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Katarzyna Rejzner

Publications and source records attributed to Katarzyna Rejzner.

15 recordsLinked to original sources
arXiv2020

QFT on curved spacetimes: axiomatic framework and examples

In this overview article we present a formalism suitable for constructing models of QFT's on curved spacetimes. The leading principle is the emphasis on local properties. It turns out that this requires a reformulation of the standard QFT framework which also yields a new perspective for the theories on Minkowski space. The aim of the present work is to provide an introduction into the framework, which should be accessible for both mathematical physicists and mathematicians.

math-ph↗
arXiv2016

Cosmological perturbation theory and quantum gravity

It is shown how cosmological perturbation theory arises from a fully quantized perturbative theory of quantum gravity. Central for the derivation is a non-perturbative concept of gauge-invariant local observables by means of which perturbative invariant expressions of arbitrary order are generated. In particular, in the linearised theory, first order gauge-invariant observables familiar from cosmological perturbation theory are recovered. Explicit expressions of second order quantities are presented as well.

gr-qc↗
arXiv2015

Quantum gravity from the point of view of locally covariant quantum field theory

We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.

math-ph↗
arXiv2014

Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization

We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. For scalar fields the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stueckelberg and Petermann.

hep-th↗
arXiv2014

Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is a proof that the renormalized time-ordered product is equivalent to the pointwise product of classical field theory. The renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and Dütsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.

math-ph↗
arXiv2013

Remarks on local symmetry invariance in perturbative algebraic quantum field theory

We investigate various aspects of invariance under local symmetries in the framework of perturbative algebraic quantum field theory (pAQFT). Our main result is the proof that the quantum Batalin-Vilkovisky (BV) operator, on-shell, can be written as the commutator with the interacting BRST charge. Up to now, this was proven only for a certain class of fields in quantum electrodynamics and in Yang-Mills theory. Our result is more general and it holds in a wide class of theories with local symmetries, including general relativity and the bosonic string. We also comment on other issues related to local gauge invariance and, using the language of homological algebra, we compare different approaches to quantization of gauge theories in the pAQFT framework.

math-ph↗
arXiv2013

The effective theory of strings

We show that the Nambu-Goto string, and its higher dimensional generalizations, can be quantized, in the sense of an effective theory, in any dimension of the target space. The crucial point is to consider expansions around classical string configurations. We are using tools from perturbative algebraic quantum field theory, quantum field theory on curved spacetimes, and the Batalin-Vilkovisky formalism. Our model has some similarities with the Lüscher-Weisz string, but we allow for arbitrary classical background string configurations and keep the diffeomorphism invariance.

math-ph↗
arXiv2013

Batalin-Vilkovisky formalism in locally covariant field theory

The present work contains a complete formulation of the Batalin-Vilkovisky (BV) formalism in the framework of locally covariant field theory. In the first part of the thesis the classical theory is investigated with a particular focus on the infinite dimensional character of the underlying structures. It is shown that the use of infinite dimensional differential geometry allows for a conceptually clear and elegant formulation. The construction of the BV complex is performed in a fully covariant way and we also generalize the BV framework to a more abstract level, using functors and natural transformations. In this setting we construct the BV complex for classical gravity. This allows us to give a homological interpretation to the notion of diffeomorphism invariant physical quantities in general relativity. The second part of the thesis concerns the quantum theory. We provide a framework for the BV quantization that doesn't rely on the path integral formalism, but is completely formulated within perturbative algebraic quantum field theory. To make such a formulation possible we first prove that the renormalized time-ordered product can be understood as a binary operation on a suitable domain. Using this result we prove the associativity of this product and provide a consistent framework for the renormalized BV structures. In particular the renormalized quantum master equation and the renormalized quantum BV operator are defined. To give a precise meaning to theses objects we make a use of the master Ward identity, which is an important structure in causal perturbation theory.

math-ph↗
arXiv2013

Perturbative algebraic quantum field theory

These notes are based on the course given by Klaus Fredenhagen at the Les Houches Winter School in Mathematical Physics (January 29 - February 3, 2012) and the course "QFT for mathematicians" given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6 -11, 2012). Both courses were meant as an introduction to modern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists.

math-ph↗
arXiv2012

The Locality Axiom in Quantum Field Theory and Tensor Products of $C^*$-algebras

The prototype of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital $C^*$-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of $C^*$-algebras.

math-ph↗
arXiv2011

Fermionic fields in the functional approach to classical field theory

In this paper, we present a formulation of the classical theory of Fermionic (anticommuting) fields, which fits into the general framework proposed by K.Fredenhagen, M.Duetsch and R.Brunetti. It was inspired by the recent developments in perturbative algebraic quantum field theory and allows for a deeper structural understanding also on the classical level. We propose a modification of this formalism that allows to treat also Fermionic fields. In contrast to other formulations of classical theory of anticommuting variables, we don't introduce additional Grassman degrees of freedom. Instead the anticommutativity is introduced in a natural way on the level of functionals. Moreover our construction incorporates the functional-analytic and topological aspects, which is usually neglected in the treatments of anticommuting fields. We also give an example of an interacting model where our framework can be applied.

math-ph↗
arXiv2011

Batalin-Vilkovisky formalism in the functional approach to classical field theory

We develop the Batalin-Vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes. A crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime. In particular, this allows a homological construction of the Poisson algebra of observables in classical gravity. Our methods heavily rely on the differential geometry of configuration spaces of classical fields.

math-ph↗
arXiv2011

Spacelike localization of long-range fields in a model of asymptotic electrodynamics

A previously proposed algebra of asymptotic fields in quantum electrodynamics is formulated as a net of algebras localized in regions which in general have unbounded spacelike extension. Electromagnetic fields may be localized in `symmetrical spacelike cones', but there are strong indications this is not possible in the present model for charged fields, which have tails extending in all space directions. Nevertheless, products of appropriately `dressed' fermion fields (with compensating charges) yield bi-localized observables.

hep-th↗
arXiv2011

Local covariance and background independence

One of the many conceptual difficulties in the development of quantum gravity is the role of a background geometry for the structure of quantum field theory. To some extent the problem can be solved by the principle of local covariance. The principle of local covariance was originally imposed in order to restrict the renormalization freedom for quantum field theories on generic spacetimes. It turned out that it can also be used to implement the request of background independence. Locally covariant fields then arise as background independent entities.

math-ph↗