H. G. Díaz Marin, R. Oeckl

UNAM-CCM-2017-6, arXiv:1712.05537

We quantize abelian Yang-Mills theory on Riemannian manifolds with boundaries in any dimension. The quantization proceeds in two steps. First, the classical theory is encoded into an axiomatic form describing solution spaces associated to manifolds. Second, the quantum theory is constructed from the classical axiomatic data in a functorial manner. The target is general boundary quantum field theory, a TQFT-type axiomatic formulation of quantum field theory.

R. Oeckl

UNAM-CCM-2017-4, to appear in the proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama" (Marseille, 2016), arXiv:1708.03047

We generalize the fermionic coherent states to the case of Fock-Krein spaces, i.e., Fock spaces with an idefinite inner product of Krein type. This allows for their application in topological or functorial quantum field theory and more specifically in general boundary quantum field theory. In this context we derive a universal formula for the amplitude of a coherent state in linear field theory on an arbitrary manifold with boundary.

R. Oeckl

UNAM-CCM-2016-2, arXiv:1610.09052

We discuss a novel framework for physical theories that is based on the principles of locality and operationalism. It generalizes and unifies previous frameworks, including the standard formulation of quantum theory, the convex operational framework and Segal's approach to quantum field theory. It is capable of encoding both classical and quantum (field) theories, implements spacetime locality in a manifest way and contains the complete modern notion of measurement in the quantum case. Its mathematical content can be condensed into a set of axioms that are similar to those of Atiyah and Segal. This is supplemented by two basic rules for extracting probabilities or expectation values for measurement processes. The framework, called the positive formalism, is derived in three completely different ways. One derivation is from first principles, one starts with classical field theory and one with quantum field theory. The latter derivation arose previously in the programme of the general boundary formulation of quantum theory. As in the convex operational framework, the difference between classical and quantum theories essentially arises from certain partially ordered vector spaces being either lattices or anti-lattices. If we add the ad hoc ingredient of imposing anti-lattice structures, the derivation from first principles may be seen as a reconstruction of quantum theory. Among other things, the positive formalism suggests a statistical approach to classical field theories with dynamical metric, provides a common ground for quantum information theory and quantum field theory, introduces a notion of local measurement into quantum field theory, and suggests a new perspective on quantum gravity by removing the incompatibility with general relativistic principles. The positive formalism as a framework for quantum theory is in conflict with various interpretations or modifications of quantum theory, including physical collapse theories, many-worlds interpretations, and non-local hidden variable theories.

M. Dohse, R. Oeckl

While the standard construction of the S-matrix fails on Anti-de Sitter (AdS) spacetime, a generalized S-matrix makes sense, based on the hypercylinder geometry induced by the boundary of AdS. In contrast to quantum field theory in Minkowski spacetime, there is not yet a standard way to resolve the quantization ambiguities arising in its construction. These ambiguities are conveniently encoded in the choice of a complex structure. We explore in this paper the space of complex structures for real scalar Klein-Gordon theory based on a number of criteria. These are: invariance under AdS isometries, induction of a positive definite inner product, compatibility with the standard S-matrix picture and recovery of standard structures in Minkowski spacetime under a limit of vanishing curvature. While there is no complex structure that satisfies all demands, we emphasize two interesting candidates that satisfy most: In one case we have to give up part of the isometry invariance, in the other case the induced inner product is indefinite.

R. Oeckl

Starting from the guiding principles of spacetime locality and operationalism, a general framework for a probabilistic description of nature is proposed. Crucially, no notion of time or metric is assumed, neither any specific physical model. Remarkably, the emerging framework converges with a recently proposed formulation of quantum theory, obtained constructively from known quantum physics. At the same time the framework also admits statistical theories of classical physics.

R. Oeckl

We construct the coherent states in the sense of Gilmore and Perelomov for the fermionic Fock space. Our treatment is from the outset adapted to the infinite-dimensional case. The fermionic Fock space becomes in this way a reproducing kernel Hilbert space of continuous holomorphic functions.

