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 Dütsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.     

Quantum relativity theory and quantum space-time

A quantum relativity theory formulated in terms of Davis' quantum relativity principle is outlined. The first task in this theory as in classical relativity theory is to model space-time, the arena of natural processes. It is shown that the quantum space-time models of Banai introduced in another paper is formulated in terms of Davis' quantum relativity. The recently proposed classical relativistic quantum theory of Prugoveki and his corresponding classical relativistic quantum model of space-time open the way to introduce, in a consistent way, the quantum space-time model (the quantum substitute of Minkowski space) of Banai proposed in the paper mentioned. The goal of quantum mechanics of quantum relativistic particles living in this model of space-time is to predict the rest mass system properties of classically relativistic (massive) quantum particles (elementary particles). The main new aspect of this quantum mechanics is that provides a true mass eigenvalue problem, and that the excited mass states of quantum relativistic particles can be interpreted as elementary particles. The question of field theory over quantum relativistic model of space-time is also discussed. Finally it is suggested that quarks should be considered as quantum relativistic particles.Supported by the Hungarian Academy of Sciences.     

The classical field limit of scattering theory for non-relativistic many-boson systems. II

We study the classical field limit of non relativistic many-boson theories in space dimensionn3, extending the results of a previous paper to more singular interactions. We prove the expected results: when tends to zero, the quantum theory tends in a suitable sense to the corresponding classical field theory, and the quantum fluctuations are governed by the equation obtained by linearizing the quantum evolution equation around the classical solution. These results hold uniformly in time and therefore apply to scattering theory. The interactions considered here are so singular as to require a change of domain in order to define the generator of the evolution of the fluctuations, but sufficiently regular so that no energy renormalization is needed.     

Geometrization of the physics with teleparallelism. II. Towards a fully geometric Dirac equation

In an accompanying paper (I), it is shown that the basic equations of the theory of Lorentzian connections with teleparallelism (TP) acquire standard forms of physical field equations upon removal of the constraints represented by the Bianchi identities. A classical physical theory results that supersedes general relativity and Maxwell-Lorentz electrodynamics if the connection is viewed as Finslerian. The theory also encompasses a short-range, strong, classical interaction. It has, however, an open end, since the source side of the torsion field equation is not geometric.In this paper, Kaehler's partial geometrization of the Dirac equation is taken as a starting point for the development of fully geometric Dirac equations via the correspondence principle given in I. For this purpose, Kaehler's calculus (where the spinors are differential forms) is generalized so that it also applies when the torsion is not zero. The point is then made that the forms can take values in tangent Clifford algebras rather than in tensor algebras. The basic Eigenschaft of the Kaehler calculus also is examined from the physical perspective of dimensional analysis.Geometric Dirac equations of great structural simplicity are finally inferred from the standard Dirac equation by using the aforementioned correspondence principle. The realm of application of the Dirac theory is thus enriched in principle, though only at an abstract level at this point: the standard spinors, which are scalar-valued forms in the Kaehler version of that theory, become Clifford-valued. In addition, the geometrization of the Dirac equation implies a geometrization of the Dirac current. When this current is replaced in the field equations for the torsion, the theory of Paper I becomes fully geometric.     

Conformal transformations of <Emphasis Type="Italic">S</Emphasis>-matrix in scalar field theory

In this paper, three methods for describing the conformal transformations of the S-matrix in quantum field theory are proposed. They are illustrated by applying the algebraic renormalization procedure to the quantum scalar field theory, defined by the LSZ reduction mechanism in the BPHZ renormalization scheme. Central results are shown to be independent of scheme choices and derived to all orders in loop expansions. Firstly, the local Callan-Symanzik equation is constructed, in which the insertion of the trace of the energy-momentum tensor is related to the beta function and the anomalous dimension. With this result, the Ward identities for the conformal transformations of the Green functions are derived. Then the conformal transformations of the S-matrix defined by the LSZ reduction procedure are calculated. Secondly, the conformal transformations of the S-matrix in the functional formalism are related to charge constructions. The commutators between the charges and the S-matrix operator are written in a compact way to represent the conformal transformations of the S-matrix. Lastly, the massive scalar field theory with local coupling is introduced in order to control breaking of the conformal invariance further. The conformal transformations of the S-matrix with local coupling are calculatedReceived: 3 June 2003, Revised: 24 July 2003, Published online: 2 October 2003Yong Zhang: Supported by Graduiertenkolleg Quantenfeldtheorie: Mathematische Struktur und physikalische Anwendungen, University Leipzig.     

