Richard Borcherds proposed an elegant geometric version of renormalized
perturbative quantum field theory in curved spacetimes, where Lagrangians are
sections of a Hopf algebra bundle over a smooth manifold. However, this
framework looses its geometric meaning when Borcherds introduces a (graded)
commutative normal product. We present a fully geometric version of Borcherds'
quantization where the (external) tensor product plays the role of the normal
product. We construct a noncommutative many-body Hopf algebra and a module over
it which contains all the terms of the perturbative expansion and we quantize
it to recover the expectation values of standard quantum field theory when the
Hopf algebra fiber is (graded) cocommutative. This construction enables to the
second quantize any theory described by a cocommutative Hopf algebra bundle.Comment: Frontiers of Fundamental Physics 14, Jul 2014, Marseille, Franc

Brouder, Christian

Dang, Nguyen Viet

Frabetti, Alessandra

English

arXiv

International audienceRichard Borcherds proposed an elegant geometric version of renormalized perturbative quantum field theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle over a smooth manifold. However, this framework looses its geometric meaning when Borcherds introduces a (graded) commutative normal product. We present a fully geometric version of Borcherds' quantization where the (external) tensor product plays the role of the normal product. We construct a noncommutative many-body Hopf algebra and a module over it which contains all the terms of the perturbative expansion and we quantize it to recover the expectation values of standard quantum field theory when the Hopf algebra fiber is (graded) cocommutative. This construction enables to the second quantize any theory described by a cocommutative Hopf algebra bundle