R. Oeckl

Hilbert spaces of states can be constructed in standard quantum field theory only for infinitely extended spacelike hypersurfaces, precluding a more local notion of state. In fact, the Reeh-Schlieder Theorem prohibits the localization of states on pieces of hypersurfaces in the standard formalism. From the point of view of geometric quantization the problem lies in the non-locality of the complex structures associated to hypersurfaces in standard quantization. We show that using a weakened version of the positive formalism puts this problem into a new perspective. This is a local TQFT type formalism based on super-operators and mixed state spaces rather than on amplitudes and pure state spaces as the one of Atiyah-Segal. In particular, we show that in the case of purely fermionic degrees of freedom the complex structure can be dispensed with when the notion of state is suitably generalized. These generalized states do localize on compact hypersurfaces with boundaries. For the simplest case of free fermionic fields we embed this in a rigorous and functorial quantization scheme yielding a local description of the quantum theory. Crucially, no classical data is needed beyond the structures evident from a Lagrangian setting. When the classical data is augmented with complex structures on hypersurfaces, the quantum data correspondingly augment to the full positive formalism. This scheme is applicable to field theory in curved spacetime, but also to field theories without metric background.

R. Oeckl

We introduce a new "positive formalism" for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.

R. Oeckl

Quantum Theory: Reconsideration of Foundations 6 (Växjö, 2012), AIP, Melville, 2012, pp. 428-432, arXiv:1210.0944

An approach to the foundations of quantum theory is advertised that proceeds by "reverse engineering" quantum field theory. As a concrete instance of this approach, the general boundary formulation of quantum theory is outlined.

R. Oeckl

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from standard quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.

R. Oeckl

We show that the Feynman path integral together with the Schrödinger representation gives rise to a rigorous and functorial quantization scheme for linear and affine field theories. Since our target framework is the general boundary formulation, the class of field theories that can be quantized in this way includes theories without a metric spacetime background. We also show that this quantization scheme is equivalent to a holomorphic quantization scheme proposed earlier and based on geometric quantization. We proceed to include observables into the scheme, quantized also through the path integral. We show that the quantized observables satisfy the canonical commutation relations, a feature shared with other quantization schemes also discussed. However, in contrast to other schemes the presented quantization also satisfies a correspondence between the composition of classical observables through their product and the composition of their quantized counterparts through spacetime gluing. In the special case of quantum field theory in Minkowski space this reproduces the operationally correct composition of observables encoded in the time-ordered product. We show that the quantization scheme also generalizes other features of quantum field theory such as the generating function of the S-matrix.

The General Boundary Formulation (GBF) is a new framework for studying quantum theories. After concise overviews of the GBF and Schrödinger-Feynman quantization we apply the GBF to resolve a well known problem on Anti-deSitter spacetime where due to the lack of temporally asymptotic free states the usual S-matrix cannot be defined. We construct a different type of S-matrix plus propagators for free and interacting real Klein-Gordon theory.

R. Oeckl

We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.

R. Oeckl

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin-Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S-matrix.

R. Oeckl

Quantum Field Theory and Gravity (Regensburg, 2010), Birkhäuser, Basel, 2012, pp. 137-156, arXiv:1101.0367

We develop a notion of quantum observable for the general boundary formulation of quantum theory. This notion is adapted to spacetime regions rather than to hypersurfaces and naturally fits into the topological quantum field theory like axiomatic structure of the general boundary formulation. We also provide a proposal for a generalized concept of expectation value adapted to this type of observable. We show how the standard notion of quantum observable arises as a special case together with the usual expectation values. We proceed to introduce various quantization schemes to obtain such quantum observables including path integral quantization (yielding the time-ordered product), Berezin-Toeplitz (antinormal ordered) quantization and normal ordered quantization and discuss some of their properties.

