FROM THE FORMALITY THEOREM OF KONTSEVICH AND CONNES-KREIMER ALGEBRAIC RENORMALIZATION TO A THEORY OF PATH INTEGRALS Abstract. Quantum physics models evolved from gauge theory on manifolds to quasi-discrete geometric-algebraic models, like lattice gauge theory on discrete lattices, Quantum Field Theory (QFT) on Feynman graphs, and Conformal Field Theory (CFT) on Riemann surfaces. These theories still use an ambient spacetime manifold to define the corresponding representations of quantum processes. The present proposed research has two goals; to formulate: 1) A theory of formal manifolds, developing the ideas of Maxim Kontsevich regarding formal pointed manifolds and the corresponding techniques used to prove the Formality Theorem, towards an intrinsic, background free, quantum dynamics. 2) A theory of path integrals on formal manifolds, generalizing QFT and CFT. Following the work of the PI and his collaborator [FI], Kontsevich s Formality Theorem is interpreted in this proposal as a natural transformation between the infinitesimal presentation of a manifold in terms of its exterior algebra of differential forms and the non-linear presentation in terms of differential operators. The PI conjectures that it is a resolution in the sense of homological algebra. This conjecture is the starting point of a theory of resolutions of manifolds, or rather of their algebras of observables and differential forms, developing Kontsevich s concept of pointed formal manifold, and using his techniques as well as our subsequent further developments [FI]. In order to define the global concept of a formal manifold, which is a problem stated by Kontsevich around 1997-1998, a gluing operation is introduced, modeled in terms of Rota-Baxter algebras and the associated algebraic framework for renormalization developed by Connes and Kreimer. What results is a groupoid and the starting point of a theory of path integrals. Specifically, a Feynman Path Integral is a functor on a fundamental groupoid. The groupoid is a Feynman category as previously defined [I-FL,I-FP], while its representations, the corresponding path integrals corresponding to a certain moduli space of the groupoid [I-LTDT], are the Feynman Processes. The classical example is that QFT as a representation of Feynman graphs (Feynman PROP [I-FP]). But the illuminating example is CFT, as a representation of Riemann surfaces (the Segal PROP); A byproduct of the theory of path integrals is the methodology of using the pointwise classical physics, carried via quantization to quantum physics, to design the modern physics of processes (states and transitions), in the spirit of Feynman s path integral method for quantization. At the level of fundamental mathematics involved, it is a process of categorification: upgrading the global algebraic approach focusing on elements, to a categorical language focusing on change (morphisms) and invariant propreties (symmetries) within a given context, a category. Further developments include a formulation of a background independent String Theory, to be implemented using the above general framework in a continuation of the proposed research. 1
2 Contents 1. Introduction 2 2. Resolutions of Manifolds 3 2.1. Haar wavelets and Multi-Resolution Analysis 3 2.2. Groups and resolutions 4 2.3. QFT: perturbations or resolutions 4 2.4. Why using resolutions? 4 3. Formal Pointed Manifolds 5 4. Formal Manifolds and Path Integrals 7 4.1. Feynman Path Integrals 7 4.2. Applications 7 4.3. Relation with Renormalization 8 5. Further developments 8 6. Acknowledgments 9 1. Introduction The mathematical models of quantum physics evolved from gauge theory on manifolds to discrete-geometric algebraic models, for example lattice gauge theory on discrete lattices, quantum field theory on Feynman graphs, and conformal theory on Riemann surfaces. These theories still use an ambient space-time manifold to define their representations of the quantum processes. The concept of an intrinsic formulation of quantum phenomena, from which space-time should arise as a moduli space, the way a geometry arises as a homogeneous space from a Klein geometry represented by an abstract group. As an example, String Theory specialists are searching for a background independent formulation, to resolve the ambiguity of the multitude of possible landscapes (vacuum states). The proposed research aims to formulate a theory of formal manifolds, developing the ideas of Maxim Kontsevich on formal pointed manifolds [K], a concept introduced while proving the Formality Theorem. The relation between pointed formal manifolds and (global) formal manifolds, is the same as between local Taylor expansions of a function and a global such function, or as the difference between a power series representation of an analytic function and its analitical continuation. In this later case the connection with a theory of path integrals is transparent. A global formal manifold should be thought of as an analog of a Riemann surface associated to a multi-valued analytic function. The formal manifold is an algebraic substitute of a manifold. The development of the theory leads to a theory of resolutions of manifolds, via the Hoschschild differential graded algebra (DGLA) of their algebra of functions.
FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS 3 The targeted mathematical framework is inspired by the use of Feynman diagrams in QFT. This is also the underlying motivation of Kontsevich model for quantizing Poisson manifolds [CF]. 2. Resolutions of Manifolds Following the work of the PI and his collaborator [FI], the universal quasi-isomorphism of Kontsevich s Formality Theorem is interpreted as a natural transformation between the infinitesimal presentation of a manifold in terms of its exterior algebra of forms and the non-linear presentation in terms of differential operators. This natural quasi-isomorphism is reminiscent of a homological resolution in the sense of homological algebra. It is the starting point of a theory of resolutions of manifolds, or rather of their algebras of observables and differential forms (a structure sheaf point of view of manifold theory can also be considered), developing Kontsevich s concept of pointed formal manifold. The physical motivation for a theory of resolutions of manifolds is the introduction of a scale of structure and a hierarchy of complexity, which is given by the grading of the resolution. The mathematical motivation is the tractability of the theory for deriving qualitative results and not just for computational purposes, which comes from the finite type character of such resolutions, i. e. being finite dimensional in each degree. 2.1. Haar wavelets and Multi-Resolution Analysis. The theory of wavelets provide one good example of how to discretize the continuum using a hierarchy of discrete approximations. First, instead of approximating the space, for example with a lattice, one would approximate an appropriate class the functions defined on it. For this purpose, the measurement unit for such functions is a wavelet, associated for instance to the characteristic function of an interval. In this case the isomorphism between the Haar decomposition of the observables on that space and the amorphic space of square integrable functions [Krantz]: H, (R n ) = L 2 (R n ) can be interpreted as a resolution of the space R n using the Z Z lattice (infinite graphs) extended by the scaling group Z. This can be intuitively interpreted as a zooming in and out procedure, to resolve more or less of the structure on that space represented by observables. Although it is an isomorphism of Hilbert spaces, the left hand side is much reacher: it is graded, i.e. it is a differential graded Hilbert space, and the grading plays the role of a scale of details. For example the MRA of R corresponds to a binary representation of the continuum line. Altough such a MRA is not canonic and depends on a choice, its equivalence class is in some sense unique, the same way homological algebra resolutions depend on a choice but are homotopic via a canonical homotopy.
4 FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS 2.2. Groups and resolutions. As a second example, groups can be presented by generators and relations. As a first approximation a group G can be approximated by the free group generated by a chosen set of generators: F(S) G, G =< S >. As a second approximation, one can zoom in to see the departure of generators from being free, i. e. take into account the relations between generators: N F(S) F(S)/N G. It turns out that this time the approximation is exact, i.e. it is an isomorphism, since any subgroup of a free group is free: Resolution : 0 N F(S) F(S)/N Object : 0 0 0 G 0. This is a resolution of the object G, and the top-down morphism of complexes is a graded quasi-isomorphism lifting an isomorphism. It is clear that the resolution introduces a hierarchy of the details of the object s structure. If the object is a system, then it is resolved into subsystems, which in turn are systems resolved into subsystems etc. Although the presentation is not unique, having such a presentation of the object is usually much more useful when studying the object, and in applications. The generators and relations become a basis in the free case, when there are no relations. 2.3. QFT: perturbations or resolutions. The same ideas apply to the other good examples: Feynman graphs, configuration spaces, Feynman Path integrals and QFT [I-pQFT]. The algebraic structure this time is that of a dg-algebra with duality. It can originate from a DGLA with a Rota-Baxter operator, which enables an involution (like a Hilbert transform), Birkhoff decomposition etc. (more details in 5). It is custom to interprete a Feynman expansion as a perturbative approach. This is due to the origin, as coming from Gaussian integrals via Wick Theorem. The PI claims that their crucial role is that of providing a resolution of the spacetime used to build configuration space integrals, as the proposed research aims to proves. The claims and conjectures in this GP are based on the PI extensive published (see bibliography) and unpublished research on this subject (see past GP available at [VI-GP]. 2.4. Why using resolutions? We already noted the benefit of Haar MRA and of presentations of groups by generators and relations 2.2. The physicists practice shows the benefit of having bases, in spaces of functions or in the space itself, e.g. Feynman graphs. In the case of QFT and CFT, this solves the background independence problem. In general, as stated in [GM], one does not need a space; all we need is a resolution of the would be space. 0
FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS 5 The modern homological algebra demonstrated by now this aspect, through the development of the theory of derived categories and derived functors; and after all, as Jim Stasheff put it, quantum physics is cohomological physics. Therefore we should expect that QFT, CFT, String Theory (ST) and topological quantum field theory (TQFT) are derived functors in some sense; or perhaps should be designed as such. 3. Formal Pointed Manifolds The structure used by Kontsevich to prove the existence of a Poisson deformation is that of a formal manifold. The expansions involved, series indexed by Kontsevich- Feynman graphs, are formally similar to Taylor series of functions of manifolds, based on a fixed point. The PI pointed out that the Feynman perturbation series are interpreted as perturbations due to their traditional way of being derived, from Gaussian integrals via Wick theorem. The resulting formalism of Feynman graphs is not perturbative, more that a homological algebra resolution is a perturbation of the corresponding object, or to be specific, more that a group is a perturbation of a free group by relations; the group, an extension of the free group by the free subgroup of relations, is a deformation of the free group, if we look at the correspondent universal enveloping algebra as a deformation of the exterior algebra (to involve Hopf algebras, as in the theory of quantum groups). Kontsevich s graph series is such an example of resolution, although it can be derived as a perturbation as in [CF]. It is a good example since it does not require renormalization, due to the compactness (finite volume) of the hyperbolic space used (the Poincare disk). Another source for the fact that Feynman graphs provide resolutions in the sense of homological algebra, is the work of the PI on cohomology of Feynman graphs [I- CFG]. In this important article relating the work on Kontsevich on graph homology and that of Kreimer on Hopf algebra renormalization, it was proved that the resulting differential graded coalgebra of graphs is a cobar construction, providing the cohomology of the Hochschild DGLA. Feynman graphs may be though of as more models more complicated than the usual simplicial models, which correspond to linear trees, and which are used in simplicial (co)homology. The main point is that one does not need a manifold as a space from which to derive an algebra of observables, vector fields and differential operators (Hochschild DGLA), but it is enough to have a finite dimensional algebra, like de Rham cohomology, or like the exterior DGLA of a Poisson manifold with Neijenhuis bracket, to obtain such a cohomology. The general idea can be read from Kontsevich s result. The DGLA of polyvectorfields represents the infinitesimal model of the manifold, while the DGLA of differential operators represents the non-linear model of the manifold. Then the Kontsevich quasi-isomorphism U G (ω) : T poly D poly
6 FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS is such a resolution, where ω is a closed form needed to construct the coefficients B Γ of U (here Γ G is a Kontsevich graph - see also [DQ]. The PI and D. Fiorenza proved the result of Kontsevich in a well-rounded form [FI]. There is an L-infinity morphism U : G Hom CE (T poly, D poly ) from the universal DGLA of graphs (or viewed as a dg-coalgebra) with Kontsevich graph homology differential and Kreimer s coproduct generalizing Gerstenhaber s composition from Hochschild DGLA, into the Eilenberg-Chevalley DGLA of the Poisson algebra of observables of R n. The universality of the DGLA of graphs and the arbitrariness of the Poisson manifold suggested the PI the following reasonable conjecture. Conjecture 3.1. There is a universal canonical natural transformation of functors G : T poly D poly, which by Kontsevich s Formality Theorem is a quasi-isomorphism. The choice of a closed form (propagator) needed to establish the actual quasi-isomorphism is a clear indication of a Yoneda Lemma corresponding to the representability of a functor, with the corresponding natural transformation. At this point a natural guess occurs: Conjecture 3.2. The Feynman graph cohomology of the Poisson algebra is representable, and represented by the de Rham cohomology functor. In this sense, we claim that Conjecture 3.3. A dg-coalgebra of Feynman graphs provide a resolution of Poisson manifolds. 3.0.1. Applications. The targeted application is to forget manifolds, since all we need is a resolution of the object [GM], Introduction). In other words, we move away from the category of manifolds to the corresponding derived category, and finally being able to formulate QFTs as derived functors, towards an intrinsic theory of representations of geometric categories like cobordism categories, Riemann surfaces (Segal PROP [I-FP]) and Feynman graphs (PROP). The ideas explaining the framework generalizing the theories mentioned above are presented in the PI s work on Feynman Processes [I-FP,I-FL]. Briefly, Feynman Processes are functorial representations of Feynman categories. Feynman categories model the causal structure of quantum phenomena, in accord with Feynman s interpretation of quantum phenomena as quantum computations. Additional details regarding the reformulation of quantum physics as a Quantum Information Dynamics are contained in the PI s books on the Digital World Theory (Vol.1 and vol.2 Q++ and a Non-standard Model ).
FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS 7 4. Formal Manifolds and Path Integrals As early as 1998, after introducing the concept of pointed formal manifolds, Kontsevich started to look for a global concept (Private communication to PI by Y. Soibelman). This is similar to looking for a global concept of function, starting from a Taylor series say at 0. In complex analysis this is achieved by analytic continuation. A way to easily understand this is to consider the case of the multi-valued logarithm. To define the logarithm as the inverse of the exponential e z : C C, one would integrate the closed form dz/z. The singularity at the origin prevents integrability, due to the non-simply connectedness of the base space (think of it as a moduli space). On the other hand, locally we have a nice theory of analytic expansion of logarithm. The solution is to accept the multi-valued logarithm, and build the Riemann surface as a path integral. We claim that Riemann surfaces are the first good example of a general Path Integral, beyond the initial example of Feynman Path Integral from QFT. The points of the Riemann surface appear as a universal branched cover of the polynomial function, interpreted as a moduli space. 4.1. Feynman Path Integrals. In general FPI appear as a universal branching cover of a groupoid π 1. The best example is that of the fundamental groupoid, like the one corresponding to the above example. Then the FPI is a functor (action functor) defined on the groupoid. The problem is to represent such a functor as a path integral. Again the guiding example is that of Riemann surfaces and voltage graphs [Stahl]. Voltage graphs are principle bundles over graphs (topologically 1-CW-complexes, but it is better to view them as discrete objects, quivers, i.e. finite concrete categories). Principle bundles with discrete structure group are also called local systems, and are equivalent to vector bundles with flat connections. Now think of the FPI on manifolds as a gauge calculus of flat connections. The algebraic theory of Path Integrals restates the gauge calculus (in need of reductions and tricks, e.g. Fadeev-Popov technique and BRST-quantization or BVformalism) into a theory of moduli spaces in the sense of Grothendieck, i.e. a theory of groupoids and their representations. 4.2. Applications. For the purpose of implementing Quantum Information Dynamics, the group of coefficients of generalized voltage graphs is SU 2. The groupoids are in fact graphs or similar objects (Riemann surfaces), the objects of a Feynman Category. To include hierarchy of structure, the Hom s are dg-coalgebras, with the usual insertion and elimination of subgraphs as operations. Fixing a graph and a Feynman functor F : Γ SU 2 models the flow of qubits on the graph. Conjecture 4.1. Representing a Feynman functor defines an action and a universal branched cover, the formal manifold of Feynman paths.
8 FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS Again the design of the theory (definitions and theorems) towards the above conjecture, will parallel the theory of Riemann surfaces, periods etc., while following the intuitive interpretation of Feynman integrals. 4.3. Relation with Renormalization. The definition of (global) formal manifolds in terms of path integrals is related with Connes-Kreimer work on renormalization, via Rota-Baxter algebras. In a similar way with the gluing of Riemann surfaces (branched covers), from presentations, formal manifolds should be obtainable by a gluing operation. At this stage we only guess that Rota-Baxter algebras will play a crucial role. Connes-Kreimer explained in [CK1] that the Birkhoff decomposition which results from fixing a Rota-Baxter operator can be pictured as the gluing of a line bundle from two halfs. The PI has extensive notes and partial results justifying the following claims. Quantization is a doubling process, leading to a J-structure (non-commutative inversion), related to Jordan algebras, inversion geometry (projective spaces) etc. For example deformation quantization can be done via a bialgebra structure, quantum groups can be obtained via a Drinfeld double (Manin triple) etc. 5. Further developments The main point is that quantization is a categorification process, transforming the pointwise picture of physics with its initial value problems standard, appropriate for a deterministic geometric world, into a categorical picture of Feynman Processes. In such a world, there are many paths joining two points (input and output). Categorification is explained by the PI in [I-cat]. It is the inverse operation to localization, which collapses all objects, and in some sense leads to the commutative world of fractions : D(s t) = ts 1 = s 1 t. It is not the place to go into further details here. The idea is that the quantum/noncommutative world is modeled by the mathematics of Hopf algebras, while the classical physics/commutative world is modeled by the mathematics of commutative fields, tangent spaces etc. Hopf algebras which include the idea of a J-structure, the antipode, do not have enough algebraic structure for a CPT-physics. The additional structure needed is a non-commutative (twisted) duality, like a Hodge duality, implemented by a reflection: the Hilbert transform (*-involution) associated to a Rota-Baxter operator. Conjecture 5.1. Graded Rota-Baxter algebras generalize Hodge theory of commutative Hopf algebras. The PI proved in [I-LTDT] that deformation theory is a higher order Lie Theory, and that The Huebschmann-Stasheff universal solution of Maurer-Cartan equation is the fixed point of an h-adic contraction, corresponding the a contraction of the DGLA onto its homology which plays the role of the Hodge dual differential [I-LTDT]. This prompts for the following conjecture.
FROM KONTSEVICH AND CONNES-KREIMER TO A THEORY OF PATH INTEGRALS 9 Conjecture 5.2. Renormalization via Birkhoff decomposition associated to a Rota- Baxter algebra structure, is a non-commutative Hodge structure on the non-commutative Feynman-de Rham Path Integral complex. To emphasize the areas vital for a deeper understanding of quantum physics and quantum mathematics, we suggest an intuitive scale of structures : field theory with Galois theory, evolved into Riemann surfaces theory and CFT, which leads to a Theory of Path Integrals in the context of Rota-Baxter graded algebras, as a non-commutative Hodge Theory for Hopf algebras. 6. Acknowledgments The PI has extensive marginal notes and unpublished manuscripts supporting the above claims and the above proposed directions of research, and addressing the issues at a concrete level. This evidence together with the past research on this subject, ensure that the goals will be achieved in the time proposed.