Topological quantum field theory

A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.On leave from Department of Physics, Princeton University. Research supported in part by NSF Grants No. 80-19754, 86-16129, 86-20266     

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.     

Quaternionic quantum field theory

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. The theory takes the form of either a functional integral with quaternion-imaginary Lagrangian, or a Schrödinger equation and transformation theory for quaternion-valued wave functions, with a quaternion-imaginary Hamiltonian. The connection between the two formulations is developed in detail, and many related issues, including the breakdown of the correspondence principle and the Hilbert space structure, are discussed.     

Point-form quantum field theory

We examine canonical quantization of relativistic field theories on the forward hyperboloid, a Lorentz-invariant surface of the form x_μx^μ = τ². This choice of quantization surface implies that all components of the 4-momentum operator are affected by interactions (if present), whereas rotation and boost generators remain interaction free—a feature characteristic of Dirac’s “point-form” of relativistic dynamics. Unlike previous attempts to quantize fields on space-time hyperboloids, we keep the usual plane-wave expansion of the field operators and consider evolution of the system generated by the 4-momentum operator. We verify that the Fock-space representations of the Poincaré generators for free scalar and spin-1/2 fields look the same as for equal-time quantization. Scattering is formulated for interacting fields in a covariant interaction picture and it is shown that the familiar perturbative expansion of the S-operator is recovered by our approach. An appendix analyzes special distributions, integrals over the forward hyperboloid, that are used repeatedly in the paper.     

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ?_μΦ, and e^nΦ. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.     

Conformal field theory representation and background-independent action for superstring field theory

The two-dimensional conformal field theory representation of Witten's open superstring field theory is discussed. We argue that the previously suggested cubic action for the superstring is actually dependent on the spacetime background, and suggest a modification which formally removes this dependence. The transformation from the new version of the superstring cubic action to Witten's action for open string states is discussed. A class of linearized solutions to the equations of motion of the open superstring cubic action are exhibited. These solutions are in one-to-one correspondence with the physical excitations (both massless and massive) of the open and closed type I superstrings.     

Zitterbewegung in quantum field theory

Traditionally, the zitterbewegung （ZB） of the Dirac electron has just been studied at the level of quantum mechanics. Seeing the fact that an old interest in ZB has recently been rekindled by the investigations on spintronic, graphene, and superconducting systems, etc., this paper presents a quantum-field-theory investigation on ZB and obtains the conclusion that, the ZB of an electron arises from the influence of virtual electron-positron pairs （or vacuum fluctuations） on the electron.     

Two-dimensional conformal quantum field theory

On leave of absence from the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria.     

Flory radius of polymers in a periodic field: An exact analytic theory

We found an exact expression for the Flory radius R _F of Gaussian polymers placed in an external periodic field. This solution is expressed in terms of the two parameters η and a that describe the reduced strength of an external field and the period of the field to the polymer gyration radius ratio, respectively. R _F is found to be a decaying function of η for any values of a . Provided that the gyration radius is of the order of the period of an external field or less, the ground-state (GS) approximation of the exact result for R _F is shown to give qualitatively incorrect results. In addition to the “ground-state” contribution, the exact solution for R _F contains an additional term that is overlooked by the GS approximation. This term gives rise to the fact that R _F as a function of η exhibits power law behavior (rather than exponential decay obtained from the GS result) once η exceeds the threshold value η_con .     

Renormalizable noncommutative quantum field theory

We discuss the Euclidean noncommutative f⁴₄{\phi^4_4}-quantum field theory as an example of a renormalizable field theory. Using a Ward identity, Disertori, Gurau, Magnen and Rivasseau were able to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger–Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalisation from this almost solvable model.     

Duality in quantum field theory

We postulate a convergent version of operator product expansions on the vacuum and explore some of their consequences. They lead to structures much reminiscent of dual resonance models.     

Noncommutative quantum mechanics from noncommutative quantum field theory

We derive noncommutative multiparticle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Particles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously and propose a way to construct noncommutative SU(5) grand unified theory.     

Continuum regularization of quantum field theory

As a complement to our earlier study of renormalization at the Langevin regularized level, we report here on equivalent renormalization programs for regularized Schwinger-Dyson systems. Both one-loop and iterated loop renormalizations of the Green functions of QCD₄ are given, and are shown to be equivalent to the Langevin results. The optional apparent ?-renormalization discussed in IV is shown to apply as well to Schwinger-Dyson systems as to Langevin systems.     

Stochastic regularization of quantum field theory

The equivalence between a D-dimensional classical field theory coupled to an external random source having Gaussian correlations and its D−2 dimensional quantum counterpart was established. Utilizing this equivalence, a regularization procedure for scalar theories is developed. The regularization amounts to a compacification of the extra two dimensions. The regularization scheme is interpreted in terms of superpropagator modifications.     

Continuum regularization of quantum field theory

As an extension of our earlier one-loop renormalization studies at the regularized Schwinger-Dyson level, we report here on equivalent renormalization programs for regularized Langevin systems. Proper structure is discussed, and proper one-loop renormalizations of the Green functions of φ ₆ ³ and QCD₄ are given. An optional apparent?-renormalization is discussed as a technical simplificaiton for gauge theories with Zwanziger's gauge-fixing.     

Category of trees in representation theory of quantum algebras

New applications of categorical methods are connected with new additional structures on categories. One of such structures in representation theory of quantum algebras, the category of Kuznetsov-Smorodinsky-Vilenkin-Smirnov (KSVS) trees, is constructed, whose objects are finite rooted KSVS trees and morphisms generated by the transition from a KSVS tree to another one.