arXiv subjects

Alexander Schenkel

Publications and source records attributed to Alexander Schenkel.

At least 73 records · Page 4Linked to original sources

arXiv2015

Homotopy colimits and global observables in Abelian gauge theory

We study chain complexes of field configurations and observables for Abelian gauge theory on contractible manifolds, and show that they can be extended to non-contractible manifolds by using techniques from homotopy theory. The extension prescription yields functors from a category of manifolds to suitable categories of chain complexes. The extended functors properly describe the global field and observable content of Abelian gauge theory, while the original gauge field configurations and observables on contractible manifolds are recovered up to a natural weak equivalence.

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arXiv2015

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.

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arXiv2014

Noncommutative connections on bimodules and Drinfeld twist deformation

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra H is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra). We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily H-equivariant. The theory canonically lifts to the tensor product structure.

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arXiv2014

Locally covariant quantum field theory with external sources

We provide a detailed analysis of the classical and quantized theory of a multiplet of inhomogeneous Klein-Gordon fields, which couple to the spacetime metric and also to an external source term; thus the solutions form an affine space. Following the formulation of affine field theories in terms of presymplectic vector spaces as proposed in [Annales Henri Poincare 15, 171 (2014)], we determine the relative Cauchy evolution induced by metric as well as source term perturbations and compute the automorphism group of natural isomorphisms of the presymplectic vector space functor. Two pathological features of this formulation are revealed: the automorphism group contains elements that cannot be interpreted as global gauge transformations of the theory; moreover, the presymplectic formulation does not respect a natural requirement on composition of subsystems. We therefore propose a systematic strategy to improve the original description of affine field theories at the classical and quantized level, first passing to a Poisson algebra description in the classical case. The idea is to consider state spaces on the classical and quantum algebras suggested by the physics of the theory (in the classical case, we use the affine solution space). The state spaces are not separating for the algebras, indicating a redundancy in the description. Removing this redundancy by a quotient, a functorial theory is obtained that is free of the above mentioned pathologies. These techniques are applicable to general affine field theories and Abelian gauge theories. The resulting quantized theory is shown to be dynamically local.

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arXiv2014

A C*-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We then show that fixing any principal U(1)-bundle, there exists a suitable category of sub-bundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

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arXiv2013

Dirac Operators on Noncommutative Curved Spacetimes

We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.

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arXiv2013

Module parallel transports in fuzzy gauge theory

In this article we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

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arXiv2013

Quantized Abelian principal connections on Lorentzian manifolds

We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group, namely the group of vertical principal bundle automorphisms. Properties of our functor are investigated in detail and, similar to earlier works, it is found that due to topological obstructions the locality property of locally covariant quantum field theory is violated. Furthermore, we prove that, for Abelian structure groups containing a nontrivial compact factor, the gauge invariant Borchers-Uhlmann algebra of the vector dual of the bundle of connections is not separating on gauge equivalence classes of principal connections. We introduce a topological generalization of the concept of locally covariant quantum fields. As examples, we construct for the category of principal U(1)-bundles two natural transformations from singular homology functors to the quantum field theory functor that can be interpreted as the Chern class and the electric charge. In this case we also prove that the electric charges can be consistently set to zero, which yields another quantum field theory functor that satisfies all axioms of locally covariant quantum field theory.

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arXiv2013

Linear bosonic and fermionic quantum gauge theories on curved spacetimes

We develop a general setting for the quantization of linear bosonic and fermionic field theories subject to local gauge invariance and show how standard examples such as linearized Yang-Mills theory and linearized general relativity fit into this framework. Our construction always leads to a well-defined and gauge-invariant quantum field algebra, the centre and representations of this algebra, however, have to be analysed on a case-by-case basis. We discuss an example of a fermionic gauge field theory where the necessary conditions for the existence of Hilbert space representations are not met on any spacetime. On the other hand, we prove that these conditions are met for the Rarita-Schwinger gauge field in linearized pure N=1 supergravity on certain spacetimes, including asymptotically flat spacetimes and classes of spacetimes with compact Cauchy surfaces. We also present an explicit example of a supergravity background on which the Rarita-Schwinger gauge field can not be consistently quantized.

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arXiv2013

Quantum field theory on affine bundles

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.

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arXiv2012

Noncommutative Gravity and Quantum Field Theory on Noncommutative Curved Spacetimes

The focus of this PhD thesis is on applications, new developments and extensions of the noncommutative gravity theory proposed by Julius Wess and his group. In part one we propose an extension of the usual symmetry reduction procedure to noncommutative gravity. We classify in the case of abelian Drinfel'd twists all consistent deformations of spatially flat Friedmann-Robertson-Walker cosmologies and of the Schwarzschild black hole. The deformed symmetry structure allows us to obtain exact solutions of the noncommutative Einstein equations in many of our models. In part two we develop a new formalism for quantum field theory on noncommutative curved spacetimes by combining methods from the algebraic approach to quantum field theory with noncommutative differential geometry. We also study explicit examples of deformed wave operators and find that there can be noncommutative corrections even on the level of free field theories. The convergent deformation of simple toy models is investigated and it is found that these theories have an improved behaviour at short distances, i.e. in the ultraviolet. In part three we study homomorphisms between and connections on noncommutative vector bundles. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of bimodules is clarified. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations.

