Coherent semiclassical states for loop quantum cosmology

IGC-11/5-6 Coherent Semiclassical States for Loop Quantum Cosmology arXiv:1105.5081v1 [gr-qc] 25 May 2011 Alejandro Corichi1, 2, ∗ and Edison Montoya3, 1, † 1 Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico 2 Center for Fundamental Theory, Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park PA 16802, USA 3 Instituto de Fı́sica y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, Mexico Abstract The spatially flat Friedman-Robertson-Walker (FRW) cosmological model with a massless scalar field in loop quantum cosmology admits a description in terms of a completely solvable model. This has been used to prove that: i) the quantum bounce that replaces the big bang singularity is generic; ii) there is an upper bound on the energy density for all states and iii) semiclassical states at late times had to be semiclassical before the bounce. Here we consider a family of exact solutions to the theory, corresponding to generalized coherent Gaussian and squeezed states. We analyze the behavior of basic physical observables and impose restrictions on the states based on physical considerations. These turn out to be enough to select, from all the generalized coherent states, those that behave semiclassical at late times. We study then the properties of such states near the bounce where the most ‘quantum behavior’ is expected. As it turns out, the states remain sharply peaked and semiclassical at the bounce and the dynamics is very well approximated by the ‘effective theory’ throughout the time evolution. We compare the semiclassicality properties of squeezed states to those of the Gaussian semiclassical states and conclude that the Gaussians are better behaved. In particular, the asymmetry in the relative fluctuations before and after the bounce are negligible, thus ruling out claims of so called ‘cosmic forgetfulness’. PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc ∗ † Electronic address: [email protected] Electronic address: [email protected] 1 I. INTRODUCTION Loop quantum cosmology has provided in past years a useful framework to ask questions about the quantum nature of the early universe [1, 2]. Closely related to loop quantum gravity [3], the formalism has shown that, for isotropic models, the big bang singularity is replaced by a quantum bounce [4–6]. These results have been obtained for closed and open FRW cosmologies [7, 8] with and without a cosmological constant [10] for a massless scalar field, and recently extended to the massive case [11]. Recently, some of these results have been extended to anisotropic [12–15] and some inhomogeneous models [16]. In all these cases, quantum gravitational effects have been shown to exert a repulsive force and halt the collapse and launch an expanding superinflationary phase before the quantum gravity effects die out and the universe then follows the standard general relativity (GR) dynamics. An exactly solvable model for the k=0 FRW with a massless scalar field has allowed to prove analytically the generic nature of the quantum bounce [17]. In particular, it was shown that the bounce occurs for all physical states, and the energy density possesses a supremum for physical states [17]. Regarding the fluctuations, it has been shown that they are bounded across the bounce implying that a semiclassical state at late times after the bounce had to come from a semiclassical state before the bounce [18]. Recently, Kaminski and Pawlowski have imposed stronger bounds on the asymmetry in relative dispersions before and after the bounce [19], thus very strongly ruling out some claims of loss of semiclassicality in [20, 21]. These robustness result have support on the fact that, within a large possibility of ‘loop quantizations’, physical considerations select a unique consistent quantization [22] (the one introduced in [6]). An interesting feature of this system is that one can approximate very well the dynamics followed by semiclassical states by an effective description [6]. That is, the classical limit of the dynamics is not that of GR + scalar field but rather is defined by an effective Hamiltonian. There have been several approaches to reach this result. For instance, in [23] the author considered kinematical coherent states and used them to compute the effective Hamiltonian as the expectation value of the quantum Hamiltonian constraint of LQC. More recently, the authors of [24] showed that, in the path integral description of the model, the paths that contribute the most are those that satisfy the effective equations and not those that follow the classical GR equations. The purpose of this paper is to construct exact physical semiclassical states for the solvable model [17]. The strategy will be to define a class of coherent initial states and, using the analytical control we have on the space of solutions to the quantum constraint equations, produce exact physical coherent states, much in the spirit of [25]. Then, we impose well motivated conditions on the behavior of the fluctuations of physical operators to select from all possible coherent states, those that exhibit a semiclassical behavior. It turns out that these conditions are sufficient to fix enough parameters of the states and to have states that are ‘peaked’ on a given (physical) phase space point. Having found semiclassical states, we explore the behavior of these states near the bounce, where one expects the state to be most quantum. In particular we consider the relative fluctuations of volume and find that, for all states and for all times, the relative fluctuations are bounded by their asymptotic value. In fact, the minimum value turns out to be precisely at the bounce. The expectation value for volume and density on semiclassical states are very well approximated by the values given by the effective theory. This leads us to conclude that the effective description is indeed a very good approximation to semiclassical physical states. A natural question is whether the 2 class of states considered is the most general one. For that purpose, we generalize the class of Gaussian states to the so called squeezed states. The idea is to explore the parameter space near the Gaussian states –that are well behaved semiclassical states– and see whether these squeezed states exhibit better semiclassical properties. Of particular interest in this regard is the asymmetry in volume fluctuations. Claims of a loss of coherence, or ‘cosmic forgetfulness’ arise due to results in an approximate framework [20] (motivated by, but not within LQC) that suggest that the fluctuations across the bounce might change significantly and spoil the semiclassical nature of the state [20, 21]. Even when strong bounds against this possibility within LQC have already appeared [19], our exact formalism is particularly well suited for addressing this question. By imposing bounds on the total asymptotic dispersion in volume (at late or early times with respect to the bounce), we are able to bound how far we can be from Gaussian states and still have a semiclassical state. The range on parameter space allowed turns out to depend on the physical requirements for the state, in such a way that, for ‘large volume universes’, it is severely constrained. This is turn sets very strong bounds on the allowed asymmetry in volume fluctuations, consistent with [19] and, in particular, disproving some of the statements of [20, 21] that claim the loss of coherence across the bounce for semiclassical states. The structure of the paper is as follows. In Sec. II we recall the solvable model in loop quantum cosmology together with the basic operators to be studied. In Sec. III we introduce the Gaussian states and compute expectation values and fluctuations for the physical operators. Semiclassicality conditions are explored in Sec. IV, together with the behavior of the states near the bounce. Squeezed states are introduced and compared to the Gaussian states in Sec. V. We end with a discussion in Sec.VI. There are four Appendices. In Appendix A we study the symplectic reduction of the ‘effective theory’ to isolate the physical degrees of freedom. In Appendix B we recall some useful integrals and formulas. In Appendix C we present an analysis of the errors introduced by considering Gaussian states, showing that they are severely suppressed and justify our analysis in the main body. In Appendix D we present in a table a comparison of different quantities for two families of Gaussian states. Some of the results reported here were summarized in the short manuscript [26], so it is natural that there is some overlap of part of the material. II. SOLVABLE LOOP QUANTUM COSMOLOGY (SLQC) In this section we shall recall the model we are considering, namely, the solvable k=0 FRW model coupled to a massless scalar field. Our presentation will be self-contained and slightly different from that of [17]. In terms of the phase space variables used in loop quantum gravity (LQG) [3], namely a connection Aia and a densitized triad Eia , the homogeneous gravitational sector can be expressed as, −(1/3) o Aia = c V0 ωai −(2/3) Eia = p V0 ; √ q0 o eai (2.1) where (o ωai , o eai ) are a set of orthonormal co-triads and triads compatible with the fiducial (flat for k = 0) metric o qab and Vo is the volume of the fiducial cell, introduced to define the symplectic structure, with respect to o qab . The phase space is characterized by conjugate variables (c, p) satisfying {c, p} = (8πGγ)/3. where γ ≈ 0.2375 is the Barbero-Immirzi parameter used in LQC. The triad p is related to the physical volume V of the fiducial cell, 2/3 