krein space quantization in curved and flat spacetimes we re examine in detail a canonical quantization method a la gupta bleuler in which the fock space is built over a so called krein space this method has already been successfully applied to the massless minimally coupled scalar field in de sitter spacetime for which it preserves covariance here it is formulated in a more general context an interesting feature of the theory is that although the field is obtained by canonical quantization it is independent of bogoliubov transformations moreover no infinite term appears in the computation of t mean values and the vacuum energy of the free field vanishes 0t000 0 we also investigate the behaviour of the krein quantization in minkowski space for a theory with interaction we show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero

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 an extended notion of the renormalized time ordered product as a binary product equivalent to the pointwise product of classical field theory originally in causal perturbation theory the time ordered product is understood merely as a sequence of multilinear maps on the space of local functionals our extended notion of the renormalized time ordered product denoted by cdot mathcal t rm r is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to cdot mathcal t rm r on this space 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 dutsch contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps

ricci flow from the renormalization of nonlinear sigma models in the framework of euclidean algebraic quantum field theory the perturbative approach to nonlinear sigma models and the associated renormalization group flow are discussed within the framework of euclidean algebraic quantum field theory and of the principle of general local covariance in particular we show in an euclidean setting how to define wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of khavkine melati and moretti for vector valued free fields as a by product of such classification we provide a mathematically rigorous proof that at first order in perturbation theory the renormalization group flow of the nonlinear sigma model is the ricci flow

local causal structures hadamard states and the principle of local covariance in quantum field theory in the framework of the algebraic formulation we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime in particular we use the properties of the exponential map to set up a local version of a bulk to boundary correspondence the bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background while the boundary is a part of the future light cone having p as its own tip in this regime we provide a novel notion for the extended algebra of wick polynomials on the aforesaid cone and on the one hand we prove that it contains the information of the bulk counterpart via an injective homomorphism while on the other hand we associate to it a distinguished state whose pull back in the bulk is of hadamard form the main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that for any two given backgrounds m and m and for any two subsets of geodesic neighbourhoods of two arbitrary points it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism this allows for the pull back of boundary states in both spacetimes and thus to set up a machinery which permits the comparison of expectation values of local field observables in m and m

the massive feynman propagator on asymptotically minkowski spacetimes we consider the massive klein gordon equation on asymptoti cally minkowski spacetimes in the sense that the manifold is r 1 d and the metric approaches that of minkowski space at infinity in a short range way jointly in time and space variables in this setup we define feynman and anti feynman scattering data and prove the fredholm property of the klein gordon operator with the associated atiyah patodi singer type boundary conditions at infinite times we then construct a parametrix with compact remainder terms for the fredholm problem and prove that it is also a feynman parametrix in the sense of duistermaat and hormander

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

stability of planar shock fronts for multidimensional systems of relaxation equations abstract we investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems our main result generalizing the work of kwon and zumbrun in the scalar relaxation case is to establish the bounds on the green s function for the linearized equation and obtain nonlinear l 2 asymptotic behavior sharp decay rate of perturbed weak shock profiles to establish green s function bounds we use the semigroup approach in the low frequency regime and use the energy method for the high frequency bounds separately for the system equilibrium case the analysis of the linearized equation is complicated due to glancing phenomena we treat this difficulty similarly as in the inviscid and viscous systems under the constant multiplicity condition

local wick polynomials and time ordered products of quantum fields in curved spacetime in order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime it is necessary to construct wick polynomials and their time ordered products for the noninteracting theory a construction of these quantities has recently been given by brunetti fredenhagen and kohler and by brunetti and fredenhagen but they did not impose any locality or covariance condition in their constructions as a consequence their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters in this paper we construct an extended wick polynomial algebra large enough to contain the wick polynomials and their time ordered products by generalizing a construction of dutsch and fredenhagen to curved spacetime we then define the notion of a local covariant quantum field and seek a definition of local wick polynomials and their time ordered products as local covariant quantum fields we introduce a new notion of the scaling behavior of a local covariant quantum field and impose scaling requirements on our local wick polynomials and their time ordered products as well as certain additional requirements such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric for a given polynomial order in powers of the field we prove that these conditions uniquely determine the local wick polynomials and their time ordered products up to a finite number of parameters these parameters correspond to the usual renormalization ambiguities occurring in minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature we also prove existence of local wick polynomials however the issue of existence of local time ordered products is deferred to a future investigation

