Clock Time in Quantum Cosmology
CClock Time in Quantum Cosmology
Marcello Rotondo ∗ and Yasusada Nambu † Department of Physics, Graduate School of Science,Nagoya University, Chikusa, Nagoya 464-8602, Japan (Dated: February 26, 2019)We consider the conditioning of the timeless solution to the Wheeler-DeWitt equation by a pre-defined matter clock state in the simple scenario of de Sitter universe. The resulting evolution ofthe geometrodynamical degree of freedom with respect to clock time is characterized by the “Berryconnection” of the reduced geometrodynamical space, which relies on the coupling of the clock withthe geometry. When the connection vanishes, the standard Schr¨odinger equation is obtained for thegeometry with respect to clock time. When one considers environment-induced decoherence in thesemi-classical limit, this condition is satisfied and clock time coincides with cosmic time. Explicitresults for the conditioned wave functions for minimal clocks made up of two quantum harmonicoscillator eigen-states are shown. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b I. INTRODUCTION
In the canonical approach to relativistic and non-relativistic quantum mechanics, where one supposes a backgroundclassical geometry, classical time sticks out in the Schr¨odinger (functional) equation as an external parameter usedby the observer. The apparent lack of external time for the dynamics of canonical quantum gravity, described by theso-called ”problem of time” put forth by the Wheeler–DeWitt (WD) equation [1] (cid:98) H Ψ = 0 , (1)has therefore been one of the elements stimulating the research concerning the nature of time in an attempt towardunderstanding how the evolution of the quantum state of the universe Ψ can be described when time and spacethemselves become dynamical variables [2–6].Different approaches to introducing an effective time variable to the canonical picture of gravity have been devised.In one class of such attempts, one tries to extract a physical variable t to be used as an effective time variable andobtain a Schr¨odinger-like structure for the Hamiltonian H : (cid:98) H Ψ = (cid:16) (cid:98) H t − i∂ t (cid:17) Ψ = 0 . (2)The structure (cid:98) H t + (cid:98) P t , with (cid:98) H t being a physical Hamiltonian describing the evolution with respect to the timevariable t and (cid:98) P t being the momentum canonically conjugated to time, is characteristic of time-reparameterizedHamiltonians in the classical theory, i.e., Hamiltonians where coordinate time t has been promoted to a dynamicalvariable t ( θ ) that is dependent on some implicit, unobservable time θ . The absence of a structure like Equation (2)with respect to time in canonical gravity, already at the classical level, is due to the fact that the theory is built onspace–time diffeomorphism invariance to begin with, and the attempts to identify a canonical momentum to use as agenerator of time translations are plagued by difficulties of various kinds and degrees (see, e.g., [7–9]).A rather general idea to extracting dynamics from a seemingly stationary system has been proposed in what issometimes called the Page-and-Wootters approach or the conditional probability interpretation (CPI) [10, 11]. In thisapproach, “time” evolution is read, under the condition that the total state Hamiltonian (cid:98) H = (cid:98) H C ⊗ R + 1 C ⊗ (cid:98) H R (3)is constrained, in the quantum correlations between the two partitions of the total state, i.e., a clock state (C) withthe physical state (R) (the “rest of the universe”) entangled with it. Time evolution emerges from the measurementof an observable clock of some