as |p| = V 2/3 = Vo a2 where a is the scale factor and, on the space of classical solutions, 3 c = γ ȧ. It is convenient to introduce new variables [17], β := ε c/p1/2 ν = ε p3/2 /(2πℓ2Pl γ) and (2.2) where ~{β, ν} = 21 . Here ε = ±1 is the orientation of the triad with respect to that of o ωai . The classical constraint, with the choice N = V , then becomes p2φ 3 2 2 C= β V − ≈0 8πGγ 2 2 (2.3) where the matter content consists of a massless scalar field φ, with canonical momenta pφ . Thus, the kinematical phase space can be though of as a four dimensional space with coordinates (β, V, φ, pφ ). The strategy for quantization is to define an operator associated to Cˆ and to look for states that are annihilated by the constraint operator. In the (β, φ) representation, the operators basic are represented as β̂ · χ(β, φ) = β χ(β, φ) ; p̂φ · χ(β, φ) = −i~ V̂ · χ(β, φ) = −i4πγℓ2Pl ∂ χ(β, φ) ; ∂β ∂ χ(β, φ) . ∂φ (2.4) (2.5) However, the operator β̂ is not well defined since β is periodic with period 2π/λ, so it has to be replaced by a well defined operator β̃ˆ := sin(λβ)/λ (see [17] for details). The quantum constraint of sLQC becomes then [17],  2 ∂2 sin(λβ) ∂ 2 · χ(β, φ) = α · χ(β, φ) (2.6) ∂φ2 λ ∂β √ with β ∈ (0, π/λ), α := 12πG. In the loop quantum cosmology literature, the value of λ is chosen such that λ2 = ∆ corresponds to the minimum eigenvalue of the area operator in loop quantum gravity (corresponding to an edge of ‘spin 1/2’). With this choice the free √ 2 2 parameter λ becomes λ = 4 3πγℓPl . By introducing x = α−1 ln | tan(λβ/2)| (2.7) the quantum constraint can be rewritten in a Klein-Gordon form ∂φ2 χ(x, φ) = ∂x2 χ(x, φ) . (2.8) As usual for this system, in its classical evolution the scalar field φ is a monotonic function and can play the role of internal clock. In the quantum theory, one can also think of the evolution of the state with respect to φ. A general solution χ(x, φ) to Eq. (2.8) can be decomposed in the left and right moving components: χ = χ+ (φ + x) + χ− (φ − x) := χ+ (x+ ) + χ− (x− ) . (2.9) The physical states that we shall consider are positive frequency solutions of (2.8). Since there are no fermions in the model, the orientations of the triad are indistinguishable and 1 Here we have adopted the notation for the variable β introduced in [22] and used thereafter. 4 χ(x, φ) satisfy the symmetry requirement χ(−x, φ) = −χ(x, φ). Thus, we can write χ(x, φ) = √ (F (x+ ) − F (x− ))/ 2, where F is an arbitrary ‘positive frequency solution’2 . The physical inner product on solutions is given as [17] Z [χ̄1 (x, φ)∂φ χ2 (x, φ) − (∂φ χ̄1 (x, φ))χ2 (x, φ)] dx (2.10) (χ1 , χ2 )phy = −i φ=φ0 Z ∞ [∂x F̄1 (x+ )F2 (x+ ) − ∂x F̄1 (x− )F2 (x− )] dx . (2.11) = i −∞ We can now compute the expectation values and fluctuations of fundamental operator such as V̂ |φo , x̂ and p̂φ , where x̂ is related to the operator β̂. For any state on the physical Hilbert space the expectation value of the volume operator at ‘time φ’ is given by hV̂ iφ := (χ, V̂ |φ χ)phy = 2πγℓ2Pl (χ, |ν̂|χ)phy (2.12) where |ν̂| is the absolute value operator obtained from ν̂ = − 2λ cosh(αx)i∂x . α (2.13) Using the inner product (2.11) the expectation value of |ν̂| is given by Z ∞ [∂x F̄ (x+ )(ν̂F (x+ )) − ∂x F̄ (x− )(−ν̂F (x− ))] dx (χ, |ν̂|χ)phy = i −∞ Z 2λ ∞ = [∂x F̄ (x+ ) cosh(αx)∂x F (x+ ) + ∂x F̄ (x− ) cosh(αx)∂x F (x− )] dx α −∞ Z 2 4λ ∞ dF = cosh(α(x − φ)) dx . (2.14) α −∞ dx Is easy to check that |ν̂| = |ν̂|† . From these expressions one can find the expectation value of V̂ 2 and the dispersion of the operator, hV̂ iφ = V+ e α φ + V− e−α φ , 2 hV̂ iφ = W0 + W+ e 2α φ + W− e (2.15) −2α φ , (∆V̂ )2φ = Y0 + Y+ e2α φ + Y− e−2α φ , (2.16) (2.17) with V± , W0 , W± , Y0 and Y± being real and positive, given by Z 2 dF 4πγℓ2Pl λ e∓α x dx , V± = α dx  Z  2 dF̄ d2 F d F̄ dF i2πγ 2 ℓ4Pl λ2 dx , − W0 = 3G dx2 dx dx dx2   2 Z iπγ 2 ℓ4Pl λ2 dF̄ d2 F ∓2α x d F̄ dF W± = dx , − e 3G dx2 dx dx dx2 2 To be precise, F (x) is a positive momentum function, i.e. with a Fourier transform that has support on the positive axis. With such a choice, the solution to the constraint equation become of positive frequency. 5 for normalized states. Here Y0 = W0 − 2V+ V− and Y± = W± − V±2 . Note that the expressions for W0 and W± correct those found in [18]. From (2.15), it follows that the expectation value of the volume V̂ |φ is large at both very early and late times and has a non-zero global minimum Vmin = 2(V+ V− )1/2 /||χ||2 . The bounce ocurrs at time φVb = (2α)−1 ln(V− /V+ ) [17]. Around φ = φVb , the expectation value of the volume hV̂ iφ is symmetric. Similarly, hV̂ 2 iφ is symmetric across the value 2 2 φVb = (4α)−1 ln(W− /W+ ) for the scalar field. A trivial observation is that if φVb = φVb , the difference in the asymptotic values of the relative fluctuation  !2 !2  (∆V̂ ) (∆V̂ )  = W+ − W− (2.18) − D := lim  φ→∞ V+2 V−2 hV̂ i hV̂ i φ −φ vanishes. It should be noted that this quantity quantifies the change in semiclassicality across the bounce as pointed out in [18]. The other observable that one might want to consider as fundamental, is an observable naturally conjugate to pφ . Given that we are considering physical states, this does not correspond to the operator φ as happens in the kinematical setting. Instead, as illustrated in the Appendix A, the quantity that is conjugate to pφ , in the reduced physical model, corresponds to the quantity x(β). Thus, it is natural to consider the family of operators x̂φ , corresponding to the observable ‘x at time φ’. The expectation value of x̂ is defined as, 1 (χ1 , x̂χ2 )phy = [(|x̂|χ1 , χ2 )phy + (χ1 , |x̂|χ2 )phy ] 2 (2.19) where |x̂|F (x+ ) = xF (x+ ) and |x̂|F (x− ) = −xF (x− ). We have included the two terms in the definition (2.19) because |x̂| is not symmetric. We can now compute the expectation values and fluctuations of x̂ for any state of the physical Hilbert space hx̂iφ = X̃1 − φ , (2.20) 2 2 hx̂ iφ = X̃2 − 2φX̃1 + φ , (∆x̂) = X̃2 − X̃12 (2.21) , (2.22) with X̃1 and X̃2 real (state dependent) constants, given by Z ∞ Z ∞ |F (x)|2 dx F̄ (x)x∂x F (x)dx − i X̃1 = −2i −∞ Z−∞ Z ∞ ∞ 2 X̃2 = −2i |F (x)|2 xdx′ , , F̄ (x)x ∂x F (x)dx − 2i −∞ (2.23) (2.24) −∞ for normalized states. Is important to note that hx̂iφ ∝ φ, hx̂2 iφ ∝ φ2 but the fluctuation ∆x̂ is independent of φ! This property indeed gives support to the proposal that x̂ be the conjugate of p̂φ . Finally, we compute the expectation values for pφ , a Dirac observable as ∞ 2 dF dx p̂φ = −i~∂φ −→ hp̂φ i = 2~ −∞ dx Z ∞ dF̄ d2 F ′ 2 2 2 2 2 dx , , p̂φ = −~ ∂φ −→ hp̂φ i = −2~ i 2 −∞ dx dx Z 6 (2.25) (2.26) for normalized states. In the following sections, it will be convenient to consider a general physical state written as, Z ∞ Z ∞ −ik(φ+x) χ(x, φ) = F̃ (k) e dk − F̃ (k) e−ik(φ−x) dk (2.27) 0 0 where the Fourier transform F̃ (k) contains all the information of the state. Positive frequency solutions to Eq. (2.8) means that F̃ (k) has support on positive k’s only. Furthermore, we can consider initial states, for an ‘initial time’ φ = 0, in order to compute some of the quantities that determine the expectation value of relevant operators. It then suffices to specify F̃ (k) and from there to compute all the quantities V± , W0 , W± , X̃1 , X̃2 , hp̂φ i and hp̂2φ i. III. GAUSSIAN INITIAL STATES In this section we shall consider physical states with initial states given generalized Gaussian states, defined by functions F̃ (k) of the form: ( 2 2 k n e−(k−k0 ) /σ eipk , for k > 0, (3.1) F̃ (k) = 0, for k ≤ 0, with σ > 0, k0 > 0, n = 0, 1, 2, .... That is, we are choosing Gaussian states centered around the point k0 , with ‘dispersion’ given by σ. The physical states one has to consider are vanishing in the negative k axis, to have positive frequency solutions. In order to gain analytical control over all the quantities at hand, we shall in what follows consider instead the Gaussian states for all values of k. Thus, the quantities we shall compute are an approximation to the real quantities but, as we shall argue in detail, this approximation is justified when the values of k0 and σ are such that the wave function has a negligible contribution from the negative axis. A detailed analysis of the errors introduced by this simplification can be found in the Appendix C. Therefore, from now on, in order to have states that approximate positive frequency solutions, we shall impose the condition k0 ≫ σ. Further consistency conditions on σ, motivated by physical considerations will be derived below. The norm of the state in k space is Z ∞ Z ∞ ¯ k|F̃ (k)|2 dk. F̃ (k)ik F̃ (k)dk = 2 (χ̃, χ̃)phy = −2i 0 0 If we take the initial states in the k-space as in (3.1), the norm is, Z ∞ 2 2 kk 2n e−2(k−k0 ) /σ dk , (χ̃, χ̃)phy = 2 0 R∞ 2 2 which can be approximated by (χ̃, χ̃)phy = 2 −∞ kk 2n e−2(k−k0 ) /σ dk (see Appendix C for a justification of this approximation). √ With the change of variables u = σ2 (k − k0 ) this last integral takes the form √ #2n+1  2n+2 Z ∞ " σ 2k0 2 kχ̃k2phy = 2 √ u+ e−u du . (3.2) σ 2 −∞ 7 For concreteness, we shall consider the state corresponding to n = 0. In this case √ # 2 Z ∞ "  √ 2k0 −u2 σ u+ e du = 2π k0 σ . kχ̃k2phy = 2 √ σ 2 −∞ (3.3) The normalized n=0 states can be used to compute explicitly the expectations values and fluctuations of several physically interesting operators. The corresponding results for n=1 are summarized in the Appendix D. A. Expectation Values for Basic Observables Let us explore the basic observables for the Gaussian states in order to gain a better understanding of the free parameters (ko , σ, p, n) that characterize the states. The first observable we shall consider is pφ , that is a Dirac observable (and therefore a ‘constant of ∂ the motion’). The operator is represented as p̂φ · χ = −i~ ∂φ · χ, and its expectation value is thus given by Z ∞ 2 dF 2~ dx′ , . hp̂φ i = kχ̃k2phy −∞ dx Using Parseval’s theorem we get 2~ hp̂φ i = kχ̃k2phy Z ∞ −∞ 2 k 2 |F̃ (k)|2 dk. (3.4) 2 If we take our initial states F̃ (k) = k n e−(k−k0 ) /σ eipk then the integral takes the form Z ∞ 2~ 2 2 hp̂φ i = k 2n+2 e−2(k−k0 ) /σ dk . (3.5) 2 kχ̃kphy −∞ With the change of variables u = √ 2 (k σ − k0 ), the expectation value (3.5) takes the form √ #2n+2 2n+3 Z ∞ "  2k0 2~ σ 2 √ hp̂φ i = e−u du . u+ 2 kχ̃kphy σ 2 −∞ In the case of pure Gaussian states, namely for n = 0, we have   σ2 hp̂φ i == ~k0 1 + 2 . 