R. Oeckl

We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein-Gordon theory.

D. Colosi and R. Oeckl

We investigate the question of unitarity of evolution between hypersurfaces in quantum field theory in curved spacetime from the perspective of the general boundary formulation. Unitarity thus means unitarity of the quantum operator that maps the state space associated with one hypersurface to the state space associated with the other hypersurface. Working in Klein-Gordon theory, we find that such an evolution is generically unitary given a one-to-one correspondence between classical solutions in neighborhoods of the respective hypersurfaces. This covers the case of pairs of Cauchy hypersurfaces, but also certain cases where hypersurfaces are timelike. The tools we use are the Schrödinger representation and the Feynman path integral.

D. Colosi and R. Oeckl

We quantize the Helmholtz equation (plus perturbative interactions) in two dimensions to illustrate a manifestly local description of quantum field theory. Using the general boundary formulation we describe the quantum dynamics both in a traditional time evolution setting as well as in a setting referring to finite disk (or annulus) shaped regions of spacetime. We demonstrate that both descriptions are equivalent when they should be.

Â. Mestre and R. Oeckl

In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-point functions.

D. Colosi and R. Oeckl

We construct a new type of S-matrix in quantum field theory using the general boundary formulation. In contrast to the usual S-matrix the space of free asymptotic states is located at spatial rather than at temporal infinity. Hence, the new S-matrix applies to situations where interactions may remain important at all times, but become negligible with distance. We show that the new S-matrix is equivalent to the usual one in situations where both apply. This equivalence is mediated by an isomorphism between the respective asymptotic state spaces that we construct. We introduce coherent states that allow us to obtain explicit expressions for the new S-matrix. In our formalism crossing symmetry becomes a manifest rather than a derived feature of the S-matrix.

D. Colosi and R. Oeckl

We provide a new method to construct the S-matrix in quantum field theory. This method implements crossing symmetry manifestly by erasing the a priori distinction between in- and out-states. It allows the description of processes where the interaction weakens with distance in space, but remains strong in the center at all times. It should also be applicable to certain spacetimes where the conventional method fails due to lack of temporal asymptotic states.

We give an introductory account of the general boundary formulation of quantum theory. We refine its probability interpretation and emphasize a conceptual and historical perspective. We give motivations from quantum gravity and illustrate them with a scenario for describing gravitons in quantum gravity.

The solution of quantum Yang-Mills theory on arbitrary compact two-manifolds is well known. We bring this solution into a TQFT-like form and extend it to include corners. Our formulation is based on an axiomatic system that we hope is flexible enough to capture actual quantum field theories also in higher dimensions. We motivate this axiomatic system from a formal Schrödinger-Feynman quantization procedure. We also discuss the physical meaning of unitarity, the concept of vacuum, (partial) Wilson loops and non-orientable surfaces.

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.

We present an implementation of Wilson's renormalization group and a continuum limit tailored for loop quantization. The dynamics of loop-quantized theories is constructed as a continuum limit of the dynamics of effective theories. After presenting the general formalism we show as a first explicit example the 2D Ising field theory, an interacting relativistic quantum field theory with local degrees of freedom quantized by loop quantization techniques.

We show that the real massive Klein-Gordon theory admits a description
in terms of states on various **timelike** hypersurfaces and
amplitudes associated to regions bounded by them. This realizes
crucial elements of the general boundary framework for quantum field
theory. The hypersurfaces considered are hyperplanes on the one hand
and timelike hypercylinders on the other hand. The latter lead to
the first explicit examples of amplitudes associated with finite
regions of space, and admit no standard description in
terms of "initial" and "final" states. We
demonstrate a generalized probability interpretation in this example,
going beyond the applicability of standard quantum mechanics.

We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces. We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate spacetime symmetries into the framework. We show how the Schrödinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.

