Coherence and Emergence of Classical Spacetime

Using the coherent-state representation we show that the classical Einstein equation for the FRW cosmological model with a general minimal scalar field can be derived from the semiclassical quantum Einstein equation. E-Mail: [email protected] E-Mail: [email protected] E-Mail: [email protected] 1 Canonical quantum gravity has been initiated by DeWitt in the seminal paper [1]. As a methodology to understand the quantum aspects of cosmology, quantum cosmology has been intensively studied, and in particular, as a great conceptual advancement, the boundary conditions have been incorporated for the Universe by Hartle-Hawking [2] and Vilenkin [3]. Semiclassical quantum gravity has also been elaborated as a methodology to include some part of quantum effects into classical gravity [4]. In order to consider the different mass scales between gravity and matter fields and to apply quantum cosmology to the early Universe, one should have the reduction scheme from canonical quantum gravity, Ĝμν = 8πT̂μν , to semiclassical quantum gravity, Gμν = 8π〈T̂μν〉, and down to classical gravity, Gμν = 8πTμν . In this Brief Report, we complete the reduction from semiclassical quantum gravity to classical gravity for a quantum FRW cosmological model with a general minimal scalar field. We find that the coherent-state representation of the semiclassical quantum Einstein equation leads to the classical Einstein equation with a quantum correction. In previous papers [5,6], we showed that in the case of a massive scalar field an exact quantum state of time-dependent Schrödinger equation gives rise to the mean energy density which has the same form as the classical one except that the field intensity is replaced by the absolute value, and that a coherent state exactly gives rise to the classical density plus an additional one from vacuum fluctuation. We extend the result of the massive scalar-field model to the general scalar-field model. As a quantum cosmological model, we consider the FRW Universe whose WheelerDeWitt equation is given by [ 2πh̄ 3mPa ∂ ∂a − 3mP 8π ka− h̄ 2a ∂ ∂φ + aV (φ) ] Ψ(a, φ) = 0. (1) Here, k takes 1, 0, and −1 for a closed, spatially flat, and open universe, respectively. The unit system is c = 1 and G = 1 mP . The corresponding classical Einstein equation is ( ȧ a ) 2 + k a = 8π 3mP ( φ̇ 2 + V (φ) ) , (2) and the classical field equation is 2 φ̈+ 3 ȧ a φ̇+ dV (φ) dφ = 0. (3) Following the reduction scheme, one obtains the semiclassical quantum gravity from the Wheeler-DeWitt equation: ( ȧ a ) 2 + k a = 8π 3mPa 〈Ĥ〉, (4) and ih̄ ∂ ∂t Φ(φ, t) = ĤΦ(φ, t) (5) where Ĥ = 1 2a π̂ + aV (φ̂). (6) We now represent the semiclassical Einstein equation in the coherent state. For the case of the massive scalar-field model, the coherent-state representation was given explicitly in terms of classical solutions [6]. For the case of a general scalar-field model, we use the coherent states constructed by Rajagopal and Marshall [7]. We follow their main idea but redefine some of variables to be suitable for the application to quantum field theory in FRW cosmology. We construct the Fock space by introducing the creation and annihilation operators Â(t) = u(t)π̂ − au̇(t)φ̂, Â(t) = u(t)π̂ − au̇(t)φ̂. (7) As in Ref. [8], if we require that  and  be the invariant operators ih̄ ∂ ∂t {   } − [{   }, Ĥ] = 0, (8) then u satisfies the equation (au̇)̇φ̂+ auV (φ̂) = 0. (9) 3 From the usual commutation relation it follows that h̄a (