# Integrable minisuperspace models with Liouville field: energy density self-adjointness and semiclassical wave packets

## Abstract

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum contexts of Wheeler–DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint, and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger’s norm and a conserved Mostafazadeh’s norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.

## 1 Introduction

Cosmological models with scalar fields have drawn a lot of attention in the last decades because of investigations on cosmological inflation [1] and dark energy [2], but few of them can be exactly integrated. A universe driven by scalar fields with an exponential potential is dubbed *Liouville cosmology*, which is one of the well-studied integrable models in cosmology. The power-law expansion of particular solutions and its applications are investigated in e.g. [3, 4, 5]. The general classical solutions have been discussed in detail under various gauge conditions in e.g. [6, 7]. The correspondence between Jordan and Einstein frame is studied in [8, 9, 10, 11, 12, 13], wherein the Liouville field in the Einstein frame is related to the power-law potential in Jordan frame through a conformal transformation combining with a parameter transformation of scalar field. The exactly solvable models with several Liouville scalar fields were developed in [14, 15]. The appearance of the Lioville cosmologies from higher-dimensional theories, in particular superstring theories and M-theory was studied in [16, 17].

General relativity is a theory with constraints, the corresponding Hamiltonian is zero [18, 19, 20, 21]. The reason for the vanishing Hamiltonian is the presence of a non-dynamical symmetry, namely diffeomorphism invariance; in other words, the gravitational theory contains redundant degrees of freedom. In the minisuperspace approximation, the redundancy appears in the form of the lapse function *N*(*t*). Therefore, to solve the dynamics of the model, it is necessary to introduce a specific gauge condition to eliminate *N*(*t*) [6, 14]. Traditionally, the lapse function is set to unity, such that the universe evolves in cosmic time [22]. However one could eliminate *N*(*t*) and avoid an explicit time parametrization to obtain exact solutions of Einstein’s equation. This fits well the Wheeler–DeWitt quantum cosmology which does not involve time.

The cosmological models driven by a scalar field with a constant potential may serve as examples of the latter approach [5, 21]. In these models, the scalar field is a cyclic coordinate, hence the conjugate momentum is integral of motion, and the conservation law can be applied to eliminate the lapse function *N*(*t*), such that the modified Friedman equation contains only minisuperspace variables. Inspired by this, we introduce a similar integral of motion in Liouville cosmology of homogeneous and isotropic models [23], in order to eliminate the redundant degrees of freedom. With the help of this integral of motion, the classical Friedman equation reduces to a time-independent nonlinear equation, the solution of which can be derived explicitly and describes the trajectory in minisuperspace. This method can also be directly extended to higher dimensional [24, 25] and anisotropic models, such as Bianchi-I cosmology considered in [12].

The physical meaning of the formal Wheeler–DeWitt equation and its correspondence with the classical theory can be derived in three steps. The first one is the selection of the space of *physical* wave functions, usually by endowing proper boundary conditions. In traditional quantum mechanics, crucial properties of the theory depend on the boundary conditions for wave functions, such as the Hermiticity of observables [26], the orthogonality of wave functions (e.g. [26, 27]) and the conservation of probability, to name a few. A similar situation holds in quantum cosmology [21, 28], in which proper boundary conditions have to be specified, such that the solutions of the Wheeler–DeWitt equation, which are not square-integrable, are eliminated from the space of physical wave functions. In this paper we address an important issue encountered at this step. The Hamiltonian operator naively constructed by the canonical quantization in some cosmological models, which are interesting from the phenomenological point of view, including phantom field, happens to be not essentially self-adjoint and its self-adjoint extension is not unique [29, 30, 31]. Namely while the clasical action fixes up to the usual ordering ambiguities how the Hamiltonian acts on the localized wavefunctions the evolution over finite amounts of time depends on its behaviour at infinity where extra ambiguity arises. Hence one classical action correspond to a family of distinct quantum models with different quantum evolutions. The cosmological models with similar self-adjointness issues were considered in [32, 33].

The second step is to define an inner product on the physical space that would give the conserved probability distribution in quantum cosmology. Since the Wheeler–DeWitt equation is of Klein–Gordon type, the ‘probability density’ defined by the so-called *Klein–Gordon norm* is not guaranteed to be positive. While one may restrict consideration to the WKB wavepackets the question arises how to interpret the wavefunction of the universe beyond the WKB region. A resolution of this problem may be provided within the pseudo-Hermitian theory by introducing the Mostafazadeh’s norm [34, 35, 36]. While we do not treat this norm as the only possible way to tackle the probability problem it may be considered as an useful tool to study the quantum cosmology as a fully consistent quantum theory within restrictions of the minisuperspace approximation.

