\rm{gr}}})$. The Yang–Baxter equations derived from this deformation theory recover new classes of quantum integrable systems in gauge field theories.
The Rota–Baxter algebra ${{\rm{RB}}}_{{\rm{core}},{\rm{gr}}}$ recovers symplectic geometry models for the spaces of Feynman graph limits (such as solutions of quantum motions) in gauge field theories.
We apply theorems
For $\lambda \in {\mathbb{K}}$, the Nijenhuis map ${{ \mathcal R }}_{\lambda }:= { \mathcal R }-\lambda \hat{{ \mathcal R }}$, with $\hat{{ \mathcal R }}:= {\rm{Id}}-{ \mathcal R }$, determines a new product ∘λ given by
$\begin{eqnarray}\begin{array}{l}{\phi }_{1}{\circ }_{\lambda }{\phi }_{2}:= {{ \mathcal R }}_{\lambda }({\phi }_{1}){* }_{{\rm{core}},{\rm{gr}}}{\phi }_{2}\\ \,\,+\,{\phi }_{1}{* }_{{\rm{core}},{\rm{gr}}}{{ \mathcal R }}_{\lambda }({\phi }_{2})-{{ \mathcal R }}_{\lambda }({\phi }_{1}{* }_{{\rm{core}},{\rm{gr}}}{\phi }_{2}),\end{array}\end{eqnarray}$
such that $\begin{eqnarray}\begin{array}{l}{\phi }_{1}{* }_{{\rm{core}},{\rm{gr}}}{\phi }_{2}({W}_{G})=\sum {\phi }_{1}({W}_{H}){\phi }_{2}({W}_{G/H}),\\ {{\rm{\Delta }}}_{{\rm{core}},{\rm{gr}}}({W}_{G})=\sum {W}_{H}\displaystyle \otimes \,{W}_{G/H}\end{array}\end{eqnarray}$
for ${\phi }_{1},{\phi }_{2}\in L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}})$. Define a new unital algebra $\begin{eqnarray}{C}_{\lambda ,{\rm{gr}},{\rm{core}}}:= (L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{A}_{{\rm{dr}}}),{\circ }_{\lambda }),\end{eqnarray}$
such that the property $\begin{eqnarray}{{ \mathcal R }}_{\lambda }({\phi }_{1}{\circ }_{\lambda }{\phi }_{2})={{ \mathcal R }}_{\lambda }({\phi }_{1}){* }_{{\rm{core}},{\rm{gr}}}{{ \mathcal R }}_{\lambda }({\phi }_{2}),\end{eqnarray}$
together with the non-cocommutativity of ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$ show that ∘λ is an associative noncommutative product. The commutator [. , . ]λ with respect to ∘λ defines a new Lie structure on ${C}_{\lambda ,{\rm{gr}},{\rm{core}}}$ such that $\begin{eqnarray}\begin{array}{l}{[{\phi }_{1},{\phi }_{2}]}_{\lambda }={[{{ \mathcal R }}_{\lambda }({\phi }_{1}),{\phi }_{2}]}_{{* }_{{\rm{core}},{\rm{gr}}}}\\ \quad +{[{\phi }_{1},{{ \mathcal R }}_{\lambda }({\phi }_{2})]}_{{* }_{{\rm{core}},{\rm{gr}}}}-{{ \mathcal R }}_{\lambda }({[{\phi }_{1},{\phi }_{2}]}_{{* }_{{\rm{core}},{\rm{gr}}}}),\end{array}\end{eqnarray}$
where the Lie bracket ${[.,.]}_{{* }_{{\rm{core}},{\rm{gr}}}}$ is the commutator with respect to the convolution product ${\ast }_{{\rm{core}},{\rm{gr}}}$. Define Derλ as the set of all linear maps ${C}_{\lambda ,{\rm{gr}},{\rm{core}}}\to {C}_{\lambda ,{\rm{gr}},{\rm{core}}}$ which satisfy the Leibniz rule, called derivations. Set Hamλ ⊂ Derλ as the $Z({C}_{\lambda ,{\rm{gr}},{\rm{core}}})$-module generated by all Hamiltonian derivations ham(φ), $\phi \in {C}_{\lambda ,{\rm{gr}},{\rm{core}}}$. The differential graded algebra on ${C}_{\lambda ,{\rm{gr}},{\rm{core}}}$ is defined by
