Regensburg Conference 2014
September 29 - October 2
Quantum Mathematical Physics
A Bridge between Mathematics and Physics
 Christian Bär (Potsdam): Characteristic Cauchy problem for wave equations on manifolds The characteristic Cauchy problem for linear wave equations consists of imposing initial values for the solution on a characteristic Cauchy hypersurface instead of initial values for the function and its normal derivative on a spacelike Cauchy hypersurface. We show that this problem is well posed on globally hyperbolic Lorentzian manifolds under suitable assumptions. This is joint work with Roger Tagne Wafo and it generalizes classical results by Hörmander. Siegfried Bethke (München): Habemus Higgsum - the story of discovering the Higgs Boson The recent discovery of a new Boson at the Large Hadron Collider sets new milestones for uncovering the basic constituents and forces that form the universe. Being the first quantum state with zero spin in the list of fundamental particles, it closely resembles the basic features expected for the long-sought Higgs Boson of the Standard Model of particle physics. The stories of its quest, its discovery and its future implications currently boost imagination - not only in the world of particle physics - and will be told in this presentation. Gerhard Börner (Garching): Cosmic puzzles-dark matter and dark energy Astronomical observations over the last decade have collected substantial evidence for a "dark side" of the universe: It seems that the baryonic matter known to us amounts to only 5 percent of the cosmic energy density, while 95 percent consist of as yet unknown components. The dynamics of galaxies and clusters of galaxies, as well as X-ray and gravitational lensing observations reveal the existence of nonluminous, nonbaryonic dark matter accounting for about 27 percent of the cosmic substrate. Precise measurements of the cosmic expansion have led to the surprising result of an accelerating expansion. A cosmological constant, or a mysterious dark energy with a contribution of 68 percent must be responsible for that. The detailed analysis of the anisotropies of the cosmic microwave background arrives independently at the same conclusion. These basic observations will be described in the talk, as well as some of the experimental and theoretical attempts to shed light on dark matter and dark energy. Claudio Dappiaggi (Pavia): On the construction of Hadamard states from null infinity In the algebraic approach to quantum field theory on curved backgrounds, there exists a distinguished class of quantum states for free field theories, called Hadamard states. These are of particular relevance since they yield finite quantum fluctuations of all observables and they can be used to implement interactions at a pertubative level. In this talk we outline a procedure to construct explicitly these states on a large class of globally hyperbolic manifolds, which are asymptotically flat at null infinity. The method calls for identifying via an injective map the algebra of observables of any free field theory with a subalgebra of a suitable counterpart built on the conformal boundary of the underlying manifold. Each state for the boundary theory identifies thus via pull-back a counterpart for the physical theory under investigation. We show that there exists a canonical choice for the boundary state which guarantees that the bulk counterpart is Hadamard and, moreover, invariant under the action of all isometries. We discuss in detail the case of linearized gravity for which a previously unknown obstruction in using this method has been recently unveiled. Dirk Deckert (Davis): Time evolution of the Dirac sea subject to an external field for initial data on Cauchy surfaces It is well-known that on a fixed fermionic Fock space a rigorous construction of the second-quantized time evolution of the Dirac sea subject to an external field is only possible if and only if the space-like components of the field equal zero. The basic obstacle is that these components act quite dramatically on the Dirac sea giving rise to excessive virtual pair creation that does not allow for a representation in Fock space. Fierz and Scharf sketched a way out of this dilemma by proposing to regard, instead of one fixed Fock space, a time evolution on time-varying Fock spaces. For such a construction it is crucial to identify polarizations w.r.t. which admissible Fock spaces can be found at each time instant. On equal time surfaces such a construction has been carried out by Mickelsson and by D., Dürr, Merkl and Schottenloher, where the latter work also identifies the degrees of freedom involved. However, the corresponding polarization classes were found, rather opaquely, with the help of perturbation theory. I will report on a recent joint work with Franz Merkl in which this construction was extended to general Cauchy surfaces in a way that also allows for a much more geometrical picture. Michael Dütsch (Göttingen): Massive vector bosons: Is the geometrical interpretation as a spontaneosuly broken gauge theory possible at all scales? A model for massive vector bosons can be interpreted as a spontaneously broken gauge theory if and only if the coupling parameters (i.e.