157 research outputs found
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[[\lambda]]. As a new result we add a proof that there exist symplectic
isomorphisms between the deformed and the undeformed symplectic
R[[\lambda]]-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.Comment: 15 pages, no figures. Talk in the Corfu Summer Institute on
Elementary Particles and Physics - Workshop on Non Commutative Field Theory
and Gravity, September 8-12, 2010 Corfu Greece. v2: Typo in Corollary 1
correcte
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.Comment: 74 pages. V2, added remark 3.7 and references therein, accepted
versio
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.Comment: v2: 42 pages; Appendix C on deformation quantization and references
added. v3: 47 pages; compatible with version to appear in Annales Henri
Poincar
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.Comment: SIGMA Special Issue on Noncommutative Spaces and Field
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.Comment: 34 pages, no figures; v2: 35 pages, compatible with version to be
published in Annales Henri Poincar
Cheeger-Simons differential characters with compact support and Pontryagin duality
By adapting the Cheeger-Simons approach to differential cohomology, we
establish a notion of differential cohomology with compact support. We show
that it is functorial with respect to open embeddings and that it fits into a
natural diagram of exact sequences which compare it to compactly supported
singular cohomology and differential forms with compact support, in full
analogy to ordinary differential cohomology. We prove an excision theorem for
differential cohomology using a suitable relative version. Furthermore, we use
our model to give an independent proof of Pontryagin duality for differential
cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125
(2003) 791]: On any oriented manifold, ordinary differential cohomology is
isomorphic to the smooth Pontryagin dual of compactly supported differential
cohomology. For manifolds of finite-type, a similar result is obtained
interchanging ordinary with compactly supported differential cohomology.Comment: 33 pages, no figures - v3: Final version to be published in
Communications in Analysis and Geometr
Quantum field theories on categories fibered in groupoids
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first examples of homotopical quantum field theories resembling some aspects of gauge theories
Algebraic field theory operads and linear quantization
We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories
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