On the cosmological constant in quantum cosmology

Quantization of a dust-like closed isotropic cosmological model with a cosmological constant is realized by the method of B. DeWitt [1]. It is shown that such quantization leads to interesting results, in particular, to a finite lifetime of the system, and appearance of the Universe in our world as penetration via the barrier. These purely quantum effects appear when Λ > 0. Quantization of cosmological models becomes an important subject on the interface of gravitation and cosmology. It was initiated by B. DeWitt [1] who performed quantization of a Friedmann universe filled with dust. Later, closed isotropic models with matter in the form of a conformal [2] and minimally coupled [3] scalar fields were quantized. Misner’s paper [4] was devoted to the problem of quantization of anisotropic cosmological models, and Yu.N. Barabanenkov’s to quantization of the Friedmann metric matched with the Kruskal one [5]. In this paper, quantization of a dust-like closed isotropic cosmological model with a cosmological constant is realized by a method similar to Ref. [1]. The system of units is chosen so that c = ~ = 16πG = 1. The corresponding Lagrangian is L = Lg + Lm = 12π αR− 12παR (R,0)2 − 4παΛR + Lm, where R is the radius (scale factor) of the Universe, Λ is the cosmological constant and−α2 = g00. Lm is the matter Lagrangian (forN non-interacting particles). Passing, by the standard recipe, to the Hamiltonian and using the constraint equation [1], we arrive at the condition Hg +Hm = 0, (1) whereHg = −Π/(48R)−12πR+48πΛR;Hm = Nm; Π is a momentum conjugate to R. In the quantum case (1) becomes a condition for the state vector Ψ (Hg +Hm) Ψ = 0. [email protected] [email protected], [email protected]