~the prefactors of the various interaction terms) are prescribed functions of the coupling constant(s) and the masses. Since, under the renormalization group (RG) flow, different interaction terms get different loop-corrections, it is uncertain, whether these functions remain fixed under this flow. We investigate this question for the \$U(1)\$-Higgs-model to 1-loop level. We also study whether BRST-invariance of the Lagrangian is maintained -- a stronger property. We proceed as follows: we generally prove that physical consistency (PC) is stable under the RG-flow; PC is the property that the \$S\$-matrix in the adiabatic limit commutes with the BRST-charge for the asymptotic free fields. To 1-loop level we work out the restrictions on the running coupling parameters coming from stability of PC. The geometrical interpretation needs precisely one additional relation among these parameters -- whether it holds depends on the renormalization scheme. Bertfried Fauser (Cookeville): From software testing to differential geometry -- Computer science methods applied to physics Differential geometry on manifolds is the mathematics underlying Einstein's gravitational theory. Recent progress was made in establishing quantum field theories on backgrounds like globally hyperbolic space times. Hence any new way to understand and possibly generalize this framework may be helpful to- wards a unification of the quantum and gravity. In this talk I will explain how a computer science approach, based on extremely general methods using test the- ories, can be used reconstruct a large part of differential calculus on manifolds. To do so, we need to apply category theory and model dynamical systems as coalgebras and model measurements, or tests, as algebras (of functors). We then specialize a rather general result of Pavlovic-Mislov-Worrell (PMW) about test- ing to the case where the modelling is done in categories of manifolds. On the technical side, we can reconstruct tangent and cotangent functors and many of their natural properties, showing explicitly how these functors, and large parts of differential geometry, emerges solely from local tangent testing. The rather special case of manifolds can be generalized along the categorical theorem of PMW, which applies to a any suitable categorical setting. This in- cludes Turing machines, hence allows to study computational and complexity aspects of theories. Using di erent categories we may gain insight how ordinary differential geometry might be generalized. I will discuss cases of interest to physicists: - A monoidal categorical setting, allowing to include generically the tensor products of higher tangents. - A serious approach to duality on tangent spaces, including the proper definition of (pseudo) metric tensors, in the sense of Lawvere. - Both items above call for an enriched category setting, to capture the monoidal structure of tensoring vector fields. - We can drop commutativity of the underlying spaces. Even more impor- tant may be that the coalgebras and algebras have not to be modelled in the same category, eg tangent structures and cotangent structures can, and possibly should, live in different categories. Keywords: Dual adjunction, algebra, coalgebra, test theories, differential calculus, manifolds, tangent functors, tangent functor monad. This talk is based on joint work with and ideas of Dusko Pavlovic, University of Hawaii, and RHUL. Chris Fewster (York): The general theory of quantum field theory in curved spacetimes Axiomatic and algebraic quantum field theory have produced many insights applicable to general theories in flat spacetimes (e.g., spin-statistics, PCT, superselection theory and so on). Until recently, most rigorous work on quantum field theory in curved spacetimes concerned specific models and general results were scarce. This has changed somewhat, following the development of locally covariant QFT by Brunetti, Fredenhagen and Verch. In this talk, I review some of what can be said in general about QFT in CST using this framework. Felix Finster (Regensburg): Causal fermion systems as an approach to quantum theory The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting space-time manifold, the general concept is to derive space-time as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. The dynamics of the system is described by the causal action principle. The aim of the talk is to give a simple introduction, with an emphasis on conceptual issues. We begin with Dirac spinors in Minkowski space and explain how to formulate the system as a causal fermion system. An example on a space-time lattice illustrates that causal fermion systems also allow for the description of discrete space-times. As an example in curved space-time, we consider spinors on a globally hyperbolic space-time. These examples lead us to the general definition of a causal fermion system. The causal action principle is introduced. We outline how for a given minimizer of the causal action, one has notions of causality, connection and curvature, which generalize the classical notions and give rise to a proposal for a "quantum geometry". In the last part of the talk, we outline how quantum field theory can be described in this framework and discuss the relation to other approaches. Christian Fleischhack (Paderborn): Configuration spaces: Loop quantization vs. symmetry reduction In loop-like quantization, the quantum configuration space is given by