4k0 (3.6) (3.7) This is telling us that, for the approximation we are taking, namely k0 ≫ σ, then hp̂φ i ≈ ~k0 . Therefore, as expected, the parameter k0 is giving us a good measure of the expectation value of pφ which can be regarded as the conjugate variable to x, on the reduced phase space of the system (See Appendix A). Let us now compute the fluctuations of this operator. Let us start by computing Z ∞ Z ∞ 2~2 i dF̄ d2 F 2~2 2 hp̂φ i = − k 3 |F̃ (k)|2 dk ′ , . (3.8) dx = kχk2phy −∞ dx dx2 kχk2phy −∞ 8 For our generalized Gaussian states the integral takes the form Z ∞ 2~2 i 2 2 2 k 2n+3 e−2(k−k0 ) /σ dk . hp̂φ i = 2 kχkphy −∞ (3.9) Which can be rewritten as, √ #2n+3  2n+4 Z ∞ " 2 σ 2k0 2~ i 2 √ e−u du . u+ hp̂2φ i = 2 kχkphy σ 2 −∞ Taking n = 0, hp̂2φ i == ~2 k02  3σ 2 1+ 2 4k0  . (3.10) Note that for large k0 , that is for those states satisfying k0 ≫ σ, then hp̂2φ i ≈ ~2 k02 . Let us now compute the dispersion of the observable p̂φ , (∆p̂φ )2 = hp̂2φ i − hp̂φ i2 . For the n = 0 case we have    2 3σ 2 σ2 2 2 2 2 2 (∆p̂φ ) = ~ k0 1 + 2 − ~ k0 1 + 2 4k0 4k0   ~2 σ 2 σ2 = (3.11) 1− 2 4 4k0 . This is telling us that which is a constant. Note that for k0 ≫ σ, the dispersion ∆p̂φ ≈ ~σ 2 the parameter σ has the interpretation one might have expected as the dispersion of the observable associated with the variable k, which in this case corresponds to pφ . Recall that the quantity that is conjugate to pφ , in the reduced model, corresponds to x(β). Thus, it is natural to consider the family of operators x̂φ , corresponding to the observable ‘x at the time φ’. It is most natural to define the symmetric operator acting on initial states F̃ , in the k-representation the operator defined in Eq. (2.20) can be written as:   i∂ i x̂F̃ := + − φ · F̃ . (3.12) ∂k 2k The first two terms arise when Eq. (2.23) is replaced into Eq. (2.20). It is now straightforward to compute the expectation value of the x̂ operator on the generalized Gaussian states. It is easy to see that, for all values of n, hx̂i = −p − φ , (3.13) which confirms the expectation that the parameter −p represents the point in x space, where the Gaussian is peaked (when φ = 0). This also corresponds to the classical dynamics (as given by the effective theory) for the variable x (with our choice pφ > 0). Furthermore, one can try to find the expectation value of the operator   2  i∂ 1 ∂ i 1 i i∂ ∂2 2 + φ2 . − 2φ + −φ =− 2 − + + x̂ = ∂k 2k ∂k k ∂k 4k 2 ∂k 2k Here it is important to note two important issues that arise when considering the operator defined by (3.12). The first one is that it involves a derivative operator and the second is the 9 factor 1/k in the operator. The derivative term implies that one needs to consider carefully the boundary conditions at k = 0, so that the boundary terms (when integrating by parts) does not contribute. In particular, we should have, for the x̂ operator, for states defined by functions f (k) and g(k), that (ḡ f k)|k=0 = 0. This condition is, of course, satisfied by all of our states n ≥ 0. The term 1/k imposes some fall-off conditions at the origin (k = 0) for the integrals to be finite. For instance, even when the expectation value of x̂ is well defined in the n = 0 Gaussians, these states are no longer in the domain of the operator. The state that results, when acting with the operator x̂ on a n = 0 state is not normalizable, and therefore, does not belong to the physical Hilbert space. We have to conclude that, in order to have the operator x̂ in the set of observables we want to consider, we have to exclude the n = 0 states and consider only n ≥ 1. In the n = 1 case, it is straightforward (if lengthy) to compute the expectation value of the operator,   3σ 2 1 + 2k02 1  + 2pφ + φ2 (3.14) hx̂2 iφ = p2 + 2  2 3σ σ 1+ 2 4k0 from which the fluctuations of the operator becomes, (∆x̂)2 = hx̂2 iφ − hx̂i2φ =  1+ 1  σ2 1 + 3σ 2 2k02 3σ 2 4k02   (3.15) which also tells us that the parameter σ has the expected interpretation of providing the inverse of the dispersion for the observable x. This is realized when k0 ≫ σ and therefore the second term in the previous expression is very close to 1. We can now examine the uncertainty relations for the observables p̂φ and x̂ (for n = 1):    9σ 4 3σ 2 3σ 2 9σ 6 1 + 1 + + − 2 4 2 6 2 16k0 4k0 64k0 2k0 ~ (∆p̂φ )2 (∆x̂)2 = (3.16)  3 4 3σ 2 1 + 4k2 0 We note that, if k0 ≫ σ, then the uncertainty relations (∆p̂φ )(∆x̂) ≥ ~ 2 are very close to being saturated. As we shall see in the next section, these conditions are necessary for the generalized Gaussian states to represent acceptable semiclassical states. B. Volume Operator Let us start by considering the operator V̂ |φ0 corresponding to the volume at ‘time’ φ0 . The expectation value of this operator is given by Eq. (2.15) so it suffices to compute the constants V± given by V± κ = αkχk2phy Z dF dx 2 e∓α x dx , 10 with κ = 4πγℓ2Pl λ which can be written as, V± 1 κ Ṽ± = = 2 kχkphy kχk2phy α 2 Z k(k ∓ iα)F̃¯ (k)F̃ (k ∓ iα) dk . (3.17) 2 If we take the state as F̃ (k) = k n e−(k−k0 ) /σ eipk , the integral is Z κ ∞ n −(k−k0 )2 /σ2 −ipk 2 2 Ṽ± = kk e e [k ∓ iα][k ∓ iα]n e−(k∓iα−k0 ) /σ eip(k∓iα) dk . α −∞ This can be rewritten as, κ Ṽ± = e±pα α Z ∞ −∞ [k(k ∓ iα)]n+1 e−(k−k0 ) 2 /σ 2 e−(k∓iα−k0 ) 2 /σ 2 dk . (3.18) √ √ It is straightforward to see that, with the change of variable u = 2k − 2k0 ∓ √iα2 , the integral (3.18) can be written as  n+1 Z ∞h in+1 √ κ 1 2 2 ±pα α2 /2σ 2 2 2 2 √ e e 2u + 4 2k0 u + 4k0 + α Ṽ± = e−u /σ du . 2α 4 −∞ For the n = 0 case we have,   Z i √ 1 ±pα α2 /2σ2 ∞ h 2 κ 2 2 Ṽ± = √ 2u + 4 2k0 u + 4k02 + α2 e−u /σ du . e e α 2 4 −∞ (3.19) From which we get Ṽ± = κ 2 2√ √ e±pα eα /2σ π(σ 3 + 4k02 σ + α2 σ) . α4 2 If we take now a normalized states using Eq. (3.3), the result is   Ṽ± πγℓ2Pl λ ±pα α2 /2σ2 σ 2 + α2 . = 4k0 + e e V± = kχ̃k2phy 2α k0 (3.20) In order to calculate the dispersion of the volume operator, it is necessary to calculate the three integrals W0 , W+ , W− . Let us start by computing the quantity W0 .  Z ∞ 2 iα2 2πγ 2 ℓ4Pl λ2 d F̄ dF dF̄ d2 F W0 = dx, with α = − , (3.21) 2 kχk2phy −∞ dx2 dx dx dx2 3G which can be written as 2α2 W̃0 = W0 = 2 kχkphy kχk2phy Z ∞ −∞ k 3 |F̃ (k)|2 dk . (3.22) This is the same integral that the one in Eq. (3.8). Then, for a normalized Gaussian state with n = 0 we only need to replace ~2 by α2 into Eq. (3.10) to obtain that   3σ 2 2 W 0 = α 2 k0 1 + 2 . (3.23) 4k0 11 Let us now compute the quantities W± = W̃± /kχk2phy , with  2  Z ∞ πγ 2 ℓ4Pl λ2 dF̄ d2 F ∓2αx d F̄ dF e dx, with α = W̃± = iα3 − , 3 dx2 dx dx dx2 3G −∞ (3.24) that can be written as, Z ∞ h i ¯ 2 ¯ k F̃ (k)(k ∓ 2iα)F̃ (k ∓ 2iα) + (k ± 2iα)F̃ (k ∓ 2iα)F̃ (k) dk . W̃± = α3 (3.25) −∞ 2 2 If we take the initial states F̃ (k) = k n e−(k−k0 ) /σ eipk , the integral is Z ∞ 2 2 ±2pα k n+2 e−(k−k0 ) /σ W̃± = α3 e −∞ n+1 −(k∓2iα−k0 )2 /σ 2 [(k ∓ 2iα) e + (k ± 2iα)n+1 e−(k±2iα−k0 ) 2 /σ 2 ]dk (3.26) In order to do the integral Z ∞ 2 2 2 2 k n+2 e−(k−k0 ) /σ (k ± 2iα)n+1 e−(k±2iα−k0 ) /σ dk . I(±) = −∞ √ It is straightforward to show that with the change of variables u = 2(k − k0 ± iα), we get n+2  n+1 Z ∞ u u 2 2 2 2 du √ + k0 ∓ iα √ + k0 ∓ iα ± 2iα e−u /σ e2α /σ √ I(±) = 2 2 2 −∞ " #n+1    Z 2 2 2 ∞ e2α /σ u u 2 2 √ + k0 + α 2 √ + k0 ∓ iα e−u /σ du . = √ (3.27) 2 2 2 −∞ The real part of I(±) is 2 e2α /σ Re[I(±)] = √ 2 2 Z ∞ −∞ " u √ + k0 2 2 +α 2 #n+1   u 2 2 √ + k0 e−u /σ du . 2 (3.28) Then Re[I(+)] = Re[I(−)], and the imaginary part of I(±) is " #n+1 2 2 2 Z ∞ u e2α /σ 2 2 √ + k0 + α 2 e−u /σ du , Im[I(±)] = ∓α √ 2 2 −∞ therefore, Im[I(+)] = −Im[I(−)]. Using the real and imaginary parts, we have that the integral W̃± takes the form W̃± = α3 e±2pα [I(+) + I(−)] = 2α3 e±2pα Re[I(+)] . (3.29) This tells us that W̃± is real valued (as expected) and that it is only necessary to calculate the integral (3.28). Taking n = 0 the integral (3.28) takes the form " # 2  2 2 Z ∞ e2α /σ u u 2 2 2 √ + k0 + α √ + k0 e−u /σ du Re[I(±)] = √ 2 2 2 −∞   Z 2 2 e2α /σ u2 u u3 2 2 3 2 −u2 /σ 2 √ = √ + 3 k + (3k + α ) + (k + α k ) e du . 