Finally one has to attribute a proper energy distribution to construct a wave packet [37, 38, 39]. For a given initial coordinate distribution of wave packet in minisuperspace, the energy distribution can be calculated, which however is not easy to realize in practice. A common compromise is to choose a Gaussian energy distribution. Then in correspondence with classical theory the probability distribution of the established wave packet should ‘centre’ at the classical path and follow it as closely as possible apart from turning points.

This paper is organized as follows. In Sect. 2 we briefly elucidate the problem of the quantum particle in the unstable potential \(V=-\mathrm {e}^{2x}\) and the ambiguity of self-adjoint extension of the Hamiltonian operator. In Sect. 3 an integral of motion is introduced for three types of Liouville cosmological models and explicit classical solutions are given in terms of minisuperspace variables. Section 4 introduces the corresponding canonical quantum cosmology and there the physical state space is constructed. As a verification of the results, in Sect. 5 the limit of potential parameter \(\lambda \) tending to zero is considered. Section 6 is devoted to the classical-quantum correspondence, in which the wave packets are implemented and the probability distributions are plotted for two kinds of norms. The conclusions Sect. 7 contain some comments on further extensions and applications of the approach adopted in this paper.

## 2 Quantum mechanics of a particle in a negative Liouville potential

*unbounded from below*, described by the Hamiltonian

In infinite dimensional Hilbert spaces it is too restrictive to demand that the domain of the operator \(\mathscr {D}(\hat{A})\) covered the whole Hilber space \(\mathscr {H}\). Therefore operators including observables are usually defined on the domains that are merely *dense* in \(\mathscr {H}\) i.e. any element in the Hilbert space can be obtained as a limit of some sequence of elements in \(\mathscr {D}(\hat{A})\). For example the operator \(\hat{p}^2\) can not be defined on the whole \(L^2(\mathbb {R})\) but is symmetric on the domain of all ‘bumps’ - infinitely differentiable functions with compact support, \(\mathscr {C}_c^{\infty }\).

*symmetric operator*\(\hat{A}\) such that,

*self-adjoint*its

*adjoint*\(\hat{A}^\dagger \) defined as,

*E*. For example for \(E_{\pm }=\pm 2\mathrm {i}\) one gets,

*E*with \({\text {Im}} E>0\) and \({\text {Im}} E<0\) are known as deficiency indices \(n_{+}\) and \(n_{-}\) respectively. If \(n_{+}=n_{-}=0\) (i.e. there are no such solutions) the operator is essentially self-adjoint, that is its self-adjoint extension is unique. If \(n_{+}\ne n_{-}\) no self-adjoint extension exists. In our case the square-integrability requires \(\tilde{C}_{\pm }=0\) however \(C_{\pm }\ne 0\) is allowed. Therefore \(n_{+}=n_{-}=1\). According to the Weyl–von Neumann theorem [29] this means that a single parameter family of self-adjoint extensions exists.

*a*is an arbitrary parameter \(a\in [0,2)\). For \(E>0\) using Eqs. (10), (11) we then get non-degenerate continuous spectrum,

*x*analogous to a reflecting wall. The classical trajectories for the particle described by

*H*reach infinity in finite time. Therefore in the first WKB approximation, the Gaussian wave packet also reaches the infinity in finite time. The subsequent motion of the particle may be described as a bounce from infinity. The non-uniqueness of the self-adjoint extension for \(\hat{H}\) may be understood intuitively in the following way. After crossing over infinity the wave function may be multiplied by an arbitrary phase factor \(\mathrm {e}^{2\uppi \mathrm {i}a}\) without losing the conservation of probability. Thus we have a family of unitary evolution operators generated by different self-adjoint extensions of \(\hat{H}\) that locally are indistinguishable however differ at finite times.

*x*playing the role of another wall,

*a*can be shown to be equal to,

## 3 Classical solutions of Liouville cosmology

*N*(

*t*) is the lapse function, and \(a(t)= \exp \alpha (t)\) the cosmological scale factor; moreover, the scalar field is a function only of time, \(\phi = \phi (t)\). With \(\varkappa = 8 \uppi G\), \(\sigma = \pm 1\) and \(\lambda \in \mathbb {R}\), the minisuperspace action reads

*quintessence model*[44], and \(\sigma = -1\) is dubbed as a

*phantom model*[45]. From Eq. (30) one readily derives the Hamiltonian density

*N*and obtain a non-linear equation

*x*is for quintessence and

*y*is for phantom.

