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.     

Resummation of Mass Terms in Perturbative Massless Quantum Field Theory

The neutral massless scalar quantum field Φ in four-dimensional space-time is considered, which is subject to a simple bilinear self-interaction. Is is well-known from renormalization theory that adding a term of the form to the Lagrangean has the formal effect of shifting the particle mass from the original zero value to m after resummation of all two-leg insertions in the Feynman graphs appearing in the perturbative expansion of the S-matrix. However, this resummation is accompanied by some subtleties if done in a proper mathematical manner. Although the model seems to be almost trivial, is shows many interesting features which are useful for the understanding of the convergence behavior of perturbation theory in general. Some important facts in connection with the basic principles of quantum field theory and distribution theory are highlighted, and a remark is made on possible generalizations of the distribution spaces used in local quantum field theory. A short discussion how one can view the spontaneous breakdown of gauge symmetry in massive gauge theories within a massless framework is presented.      

Star Products and Quantum Algebras

I show explicitly that the star product on atriangular Poisson Lie group leads to a quantum algebrastructure (triangular Hopf algebra) on the quantizedenveloping algebra of the Lie algebra of the Lie group, and that equivalent star-productsgenerate isomorphic quantum algebras.     

The Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes

The time slice axiom states that the observables which can be measured within an arbitrarily small time interval suffice to predict all other observables. While well known for free field theories where the validity of the time slice axiom is an immediate consequence of the field equation it was not known whether it also holds in generic interacting theories, the only exception being certain superrenormalizable models in 2 dimensions. In this paper we prove that the time slice axiom holds at least for scalar field theories within formal renormalized perturbation theory.     

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 μ appearing in the Hadamard parametrix, used to regularize the fields. Finally, we briefly discuss some further examples of more involved fields.     

The Operator Product Expansion Converges in Perturbative Field Theory

We show, within the framework of the massive Euclidean j⁴{\varphi^4} -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.     

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

Some Perturbative and Non-Perturbative Results Exposing Certain Field Aspects of Reggeon Field Theory

The present work is to discuss certain aspects of Reggeon field theory. Reggeon field theory demands special interest since the Reggeons are considered to be quasi particles of non-relativistic world. Scale symmetry has been applied in RFT effecting modification in the Re-normalization group equation (RGE) and the possible break down in the exact symmetry has been pointed out. The non-perturbative functional method has been applied to RFT under the Eikonal approximation resulting in an expected high energy behaviour.     

Braided Quantum Field Theory

We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for n-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to φ⁴-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. Received: 3 July 1999 / Accepted: 10 November 2000     

PT-Symmetric Quantum Field Theory

In the context of the PT-symmetric version of quantum electrodynamics, it is argued that the C-operator introduced in order to define a unitary inner product has nothing to do with charge conjugation.     

Shuffling Quantum Field Theory

We discuss shuffle identities between Feynman graphs using the Hopf algebra structure of perturbative quantum field theory. For concrete exposition, we discuss vertex function in massless Yukawa theory.     

Relative Quantum Field Theory

We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions.     

Relativistic Quantum Mechanics and Field Theory

The problems which arise for a relativistic quantum mechanics are reviewed and critically examined in connection with the foundations of quantum field theory. The conflict between the quantum mechanical Hilbert space structure, the locality property and the gauge invariance encoded in the Gauss' law is discussed in connection with the various quantization choices for gauge fields.     

Quantum Field Theory and Dense Measurement

We show, using quantum field theory (QFT), that performing a large number of identical repetitions of the same measurement does not only preserve the initial state of the wave function (the Zeno effect), but also produces additional physicaleffects. We first discuss the Zeno effect in the framework of QFT, that is, as a quantum field phenomenon. We then derive it from QFT for the general case in which the initial and final states are different. We use perturbation theory and Feynman diagrams and refer to the measurement act as an external constraint upon the system that corresponds to the perturbative diagram that denotes this constraint. The basic physical entities dealt with in this work are not the conventional once-perfomed physical processes, but their n times repetition where n tends to infinity. We show that the presence of these repetitions entails the presence of additional excited state energies, and the absence of them entails the absence of these excited energies.     

Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.     

Ultraviolet Finite Quantum Field Theory on Quantum Spacetime

 We discuss a formulation of quantum field theory on quantum space time where the perturbation expansion of the S-matrix is term by term ultraviolet finite. The characteristic feature of our approach is a quantum version of the Wick product at coinciding points: the differences of coordinates q _j −q _k are not set equal to zero, which would violate the commutation relation between their components. We show that the optimal degree of approximate coincidence can be defined by the evaluation of a conditional expectation which replaces each function of q _j −q _k by its expectation value in optimally localized states, while leaving the mean coordinates invariant. The resulting procedure is to a large extent unique, and is invariant under translations and rotations, but violates Lorentz invariance. Indeed, optimal localization refers to a specific Lorentz frame, where the electric and magnetic parts of the commutator of the coordinates have to coincide [11]. Employing an adiabatic switching, we show that the S-matrix is term by term finite. The matrix elements of the transfer matrix are determined, at each order in the perturbative expansion, by kernels with Gaussian decay in the Planck scale. The adiabatic limit and the large scale limit of this theory will be studied elsewhere. Received: 15 January 2003 / Accepted: 20 March 2003 Published online: 5 May 2003 RID="*" ID="*" Research supported by MIUR and GNAMPA-INDAM RID="*" ID="*" Research supported by MIUR and GNAMPA-INDAM Communicated by H. Araki and D. Buchholz