0 0 0 0 22/3 2 2 2 12 Using the integrals from Appendix B, the integral is   2 2√ e2α /σ π 3 3 3 2 √ Re[I(±)] = k0 σ + k 0 σ + α k0 σ , 4 2 (3.30) from which we have that, W̃± = α3 e ±2pα 2α2 /σ 2 e √ 2π  3 k0 σ 3 + k03 σ + α2 k0 σ 4  Including now the normalization of the state we have   πγ 2 ℓ4Pl λ2 ±2pα 2α2 /σ2 2 3 2 2 W± = k0 + σ + α . e e 3G 4 . (3.31) (3.32) From the last equation and Eq. (3.20) we can observe that the difference in the asymptotic relative fluctuations after and before the bounce, as defined by Eq. (2.18), vanishes. That is, D = 0. It is important to note that this is a general result for any Gaussian state of the form of Eq. (3.1), for all values of n = 0, 1, 2, ..., as was expected to happen [18]. IV. SEMICLASSICALITY CONDITIONS So far we have considered generalized Gaussian states as initial states for physical states in the exactly solvable k=0 LQC model with a massless scalar field. We have seen that the parameters that define the states, in the pure Gaussian case with n = 0, have the expected interpretation: The parameter k0 is related to the expectation value of p̂φ , −p to the expectation value of x̂, and ~σ/2 to the dispersion of pφ . The only condition that we have imposed so far is k0 ≫ σ, which guaranties the validity of our approximation. We have also seen that, for states which satisfy this consistency condition, there are at least two results that follow. First, it warranties that the relative dispersion of pφ is small. Second, this conditions implies that the uncertainty relations become saturated in the sense that (∆p̂φ )(∆x̂) ≈ ~/2. From this perspective, it could seem that any value of k0 and σ, provided they satisfy k0 ≫ σ, might be acceptable to define semiclassical states. The purpose of this section is to explore this issue further and answer the following question: Are there more stringent conditions that one must impose in order to have semiclassical states that will further restrict the possible values of k0 and σ? As we shall see, the answer is in the affirmative. A. Asymptotic Volume In order to answer this question we shall first consider the volume operator. Since all physical states have the property that the expectation value of the volume operator V̂ |φ at time φ follow the same functional form and, therefore, follow for large φ the same dynamics of the classical dynamics (i.e. Einstein’s equations) V (φ) ∼ e±αφ , we need more criteria to select those states that are semiclassical. The obvious strategy is to consider the state’s relative  dispersion (∆V̂ )/hV̂ i 2 φ of the volume operator at time φ. One expects that semiclassical states will have a very small relative dispersion ‘at late times’ when the dynamics approaches 13 the classical dynamics. It is then natural to consider the asymptotic relative dispersions given by ! ∆± := lim φ→±∞ 2 ∆V̂ = hV̂ i φ W± − 1. V±2 (4.1) It is straightforward to find the analytical expression for these quantities in our states (in the case n = 0) to be   3σ 2 α2 + 1 + 4k02 k02 W± 2 2 ∆± = 2 − 1 = eα /σ  (4.2) 2 − 1 . V± σ 2 +α2 1 + 4k2 0 It is interesting to note that, in the previous expression, there is some competition between 2 2 the factor eα /σ and the second part. The exponential becomes close to one if σ is large compared to α, but can become very large if σ becomes too small. On the other hand, the larger σ becomes, the larger the second term. Thus, that there must be an optimal value σ̃ for which ∆± is the smallest. The form of ∆± can be seen in Fig. 1. Another important aspect to consider is the following. So far, we have not imposed any condition on pφ . That is, we have not said that it has to be ‘large’, mainly because in the classical theory there is no dimension-full quantity with respect to which one can compare it. In the quantum theory, with the introduction of ~ we do have such scale. As we can see from Eq.(4.2) and the expression for hp̂φ i, the quantity ~α has the same dimensions of pφ . Furthermore, if we now impose the condition k0 ≫ α, all the terms in parenthesis in Eq.(4.2) are small, so we can indeed try to find the optimum value for σ. Note also that the condition k0 ≫ α implies hp̂φ i ≫ ~α. It is in this sense that we can call the momentum pφ ‘large’. The condition that the expression (4.2) be an extrema is a cubical equation, and the only physically interesting solution can be approximated as, σ̃ 2 ≈ 2 α k0 + 2 2 α . 7 (4.3) Since we require that k0 ≫ α, √ then the second term is very small compared to the first one and we can approximate σ̃ ≈ 2 α k0 . It is important to stress that this condition selects a preferred value for σ for which the state is most semiclassical. As can be seen from the Figure 1, if we make σ slightly smaller (in an attempt to ‘make the dispersion small’) the asymptotic relative dispersion in volume becomes very large, making the state not a good candidate for a semiclassical state. If we now introduce this value for σ̃ into the relative fluctuation function, then this can be approximated by " # (∆V̂ )2 α lim (4.4) (k0 ) ≈ 2 φ→±∞ k0 hV̂ i σ̃ This approximation has an error of 0.0002% when k0 = 30000 (in Planck units), and becomes better as k0 grows. This tells us the precise relation between the parameter k0 that controls the expectation value of pφ and the minimum value of the asymptotic relative fluctuation of the volume operator. The σ dependence of the asymptotic relative dispersion of the volume can be seen in Figure 1, with k0 = 30000. Two features characterize this plot, namely, the first one is that for σ 2 < σ̃ 2 the function is exponential so it grows very quickly as 14 FIG. 1: The asymptotic relative fluctuation ∆± is plotted as a function of σ 2 with k0 = 30000 in Planck units, where α ≈ 6.14. Here we have chosen n = 0 and p = 0. Two features characterize the behavior of the function near the minimum σ̃. The first one is that for σ 2 < σ̃ 2 the function is exponential; the second one is that for σ 2 > σ̃ 2 the function has a polynomial behavior. one approaches zero; the second feature is that for σ 2 > σ̃ 2 the function has a polynomial behavior, so one can have values of σ > σ̃ without ∆± changing too much. To summarize, we have seen that not any Gaussian state will be an admissible semiclassical state. In order to minimize the asymptotic relative fluctuation ∆± , one has to choose the parameter σ carefully as a function of the other relevant parameter (in this case, the parameter k0 ). That is, if we choose σ to be of the order of σ 2 ∼ 2αk0 , we can then minimize the asymptotic relative dispersion of volume after and before the bounce (which are the same, as we have seen). Gaussian states for which this condition is not satisfied will have, in general, large quantum fluctuations in the regime where we expect them to be small –for large times after and before the bounce– so they can not be regarded as semiclassical. Let us then assume that we have chosen the value σ̃ for σ. The question we want to ask now is the following: Can we reproduce some of the features that are characteristic of the so called “effective dynamics”? As we have discussed before, the ‘classical’ Hamiltonian constraint one obtains from replacing classical quantities like connection and curvature by a corresponding holonomy functions with a finite parameter λ, can be seen as either a starting point for ‘polymer quantization’, or as the ‘classical limit’ of a loop quantized theory. In any case, we expect that the semiclassical states of the quantum theory approach this ‘classical theory’ in the appropriate regime. In other words, we want to know when the effective dynamics is a good approximation to the full quantum dynamics. Is there a choice of parameters in the Gaussian states for which the effective dynamics is not a good approximation? We shall answer these questions in the remainder of this section. B. How Semiclassical is the Bounce? Another important issue is the question of how semiclassical the state is at the bounce. For instance, if the state has a very small relative asymptotic dispersion ∆± at late times, 15 one might want to know what the relative dispersion is at the bounce and how it compares to ∆± . A plot of the relative fluctuation, as a function of internal time φ can be seen in Fig. 2, where it is shown that the relative fluctuation is bounded, it attains its minimum value at the bounce and approaches ∆± asymptotically. Thus, somewhat surprisingly, the state does not become very “quantum-like” at the Planck scale but rather preserves its coherence across the bounce. Let us now try to estimate the value of the relative dispersion in comparison to its 1 1 − ln VV−+ = 4α ln W , then hV̂ iφb = asymptotic value. We know that, at the bounce, φb = 2α W+ √ 2 V+ V− , and p (∆V̂ )2φ = W0 − 4V+ V− + 2 W− W+ . (4.5) b Then the relative fluctuation in the bounce is (∆V̂ )2 hV̂ i2 φb √ W0 + 2 W− W+ − 1. = 4V+ V− (4.6) For n = 0 in the Gaussian states we have (∆V̂ ) 2 hV̂ i2 φb − 1 (∆V̂ ) 2 hV̂ i2 2 φ→±∞   2 2 2 e−α /σ 1 + 3σ 4k02 1 =  2 − . 2 2 2 2 1 + σ 4k+α 2 (4.7) 0 Now, if we impose k0 ≫ σ ≫ α, which is satisfied if we choose σ = σ̃ = √ 2αk0 , then the FIG. 2: The relative fluctuation (∆V̂ )2 /hV̂ i2 is plotted as a function of internal time φ with k0 = 30000, in Planck units where α ≈ 6.14. Note that the minimum of this quantity is at the bounce (here corresponding to φ = 0). It is found that the value at the bounce is approximately 1/2 of its asymptotic value for the generalized Gaussian states. last equation has the form, (∆V̂ )2 hV̂ i2 φb − 1 (∆V̂ )2 2 hV̂ i2 = O(α2 /σ 2 ) + O(σ 2 /k02 ) . φ→±∞ 16 This tells us that when k0 ≫ σ ≫ α, the relative volume fluctuation at the bounce becomes (∆V̂ )2 hV̂ i2 φb ≈ 1 (∆V̂ )2 2 hV̂ i2 φ→±∞ 1 = ∆± . 2 (4.8) This approximation, for instance, has an error of 0.007% when k0 = 30000, and it becomes smaller as k0 grows. If we use the Eqs. (4.8) and (4.4) and choose the value of σ that minimized the relative fluctuation we get (∆V̂ )2 hV̂ i2 φb ≈ α . 