HAL-UJM

Noncommutative version of Borcherds’ approach toquantum field theoryChristian Brouder, Nguyen Viet Dang, Alessandra FrabettiTo cite this version:Christian Brouder, Nguyen Viet Dang, Alessandra Frabetti. Noncommutative version ofBorcherds’ approach to quantum field theory. Frontiers of Fundamental Physics 14, Jul 2014,Marseille, France. <hal-01111764>HAL Id: hal-01111764https://hal.archives-ouvertes.fr/hal-01111764Submitted on 30 Jan 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Noncommutative version of Borcherds’ approach toquantum field theoryChristian Brouder∗Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie,Sorbonne Universités – UPMC, Université Paris 6, UMR CNRS 7590,Muséum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France.E-mail: christian.brouder@upmc.frNguyen Viet DangLaboratoire Paul Painlevé, UMR CNRS 8524,Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France.Alessandra FrabettiInstitut Camille Jordan, Université Lyon1, UMR CNRS 5208,Bâtiment Jean Braconnier, 43 bd. du 11 novembre 1918, 69622 Lyon, France.Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantumfield theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle overa smooth manifold. However, this framework looses its geometric meaning when Borcherdsintroduces a (graded) commutative normal product. We present a fully geometric version ofBorcherds’ quantization where the (external) tensor product plays the role of the normal product.We construct a noncommutative many-body Hopf algebra and a module over it which containsall the terms of the perturbative expansion and we quantize it to recover the expectation values ofstandard quantum field theory when the Hopf algebra fiber is (graded) cocommutative. This con-struction enables to the second quantize any theory described by a cocommutative Hopf algebrabundle.Frontiers of Fundamental Physics 14 - FFP14,15-18 July 2014Aix Marseille University (AMU) Saint-Charles Campus, Marseille∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/Noncommutative Borcherds’ QFT Christian Brouder1. IntroductionIn an article entitled “Renormalization and quantum field theory” [1], Richard Borcherds de-scribed a rigorous approach to renormalized perturbative quantum field theory in curved space-times. Borcherds’ approach is closely related to the causal algebraic formalism [2], and it employssheaf theory and Hopf algebras to achieve a particularly elegant and compact picture of quantumfield theory (QFT). In particular, the combinatorial aspects of quantization and renormalization arecompletely taken care of by a Hopf algebraic structure. Moreover, Borcherds’ approach has definiteadvantages when it comes to generalization. For example, the use of Hopf algebras is particularlypowerful to deal with systems involving an initial state which is not quasi-free [3] and many of itstools (for example vector bundles and Hopf algebras) have natural noncommutative analogues thatcan be used to investigate noncommutative versions of quantum field theory.In the present paper, which is a sketch of a more detailed article in preparation, we extend partsof Borcherds’ approach by replacing his graded commutative normal product of classical fields bya tensor product which (i) allows us to formulate a fully geometric version of second quantization,(ii) provides a manageable topology for the many-body algebra, (iii) enables us to second quantizeany cocommutative Hopf algebra bundle.2. Hopf algebra bundlesIn this section we introduce some concepts that are used in Borcherds’ approach to QFT.Classical fields are sections of vector bundles over the space-time manifoldM. We first reformulateBorcherds’ sheaves into more familiar sections of vector bundles.Let M be a smooth manifold and Fpi→M a smooth vector bundle over M [4]. We denote byφα : pi−1(Uα)→Uα×V the local trivializations (whereV is a vector space) and by tαβ the transitionfunctions such that φα ◦φ−1β (x,v) = (x, tαβ (x)v), where φα ◦φ−1β : (Uα ∩Uβ )×V → (Uα ∩Uβ )×Vand where the isomorphism tαβ (x) is an element of GL(V ). A vector bundle is an algebra bundleif the fiber model V is an algebra over K (where K is R or C) and if the transition functions arealgebra isomorphisms: tαβ (x)(u · v) = tαβ (x)(u) · tαβ (x)(v). An algebra bundle is a Hopf algebrabundle if V is a Hopf algebra over K and the transition functions are Hopf algebra morphisms. Inparticular, the coproduct sends V to V ⊗V , which is the fiber of the (internal) tensor product ofvector bundles F⊗Fpi ′→M [4].The space of sections Γ(M,F) is an infinite-dimensional vector space, but it is also a moduleover the ringC∞(M) of K-valued smooth functions: as such, it admits a (locally) finite basis whichallows to use simple linear algebra tools. If F is an algebra bundle, then the space of sectionsΓ(M,F) is an algebra over the ringC∞(M): if σ1 and σ2 are such sections with φα(σ1(x))= (x,v1)and φα(σ2(x))= (x,v2), then φα(σ1 ·σ2(x))= (x,v1 · v2). Similarly, the space of sections of aHopf algebra bundle is a Hopf algebra over the ring C∞(M). In particular, the coproduct is now amap from Γ(M,F) to Γ(M,F⊗F)∼= Γ(M,F)⊗ˆC∞(M)Γ(M,F).Borcherds starts from a vector bundle Epi→ M of finite rank whose sections are the classicalfields of the model. To define Lagrangian densities as polynomials in the field and its derivatives,he considers the infinite jet bundle JEpi→ M and the Hopf algebra bundle S(JE∗)pi→ M, whichdescribes the polynomial functions on JE .2Noncommutative Borcherds’ QFT Christian BrouderFor example, the element L= f +gµ +hµν + k, where f ∈ Γ(M,E∗), gµ ∈ Γ(M,J1E∗), hµν ∈Γ(M,S2(J1E∗)) and k ∈ Γ(M,S4(E∗)), corresponds to the Lagrangian density L(ϕ) = 〈 f ,ϕ〉+〈gµ ,ϕµ〉+ 〈hµν,ϕµϕν〉+ 〈k,ϕϕϕϕ〉, where ϕ is a field, ϕµ its derivatives and 〈·, ·〉 is the dualitypairing between Γ(M,S(JE∗)) and Γ(M,S(JE)) induced by the duality pairing between JE∗ andJE . This Hopf algebra is commutative and cocommutative. Note that the topological propertiesof this algebra must be carefully taken into account because JE∗ is an infinite-dimensional Fréchetmanifold.In the next section, we shall consider a general Hopf algebra bundle Fpi→M whose sectionsplay the role of Lagrangian densities, where F = S(JE∗) in Borcherds’ case.3. The Fock Hopf algebra of classical fieldsSecond quantization starts from the construction of an algebra containing classical fields de-fined on any number of spacetime points. The commutative product of this many-body alge-bra is called the normal product, and it will be deformed to define a quantum field algebra. InBorcherds’ paper, the algebra corresponding to the normal product of QFT is the symmetric al-gebra SK(Γ(M,F)) on the space of sections, which is too big to have a reasonable topology andwhich is no longer geometric, in the sense that SK(Γ(M,F)) is not the space of sections of a bundleover a manifold. This is because this manifold should be the quotient of Mn by the action of thesymmetric group on n elements, which is generally not a topological manifold [5].To solve that problem, note that for any bundle Fpi→M there exists an external tensor productof bundles FFpi×pi−→M×M whose space of sections describes the (completed) tensor product ofsections (over K), Γ(M×M,FF)∼= Γ(M,F)⊗ˆKΓ(M,F), that is, σ(x1,x2) = ∑σ1(x1)⊗σ2(x2).Moreover, since Γ(M,F) is a Hopf algebra over C∞(M), then Γ(M×M,FF) is a Hopf algebraover C∞(M2). Similarly,Definition 1. If Fpi→ M is a Hopf algebra bundle, the normal product of classical fields over nspacetime points is described by the normal product algebra Γ(Mn,Fn), which is a Hopf algebraover C∞(Mn).Therefore, our normal product is encoded in the tensor product of sections, corresponding tothe external tensor product of bundles. From a physical point of view, if F = S(JE∗) is the bundle ofpolynomial Lagrangians of E-valued fields, the external tensor product  describes exactly the nor-mal product of field polynomials at 2 points of M: e.g. the normal productϕ4(x1)∂µϕ(x2)∂µ ϕ(x2)corresponds to the section σ(x1,x2) =((x1,x2),ϕ4⊗∂µϕ∂µϕ)of the bundle FF over the point(x1,x2) ∈M×M. The exterior tensor product can be performed on any number n of copies of thebundle F , giving the Hopf bundle Fnpin−→Mn.To describe QFT, then, we need to define a single algebra which contains all numbers of points.The difficulty is that the algebras Γ(Mn,Fn) are defined over different ringsC∞(Mn), one for eachn. It turns out that this problem was solved a long time ago by Bourbaki. The first step is to builda ring R = lim−→C∞(Mn) [6], which is the inductive limit of the rings C∞(Mn) corresponding to themap φmn : C∞(Mm)→ C∞(Mn), with m ≤ n, defined by φmn( f )(x1, . . . ,xn) = f (x1, . . . ,xm). Theinductive limit of algebras over different rings is also defined by Bourbaki [6] and its extension to3Noncommutative Borcherds’ QFT Christian BrouderHopf algebras is straightforward. Thus, we obtain a Hopf algebra which is reminiscent of the Fockspace in the sense that it contains any number of points.Definition 2. If Fpi→M is a Hopf algebra bundle, the Fock Hopf algebra is the inductive limit ofHopf algebras HFock = lim−→Γ(Mn,Fn), which is a Hopf algebra over the ring RFock = lim−→C∞(Mn).Note that the Fock Hopf algebra is commutative iff F is commutative. The Hopf algebrastructure on the Fock algebra is used to perform its deformation quantization.We can now wonder whether the Fock Hopf algebra is a space of sections of a bundle oversome infinite dimensional manifold. When M can be described by a single chart to Rd , then theanswer is yes and the manifold