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 Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler 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. Received: 31 March 1999 / Accepted: 10 June 1999     

Polarized vector bosons on the de Sitter expanding universe

The quantum theory of the vector field minimally coupled to the gravity of the de Sitter spacetime is built in a canonical manner starting with a new complete set of quantum modes of given momentum and helicity derived in the moving chart of conformal time. It is shown that the canonical quantization leads to new vector propagators which satisfy similar equations as the propagators derived by Tsamis and Woodard (J Math Phys 48:052306, 2007) but having a different structure. The one-particle operators are also written down pointing out that their properties are similar with those found already in the quantum theory of the scalar, Dirac and Maxwell free fields.     

Linear spin-zero quantum fields in external gravitational and scalar fields

The quantum theory of both linear, and interacting fields on curved space-times is discussed. It is argued that generic curved space-time situations force the adoption of the algebraic approach to quantum field theory: and a suitable formalism is presented for handling an arbitrary quasi-free state in an arbitrary globally hyperbolic space-time.For the interacting case, these quasi-free states are taken as suitable starting points, in terms of which expectation values of field operator products may be calculated to arbitrary order in perturbation theory. The formal treatment of interacting fields in perturbation theory is reduced to a treatment of free quantum fields interacting with external sources.Central to the approach is the so-called two-current operator, which characterises the effect of external sources in terms of purely algebraic (i.e. representation free) properties of the source-free theory.The paper ends with a set of Feynman rules which seems particularly appropriate to curved space-times in that it takes care of those aspects of the problem which are specific to curved space-times (and independent of interaction). Heuristically, the scheme calculates in-in rather than in-out matrix elements. Renormalization problems are discussed but not treated.Work partly supported by the Schweizerische Nationalfonds     

The hypergeneralized Heun equation in quantum field theory in curved space-times

We show for the first time the role played by the hypergeneralized Heun equation (HHE) in the context of quantum field theory in curved space-times. More precisely, we find suitable transformations relating the separated radial and angular parts of a massive Dirac equation in the Kerr-Newman-deSitter metric to a HHE.     

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin–Marinov action

It is known that the actions of field theories on a noncommutative space-time can be written as some modified (we call them θ-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and the usual quantum mechanical features of the corresponding field theory. In the present article, we discuss the problem of constructing θ-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract θ-modified actions of the relativistic particles from path-integral representations of the corresponding noncommutative field theory propagators. We consider the Klein–Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as θ-modified actions of the relativistic particles. To confirm the interpretation, we canonically quantize these actions. Thus, we obtain the Klein–Gordon and Dirac equations in the noncommutative field theories. The θ-modified action of the relativistic spinning particle is just a generalization of the Berezin–Marinov pseudoclassical action for the noncommutative case.     

Quantum Fields Obtained from Convoluted Generalized White Noise Never Have Positive Metric

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (Lévy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by Baumann, based on the Dell’Antonio–Robinson–Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.     

Path integration in non-relativistic quantum mechanics

A new method for computing path integrals explicitly is developed and applied to problems in non-relativistic quantum mechanics, such as: wave functions, propagators on configuration spaces and on phase space, caustic problems, bound states. Path integrals for paths on curved spaces and for paths on multiply-connected spaces are computed.     

Gravitational interaction and spontaneous breaking of symmetries

An explicit example of spontaneous symmetry breaking due to gravitational interaction is given. It is shown that, in the framework of quantum field theory in curved space-times, the dragging of inertial frame effects lead to a spontaneous symmetry breaking in the ultrarelativistic regime. This situation is compared with those that arise in other interactions. It is pointed out that spinning-up of relativistic configurations is analogous to cooling-down of systems in solid state physics-especially in magnetism-and in high-energy physics. This analogy may turn out to be significant in the investigation of thermodynamical properties of the gravitational field.This essay was awarded the first prize for 1977 by the Gravity Research Foundation.Supported in part by the NSF contract PHY 76-81102 with the University of Chicago.     

A Relativistic Version of the Ghirardi–Rimini–Weber Model

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.     

Higher-spin fields in a curved space-time

Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.     

Multivalued fields on the complex plane and conformal field theories

In this paper a class of conformal field theories with nonabelian and discrete group of symmetry is investigated. These theories are realized in terms of free scalar fields starting from the simpleb-c systems and scalar fields on algebraic curves. The Knizhnik-Zamolodchikov equations for the conformal blocks can be explicitly solved. Besides the fact that one obtains in this way an entire class of theories in which the operators obey nonstandard statistics, these systems are interesting in exploring the connection between statistics and curved space-times, at least in the two dimensional case.Work supported by the Consiglio Nazionale Ricerche, P.le A. Moro 7, Roma, Italy     

Renormalization and asymptotic behavior in a finite quantum field theory with shadow states

In this paper we present a study of the renormalization problem in a finite quantum field theory with shadow states for a system of a physical scalar field interacting with a physical fermion field. In order to make the theory finite, two fermion shadow fields are introduced. We observe that the stability criterion of renormalization can not be satisfied simultaneously by both physical fields and shadow fields, if the finiteness of the theory is to be maintained. A physical interpretation of this result is given. Furthermore, we find that the effective complete propagators for large space-like momenta behave like free field propagators without the logarithmic factors observed in the non-abelian gauge theory.     

Deformations of Quantum Field Theories on Spacetimes with Killing Vector Fields

The recent construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of curved spacetimes. These spacetimes carry a family of wedge-like regions which share the essential causal properties of the Poincaré transforms of the Rindler wedge in Minkowski space. In the setting of deformed quantum field theories, they play the role of typical localization regions of quantum fields and observables. As a concrete example of such a procedure, the deformation of the free Dirac field is studied.     

Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and the role of physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.     

Small cosmological constant from the QCD trace anomaly?

According to recent astrophysical observations the large scale mean pressure of our present Universe is negative suggesting a positive cosmological constant-like term. The issue of whether nonperturbative effects of self-interacting quantum fields in curved space-times may yield a significant contribution is addressed. Focusing on the trace anomaly of quantum chromodynamics, a preliminary estimate of the expected order of magnitude yields a remarkable coincidence with the empirical data, indicating the potential relevance of this effect.     

Thermal representation of the energy density in bounded spaces

The energy-momentum tensor of a quantum massless free field in a curved spacetime can be written in many cases as an integral with a thermal denominator and a modified phase-space numerator. It is shown that in general the thermal denominator is related to the bounded nature of the system, which in turn implies a representation of the energy density as an infinite numerable sum in Fock space. The modification of the phase-space density is related to the absence of long-wave contributions for nonzero values of the spin.     

Twistor theory: An approach to the quantisation of fields and space-time

Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity. Twistors for flat space-time are the SU(2,2) spinors of the twofold covering group O(2,4) of the conformal group. They describe the momentum and angular momentum structre of zero-rest-mass particles. Space-time points arise as secondary concepts corresponding to linear sets in twistor space. They, rather than the null cones, should become “smeared out” on passage to a quantised gravitational theory. Twistors are represented here in two-component spinor terms. Zero-rest-mass fields are described by holomorphic functions on twistor space, on which there is a natural canonical structure leading to a natural choice of canonical quantum operators. The generalisation to curved space can be accomplished in three ways; i) local twistors, a conformally invariant calculus, ii) global twistors, and iii) asymptotic twistors which provide the framework for an S-matrix approach in asymptotically flat space-times. A Hamiltonian scattering theory of global twistors is used to calculate scattering cross-sections. This leads to twistor analogues of Feynman graphs for the treatment of massless quantum electrodynamics. The recent development of methods for dealing with massive (conformal symmetry breaking) sources and fields is briefly reviewed.