$\begin{eqnarray}{{\rm{\Omega }}}^{\bullet }({C}_{\lambda ,{\rm{gr}},{\rm{core}}}):= ({\oplus }_{n\geqslant 0}{{\rm{\Omega }}}_{\lambda ,{\rm{gr}},{\rm{core}}}^{n},{{\rm{d}}}_{\lambda }),\end{eqnarray}$
such that for each n≥1, ${{\rm{\Omega }}}_{\lambda ,{\rm{gr}},{\rm{core}}}^{n}$ is the space of all $Z({C}_{\lambda ,{\rm{gr}},{\rm{core}}})$-multilinear antisymmetric mappings from Hamλ × ... n × Hamλ into ${C}_{\lambda ,{\rm{gr}},{\rm{core}}}$ and ${{\rm{\Omega }}}_{\lambda ,{\rm{gr}},{\rm{core}}}^{0}={C}_{\lambda ,{\rm{gr}},{\rm{core}}}$. For each $\omega \in {{\rm{\Omega }}}_{\lambda ,{\rm{gr}},{\rm{core}}}^{n}$ and θi ∈ Hamλ, we have $\begin{eqnarray*}\begin{array}{l}{{\rm{d}}}_{\lambda }\omega ({\theta }_{0},\ldots ,{\theta }_{n}):= \displaystyle \sum _{k=0}^{n}{(-1)}^{k}{\theta }_{k}\omega ({\theta }_{0},\ldots ,\hat{{\theta }_{k}},\ldots ,{\theta }_{n})\\ +\displaystyle \sum _{0\,\leqslant \,r\lt s\,\leqslant \,n}{(-1)}^{r+s}\omega ({[{\theta }_{r},{\theta }_{s}]}_{\lambda },{\theta }_{0},\cdots ,\hat{{\theta }_{r}},\cdots ,\hat{{\theta }_{s}},\ldots ,{\theta }_{n}),\\ \quad \,{{\rm{d}}}_{\lambda }^{2}=0.\end{array}\end{eqnarray*}$
The symplectic form ωλ in this differential graded algebra is defined by $\begin{eqnarray}\begin{array}{l}{\omega }_{\lambda }:{{\rm{Ham}}}_{\lambda }\times {{\rm{Ham}}}_{\lambda }\to {C}_{\lambda ,{\rm{gr}},{\rm{core}}},\\ {\omega }_{\lambda }(\theta ,{\theta }^{{\prime} }):= \displaystyle \sum _{i,j}{u}_{i}{\circ }_{\lambda }{v}_{j}{\circ }_{\lambda }{[{f}_{i},{h}_{j}]}_{\lambda },\end{array}\end{eqnarray}$
such that $\{{f}_{1},\ldots ,{f}_{m},{h}_{1},\ldots ,{h}_{n}\}\subset {C}_{\lambda ,{\rm{gr}},{\rm{core}}}$, $\{{u}_{1},\ldots ,{u}_{m},{v}_{1},\ldots ,{v}_{n}\}\,\subset \,Z({C}_{\lambda ,{\rm{gr}},{\rm{core}}})$ and $\begin{eqnarray}\theta =\displaystyle \sum _{i}{u}_{i}{\circ }_{\lambda }{\rm{ham}}({f}_{i}),\,\,\,{\theta }^{{\prime} }=\displaystyle \sum _{j}{v}_{j}{\circ }_{\lambda }{\rm{ham}}({h}_{j}).\end{eqnarray}$
Its corresponding Poisson bracket on ${C}_{\lambda ,{\rm{gr}},{\rm{core}}}$ is given by $\begin{eqnarray}{\{f,g\}}_{\lambda }:= {i}_{{\theta }_{f}^{\lambda }}({{\rm{d}}}_{\lambda }g),\end{eqnarray}$
such that ${\theta }_{f}^{\lambda }$ is the symplectic vector field associated to f, and $\begin{eqnarray}\begin{array}{l}{i}_{\theta }({\omega }_{0}{{\rm{d}}}_{\lambda }{\omega }_{1}...{{\rm{d}}}_{\lambda }{\omega }_{n})=\displaystyle \sum _{j=1}^{n}{(-1)}^{j-1}{\omega }_{0}{{\rm{d}}}_{\lambda }{\omega }_{1}...\theta ({\omega }_{j})...{{\rm{d}}}_{\lambda }{\omega }_{n}\\ \,\Rightarrow {\{f,g\}}_{\lambda }={i}_{{\theta }_{f}^{\lambda }}{i}_{{\theta }_{g}^{\lambda }}{\omega }_{\lambda }.\end{array}\end{eqnarray}$