is lim←−Mn, which is a Fréchet manifold built on lim←−(Rd)n. If Mneeds several charts, then the projective limit topology is not compatible with the structure of aFréchet manifold and we need more general concepts of infinite-dimensional manifolds. We canalso wonder whether the definition of φmn is not too arbitrary. Instead of picking up the m firstpoints of (x1, . . . ,xn), we can define an inductive limit corresponding to any subset of m elements,but by doing so we recover exactly HFock and RFock (because the family of sets {1, . . . ,n} is cofinalin the family of subsets of N [7]) so we stick to the simpler definition because countable inductivelimits have better properties than uncountable ones.4. Deformation quantization of HFockIt remains to quantize the Fock Hopf algebra to recover the operator product of standard quan-tum field theory as a special case. A convenient method to do so is to use quantum groups, thatDrinfeld created as a quantization of algebras [8]. His foundation paper even cites the quantizationmethod of Berezin, Vey, Lichnerowicz, Flato and Sternheimer (i.e. deformation quantization or starproduct). However, the quantization of fields does not use Drinfeld’s quasitriangular structure butits dual, the Laplace pairing, which was first defined by Lyubashenko [9]. Rota and Stein called ita Laplace pairing because, for anticommuting variables, its definition is equivalent to the Laplaceidentity of determinants [10]. Borcherds calls it a bicharacter.4.1 Laplace pairingThe problem is now that the Fock Hopf algebra is made of products of polynomials of smoothsections and their derivatives, whereas the quantum field amplitudes are distributions. Therefore,we need to introduce the space D ′Fock = lim−→D ′(Mn), which is the inductive limit of the spaces ofdistributions on Mn. The Laplace pairing is an RFock-linear map (·|·) : HFock⊗RFock HFock →D′Fock,such that, for a, b and c in HFock, (1|a) = (a|1) = ε(a) and (a|bc) = ∑(a(1)|b)(a(2)|c) and (ab|c) =∑(a|c(1))(b|c(2)). Since the terms (a(1)|b)(a(2)|c) and (a|c(1))(b|c(2)) involve distributions, the productis only done when wavefront set conditions are satisfied [11].In the case of standard quantum field theory, where HFock is built from the fiber F = S(JE∗),the Laplace pairing is determined for f and g in Γ(M,E∗) by ( f ⊗ 1|1⊗ g) = 〈 f ⊗ g,D+〉, whereD+ ∈D′(M2,E2) is the Wightman propagator. This can also be written in a more physical way as(ϕ ⊗1|1⊗ϕ) = D+ or in a non-rigorous way (ϕ(x)|ϕ(y)) = D+(x,y) in the fiber over (x,y). Thisdefinition is extended to derivatives of fields by (∂ α ϕ ⊗ 1|1⊗ ∂ β ϕ) = ∂ α∂ βD+, where α and β4Noncommutative Borcherds’ QFT Christian Brouderare multi-indices. This pairing is well defined because of the structural theorem [12]D′(M2,E2) =(Γc(M2,(E∗)2))′∼= D ′(M2)⊗C∞(M2) Γ(M2,E2)∼= LC∞(M2)(Γ(M2,(E∗)2),D ′(M2)).4.2 Star productQuantum group quantization was first defined by Rota and Stein [10], then developed byFauser and coworkers [13, 14, 15]. Its equivalence with the star product was proved by Hirsh-feld [16]. Borcherds does not define this product.Definition 3. Let Fpi→M be a Hopf algebra bundle and HFock the corresponding Fock Hopf alge-bra. Then, CFock = lim−→D ′(Mn)⊗C∞(Mn) Γ(Mn,Fn) is a HFock-Hopf module where the coaction βis defined on c= u⊗h by βc = ∑c′⊗ c′′ = ∑(u⊗h(1))⊗h(2). The star product on CFock is definedbyc?d = ∑c′d′(c′′|d′′), (4.1)where (c′′|d′′) is identified with (c′′|d′′)⊗1. If the Hopf algebra is cocommutative, the star productis associative.If we consider the example c= u⊗h and d = v⊗k we find c?d = ∑uv(h(2)|k(2))⊗h(1)k(1). Theproduct ab(h(1)|k(2)) is a product of three distributions which is well-defined by the wavefront setcondition [11] for standard quantum field theory [2]. Note thatCFock equipped with the star productis a sort of generalized Frobenius algebra, in the sense that (c?d|e) = (c|d ? e) [15].For example if c= (1⊗1)⊗(ϕ⊗1) and d = (1⊗1)⊗(1⊗ϕ), then c?d = (1⊗1)⊗(ϕ⊗ϕ)+D+⊗(1⊗1) and we recover Wick’s theorem usually written ϕ(x)?ϕ(y) = :ϕ(x)ϕ(y):+D+(x,y) in QFTtextbooks. This completes the quantization of the Fock Hopf algebra, i.e. the second quantizationof the Hopf algebra bundle F .4.3 The time-ordered productThe last step to obtain Green functions of QFT is to define time-ordered products. We dothis by following the causal approach developed by Stueckelberg, Bogoliubov, Epstein, Glaser [18]and finally Brunetti and Fredenhagen [2]. Then, the time-ordered product becomes a comodulemorphism T : CFock → CFock and the Wick expansion of time-ordered products takes the simpleform T (c) = ∑t(c′)c”, where t(c) = (1⊗ ε)(T (c)) [15]. The time-ordered product is definedrecursively by the causality relation1 saying that T (cd) = T (c) ?T (d) if the spacetime support ofc