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k|g²T. Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: <hep-ph/9810313>; G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: <hep-ph/0001216>]. In this work we provide a complementary, more analytic approach based on Dyson–Schwinger equations. Using methods known from stochastic quantitation, we recast Bödeker’s Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally, Dyson–Schwinger equations are derived.     

Local approach to dilatation invariance

Scale invariance is analyzed locally by coupling the energy-momentum tensor to a source which is the metric field of curved space-time. The resulting theory at the classical level has no mass parameters only if the general coordinate transformation group can be represented in Weyl's scheme. We further discuss the quantum extension of the theory; the Ward identities become anomalous under radiative corrections and the anomaly is shown to be connected with the instability of the classical metric field representation. The anomalies, recognized as the well-known trace anomalies for the energy-momentum tensor, are then reabsorbed by a perturbative alteration of the original metric field transformation law and we prove the modified Ward identities to be renormalizable in the flat limit. Finally we show that our approach is equivalent to the well-known parametric equations of the Callan-Symanzik type only if the dilatation invariance is not spontaneously broken. In the presence of spontaneous scale breaking we derive a functional equation which will be applied to cases of physical interest in a forthcoming paper.     

The classical field limit ofP(ϕ)2 quantum field theory

It will be shown that, for a convex polynomialP, theP()₂ quantum field theory without cutoff has a classical field limit as Planck's constanth tends to zero. This extends work of Hepp [1], who considered theories with a space cutoff.     

Quantum mechanics in a discrete model of classical physics

The role of probability theory in classical physics is examined. It is found that the probabilities for the outcomes of typical experiments depend strongly on the assumed behavior of given classical models at infinity. A discrete classical model is introduced and it is shown that the resulting probabilities are similar to those in the usual theory of quantum mechanics.     

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then construct a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.The author is supported by NSF grant DMS-8805684, a Presidential Young Investigators award DMS-9057144, and by the O'Donnell Foundation. He warmly thanks the Geometry Center at the University of Minnesota for their hospitality while this work was undertaken     

Symplectic approach to the theory of quantized fields. I

Our aim in this paper the first one of a series concerned with the problem of field quantization starting from the symplectic structure underlying the classical theory, is to build up the variational theory necessary to all further constructions. The basic notions are the vertical bundle and thestructure 1-form used to define thegeneralized infinitesimal contact transformation which allows us to state and solve the variational problem related to field physics.Giving a system of modulevalued differential forms of different degree on the vertical bundle which solutions are the stationary cross sections is the main result in the paper. In this scheme the Euler-Lagrange classical equations are the expressions induced by such a system of differential forms on any cross section of the vertical bundle. This gives us a complete linearization of the Euler-Lagrange equations and, starting from it, a natural globalization of these equations. Finally, the notion of variational problem invariant by a Lie group is defined in this scheme, Noether's theorem related to such invariant problem is formulated and an intrinsic version of the so-called Noether invariants of classical variational calculus is obtained.This work has been realized in the Seminar of Mathematical Physics, directed by ProfessorJ. Sancho, in the Faculty of Science at the University of Barcelona (Spain).     

Gauge theories on four dimensional Riemannian manifolds

This paper develops the Riemannian geometry of classical gauge theories — Yang-Mills fields coupled with scalar and spinor fields — on compact four-dimensional manifolds. Some important properties of these fields are derived from elliptic theory: regularity, an energy gap theorem, the manifold structure of the configuration space, and a bound for the supremum of the field in terms of the energy. It is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities (this extends a theorem of K. Uhlenbeck).The author holds an A.M.S. Postdoctoral Fellowship     

WKB properties of the time-dependent Schrödinger system

It is shown that the time-dependent WKB expansion highlights some of the hidden properties of the Schrödinger equation and forms a natural bridge between that equation and the functional integral formulation of quantum mechanics. In particular it is shown that the leading (zero- and first-order in ) terms in the WKB expansion are essentially classical, and the relationship of this result to the classical nature of the WKB partition function, and of the anomalies in quantum field theory, is discussed.     