kind, whose reading conditions the physical state. Attention to this approach seems tohave undergone a recent revival, especially after some of the most important criticisms levied against it in the pasthave been addressed, together with an experimental illustration of the mechanism [12–16]. The actual application ofthe CPI to timelessness in canonical quantum gravity, which originally motivated it, still seems to be quite lacking.One example is [17], which may have implications for the present work because it also presents a treatment of quantumdecoherence, and recent contributions that take into account the gravitational coupling of the clock are provided ina series of works [18–20]. In the latter references, the importance of accounting for the coupling of quantum clockswith gravity in the CPI is addressed. Soon after the first version of this work appeared, one of the authors made usaware of another recent contribution to the general case of coupled clocks in the context of non-relativistic quantummechanics [21]. We redirect the reader to this and the previously cited works for a good introduction to the CPIand some aspects of the effects of coupling quantum clocks with gravity. Indeed, one important characteristic that isgenerally put forth as a requirement for a good clock is that it weakly interacts with the system whose evolution itdescribes. When we take gravity as partaking in the dynamics, though, it couples to all forms of energy, including theclock, and while the coupling might become weak for some chosen cases (e.g., when the clock has conformal couplingto a conformally invariant geometry) or approximations, it may generally be strong or even dominant in the quantumregime. In this regard, note that the results of the mentioned study [21] do not apply straightforwardly to the caseof canonical quantum gravity considered here in that it is not sufficient to just include a contribution to the totalHamiltonian coming from the coupling: the completely free Hamiltonian contribution from the clock must also beexplicitly excluded. Concerning the importance of the coupling, note also that when we consider how classical timeemerges from the timeless WD Equation (1) in the semiclassical limit, we see that it comes to be defined as theparameter along the classical trajectories of the gravitational degrees of freedom in the WKB approximation. Theorigin of the classical time of the (functional) Schr¨odinger equation therefore lies precisely in the coupling betweengeometry and matter fields.In the present work, rather than trying to recover an equation of the Schr¨odinger type (Equation (2)) or a notionof time through dynamical observables, we study how, starting from a definition of the clock time, the resultingconditioned state for the geometry evolves with respect to it in the general case where coupling cannot be neglected.We treat the simple example of the FLRW mini-superspace model with a homogeneous massive scalar field minimallycoupled to gravity. In Section II, we introduce the model and expand the solution of the WD equation in the eigenstatesof the matter Hamiltonian. In Section III, we discuss how time emerges in the semiclassical limit of gravity and howit appears for conditioning by a predefined matter clock in the quantum regime. In Section IV, we show the explicitsolution for minimal working clocks. Final observations and conclusions are provided in Section V. II. MINI-SUPERSPACE MODEL
We consider the simple scenario of the mini-superspace for a spatially flat FLRW universe of scale factor a , ds = − N ( t ) dt + a ( t ) δ ij dx i dx j (4)with a minimally coupled scalar field φ and a cosmological constant Λ. The classical action is S = 12 κ (cid:90) d x √− g ( R − (cid:90) d x √− g (cid:18) −