2k0 This approximation have an error of 0.01% when k0 = 30000, and becomes smaller as we increase k0 . We have performed intensive numerical explorations and have seen that, for n ≥ 1, the relative fluctuation is again smaller at the bounce and approaches 1/2 of its asymptotic value. Thus, we can conclude that states that satisfy the conditions k0 ≫ σ ≫ α, imposed by our requirements on the asymptotic relative dispersion of the volume through the choice σ = σ̃, exhibit also semiclassical behavior at the bounce, in the sense that the relative dispersion in volume is bounded and small. Even more, the relative fluctuation at the bounce is approximately 1/2 of the asymptotic value. These analytical results give support to the numerical explorations reported in [5, 6]. C. Volume and Energy Density at the Bounce Having shown that a semiclassical quantum state behaves also semiclassically at the bounce, it is then natural to ask whether the state yields observables –through their expectation values– that follow some classical trajectories. It turns out that it does, but the ‘classical theory’ it approaches is not the standard classical theory (GR+ massless scalar field), but rather an effective theory, as defined by an effective Hamiltonian constraint Cλ [23], which contains the quantum geometry scale λ. The effective dynamics, is generated by the effective Hamiltonian constraint p2φ 3 2 2 Cλ = − V sin (λβ) + ≈0 8πGγ 2 λ2 2 and has several important features. The first one is that all trajectories have a bounce that occurs at the same critical density ρcrit = 8πGγ3 2 λ2 . The volume reaches a minimum value at the bounce that depends on the observable pφ : r 8πGγ 2 λ2 pφ . (4.9) Vmin = 6 In the quantum theory, we have seen that √ every physical states has a bounce where the minimum volume is given by Vbounce = 2 V+ V− . It is then natural to find the corresponding volume at the bounce for our family of Gaussian states. It is straightforward to see that the minimal value of the volume, for the case n = 0, is   4πγℓ2Pl λ α2 /2σ2 σ 2 + α2 Vbounce = . (4.10) e k0 1 + α 4k02 17 which is of the form, Vbounce = 4πγℓ2Pl λ k0 + O(α2 /σ 2 ) + O(σ 2 /k02 ) + O(α2 /k02 ) . α If we recall our previous result that hp̂φ i = ~ k0 + O(σ 2 /k02 ), and assume the semiclassically conditions √ found in previous sections, namely that σ ≫ α and k0 ≫ σ (as is the case when σ = σ̃ =∼ 2αk0 and k0 ≫ α) in Eq. (4.10), we can compare the last two equations to see √ 4πγℓ2 λ hp̂ i that Vbounce ≈ αPl ~φ . Using the value α = 12πG, we conclude that Vbounce ≈ Vmin , (4.11) which is what we wanted to show. Note that the higher the value of k0 , the better the approximation becomes. In other words, if k0 were not large enough (compared to α), not only would the asymptotic relative fluctuation of volume be large, but the effective equations would fail to be a good approximation to the quantum dynamics. This seems to indicate that quantum Gaussian states with a higher value of k0 behave more classically, in contradiction with the classical intuition that tells us that ‘rescaling of k0 ’ is physically irrelevant3 . Thus, we have to conclude that this rescaling symmetry of the classical theory is broken in the quantum theory. Note however, that even when the exact symmetry is broken, one might expect to regain it in the limit of large k0 which, as we have seen, corresponds to hp̂φ i ≫ ~α. For an in-depth discussion regarding this scaling freedom see [26]. Another important result for this solvable model is that the energy density ρ is absolutely bounded by the critical density ρcrit [17]. It is then natural to ask what the behavior of the energy density at the bounce is, for semiclassical states. One might imagine that, for instance, the more semiclassical the state, the higher the density at the bounce. In this part we shall explore this question by taking several quantities to measure density. For instance, hp̂ i2 the simplest one would be ρ̃ = 2hV̂φ i2 , a quantity shown to be bounded by ρcrit [17]. It is straightforward to find this quantity for the n = 0 states, 2  2 2 σ2 e−α /σ 1 + 4k 2 0 (4.12) ρ̃ = ρcrit  2 σ 2 +α2 1 + 4k2 0 Note√ that ρ̃ < ρcrit and one approaches it as k0 → ∞, assuming again that we have chosen σ ∼ 2α k0 and k0 ≫ α. One can also see that, if we were to fix k0 and leave σ free, the density grows and approaches ρcrit as σ grows. Thus, asking for the density to approach the critical density is not a very stringent condition on σ as the relative fluctuations in volume was. Note also that, if k0 was small enough to be of the order of α, the density at the bounce would be far from the critical density giving yet another indication that those states are not semiclassical. The other quantities representing density that one can build, by taking for instance hp̂2φ i and hV̂ 2 i, give expressions that have the same qualitative behavior, approaching ρcrit as k0 grows. 3 Recall that the k=0 FRW model has a rescaling symmetry of the equations of motion, and of the underlying spacetime metric given by (V, β, φ, pφ ) → (ℓV, β, φ, ℓpφ ) for ℓ a constant. This can be understood as coming from the freedom of choosing arbitrary fiducial cells for the formulation of the theory [22]. 18 D. Where is the Classical Region? Φ 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0 100 000 200 000 300 000 400 000 k0 500 000 FIG. 3: The values φ̃ of φ for which one reaches the classical region are plotted as a function of k0 , where n=0, σ = σ̃ and p = 0 at φ = 0. In the plot, δ = 1 dotted, δ = 0.5 dashed, δ = 0.25 dotdashed, δ = 0.125 down line, δ = 0.0625 middle line, and δ = 0.03125 the top line. Note that for high values of k0 , the time φ̃ remains a constant. Ρ Ρcrit 0.020 0.015 0.010 0.005 0 100 000 200 000 300 000 400 000 k0 500 000 FIG. 4: Values of density are plotted for different values of δ(φ̃), as function of k0 , where n=0, σ = σ̃ and p = 0 at φ = 0. Here, ρ1,2,3,4 /ρcrit for which δ = 1 dotted, δ = 0.5 dashed, δ = 0.25 dotdashed, δ = 0.125 up line, δ = 0.0625 middle line, and δ = 0.03125 the bottom line. In the LQC description of the dynamics as governed by the effective Hamiltonian, one can unambiguously ask when one recovers the GR classical dynamics. One way of measuring this could be in terms of the variable β (when λβ ≪ 1), or in terms of the energy density (ρ ≪ ρcrit ). However if one wants to ask a similar question in the quantum realm, one 19 immediately faces the following dilemma. As we have seen in the previous part, a Gaussian state remains semiclassical across and after the bounce. In fact, the closer to the bounce, the more semiclassical it is. We can not therefore ask when does the state become semiclassical, based on the behavior of fluctuations of the relevant operators. Still, we would like to have the notion of a transition from a quantum gravity dominated regime to a the regime where the quantum gravity effects are small and one is in the “classical regime”. For this purpose, we want to propose such a criteria, and define the (percentual) error between the relative fluctuation and the asymptotic relative fluctuation, δ(φ) := (∆V )2 hV i2 φ − ∆+ ∆+ × 100 . (4.13) We expect that this error will be small when we are approaching the classical region. Let us now motivate this proposal. While we can not define a transition in terms of a change in the relative fluctuations of the basic operators, we could compare them to those of the standard ‘Wheeler DeWitt’ model. For we know that the expectation values in the WDW theory follow closely the classical dynamics of general relativity. There is indeed a canonical way of mapping a LQC state to a WDW state, so the comparison of expectation values is well defined [17]. We have done that for our Gaussian states and computed the corresponding relative fluctuation for the volume. It turns out that, for the WDW dynamics, the relative fluctuation is constant in φ evolution, and corresponds precisely to the asymptotic relative dispersion of LQC. Thus, the LQC evolution not only approximates the WDW one in terms of expectation values but also in terms of the relative fluctuations. It is then justified to measure the transition to the ‘classical regime’ in terms of how close the LQC evolution is to the corresponding WDW dynamics. In order to study where the classical region begins, we found the value of φ for which δ = 1/2m with m = 0, ..., 5, taking φ = 0 at the bounce. The values φ̃ of φ that satisfy these hp̂ i2 hp̂ i2 conditions are plotted in figure 3. Then we evaluated the densities ρ1 = 2hV̂φ i2 , ρ2 = 2hV̂φ 2 i hp̂2 i hp̂2 i , ρ3 = 2hV̂φi2 , ρ4 = 2hV̂φ2 i , at this values of φ. The density is plotted in figure 4. This plot does not depend of the used density (ρ1 , ρ2 , ρ3 , ρ4 ). From figure 4 we can see that ρ(φ̃) δ(φ̃) ∼ , ρcrit 5 (4.14) √ for k0 ≫ α, σ = σ̃ = 2αk0 , p = 0 and n = 0. These results are similar if n > 0. Furthermore, if we choose p 6= 0, the only change is in the value of φ at the bounce. If we change the value of σ for a fixed k0 , then for larger (than σ̃) values, the time φ̃ of arrival to the classical region is smaller. In that case, the value of the density is closer to the critical one, and in the limit of large σ the classical region is ‘at the bounce’, which does not make much sense. In this way the need to bound σ to be near the value σ̃ is manifested. The relation given by Eq. (4.14) is telling us the order of the quantum corrections. Thus, if we say that the quantum corrections are negligible when, say, δ ∼ 10−10 then the density will be of the order ρ/ρcrit ∼ 2 × 10−11 . Recall that Gaussian states are symmetric about the bounce, namely ∆+ = ∆− , therefore all the conclusions of this part equally apply before the bounce. Let us now summarize this section. We have seen that appropriate physical conditions imposed on the behavior of basic physical observables are enough to constraint the parameters that characterize the generalized Gaussian states (3.1), and obtain truly semiclassical 20 states. Of the three continuous parameters (k0 , σ, p) that characterize the states, two of them are fixed by semiclassicality considerations, while the choice of p represents a true choice of initial condition. We have seen that asking that the relative fluctuations of the observable p̂φ be small imposes that k0 ≫ σ. Smallness of