Recent developments in the formulation of algebraic quantum field theory led Fredenhagen et al. to promote to formulate quantum field theory as a functor from globally hyperbolic manifolds into star operator algebras. This approach should be compared with the notion of a Tannakian representation category as used for example in the work of Connes-Marcolli. I will give arguments that these two notions form a pair of adjoint functors (Tannaka-Krein duality). To exemplify this claim, we will study combinatorial representation theory and recall that an algebraic variety can be defined as a functor. This approach also provides the notion of an algebraic group scheme and implements the Hopf algebraic structures which finally pup up (dualized) in the quantum field theory. The notion of adjointness imposes restrictions to the source and target categories, which may help to identify (non-commutative) geometries which can carry a quantum field theory and so complements a functorial approach to quantum gravity as proposed recently by Verch and Paschke.
The idea of mathematical beauty in the fundamental equations of nature has driven the unification of various aspects of physics over the last centuries. In this spirit Maxwell’s equations were born and more recently the electroweak unification in the standard model of elementary particle physics. Even with the spectacular success of quantum field theory in the standard model the conceptual difficulties with general relativity and thus understanding the gravitational force at the quantum level are still unsolved. How can these difficulties be overcome? This paper proposes a radically different approach on how to derive the fundamental equations of nature based on a top-down approach,
starting only with a single simple mathematical concept, the Cayley algebra. This compares to the traditional bottom-up approach: Moving from a set of physical model equations derived from experiment up to a more condensed set of equations with the aim to reach a unified view of nature.Following the top-down approach based on a simple field theory over the nonassociative Cayleynumbers leads to mathematical structures that allow a surprisingly tight embedding of particle and interaction concepts found in the standard model of elementary particle physics. But even more important the nonassociative quantum field theory suggests extensions of the standard model to include gravity and puts the problem of quark confinement in a completely new light where it can be seen as pure consequence of the nonassociativity of the theory.
With special variants of quaternionic automorphic forms we set up a closed formula for the fundamental solution of the Dirac equation on some conformally flat spin manifolds in R4, for example on the 4-torus, the Hopf manifold S1 x S3 and on k-handled 4-tori. These in turn can be interpreted as self-dual solutions to the Yang-Mills equation for SU(2)-instantons on thesemanifolds. With a special quaternionic argument principle one obtains an explicit link between these solutions and the Chern index of the SU(2)-principal bundles over these manifolds.
Cosmology is the study of the structure of our universe on the largest scales which have been observed. The dominant physical force involved is gravity and the appropriate theoretical description is given by general relativity. The basic equations are the Einstein equations coupled to equations describing matter. These form a system of PDE whose character is essentially hyperbolic. Of particular interest from both the physical and mathematical points of view is the spatially homogeneous case where the equations reduce to a system of ODE.
There are various motivations, such as the recent discovery that the cosmological expansion is accelerated, which have led to a proliferation of exotic matter models which go under the name of dark energy. I will explain what is known about these mathematically and sketch the mechanism by which they can lead to homogeneity of the universe. The standard mechanism used to explain the generation of inhomogeneities and hence structures such as galaxies is the Jeans instability and I will present an attempt to put this on a better mathematical footing.
Knot invariants play a central role at various places in mathematics and physics. Furthermore, the classification of knots is still an open mathematical problem. Therefore new approaches to knot invariants are always desirable. We develop knot invariants from the character Hopf algebras of centralizer subgroups of the $GL(n)$ groups in the stable limit $n\rightarrow \infty$. This invariants are induced by plethystic branchings. Usually one exploits the homomorphisms $G\righarrow U(SL(2))$ or $G\rightarrow U_q(SL(2))$ to derive e.g. the Jones polynomial. Our method draws invariants directly from the character rings of infinitely many centralizer subgroups of $GL(\infty)$ and does not need mandatorily a $q$-deformation. We show how state models can be derived from our method and how it related for example to the Kauffman bracket and Jones polynomial. Joint work with: Peter D. Jarvis, Hobart, and Ronald C. King, Southampton.
