**1. Introduction**

In the framework of linear cosmological perturbation theory the Mukhanov-Sasaki equation plays a central role. It encodes the dynamics of the Mukhanov-Sasaki variable, which is a linearized and gauge invariant quantity that is built from a specific combination of matter and gravitational perturbations such that the resulting expression is gauge invariant up to linear order. A way to derive this equation is to consider the Einstein-Hilbert action together with a scalar field minimally coupled to gravity and expand this action up to second order in the perturbations around an FLRW background. One decomposes the perturbations into scalar, vector and tensor perturbations since these decouple at linear order. In the scalar sector, we are left with one physical degree of freedom that can for instance be expressed in terms of the Mukhanov-Sasaki variable denoted by *<sup>v</sup>*(*η*, **<sup>x</sup>**). Given this, we can express

*Universe* **2019**, *5*, 170

the scalar part of the perturbed action entirely in terms of the Mukhanov-Sasaki variable and the corresponding equation of motion takes the following form [1]:

$$v''(\eta, \mathbf{x}) - \left(\Delta + \frac{z''(\eta)}{z(\eta)}\right) v(\eta, \mathbf{x}) = 0, \qquad z(\eta) = \frac{a}{\mathcal{H}} \frac{d\bar{\phi}}{d\eta}, \qquad \eta := \int^t \frac{d\tau}{a(\tau)},$$

where Δ is the spatial Laplacian, *η* denotes conformal time, *a* the scale factor, *φ*¯(*η*) the isotropic background scalar field and H := *aa* the Hubble parameter with respect to conformal time. Contrary to the background quantities, the linear perturbations carry a position dependence breaking the spatial symmetries of the FLRW background spacetime. Throughout this article we will work with the Fourier transform of this differential equation. For each Fourier mode *<sup>v</sup>***k**(*η*), this leads to a differential equation given by:

$$v\_{\mathbf{k}}''(\eta) + \left(\|\mathbf{k}\|\|^2 - \frac{z''(\eta)}{z(\eta)}\right) v\_{\mathbf{k}}(\eta) = 0,\tag{1}$$

where quantities with **k**-label corresponds to the associated Fourier transforms. The quantity in the brackets of the Fourier transformed equation is called the Mukhanov-Sasaki frequency *<sup>ω</sup>***k**(*η*) and reflects the backreaction of the matter degrees of freedom with the background spacetime. Further commonly used gauge invariant quantities in the context of linear cosmological perturbation theory are the Bardeen potential Φ*B* as well as the comoving curvature perturbation R. The latter is related to the Mukhanov-Sasaki variable *v* by *v* = *<sup>z</sup>*R. Whether one considers a specific gauge invariant quantity is often influenced by the choice of a particular gauge in which these variables simplify and have an obvious physical interpretation. For the Bardeen potential this is the longitudinal gauge, whereas the Mukhanov-Sasaki variable naturally arises in the spatially flat gauge, where it is directly related to the perturbations of the inflaton scalar field. More details about the construction of these gauge invariant variables as well as the derivation of their dynamics from the perturbed Einstein equations in the Lagrangian framework can for instance be found in Reference [2]. A similar derivation in the canonical approach is for example presented in References [3–6]. Here we will not work in a particular gauge but take the form of the Mukhanov-Sasaki equation in (1) as our starting point. As far as a comparison with experimental data is concerned, the relevant quantity is the power spectrum that is defined as the (dimensionless) Fourier transform of the real space two-point correlation function, that is in the case of the quantized Mukhanov-Sasaki variable 0| *<sup>v</sup>*<sup>ˆ</sup>(*η*, **<sup>x</sup>**), *<sup>v</sup>*<sup>ˆ</sup>(*η*, **y**)|<sup>0</sup> .

Obviously the power spectrum can only be determined if some initial state has been chosen with respect to which the correlation functions are defined. The most common choice for the initial state is the Bunch-Davies vacuum that can be uniquely selected by the conditions that it is de Sitter invariant and satisfies the Hadamard condition. The latter requires that the corresponding two-point function has a specific behavior in the ultraviolet, that is for short distances. If we drop the Hadamard condition, we obtain the family of so-called *α*-vacua that include the Bunch-Davies vacuum. Other choices for the initial conditions than the ones for the Bunch-Davies vacuum have been considered and their possible fingerprints on the power spectrum have been investigated, see for instance References [7–9] and references therein. The Bunch-Davies vacuum is selected by requiring that in the limit of *η* → −∞ the mode functions take the form of the usual Minkowski mode functions. Another method to choose an initial state is the so-called Hamiltonian diagonalization method, where one minimizes the expectation value <sup>0</sup>*η*0 | *H*ˆ (*η*0)|<sup>0</sup>*η*0 of the Mukhanov-Sasaki Hamiltonian at one moment in time, say *η*0. Hamiltonian diagonalization refers to the fact that at *η*0 the coefficients of the off-diagonal terms involving second powers of annihilation and creation operators, respectivley, vanish for all modes. That is, at *η*0 the Mukhanov-Sasaki Hamiltonian is given by the field theoretical generalization of the standard harmonic oscillator. Considering this, a natural question to ask is whether such a Hamiltonian diagonalization can be obtained not only instantaneously but for each moment in time and particularly how this aspect is related to the choice of initial states. The usual form of Hamiltonian diagonalization has been critizised in the literature, see for instance Reference [10]. In the framework considered in our work this corresponds to the question whether there exists a canonical transformation that maps the