For $\lambda \in {\mathbb{K}}$, the pair $({C}_{\lambda ,{\rm{gr}},{\rm{core}}},{\omega }_{\lambda })$ is a symplectic geometry model for ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{rg}}}^{{\rm{cut}}}$ which can be lifted onto $((L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}),{* }_{{\rm{ren}},{\rm{gr}}}),{\omega }_{\lambda })$ as a symplectic model for ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ where solutions of quantum motions are represented by stretched Feynman graphons in ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$. ☐
For $\lambda \in {\mathbb{K}}$, the Poisson algebra $({C}_{\lambda ,{\rm{gr}},{\rm{core}}},{\{.,.\}}_{\lambda })$ is a deformation quantization of the algebra $(L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{rg}}}^{{\rm{cut}}},{A}_{{\rm{dr}}}),{* }_{{\rm{core}},{\rm{gr}}})$. Therefore, for any gauge field theory Φ, the Poisson algebra $({C}_{\lambda ,{\rm{gr}},{\rm{core}}},{\{.,.\}}_{\lambda })$ recovers the deformation quantization of $(L({H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}}),{* }_{{\rm{ren}}})$.
For the Rota–Baxter algebra RBgr,Φ and $\lambda \in {\mathbb{K}}$, the Rota–Baxter map ${{ \mathcal R }}_{\lambda }$ as the solution of the Yang–Baxter equation
$\begin{eqnarray}\begin{array}{l}{[F({\phi }_{1}),F({\phi }_{2})]}_{\lambda }+F({[{\phi }_{1},{\phi }_{2}]}_{\lambda })\\ \,=\,F({[F({\phi }_{1}),{\phi }_{2}]}_{\lambda }+{[{\phi }_{1},F({\phi }_{2})]}_{\lambda }),\end{array}\end{eqnarray}$
${\phi }_{1},{\phi }_{2}\in L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),{A}_{{\rm{dr}}})$, determines integrals of motion for a new class of quantum integrable systems in a gauge field theory Φ.It is a result of [11] and theorem
The equation (44 ) is a ‘quantized version’ of the Yang–Baxter equation in RBgr,Φ where the Lie bracket ${[.,.]}_{{* }_{{\rm{ren}},{\rm{gr}}}}$ with respect to the product *ren,gr is replaced by the deformed Lie bracket [. , . ]λ with respect to the product ∘λ. In other words, theorem 3.1 recovers quantum integrability of motion integrals in quantum field theory on the basis of the BPHZ renormalization program.
A min-plus semiring ${\mathbb{S}}={\mathbb{R}}\cup \{\infty \}$ is equipped with the associative commutative operations ⊕ and ⊙ given by 30 ), defines a convolution product * on the space of min-plus characters given by
$\begin{eqnarray}x\oplus y={\rm{\min }}\{x,y\},\,x\odot y=x+y,\end{eqnarray}$
such that ∞ is the identity element for ⊕, 0 is the identity element for ⊙ and ⊙ is distributive over ⊕. It is called a Rota–Baxter semiring of weight λ > 0 if there exists a ⊕-additive map $T:{\mathbb{S}}\to {\mathbb{S}}$ such that $\begin{eqnarray}\begin{array}{l}T({s}_{1})\odot T({s}_{2})=T(T({s}_{1})\odot {s}_{2})\displaystyle \oplus T({s}_{1}\odot T({s}_{2}))\\ \,\displaystyle \oplus T({s}_{1}\odot {s}_{2})\odot \,{\rm{log}}\,\lambda ,\end{array}\end{eqnarray}$
for any ${s}_{1},{s}_{2}\in {\mathbb{S}}$. It is called a Rota–Baxter semiring of weight λ < 0 if there exists a ⊕-additive map $T:{\mathbb{S}}\to {\mathbb{S}}$ such that $\begin{eqnarray}\begin{array}{l}T({s}_{1})\odot T({s}_{2})\displaystyle \oplus T({s}_{1}\odot {s}_{2})\odot \,{\rm{log}}\,(-\lambda )\\ \quad =\,T(T({s}_{1})\odot {s}_{2})\displaystyle \oplus T({s}_{1}\odot T({s}_{2})),\end{array}\end{eqnarray}$
