Introduction to string theory and conformal field theory

A concise survey of noncritical string theory and two-dimensional conformal field theory is presented. A detailed derivation of a conformal anomaly and the definition and general properties of conformal field theory are given. Minimal string theory, which is a special version of the theory, is considered. Expressions for the string susceptibility and gravitational dimensions are derived.     

Generalized measure theory

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic of quantum mechanics are given.     

Time-dependent density-matrix theory

The mass fluctuations in damped reactions of¹⁶O+¹⁶O are studied in an extended time-dependent Hartree-Fock theory. The theory determines the time evolution of a two-body density matrix as well as that of a one-body density matrix, providing us with a microscopic way to calculate the fluctuations of one-body quantities. The results of the theory are compared with those obtained in a transport model. It is found that the dispersions in fragment mass calculated in the two models are of the same order of magnitude and much larger than those calculated in the time-dependent Hartree-Fock theory. The differences between the microscopic theory and the transport model are also discussed.     

On the relation between Brueckner theory and Jastrow theory of nuclear matter

The two- and three-hole-line contributions to the ground state energy as calculated from Brueckner theory are derived from a cluster expansion followed by variation of the trial function. The implications of that derivation both for Brueckner theory and for Jastrow theory are worked out in detail. It is argued that the Jastrow theory is able to give simpler methods to calculate the ground state energy which may be of the same accuracy as current Brueckner calculations. It is shown that the single-particle potential of Brueckner theory is intimately related to a subsidiary condition used in the variation of the trial function. The main steps which have to be taken in a derivation of the general hole-line expansion from Jastrow theory are indicated. It is shown that the hole-line expansion is not a cluster expansion in the sense of Jastrow theory, and an interpretation is given of the “self-consistent choice” of the single-particle potential advocated in Brueckner theory.     

Discrete quantum theory

This paper is concerned with tracing the implications of two ideas as they affect quantum theory. One, which descends from Leibniz and Mach, is that there is no space-time continuum, but that which are involved are spacial and temporal relations involving the distant matter of the universe. The other is that our universe is finite. The picture of the world to which we are led is that of an enormous space-time Feynman diagram whose vertices are events. A consequence of finiteness is that between each pair of events, along a world line, there can be only finitely many intermediate events. A further change is that we are no longer required to believe that particles need be anywhere between events. The paper takes up nonrelativistic quantum theory in a way that is consistent with these ideas. By considering analogies between the Wiener and the Feynman integrals, and between the Wiener process and related discrete processes, there is obtained a straightforward theory for the Feynman integral. Propagators are worked out for many of the cases relevant to the nonrelativistic theory.The paper shows that, even when there are, along each world-line, no more than one event per Compton wavelength, agreement is good with the usual Schrödinger theory.Research supported in part by the NSF.     

Scalar-tensor theory and propagating torsion theory

We prove that scalar conformal transformations can convert the variational principle of the propagating torsion theory into the variational principle of general scalar-tensor theory, and show that scalar-tensor theory is conformally equivalent to propagating torsion theory.     

On second-quantized open superstring theory

The SO(32) theory, in the limit where it is an open superstring theory, is completely specified in the light-cone gauge as a second-quantized string theory in terms of a “matrix string” model. The theory is defined by the neighborhood of a 1 + 1-dimensional fixed point theory, characterized by an Abelian gauge theory with type IB Green-Schwarz form. Non-orientability and SO(32) gauge symmetry arise naturally, and the theory effectively constructs an orientifold projection of the (weakly coupled) matrix type IIB theory (also discussed herein). The fixed point theory is a conformal field theory with boundary, defining the free string theory. Interactions involving the interior of open and closed strings are governed by a twist operator in the bulk, while string endpoints are created and destroyed by a boundary twist operator.     

Thermodynamic-like transformations in information theory

This article explores the application of thermodynamic and statistical thermodynamic formalism to information theory problems. In particular, the applicability of the transformation theory of thermodynamics is investigated. After a brief tutorial discussion of thermodynamic and statistical thermodynamic methods and concepts, their information theory analogues are developed. Besides information theory entropy, the information theory counterparts of temperature, chemical potential, Helmholtz free energy, etc., are developed and related to conventional information theory concepts such as channel capacity, matching of source and channel, etc. Information theory theorems are proved via the statistical thermodynamic analogue method; and, finally, several problems are formulated and solved using thermodynamic-like transformations. This article is aimed chiefly at bridging the interface between the two disciplines, and is intended to be provocative. Therefore, no attempt has been made to have it be all inclusive.     

Quantum theory of Ur-objects as a theory of information

The quantum theory of ur-objects proposed by C. F. von Weizsäcker has to be interpreted as a quantum theory of information. Ur-objects, or urs, are thought to be the simplest objects in quantum theory. Thus an ur is represented by a two-dimensional Hilbert space with the universal symmetry groupSU(2), and can only be characterized asone bit of potential information. In this sense it is not a spatial but aninformation atom. The physical structure of the ur theory is reviewed, and the philosophical consequences of its interpretation as an information theory are demonstrated by means of some important concepts of physics such as time, space, entropy, energy, and matter, which in ur theory appear to be directly connected with information as the fundamental substance. This hopefully will help to provide a new understanding of the concept of information.     