Let G be a compact Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy be the multiplication operator. The associated fibration m:GxG->G is a Riemannian submersion with totally geodesic fibers. The associated spectral geometry of the submersion is studied. Eigen functions on G pull back to eigen funtions on GxG with the same eigenvalue. Eigen p-forms for p>0 on the base pull back to eigen p-forms on GxG with finite Fourier series; there are examples where the number of eigenvalues in the Fourier series of the pull back on GxG is arbitrarily large. If w is an harmonic p-form on the base, necessary and sufficient conditions are given to ensure the pull back of w is harmonic on GxG.
This is joint work with Corey Dunn (Cal State San Bernadino USA) and JeongHyeong Park (SungKyungKwan University Korea).
In contrast to the quadratic Yang-Mills action, Einstein's theory of gravity is based on a functional which is linear in the curvature. This is known to have far reaching consequences. Many attempts have been made over the last decades to recast Einstein's theory of gravity into a Yang-Mills like form. In the following we up-side-down this point of view and show how to express Yang-Mills gauge theories linearly in the curvature of Dirac bundles and thereby recast gauge theories into a "gravity like form". For this we discuss a canonical action functional within the geometrical setting of Dirac bundles in the light of a new global Lichnerowicz like formula.
Groebner bases are useful in practical problems when solving systems of polynomial equations. Applications of these bases to finding parallel lines (envelopes) and parallel surfaces at the given offset will be shown. As examples, parallel lines to nondegenerate quadrics, Bezier cubics and Bezier surfaces will be presented. Singularities of the parallel lines will be discussed.
I will outline the most recent developments in the quest to show a spectral triple (A,H,D) satisfying certain additional conditions is in fact the Dirac triple of a spin^c manifold. This is joint work with Prof. Joe Varilly.
(Joint work with Jesper Grimstrup)
In this talk I will propose an intersection of noncommutative geometry and loop quantum gravity. Alain Connes' noncommutative geometry provides a framework in which the standard model of particle physics coupled to general relativity is formulated as a unified, gravitational theory. However, to this day no quantization procedure compatible with this framework is known. In this talk I will consider the noncommutative algebra of holonomy loops on a functional space of certain spin-connections. Construction of a spectral triple is outlined and ideas on interpretation and classical limit will be presented.
The concept of an R-Matrix has its origins in the theory of integrable systems and encountered an adequate mathematical interpretation in the theory of Hopf algebras and quantum groups where it plays a fundamental role in stimulating decisively the development of this area. Here we present recent results about an extension of this concept, the so-called dynamical R-matrix discussing it in the context of one of the most traditional classes of integrable models in mechanics - the Calogero models.
We give a comprehensive review of various methods to define currents and the energy-momentum tensor in classical field theory, with emphasis on a geometric point of view. The necessity of "improving" the expressions provided by the canonical Noether procedure is addressed and given an adequate geometric framework. The main new ingredient is the explicit formulation of a principle of "ultralocality" with respect to the symmetry generators, which is shown to fix the ambiguity inherent in the procedure of improvement and guide it towards a unique answer: when combined with the appropriate splitting of the fields into sectors, it leads to the well-known expressions for the current as the variational derivative of the matter field Lagrangian with respect to the gauge field and for the energy-momentum tensor as the variational derivative of the matter field Lagrangian with respect to the metric tensor. In the second case, the procedure is shown to work even when the matter field Lagrangian depends explicitly on the curvature, thus establishing the correct relation between scale invariance, in the form of local Weyl invariance "on shell", and tracelessness of the energy-momentum tensor, required for a consistent definition of the concept of a conformal field theory.