the asymptotic relative fluctuation of the volume implies then that k0 ≫ σ ≫ α. If these conditions are satisfied, we saw that the semiclassical states at ‘late times’ remain semiclassical across the bounce and that the volume and density at the bounce are very well approximated by the effective theory. We have also seen that the scaling symmetry of the classical theory is not present, within the class of states under consideration, when k0 is of the order of α. However, in the limit of large k0 , all the properties of the state remain invariant –in terms of being well approximated by the classical effective theory–. This leads us to conclude that the scaling symmetry is approximately recovered for “large pφ ” (for details regarding this issue, see [26]). Still, the Gaussian states we have considered are not the most general states one can consider. In the following section we shall explore the so called squeezed states, a generalization of the Gaussian states and compare their semiclassicality properties to those of the Gaussian states. V. SQUEEZED STATES In this section we consider states that generalize the Gaussian states considered so far. The strategy will be to consider squeezed states that are nearby the Gaussian semiclassical states and compare their properties. In particular, we would like to know if the squeezed states can improve the samiclassicality properties such a better behavior of relative fluctuations of volume or a better approximation to the ‘effective theory’. The generalized Gaussian state we have considered so-far had three free (real) parameters (p, k0 , σ) and one discrete parameter (n). We saw that, given a point of the physical phase space, we could approximate the internal dynamics of the theory by a choice of p, and that semiclassicality imposes conditions on the two other parameters k0 and σ. Let us now take the initial states as ( 2 k n e−η(k−β) , for k > 0, η, β ∈ C F̃ (k) = (5.1) 0, for k ≤ 0, depending on two complex parameters (η, β). One can reduce to a Gaussian state from a squeezed state by setting Im(η) =: ηI = 0, Re(η) =: ηR = 1 , σ2 Re(β) =: βR = k0 , 2ηR βI = p . (5.2) Thus, we see that the extra parameter in the definition of the states if given by the imaginary part of η. The norm of this state in k space is Z ∞ 2 2 kk 2n e−η(k−β) e−η̄(k−β̄) dk . (χ̃, χ̃)phy = 2 −∞ Using relation (B9) for t = 0 the last integral can be written as Z ∞ 2 2 +β 2 ) 4η β β 2 2ηR (a2 −βR I I R I e k 2n+1 e−2ηR (k−a) dk , kχ̃kphy = 2e −∞ 21 (5.3) where ηI . (5.4) ηR If we want the convergence of the integral then ηR > 0 and if we want to select the positive frequency then a ≫ √1ηR i.e. βR − βI ηηRI ≫ √1ηR . With this assumptions the integral is well defined, and the error is small, as shown in Appendix C. If we take n = 0 and use the change of variables u = k − a we have Z ∞ 2 +β 2 ) 4η β β 2 2 2ηR (a2 −βR I I R I e (u + a)e−2ηR u dk . (5.5) kχ̃kphy = 2e a = βR − βI −∞ Then kχ̃k2phy A. r 2π 2ηR (a2 −βR2 +βI2 ) 4ηI βI βR =a . e e ηR (5.6) Elementary Observables We can compute the expectation value and dispersion of the fundamental observable p̂φ . Z ∞ 2~ 2 2 k 2n+2 e−η̄(k−β̄) e−η(k−β) dk. (5.7) hp̂φ i = 2 kχ̃kphy −∞ If we use Eq. (B9), with t = 0, then we obtain 2~ 2ηR (a2 −βR2 +βI2 ) 4ηI βI βR hp̂φ i = e e kχ̃k2phy Z ∞ 2 k 2n+2 e−2ηR (k−a) dk . (5.8) −∞ If we take n = 0, 2~ 2ηR (a2 −βR2 +βI2 ) 4ηI βI βR hp̂φ i = e e kχ̃k2phy √ π (2ηR )−3/2 + a2 2 r π 2ηR  . (5.9) This becomes, using Eq. (5.6),   ~ 1 2 hp̂φ i = +a . a 4ηR (5.10) Now we want to calculate the expectation value of p̂2φ . Z ∞ 2~2 2 k 3 |F̃ (k)|2 dk. hp̂φ i = kχ̃k2phy −∞ (5.11) In the squeezed states (5.1) we get hp̂2φ i 2~2 = kχ̃k2phy Z ∞ 2 2 k 2n+3 e−η̄(k−β̄) e−η(k−β) dk. (5.12) −∞ If we now use Eq. (5.6), the result is for n = 0   3 2 2 2 hp̂φ i = ~ +a . 4ηR 22 (5.13) From this we can find the dispersion and get, (∆p̂φ )2 = ~2 ~2 . − 4ηR 16ηR2 a2 Writing this explicitly in the original variables is −2  ~2 ~2 ηI 2 (∆p̂φ ) = βR − βI − . 4ηR 16ηR2 ηR B. (5.14) (5.15) Volume We shall compute the quantities V± and W0,± from which one can find the expectation value and dispersion of the volume operator. The relevant coefficients are Z κ ∞ 2 2 (−ik)k n e−η̄(k−β̄) i[k ∓ iα][k ∓ iα]n e−η(k∓iα−β) dk Ṽ± = α −∞ Z κ ∞ n+1 2 2 k [k ∓ iα]n+1 e−η̄(k−β̄) e−η(k∓iα−β) dk. (5.16) = α −∞ with κ = 4πγℓ2Pl λ. As in previous subsections we will only write down explicitly the n = 0 case. This expression can be put on the form,   2    b2± α2 ηI2 α ηR 1 κ 1+ 2 exp ± 2αβI ηR + + (5.17) Ṽ± = aα 2 ηR 8ηR 2 8
where b± = a ∓ α2 ηηRI = βR − ηηRI βI ± α2 . In the Gaussian variables it takes the form Ṽ±   −1   2
κ pσ 4 α 2 4 = k 0 − ηI ± pα 1 + ηI σ × exp α 2 2σ 2 # "  2  2 2 pσ α α σ2 1 k 0 − ηI σ 2 + + ± 8 2 2 2 8 (5.18) which reduces to the Gaussian case when ηI = 0. The next quantities we need to compute in order to find the dispersion of the volume operator are the coefficients (W0 , W± ). The first one is given by Z ∞ 2πγ 2 ℓ4Pl λ2 k 3 |F̃ (k)|2 dk, with α2 = W̃0 = 2α2 . (5.19) 3G −∞ This is the same integral that the one in Eq. (5.11). Then, in order to consider the proper normalization for the squeezed state with n = 0, we only need to replace ~2 by α2 into Eq. (5.13), the result is " 2 #  ηI 3 W0 = α 2 + βR − βI . (5.20) 4ηR ηR 23 The other quantities are given by the expression, Z ∞ h i k 2 F̃¯ (k)(k ∓ 2iα)F̃ (k ∓ 2iα) + C.C. dk , W̃± = α3 −∞ = 2α3 Re [I± ] , (5.21) R∞ πγ 2 ℓ4Pl λ2 with α3 = and I± = −∞ k 2 F̃¯ (k)(k ∓ 2iα)F̃ (k ∓ 2iα)dk. This integral can be 3G performed for n = 0 and we get     ′ 
b± 1 ηI2 ′2 2 2 (5.22) × + + b± + α , W̃± = α3 exp 2α ηR ± 4αηR βI 1 + 2 ηR 2ηR a 4ηR where b′± = a∓α ηηRI = βR − ηηRI (βI ± α). It is now straightforward to find ∆± , the asymptotic relative dispersion for volume, i h   b′± + 1 + b′ 2 + α2  2 2 ± 2ηR a 4ηR η a − 1′ , . (5.23) ∆± = exp α2 ηR 1 + 2I h i2 2 b 2 4 ηR 1 α ± + + 8ηR 2 8 Let us now explore those conditions that select semiclassical states. C. Semiclassicality Conditions In this part, we want to see how the semiclassicality conditions are satisfied if we move along the ‘squeezing parameter’ ηI , off the Gaussian semiclassical states, corresponding to ηI = 0. Instead of performing an exhaustive analysis of this question, as we have done in previous sections for Gaussian states, we have computed the relevant quantities and plotted them. Recall that for Gaussian states, asking that the asymptotic relative dispersion in volume √ be a minimum, selected an optimal ‘width’ σ = σ̃ ≈ 2αk0 for the Gaussian. This value had also the property that the density at the bounce was very close to the critical density ρcrit . In Fig. 5 we have plotted the asymptotic relative dispersion and density at the bounce as functions of σ and ηI . The ηI = 0 slice corresponds then to the values found for Gaussian states. It can not be appreciated in the figure but, for a fixed σ, the asymptotic relative dispersion ∆± , as a function of ηI does not attain its minimum at zero, but rather at some value η̃I 6= 0 for ∆+ (and ηI = −η̃I for ∆− ). Thus, if one chooses, for the parameter of the squeezed state the value ηI = η̃I , one is ‘improving’ the asymptotic relative dispersion after the bounce, and, at the same time, increasing the value of the asymptotic dispersion before the bounce. In this process, one is introducing an asymmetry in the fluctuations. This fact has been used to suggest the possibility, for example, that the asymmetry could be so large to spoil the semiclassicality properties across the bounce. As we shall see below, this possibility is however, not realized. What one can indeed see from the figure is that, as we move further away from the Gaussian states, that is, as we increase to values |ηI | > |η̃I |, the asymptotic relative dispersion increases steeply, for both signs of ηI . As can be seen from the figure, the density at the bounce (in blue) decreases as we go away from ηI = 0 in both directions, reaching densities much less than the critical value very fast. 24 FIG. 5: Density and asymptotic relative fluctuation ∆+ . In the plot 1/ηR = σ 2 : (0, 100000) , ηI : (−0.001, 0.001) , βR = k0 = 30000 , βI = 0 and n = 0. The maximal value of the density correspond to the Gaussian states. Is clear that the squeezed states have a maximal density close to ρcrit only if the state is near to the Gaussian states. FIG. 6: Volume at the Bounce. In the plot 1/ηR = σ 2 : (0, 8 × 106 ) , ηI : (−0.0001, 0.0001) , βR = k0 = 30000 , βI = 0 and n = 0. The global minimum is over the Gaussian states and has an exponential behavior if we move far away from the Gaussian state or if σ 2 is smaller that σ̃ 2 = 2αk0 . 