for any ${s}_{1},{s}_{2}\in {\mathbb{S}}$ [36–38]. For the topological core Hopf algebra ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$, a min-plus character is a map $\psi :{{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}\to {\mathbb{S}}$ such that $\begin{eqnarray}\psi ({W}_{{\mathbb{I}}})=0,\,\psi ({W}_{{G}_{1}}{W}_{{G}_{2}})=\psi ({W}_{{G}_{1}})+\psi ({W}_{{G}_{2}}),\end{eqnarray}$
for any ${W}_{{G}_{1}},{W}_{{G}_{2}}$ in ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$. The coproduct ${{\rm{\Delta }}}_{{\rm{core}},{\rm{gr}}}$, given by ( $\begin{eqnarray}\begin{array}{l}({\psi }_{1}* {\psi }_{2})({W}_{G})={\rm{\min }}\{{\psi }_{1}({W}_{H})\\ \,+\,{\psi }_{2}({W}_{G/H})\}=\displaystyle \oplus ({\psi }_{1}({W}_{H})\odot {\psi }_{2}({W}_{G/H})),\end{array}\end{eqnarray}$
such that the minimum is taken over all pairs (WH, WG/H) which contributes to the coproduct ${{\rm{\Delta }}}_{{\rm{core}},{\rm{gr}}}$. For any Rota–Baxter semiring $({\mathbb{S}},T)$ of weight λ > 0 define $\begin{eqnarray}\begin{array}{l}{{\rm{RB}}}_{({\mathbb{S}},{\rm{T}}),{\rm{core}},{\rm{gr}}}\\ := \,(L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{\mathbb{S}}),* ,{ \mathcal T }:L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{\mathbb{S}})\to L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{\mathbb{S}})).\end{array}\end{eqnarray}$
It deforms the convolution product * to define a new associative product *λ on the space of min-plus characters given by $\begin{eqnarray}{\psi }_{1}{\ast }_{\lambda }{\psi }_{2}=T({\psi }_{1})\ast {\psi }_{2}\oplus {\psi }_{1}\ast T({\psi }_{2})\oplus T({\psi }_{1}\ast {\psi }_{2})\odot \,{\rm{log}}\,\lambda .\end{eqnarray}$
Theorem 3.1 provides a symplectic geometry model for the space of stretched graphons from the perspective of the min-plus semiring ${\mathbb{S}}$. Thanks to corollary 3.2 , Rota–Baxter semiring $({\mathbb{S}},T)$ of weight λ > 0 determines integrals of motion for a quantum integrable system originated from topological core Hopf algebra ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$. The case with any Rota–Baxter semiring $({\mathbb{S}},T)$ of weight λ < 0 is analogous where the deformation of the convolution product * is performed in terms of the equation (47 ). In addition, a thermodynamic semiring is given by a deformation of any min-plus algebra where ⊙ is unchanged but ⊕ is deformed by a binary entropy functional together with a deformation parameter [36–38]. Let ${{\mathbb{S}}}_{h,S}$ be a thermodynamic deformation of ${\mathbb{S}}$ with operations ⊕h,S and ⊙ with respect to the Shannon entropy S and the deformation parameter h≥0. For any thermodynamic Rota–Baxter structure $({{\mathbb{S}}}_{h,S},T)$, the new Rota–Baxter structure 3.1 provides a symplectic geometry model for the space of stretched graphons from the perspective of $({{\mathbb{S}}}_{h,S},T)$.
