scholarly journals PROPER TIME FORMALISM AND GAUGE INVARIANCE IN OPEN STRING INTERACTIONS

1994 ◽  
Vol 09 (18) ◽  
pp. 1681-1693
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Sigma Model ◽  
Gordon Equation ◽  
Proper Time ◽  
Open String ◽  
Gauge Invariant ◽  
Point Particles ◽  
Boundary Terms ◽  
Time Formalism ◽  
Klein Gordon ◽  
Model Formalism

The issue of gauge invariances in the sigma model formalism is discussed at the free and interacting level. The problem of deriving gauge invariant interacting equations can be addressed using the proper time formalism. This formalism is discussed, both for point particles and strings. The covariant Klein-Gordon equation arises in a geometric way from the boundary terms. This formalism is similar to the background independent open string formalism introduced by Witten.

scholarly journals THE PROPER TIME FORMALISM, GAUGE INVARIANCE AND THE EFFECTS OF A FINITE WORLD SHEET CUTOFF IN STRING THEORY

1995 ◽  
Vol 10 (31) ◽  
pp. 4501-4519 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Transformation ◽  
Gauge Invariance ◽  
Proper Time ◽  
World Sheet ◽  
Massive Mode ◽  
Yang Mills ◽  
Previous Discussion ◽  
Time Formalism ◽  
Klein Gordon ◽  
Type Interaction

We discuss the issue of going off-shell in the proper time formalism. This is done by keeping a finite world sheet cutoff. We construct one example of an off-shell covariant Klein-Gordon type interaction. For a suitable choice of the gauge transformation of the scalar field, gauge invariance is maintained off-mass-shell. However, at the second order in the gauge field interaction, one finds that [U(1)] gauge invariance is violated due to the finite cutoff. Interestingly, we find, to the lowest order, that by adding a massive mode with appropriate gauge transformation laws to the sigma model background, we can restore gauge invariance. The gauge transformation law is found to be consistent, to the order calculated, with what one expects from the interacting equation of motion of the massive field. We also extend some previous discussion on applying the proper time formalism for propagating gauge particles, to the interacting (i.e. Yang-Mills) case.

scholarly journals THE PROPER TIME EQUATION AND THE ZAMOLODCHIKOV METRIC

1996 ◽  
Vol 11 (16) ◽  
pp. 2887-2906 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Electromagnetic Field ◽  
Renormalization Group ◽  
Vector Field ◽  
Proper Time ◽  
Beta Function ◽  
Open String ◽  
Point Function ◽  
Gauge Invariant ◽  
Electromagnetic Field Strength ◽  
Uniform Electromagnetic Field

The connection between the proper time equation and the Zamolodchikov metric is discussed. The connection is twofold. First, as already known, the proper time equation is the product of the Zamolodchikov metric and the renormalization group beta function. Second, the condition that the two-point function is the Zamolodchikov metric implies the proper time equation. We study the massless vector of the open string in detail. In the exactly calculable case of a uniform electromagnetic field strength we recover the Born-Infeld equation. We describe the systematics of the perturbative evaluation of the gauge-invariant proper time equation for the massless vector field. The method is valid for nonuniform fields and gives results that are exact to all orders in derivatives. As a nontrivial check, we show that in the limit of uniform fields it reproduces the lowest order Born-Infeld equation.

scholarly journals Proper time and conformal problem in Kaluza–Klein theory

2015 ◽  
Vol 12 (05) ◽  
pp. 1550063
Author(s):  
E. Minguzzi
Keyword(s):  
Scalar Field ◽  
Gordon Equation ◽  
Proper Time ◽  
Inertial Mass ◽  
Scalar Fields ◽  
Conformal Factor ◽  
Klein Theory ◽  
Klein Gordon ◽  
Klein Gordon Equation ◽  
Kaluza Klein

In the traditional Kaluza–Klein theory, the cylinder condition and the constancy of the extra-dimensional radius (scalar field) imply that time-like geodesics on the five-dimensional bundle project to solutions of the Lorentz force equation on spacetime. This property is lost for nonconstant scalar fields, in fact there appears new terms that have been interpreted mainly as new forces or as due to a variable inertial mass and/or charge. Here we prove that the additional terms can be removed if we assume that charged particles are coupled with the same spacetime conformal structure of neutral particles but through a different conformal factor. As a consequence, in Kaluza–Klein theory the proper time of the charged particle might depend on the charge-to-mass ratio and the scalar field. Then we show that the compatibility between the equation of the projected geodesic and the classical limit of the Klein–Gordon equation fixes unambiguously the conformal factor of the coupling metric solving the conformal ambiguity problem of Kaluza–Klein theories. We confirm this result by explicitly constructing the projection of the Klein–Gordon equation and by showing that each Fourier mode, even for a variable scalar field, satisfies the Klein–Gordon equation on the base.

scholarly journals THE DIRAC FIELD IN TAUB–NUT BACKGROUND

2001 ◽  
Vol 16 (10) ◽  
pp. 1743-1758 ◽  
Author(s):  
ION I. COTĂESCU ◽  
MIHAI VISINESCU
Keyword(s):  
Invariant Theory ◽  
Gordon Equation ◽  
Dirac Field ◽  
Dirac Fermions ◽  
Large Collection ◽  
Scalar Theory ◽  
Gauge Invariant ◽  
Commuting Operators ◽  
Klein Gordon ◽  
Kaluza Klein

We investigate the SO (4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza–Klein monopole, pointing out that the quantum modes can be recovered from a Klein–Gordon equation analogous to the Schrödinger equation in the Taub–NUT background. Moreover, we show that there is a large collection of observables that can be directly derived from those of the scalar theory. These offer many possibilities of choosing complete sets of commuting operators which determine the quantum modes. In addition there are some spin-like and Dirac-type operators involving the covariantly constant Killing–Yano tensors of the hyper-Kähler Taub–NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form.

The kinematics of a point particle

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Internal Structure ◽  
Proper Time ◽  
World Line ◽  
Point Particle ◽  
Material Point ◽  
Spatial Extent ◽  
Geometric Representation ◽  
Mathematical Result ◽  
Point Particles ◽  
The World

This chapter discusses the kinematics of point particles undergoing any type of motion. It introduces the concept of proper time—the geometric representation of the time measured by an accelerated clock. It also describes a world line, which represents the motion of a material point or point particle P, that is, an object whose spatial extent and internal structure can be ignored. The chapter then considers the interpretation of the curvilinear abscissa, which by definition measures the length of the world line L representing the motion of the point particle P. Next, the chapter discusses a mathematical result popularized by Paul Langevin in the 1920s, the so-called ‘Langevin twins’ which revealed a paradoxical result. Finally, the transformation of velocities and accelerations is discussed.

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

scholarly journals Fractional Klein-Gordon equation with singular mass

2021 ◽  
Vol 143 ◽  
pp. 110579
Author(s):  
Arshyn Altybay ◽  
Michael Ruzhansky ◽  
Mohammed Elamine Sebih ◽  
Niyaz Tokmagambetov
Keyword(s):  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation
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