25 Both phenomena suggest that we can indeed consider squeezed states as semiclassical states provided we remain very close to the Gaussian states. The question is then how close we can be in order to still have semiclassical behavior as exhibited by the Gaussian states. For this purpose, we have plotted in Fig. 6 the volume at the bounce as function of σ and ηI . We again see that, as was the case for the density at the bounce, the volume at the bounce has indeed a minimum at the Gaussian states, namely, for ηI = 0. Using both relative volume and density at the bounce, we can then define an interval, as function of σ, in which the parameter ηI can take values and the state still be considered as semiclassical. ˜ and More precisely, we define some tolerance for the asymptotic relative dispersion ∆+ = ∆, max ˜ We find the maximum value of ηI for which the relative dispersion is below the value ∆. max then find the value of the density at the bounce for the value ηI . We have made extensive ˜ and k0 , and have found that, for a fixed value of ∆, ˜ explorations for different values of ∆ ˜ the larger the value of k0 , the smaller the allowed interval in ηI . For example, for ∆ = 0.01, the dependence of ηImax on k0 can be approximated (for large k0 , σ = σ̃, n = 0 and p = 0) √ ˜ = 0.001, as ηImax ∼ √ 1/ 47000 k0 . The value of ρb at that point is equal to 0.99ρcrit . (For ∆ max ηI ∼ 1/ 470000 k0 and ρb = 0.999ρcrit .) − . FIG. 7: Relative error of the difference in the asymptotic values of the relative fluctuation ∆+∆−∆ − 2 6 In the plot 1/ηR = σ : (0, 10 ) , ηI : (−0.0003, 0.0003) , βR = k0 = 30000 , βI = 0 and n = 0. Outside of the vertical one the squeezed state have an asymptotic relative fluctuations (∆+ and ∆− ) bigger that 1. One line mark the Gaussian states (ηI = 0) that is just when the curve is zero and the other line is for states with σ̃ 2 = 2αk0 . Let us now return to the issue of the asymmetry across the bounce of the relative dispersion in volume. As we have mentioned, without any control on the value of the parameters, one could in principle have a large asymmetry that translates into a loss of semiclassicality across the bounce. This scenario was suggested in [21], but where the analysis was made for an effective formalism not directly derived from LQC and with little control on the semiclassicality conditions. Here we are better equipped to make precise statements about this 26 asymmetry based on the control we have on analytical expressions and the possible values the parameters of the states can have. In Fig. 7 we have plotted the relative difference ∆+ /∆− − 1 in asymptotic relative dispersion, as a function of σ and ηI . We have included, as vertical surfaces, the boundary of the allowed region in parameter space. Two slices in the figure are worth mentioning. The curve ηI = 0, which is viewed almost vertical in the figure corresponds to the Gaussian states. On that curve, ∆+ = ∆− so the relative difference vanishes. The second highlighted curve corresponds to the choice σ = σ̃, which is the extension to ηI 6= 0 of the optimal Gaussian states. Note that this curve starts at zero, increases (for positive ηI ), reaches a maximum and then decreases tending to zero. The curve is antisymmetric with respect to ηI . Thus, there are two points for which |∆+ /∆− − 1| reaches a maximum. What we have seen, by taking many different values of k0 , is that this maximum value is indeed, extremely small. For instance, for k0 = 105 , the maximum is about 10−4 . As we increase k0 this value decreases even further. These results show that indeed, semiclassical states are very symmetric and that semiclassicality is preserved across the bounce. One should also note that this quantitative analysis invalidates the claims of [20], and [21] regarding the allowed asymmetry in fluctuations across the bounce and ‘cosmic forgetfulness’, and supports the results of [18] and [19]4 . For instance, in the first reference of [21], it is claimed that the relative change in relative asymptotic dispersion in volume can be as large as 20 for semiclassical states, while we have demonstrated that this quantity is several orders of magnitude smaller for realistic values of the parameter k0 (of the order of 10−4 for k0 = 105 , and much smaller for the values expected to represent realistic universes (k0 ∼ 10100 or larger)). VI. DISCUSSION Let us summarize our results. We have defined coherent Gaussian states as candidates for semiclassical states ‘peaked’ around points of the classical (physical) phase space. We have imposed consistency conditions on such states by asking that the relative dispersion of the volume be small at late times after the bounce. This condition implies that one of the parameters, yielding the momentum of the scalar field, be large and fixes one of the other parameters. We can therefore find a canonical Gaussian state for a given point of the classical phase space corresponding to a classical solution at late times. When exploring the properties of these states near the quantum bounce we found that they behave also semiclassically in the deep quantum region; the relative fluctuations of volume are of the same order and smaller than in the asymptotic, large volume, region. Furthermore, for those states that have small fluctuations, the expectation value of the volume and energy density at the bounce are very well approximated by the so called effective theory, thus giving support to the claim that this theory is a good approximation to the quantum dynamics, even in the deep Planck regime. Next, we introduced squeezed states, a one parameter generalization of Gaussian states and studied their properties in a vicinity of the semiclassical Gaussian 4 One should recall that the model defined in [20], even though using a quantization quite different than that in [6] could be seen, in a loose sense, as a different factor ordering from the choice made in LQC and here. For a discussion on the differences between the assumptions made in the model of [20] and standard LQC see [17], and for a detailed criticism of some claims made in [20], see [18]. Our results here can be seen as giving further validity to the arguments presented in [18] against the claims of [20, 21]. 27 states. As we showed, the range of this extra parameter is severely restricted if we want to maintain a small relative fluctuation for the volume. Furthermore, if one departs from the Gaussian states too much, both the volume and density at the bounce differ very rapidly from the value on the Gaussian states (and also the ‘effective theory’). Thus, we are lead to conclude that the Gaussian states exhibit a better semiclassical behavior than squeezed states. An important issue that can be studied quantitatively in this solvable model is that of the asymmetry on the volume fluctuations. On Gaussian states the fluctuations are the same before and after the bounce. As one introduces the squeezing parameter, the fluctuations are no longer symmetric. In fact, depending on the choice of sign of the parameter, either one of the fluctuations becomes smaller and reaches an absolute minimum in the vicinity of the Gaussian states. At the same time, the relative fluctuation on the opposite side of the bounce increases. The question is then how big can this asymmetry be? What we saw is that for a given, somewhat arbitrary choice of parameters, that is still far from those giving rise to a realistic ‘large’ universe, the relative difference in relative dispersion is very small (of the order of 10−4 ), and decreases for more realistic values of the parameters. Thus, even if we were to choose the squeezing parameter such that this asymmetry is maximized, the relative change is so small that, for all practical purposes, the state is symmetric. A state that is semiclassical on one side of the bounce not only remains semiclassical on the other side, but maintains its coherence. This quantitative analysis therefore invalidates claims of ‘loss of coherence’ across the bounce [21], where it was claimed that the relative change in (relative) dispersion can be several orders of magnitude larger. One particular feature of the classical description of the system under consideration is that it possesses a rescaling symmetry. In the spacetime description, we can rescale the volume by a constant and the physical properties of the spacetime remain invariant. In the phase space description this is manifested by a symmetry in the equations of motion. This symmetry, however, can not be treated as gauge, in the same sense that Hamiltonian symmetries are. One can understand the origin of this symmetry by recalling that the k=0 model needs a fiducial volume for its Hamiltonian description, and this choice is completely arbitrary. In turn, this means that there is no scale with respect to which one could compare the momentum pφ of the scalar field; there is no meaning to the statement that ‘pφ is large’. As we have seen, in the quantum theory Planck constant introduces such a scale and this is manifested in the behavior of certain quantum observables. One of the semiclassicality conditions we have found is that the larger the expectation value of hp̂φ i, the more semiclassical the state is. What we have also seen is that the rescaling symmetry is only approximately recovered, for our class of states, in the ‘very large hp̂φ i limit’, which can be taken as the classical limit of the system. For further details on this issue, see [26]. Acknowledgments We thank A. Ashtekar, Y. Ma, T. Pawlowski and P. Singh for helpful discussions and comments. This work was in part supported by DGAPA-UNAM IN103610, by NSF PHY0854743 and by the Eberly Research Funds of Penn State. 