12 ( ∂ µ φ ) − U ( φ ) (cid:19) . (5)The Lagrangian is L = Vκ (cid:18) − a ˙ a N − N a Λ (cid:19) + V a (cid:32) ˙ φ N − N U (cid:33) (6)= 1 κ (cid:18) − ˙ ρ N ρ − N ρ Λ (cid:19) + ρ (cid:32) ˙ φ N − N U (cid:33) , (7)where ˙ := d/dt , and V denotes the comoving volume of the universe. We also introduce the physical volume of theuniverse ρ := V a as a dynamical variable. The canonical momenta are π ρ = ∂L∂ ˙ ρ = − κ ρ N ρ , π φ = ∂L∂ ˙ φ = ρ ˙ φN . (8)The Hamiltonian reads H = π ρ ˙ ρ + π φ ˙ φ − L = N ( H ρ + H φ ) , (9)where H ρ := ρ (cid:18) − κ π ρ + Λ κ (cid:19) , H φ := π φ ρ + ρ U ( φ ) . (10)The metric is ds = − N dt + ( ρ/ρ ) / δ ij dx i dx j , ρ := V . (11)The first-class Hamiltonian constraint δH/δN = H ρ + H φ = 0 (12)becomes 13 (cid:18) ˙ ρN ρ (cid:19) = Λ + κ ρ (cid:32) ˙ φN (cid:33) + U ( φ ) , (13) (cid:126) = 1 , κ = 8 πG = 1 M p . which corresponds to the Friedmann equation for the universe with a minimal scalar field if we choose N = 1.Constraint quantization of Equation (12) for a physical state Ψ gives the WD equation,( (cid:98) H ρ + (cid:98) H φ )Ψ = 0 . (14)In the representation diagonalizing ( ρ, φ ), we have the WD equation for the wave function Ψ( ρ, φ ): (cid:20) ρ (cid:18) ∂ ∂ρ + λ (cid:19) + 43 κ (cid:18) − ρ ∂ ∂φ + ρ U ( φ ) (cid:19)(cid:21) Ψ( ρ, φ ) = 0 , (15)where λ = 4Λ / (3 κ ), and we have assumed the appropriate ordering of the operator (cid:98) π ρ for simplicity of analysis.For the free massive field U ( φ ) = µ φ /
2, the matter Hamiltonian is (cid:98) H φ = − ρ ∂ ∂φ + 12 ρ µ φ . (16)When we consider the eigenstates of the Hamiltonian (cid:98) H φ , the variable ρ contained in (cid:98) H φ can be treated as anunknown external parameter, and the eigenvalue equation is (cid:98) H φ χ n ( φ | ρ ) = E n χ n ( φ | ρ ) , (17)where χ n ( φ | ρ ) are the energy eigenstates of (cid:98) H φ for any value of ρ , and the energy eigenstates can be determined by χ n ( φ | ρ ) = 1 (cid:112) n n ! √ π ( µρ ) e − µρ φ / H n ( √ µρ φ ) . (18) H n ( x ) are the Hermite polynomials, and the associated eigenvalues are E n = µ (cid:18) n + 12 (cid:19) , n = 0 , , , · · · . (19)The eigenmodes χ n satisfy the orthogonality and completeness relations( χ m , χ n ) := (cid:90) ∞−∞ dφ χ ∗ m ( φ | ρ ) χ n ( φ | ρ ) = δ mn , (20) (cid:88) k χ ∗ k ( φ | ρ ) χ k ( φ | ρ ) = δ ( φ − φ ) . (21)Then, we can generally expand the universal wave function Ψ( ρ, φ ) asΨ( ρ, φ ) = (cid:88) n ψ n ( ρ ) χ n ( φ | ρ ) . (22)where the components ψ n ( ρ ) of the expansion encode the information on the quantum state of the geometry for agiven choice of the clock. These components are determined by substituting (22) into the WD Equation (15): (cid:88) n (cid:20) ρ (cid:0) ∂ ρ ψ n + λ ψ n (cid:1) χ n + 43 κ ψ n (cid:98) H φ χ n (cid:21) = − (cid:88) n ρ (cid:2) ∂ ρ ψ n ∂ ρ χ n + ψ n ∂ ρ χ n (cid:3) . (23)After taking the inner product with χ m , the wave function of the universe ψ n obeys (cid:88) n (cid:34) ρ (cid:88) k (cid:98) D mk (cid:98) D kn + (cid:18) ρλ + 4 E m κ (cid:19) δ mn − ( ∂ ρ χ m , ∂ ρ χ n ) + (cid:88) k ( ∂ ρ χ m , χ k ) ( χ k , ∂ ρ χ n ) (cid:35) ψ n = 0 , (24)where the covariant derivative is introduced as (cid:98) D mn := δ mn ∂ ρ − iA mn , (25)with the “Berry” connection [22, 23] A mn := i ( χ m , ∂ ρ χ n ) . (26)If we take into account the completeness of the energy eigenbasis { χ n } , the last two terms of Equation (24) vanish,and we obtain (cid:88) n (cid:34) ρ (cid:88) k (cid:98) D mk (cid:98) D kn + (cid:18) ρλ + 4 E m κ (cid:19) δ mn (cid:35) ψ n = 0 , (27)As a result of the connection, different components ψ n of the expansion in (22) become generally coupled to eachother. For each component, the connection leads to a geometric phase, and the formal solution of Equation (27) isgiven by ψ n ( ρ ) = (cid:88) m (cid:20) P exp (cid:18) i (cid:90) dρA ( ρ ) (cid:19)(cid:21) nm b m G m ( ρ ) , (28)where b m represents constants, the symbol P denotes a path ordering, and the functions G n satisfy − (cid:98) H ρ G n = E n G n (29)which gives G n = ρ e − i √ λ ρ F [1 + iβ n / √ λ, , i √ λρ ] , β n = 4 µ κ ( n + 1 / , n = 0 , , , . . . (30)with F being Kummer’s confluent hypergeometric function. III. EMERGENCE OF TIMEA. WKB Time
In the semiclassical limit, time naturally appears as a parameter along the superspace trajectories of the spatialgeometry. One considers the WKB ansatz for the wave function [24, 25]Ψ( ρ, φ ) = exp (cid:34) i ∞ (cid:88) n =0 (cid:18) κ (cid:19) n − S n ( ρ, φ ) (cid:35) . (31)By substituting in Equation (15) and equating each order of (cid:0) κ / (cid:1) p , one obtains[ p = −
2] : ∂ φ S = 0 , (32)[ p = −
1] : ( ∂ ρ S ) = 3Λ , (33)[ p = 0 ] : (cid:18) − ρ ∂ ∂φ + ρU (cid:19) e iS = ρ∂ ρ S ∂ ρ S , (34) . . . Equations (32) and (33) give, respectively, S = S ( ρ ) and the Einstein–Hamilton–Jacobi equation for a de Sitterspace, where matter contributions to the action are taken to be perturbative. Then, introducing WKB time as aparameter along the classical trajectories of ρ ( τ ), ∂ τ := − ρ ( ∂ ρ S ) ∂ ρ = − ρπ ρ ∂ ρ , (35)Equation (34) gives the functional Schr¨odinger equation ∓ i∂ τ χ = (cid:98) H φ χ (36)for the matter wave functional χ := e iS on the classical de Sitter background. Ambiguity in the sign of (35), whichdetermines the direction of the cosmological arrow of time, corresponds to the choice of sign in Equation (33), wherea positive sign gives the contracting de Sitter universe, and a negative sign yields the expanding one. Our definitionof G n ( ρ ) (Equation (29)) as a fundamental solution corresponds to a choice for fixing the sign ambiguity. In thesemiclassical approximation, the total wave function isΨ( ρ, φ ) ∝ e ± i √ λρ χ ( ρ, φ ) . (37) For a function ψ = BG , ( ∂ ρ − iA ) ψ = ( ∂ ρ B − iAB ) G + B∂ ρ G = B∂ ρ G for ∂ ρ B = iAB . The solution of this condition is B = P exp (cid:0) i (cid:82) dρA ( ρ ) (cid:1) . B. Scalar Field as a Clock
Classical (WKB) time (35) relies on the coupling between geometry and matter and on the existence of trajectoriesfor the geometrodynamical degrees of freedom. Therefore, it cannot be extended straightforwardly to the full quantumregime. In the present approach, rather than trying to extract a viable time variable from the dynamics of matterand geometry, we follow the original CPI proposal and start with the definition of the clock time through the scalarfield and discuss the results of using the thus-defined time to track evolution. More specifically, given the Hamiltonian (cid:98) H φ , we define a “clock time” T as a parameter of the following normalized clock wave function (cid:101) χ ( T, φ | ρ ) = (cid:88) n c n e − iE n T χ n ( φ | ρ ) , (cid:88) n c ∗ n c n = 1 , (38)where the sum is extended over non-vanishing coefficients c n (cid:54) = 0 only. The state (38) formally satisfies the standardSchr¨odinger equation, i∂ T (cid:101) χ = (cid:98) H φ (cid:101) χ , (39)and the time variable T in this sense is well-defined to begin with. Just as for quantum clocks in non-relativisticquantum mechanics, the time parameter T can be estimated by measuring some physical observable of the clock (cid:101) χ ( T, φ | ρ ) and applying quantum estimation theory. With