We investigate the possibility of defining states on timelike hypersurfaces in quantum field theory. To this end we consider hyperplanes in the real massive Klein-Gordon theory using the Schrödinger representation. We find a well defined vacuum wave functional, existing on any hyperplane, with the remarkable property that it changes smoothly even under Euclidean rotation through the light-cone. Moreover, particles on timelike hyperplanes exist and occur in two variants, incoming and outgoing, distinguished by the sign of the energy. Multi-particle wave functionals take a form similar to those on spacelike hypersurfaces. The role of unitarity and the inner product is discussed.

We use a coproduct on the time-ordered algebra of field operators to derive simple relations between complete, connected and 1-particle irreducible n-point functions. Compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations. It may also be used to efficiently perform tree level computations.

Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity (Rio de Janeiro, 2003), World Scientific, Singapore, 2006, pp. 2296-2300, gr-qc/0401087

We give an introductory account to the renormalization of models without metric background. We sketch the application to certain discrete models of quantum gravity such as spin foam models.

Proceedings of the First International Conference on Physics, Amirkabir University, Tehran, 2004, pp. 257-265, gr-qc/0312081

We present an approach to quantum gravity based on the general boundary formulation of quantum mechanics, path integral quantization, spin foam models and renormalization.

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.

We consider a local formalism in quantum field theory, in which no reference is made to infinitely extended spacial surfaces, infinite past or infinite future. This can be obtained in terms of a functional W[f,S] of the field f on a closed 3d surface S that bounds a finite region R of Minkowski spacetime. The dependence of W on S is governed by a local covariant generalization of the Schroedinger equation. Particles' scattering amplitudes that describe experiments conducted in the finite region R --the lab during a finite time-- can be expressed in terms of W. The dependence of W on the geometry of S expresses the dependence of the transition amplitudes on the relative location of the particle detectors. In a gravitational theory, background independence implies that W is independent from S. However, the detectors' relative location is still coded in the argument of W, because the geometry of the boundary surface is determined by the boundary value f of the gravitational field. This observation clarifies the physical meaning of the functional W defined by non perturbative formulations of quantum gravity, such as the spinfoam formalism. In particular, it suggests a way to derive particles' scattering amplitudes from a spinfoam model. In particular, we discuss the notion of vacuum in a generally covariant context. We distinguish the nonperturbative vacuum |0_S>, which codes the dynamics, from the Minkowski vacuum |0_M>, which is the state with no particles and is recovered by taking appropriate large values of the boundary metric. We derive a relation between the two vacuum states. We propose an explicit expression for computing the Minkowski vacuum from a spinfoam model.

We show how super BF theory in any dimension can be quantized as a spin foam model, generalizing the situation for ordinary BF theory. This is a first step toward quantizing supergravity theories. We investigate in particular 3-dimensional (p=1,q=1) supergravity which we quantize exactly. We obtain a super-Ponzano-Regge model with gauge group OSp(1|2). A main motivation for our approach is the implementation of fermionic degrees of freedom in spin foam models. Indeed, we propose a description of the fermionic degrees of freedom in our model. Complementing the path integral approach we also discuss aspects of a canonical quantization in the spirit of loop quantum gravity. Finally, we comment on 2+1-dimensional quantum supergravity and the inclusion of a cosmological constant.

I propose to formalize quantum theories as topological quantum field theories in a generalized sense. This implies the unusual feature of associating state spaces with boundaries of arbitrary (and possibly finite) regions of space-time. The formulation is "holographic" in the sense that these boundary state spaces carry all the available information about the interior. I further propose to obtain such "general boundary" quantum theories through a generalized path integral quantization. I show, using this quantization scheme, how both non-relativistic quantum mechanics and quantum field theory can be given a "general boundary" formulation. Surprisingly, in the first case, features normally associated with quantum field theory emerge from consistency conditions. In particular, it is necessary to include states with any number of particles and an identical treatment of particles induces the occurrence of pair creation. I also note how three dimensional quantum gravity is an example for a realization of both proposals. The proposals are mainly designed to obtain in the case of quantum gravity a description that is both local and allows a straightforward interpretation of amplitudes in terms of measurement processes.