- 1.When \(m_x\) and
*V*are of different sign, one obtainswhere \(c_1\) is an integration constant associated with the initial conditions. Equation (39) contains two distinct solutions separated by \(\lambda \sqrt{\frac{3}{2 \varkappa }} \alpha +\sqrt{\frac{3 \varkappa }{2}} \phi +c_1 = 0\) due to the divergence of \({{\mathrm{csch}}}x\) for \(x \rightarrow 0\). Both of the solutions can be interpreted as an expansion model, see e.g. Fig. 2. For \(\omega =0\), one recovers the power-law special solution or \(\alpha \propto \phi \) in [5].$$\begin{aligned} \mathrm {e}^{6 \alpha +\lambda \phi } =\frac{3 \varkappa \omega ^2}{- V m_x} {{{\mathrm{csch}}}^2}\mathopen {}\left( \lambda \sqrt{\frac{3}{2 \varkappa }} \alpha +\sqrt{\frac{3 \varkappa }{2}} \phi +c_1\right) \mathclose {}, \end{aligned}$$(39) - 2.When \(m_x\) and
*V*are of the same sign, one hasthis trajectory contains a single turning point in finite domain of minisuperspace.$$\begin{aligned} \mathrm {e}^{6 \alpha +\lambda \phi } =\frac{3\varkappa \omega ^2}{ V m_x} {{{\mathrm{sech}}}^2}\mathopen {}\left( \lambda \sqrt{\frac{3}{2 \varkappa }} \alpha +\sqrt{\frac{3 \varkappa }{2}} \phi +c_1\right) \mathclose {}, \end{aligned}$$(40)

## 4 Dirac quantization of Liouville cosmology

### 4.1 Inner product and probabilities

On of the basic building blocks of any quantum model is the inner product that allows to assign probabilities. However this is a long standing problem in quantum cosmology due to the Wheeler–DeWitt equation being of the Klein–Gordon type. The naturally conserved Klein–Gordon inner product corresponds to the indefinite norm [20, ch. 5]. Pseudo-Hermitian quantum mechanics [36] provides a cure and will be applied here to reconstruct wave packets based on consistent norms.

### 4.2 Quintessence field

*time of a Klein–Gordon-type equation*and \(\omega \) as its Fourier conjugate. The Hamiltonian in Eq. (31) then becomes

### 4.3 Phantom field

*a*can be fixed by the condition,

## 5 The limit \(\lambda \rightarrow 0\)

As a verification of our approach to the minisuperspace trajectory, the limit \(\lambda \rightarrow 0\) will be considered, which have been extensively studied as a pedagogic model, see e.g. [5, 13, 21]. This limit enforces \(m_x<0\) which will be assumed for the rest of the section.

## 6 Semiclassical wave packets and comparisons with classical solutions

With the explicit form of minisuperspace trajectories at hand, its comparison with the quantum solutions becomes more transparent, since the latter does not depend on any time parameter, but only the minisuperpace coordinates. It is expected that a classical trajectory could be restored from the wave functions at the limit \(\hslash \rightarrow 0\), which must be consistent with the results in Sect. (3); furthermore, the cosmological wave packets are expected to go along the classical trajectories in minisuperspace, which can be visualized in plots.

### 6.1 WKB limit as \(\hslash \rightarrow 0\)

The minisuperspace Wheeler–DeWitt wave functions can be compared with the classical trajectories by taking the WKB limit, i.e. expanding at \(\hslash \rightarrow 0\).

### 6.2 WKB Gaussian wave packet

### 6.3 Numerical matching

The integral with Gaussian distribution in Eq. (98) cannot be implemented analytically. Even though the WKB approximation Sect. (6.2) is effective, its precision is poor in regions where semiclassical approach does not hold, for instance near the classical turning point. Instead, one can turn to numerical approaches.

With the wave functions normalized, one may construct wave packets for the quintessence and phantom models. The corresponding plots are in (2), (3) and (4). The parameters are specified in Planck units \(\hbar =\varkappa =1\). The common feature of the plots is that the wave packets coincide with classical trajectories and follow them as closely as possible. The height of the wave ‘tube’ is negatively correlated to the ‘speed’ of the classical trajectory with respect to the Klein–Gordon time \(\tau \), i.e. the higher the ‘speed’ is, the lower the amplitude of the wave ‘tube’ is [53]. It is interesting to note that for all models the naive inner product Eq. (46) happen to approximate the conserved norm Eq. (45) very well so that there’s no noticeable difference in plots.