respect to the parameter T , the quantum state of thegeometry conditioned by the clock reading T is then effectively described by (cid:101) ψ ( ρ, T ) : = ( (cid:101) χ ( T, φ | ρ ) , Ψ( ρ, φ, T )) = (cid:88) m c ∗ m e iE m T ψ m ( ρ ) = (cid:88) m,n c ∗ m b n e iE m T B mn G n ( ρ ) (40)where for simplicity of notation we define the matrix of elements, B mn ( ρ ) := (cid:20) P exp (cid:18) i (cid:90) dρA ( ρ ) (cid:19)(cid:21) mn . (41)The exact form of the connection is A mn ( ρ ) = i ( χ m , ∂ ρ χ n ) = i α mn ρ , (42)where α mn : = 4 ρ (cid:90) ∞−∞ dφ χ ∗ m ( φ | ρ ) ∂ ρ χ n ( φ | ρ )= (cid:112) n ( n − δ m,n − − (cid:112) ( n + 2)( n + 1) δ m,n +2 . (43)Notice that for the present choice of Hamiltonian, the energy eigenvalues are independent of the parameter ρ andare therefore not affected by the derivative in the connection. The integral of the connection is (cid:90) ρρ dρ (cid:48) A mn ( ρ (cid:48) ) = i α mn ρ/ρ ) . (44)where we can identify the arbitrary scale ρ with the comoving volume of the universe introduced before. If we setthis scale, for example, as the Planck scale, then ρ < ρ belongs to the sub-Planckian, strong quantum regime. C. Evolution Equation
The time evolution of state (cid:101) ψ ( ρ, T ) = (cid:80) n c ∗ n e iE n T ψ n is determined implicitly through its conditioning by the clockand is not generally of the Schr¨odinger type with the pure geometrodynamical Hamiltonian H ρ . To derive the explicitdynamic law, one may start by noticing that1∆ T (cid:90) T +∆ TT dT (cid:48) e iµnT (cid:48) = δ n , ∆ T := 2 πµ , (45)and, therefore, c ∗ n ψ n ( ρ ) = 1∆ T (cid:90) T +∆ TT dT (cid:48) e − iE n T (cid:48) (cid:101) ψ ( ρ, T (cid:48) ) , (46) c ∗ n E n ψ n ( ρ ) = − i ∆ T (cid:90) T +∆ TT dT (cid:48) e − iE n T (cid:48) (cid:101) ψ ( ρ, T (cid:48) ) ∂ T (cid:48) (cid:101) ψ ( ρ, T (cid:48) ) . (47)This allows us to write (cid:101) ψ ( ρ, T ) = (cid:88) n c ∗ n e iE n T ψ n (48)= 1∆ T (cid:88) n (cid:90) T +∆ TT dT (cid:48) e − iE n ( T − T (cid:48) ) (cid:101) ψ ( ρ, T (cid:48) ) . (49)A representation of the delta function in the interval [ T, T + ∆ T ] is therefore δ ( T (cid:48) − T ) := 1∆ T (cid:88) n e − iE n ( T (cid:48) − T ) . (50)Using the previous relations and assuming c n (cid:54) = 0 ∀ n , the WD Equation (27) gives the evolution equation of (cid:101) ψi∂ T (cid:101) ψ ( ρ, T ) = (cid:98) H ρ (cid:101) ψ ( ρ, T ) + 3 κ T ρ (cid:88) m (cid:54) = n c ∗ m c ∗ n (cid:90) T +∆ TT dT (cid:48) e i ( E m T − E n T (cid:48) ) [ (cid:98) D ] mn (cid:101) ψ ( ρ, T (cid:48) ) . (51)Although the obtained equation has the structure of a differential-integral equation, when [ D ] mn has only diagonalnon-vanishing components (the connection is zero), the Schr¨odinger equation with Hamiltonian (cid:98) H ρ is recovered for (cid:101) ψ .The first term of the RHS of Equation (51) is inherited by the conditioned state independent of the couplingbetween the clock and the geometry and corresponds to the mechanism discussed by Page and Wootters [10, 11]. Onthe other hand, the second term depends on the coupling through the connection in the geometric phase term. Noticethat the vanishing of this term does not necessarily require the coupling to be absent, and one may obtain for (cid:101) ψ theSchr¨odinger evolution generated by (cid:98) H ρ also when the coupling is strong. On the other hand, recovery of semiclassicaltime (WKB time) itself requires that a coupling between matter and geometry exists.In the semiclassical expansion of Ψ( ρ, φ ) for solving the WD equation, the geometrodynamical variable ρ and mattervariable φ are treated, respectively, as “heavy” and “fast” degrees of freedom of the system. The evolution of the heavydegrees of freedom is described by the classical Einstein–Hamilton–Jacobi Equation (33) for the chosen metric, whilethe matter field describes quantum perturbations of the system. For a cyclic evolution of ρ , the various componentsof (44) become contributions to the so-called Berry phase acquired by the system, which has been discussed both inthe context of non-relativistic quantum mechanics, where it was first introduced, as well as in quantum cosmology[26–28]. For discussions of the relation between the Berry connection and the emergence of semiclassical time, see also[29–31]. In the present case, though, the relevance of the phase (44) originates from the fact that it determines thecoupling of different components of the expansion (22) and the time evolution law of the geometrodynamical state.When one takes into account the necessary coupling of gravity with “environmental” degrees of freedom [32], thefast energy eigenmodes in the expansion (22) decohere from each other, and one can neglect the off-diagonal elementsof the connection (26), in which consists the so-called “Born–Oppenheimer approximation”. Since the coherencebetween different components of the clock decays, the clock cannot be used any longer to track time because thequantum superposition is destroyed, and the time-dependent relative phases between distinct energy eigenstates arelost. This is reflected in the decoupling of the different indexes of ψ n in Equation (27) and the diagonalization of thematrix (41). In this limit, we obtain an effective Schr¨odinger equation for each component of (cid:101) ψ (cid:98) H ρ ψ n = − E n ψ n (52)which is equivalent to the equation defining G n in the timeless picture (Equation (29)). For a given n -branch with E n := E , the solution is (cid:101) ψ ≈ exp [ iET + iS ( E, ρ )], and the classical trajectory is recovered by the conditionconst. = ∂ E ( ET + S ) = T + ∂ E S ( E, ρ ) , (53)from which the kinematic expression for ρ = ρ ( T ) can be obtained. For the present case, it is possible to check that weobtain the classical equation for ρ with a pure cosmological constant, with N = 1 slice and clock time T correspondingto cosmic time. It is important to observe that the correlation between ρ and T , as described by the resulting ρ ( T ),emerges only in the classical limit and, due to its conditional nature with respect to the clock choice, it describes theclassical “rest of the universe” alone and does not enter the definition of clock itself. IV. SOLUTION OF THE WD EQUATION WITH CLOCK
We consider different clock models and show the behavior of the corresponding conditioned state (cid:101) ψ . A. Clock with a Single Eigenstate
To start, let us consider the total wave function with only the m th component,Ψ( ρ, φ ) = ψ m ( ρ ) χ m ( ρ, φ ) . (54)For this state, the connection vanishes A = A mm = 0, and the wave function of the universe conditioned by theclock is simply (cid:101) ψ ( T, ρ ) = e iµ ( m +1 / T G m ( ρ ) . (55)Thus, the wave function has a trivial time dependence given by an overall phase factor, which is not measurable.The disappearance of clock time for the case of a single energy eigenstate reflects the fact that a working clock needsa superposition of at least two energy eigenstates to track the time evolution. B. Clock with Two Eigenstates