the spectrum of a \$C^\ast\$-algebra, consisting of appropriate bounded functions on the respective classical configuration space. Originally, it had been taken for granted that the classical-level embedding of symmetric configurations into the full space can be continued to the quantum level. In standard loop quantum cosmology (LQC), this turned out to be wrong. In the talk, we first present an alternative definition of LQC configuration spaces, avoiding this problem. Then we present how to implements symmetries directly on quantum level. Moreover, we show that it does matter whether one first imposes symmetries and then quantizes, or whether one does it the other way round. Finally, we discuss measures on the quantum configuration spaces. Domenico Giulini (Hannover): Models of self-gravitating quantum-mechanical systems In the classical realm, Einstein's equivalence principle enforces matter-gravity couplings to follow a universal pattern. Already for non-relativistic quantum-mechanical systems, however, this pattern is not as obvious as one might think, albeit for *weak* and *external* gravitational fields the interaction is easy to guess. This guess has, e.g., been tested for Quantum systems in the gravitational field of the Earth. In the general case, what remains unclear is how quantum-matter (i.e. matter in non-classical states) sources gravitational fields. I will discuss a particular interpretation of what it means to treat gravitation "semi-classically" and show how it gives rise to non-linear Schroedinger evolutions due to gravitational self-interaction. Prospects for experimental tests will be mentioned. If time permits, I will end by comparing this way of dealing with gravitation to the way electromagnetic self interactions are usually considered. Jose Gracia-Bondia (Zaragoza): On the recursive evaluation of Feynman amplitudes in x-space: Differential versus Epstein—Glaser methods Causal perturbation theory in configuration space is reputed more for logical clarity and rigor than for computational convenience. It is however a matter of principle to show how it is able to deal with multi-loop graphs. For the massless quartic model of Euclidean scalar field theory, widely used in the theory of critical exponents, I demonstrate a simple recursive method of calculation, superseding the causal factorization property by Nikolov, Stora and Todorov. In particular the 1PI four-point function for the model is computed order by order. This calls for a convolution-like machine in distribution theory, to deal with the mild infrared problems posed by graphs with internal vertices. Michael Gransee (Leipzig): Local thermal equilibrium states in QFT and a generalization of the KMS condition It is well-known that thermal equilibrium states in quantum statistical mechanics and quantum field theory can be mathematically rigourously desricbed by means of the KMS-condition. This condition is based on certain analytical and periodicity properties of correlation functions. On the other hand, the characterization of non-equilibrium states which locally still have thermal properties constitutes a challenge in quantum field theory. Analyzing the analyticity properties of KMS states, a proposal for a generalized KMS condition is made. The relations of that condition to a related proposal by D. Buchholz, I. Ojiama and H.-J. Roos for characterizing local thermal equilibrium states in quantum field theory are investigated. Harald Grosse (Wien): A nontrivial four dimensional QFT Together with Raimar Wulkenhaar we showed that the quartic matrix model with an external matrix is exactly solvable in terms of the solution of a non-linear integral equation. The interacting scalar model on four dimensional Moyal space is of this type, and our solution leads to the construction of Schwinger functions. Taking a special limit leads to a QFT on ${4}^{}$, which satisfies growth property, covariance and symmetry and there is a numerical evidence for reflection positivity of the 2-point function for a certain range of the coupling constant. Christian Hainzl (Tübingen): Mathematical aspects of many-particle quantum systems Stephan Hollands (Leipzig): Dynamical vs. thermodynamical (in)stability of black objects in gravity Understanding the stability properties of black objects is both a very important, but also a very complex problem in general relativity and its higher dimensional generalizations. Based on the well-known dictionary between black objects and thermodynamics, it is natural to come up with criteria for the (in)stability of black objects akin to the standard criteria involving the "specific heat" in the context of phenomenological thermodynamics. A key question is what such notions have to do with notions of instability based on the existence of "growing modes" of the corresponding perturbed Einstein equations. In this talk, I review the "canonical energy method", which, as I argue, provides a beautiful and clear link between these different concepts of stability (and also a direct connection to the recently proposed approach via a "local Penrose inequality"). I outline some applications of the canonical energy method, such as (1) a proof of the Gubser-Mitra conjecture for black branes, and (2) a connection between the stability of rotating higher dimensional black holes and that of their