Mukhanov-Sasaki Hamiltonian to the time-independent harmonic oscillator for each moment in time. In order to work into that direction we take into account that the Mukhanov-Sasaki equation represents a time-dependent harmonic oscillator in each Fourier mode, whereas the specific form of the time dependence reflects the properties of the expanding background spacetime. What we are aiming at is a transformation that maps the time-dependent harmonic oscillator to the time-independent harmonic oscillator for each mode and all times. Defining such a transformation will only work if we consider time-depedendent canonical transformations, that are adapted specifically to the two systems of the time-dependent and time-independent harmonic oscillator, respectively. This is conveniently done in the extended phase space framework outlined below.

There has been considerable interest in the study of the time-dependent harmonic oscillator, both in a purely classical and quantum mechanical context. A distinct role in all of these considerations is played by the Lewis-Riesenfeld invariant, which is a constant of motion with respect to the evolution governed by a time-dependent harmonic oscillator. At the classical level, this invariant has been considered in the context of a canonical transformation in the extended phase space [11,12] that involves time and its momentum as canonical phase space variables among the usual position and momentum variables. Such an extended phase space provides a convenient platform to implement time-dependent canonical transformations. The obtained canonical transformation allows to map the system of the time-dependent harmonic oscillator onto the system of a harmonic oscillator with constant frequency and thus completely removes the time dependence of the Hamiltonian, which drastically simplifies the task of finding solutions of the equations of motion after applying the transformation. As shown in Reference [13], the invariant can also be defined in the context of quantum mechanics. In this case the eigenstates of the invariant can be used to construct solutions of the Schrödinger equation involving the original time-dependent Hamiltonian. Further application are to construct coherent states of the time-dependent harmonic oscillator by means of the eigenstates of the Lewis-Riesenfeld invariant as for instance discussed in References [13,14].

If we aim at relating the framework of the Lewis-Riesenfeld invariant to the notion of initial states associated with the Mukhanov-Sasaki equation, we need to generalize this approach to the field theory context. There exists already some work in this direction, see for example in References [14,15] and references therein, although with a slightly different focus than we want to consider here, because both of them do not apply this techniques directly to the Mukhanov-Sasaki equation in the framework of the extended phase space, meaning that they consider different time-dependent frequencies in general and particularly the generalization to field theory was not analyzed in very much detail in Reference [15]. The strategy we want to follow in our work is that first we consider the Lewis-Riesenfeld invariant and the corresponding canonical transformation at the classical level for finitely many degrees of freedom in the extended phase space, building on former work of References [11,12], who however did not consider the quantization of the canonical transformation. In order to be able to implement the corresponding unitary map at the quantum level, we also construct the corresponding generator of the canonical transformation. For the reason that in the extended phase space the physical system of the time-dependent harmonic oscillator is described as a constrained system, we construct Dirac observables and use the technique of reduced phase space quantization to implement this unitary map on the physical Hilbert space in a quantum mechanical setting, where it can also be formulated in terms of a time-dependent Bogoliubov transformation. Given this setup, we could take the vacuum of the time-independent harmonic oscillator, apply the constructed unitary map to it and obtain a in this sense natural candidate for a vacuum state for the time-dependent harmonic oscillator, that has then been determined directly by means of the unitary map.

The question we want to address in this article is whether we can carry this idea over from finitely many degrees of freedom to field theory and use the Lewis-Riesenfeld invariant approach to obtain possible candidates for initial states. In particular, we are interested in the physical properties of such initial states and their relation to the Bunch-Davies vacuum and other adiabatic vacua. As we will show, the most straightforward generalization to field theory is not possible because the so constructed

map involves an infrared divergence, hence the Shale-Stinespring condition is violated. As we will discuss, a suitable modification of the map in the infrared range can be obtained to cure the infrared divergenes. Furthermore, as we will show, if this map is not chosen carefully for all but the infrared modes it can also involve ultraviolet divergences strictly permitting a unitary implementation on Fock space. Interestingly, in the context of the Mukhanov-Sasaki Hamiltonian, different choices at this level can be related to different choices for the initial conditions of the associated mode functions. Moreover it becomes clear that we can recover the defining differential equation for adiabatic vacua from the Ermakov equation, where the latter is an auxiliary differential equation whose solution is needed to explicitly construct the Lewis-Riesenfeld invariant and the corresponding canonical transformation. This allows us to interpret the initial conditions and the result for the Fourier modes we obtain using the method of the Lewis-Riesenfeld invariant in the context of adiabatic vacua.

This article is structured as follows: In Section 2 we introduce the framework of the extended phase space and rederive the canonical transformation that maps the system of the time-dependent harmonic oscillator to the time-independent one generalizing the approach in Reference [12]. The time-rescaling that is involved in this canonical transformation naturally occurs in the extended phase space and the physical interpretation of the Lewis-Riesenfeld invariant can be easily understood. In order to deal with the constrained system in the extended phase space later on, we want to choose reduced phase space quantization and thus derive the reduced phase space in terms of Dirac observables. Their dynamics is generated by the Dirac observable associated with the time-dependent Hamiltonian. As the next step in Section 3, we consider the quantization of the system and show that the canonical transformation can be implemented as a unitary map on the one-particle physical Hilbert space, where our results agree with already existing results in the literature for finitely many degrees of freedom. In order to simplify the actual application of the unitary operator we perform a generalized Baker-Campbell-Hausdorff decomposition by means of which we then rewrite the unitary transformation as a time-dependent Bogoliubov map.