Measurement theory for physics

A highly abstracted theory of measurement is synthesized from classical measurement theory, fuzzy set theory, generalized information theory, and predicate calculus. The theory does not require specific truth value concepts, nor does it specify what subsets of the reals can be observed, thus avoiding the usual fundamental difficulties. Problems such as the definition of systems, the significance of observations, numerical scales and observables, etc. are examined. The general logico-algebraic approach to quantum/classical physics is justified as a special case of measurement theory.     

Superstring theory

Dual string theories, initially developed as phenomenological models of hadrons, now appear more promising as candidates for a unified theory of fundamental interactions. Type I superstring theory (SST I), is a ten-dimensional theory of interacting open and closed strings, with one supersymmetry, that is free from ghosts and tachyons. It requires that an SO(n) or Sp(2n) gauge group be used. A light-cone-gauge string action with space-time supersymmetry automatically incorporates the superstring restrictions and leads to the discovery of type II superstring theory (SST II). SST II is an interacting theory of closed strings only, with two D = 10 supersymmetries, that is also free from ghosts and tachyons. By taking six of the spatial dimensions to form a compact space, it becomes possible to reconcile the models with our four-dimensional perception of spacetime and to define low-energy limits in which SST I reduces to N = 4, D = 4 super Yang-Mills theory and SST II reduces to N = 8, D = 4 supergravity theory. The superstring theories can be described by a light-cone-gauge action principle based on fields that are functionals of string coordinates. With this formalism any physical quantity should be calculable. There is some evidence that, unlike any conventional field theory, the superstring theories provide perturbatively renormalizable (SST I) or finite (SST II) unifications of gravity with other interactions.     

Segal's chronometric theory as a completion of the special theory of relativity

The main ideas and conclusions of the chronometric theory of I. Segal are surveyed. This theory differs from the special theory of relativity in that it replaces Minkowski space by a modelM=R ¹ ×S ³, but without specifying in the latter a Lorentz metric. In modelM a 15-dimensional conformal group acts globally. Some successful applications of the chronometric theory to cosmology are mentioned, and a number of examples are cited to show the prospects for its use in the physics of elementary particles. The chronometric theory is briefly compared with the twistor program of Penrose.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 84–89, August, 1993.     

An introduction to percolation theory

Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation. In this review the concepts of percolation theory and the general features associated with the critical region about the onset of percolation are developed in detail. In particular, several dimensional invariants are examined which make it possible to unify much of the available information, and to extend the insights of percolation theory to processes which have not yet received numerical study. The compilation of the results of percolation theory, both exact and numerical, is believed to be complete through 1970. A selective bibliography is given. In a concluding chapter several recent applications of percolation theory to classical and to quantum mechanical problems are discussed.     

Quantum theory as a theory in a classical propositional calculus

Classical logic and Boolean algebras are, of course, very intimately related. It is, however, possible to show that lattices of propositions isomorphic to the lattice of all the closed subspaces of a separable Hilbert space arise quite naturally within the classical propositional logic. This was first shown by the author in 1987 in connection with a certain type of theories calledtheories with orthocomplementation. These theories are not easy to interpret physically and it is shown that simpler theories, which are more amenable to physical interpretation, can also be used. It is then possible to assume that quantum theory is such a theory and, as a result, to formulate a new approach that provides a way of looking at the wave-particle duality and touches upon the foundations of quantum field theory.     

Generalized theory of gravitation

The mathematical formulation of the nonsymmetric gravitation theory (NGT) as a geometrical structure is developed in a higher-dimensional space. The reduction of the geometrical scheme to a dynamical theory of gravitation in four-dimensional space-time is investigated and the basic physical laws of the theory are reviewed in detail.     

Conformal field theory and the symmetries of string field theory

The two-dimensional conformal field theory representation of Witten's bosonic string field theory is discussed. The basic overlap equations, K_n symmetry and BRST invariance are proved directly, without the usual expansion in oscillators. The conformal field theory approach naturally provides local overlap identities which (when integrated over half the string) can be used to verify properties of the cubic action. In particular, a recently proposed diffeomorphism invariance is shown to be free of anomalies. Finally, a new class of symmetries, including generalizations of the K_n symmetries which are local in spacetime, are presented.     

Combustion theory and modeling

In honor of the fiftieth anniversary of the Combustion Institute, we are asked to assess accomplishments of theory in combustion over the past fifty years and prospects for the future. The title of our article is chosen to emphasize that development of theory necessarily goes hand-in-hand with specification of a model. Good conceptual models underlie successful mathematical theories. Models and theories are discussed here for deflagrations, detonations, diffusion flames, ignition, propellant combustion, and turbulent combustion. In many of these areas, the genesis of mathematical theories occurred during the past fifty years, and in all of them significant advances are anticipated in the future. Increasing interaction between theory and computation will aid this progress. We hope that, although certainly not complete in topical coverage or reference citation, the presentation may suggest useful directions for future research in combustion theory.