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arXiv2012

Twist deformations of module homomorphisms and connections

Let H be a Hopf algebra, A a left H-module algebra and V a left H-module A-bimodule. We study the behavior of the right A-linear endomorphisms of V under twist deformation. We in particular construct a bijective quantization map to the right A_\star-linear endomorphisms of V_\star, with A_\star,V_\star denoting the usual twist deformations of A,V. The quantization map is extended to right A-linear homomorphisms between left H-module A-bimodules and to right connections on V. We then investigate the tensor product of linear maps between left H-modules. Given a quasitriangular Hopf algebra we can define an H-covariant tensor product of linear maps, which restricts for left H-module A-bimodules to a well-defined tensor product of right A-linear homomorphisms on tensor product modules over A. This also requires a quasi-commutativity condition on the algebra and bimodules. Using this tensor product we can construct a new lifting prescription of connections to tensor product modules, generalizing the usual prescription to also include nonequivariant connections.

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arXiv2012

Quantization of the massive gravitino on FRW spacetimes

In this article we study the quantization and causal properties of a massive spin 3/2 Rarita-Schwinger field on spatially flat Friedmann-Robertson-Walker (FRW) spacetimes. We construct Zuckerman's universal conserved current and prove that it leads to a positive definite inner product on solutions of the field equation. Based on this inner product, we quantize the Rarita-Schwinger field in terms of a CAR-algebra. The transversal and longitudinal parts constituting the independent on-shell degrees of freedom decouple. We find a Dirac-type equation for the transversal polarizations, ensuring a causal propagation. The equation of motion for the longitudinal part is also of Dirac-type, but with respect to an `effective metric'. We obtain that for all four-dimensional FRW solutions with a matter equation of state p = w rho and w in (-1,1] the light cones of the effective metric are more narrow than the standard cones, which are recovered for the de Sitter case w=-1. In particular, this shows that the propagation of the longitudinal part, although non-standard for w different from -1, is completely causal in cosmological constant, dust and radiation dominated universes.

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arXiv2011

QFT on homothetic Killing twist deformed curved spacetimes

We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric spacetimes, which are deformed by an abelian Drinfel'd twist constructed from a Killing and a homothetic Killing vector field. In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion and Green's operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive physical picture that noncommutative geometry prevents arbitrary localization in spacetime.

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arXiv2011

Quantum Field Theory on Curved Noncommutative Spacetimes

We summarize our recently proposed approach to quantum field theory on noncommutative curved spacetimes. We make use of the Drinfel'd twist deformed differential geometry of Julius Wess and his group in order to define an action functional for a real scalar field on a twist-deformed time-oriented, connected and globally hyperbolic Lorentzian manifold. The corresponding deformed wave operator admits unique deformed retarded and advanced Green's operators, provided we pose a support condition on the deformation. The solution space of the deformed wave equation is constructed explicitly and can be canonically equipped with a (weak) symplectic structure. The quantization of the solution space of the deformed wave equation is performed using *-algebras over the ring C[[λ]]. As a new result we add a proof that there exist symplectic isomorphisms between the deformed and the undeformed symplectic R[[λ]]-modules. This immediately leads to *-algebra isomorphisms between the deformed and the formal power series extension of the undeformed quantum field theory. The consequences of these isomorphisms are discussed.

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arXiv2010

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel'd twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green's operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.

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arXiv2010

High energy improved scalar quantum field theory from noncommutative geometry without UV/IR-mixing

We consider an interacting scalar quantum field theory on noncommutative Euclidean space. We implement a family of noncommutative deformations, which -- in contrast to the well known Moyal-Weyl deformation -- lead to a theory with modified kinetic term, while all local potentials are unaffected by the deformation. We show that our models, in particular, include propagators with anisotropic scaling z=2 in the ultraviolet (UV). For a Φ^4-theory on our noncommutative space we obtain an improved UV behaviour at the one-loop level and the absence of UV/IR-mixing and of the Landau pole.

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arXiv2010

Field Theory on Curved Noncommutative Spacetimes

We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave equations. We provide explicit examples of deformed Klein-Gordon operators for noncommutative Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve the noncommutative Einstein equations. We study the construction of deformed Green's functions and provide a diagrammatic approach for their perturbative calculation. The leading noncommutative corrections to the Green's functions for our examples are derived.

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