$\begin{eqnarray}\begin{array}{l}{{\rm{RB}}}_{({{\mathbb{S}}}_{h,S},T),core,gr}:= (L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{{\mathbb{S}}}_{h,S}),{\ast }_{h,S},\\ \,{ \mathcal T }:L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{{\mathbb{S}}}_{h,S})\to L({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}},{{\mathbb{S}}}_{h,S})),\end{array}\end{eqnarray}$
deforms *h,S to define a new associative product *λ,h,S on the space of min-plus characters. Theorem We are going to show that it is possible to formulate this setting of symplectic geometry model independent of renormalization scheme where the regularization algebra can be replaced by an arbitrary commutative associative algebra. For a given unital free commutative algebra A with the multiplication m(f, g) = [fg], consider the graded tensor module T(A) ≔ ⊕n≥0A⨂n generated by a1 ⨂ a2 ⨂ ... ⨂ an such that each element in A is called letter and each sequence U ≔ a1a2 ... an is called word with the length n where the empty word e is the unit object in T(A). Define a new shuffle product on T(A) given by
$\begin{eqnarray}\begin{array}{l}fU \circledcirc gV:= f(U \circledcirc gV)+g(fU \circledcirc V)-e[fg](U \circledcirc V),\end{array}\end{eqnarray}$
which is unital and associative. This product defines a new quasi-shuffle product on $\overline{T}(A):= {\oplus }_{n\geqslant 1}{A}^{\otimes n}$ given by $\begin{eqnarray}fU \circleddash gV:= [fg](U \circledcirc V),\end{eqnarray}$
which is unital and associative. Consider the linear map ${B}_{e}^{+}$ on $\overline{T}(A)$ which sends each word U of length n to the new word eU of length n + 1. The pair $(\overline{T}(A), \circleddash ,{B}_{e}^{+})$ is the universal Nijenhuis algebra in a category of Nijenhuis algebras generated by A [52].For the commutative algebra ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}$ defined in theorem
The operator ${B}_{{W}_{{\mathbb{I}}}}^{+}$ can be lifted onto the unital noncommutative associative algebra $(L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),\overline{T}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}})),{* }_{{\rm{ren}},{\rm{gr}}})$ to determine the Nijenhuis map ${{ \mathcal N }}_{{\rm{\Phi }}}(\psi ):= {B}_{{W}_{{\mathbb{I}}}}^{+}\circ \psi $. It defines a new noncommutative associative product ∘u given by
$\begin{eqnarray}\begin{array}{l}{\psi }_{1}{\circ }_{u}{\psi }_{2}:= {{ \mathcal N }}_{{\rm{\Phi }}}({\psi }_{1}){* }_{ \circleddash }{\psi }_{2}+{\psi }_{1}{* }_{ \circleddash }{{ \mathcal N }}_{{\rm{\Phi }}}({\psi }_{2})-{{ \mathcal N }}_{{\rm{\Phi }}}({\psi }_{1}{* }_{ \circleddash }{\psi }_{2}),\end{array}\end{eqnarray}$
such that $\begin{eqnarray}\begin{array}{l}{\psi }_{1}{* }_{ \circleddash }{\psi }_{2}({W}_{X})=\sum {\psi }_{1}({W}_{{\rm{\Gamma }}}) \circleddash {\psi }_{2}({W}_{X/{\rm{\Gamma }}}),\\ \,{{\rm{\Delta }}}_{{\rm{ren}},{\rm{gr}}}({W}_{X})=\sum {W}_{{\rm{\Gamma }}}\displaystyle \otimes {W}_{X/{\rm{\Gamma }}}.\end{array}\end{eqnarray}$
Define ${C}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}:= (L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),\overline{T}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}})),{\circ }_{u})$ such that the commutator [. , . ]u with respect to ∘u defines a Lie structure on $L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),\overline{T}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}))$ where $\begin{eqnarray}\begin{array}{l}{[{\psi }_{1},{\psi }_{2}]}_{u}=[{{ \mathcal N }}_{{\rm{\Phi }}}({\psi }_{1}),{\psi }_{2}]+[{\psi }_{1},{{ \mathcal N }}_{{\rm{\Phi }}}({\psi }_{2})]-{{ \mathcal N }}_{{\rm{\Phi }}}([{\psi }_{1},{\psi }_{2}]).\end{array}\end{eqnarray}$