is not earlier than the spacetime support of d. By Stora’s lemma2, the causality relation and thepartial order imply that T is defined recursively except on the diagonals, where the distributionshave to be extended [2]. The ambiguity of this extension is organized by the renormalization group.1Borcherds’ Gaussian property is a consequence of the causality relation [18].2It can easily be inferred from a remark by Bergbauer [17] that Stora’s lemma only requires a (closed) partial orderon M, which is taken to be the causal order in applications to Lorentzian manifolds.5Noncommutative Borcherds’ QFT Christian Brouder5. ConclusionA second quantization method was described for any theory whose Lagrangian density isan element of a cocommutative Hopf algebra bundle. Fermions can be taken into account byusing a graded cocommutative Hopf algebra [14]. Since we do not require the Hopf algebra to becommutative, we expect this approach to play a role in the second quantization of noncommutativegeometry.References[1] R. E. Borcherds. Renormalization and quantum field theory. Algebra & Number Theory, 5:627–58,2011.[2] R. Brunetti and K. Fredenhagen. Microlocal analysis and interacting quantum field theories:renormalization on physical backgrounds. Commun. Math. Phys., 208:623–61, 2000.[3] Ch. Brouder and F. Patras. One-particle irreducibility with initial correlations. In K. Ebrahimi-Fard,M. Marcolli, and W. D. van Suijlekom, editors, Combinatorics and Physics, volume 539 of Contemp.Math., pages 1–25. Amer. Math. Soc., 2011.[4] J. M. Lee. Manifolds and Differential Geometry, volume 82 of Graduate Studies in Mathematics.American Mathematical Society, Providence, 2009.[5] C. H. Wagner. Symmetric, cyclic and permutation products of manifolds. Dissert. Math., 182:1–48,1980.[6] N. Bourbaki. Elements of Mathematics. Algebra I Chapters 1-3. Springer, Berlin, 1989.[7] G. Köthe. Topological Vector Spaces I. Springer Verlag, New York, 1969.[8] V. G. Drinfel’d. Quantum groups. In A. M. Gleason, editor, Proceedings of the InternationalCongress of Mathematicians (Berkeley, 1986), pages 798–820, Providence, 1988. Amer. Math. Soc.[9] V. V. Lyubashenko. Hopf algebras and vector symmetries. Russian Math. Surv., 41:153–4, 1986.[10] G.-C. Rota and J. A. Stein. Plethystic Hopf algebras. Proc. Natl. Acad. Sci USA, 91:13057–61, 1994.[11] L. Hörmander. The Analysis of Linear Partial Differential Operators I. Distribution Theory andFourier Analysis. Springer Verlag, Berlin, second edition, 1990.[12] M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer. Geometric Theory ofGeneralized Functions with Applications to General Relativity. Springer, Berlin, 2001.[13] B. Fauser. On the Hopf algebraic origin of Wick normal-ordering. J. Phys. A: Math. Gen.,34:105–116, 2001.[14] Ch. Brouder, B. Fauser, A. Frabetti, and R. Oeckl. Quantum field theory and Hopf algebracohomology. J. Phys. A: Math. Gen., 37:5895–927, 2004.[15] Ch. Brouder. Quantum field theory meets Hopf algebra. Math. Nachr., 282:1664–90, 2009.[16] A.C. Hirshfeld and P. Henselder. Star products and quantum groups in quantum mechanics and fieldtheory. Ann. Phys., 308:311–28, 2003.[17] C. Bergbauer. Epstein-Glaser renormalization, the Hopf algebra of rooted trees and theFulton-MacPherson compactification of configuration spaces. Diplomarbeit, Freie Universität Berlin,2004.6Noncommutative Borcherds’ QFT Christian Brouder[18] H. Epstein and V. Glaser. The role of locality in perturbation theory. Ann. Inst. Henri Poincaré,19:211–95, 1973.7

Noncommutative version of Borcherds' approach to quantum field theory

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International audienceRichard Borcherds proposed an elegant geometric version of renormalized perturbative quantumfield theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle overa smooth manifold. However, this framework looses its geometric meaning when Borcherdsintroduces a (graded) commutative normal product. We present a fully geometric version ofBorcherds’ quantization where the (external) tensor product plays the role of the normal product.We construct a noncommutative many-body Hopf algebra and a module over it which containsall the terms of the perturbative expansion and we quantize it to recover the expectation values ofstandard quantum field theory when the Hopf algebra fiber is (graded) cocommutative. This constructionenables to the second quantize any theory described by a cocommutative Hopf algebrabundle

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Noncommutative version of Borcherds’ approach toquantum field theory

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Noncommutative version of Borcherds' approach to quantum field theory

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