Let us consider the minimal case of a working clock, made up of the only two eigenstates 0 and 1 ( c = c = 1 / √ ρ, φ ) = 1 √ ψ ( ρ ) χ ( φ | ρ ) + ψ ( ρ ) χ ( φ | ρ )) . (56)Notice that when looking for the solutions ψ n (Equation (28)) satisfying Equation (27), we have assumed anexpansion that includes all the eigenstates. This enables the use of the completeness relation to grant the vanishingof the extra ρ − -order term (cid:80) k ( ∂ ρ χ m , χ k )( χ k , ∂ ρ χ n ) − ( ∂ ρ χ m , ∂ ρ χ n ). We may consider the present case of a finitenumber of energy eigenstates as an approximation wherein all other coefficients c m are negligibly small and can bedropped from the equation. Also, in this case, the connection A nm ( n, m = 0 ,
1) vanishes and [ e i (cid:82) dρA ] nm = nm .The conditional state becomes (cid:101) ψ ( T, ρ ) = e iµT/ √ (cid:0) G ( ρ ) + e iµT G ( ρ ) (cid:1) , (57)where the relative phase makes the time dependence observable.As discussed in the previous section, the simple time dependence of (cid:101) ψ ( T, ρ ) follows from the diagonality of thematrix (41). For the choice of matter Hamiltonian (16), diagonality is granted for m (cid:54) = n ±
2. The simplest examplein which this condition is not satisfied and (cid:101) ψ ( T, ρ ) takes a more complex time evolution is the case of eigenstates n, m = 0 ,
2: Ψ = 1 √ ψ ( ρ ) χ ( φ | ρ ) + ψ ( ρ ) χ ( φ | ρ )) . (58)The integral of the connection becomes (cid:90) ρρ dρ (cid:48) A ( ρ (cid:48) ) = i √ ρ/ρ ) (cid:18) − (cid:19) , (59)and B = cos (cid:16) ln( ρ/ρ ) √ (cid:17) − sin (cid:16) ln( ρ/ρ ) √ (cid:17) sin (cid:16) ln( ρ/ρ ) √ (cid:17) cos (cid:16) ln( ρ/ρ ) √ (cid:17) , (60)which is independent of the path ordering since the dependence on ρ appears in the connection integral as an overallmultiplicative factor. The wave function of the universe conditioned by the clock (taking b n = 1 / √
2) becomes (cid:101) ψ ( T, ρ ) = e iµT/ (cid:2)(cid:0) B ( ρ ) + e iµT B ( ρ ) (cid:1) G ( ρ ) + (cid:0) B ( ρ ) + e iµT B ( ρ ) (cid:1) G ( ρ ) (cid:3) . (61)For the chosen clock Hamiltonian, the general time evolution of Equation (51) is simply i∂ T (cid:101) ψ ( ρ, T ) = (cid:98) H ρ (cid:101) ψ ( ρ, T ) + 3 κ (cid:18) ∂ ρ − ρ (cid:19) (cid:88) m (cid:54) = n c ∗ m e iE m T α mn ψ n ( ρ ) (cid:124) (cid:123)(cid:122) (cid:125) X ( ρ,T ) . (62)While the state (57) exactly follows the Schr¨odinger equation with Hamiltonian (cid:98) H ρ , for the time evolution of thestate (61), the second term X of Equation (62) will generally not vanish.The behavior of the ratio X/H ρ (cid:101) ψ for the { , } -clock is shown in Figure 1. The values of X ( ρ, T ) and H ρ (cid:101) ψ areundetermined at ρ = 0 ∀ T because of the “rotation” B (ln( ρ/ρ ) / √ ρ = 0. In thisregime, Equation (62) diverges from the Schr¨odinger evolution. On the other hand, for ρ (cid:29) λ − = (cid:96) P (cid:96) H (63)where (cid:96) − H := 2 (cid:112) Λ / (cid:96) P = κ is the Planck length, the ratio becomes negligiblysmall. In this limit, G m ( ρ ) can be approximated as G m ( ρ ) ≈ g m e i ( √ λρ + βm √ λ log ( √ λρ ) ) , g m = 14 √ λ (2 i ) i βm √ λ Γ (cid:16) i β m √ λ (cid:17) . (64) - - FIG. 1. Plot of Re[
X/H ρ (cid:101) ψ ] ( left ) and | X/H ρ (cid:101) ψ | ( right ) for unitary values of the physical constants λ = µ = κ = ρ = 1 whenthe { , } -clock is employed. ∂ ρ B can be neglected in this limit and, neglecting the other contributions of order ρ − , X ( ρ, T )will be approximated as an oscillatory term of order κ : X ( ρ, T ) ≈ i κ (cid:114) λ (cid:16)(cid:0) B − e i µT B (cid:1) e i β √ λ log ( √ λρ ) g + (cid:0) B − e i µT B (cid:1) e i β √ λ log ( √ λρ ) g (cid:17) e i ( µ T + √ λρ ) , (65)which can be neglected compared with the growing contribution of the geometrodynamical Hamiltonian. Therefore,in the limit (63), the Schr¨odinger equation is effectively recovered using this clock. V. CONCLUSIONS
