Quantum Field Theory - Quantum String Duality in Curved Backgrounds

We introduce a R “Dual” transform which relates Quantum Field Theory and Quantum String regimes, both in a curved background. This operation maps the characteristic len gth of one regime into the other and, as a consequence, mass domains as well. The Hawking-Gibbons tempe rature and the string maximal or critical temperature are dual of each other. If back reaction of quan tum matter is included, Quantum Field and Quantum String phases appear, and duality relations between them manifest as well. This Duality is shown in two relevant examples: Black Hole and de Sitte r space times. Departamento de F́ısica Teórica, Facultad de Ciencias F́ısicas, Universidad Complutense, E-28040, Madrid, Spain. Observatoire de Paris, Demirm (Laboratoire Associé au CNRS UA 336, Observatoire de Paris et Ecole Normale Supérieure), 61 Avenue de l’Observatoire, 75014 Paris, France. 1 The combined study of Quantum Field Theory (QFT) and Quantum String Theory (QST) in curved space times allows to go further in the understanding of quantum gravity effects Ref. [1,2,3]. The string “analogue model” (or thermodynamical approach) is a suitable framework for this purpose. Strings are considered as a collection of quantum fields φn coupled to the curved background, and whose masses are given by the degenerated string spectrum in this background. The higher mass spectrum is described by the density of string mass levels ρ(m) in the space time considered Ref. [3]. We define a R “dual” transformation over a length L, as following L̃ = RL = LR L −1 (1) where LR has dimensions of (length) . For physical theories, R maps classical length scales Lcl into quantum string length scales Lq, and conversely L̃cl ≡ RLcl = LR L −1 cl = Lq (2.a) and L̃q ≡ RLq = LR L −1 q = Lcl (2.b) We say then that Lcl is “dual” to Lq and viceversa. LR depends on the dimensional parameters of the theory and is given by LR ≡ Lcl Lq (3) Lcl sets up a length scale for the semiclassical QFT regime and Lq is the length that characterizes the string domain; Lq depends on the dimensional string constant α ′ (T = c/2πT , where T is the string tension) and on the specific background considered. QFT in curved backgrounds with event horizons posseses an intrinsic (Hawking-Gibbons) temperature TH Ref. [1,4,5], which can be expressed, in general, as a function T of Lcl (and of the fundamental constants h̄, c and kB) TH = T (Lcl) (4)