Afterwards we consider the generalization of our results obtained so far to field theory, discussing the two most common cases in the literature, where one maps from a time-dependent harmonic oscillator to a harmonic oscillator with either frequency *ω***k** = *k* or *ω***k** = 1. As far as the implementation on Fock space is considered, the first choice can be implemented unitarily, whereas the second cannot due to an ultraviolet divergence. This ultraviolet divergence is caused by a residual squeezing operation by which the two maps differ. To avoid issues that occur for the infrared modes, we discuss a possible modification of the map using the Arnold transformation discussed in Reference [16]. Section 6 presents practical applications of this formalism by considering the case of a quasi-de Sitter spacetime and the corresponding Mukhanov-Sasaki equation in a slow-roll approximation. We construct the Lewis-Riesenfeld invariant and, in the context of a quantum mechanical toy model, compute the lowest and next to lowest eigenvalue eigenstates associated to it and analyze their properties. Finally we summarize and conclude in Section 7.

### **2. Extended Phase Space Formulation and Time-Dependent Canonical Transformations**

A convenient framework for implementing time-dependent canonical transformations is the extended phase space in which also time and its conjugate momentum are treated as phase space variables and thus transformations of them can be naturally formulated. Motivated by the Mukhanov-Sasaki equation, we first investigate a single mode of the equation in a classical context. This corresponds to a harmonic oscillator with time-dependent frequency. We will consider the single mode Mukhanov-Sasaki Hamiltonian as a mechanical toy model and later generalize the results obtained in this case to the field theory context. Our goal is to remove this explicit time dependence by a time-dependent canonical transformation. This transformation will be defined on the extended phase space as a symplectic map that also includes the time variable and its associated conjugate momentum as phase space degrees of freedom.

### *2.1. Time-Dependent Hamiltonians on Extended Phase Space*

As a first step we reformulate the dynamics encoded in the single mode Mukhanov-Sasaki Hamiltonian on the extended phase space, where it becomes a constrained system. Let us consider a system with finitely many degrees of freedom where we denote all configuration variables as **q** = (*q*1, ··· , *q<sup>n</sup>*) and the configuration space by Σ. A time-dependent Lagrangian is then defined as a function *L* : *T*Σ × R → R. Because we want to include time among the elementary configuration variables, closely following the work in References [11,12], we extend the configuration manifold Σ to *M* := Σ × R and rewrite the action as

$$S[L] = \int\_{\mathbb{R}} \mathrm{d}s \, L\Big(\bar{\mathbf{q}}(s), t(s), \left(\frac{\mathrm{d}t}{\mathrm{d}s}\right)^{-1} \frac{\mathrm{d}\bar{\mathbf{q}}}{\mathrm{d}s}\Big) \frac{\mathrm{d}t(s)}{\mathrm{d}s} \tag{2}$$

$$:= \int\_{\mathbb{R}} \mathrm{d}s \, \mathfrak{L}\Big(\bar{\mathbf{q}}(s), t(s), \left(\frac{\mathrm{d}t}{\mathrm{d}s}\right)^{-1} \frac{\mathrm{d}\bar{\mathbf{q}}}{\mathrm{d}s}, \frac{\mathrm{d}t(s)}{\mathrm{d}s}\Big) =: S[\mathfrak{L}],$$

where we will refer to L as the *extended* Lagrange function now understood as a function on the extended tangent bundle *TM* that is even-dimensional and associated to the extended configuration manifold, including the former system evolution parameter commonly referred to as time. A non-degenerate symplectic structure on the corresponding cotangent bundle *T*<sup>∗</sup>*M*, whose elementary variables are (**q**˜, *t*, **p**˜, *pt*) can be defined as usual. This allows to establish a one-to-one correspondence between smooth phase space functions and Hamiltonian vector fields. In complete analogy to the conventional case, one can formulate the Euler-Lagrange equations in terms of the extended variables by means of the variational principle, which results in equivalent equations of motion as derived from the original action *<sup>S</sup>*[*L*]. The equations of motion for the time variable are just given by d*t* d*s* = *<sup>λ</sup>*(*s*) where *<sup>λ</sup>*(*s*) is an arbitrary real parameter reflecting the rescaling symmetry of the action, this reflects the arbitrary parametrization of time and has no physical significance. If we perform a Legendre transform, we realize that *pt* = −*<sup>H</sup>*(**q**˜, *p*˜, *t*) becomes a primary constraint since it cannot be solved for the velocities *dt ds*with<sup>1</sup>

$$H(\mathbf{\bar{q}}, \mathbf{\bar{p}}, t) = \frac{\mathbf{\bar{p}}^2}{2} + \frac{1}{2} \omega^2(t) \mathbf{\bar{q}}^2 \omega$$