Field theoretical construction of an infinite set of quantum commuting operators related with soliton equations

The quantum version of an infinite set of polynomial conserved quantities of a class of soliton equations is discussed from the point of view of naive continuum field theory. By using techniques of two dimensional field theories, we show that an infinite set of quantum commuting operators can be constructed explicitly from the knowledge of its classical counterparts. The quantum operators are so constructed as to coincide with the classical ones in the 0 limit (; Planck's constant divided by 2). It is expected that the explicit forms of these operators would shed some light on the structure of the infinite dimensional Lie algebras which underlie a certain class of quantum integrable systems.     

Neutrino radiation in spherically-symmetric gravitational fields I. The energy-momentum tensor for the one-neutrino and many-particle neutrino field

The physical situation of a star emitting neutrinos is considered. Some difficulties in the classical theory are mentioned, and a more detailed approach to the properties of neutrino radiation in general relativity is given. The classical theory is here regarded as a one-particle theory, and by summing over many particles propagating in randon radial directions, the energy-momentum tensor of the total radiation field is shown to approximate to the geometrical optics type satisfying other conditions defining its radial and time dependence.     

Renarks on the classical limit of quantum field theories

Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called constructive approach to quantum field theory.After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limith0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimmet al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared.     

The loop expansion for the effective potential in theP(φ)2 quantum field theory

We study the loop expansion for the effective potential, defined as the Fenchel transform (convex conjugate) of the pressure in an external field, in theP()₂ quantum field theory. For values of the classical fielda for which the classical potentialU ₀(a)=P(a)+1/2m ² a ² equals its convex hull and has nonvanishing curvature we prove that the 1-PI loop expansion is asymptotic as 0. We also give an example of a double well classical potential for which the 1-PI loop expansion fails to be asymptotic, and find the true asymptotics.This paper is a condensed version of the author's Ph.D. thesis for the Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Y4     

Electrodynamics of moving particles

A consistent relativistic theory of the classical Maxwell field interacting with classical, charged, point-like particles is proposed. The theory is derived from a classical soliton-like model of an extended particle. An approximation procedure for such a model is developed, which leads to an already renormalized formula for the total four-momentum of the system composed of fields and particles. Conservation of this quantity leads to a theory which is universal (i.e. does not depend upon a specific model we start with) and which may be regarded as a simple and necessary completion of special relativity. The renormalization method proposed here may be considered as a realization of Einstein's idea of deriving equations of motion from field equations. It is shown that the Dirac's 3-dots equation does not describe a fundamental law of physics, but only a specific family of solutions of our theory, corresponding to a specific choice of the field initial data.     

Classical theory of laser linewidth

The classical theory of light fluctuations rests on the intuitive concept that jumps between atomic states occur at independent times when the optical field has a prescribed value. The statistical properties of phase-noise sources are obtained in the present paper by applying this principle to detuned atoms. Formulae for amplitude and phase fluctuations coincide with quantum-theory results even when non-classical states of light are generated. Theories employing semiclassical or quantum concepts are reviewed. We consider particularly the linewidth of laser oscillators operating below and above threshold when the atomic polarization cannot be adiabatically eliminated. Quantum-theory results by Lax (1966) are recovered from classical theory in a straightforward manner. More general results are given for dispersive loads, applicable to external-cavity lasers and relevant to gain guidance. It is emphasized that the K-factor as calculated by Petermann is applicable only below threshold. When more than one emitting element is present, population rate equations need to be considered and the linewidth decreases when the pump fluctuations are suppressed. The role of gain compression relating to semiconductor lasers is discussed. It is shown that at low and moderate powers gain compression reduces the effective phase-amplitude coupling factor, . But at high power a number of mechanisms contribute to linewidth rebroadening. One of them is the statistical (quasi-thermal equilibrium) fluctuation of the refractive index. General concepts applicable to broadband light are outlined in an appendix.     

Interaction in scalar theories of gravitation

It is shown that O. Bergmann's (1956) scalar field theory is similar to G. Nordström's (1912). The interaction term in the former's theory is equivalent to non-linearising the Nordström theory by including twice the energy density of the field as a source term in the Poisson-like equation. It is further shown that, if the interaction term (1+v) in Bergmann's theory is replaced by (1+v)², then the subsequent field equation appears more reasonable in that the energy density (not twice) appears as a source term.