A Rota-Baxter algebra is an algebra with a linear endomorphism P that satisfies the relation P(x)P(y)=P(xP(y))+P(P(x)y)+lambda P(xy) for all x, y. Here lambda is a fixed constant. After a brief summary of its basic properties and main applications, we will focus on the construction of free Rota-Baxter algebras. In the commutative case, the construction is related to the shuffle product and quasi-shuffle product with applications to multiple zeta values and symmetric functions. In the noncommutative case, it is related to planar rooted trees with decorations. As an application, we study the adjoint functor of the functor from Rota-Baxter algebras to dendriform algebras.
Linear models of change in stochastic systems often entail assumptions of independence amongst subsystems. In an appropriate tensor setting, the analysis of this independence is the same as that of the analysis of the entanglement of state vectors in quantum systems, and is at base an application of classical invariant theory.
The talk will review continuous time Markov models on trees as commonly used in phylogenetics, and will explain how various entanglement measures (some of which can be lifted from the physics literature on the analysis of multi-qubit systems) are informative about phylogenetic relationships. Specifically they are relevant to distance-based tree reconstruction methods, and have been shown (under simulation) to improve some algorithms.
The talk will also introduce the major group theoretical tool applied to this problem, namely the theory of plethysms, for the enumeration of multilinear invariants. In the biological context the appropriate invariance group is the (finite dimensional) affine group.
(Joint work with Jeremy Sumner, School of Mathematics and Physics, University of Tasmania)
I report on recent work which uses passivity, an expression of the Second Law of Thermodynamics, as a selection criterion for reference (vacuum) states of quantum field theories over (certain) curved space-times. The representations associated with such states manifest properties which are physically desirable. Among others, causality and (weak) locality properties are predicted. Though the quantum fields in such representations are local in the selected state, they need not be local on the entire representation space. The curious properties of such nonlocal but weakly local models will be discussed from the point of view of a new approach to constructing quantum field models.
The canonical and microcanonical ensemble approach to BEC is very closely related to general partitioning problems of number theory. First this relationship is explained in detail. Then techniques from analytical number theory are used to evaluate ground state occupation numbers and condensate fluctuations of Bose gases in external traps.
Various notions of symmetry are important in mathematical physics. Sometimes this symmetry arises from a transitive isometric group action. But there are other notions of interest. One says a manifold is curvature homogeneous if there is an isometry between the tangent spaces of any two points of the manifolds preserving the curvature tensor. We exhibit Lorentz manifolds which are curvature homogeneous but not homogeneous and discuss their geometric properties. As pseudo-Riemannian manifolds which have dimension greater than $4$ and signatures other than Riemannian or Lorentzian are important in many physical applications (Kaluza-Klein gravity and brane world cosmology, we shall also discuss higher signature examples with interesting geometrical properties.
The concept of typical states is central to classical and quantum information theory. This will be explained in the context of the Theorem of Shannon-McMillan-Briman and the Theorem of Sanov.
We use newly developed techniques from Hopf algebra theory to provide (formal) group characters for subgroups of $GL(n)$ fixing a tensor $T_\pi$ of Young symmetry type $\{\pi\}$. New character formulae, branching rules, and generalized Newell-Littlewood theorems follow. The groups $H_{\{13\}}(n)$, being non classical in general, which stabilize an antisymmetric tensor of degree 3 will serve as an example showing furthermore the development of modification rules.
We present a framework for constructing gauge theories where the structural Lie algebra is generalized to a Lie algebroid (and, in the integrable case, the structural Lie group to the corresponding Lie groupoid). Besides examples such as the Poisson sigma model, we will present a generalization of ordinary YM theories in arbitrary spacetime dimension. From the point of view physics, the resulting theory turns out to effectively describe a bundle of ordinary Yang-Mills theories, defined over a finite dimensional moduli space. The gluing of different fibers is dictated by the originally chosen Lie algebroid, such as the gluing of fibers over a base manifold in a fiber bundle is prescribed by the global construction of the total space---with the exception that there is no typical fiber: critical orbits in the Lie algebroid give rise to points of symmetry enhancement in the moduli space, i.e. to a YM theory defined over the cricitcal point in the moduli space which has a higher dimensional structure group.