I review basic principles of the quantum mechanical measurement process in view of their implications for a quantum theory of general relativity. It turns out that a clock as an external classical device associated with the observer plays an essential role. This leads me to postulate a "principle of the integrity of the observer". It essentially requires the observer to be part of a classical domain connected throughout the measurement process. Mathematically this naturally leads to a formulation of quantum mechanics as a kind of topological quantum field theory. Significantly, quantities with a direct interpretation in terms of a measurement process are associated only with amplitudes for connected boundaries of compact regions of space-time. I discuss some implications of my proposal such as in-out duality for states, delocalization of the "collapse of the wave function" and locality of the description. Differences to existing approaches to quantum gravity are also highlighted.

Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow.

We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two.

A main motivation for our approach are discrete models of quantum gravity. We investigate both the Reisenberger and the Barrett-Crane spin foam model in view of their amenability to a renormalization treatment. In the second case a lack of tunable local parameters prompts us to introduce a new model. For the Reisenberger and the new model we discuss qualitative aspects of the renormalization groupoid flow. In both cases quantum BF theory is the UV fixed point.

Mathematical physics research on the leading edge (C.V. Benton, ed.), Nova Science, Hauppauge NY, 2004, pp. 63-90, hep-th/0208118

The quantum field algebra of real scalar fields is shown to be an example of infinite dimensional quantum group. The underlying Hopf algebra is the symmetric algebra S(V) and the product is Wick's normal product. Two coquasitriangular structures can be built from the two-point function and the Feynman propagator of scalar fields to reproduce the operator product and the time-ordered product as twist deformations of the normal product. A correspondence is established between the quantum group and the quantum field concepts. On the mathematical side the underlying structures come out of Hopf algebra cohomology.

CPT-2002/P.4361, hep-ph/0206054

Quantum fields are shown to provide an example of infinite-dimensional quantum groups. A dictionary is established between quantum field and quantum group concepts: the expectation value over the vacuum is the counit, Wick's theorem is the definition of a twisted product, operator and time-ordered products are examples of twisted products. Through this dictionary, coquasitriangular structures are introduced in quantum field theory. These structures are the origin of Wick's theorem and quasifree states. Renormalization becomes the replacement of a coquasitriangular structure by a 2-coboundary. Quantum groups provide a second quantization without commutators which can second-quantize noncommutative algebras.

We provide a simple proof of the topological invariance of the Turaev-Viro model (corresponding to simplicial 3d pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF-theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, being proposed as models of quantum gravity in 4d, under refinement of the cellular decomposition.

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold.

In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension).

Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover.

The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks.

Nucl. Phys. B Proc. Suppl. 106-107 (2002) 1010-1012, (hep-lat/0110034)

Non-Abelian Lattice Gauge Theory in Euclidean space-time of dimension d>=2 whose gauge group is any compact Lie group is related to a Spin Foam Model by an exact strong-weak duality transformation. The group degrees of freedom are integrated out and replaced by combinatorial expressions involving irreducible representations and intertwiners of the gauge group. This transformation is available for the partition function, for the expectation value of observables (spin networks), and for the correlator of centre monopoles which is a ratio of partition functions in the original model and an ordinary expectation value in the dual formulation.

The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention has so far been focused on the case with maximal symmetry -- where the base space is a quantum group and the bimodules are bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules.

We investigate the "next best" case -- where the base space is a quantum homogeneous space and the bimodules are covariant. We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant bimodules and a new kind of "crossed" modules which we define. The latter are attached to the pair of quantum groups which defines the quantum homogeneous space.

We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced differential calculus.

We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation.