28 Appendix A: Effective Reduced Phase Space The kinematical phase space of the model is Γ = R4 with coordinates (β, V, φ, pφ ). Here φ is the matter scalar field, pφ its canonical momentum, V the volume (of the fiducial cell) and β its canonical momentum. The effective Hamiltonian constraint of the system is [23]: p2φ 3 2 2 V sin(λβ) − ≈0 Hλ = 8πGγ 2 λ2 2 which is labeled by a real parameter λ with dimensions of length. The symplectic structure is given by: 1 Ω0 = dV ∧ dβ + dpφ ∧ dφ (A1) 4πGγ which induces the following Poisson brackets {β, V }0 = 4πGγ , {φ, pφ }0 = 1 . The Hamiltonian constraint defines a hypersurface Γ̄ in phase space. It can be taken as a disjoint union of hypersuperfaces where β ∈ (n πλ , (n + 1) πλ ) with n ∈ Z. In what follows we shall only consider the connected component where β ∈ (− πλ , 0). Note that the points where β = n πλ , n ∈ Z are not in the constrained surface. These are degenerate points (hyperplanes) where the system does not have a well defined dynamics. We shall see that physically, these points should be excluded. We shall now consider the pullback Ω̄ of the symplectic structure to the hypersurface Γ̄. That is, we find Ω̄ such that Ω0 = ι ∗ Ω̄, for ι : Γ̄ → Γ the embedding of Γ̄ into Γ. Let us write 3 Hλ = η 2 V 2 sin(λβ)2 − p2φ ≈ 0 , with η 2 = 4πGγ 2 λ2 then, on the constrained surface we have pφ ≈ ±ηV sin(λβ) (A2) The pullback of the gradient of Hλ to Γ̄ yields, η 2 2V dV sin(λβ)2 + 2η 2 V 2 sin(λβ) cos(λβ)λdβ ≈ 2pφ dpφ ±η[sin(λb)dV + V cos(λβ)λdβ] ≈ dpφ (A3) (A4) which can be used for solving dpφ , dV or dβ and inserting back in Eq. (A1). Each of this choices will depend on our choice of coordinates for the constrained surface. Let us study two of such parametrizations. In the first one we solve for pφ , which means that we use (V, β, φ) as coordinates. Then Ω̄1 = 1 dV ∧ dβ ± η[sin(λβ)dV + V cos(λβ)λdβ] ∧ dφ 4πGγ which gives us the expression for the two disconnected hypersuperfaces parametrized by positive and negative values for pφ . The value pφ = 0 is excluded if β 6= n πλ or V 6= 0, with n ∈ Z. If we take pφ > 0 we get Ω̄1 = 1 dV ∧ dβ + η sin(λβ)dV ∧ dφ + η λ V cos(λβ) dβ ∧ dφ 4πGγ 29 If we now solve for dV , we get the expression for the pre-symplectic structure in the (β, φ, pφ ) parametrization as, Ω̄2 = ± 1 1 dpφ ∧ dβ + dpφ ∧ dφ 4πGγ sin(λβ)η We can again restrict ourselves to pφ > 0, which yields 1 1 dpφ ∧ dβ + dpφ ∧ dφ 4πGγη sin(λβ) Ω̄2 = on the corresponding connected component. Regardless of the parametrization chosen, the pre-symplectic form Ω̄ has a degenerate direction corresponding to the Hamiltonian Vector field XHλ of the Hamiltonian constraint Hλ . Physically the integral curves represent gauge direction along which points are physically indistinguishable. We can find the gradient of the constraint, ∇a H = 2η 2 V sin(λβ)2 ∇a V + 2η 2 λV 2 sin(λβ) cos(λβ)∇a β − 2pφ ∇a pφ a and, using that XH = Ωab ∇b H, and ∂ [a ∂ b] ∂ [a ∂ b] + Ω = 2 4πGγ ∂β ∂V ∂φ ∂pφ ab we get, a XH |Γ̄ 2 = 16πGγη V sin(λβ) 2  ∂ ∂β a  2  2 − 16πGγη λ V sin(λβ) cos(λβ)  ∂ ∂V a − 4pφ  ∂ ∂φ a as the restriction of the vector field to the constrained hypersurface with coordinates (V, β, φ), using Eq. (A2). It is easy to check that this is indeed the degenerate direction of Ω̄: Ω̄1 (XH , ·) = 0. We are now interested in understanding qualitatively the structure of the reduced phase space, for which we would need to take the quotient of Γ̄ by the gauge directions generated by XH . In this case, it turns out to be easier to perform a gauge fixing which, as we shall see, is well defined everywhere. Let us define C = φ − c, with c = constant. It is direct to see that it forms, with Hλ a second class pair, {H, C}0 = {H, φ}0 = {η 2 V 2 sin(λβ)2 − p2φ , φ}0 = −{p2φ , φ}0 = 2pφ provided pφ 6= 0, which is a condition we had asked before. The gauge fixed phase space Γ̂ is therefore parametrized by (β, pφ ) (or (V, β)) with φ = constant, pφ > 0 or pφ < 0 and β ∈ (− πλ , 0). The corresponding symplectic structures on each of this two different parametrizations are 1 1 dV ∧ dβ , Ω̂2 = ± dpφ ∧ dβ Ω̂1 = 4πGγ 4πGγη sin(λβ) If we introduce the new variable (as was done in the main text),   1 λβ ln tan x= √ 2 12πG then the symplectic form Ω̂2 becomes Ω̂2 = ±dpφ ∧ dx which indicates that (x, pφ ) can be seen as conjugate variables in the reduced theory. 30 Appendix B: Some Integrals Here we list some integrals that are useful in the main part of the manuscript. r Z ∞ π −ηx2 e dx = η −∞ Z Some error integrals. Z ∞ 2 x2n+1 e−x dx = 0, x e −∞ ∞ with n ∈ Z −x0 2n −ηx2 Z Z x0 ∞ e −x2 z 2 −x2 xe z dn dx = (−1) dη n n dx = r π , η √ π erf c(z), 2 with n ∈ Z with z ∈ C √ π z −z2 + erf c(z), dx = e 2 4 with z ∈ C (B1) (B2) (B3) (B4) (B5) where erf c(z) is the Complementary Error Function 2 erf c(z) = 1 − erf (z) = 1 − √ π Z z 2 e−x dx (B6) 0 which can be bounded using its asymptotic expansion [27] as 2 e−|z| |erf c(z)| < √ |z| π (B7) with |arg z| < π/4 and z → ∞. When z = x is real then 2 e−x erf c(x) < √ x π 1. (B8) Some Relations 2 iηt − η̄(k − β̄) − η(k ∓ it − β) = −2ηR k − a ∓ 2ηR    2 ηI2 t ηR 1+ 2 ∓ 2tβI ηR + 2 ηR 2 2 2 +2ηR (a − βR + βI ) + 4ηI βI βR 2  iηα u+a± 2ηR  2   iηα α2 u+a± ∓ iα = u2 + 2ub± + b2± + 2ηR 4 31 (B9) (B10) Appendix C: Errors We show in explicit form the errors for the norm and volume and explain why the approximations we take throughout the manuscript are completely justified. 1. Norm Error The norm of the Gaussian states is given by Z ∞ 2 2 2 kk 2n e−2(k−k0 ) /σ dk kχ̃kphy = 2 0 " √ #2n+1 2n+2 Z ∞  2k0 σ 2 e−u du u+ = 2 √ √ σ 2 − 2k0 /σ which for the n = 0 case becomes,  2 Z ∞ σ 2 2 kχ̃kphy = 2 √ [u + y] e−u du , 2 −y (C1) (C2) √ with y = 2k0 . σ (C3) ! (C4) Then kχ̃k2phy = σ 2 Z ∞ ue −u2 du + σ −y 2 Z ∞ 2 ye−u du −y √ √ π σ −y2 2 2 e +σ y π−σ y erf c(y) . = 2 2 2 Using the form of y, we get kχ̃k2phy = √ σ2 2πk0 σ + e−( 2 √ 2k0 2 ) σ − σk0 r π erf c 2 √ 2k0 σ The norm that we used in the manuscript was √ kχ̃k2phy ≈ 2πk0 σ. . (C5) Then the error in this approximation is just σ2 Error(kχ̃k2phy ) = e−( 2 √ 2k0 2 ) σ − σk0 r π erf c 2 √ 2k0 σ ! . (C6) This tells us that the error in the norm for n = 0 is the error in the approximation of the −x2 function erf c(x) by the function ex√π . If we use Eq. (B8) to bound the function erf c(y) then the error can be bounded by √ σ 2 −( 2k0 )2 . (C7) e σ 2 Since we have seen in the main text that semiclassicallity implies that we take k0 ≫ σ > 0 then this error is indeed very small. These errors were also studied numerically for n > 0 and present similar behavior. We can then conclude that the approximation made is very good when computing the norm of the states. Error(kχ̃k2phy ) < 32 2. Volume Error The coefficients in the expectation value for volume are, Z κ ±pα ∞ 2 2 2 2 [k(k ∓ iα)]n+1 e−(k−k0 ) /σ e−(k∓iα−k0 ) /σ dk Ṽ± = e (C8) α 0  n+1 Z ∞h in+1 √ 1 κ 2 2 2 2 ±pα α2 /2σ 2 2 e−u /σ du , (C9) e e 2u + 4 2k0 u + 4k0 + α = √ 2α 4 −u0 √ √ with u0 = 2k0 ± iα/ 2. Then for n = 0 are Z i √ κ 1 ±pα α2 /2σ2 ∞ h 2 2 2 e e 2u + 4 2k0 u + 4k02 + α2 e−u /σ du . (C10) Ṽ± = √ 2α 4 −u0 Now we introduce other change of variables t = σu , and then the integral takes the form Z i 2 √ κσ ±pα α2 /2σ2 ∞ h 2 2 (C11) 2σ t + 4 2k0 σt + 4k02 + α2 e−t / dt , Ṽ± = √ e e 4 2α −z √ α 0 with z = x ± iy = 2k . The integral is ± i σ√ σ 2 Z ∞ Z −z   2  2 √ √ 2 2 2 2 −t ˜ 2σ t + 4 2k0 σt + 4k0 + α e dt − I± = 2σ 2 t2 + 4 2k0 σt + 4k02 + α2 e−t dt −∞ −∞ # "   2 2 2 2 √ e−z2 √ σ +α zσ −z2 σ α2 erf c(z) . −√ e − √ erf c(z) − 2 2σ √ − 2k0 + = πk0 4k0 + k0 2k0 π 2 πk0 π Then Ṽ± takes the form Ṽ±  κ ±pα α2 /2σ2 √ σ 2 + α2 = e e 2πσk0 4k0 + 8α k0 #   √ e−z2 zσ 2 −z2 σ2 α2 −√ e − √ erf c(z) − 2 2σ √ − 2k0 + erf c(z) . 2k0 π 2 πk0 π If we normalize using Eq. (C5) then  σ 2 + α2 κ ±pα α2 /2σ2 4k0 + e e V± = 8α k0 #   √ e−z2 zσ 2 −z2 σ2 α2 −√ e − √ erf c(z) − 2 2σ √ − 2k0 + erf c(z) 2k0 π 2 πk0 π The values that we used in the manuscript was   κk0 ±pα α2 /2σ2 σ 2 + α2 V± ≈ . 1+ e e 2α 4k02 (C12) Then the error in the integration is " 2 #  2 2 √ σ α κ ±pα α2 /2σ2 e−z √ (zσ 2 + 2 2σ) + erf c(z) √ + 2k0 + Error(V± ) = − e e 8α 2k0 π 2 πk0 33 Using the triangular inequality " #  2 2 2 √ α κ ±pα α2 /2σ2 e−|z| σ √ (|z|σ 2 + 2 2σ) + |erf c(z)| √ e e + 2k0 + |Error(V± )| ≤ 8α 2k0 π 2 πk0 Using the relation (B7) (where |argz| = arctan xy = arctan kα0 < π 4 because α ≪ k0 )   √ σ2 κ 2k0 α2 2 ±pα α2 /2σ 2 −|z|2 |z|σ + 2 2σ + √ |Error(V± )| < √ e e e + + |z| 2k0 |z| 8 πα 2 πk0 |z|  √ 1 κ 1 2 2 2 √ < √ 2 2σ 2 α2 k0 |z|e±pα eα /2σ e−|z| + 2 8 πα 2 2α2 k0 α σk0 |z|  1 1 1 √ + √ + + √ 4 2πk02 |z|2 α2 |z|2 2σ 2 α2 4 2k02 |z|2 σ 2 As all quantities in the brackets are positive and less than one, we can bound the error as |Error(V± )| < 5κσ 2 αk0 |z| ±pα α2 /2σ2 −|z|2 √ e e e 2 2π Given that V± can be too small or large then the quantity that really gives a measure of the ± )| error is the relative error |Error(V . Using Eq. (C12), this becomes V± −1  5α2 σ 2 |z| −|z|2 |Error(V± )| σ 2 + α2 < √ 1+ e V± 4k02 2π  1/2 α2 5α2 k0 σ −2k02 /σ2 −α2 /2σ2 1 + 4k02   e e < √ 2 2 2π 1 + σ +α 2 4k0 The relation between the parentheses is less than one, then |Error(V± )| 5α2 k0 σ −2k02 /σ2 −α2 /2σ2 e e < √ V± 2π 2 2 Since σ ≫ α, then e−α /2σ ≈ 1. Furthermore, as k0 ≫ σ ≫ α, then the negative exponential wins over the polynomial factor, which show that the error is small. These errors were also studied numerically for n > 0 and present similar behavior. Appendix D: Tables In this appendix we summarize the main quantities that were found in the main text, for the generalized Gaussian states for n = 1 and compare them to the n = 0 case. It should be noted, that for higher values of n, the structure of the terms is similar, with terms of the  m  l and kσ0 . form kα0 34 n=0 kχ̃k2phy V± √ 2πk0 σ  2 2 αe±pα eα /2σ 21 k0 +  α2 k02 + W0 W± α3 e±2pα e2α 2 /σ 2 hp̂2φ i  ~2 k02 1 + ~2 σ 2 4 Vbounce α2 /2σ 2 )2 (∆V hV i2 αe  3σ 2 4  σ2 4k02  αe±pα eα 2 /2σ 2 
3σ 4 +24k02 σ 2 +2α2 σ 2 +16k04 +8k02 α2 +α4 24k0 σ 2 +32k03 α2 1−   3σ 2 4k02  σ2 4k02   15σ 4 +40k02 σ 2 +16k04 12σ 2 +16k02  k0 1 +    4 2 1+ 3σ2 + 3σ 4 2k0 16k0   0 2 1+ 3σ2 4k0   4 5σ 2 1+ 2 + 15σ4 2k 16k0 2 2   0 0 2 1+ 3σ2 4k0   3σ 2 9σ 4 9σ 6 1+ + − 2 4 6 4k 16k0 64k0 ~2 σ 2  0  4 3σ 2 9σ 4 1+ 2 + 4 2k0 16k0 3σ 4 3σ 2 α2 σ 2 α2 α4 1+ 4 + 2k2 + 8k4 + 2k2 + 16k4 2 2 16k α /2σ 0 0 0 0 0 0 2 1+ 3σ2 4k0   5σ 2 3α2 σ 2 2α2 α4 15σ 4 + + + + 1+ 4 2 4 2 4 16k0 2k0 2k0 k0 k0 3σ 2  2 4k02 3σ 2 α2 σ 2 α2 α4 3σ 4 + + + + 1+ 4 2 4 2 4 16k0 2k0 8k0 2k0 16k0 ~k ~ k σ 2 +α2 4k02  2 2 1+ 3σ2 + α2 2 2 4k0 k0 2 eα /σ  σ 2 +α2 1+ 2  σ 2 +α2 8k0 2πk0 σ k02 + 34 σ 2 0  ~k0 1 + )2 √  4 
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