We study the coalgebraic counterparts of addition and multiplication. This allows to construct two Hopf convolutions, also called Hopf gebras, for both addition and multiplication. Neither of this convolutions is forming a Hopf algebra, however, the multiplicative convolution embodies the Dirichlet convolution of number theoretic functions. There is an opportunity to introduce a new coalgebra structure, called renormalized, such that a nice Hopf algebra structure emerges in such a way that the primitive elements are identical. A subtraction scheme, which might be related to renormalization in quantum field theory, allows to use the nice algebra for computations while actually dealing with the original Hopf convolution. There is a deeper relation of addition and multiplication which relies on n-categories. We give as examples the normal ordering in quantum mechanics and its relation to Stirling numbers and Baxter operators as also the construction of the renormalization coproduct employed in renormalization of quantum fields. An outlook will show how quantum field theoretic methods may be used in number theory.
We study the Lip-normed C*-algebras introduced by M. Rieffel, showing that the family of equivalence classes up to isomorphism preserving the Lip-seminorm is not complete w.r.t. the complete quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, necessary and sufficient conditions are given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebras. Such conditions can be considered as a generalisation of the f-Leibnitz conditions considered by Li and Kerr. Furthermore, it is shown that our conditions are not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited from the approximating C*-algebras.
In this talk I will consider the flat phase space $C^n$ with its usual canonical Poisson bracket. Using the Kähler structure one can define a normal ordered (= Wick ordered) quantization for polynomials on $C^n$ and use this to define a corresponding (formal in $\hbar$) star product, the Wick star product. I will explain the crucial positivity properties of this star product which allow to show that any classical state (positive functional) is also a state in the quantum theory without quantum corrections. This (very strong) positivity property is used to define a convergence scheme for the formal power series in $\hbar$. The result will be a Frechet *-algebra of real-analytic functions on which the Wick star product converges and which contains the polynomials as a dense subspace.
A generalization of the concept of a local counterterm suitable for field theories on noncommutative spacetimes leads to a modified definition of well-defined products of fields, called quasiplanar Wick products. The drastic physical consequences of this definition are mentioned.
The Hopf algebra structure of quantum field theory was investigated recently by several groups. Using the functional approach to quantum field theory, the effect of the interaction Hamiltonian can be calculated using the Hopf algebra of functional derivations. This enables us to obtain nonperturbative results for quantum field theory with initial correlations, which is important in solid-state and molecular physics.
In particular, the hierarchy of Green functions with initial correlations will be given explicitly.
We discuss a new general conjecture on attractors and soliton-type asymptotics of the solutions to nonlinear wave and Klein-Gordon Eqns, with a general Lie group $G$, in an infinite space. For the case of relativistic invariant equations, the conjecture reads as follows: every finite energy solution decays to a finite number of solitons combined with a dispersive wave. The asymptotics are inspired by the Bohr Quantum Transitions and de Broglie's Wave-Particle Duality.We explain the physical motivations and suggestions, list known results and describe our numerical experiments.
Let $A$ be an associative $k$-algebra and $B$ a bialgebra. A $B$-module algebra structure on $A$ is an action of $B$ on $A$ such that the comultiplication of $B$ is compatible with the multiplication of $A$, i.e.\ $b\circ\mu_a=\mu_a\circ\Delta_B(b)$. We discuss formal deformations of such structures, i.e.\ their behavior under the extension $k\rightarrow k\ha$. For $B'=B\ha$ an element $P\in B'\otimes B'$ defines a universal deformation formula $\Delta_h=P^{-1}\circ\Delta\circ P$ for the comultiplication on $B'$ and $\mu_h=\mu\circ P$ for the multiplication on $A'=A\ha$ if it is a solution of the compatibility condition $\Delta_1 P \cdot P_{12}=\Delta_2 P \cdot P_{23}$. For $B={\cal D}(H)$, the quantum double of the Hopf algebra $H$, the Quantum Yang-Baxter-Equation is a special case of this equation. Many special algebra deformations (e.g., quantizations) considered in the literature can be obtained this way, i.e.\ there is a (often hidden) $B$-module algebra structure and a universal deformation formula $P$ for them. The compatibility condition has the form of a Maurer-Cartan equation. It turns out that $\mu_h$ can be extended to a trivial deformation of the smash product $A \# B'$. Most of the equations arize as certain commutativity conditions of diagrams, that are best formulated in the language of categories. It is well known that the basic facts about quantum group theory or topological quantum field theory (TQFT) can be formulated in the language of monoidal categories. In this framework we will more generally discuss deformations of the structure of categories.