In this article, we discuss the time dependence of the quantum state of the geometry obtained by conditioningthe timeless solution of the Wheeler–DeWitt equation with a predefined clock state. The resulting time dependenceis generally not trivial, and the coupling between the clock and the geometry affects the quantum state through ageometric phase that couples different components of the expansion of the timeless state on the basis of the energyeigenstates of the clock. We derived an evolution law for the geometry with respect to the clock time that alsoholds when the coupling between the clock and the geometry cannot be neglected. A standard Schr¨odinger-typeevolution generated by the geometrodynamical Hamiltonian is always recovered when the off-diagonal elements of theBerry connection, which determines the geometric phase, vanish. In the semiclassical limit when environment-induceddecoherence is taken into account, the off-diagonal elements of the connection can effectively be neglected as well, andthe different components of the clock decohere from each other. The disappearance of quantum superposition betweendifferent energy eigenstates of the clock makes it impossible to use it to track time: in this limit, observers will relyon the classical time parameter, which emerges in the WKB approximation of the total wave function. We show theexplicit result for minimal working clocks made up of two harmonic oscillators. For a clock made by a superpositionof the ground state and the first energy level (or, indeed, any two states that are not separated by two energy levels),the Schr¨odinger evolution is retrieved exactly. Using a superposition of the ground state and the second energy levelharmonic oscillator, the evolution of the quantum wave function of the geometry presents a non-trivial deviation fromthe Schr¨odinger-type evolution generated by the pure geometrodynamical Hamiltonian. The Schr¨odinger equation isapproximately recovered also for this clock for a physical volume much larger than the scale (cid:96) P (cid:96) H . This property isindependent of the details of the decoherence mechanism.Although we consider the simple case of an FLRW mini-superspace, the extension to other geometries and differentmatter Hamiltonians for the clock may proceed along the same lines. We focus on the effective time evolutionresulting from the conditioning of the geometry by the clock state, but some important issues are not addressed andpostponed to future work. For example, we do not consider the time evolution of the expectation values for ˜ ψ (say, (cid:68) ˜ ψ (cid:12)(cid:12)(cid:12)(cid:98) ρ (cid:12)(cid:12)(cid:12) ˜ ψ (cid:69) ), which would allow a discussion of the physical significance and applicability of the CPI for the study ofearly cosmology. Addressing this issue is conditioned to solving the problem of normalization for the wave function ˜ ψ and recovering an interpretation for it as a probability amplitude density. Furthermore, we do not explicitly discussthe effect of the back-reaction of the clock on the state of the geometry. We expect that this would result in theclassical limit in a modification of the Einstein–Hamilton–Jacobi equation, which may be derived from the effectiveSchr¨odinger evolution for ρ . ACKNOWLEDGMENTS
Y.N. was supported in part by JSPS KANENHI Grant No. 15K0573. M.R. gratefully acknowledges support fromthe Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The authors would like tothank G. Venturi and A.R. Smith for suggesting useful references. [1] DeWitt, B.S. Quantum theory of gravity. I. The canonical theory.
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