A FIELD THEORY MODEL FOR CLOSED BOSONIC STRINGS

Besed on the BRST approach to the first quantization of closed bosonic strings, we propose a gauge covariant field theory model for both free and interacting closed bosonic strings. The action is of the Chexn-Simons type and coincides with the cohomology of the BRST operators.     

SYMPLECTIC STRUCTURES AND QUANTIZATION FOR BOSONIC STRINGS IN BRST FORMALISM

The sympletic structures for bosonic strings are given by the method restricting J^'Y to its Lagrangian submanifold and geometric quantization in BRST formulation for strings is discussed.It is found that conformal anomaly is cancelled when curvature of BRST vacuum bundle on G₀=G₀/H vanishes.     

Symplectic geometry and geometric quantization on supermanifold withU numbers

The usual theory of supermanifold is extended to the case that contains (anti)commuting variable pairs with oppositeU numbers. The symplectic geometry and geometric quantization on such a special manifold are discussed in detail. As applications, the BRST system with finite dimensional first class constraints and bosonic strings are investigated systematically.     

Schrödinger representation in the bosonic string theory: Quantization without anomalous terms

Using the theory of infinite-dimensional pseudodifferential operators in superspace, a Schrödinger representation is introduced to the theory of bosonic strings (the superstucture emerges under BRST quantization). qp-Symbol quantization considered in this paper, possesses no anomalies (unlike quantization using normal ordering). In particular, under BRST quantization, the anomalous term vanishes in space of arbitrary dimension, so that there is no critical dimension in this theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 19–22, October, 1990.The author thanks V. S. Vladimirov, I. V. Volovich, and A. A. Slavnov for useful advice, and all the participants of the seminars in mathematical physics and quantum field theory of MIAN for useful discussions of mathematical aspects of field and string theory.     

The algebraic structure of BRST quantization

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.     

A Consideration of BRST Cohomology in the First Quantization of RNS String

The difference of Grassmanian even-odd and degree even-odd of (super-) conformal ghosts is stressed. With some modifications in the BRST charges and the commutation relations, it can be shown that the first quantization of RNS strings allows the BRST cohomology analysis.     

BRST quantization of superstring field theory

We write the gauge fixed action which arises in the quantization of Witten's string field theory in a linear gauge, in a form which applies to both the superstring and the bosonic string. The corresponding BRST transformation is nilpotent only on-shell. We construct also an off-shell nilpotent BRST transformation which formally leaves invariant the quantum effective action. This BRST transformation has a geometrical interpretation which could allow to describe the gauge anomalies of the superstring field theory as the nontrivial cohomology of the BRST charge via the Wess-Zumino consistency condition.     

THE BKS DERNEL OF BOSONIC STRINGS AND PATH INTEGRAL

The connections between geometric quantization and path integral quantization of bosonic strings are investigated.The Polyakov path integral formulation and its measure are manifestly deduced from the Blattner-Kostant-Sternberg(BKS) kernel of geometric quantization.     

BRST quantization in the Siegel gauge

From a formal generalization to N copies of the free open string field theory BRST-quantized in the Siegel gauge we reproduce the BRST quantization of the free closed bosonic string field theory and obtain the one of massless higher spin field theories.     

BRST invariance and gauge independence of string path-integrals

The covariant quantization of relativistic strings is performed in the BRST formulation of path-integrals. BRST-invariant measures are constructed in the critical number of dimensions and the gauge independence of the resulting path-integrals is proved.     

Non-commutative geometry and string field theory

An attempt is made to interpret the interactions of bosonic open strings as defining a non-cummulative, associative algebra, and to formulate the classical non-linear field theory of such strings in the language of non-commulative geometry. The point of departure is the BRST approach to string field theory. A setting is given in which there is a unique gauge invariant action, whose linearized approximation reproduces the conventional Veneziano spectrum. A derivation of conventional Veneziano model amplitudes from this gauge invariant action is sketched. Some brief comments are made about attempts to extend these results to open superstrings and to closed strings.     

BRST quantization and computation of static potentials for bosonic membranes

The static potentials for both open and closed bosonic membranes are derived using the extended phase-space functional integral. It is shown that the BRST quantization scheme in the case of background gauge coincides with the ordinary phase-space quantization. The results for the mass of a rectangular pointlike membrane for the critical radius of the spherical membranes (under which appear tachyons) as well as for the tachyonic mass differ by numerical factors from those found using configuration-space functional methods. The latter is a consequence of the noncorrectness of the configuration-space quantization for the membrane theory.     

BRST Extension of Geometric Quantization

Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathematically rigorous BRST quantization of the constrained system whose cohomology at ghost number zero recovers the constrained quantum states. Moreover this space of constrained states has a well-defined Hilbert space structure inherited from that of the original system. Treatments of these ideas in the physics literature are more general but suffer from having states with infinite or zero "norms" and thus are not admissible as states. Also BRST operators for many systems require regularization to be well-defined. In our more restricted context, we show that our treatment does not suffer from any of these difficulties.     

Sp(2) BRST invariant quantization of strings: The harmonic gauge

We analyze the mixed algebra of local diffeomorphisms and Weyl transformations for bosonic strings. BRST and anti-BRST operators are then constructed keeping a manifest Sp(2) invariance. The harmonic gauge arises as a natural gauge choice. All this work is redone in the presence of a two-dimensional background metric. We manage to write down a simple action, to compute the stress tensor and to work out the critical dimensions.     

A reparametrization-invariant approach to string field theory

Using reparametrization invariance as the only requirement, we show, by a series of algebraic steps, how the only reparametrization-invariant generalization of d'Alembert's operator for both open and closed bosonic strings, is in fact nilpotent and thus the BRST charge. The construction follows a proposal made many years ago by one of the authors.     

Bosonic strings and complex structures

We write the action of the bosonic string in terms of two-dimensional complex structures rather than of two-dimensional metrics. We describe in some detail the behaviour under reparametrization of the world sheet, and in particular we give an expression for the two-dimensional diffeomorphism anomaly. We describe a possible gauge-fixing procedure for the BRST quantization.     

OSp[D+1,3 | 2] unifies the conformal symmetry and BRST

We show that the conformal transformations on the invariant metric of OSp[D,2 | 2], which is the group unifying Lorentz and BRST symmetries, are generated by OSp[D+1, 3 | 2], which thus unifies the conformal and BRST symmetries. We explicitly construct the generators for the scalar and for the spinning particle. We express the generators in terms of the energy-momentum tensor and study their action on the fields. We also study the linear realization of OSp[D+1, 3 | 2] on a space of D+4 bosonic and 2 fermionic variables. Field quantization is also examined.     

Strings with zero tension

We describe bosonic strings by using a kind of Lagrangian compatible with the zero tension limit. The work is developed on an extended configuration space and the quantization is carried out with details.     

From points to gauge fields

We show that several classes of free field theories with local gauge invariance (e.g. the Yang-Mills, Einstein and p-forms linearized actions) can be constructed from classical actions for a finite number of points by applying the BRST quantization initiated by Siegel. We briefly outline the generalization of this construction for strings.     

Strings, world-sheet covariant quantization and Bohmian mechanics

The covariant canonical method of quantization based on the De Donder–Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics. PACS 11.25.-w; 04.60.Ds; 03.65.Ta