The talk will survey standard symmetric function theory, with emphasis on the underlying Hopf algebra(s). It will be shown how, in combination with Sweedler's cohomology and natural Laplace pairings, these are able to unify several structural aspects of symmetric functions and generalisations. Applications to group branching rules, combinatorics and quantum physics are suggested.
Joint work with B Fauser (MPI), math-ph/0308043
A surprising property of quantum fields is that their local energy densities need not be positive (and are in fact unbounded from below as a function of the quantum state). In principle, negative energy densities, especially when coupled to gravity, permit many pathological phenomena: it is therefore important to place constraints on their magnitude and duration. This talk will describe bounds on negative energy densities known as Quantum Energy Inequalities (QEIs) and some of their applications.
A variety of QEIs have been proved for free quantum fields. This talk will present the first examples of QEIs to be established for interacting quantum fields, namely for a class of two-dimensional conformal field theories (joint work with S Hollands). In addition, analogues of QEIs in quantum mechanics will be described (joint work with SP Eveson & R Verch).
The renormalisation of field theories on noncommutative $\mathbb{R}^4$ to all orders is still an open question. I propose to take the UV/IR-entanglement observed in one-loop Feynman graph calculations as a message to extend the free-field action by an oscillator potential. In a base where the $\phi^4$ $\star$-product interaction is realised as a matrix product, the free action can be diagonalised via Meixner polynomials. The resulting propagator shows an asymptotic and local behaviour which according to a power-counting theorem derived by renormalisation group techniques leads to only four relevant or marginal base interactions. Thus, the model is due to properties of the Meixner polynomials renormalisable to all orders by imposing normalisation conditions for the mass, the field amplitude, the coupling constant and the oscillator frequency. The oscillator potential leads to quantised momenta in agreement with recent data on the cosmic microwave background.
Let $Y$ be a cell complex with a single 0-cell, let $K$ be its Kan group, a free simplicial group whose combinatorial structure reflects the incidence structure of $Y$ and which is a model for the based loop space $\Omega Y$ of $Y$, and let $G$ be a Lie group. We will describe the construction of a weak $G$-equivariant homotopy equivalence from the geometric realization $|\textnormal{Hom}(K,G)|$ of the cosimplicial manifold $\textnormal{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to the space $\textnormal{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$ of $G$ where $G$ acts on $BG$ by conjugation. In this fashion, $|\textnormal{Hom}(K,G)|$ appears as a model for the space of based gauge equivalence classes of connections. Combined with an explicit purely finite dimensional construction of generators of the equivariant cohomology of the geometric realization of $\textnormal{Hom}(K,G)$ and hence of $\textnormal{Map}^o(Y,BG)$, this construction, when carried out for the special case where $Y$ underlies a smooth manifold, may be viewed as a rigorous approach to lattice gauge theory. Under these circumstances it yields, (i) when {$\textnormal{dim}(Y)=2$,} equivariant de Rham representatives of generators of the equivariant cohomology of certain moduli spaces and (ii) when {$\textnormal{dim}(Y)=3$,} equivariant cohomology generators including a rigorous combinatorial description of the Chern-Simons function for a closed 3-manifold. More details may be found in:J. Huebschmann: Extended moduli spaces, the Kan construction, and lattice gauge theory; Topology, 38, 1999, 555--596The Kan group may be viewed as a combinatorial version of the hoop group\/} introduced by Ashtekar et al, and $|\textnormal{Hom}(K,G)|$ is a combinatorial version of the space of homomorphims from the hoop group to the structure group explored by Ashtekar et al.
