KNUTH-26,March1995 Unitarity Condition on Quantum Fields in Semiclassical Gravity Sang Pyo Kim ∗ Department of Physics Kunsan National University Kunsan 573-701, Korea Abstract The condition for the unitarity of a quantum field is investigated in semiclas- sical gravity from the Wheeler-DeWitt equation. It is found that the quantum field preserves unitarity asymptotically in the Lorentzian universe, but does not preserve unitarity completely in the Euclidean universe. In particular we obtain a very simple matter field equation in the basis of the generalized invariant of the matter field Hamiltonian whose asymptotic solution is found explicitly. Published in Physics Letters A 205, 359 (1995) Unitarity of quantum field theory in curved space-time has been a problem long debated but sill unsolved. In particular the issue has become an impassioned altercation with the discovery of the Hawking radiation [1] from black hole in relation to the information loss problem. Recently there has been a series of active and intensive investigations of quantum ∗E-mail : [email protected] 1 effects of matter field through dilaton gravity and resumption of unitarity and information loss problem (for a good review and references see [2]). In this letter we approach the unitarity problem and investigate the condition for the unitarity of a quantum field from the point of view of semiclassical gravity based on the Wheeler-DeWitt equation [3]. By developing various methods [4–18] for semiclassical gravity and elaborating further the new asymptotic expansion method [19] for the Wheeler-DeWitt equation, we derive the quantum field theory for a matter field from the Wheeler-DeWittt equation for the gravity coupled to the matter field, which is equivalent to a gravitational field equation and a matrix equation for the matter field through a definition of cosmological time. The full field equation for the matter field is found to preserve unitarity asymptotically in the limit ¯h 0 in the Lorentzian universe with an oscillatory gravitational wave function M → but does not preserve unitarity completely for an exponential gravitational wave function in the Euclidean universe. We find the exact quantum state for the asymptotic field equation in terms of the eigenstates of the generalized invariant of the matter field Hamiltonian. We consider the Wheeler-DeWitt equation for a quantum cosmological model 2 h¯ 2 ˆ i δ MV(ha)+H( ,φ;ha) Ψ(ha,φ)=0: (1) "−2M ∇ − h¯ δφ # Here M is the Planck mass squared, 2 = G (δ2)=(δh δh ), V denotes the superpotential ∇ ab a b of three-curvature with or without the cosmological constant on superspace with the DeWitt metric G with the signature ( ; +; ; +), and Hˆ represents the matter field Hamiltonian. ab − ··· We find the wave function of the form Ψ(ha,φ)= (ha)Φ(φ, ha); (2) where Φ(φ, ha) is a gravitational field-dependent quantum state of the matter field. We expand the quantum state by some orthonormal basis Φ(φ, h )= c (h ) Φ (φ, h ) ; Φ Φ = δ : (3) a k a | k a i h k| ni kn Xk Substituting Eqs. (2) and (3) into Eq. (1) and acting Φ on both sides, one obtains the h n| following matrix equation equivalent to the Wheeler-DeWitt equation 2 2 2 h¯ 2 h¯ cn(ha) MV (ha)+Hnn(ha) (ha) (ha) cn(ha) −2M ∇ − ! − M ∇ ·∇ h¯2 +i (ha) Ank(ha)ck(ha)+ (ha) Hnk(ha)ck(ha) M ∇ · k k=n X X6 h¯2 (h ) Ω (h )c (h )=0; (4) −2M a nk a k a Xk where H (h )= Φ (h ) Hˆ Φ (h ) ; nk a h n a | | k a i A (h )=i Φ (h ) Φ (h ) ; nk a h n a |∇| k a i Ω (h )= 2δ 2iA +Ω(2); (5) nk a ∇ nk − nk ·∇ nk where Ω(2)(h )= Φ (h ) 2 Φ (h ) : (6) nk a h n a |∇ | k a i By noting that i acts on c (h ) as a hermitian operator, it follows that H, A,Ω(2),and ∇ k a thereby Ω are hermitian matrices. We separate the matrix equation equivalent to the Wheeler-DeWitt equation into the gravitational part 2 h¯ 2 MV(ha)+Hnn(ha) (ha)=0; (7) −2M ∇ − ! and the matter field part h¯2 h¯2 (h ) c (h )+i (h ) A (h )c (h ) −M ∇ a ·∇ n a M ∇ a · nk a k a Xk h¯2 + (ha) Hnk(ha)ck(ha) (ha) Ωnk(ha)ck(ha)=0: (8) k=n − 2M k X6 X The gravitational wave function has an effective potential MV H . The gravitational − nn wave function can have either a real action in a region of the