Thanks to [51], the differential graded algebra on Cu,gr,Φ is defined by $\begin{eqnarray}{{\rm{\Omega }}}^{\bullet }({C}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}):= ({\oplus }_{n\geqslant 0}{{\rm{\Omega }}}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}^{n},{{\rm{d}}}_{u}),\end{eqnarray}$
such that For each n≥1, ${{\rm{\Omega }}}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}^{n}$ is the space of all Z(Cu,gr,Φ)-multilinear antisymmetric mappings from Hamu × ⋯n × Hamu into Cu,gr,Φ and ${{\rm{\Omega }}}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}^{0}={C}_{{\rm{u}},{\rm{gr}},{\rm{\Phi }}}$. The symplectic form ωu, determined by [. , . ]u identifies the Poisson bracket $\begin{eqnarray}{\{f,g\}}_{u}={i}_{{\theta }_{f}^{u}}({{\rm{d}}}_{u}g)={i}_{{\theta }_{f}^{u}}{i}_{{\theta }_{g}^{u}}{\omega }_{u},\end{eqnarray}$
on Cu,gr,Φ. The pair (Cu,gr,Φ, ωu) is a symplectic geometry model for ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ where solutions of quantum motions are represented by stretched Feynman graphons in ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$. ☐
For the universal Nijenhuis algebra $(\overline{T}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}), \circleddash ,{B}_{{W}_{{\mathbb{I}}}}^{+})$, the Nijenhuis operator ${{ \mathcal N }}_{{\rm{\Phi }}}$ as the solution of the Yang–Baxter equation
$\begin{eqnarray}\begin{array}{l}{[F({\psi }_{1}),F({\psi }_{2})]}_{u}+{F}^{2}({[{\psi }_{1},{\psi }_{2}]}_{u})\\ \quad =\,F({[F({\psi }_{1}),{\psi }_{2}]}_{u}+{[{\psi }_{1},F({\psi }_{2})]}_{u}),\end{array}\end{eqnarray}$
${\psi }_{1},{\psi }_{2}\in L({{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }}),\overline{T}({{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}})),$ determines integrals of motion for a new class of quantum integrable systems in a gauge field theory Φ.The equation (60 ) is a ‘quantized version’ of the Yang–Baxter equation in RBgr,Φ where the Lie bracket ${[.,.]}_{{* }_{{\rm{ren}},{\rm{gr}}}}$ with respect to the product *ren,gr is replaced by the deformed Lie bracket [. , . ]u with respect to the product ∘u. In other words, theorem 3.3 recovers quantum integrability of motion integrals in quantum field theory independent of renormalization schemes.
The Connes–Kreimer Renormalization Group associated to the topological Hopf algebra of renormalization ${{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ is an infinite dimensional quantum integrable system which encodes the evolution of Feynman graph limits in a gauge field theory Φ.
Consider an infinitesimal punctured disk Δ⋆ around z = 0 in the complex plane with the boundary C = ∂Δ⋆, a loop $\alpha :C\to {{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({A}_{{\rm{dr}}})$ and a one-parameter group ${\{{\theta }_{t}\}}_{t\in {\mathbb{C}}}$ of automorphisms on ${{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({A}_{{\rm{dr}}})$ given by
$\begin{eqnarray}{\theta }_{t}(\varphi )={{\rm{e}}}^{nt}\varphi ,\,\lt {\theta }_{t}(\varphi ),{W}_{{\rm{\Gamma }}}\gt =\lt \varphi ,{\theta }_{t}({W}_{{\rm{\Gamma }}})\gt ,\end{eqnarray}$
for n≥1, $\varphi \in {{ \mathcal H }}_{{\rm{gr}}}^{(n),\vee }$ and ${W}_{{\rm{\Gamma }}}\in {{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$. The Rota–Baxter algebra RBgr,Φ generates the Hopf–Birkhoff factorization (α−, α+) such that $\alpha ={\alpha }_{-}^{-1}{\ast }_{{\rm{ren}},{\rm{gr}}}{\alpha }_{+}$. Following [21], let ${\{{F}_{t}\}}_{t\in {\mathbb{C}}}$ be the Renormalization Group given by