The old question of extending the Poincaré group to unify external and internal symmetries (solved by supersymmetry) is reexamined in the new framework. The reason why we should allow supergroups in this case becomes completely transparent. However, the true symmetries are not expressed by groups or supergroups here but by ordinary (not super) quantum groups. We show in this generalized framework that supersymmetry is the most general unification of internal and space-time symmetries provided that all particles are either bosons or fermions.

Finally, we demonstrate with some examples how quantum geometry provides a natural setting for the construction of super-extensions, super-spaces, super-derivatives etc.

We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group G as the gauge group and on Euclidean space-time lattices of dimension d>=2. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finite-dimensional unitary representations, intertwiners and characters of G. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice.

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin-statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.

Indications from various areas of physics point to the possibility that space-time at small scales might not have the structure of a manifold. Noncommutative geometry provides an attractive framework for a perhaps more accurate description of nature. It encompasses the generalisation of spaces to noncommutative spaces and of symmetry groups to quantum groups. This motivates efforts to extend quantum field theory to noncommutative spaces and quantum group symmetries. One also expects that divergences of conventional theories might be regularised in this way.

We show that there is a duality exchanging noncommutativity and non-trivial
statistics for quantum field theory on **R**^{d}. Employing
methods of quantum groups, we observe that ordinary and noncommutative
**R**^{d}
are related by twisting. We extend the twist to an equivalence for quantum
field theory using the framework of braided quantum field theory. The twist
exchanges both commutativity with noncommutativity and ordinary with non-trivial
statistics. The same holds for the noncommutative torus.

We develop a general framework for quantum field theory on noncommutative
spaces, i.e., spaces with quantum group symmetry. We use the path integral
approach to obtain expressions for n-point functions. Perturbation theory
leads us to generalised Feynman diagrams which are braided, i.e., they
have non-trivial over- and under-crossings. We demonstrate the power of
our approach by applying it to phi^{4}-theory on the quantum 2-sphere.
We find that the basic divergent diagram of the theory is regularised.

We show that the crossed modules and bicovariant different calculi on
two Hopf algebras related by a cocycle twist are in 1-1 correspondence.
In particular, for quantum groups which are cocycle deformation-quantisations
of classical groups the calculi are obtained as deformation-quantisation
of the classical ones. As an application, we classify all bicovariant differential
calculi on the Planck scale Hopf algebra **C**[x]×_{h,G}**C**[p].
This is a quantum group which has an h -> 0 limit as the functions on a
classical but non-Abelian group and a G -> 0 limit as flat space quantum
mechanics. We further study the noncommutative differential geometry and
Fourier theory for this Hopf algebra as a toy model for Planck scale physics.
The Fourier theory implements a T-duality-like self-duality. The noncommutative
geometry turns out to be singular when G -> 0 and is therefore not visible
in flat space quantum mechanics alone.

We give a complete classification of bicovariant first order differential
calculi on the quantum enveloping algebra U_{q}(b_{+})
which we view as the quantum function algebra C_{q}(B_{+}).
Here, b_{+} is the Borel subalgebra of sl_{2}. We do the
same in the classical limit q -> 1 and obtain a one-to-one correspondence
in the finite dimensional case. It turns out that the classification is
essentially given by finite subsets of the positive integers. We proceed
to investigate the classical limit from the dual point of view, i.e. with
"function algebra" U(b_{+}) and "enveloping algebra" C(B_{+}).
In this case there are many more differential calculi than coming from
the q-deformed setting. As an application, we give the natural intrinsic
four-dimensional calculus of kappa-Minkowski space and the associated formal
integral.

Rapidity gaps between two hard jets at the Fermilab Tevatron have been
interpreted as being due to the exchange of two gluons which are in an
overall color-singlet state. We show that this simple picture involves
unitarity violating amplitudes. Unitarizing the gluon exchange amplitude
leads to qualitatively different predictions for the fraction of t-channel
color singlet exchange events in forward *qq*, *qg *or *gg*
scattering, which better fit Tevatron data.