Lorentzian universe or an imaginary action in a region of the Euclidean universe. First, we consider a region of the Lorentzian universe, in which the gravitational wave function takes the form 3 i (ha)=f(ha)exp Snn(ha) : (9) h¯ In the semiclassical limith ¯ 0, the gravitational action satisfies the Einstein-Hamilton- → Jacobi equation with the quantum back-reaction 1 ( S (h ))2 MV(h )+H (h )=0: (10) 2M ∇ nn a − a nn a There are possibly an infinite number of the gravitational wave functions depending on the mode number due to the quantum back-reaction of the matter field. Each gravitational wave function, whose peak corresponds to a classical solution with the matter field, describes a history of evolution of the universe and contains the whole information of the universe. So along each wave function in this region we may introduce a cosmological time @ 1 := S (h ) : (11) ∂τ(n) M ∇ nn a ·∇ Substituting (9) into (8) and dividing (ha), we obtain the matter field equation @ h¯2 ih¯ c +Ω(1)c + Ω(1) H c + Ω(3)c =0; (12) (n) n nn n nk nk k nk k ∂τ k=n − 2M k X6 X where @ h¯ Ω(1) = ih¯ Φ Φ = S A ; (13) nk h n| ∂τ(n) | ki M ∇ nn · nk and 1 1 Ω(3) =Ω +2 f δ 2i f A : (14) nk nk f ∇ ·∇ nk − f ∇ · nk The full field equation (12) for the matter field is not unitary due to the terms 2 1 f f ∇ · δ 2i 1 f A which do not obviously act as unitary operators, even though all the ∇ nk − f ∇ · nk (1) other terms Ω , Hnk,andΩnk are hermitian matrices. The unitarity violating terms were discovered using different method in Ref. [13]. In a previous paper [19], we also obtained a small unitarity violating term, i ¯h 2S (h ), which originated from the gravitational − 2M ∇ nn a i wave function in the form (ha)=exp(¯h Snn(ha)). 4 However, it is only in the asymptotic limit ¯h 0 that for any gravitational wave M → function satisfying (7) these unitarity violating terms are suppressed and one gets @ ih¯ c +Ω(1)c + Ω(1) H c =0: (15) (n) n nn n nk nk k ∂τ k=n − X6 (1) (1) In this case, since H† = H,Ω † =Ω ,andΩ† = Ω, it follows that @ c† c =0; (16) ∂τ(n) · where c denotes a column vector of c . The norm of vector, c , is preserved. This implies k | | a unitary operator such that (n) (n) (n) (n) c(τ )=Uc(τ ,τ0 )c(τ0 ): (17) And since the eigenstates of the generalized invariant form an orthonormal basis, there is also a unitary operator (n) (n) (n) (n) Φ(τ ) = UΦ(τ ,τ0 ) Φ(τ0 ) (18) E E for the evolution of the column vector of Φ . The physical implication is that the quantum | ki field theory of the matter field is asymptotically unitary in this sense. In particular, in terms of the eigenstates of the generalized invariant @ i Iˆ I;ˆ Hˆ =0; ∂τ(n) − h¯ h i Iˆ Φ = λ Φ (19) | ki k | ki there is a well-known decoupling theorem [20] such that H =Ω(1);n= k: (20) nk nk 6 In the basis of the eigenstates of the generalized invariant the matter field equation takes the simpler form @ h¯2 ih¯ c +Ω(1)c + Ω(3)c =0: (21) ∂τ(n) n nn n 2M nk k Xk 5 In the asymptotic limit of ¯h 0 we obtain the asymptotic quantum state of matter field M → [18] (n) i (1) (n) Φ(φ, ha)=cn(τ0 )exp Ωnn (ha)dτ Φn(φ, ha) : (22) h¯ Z | i With an additional phase factor exp i H dτ (n) (22) is also one of the exact quantum − ¯h nn states of time-dependent Schr¨odinger equationR @ i δ ih¯ Φ(φ, h )=Hˆ ( ,φ;h )Φ(φ, h ): (23) ∂τ(n) a h¯ δφ a a The exact quantum state (3) can be obtained solving pertubativley (21), which is a linear superposition of eigenstates of the generalized invariant with gravitational field-dependent coefficient functions. Second, we consider an exponential gravitational wave function of the form in the Eu- clidean universe 1 (h )=f(h )exp( S (h )): (24) a a −h¯ nn a The cosmological time is imaginary and given by @ 1 = i S ∂τ M ∇ nn ·∇ @ := i (n) : (25) − ∂τim The matter field equation becomes @ h¯2 h¯ c +Ω(1)c + Ω(1) H c + Ω(3)c =0: (26) (n) n nn n nk nk k 2M nk k ∂τim k=n − k X6 X Even though the basis evolves unitarily, the coefficient functions in (3) with respect to which the exact quantum state is expanded do not. This violates completely the unitarity of exact quantum state. As an application, we consider a free massive scalar field coupled to the gravity.