Canonical quantization and expanding metrics

The canonical formalism for expanding metrics scenarios is presented. Nonunitary time evolution implied by expanding geometry is described as a trajectory over unitarily inequivalent representations at different times of the canonical commutation relations. Thermal properties of inflating Universe are also discussed. PACS: 04.60.d, 98.80.c In this paper we study the quantum field theory (QFT) formalism for non-unitary time evolution implied by inflationary models and in general by expanding metrics scenarios. As well known, non-unitary time evolution cannot be handled in the canonical formalism and for this reason many semi-classical techniques are currently used; the problem of the canonical quantization of inflationary time evolution is thus still at an unsatisfactory stage. In this paper we show that the operator formalism for the canonical quantization of time evolution in expanding geometry can be formulated provided the full set of unitarily inequivalent representations of the canonical commutation relations are considered. Let us start our discussion, which will be limited to gravitational wave modes, by denoting, as customary, the flat time-dependent metrics by gμν(t) = g 0 μν(t) + hμν(t). As well known, use of the De Donder gauge condition ∂μhμν = 0 and the Einstein equations give 2 hμν = 0. (1) The field hμν may be then decomposed into harmonic modes uk obeying the equation: ük(t) +Hu̇k(t) + ω 2 k(t)uk(t) = 0 (2) with ωk (t) = kc a2(t) , a(t) = a0e 1 3 Ht . (3) E-mail:[email protected] E-mail:[email protected]