Moduli spaces of semistable holomorphic vector bundles on a Riemann surface and generalizations thereof to moduli spaces of semistable holomorphic principal bundles for a complex reductive Lie group may be obtained via extended moduli spaces as well as by a geometric invariant theory construction. Both constructions are finite dimensional; the former leads to the stratified symplectic structure while the latter yields the complex analytic one. We will explain a finite dimensional approach which involves suitable extended moduli spaces arising from spaces of holomorphic maps; this approach is aimed at providing the stratified symplectic and complex analytic structures at the same time, thereby establishing the fact that the two structure combine to a stratified Kähler structure. Our approach includes a construction of the familiar line bundle on such a moduli space.
Large random band matrices may prove useful for the study of the random Schrödinger operators, as they seem to have similar spectral properties. In this context we studied the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed large band width.
Die Riemannsche Vermutung besagt, dass alle nichttrivialen Nullstellen der Riemannschen Zetafunktion Realteil 1/2 besitzen. Unter der Annahme dieser Vermutung kann man genauere Fragen nach der empirischen lokalen Verteilung dieser Nullstellen stellen: Es gibt starke numerische und theoretische Hinweise - aber immer noch keinen vollstaendigen Beweis - dass die Nullstellenverteilung nach geeigneter Skalierung durch die lokale Verteilung von Eigenwerten von GUE Zufallsmatrizen beschrieben wird. Der gleiche Punktprozess tritt auch in der Quantenfeldtheorie als raeumliche Verteilung freier Fermionen im Grundzustand auf. In dem Vortrag will ich einige Resultate von Montgomery, Hejhal, Rudnick und Sarnak beschreiben, die Teile dieser GUE Vermutung beweisen. Darueber hinaus will ich ueber ein laufendes Projekt berichten, das die Verteilung der Nullstellen der Zetafunktion auf anderen Skalen als dem "typischen" Abstand zwischen zwei Nullstellen untersucht.
In 1981 Parisi and Sourlas conjectured values for the critical exponents for self-avoiding branched polymers in 2, 3 and 4 dimensions. We prove these conjectures and other results as a corollary of a combinatoric identity that identifies the generating function for branched polymers in D+2 dimensions with the pressure of the hard core gas in D dimensions.
A first principle formulation of quantum mechanics on smooth (or discrete) configuration spaces is studied with additive (a- ) or multiplicative (q-) difference operators instead of differential operators. The Borel quantisation is used as a guiding principle, shortly discussed and translated to a framework based on difference operators step by step with certain additional assumptions. q-difference operators are discussed in detail. The framework is explained for the configuration space S1 and for its N-point discretisation; the connection with q-deformations of the Witt-algebra and its inhomogenisation is elaborated. For a 'natural' choice of the additional assumptions a corresponding q-evolution (Schrödinger-)equation is obtained. The study shows some of the difficulties to generalise a physical theory from a known to a 'new' mathematical formalism.
A prerequisite for the satisfactory resolution of many problematical aspects of 'entanglement' in quantum physics is a definitive formulation of intrinsic coordinates for composite quantum systems. In the talk, the group theoretical aspects of the problem will be considered, based on Hilbert series.
After an elementary introduction to Yang-Mills theory on quantum spaces and the Seiberg-Witten map, cohomological techniques are used to construct these maps. A generalization to generic gauge theories is proposed and illustrated on a new noncommutative model.
Reference:
SEIBERG-WITTEN MAPS FROM THE POINT OF VIEW OF CONSISTENT DEFORMATIONS OF GAUGE THEORIES. By G. Barnich, M. Grigoriev, M. Henneaux. ULB-TH-01-17, Jun 2001. 18pp.