associated near horizon geometries, thereby proving a recent conjecture of Durkee-Reall. Jürgen Jost (Leipzig): Mathematical aspects of variational problems from QFT I shall describe how the action functionals of supersymmetric quantum field theory can be converted into variational problems that have deep connections with the geometry of Riemannian manifolds and pose challenging new problems for geometric analysis. Enno Keßler (Leipzig): A superconformal action functional for super Riemann surfaces There is a generalization of the harmonic action fucntional on Riemann surfaces to supergeometry. This involves the definiton of Super Riemann Surfaces or superconformal structures. In this talk I am going to explain how to express superconformality in terms of metrics. It will turn out, that a superconformal transformation is a bit more than just the expected rescaling. Never the less the action is invariant under all superconformal transformations. Michael Kiessling (Rutgers): The Dirac equation and the Kerr-Newman spacetime Dirac's wave equation for a point electron in the electromagnetic Kerr-Newman spacetime is studied in the zero-gravity limit; here, ``zero-gravity'' means G-->0, where G is Newton's constant of universal gravitation. The zero-G limit eliminates the troublesome Cauchy horizon of the Kerr-Newman spacetime and also its physically problematic acausal region of closed timelike loops. While the gravitational features of the Kerr--Newman manifold vanish as well when G-->0, this limit retains the nontrivial topology associated with its ring singularity, and all its electromagnetic features. It is first shown that the formal Dirac Hamiltonian on a static spacelike slice of the maximal analytically extended zero-G Kerr-Newman spacetime is essentially self-adjoint for small coupling constant, and that the spectrum of its self-adjoint extension is symmetric about zero. It is next shown that this Dirac operator has a continuous spectrum with a gap about zero that contains a pure point spectrum. The pure point spectrum is associated with time- periodic normalizable solutions, representing bound states of Dirac's point electron in the electromagnetic field of the ring singularity of the zero-G Kerr-Newman spacetime. This is joint work with A. Shadi Tahvildar-Zadeh. Gandalf Lechner (Leipzig): The structure of the field algebra in non-commutative QFT and uniqueness of its KMS states In this talk, the structure of a typical field algebra of a non-commutative fully Poincare covariant quantum field theory model on Moyal Minkowski space is analyzed. The essential differences to both, the commutative version and also the non-covariant non-commutative version with fixed deformation parameter, are explained. These differences manifest itself in particular in a different structure of ideals, which result from non-trivial mixing between different deformation parameters. It is then explained how this phenomenon leads to uncountably many normalized KMS functionals at fixed temperature (whereas one has a unique such KMS functional in the commutative and non-covariant non-commutative case). However, there exists a unique normalized positive KMS state (at fixed temperature) which essentially decouples all deformation parameters. The connection of this result to thermal states of integrable models and crossed products of deformed C*-algebras is discussed. Frédéric Paugam (Paris): Categorical methods in quantum field theory We will present the approach to the geometry of quantum field theory used in the book “Towards the mathematics of quantum field theory” and explain its usefulness in the study of quantum gauge theories. Martin Reuter (Mainz): Quantum gravity, background independence and asymptotic safety We briefly review the various components of the Asymptotic Safety Program which aims at finding a non-perturbative infinite-cutoff limit of a regularized functional integral for a quantum field theory of gravity. It is explained why in the continuum formulation based on the Effective Average Action the key requirement of background independence leads to a "bi-metric" framework, and recent results on truncated RG flows of bi-metric actions are presented. As an application, a method which can be used to characterize and count physical states is described. Israel Sigal (Toronto): Asymptotic completeness of Rayleigh scattering Experiments on scattering of photons on atoms (Rayleigh scattering) and on free electrons (Compton scattering) led, in the beginning of 20th century, to our understanding of composition of matter and eventually to creation of quantum mechanics. The mathematical framework for describing these processes was developed soon after discovery of quantum mechanics and is given by the Schroedinger equation of the non-relativistic quantum electro-dynamics. Nevertheless, the mathematical theory of these physical phenomena has begun being constructed only recently. In this talk I will report on the recent works, jointly Jean-Francois Bony and Jeremy Faupin, on photon velocity bounds and on the proof of asymptotic completeness of Rayleigh scattering. Christoph Stephan (Potsdam): Noncommutative geometry in the LHC-era Noncommutative geometry (NCG) allows to unify the basic building blocks ofparticle physics, Yang-Mills-Higgs theory and General relativity, into a single geometrical