where the Hamiltonian is a function *H* : *T*∗*M* → R that is independent of *pt*. We denote this constraint by *C* := *pt* + *<sup>H</sup>*(**q**˜, **p**˜, *t*). Therefore, we apply the Legendre transform for singular systems and obtain the following Hamiltonian on the extended phase space *T*<sup>∗</sup>*M*:

$$\mathfrak{H} = \mathfrak{p}\_a \frac{\mathrm{d}\mathfrak{p}^t}{\mathrm{d}\mathbf{s}} + p\_t \frac{\mathrm{d}t}{\mathrm{d}\mathbf{s}} - \mathfrak{L} \Big|\_{\mathfrak{q}^s(\mathfrak{q}, \mathfrak{p}, t \lambda), \frac{\mathrm{d}t}{\mathrm{d}\mathbf{s}} = \lambda} = \left( H(\mathfrak{q}, \mathfrak{p}, t) + p\_t \right) \lambda(\mathbf{s}) = \lambda(\mathbf{s}) \mathbb{C} \approx 0,$$

where we used ≈ to denote weak equivalence and used the definition of *<sup>λ</sup>*(*s*) from above. Due to reparametrization invariance of the extended action, there is no true Hamiltonian but a Hamiltonian *constraint C*. From now on we will neglect the tilde on the top of the variables **q**, **p** to keep our notation more compact. For the time-dependent harmonic oscillator the so-called Lewis-Riesenfeld invariant *ILR* has played a pivotal role, particularly in the construction of solutions for the corresponding equations of motion. *ILR* is a phase space function being quadratic in the elementary variables (**<sup>q</sup>**, **p**) and its time dependence is encoded in a function *ξ* : I ⊆ R → R. Explicitly, it is given by:

$$I\_{LR}\left(\mathbf{q}, \mathbf{p}, t\right) := \frac{1}{2} \left( \left( \xi(t)\mathbf{p} - \xi(t)\mathbf{q} \right)^2 + \frac{\omega\_0^2 \mathbf{q}^2}{\xi^2(t)} \right). \tag{3}$$

<sup>1</sup> In general we could also take into account a time dependent mass in the Hamiltonian, however in the case of the Mukhanov-Sasaki equation it is sufficient to set the mass parameter *m* equal to *m* = 1.

Since *ILR* is an invariant, it has to commute with the constraint *C* on the extended phase space2:

$$\{I\_{LR}, \mathbb{C}\}\_{ext} = \{I\_{LR}, H(t)\} + \frac{\partial I\_{LR}}{\partial t} = 0. \tag{4}$$

This carries over to a condition on the function *ξ* that has to satisfy the following non-linear, ordinary, second-order differential equation

$$
\left(\frac{\mathrm{d}^2}{\mathrm{d}t^2} + \omega(t)^2\right)\xi(t) - \omega\_0^2\,\xi(t)^{-3} = 0,\tag{5}
$$

known as the Ermakov equation. It has been shown that *ILR* is an invariant both at the classical level [11] and at the quantum level [13,17]. In the following, the explicit form of *ILR* will be our guiding line for finding an extended canonical transformation that removes the time dependence from the Hamiltonian of the harmonic oscillator with a time-dependent frequency.

### *2.2. Extended Canonical Transformations and Hamiltonian Flows*

In the framework of the extended phase space formalism we can now regard time as a configuration degree of freedom and consequently apply a canonical transformation to implement a time-rescaling. It is worth noting that the Mukhanov-Sasaki equation in *conformal* time (commonly denoted *η*) takes the form of a time-dependent harmonic oscillator, however we shall refer to the time variable as *t* in the context of the classical and one-particle quantum theory, respectively. We aim at finding a symplectic map Φ such that the explicitly time-dependent Mukhanov-Sasaki Hamiltonian is mapped into an autonomous one, that is, one with time-independent frequency that we denote by *ω*0. During this procedure, the Hamiltonian constraint together with the Poisson structure on the extended phase space remain invariant by construction, that is:

$$\Phi: T^\*M \to T^\*M, \quad \mathfrak{H} \mapsto \Phi^\*\mathfrak{H} = \mathfrak{H}.\tag{6}$$

Correspondingly, the symplectic form Ω on *T*∗*M* is invariant under the Hamiltonian flow of the associated Hamiltonian vector field of Φ that infinitesimally generates this transformation. In order to apply this procedure to the case of the single-mode Mukhanov-Sasaki Hamiltonian, we need to impose conditions on the explicit form of Φ. We would like to preserve the functional dependence of the Hamiltonian constraint on the one hand and keep the quadratic order in both momentum and configuration variables on the other hand. For this purpose, we make the following ansatz, closely related to the work presented in Reference [12]:

$$\Phi: \begin{pmatrix} q^a \\ p\_a \\ t \\ p\_t \end{pmatrix} \quad \mapsto \quad \begin{pmatrix} Q^a(\mathbf{q}, t) \\ F(\mathbf{q}, t)p\_a + \mathbf{G}\_d(\mathbf{q}, t) \\ T(\mathbf{q}, t) \\ \mathbf{P}\_T(\mathbf{q}, t, \mathbf{p}\_r, p\_t) \end{pmatrix} \quad \text{s.t.} \quad \Phi^\*\Omega = \Omega \tag{7}$$

