Power Spectrum in Krein Space Quantization

Abstract

The power spectrum of scalar field and space-time metric perturbations produced in the process of inflation of universe, have been presented in this paper by an alternative approach to field quantization namely, Krein space quantization [1,2]. Auxiliary negative norm states, the modes of which do not interact with the physical world, have been utilized in this method. Presence of negative norm states play the role of an automatic renormalization device for the theory.Comment: 8 pages, appear in Int. J. Theor. Phy

Spectrum of Scalar Curvature Perturbations in Krein Space Quantization

“…From (13) we have = ∂$ ∂u = u , which using (14) and (15) translates to u k (η) = k (η) for the Fourier transforms. Also the canonical commutation relations are…”

“…The operators a(k), a † (k), b(k) and b † (k) are defined in [13]. At very short wave length, k aH 1, solutions are…”

“…Accordingly to the second perspective of [13] and using (34), to leading order of slow roll parameter, we have…”

“…On the other hand, accordingly to the first perspective of [13], using the above physical mode of (36) and also (14) of [13] as the non-physical mode, we have…”

Trans-Planckian Scale and Krein Space Quantization

“…Let us make this remark that there is a similarity between the Krein quantization method and usual re-normalization. In the curved space-time, the standard re-normalization of the Ultraviolet divergence of the vacuum energy is accomplished by subtracting the local divergence of Minkowski space [4,5] In which | is the vacuum state in curved space and |0 is the vacuum state in Minkowski space. The minus sign in equation (1) can be interpreted as the negative norm states which is added to the positive norm states [2].…”