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 luscher weisz string but we allow for arbitrary classical background string configurations and keep the diffeomorphism invariance

renormalization proof for massive phi_4 4 theory on riemannian manifolds in this paper we present an inductive renormalizability proof for massive phi 4 4 theory on riemannian manifolds based on the wegner wilson flow equations of the wilson renormalization group adapted to perturbation theory the proof goes in hand with bounds on the perturbative schwinger functions which imply tree decay between their position arguments an essential prerequisite is precise bounds on the short and long distance behaviour of the heat kernel on the manifold with the aid of a regularity assumption often taken for granted we also show that for suitable renormalization conditions the bare action takes the minimal form that is to say there appear the same counterterms as in flat space apart from a logarithmically divergent one which is proportional to the scalar curvature

algebraic quantum field theory perturbation theory and the loop expansion the perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory we show that the algebra of interacting fields is additive i e fully determined by its subalgebras associated to arbitrary small subregions of minkowski space we also give an algebraic formulation of the loop expansion by introducing a projective system n of observables up to n loops where 0 is the poisson algebra of the classical field theory finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of green s functions

conformal generally covariant quantum field theory the scalar field and its wick products in this paper we generalize the construction of generally covariant quantum theories given in bfv03 to encompass the conformal covariant case after introducing the abstract framework we discuss the massless conformally coupled klein gordon field theory showing that its quantization corresponds to a functor between two certain categories at the abstract level the ordinary fields could be thought of as natural transformations in the sense of category theory we show that the wick monomials without derivatives wick powers can be interpreted as fields in this generalized sense provided a non trivial choice of the renormalization constants is given a careful analysis shows that the transformation law of wick powers is characterized by a weight and it turns out that the sum of fields with different weights breaks the conformal covariance at this point there is a difference between the previously given picture due to the presence of a bigger group of covariance it is furthermore shown that the construction does not depend upon the scale appearing in the hadamard parametrix used to regularize the fields finally we briefly discuss some further examples of more involved fields

anomalies in time ordered products and applications to the bv brst formulation of quantum gauge theories we show that every graded derivation on the algebra of free quantum fields and their wick powers in curved spacetimes gives rise to a set of anomalous ward identities for time ordered products with an explicit formula for their classical limit we study these identities for the koszul tate and the full brst differential in the bv brst formulation of perturbatively interacting quantum gauge theories and clarify the relation to previous results in particular we show that the quantum brst differential the quantum antibracket and the higher order anomalies form an l infty algebra the defining relations of this algebra ensure that the gauge structure is well defined on cohomology classes of the quantum brst operator i e observables furthermore we show that one can determine contact terms such that also the interacting time ordered products of multiple interacting fields are well defined on cohomology classes an important technical improvement over previous treatments is the fact that all our relations hold off shell and are independent of the concrete form of the lagrangian including the case of open gauge algebras

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 this closed expression which we call the epstein glaser forest formula is analogous to zimmermann s forest formula for bph renormalization 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 stuckelberg and petermann

supergeometry in locally covariant quantum field theory in this paper we analyze supergeometric locally covariant quantum field theories we develop suitable categories sloc of super cartan supermanifolds which generalize lorentz manifolds in ordinary quantum field theory and show that starting from a few representation theoretic and geometric data one can construct a functor mathfrak a sloc to s alg to the category of super algebras which can be interpreted as a non interacting super quantum field theory this construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small we then solve this problem by using techniques from enriched category theory which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints we construct super quantum field theories in terms of enriched functors mathfrak e mathfrak a esloc to es alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework as examples we analyze the superparticle in 1 1 dimensions and the free wess zumino model in 3 2 dimensions

on the renormalization group in curved spacetime we define the renormalization group flow for a renormalizable interacting quantum field in curved spacetime via its behavior under scaling of the spacetime metric g 2g we consider explicitly the case of a scalar field with a self interaction of the form 4 although our results should generalize straightforwardly to other renormalizable theories we construct the interacting field as well as its wick powers and their time ordered products as formal power series in the algebra generated by the wick powers and time ordered products of the free field and we determine the changes in the interacting field observables resulting from changes in the renormalization prescription our main result is the proof that for any fixed renormalization prescription the interacting field algebra for the spacetime m 2g with coupling parameters p is isomorphic to the interacting field algebra for the spacetime m g but with different values p of the coupling parameters the map p p yields the renormalization group flow the notion of essential and inessential coupling parameters is defined and we define the notion of a fixed point as a point p in the parameter space for which there is no change in essential parameters under renormalization group flow

analytic dependence is an unnecessary requirement in renormalization of locally covariant qft finite renormalization freedom in locally covariant quantum field theories on curved spacetime is known to be tightly constrained under certain standard hypotheses to the same terms as in flat spacetime up to finitely many curvature dependent terms these hypotheses include in particular locality covariance scaling microlocal regularity and continuous and analytic dependence on the metric and coupling parameters the analytic dependence hypothesis is somewhat unnatural because it requires that locally covariant observables which are simultaneously defined on all spacetimes depend continuously on an arbitrary metric with the dependence strengthened to analytic on analytic metrics moreover the fact that analytic metrics are globally rigid makes the implementation of this requirement at the level of local algebras of observables rather technically cumbersome we show that the conditions of locality covariance scaling and a naturally strengthened microlocal spectral condition are actually sufficient to constrain the allowed finite renormalizations equally strongly thus eliminating both the continuity and the somewhat unnatural analyticity hypotheses the key step in the proof uses the peetre slovak theorem on the characterization of in general non linear differential operators by their locality and regularity properties