framework. The resulting effective theory constrains the couplings of the Standard Model (SM) and reduces the number of degrees of freedom. After briefly introducing the basic ideas of NCG, I will present its predictions for the SM and the few known models beyond the SM based on a classification scheme for finite spectral triples. Most of these models, including the Standard Model, are now ruled out by LHC data. But interesting extensions of the SM which agree with the presumed Higgs mass, predict new particles (Fermions, Scalars and Bosons) and await further experimental data. Alexander Strohmaier (Loughborough): The quantization of the electromagnetic field and its relation to spectral geometry and topology I will show how the Gupta-Bleuler quantization scheme for the quantization of the electromagnetic field can be formulated on globally hyperbolic space-times without reference to any frequency splitting. I will then discuss how the construction of ground states on ultra-static space-times is related to certain cohomology theories and the topology of the Cauchy surface. (joint work with Felix Finster) Stefan Teufel (Tübingen): Dimensional reduction for the Laplacian I will review a number of related results on the dimensional reduction of wave equations containing the Laplacian in slightly different contexts. Dimensional reduction here means to approximate the Laplace-Beltrami operator on a high-dimensional space with „small“ directions by an effective operator on a lower-dimensional space without the „small“ directions. Starting from the problem of constraining a quantum particle to a submanifold of configuration space by steep potentials and the related problem of thin quantum wave guides, I will explain the adiabatic structure of the dimensional reduction problem. Then I discuss how the methods of adiabatic perturbation theory can be used to address the question in much more generality: the high-dimensional space is a fibre-bundle with compact fibres that are small with respect to a given Riemannian metric and the Laplacian is the connection Laplacian on some vector-bundle over this space. It turns out that the leading order terms in the effective operator depend on the energy scale on which the approximation should hold and that they are, in general, not given by the „naive“ guess for such a reduced Laplacian. This is based on joint work with Stefan Haag, Jonas Lampart and Jakob Wachsmuth. Roderich Tumulka (Rutgers): Novel type of Hamiltonians without ultraviolet divergence for quantum field theories In quantum field theories, the terms in the Hamiltonian governing particle creation and annihilation are usually ultraviolet (UV) divergent. The problem can be circumvented by either discretizing space or attributing a nonzero radius to the electron (or other particles). I describe a novel way of defining a Hamiltonian, to our knowledge not previously considered in the literature; these Hamiltonians are well defined, involve particle creation and annihilation, treat space as a continuum, and give radius zero to electrons. They are defined in the particle-position representation of Fock space by means of a new kind of boundary condition on the wave function, which we call an interior-boundary condition (IBC) because it relates values of the wave function on a boundary of configuration space to values in the interior. Here, the relevant configuration space is that of a variable number of particles, the relevant boundary consists of the collision configurations (i.e., those at which two or more particles meet), and the relevant interior point lies in a sector with fewer particles. I will describe results about Schrodinger and Dirac operators with IBCs. This is joint work with Stefan Teufel, Julian Schmidt, and Jonas Lampart. Rainer Verch (Leipzig): Linear hyperbolic PDEs with non-commutative time Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form (D + sW)f= 0 are studied, where D is a normal or prenormal hyperbolic differential operator on Minkowski spacetime, s is a coupling constant, and W is a regular integral operator with compactly supported kernel. In particular, W can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small s, the hyperbolic character of the PDE is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in s, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of W on the space of solutions of D. It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces. This is joint work with Gandalf Lechner (see arXiv:1307.1780). Stefan Waldmann (Würzburg): Recent developments in formal deformation quantization In my talk I will discuss some of the more recent developments in deformation quantization, mainly focusing on the representation theory and classification results: Star product algebras can be classified up to Morita equivalence including symmetries. Several applications to quantum physics will be outlined. If time permits, I will also indicate some more recent developments concerning analytical properties of star products and convergence of the formal series. Raimar Wulkenhaar (Münster): A nontrivial four dimensional QFT II Together with Harald Grosse we construct a quantum field theory on ${ℝ}^{4}$ as a particular limit of an exactly solvable quantum field theory model on a noncommutative geometry. The talk covers further aspects of this construction.