Entropy of bosonic open string and boundary conditions

Abstract

The entropy of the states associated to the solutions of the equations of motion of the bosonic open string with combinations of Neumann and Dirichlet boundary conditions is given. Also, the entropy of the string in the states A i = α i −1 |0 and |φ a = α a −1 |0 that describe the massless fields on the world-volume of the Dp-brane is computed. *

Thermal D-branes States from Superstrings in Light-Cone Gauge

“…The TFD was applied previously to the string theory, e. g. in [8,9]. More recent results were obtained in [10,11,12]. It deserves to emphasize two recent results from [13] where the equivalence between the Matsubara and the TFD formalisms was proved and [14] where the extension of the formalism to the superstring in the pp-background was done.…”

Closed string thermal torus from thermo field dynamics

“…The idea of using the Fock space formulation in string theory came up in Refs. [23,24,25,26,27,28], where the thermal space was used to construct bosonic thermal boundary states interpreted as D-branes at finite temperature. To further explore the algebraic characteristics of the TFD, and to upgrade it to a powerful tool to understand string theory at finite temperature, it is first necessary to set up the connections between the TFD and the Imaginary Time formalisms, when both are applied to strings at thermal equilibrium.…”

Tensionless strings from worldsheet symmetries

“…In this formalism, the thermal energy is given by computing the matrix elements of the T = 0 Hamiltonian in the thermal vacuum. The function S develops a natural interpretation of being thermal entropy of the system once it is computed by using the entropy operatorŜ defined for the left-movers of the closed string [49,51,52]. Explicit form of the entropy operator is given by,…”