$\begin{eqnarray}{F}_{t}={{\rm{lim}}}_{z\to 0}{\alpha }_{-}(z){\theta }_{tz}({\alpha }_{-}{(z)}^{-1})\in {{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({A}_{{\rm{dr}}}),\end{eqnarray}$
such that for each ${W}_{{\rm{\Gamma }}}\in {{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$, Ft(WΓ) is a polynomial in t. For a Cauchy sequence ${\{{{\rm{\Gamma }}}_{n}\}}_{n\geqslant 1}$ of Feynman diagrams with respect to the metric (We apply theorem
$\begin{eqnarray}{\{f,{F}_{{t}_{0}}\}}_{\lambda ,{* }_{{\rm{gr}}}}={i}_{{\theta }_{{F}_{{t}_{0}}}^{\lambda }}{i}_{{\theta }_{f}^{\lambda }}{\omega }_{\lambda }={w}_{\lambda }({\theta }_{{F}_{{t}_{0}}}^{\lambda },{\theta }_{f}^{\lambda })=0.\end{eqnarray}$
For each ${F}_{t}\in {\{{F}_{t}\}}_{t\in {\mathbb{C}}}$, we have $\begin{eqnarray}\begin{array}{rcl}{\{{F}_{t},{F}_{{t}_{0}}\}}_{\lambda ,{* }_{{\rm{gr}}}} & = & {[{{ \mathcal R }}_{\lambda }({F}_{t}),{F}_{{t}_{0}}]}_{\lambda ,{* }_{{\rm{gr}}}}\\ & & +{[{F}_{t},{{ \mathcal R }}_{\lambda }({F}_{{t}_{0}})]}_{\lambda ,{* }_{{\rm{gr}}}}-{{ \mathcal R }}_{\lambda }({[{F}_{t},{F}_{{t}_{0}}]}_{\lambda ,{* }_{{\rm{gr}}}}),\end{array}\end{eqnarray}$
such that ${{ \mathcal R }}_{\lambda }({F}_{t})=(1-\lambda ){ \mathcal R }({F}_{t})-\lambda {F}_{t}$. Since Rms(Ft(WΓ)) = 0 for each ${W}_{{\rm{\Gamma }}}\in {{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})$ and Ft*ren,grFs = Ft+s, $s,t\in {\mathbb{C}}$, For the case λ = 0, we have $\begin{eqnarray}{\{{F}_{t},{F}_{{t}_{0}}\}}_{0,{* }_{{\rm{gr}}}}=0.\end{eqnarray}$
A given combinatorial Dyson–Schwinger equation DSE with the unique solution X = ∑n≥0cnXn and the sequence ${\{{Y}_{m}\}}_{m\geqslant 1}$ of partial sums determine a graded connected commutative Hopf subalgebra HDSE in HFG(Φ) [30] such that the large Feynman graph X is in ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{\Phi }})$ [31, 39, 40]. Therefore, thanks to lemmas
We have
$\begin{eqnarray}X={{\rm{lim}}}_{m\to \infty }{Y}_{m}\,\iff \,{{\rm{lim}}}_{m\to \infty }{W}_{{Y}_{m}}={W}_{X},\end{eqnarray}$
which leads us to show that $\begin{eqnarray*}\begin{array}{l}{F}_{t}(X)={F}_{t}({{\rm{lim}}}_{m\to \infty }{Y}_{m})={F}_{t}({{\rm{lim}}}_{m\to \infty }\displaystyle \sum _{n=1}^{m}{c}^{n}{X}_{n})\\ \quad =\,{{\rm{lim}}}_{m\to \infty }\displaystyle \sum _{n=1}^{m}{F}_{t}({c}^{n}{X}_{n})={{\rm{lim}}}_{m\to \infty }\displaystyle \sum _{n=1}^{m}\\ \quad {{\rm{lim}}}_{z\to 0}{\tilde{\alpha }}_{{\rm{DSE,-}}}(z)({c}^{n}{X}_{n}){\theta }_{tz}({\tilde{\alpha }}_{{\rm{DSE,-}}}^{-1}(z)({c}^{n}{X}_{n}))\\ \quad =\,{{\rm{lim}}}_{m\to \infty }{{\rm{lim}}}_{z\to 0}\displaystyle \sum _{n=1}^{m}{\tilde{\alpha }}_{{\rm{DSE}},-}(z)\\ \quad ({c}^{n}{X}_{n}){\theta }_{tz}({\tilde{\alpha }}_{{\rm{DSE}},-}^{-1}(z)({c}^{n}{X}_{n})).\end{array}\end{eqnarray*}$
Therefore, for a fixed ${F}_{{t}_{0}}$ from the Renormalization Group, the evolution of X at the stage m≥1 from the perspective of the quantum integrable system ${\{{F}_{t}\}}_{t}$ with the Hamiltonian ${F}_{{t}_{0}}$ is given by the Poisson bracket ${\{{F}_{{t}_{0}},{\tilde{\alpha }}_{{\rm{DSE}}}(z)\}}_{0,{* }_{{\rm{gr}}}}$. ☐