Note the ansatz **p** ∝ **P** + **<sup>G</sup>**(**<sup>q</sup>**, *t*) ensures that the transformed Hamiltonian is again quadratic in the new momentum, whereas the prefactor allows for a time-rescaling of the momentum variable. Additionally, the only variable that carries a dependence on *pt* is the new momentum conjugate to *T* denoted by *PT*, which is a choice that preserves the form of the Hamiltonian constraint being linear in the conjugate momentum of the time variable. We employ the ansatz in (7) for the symplectic map Φ and from subsequent comparison of coefficients of the two-form basis elements we obtain a set

<sup>2</sup> The symplectic form associated with the Poisson bracket {., .}ext on the extended phase space has the form Ω = *dq*˜*<sup>a</sup>* ∧ *dp*˜*a* + *dt* ∧ *dpt*.

of five coupled differential equations that determine the form of Φ to be a canonical transformation. This system of differential equations corresponds to a generalization to *n* + 1 configuration degrees of freedom of the set of equations presented in Reference [12], where only the case for *n* = 1 was presented. It explicitly reads:

$$\frac{\partial Q^a}{\partial t} \left( p\_a \frac{\partial F}{\partial q^b} + \frac{\partial G\_a}{\partial q^b} \right) + \frac{\partial P\_T}{\partial q^b} \frac{\partial T}{\partial t} = \frac{\partial Q^a}{\partial q^b} \left( p\_a \frac{\partial F}{\partial t} + \frac{\partial G\_a}{\partial t} \right) + \frac{\partial P\_T}{\partial t} \frac{\partial T}{\partial q^b},$$

$$F \frac{\partial Q^a}{\partial t} + \frac{\partial P\_T}{\partial p\_a} \frac{\partial T}{\partial t} = 0, \qquad \frac{\partial P\_T}{\partial p\_a} \frac{\partial T}{\partial q^b} + F \frac{\partial Q^a}{\partial q^b} = \delta^a\_{b'}. \tag{8}$$

$$\frac{\partial P\_T}{\partial p\_t} \frac{\partial T}{\partial q^a} = 0, \qquad \frac{\partial P\_T}{\partial p\_t} \frac{\partial T}{\partial t} = 1.$$

We can ge<sup>t</sup> a first hint how a solution could look like when we consider the Lewis-Riesenfeld invariant *ILR* from Equation (3) in Section 2 above. Hence, there is a natural starting point for finding the favored canonical transformation we are aiming at, by fixing the transformations of **q** and **p** according to a factorization of *ILR*. This leads to

$$Q^a(\mathbf{q}, t) := \frac{q^a}{\tilde{\xi}(t)} \quad \Longleftrightarrow \quad q^a(\mathbf{Q}, T) = \tilde{\xi}(t(T))Q^a,\tag{9}$$

$$P\_{\mathbf{d}}(\mathbf{q}, \mathbf{p}, t) := \xi(t)p\_{\mathbf{d}} - \dot{\xi}(t)q\_{\mathbf{d}} \quad \Longleftrightarrow \quad p\_{\mathbf{d}}(\mathbf{Q}, \mathbf{P}, T) = \frac{P\_{\mathbf{d}}}{\xi(t(T))} + \dot{\xi}(t(T))Q\_{\mathbf{d}},\tag{10}$$

where ˙ *ξ* = *∂tξ* is the derivative with respect to the dynamical time variable. In order to proceed, we need to find a suitable transformation for the time variable *T*(*t*) that is consistent with the system of Equation (8) previously found. A convenient possibility is to use Euler's time scaling transformation for the three-body problem, recently introduced by Struckmeier [11] in the context of the time-dependent harmonic oscillator. However, the approach in Reference [11] differs from the one outlined in this work in the sense that we derive the explicit form of the transformation instead of making use of the corresponding generating function. The relevant transformation of *t* is given by:

$$T(t) := \int\_{t\_0}^t \frac{\mathbf{d}\tau}{\xi^2(\tau)} \quad \Longleftrightarrow \quad \frac{\partial T}{\partial t} = \frac{1}{\xi^2(t)},\tag{11}$$

where *ξ*(*t*) ∈ *C*<sup>2</sup>(R) is an up-to-now arbitrary function with the only restriction that the above integral needs to be well-defined. Given the explicit form of (11), we can require mutual consistency of the transformations in (8) in order to fix the form of the transformed canonical momentum *PT*. Solving the first equation in (8) for *∂bPT* and subsequently integrating the obtained expression yields the following result:

$$P\_T(\mathbf{q}, \mathbf{p}, t, p\_t) = \zeta^2(t)p\_t + \zeta(t)\zeta(t)\mathbf{q} \cdot \mathbf{p} - \frac{1}{2} \left(\zeta(t)\zeta(t) + \zeta^2(t)\right)\mathbf{q}^2,\tag{12}$$

with the term *ξ*<sup>2</sup>(*t*)*pt* arising from an arbitrary additive constant with respect to **q** and the requirement of inverse scaling behavior between *t* and *pt* according to (8). Now that we have fixed the transformation to the new canonical coordinates, we can use the invariance of the Hamiltonian constraint H under the change of canonical coordinates to derive an autonomous Hamiltonian from the original, time-dependent one:

$$\boldsymbol{\Phi}^{\*}\mathfrak{H} = \left(\boldsymbol{\Phi}^{\*}\boldsymbol{H} + \boldsymbol{P}\_{\Gamma}\right)\frac{\mathrm{d}\boldsymbol{T}}{\mathrm{d}\boldsymbol{s}} = \left(\boldsymbol{\Phi}^{\*}\boldsymbol{H} + \boldsymbol{P}\_{\Gamma}\right)\frac{\partial\boldsymbol{T}}{\partial\boldsymbol{t}}\frac{\mathrm{d}\boldsymbol{t}}{\mathrm{d}\boldsymbol{s}} = \left(\boldsymbol{H}(\boldsymbol{\mathsf{q}},\boldsymbol{p},\boldsymbol{t}) + \boldsymbol{p}\_{l}\right)\frac{\mathrm{d}\boldsymbol{t}}{\mathrm{d}\boldsymbol{s}} = \mathfrak{H}.\tag{13}$$

In fact, using the one before the last equality sign in (13) we find an expression for Φ<sup>∗</sup>*H*:

$$H\_0 := \Phi^\* H = \mathfrak{J}^2(t) \left( H(\mathbf{q}, \mathbf{p}, t) + p\_t \right) \Big|\_{\left(\Phi\right) \left(\mathbf{q}, \mathbf{p}, t\right)} - P\_{T\_\prime} \tag{14}$$

with **q**, **p** and *t* considered as functions of the new variables **Q**, **P** and *T* via the extended canonical transformation <sup>Φ</sup>(**<sup>q</sup>**, **p**, *t*) defined in Equation (7). Analogous to the treatment displayed in Reference [11], we would also like to point out the crucial property that not the bare constraint *C* but the product with the Lagrange multiplier H = *<sup>λ</sup>C*(**<sup>q</sup>**, **p**, *t*, *pt*) is invariant under this transformation by construction. As a consequence, the canonical momentum *PT* drops out in *H*0. If we evaluate all expressions using the inverse of Φ to express **q**, **p** in terms of **Q**, **P**, we finally obtain:

$$H\_{0}(\mathbf{Q},\mathbf{P},T) = \left[\frac{\xi^{2}(t)}{2}\left(\mathbf{p}^{2} + \omega(t)^{2}\mathbf{q}^{2}\right) - \xi(t)\dot{\xi}(t)\mathbf{q}\cdot\mathbf{p} + \frac{1}{2}\Big{(}\xi(t)\ddot{\xi}(t) + \dot{\xi}^{2}(t)\Big{)}\mathbf{q}^{2}\right]\bigg{)}\_{\left(\mathbf{p}^{-1}\right)\left(\mathbf{Q},T\right)}$$

$$= \frac{\xi^{2}}{2}\left(\frac{\mathbf{p}^{2}}{\xi^{2}} + 2\frac{\dot{\xi}}{\xi}\mathbf{Q}\cdot\mathbf{P} + \dot{\xi}^{2}\mathbf{Q}^{2} + \omega(t(T))^{2}\dot{\xi}^{2}\mathbf{Q}^{2}\right) - \xi\dot{\xi}\mathbf{Q}\cdot\mathbf{P} - \frac{1}{2}\left(\xi^{2}\dot{\xi}^{2} - \xi^{3}\frac{\dot{\varkappa}}{\xi}\right)\mathbf{Q}^{2} \quad \text{(15)}$$

$$= \frac{1}{2}\left(\mathbf{P}^{2} + \xi^{3}\left(\ddot{\xi} + \omega\left(t(T)\right)^{2}\dot{\xi}\right)\mathbf{Q}^{2}\right) = \frac{1}{2}\left(\mathbf{P}^{2} + \omega\_{0}^{2}\mathbf{Q}^{2}\right),$$

where we designed the symplectic map Φ in such a way that the requirement that the term in the brackets multiplying **Q**<sup>2</sup> in Equation (15) equals *ω*20 ∈ R is respected. This leads to the condition that *ξ*(*t*) needs to satisfy the Ermakov differential equation, which we already encountered during the discussion of the Lewis-Riesenfeld invariant *ILR* in Section 2 in (5). The so constructed map Φ describes a time-dependent canonical transformation that maps a harmonic oscillator with time-dependent frequency *ω*(*t*) onto a time-independent harmonic oscillator with constant frequency *ω*0. The explicit form of the map of course depends on the time dependence of *ω*(*t*) but can be determined from the Ermakov equation once *ω*(*t*) is given. While in principle we could fix the frequency *ω*0 to one, as it has been done for the form of the Ermakov equation for instance in References [13,14], we would like our transformation Φ to correspond to the identity for an already time-independent harmonic oscillator Hamiltonian. This can only be achieved if not all time-dependent frequencies are mapped to unity, as even a constant *ω*0 would then be transformed non trivially, resulting in a residual transformation analogous to a squeezing operation in quantum theory.