*a*corresponds to slightly different wave packets.

## 7 Conclusions

In this paper, by using the integral of motion to eliminate the lapse function in Friedmann equation, we have solved the cosmological model with Liouville field for homogeneous isotropic metrics. The general classical solutions are obtained and represented in terms of minisuperspace variables only, such that the correspondence between classical and quantum theory can be demonstrated manifestly. The quantum wave packets reproduce the classical limit in a sense that the distributions of traditional Schrödinger’s norm and the Mostafazadeh’s inner product are maximized near the classical trajectories.

Generally if ordinary matter is added, the model loses integrability and it is not possible to find the analytical solution even in terms of minisuperspace variables. Nevertheless one may consider the integrable model with multiple scalar fields considered in [14]. The homogeneous Wheeler–DeWitt equation for this model is separated into a system of the Schrödinger equations for one-dimensional particle in the exponential potential. Each of the fields could be then treated independently in the same way as the one-field model considered in this paper. Thus our results are trivially generalized to this integrable model.

The classical models of quintessence with potential unbounded below and the phantom fields give rise to the appearance of a family of non-equivalent quantum models, because the energy density operators are not essentially self-adjoint operator. In order to preserve unitarity and correct classical limit one has to omit half of the spectrum. While this requires that the wave packet at some fixed \(\tau \) belongs to much narrower class than \({L^2}\mathopen {}\left( \mathbb {R}\right) \mathclose {}\), it is enough to produce wave packets in the vicinity of the classical trajectories.

For the phantom field the resulting spectrum is discrete. It is associated with the fact that at the classical level the universe exists in a finite interval between two singularities and non-singular unitary evolution is accessible through the periodicity of wave function. This periodicity may be regarded as a fundamental condition not only for the homogeneous but also on inhomogeneous modes. On the other hand, if the minisuperspace wave packet contains multiple semiclassical branches they may be associated with coherent superposition of different universes. This Schrödinger-cat-like effect at the cosmic scale might be an artifact of the model in minisuperspace. In the full theory in Wheeler’s superspace [54], inhomogeneity is involved, which may serve as an unobservable environment, in contrast with the scale factor [55]. The observable effects are then fully described by the density matrix of the scale factor only, whose off-diagonal elements characterize the superposition of universes with different scale factors. Calculation suggests that those elements are highly-suppressed in the above-mentioned decoherence scheme [56, 57]; hence the cosmic Schrödinger cat might be fictitious, and the superposition of distinct semiclassical branches might be decohered to vanish. The approach developed in the paper can also be extended to Higher dimensional [24, 25] and anisotropic models, such as Bianchi-I cosmology considered in [12].

It is known that the quantum field theory with the phantom fields considered on the classical cosmological background suffers from the vacuum instability problem [58, 59]. However the self-adjoint issues in the homogeneous modes quantization considered in this paper may influence the applicability of the mean field approach. To address this question one should combine the Wheeler-DeWitt equation on the homogeneous minisuperspace with inhomogeneous perturbations. This could be done for instance starting with the Born-Oppenheimer approximation of the Wheeler–DeWitt equation for the free inhomogeneous perturbations [60, 61, 62, 63]. We leave this problem for future work.

As the different self-adjoint extensions lead to different quantum evolution and require the wavefunction to belong to the different restricted functional class they may produce different observable results. The leading order of WKB approximation is insensitive however one may expect that the choice of self-adjoint extension should be important for the NLO corrections to the spectra of perturbations [64, 65, 66, 67].

## Notes

### Acknowledgements

The funding for this work was provided by the RFBR projects 16-02-00348 and 18-02-00264 (A. A. and O. N.) and the Spanish MINECO under project MDM-2014-0369 of ICCUB (Unidad de Excelencia ‘Maria de Maeztu’), Grant FPA2016-76005-C2-1-P, Grant 2014-SGR-104 (Generalitat de Catalunya) (A. A.). Y. -F. Wang is grateful to the Bonn-Cologne Graduate School for Physics and Astronomy (BCGS) for financial support. The authors are grateful to Claus Kiefer for illuminating comments and suggestions as well as to Maxim Kurkov for valuable remarks. Y. -F. Wang is also grateful to Nick Kwidzinski, Dennis Piontek and Tim Schmitz for inspiring discussions.

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