Thanks to [37, 38], for any Rota–Baxter semiring $({\mathbb{S}},T)$ of weight +1, the Hopf–Birkhoff factorizations of min-plus characters $\phi :{{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})\to {\mathbb{S}}$ exist. Therefore the Connes–Kreimer Renormalization Group ${\{{F}_{t}\}}_{t\in {\mathbb{C}}}\subset {{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({\mathbb{S}})$ is well-defined and it leads us to search for possible extensions of corollary 3.5 for the formulation of quantum integrable systems in the physical theory from the perspective of $({\mathbb{S}},T)$. In addition, thanks to [36–38], the Hopf–Birkhoff factorizations of min-plus characters $\phi :{{ \mathcal H }}_{{\rm{gr}}}^{{\rm{cut}}}({\rm{\Phi }})\to {{\mathbb{S}}}_{h,S}$ for any thermodynamic Rota–Baxter structure $({{\mathbb{S}}}_{h,S},T)$ exist where ${{\mathbb{S}}}_{h,S}$ is a thermodynamic deformation of ${\mathbb{S}}$ with operations ⊕h,S and ⊙ with respect to the Shannon entropy S and the deformation parameter h≥0. Therefore the Connes–Kreimer Renormalization Group ${\{{F}_{t}\}}_{t\in {\mathbb{C}}}\subset {{\mathbb{G}}}_{{\rm{\Phi }},{\rm{gr}}}({{\mathbb{S}}}_{h,S})$ is well-defined and it leads us to search for possible extensions of corollary 3.5 for the formulation of quantum integrable systems in the physical theory from the perspective of $({{\mathbb{S}}}_{h,S},T)$.
The Banach space ${{ \mathcal S }}_{\approx }^{{\rm{\Phi }},{\rm{\Omega }}}$ of stretched Feynman graphons associated to a gauge field theory Φ is embedded as a subspace of the Banach space ${{ \mathcal W }}_{\approx }^{{\rm{\Omega }}}$ of real valued stretched graphons defined on a σ-finite measure space (Ω ⫅ [0, ∞), μ). Therefore quantum integrable systems determined by the symplectic geometry model of ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$ recover quantum integrable systems given by Corollaries 3.2 and 3.4 . In addition, thanks to theorems 1.2 , 2.2 , lemma 2.1 and corollary 3.5 , the Connes–Kreimer Renormalization Group associated to ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$ is an infinite dimensional quantum integrable system which recovers the evolution of solutions of quantum motions in gauge field theories. Thanks to the topological enrichment of the quotient Hopf subalgebras HFG(QED)/IWT and HFG(QCD)/IST studied in [24, 31], the formal setting of corollary 3.5 can be applied to formulate quantum integrable systems in QED and QCD as gauge field theories on (D + 1)-space-time background by the method of Renormalization Group. In addition, Corollaries 3.2 and 3.4 can be lifted onto ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{QED}})/{I}_{{\rm{WT}}}^{{\rm{cut}}}$ and ${H}_{{\rm{FG}}}^{{\rm{cut}}}({\rm{QCD}})/{I}_{{\rm{ST}}}^{{\rm{cut}}}$ to recognize those quantum Yang–Baxter equations which encode integrals of motion of integrable systems governed by Dyson–Schwinger equations in QED and QCD.
While the deformation theory of classical groups defines some quantum integrable systems in terms of Yang–Baxter equation associated to the corresponding quantum groups, here we showed that the deformation theory of a certain class of noncommutative associative algebras determines a new class of quantum integrable systems which contribute to non-perturbative domain of physical theories. Our investigation in this study shows the universal role of the topological core Hopf algebra ${{ \mathcal H }}_{{\rm{FGrp}},{\rm{gr}}}^{{\rm{cut}}}$ for the study of quantum integrable systems in quantum field theory.
4. Declarations
| • | The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. |
| • | ASF: conceptualization, original motivation, methodology, research, writing/reviewing/editing: original draft and final preparation. |