Effective theories of two T-dual theories are also T-dual

We investigate how T-duality and solving the boundary conditions of the open bosonic string are related. We start by considering the T-dualization of the open string moving in the constant background. We take that the coordinates of the initial theory satisfy either Neumann or Dirichlet boundary conditions. It follows that the coordinates of T-dual theory satisfy exactly the opposite set of boundary conditions. We treat the boundary conditions of both theories as constraints, and apply the Dirac procedure to them, which results in forming $\sigma$-dependent constraints. We solve these constraints and obtain the effective theories for the solution. We show that the effective closed string theories are also T-dual

T-duality diagram for a weakly curved background

In one of our previous papers we generalized the Buscher T-dualization procedure. Here we will investigate the application of this procedure to the theory of a bosonic string moving in the weakly curved background. We obtain the complete T-dualization diagram, connecting the theories which are the result of the T-dualizations over all possible choices of the coordinates. We distinguish three forms of the T-dual theories: the initial theory, the theory obtained T-dualizing some of the coordinates of the initial theory and the theory obtained T-dualizing all of the initial coordinates. While the initial theory is geometric, all the other theories are non geometric and additionally nonlocal. We find the T-dual coordinate transformation laws connecting these theories and show that the set of all T-dualizations forms an Abelian group

Twisted C-brackets

We consider the double field formulation of the closed bosonic string theory, and calculate the Poisson bracket algebra of the symmetry generators governing both general coordinate and local gauge transformations. Parameters of both of these symmetries depend on a double coordinate, defined as a direct sum of the initial and T-dual coordinate. When no antisymmetric field is present, the $C$-bracket appears as the Lie bracket generalization in a double theory. With the introduction of the Kalb-Ramond field, the $B$-twisted $C$-bracket appears, while with the introduction of the non-commutativity parameter, the $\theta$-twisted $C$-bracket appears. We present the derivation of these brackets and comment on their relations to analogous twisted Courant brackets and T-duality

Properties of the quantum state arising after the L-photon state has passed trough a linear quantum amplifier

We consider the system of N two-level atoms, of which N0 atoms are unexcited and N1 are excited. This system of N two-level atoms, which forms a linear quantum amplifier, interacts with a single-mode electromagnetic field. The problem of amplification of the L-photon states using such an amplifier is studied. The evolution of the electromagnetic field density matrix is described by the master equation for the field under amplification. The dynamics of this process is such that it can be described as the transformation of the scale of the phase space. The exact solution of the master equation is expressed using the transformed Husimi function of the L-quantum state of the harmonic oscillator. The properties of this function are studied and using it the average photon number and its fluctuations in the amplified state are found. © 2021, Editura Academiei Romane. All rights reserved

CANONICAL APPROACH TO THE CLOSED STRING NON-COMMUTATIVITY

We consider the propagation of the closed bosonic string in the weakly curved background. We show that the closed string non-commutativity is essentially connected to the T-duality and nontrivial background. From the T-duality transformation laws, connecting the canonical variables of the original and T-dual theory, we nd the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. We nd that the commutative original theory is equivalent to the non-commutative T-dual theory, in which Poisson brackets close on winding and momenta numbers and the coecients are proportional to the background uxes

T-duality in the weakly curved background

We consider the closed string propagating in the weakly curved background which consists of constant metric and Kalb-Ramond field with infinitesimally small coordinate dependent part. We propose the procedure for constructing the T-dual theory, performing T-duality transformations along coordinates on which the Kalb-Ramond field depends. The obtained theory is defined in the non-geometric double space, described by the Lagrange multiplier $y_\mu$ and its $T$-dual $\tilde{y}_\mu$. We apply the proposed T-duality procedure to the T-dual theory and obtain the initial one. We discuss the standard relations between T-dual theories that the equations of motion and momenta modes of one theory are the Bianchi identities and the winding modes of the other

The T-dual symmetries of a bosonic string

Abstract We investigate whether the symmetry transformations of a bosonic string are connected by T-duality. We start with a standard closed string theory. We continue with a modified open string theory, modified to preserve the symmetry transformations possessed by the closed string theory. Because the string theory is conformally invariant world-sheet field theory, in order to find the transformations which preserve the physics, one has to demand the isomorphism between the conformal field theories corresponding to the initial and the transformed field configurations. We find the symmetry transformations corresponding to the similarity transformation of the energy-momentum tensor, and find that their generators are T-dual. Particularly, we find that the general coordinate and local gauge transformations are T-dual, so we conclude that T-duality in addition to the well-known exchanges, transforms symmetries of the initial and its T-dual theory into each other