functional properties of hormander s space of distributions having a specified wavefront set the space mathcal d gamma prime of distributions having their wavefront sets in a closed cone gamma has become important in physics because of its role in the formulation of quantum field theory in curved spacetime in this paper the topological and bornological properties of mathcal d gamma prime and its dual mathcal e lambda prime are investigated it is found that mathcal d gamma prime is a nuclear semi reflexive and semi montel complete normal space of distributions its strong dual mathcal e lambda prime is a nuclear barrelled and ultra bornological normal space of distributions which however is not even sequentially complete concrete rules are given to determine whether a distribution belongs to mathcal d gamma prime whether a sequence converges in mathcal d gamma prime and whether a set of distributions is bounded in mathcal d gamma prime

micro local approach to the hadamard condition in quantum field theory on curved space time for the two point distribution of a quasi free klein gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space time we prove the equivalence of 1 the global hadamard condition 2 the property that the feynman propagator is a distinguished parametrix in the sense of duistermaat and hormander and 3 a new property referred to as the wave front set spectral condition wfssc because it is reminiscent of the spectral condition in axiomatic quantum field theory on minkowski space results in micro local analysis such as the propagation of singularities theorem and the uniqueness up toc of distinguished parametrices are employed in the proof we include a review of kay and wald s rigorous definition of the global hadamard condition and the theory of distinguished parametrices specializing to the case of the klein gordon operator on a globally hyperbolic space time as an alternative to a recent computation of the wave front set of a globally hadamard two point distribution on a globally hyperbolic curved space time given elsewhere by kohler to correct an incomplete computation in 32 we present a version of this computation that does not use a deformation argument such as that used in fulling narcowich and wald and is independent of the cauchy evolution argument of fulling sweeny and wald both of which are relied upon in kohler s proof this leads to a simple micro local proof of the preservation of hadamard form under cauchy evolution first shown by fulling sweeny and wald relying only on the propagation of singularities theorem in another paper 33 the equivalence theorem is used to prove a conjecture by kay that a locally hadamard quasi free klein gordon state on any globally hyperbolic curved space time must be globally hadamard

boundary regularity of proper holomorphic mappings it s main result is formulated in theorem 2 below although the answer to the conjecture in the general case is still unknown it has been shown to hold in many special cases a fairly detailed history of this subject will be contained in 14 one class of positive answers deals with cases in which one knows that f21 and 2 posses fairly large classes of holomorphic automorphisms that extend holomorphically across the boundaries like for instance reinhardt domains see e g h cartan 8 w kaup 16 and st bell 6 in these situations one even often does not need a smoothness assumption on the boundaries of 1 and 22 on the other hand the first result without a priori knowledge on automorphisms which gave c extendability was proved by ch fefferman in 1974 in 15 and says

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

supersymmetric field theoretic models on a supermanifold we propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general space time manifold so as to include superfield models on a supermanifold we only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure our considerations are based on the catenacci reina teofillatto bryant approach to supermanifolds in particular we show that the class of supermanifolds constructed by bonora pasti tonin satisfies the criteria which guarantee that a supermanifold admits a hausdorff body manifold this construction is the closest to the physicist s intuitive view of superspace as a manifold with some anticommuting coordinates where the odd sector is topologically trivial the paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects moreover a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions which is equivalent to the requirement that all of the component fields satisfy on the body manifold a microlocal spectral condition proposed by brunetti fredenhagen kohler

quantum energy inequalities and local covariance i globally hyperbolic spacetimes we begin a systematic study of quantum energy inequalities qeis in relation to local covariance we define notions of locally covariant qeis of both absolute and difference types and show that existing qeis satisfy these conditions local covariance permits us to place constraints on the renormalized stress energy tensor in one spacetime using qeis derived in another in subregions where the two spacetimes are isometric this is of particular utility where one of the two spacetimes exhibits a high degree of symmetry and the qeis are available in simple closed form various general applications are presented including a priori constraints depending only on geometric quantities on the ground state energy density in a static spacetime containing locally minkowskian regions in addition we present a number of concrete calculations in both two and four dimensions that demonstrate the consistency of our bounds with various known ground and thermal state energy densities examples considered include the rindler and misner spacetimes and spacetimes with toroidal spatial sections in this paper we confine the discussion to globally hyperbolic spacetimes subsequent papers will also discuss spacetimes with boundary and other related issues

existence of local covariant time ordered products of quantum fields in curved spacetime we establish the existence of local covariant time ordered products of local wick polynomials for a free scalar field in curved spacetime our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem so our construction essentially completes the analysis of the existence uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime as a byproduct of our analysis we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times lorentz invariant distributions plus a remainder term of arbitrarily low scaling degree

the microlocal spectrum condition initial value formulations and background independence we analyze implications of the microlocal spectrum hadamard condition for states in a linear quantum field theory on a globally hyperbolic spacetime m in the context of a distributional initial value formulation more specifically we work in 3 1 split m and give a bound independent of the spacetime metric on the wave front sets of the initial data for a quasi free hadamard state in a quantum field theory defined by a normally hyperbolic differential operator p acting in a vector bundle e m this aims at a possible way to apply the concept of hadamard states within approaches to quantum field theory gravity relying on a hamiltonian formulation potentially without a classical background metric g