#### *2.3. The Reduced Phase Space Associated with T*∗*M and the Infinitesimal Generator of* Φ

In this section we want to derive the infinitesimal generator corresponding to the finite canonical transformation Φ on the extended phase space *T*∗*M* that we presented in the last section. This will be relevant later on when we discuss the implementation of Φ in the quantum theory. As we have discussed, the system under consideration can be understood as a constrained system in the context of the extended phase space. Consequently, we have two options for handling the constraint, either we solve it in the quantum theory via Dirac quantization or we reduce with respect to this constraint already classically and quantize the reduced phase space only. In the first place, both approaches are equally justified from the physical perspective, so this is a choice one makes for each given model. In our case this goes along with the selection whether we want to implement the canonical transformation Φ on the extended or reduced phase space, respectively. Firstly, as the transformation from *t* to *T*(*t*) in (11) involves a time-rescaling in form of an integral, if we are not able to obtain the antiderivative of the integrand in closed form, it will be problematic to formulate this kind of canonical transformation in the quantum theory based on the extended phase space where *t* becomes an operator. Secondly, following Dirac quantization, we need to construct a physical inner product for physical states and this is non-trivial if the constraint is of the form *C* = *pt* + *<sup>H</sup>*(**<sup>q</sup>**, **p**, *t*) with *H* being explicitly time-dependent, a similar situation that occurs in loop quantum cosmology if we consider the inflaton as reference matter. The final physical sector of the theory should be related in both approaches and in the best case yield the same physical predictions. This might not be the case in general but yields some restrictions on possible choices in the quantization procedure to match the models based on Dirac and reduced quantization respectively. In the following we choose the reduced phase space approach for which the initial phase space *T*∗Σ can be naturally identified with the reduced phase space of our system. In order to show this we construct Dirac observables for our constrained system by means of the formalism presented in References [18,19] and references therein, that is based on the relational formalism originally introduced in References [20,21]. In the extended phase space, we consider the configuration variable *t* as the reference field (clock) for time and introduce the following gauge fixing condition *Gτ* := *t* − *τ* ≈ 0. *Gτ* together with the first class constraint *C* build a second class pair since {*<sup>G</sup><sup>τ</sup>*, *C*} = 1. The Dirac observables for all degrees of freedom except the clock degrees of freedom (*t*, *pt*) are given by

$$\mathcal{O}\_{q^a,t}^{\mathbb{C}}(\tau) = \sum\_{n=0}^{\infty} \frac{G\_{\tau}^n}{n!} \{ \mathbb{C}(\mathbf{q}\_\prime \mathbf{p}, t), q^a \}\_{\{\mathbf{u}\}'} \quad \mathcal{O}\_{p\_a,t}^{\mathbb{C}}(\tau) = \sum\_{n=0}^{\infty} \frac{G\_{\tau}^n}{n!} \{ \mathbb{C}(\mathbf{q}\_\prime \mathbf{p}, t), p\_a \}\_{\{\mathbf{u}\}'} \tag{16}$$

where {*<sup>A</sup>*, *<sup>B</sup>*}(*n*) denotes the iterated Poisson bracket defined via {*<sup>A</sup>*, *<sup>B</sup>*}(*n*) := {*<sup>A</sup>*, {*<sup>A</sup>*, *<sup>B</sup>*}(*<sup>n</sup>*−<sup>1</sup>)} and {*<sup>A</sup>*, *<sup>B</sup>*}(0) := *B* and we have used that *qa* and *pa* both commute with the conjugate momentum *pt*. The observable map can also be applied to the clock degrees of freedom, leading to

$$\mathcal{O}\_{t,l}^{\mathbb{C}}(\tau) = \sum\_{n=0}^{\infty} \frac{G\_{\tau}^{n}}{n!} \{ \mathbb{C}(\mathbf{q}\_{\tau} \mathbf{p}, t, p\_{l}), t \}\_{(n)} = \tau, \quad \mathcal{O}\_{p\_{l}, l}^{\mathbb{C}}(\tau) = \sum\_{n=0}^{\infty} \frac{G\_{\tau}^{n}}{n!} \{ \mathbb{C}(\mathbf{q}\_{\tau} \mathbf{p}, t, p\_{l}), p\_{l} \}\_{(n)}.\tag{17}$$

We realize that the clock *t* is mapped to the parameter *τ* as expected, whereas contrary to the deparametrized models presented in References [22–29], the physical Hamiltonian retains its time dependence, hence *pt* is not ye<sup>t</sup> a Dirac observable by itself. Using the properties of the observable map we have that *pt* = −O*CH*(**<sup>q</sup>**,**p**,*<sup>t</sup>*),*<sup>t</sup>* = <sup>−</sup>*<sup>H</sup>*(O*C***q**,*t*, <sup>O</sup>*C***p**,*t*, *τ*) and hence *pt* can be expressed as a function of <sup>O</sup>*C***q**,*t*, <sup>O</sup>*C***p**,*<sup>t</sup>* only, where we introduced the abbreviation <sup>O</sup>*C***q**,*<sup>t</sup>* := (O*Cq*1,*t*, ··· , <sup>O</sup>*Cqn*,*<sup>t</sup>*) and likewise for the momenta. This shows that (O*C***q**,*t*, <sup>O</sup>*C***p**,*<sup>t</sup>*) are the elementary variables of the reduced phase space and the degrees of freedom encoded in (*t*, *pt*) have been reduced, which leaves us with 2*n true* degrees of freedom in the physical sector of the phase space. As a consequence, we can identify the reduced phase space with *T*∗Σ and the Hamiltonian can be understood as a function from *T*∗Σ × R to the real numbers. In order to analyze the Poisson algebra of the observables we have to construct the corresponding Dirac bracket, denoted by {., .}∗, associated to the second class system (*<sup>G</sup><sup>τ</sup>*, *<sup>C</sup>*). However, for the reason that all variables (**<sup>q</sup>**, **p**) commute with the gauge fixing condition, their Dirac bracket reduces to the usual Poisson bracket. Given this and considering the result in Reference [19], the algebra of our Dirac observables reads:

$$\{\mathcal{O}\_{q^{\mathfrak{s}},t}^{\mathbb{C}}(\tau),\mathcal{O}\_{p\_{\mathfrak{b}},t}^{\mathbb{C}}(\tau)\} = \mathcal{O}\_{\{q^{\mathfrak{s}},p\_{\mathfrak{b}}\}^{\*},t}^{\mathbb{C}}(\tau) = \delta\_{\mathfrak{b}}^{\mathfrak{s}}.$$

Thus, the kinematical Poisson algebra of (**<sup>q</sup>**, **p**) and the algebra of their corresponding Dirac observables are isomorphic, which is a big advantage for finding representations of the observable algebra in the context of the quantum theory in Section 3. The observable map applied to a generic phase space function *f* returns the values of *f* at those values where the clock takes the value *τ*. Therefore, the natural evolution parameter for these Dirac observables is *τ*. If the constraint is linear in the clock momenta as in our case where *C* = *pt* + *<sup>H</sup>*(**<sup>q</sup>**, **p**, *t*), then as shown in References [19,26] the so-called physical Hamiltonian generating the *τ*-evolution is given by the Dirac observable

corresponding to *<sup>H</sup>*(**<sup>q</sup>**, **p**, *t*). Thus, in our case the evolution on the reduced phase space is given by the following Hamilton's equations:

$$\frac{\mathbf{d}}{\mathbf{d}\tau}\mathcal{O}^{\mathbb{C}}\_{q^{\mathfrak{s}},t} = \{\mathcal{O}^{\mathbb{C}}\_{q^{\mathfrak{s}},t'}H(\mathcal{O}^{\mathbb{C}}\_{\mathbf{q},t'}\mathcal{O}^{\mathbb{C}}\_{\mathbf{p},t'}\tau)\}, \quad \frac{\mathbf{d}}{\mathbf{d}\tau}\mathcal{O}^{\mathbb{C}}\_{p\_{\mathfrak{s}},t} = \{\mathcal{O}^{\mathbb{C}}\_{p\_{\mathfrak{s}},t'}H(\mathcal{O}^{\mathbb{C}}\_{\mathbf{q},t'}\mathcal{O}^{\mathbb{C}}\_{\mathbf{p},t'}\tau)\}.\tag{18}$$

Lastly, by an abuse of notation we replace *τ* by *t* as well as O*<sup>C</sup> q<sup>a</sup>*,*<sup>t</sup>* by *qa* and O*<sup>C</sup> pa*,*t* by *pa* in order to be closer to the notation used in previous works in the literature and emphasize that the generator of Φ acts as a one-parameter family of transformations on configuration and momentum degrees of freedom in *T*<sup>∗</sup>Σ. When we have a look at the form of Φ, we immediately recognize that the generator G ∈ *C* ∞(*T*∗Σ × R) needs to be a polynomial of second order in the original configuration and momentum variables, where *T*∗Σ × R corresponds to the presymplectic space for explicitly time-dependent systems as for instance used in Reference [11]. This ensures that the action of the associated Hamiltonian vector field *X*G with *X*G (*f*) := {G, *f* } onto the elementary phase space variables **q** and **p** results in a linear combination of those quantities. The explicit form of Φ suggests an ansatz in order to find G, which naturally depends on *ξ*, ˙ *ξ*, incorporating the parametric dependence on *t*:

$$\mathcal{G}(\ulcorner, \dot{\zeta}, \mathbf{q}, \mathbf{p}) := f(\ulcorner, \dot{\zeta}, \dot{\ulcorner}) \mathbf{q} \cdot \mathbf{p} + \frac{1}{2} \mathbf{g}(\ulcorner, \dot{\zeta}, \dot{\ulcorner}) \mathbf{q}^2,\tag{19}$$

where the factor in front of *g*(*ξ*, ˙ *ξ*) was introduced for later convenience. Application of the exponentiated Hamiltonian vector field *X*G onto **q** and **p** leads to the following results:

$$\exp\{X\_{\mathcal{G}}\}q^{a} := \sum\_{n=0}^{\infty} \frac{1}{n!} \{\mathcal{G}, q^{a}\}\_{(n)} = \sum\_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \{f(\xi^{x}, \dot{\xi})\}^{n} q^{a} = e^{-f(\xi, \dot{\xi})} q^{a},\tag{20}$$

$$\exp\{X\_{\mathcal{G}}\}p\_{a} := \sum\_{n=0}^{\infty} \frac{1}{n!} \{\mathcal{G}, p\_{a}\}\_{(n)} = e^{f\left(\overline{\xi}, \overline{\xi}\right)} p\_{a} + \frac{1}{2} \left(e^{f\left(\overline{\xi}, \overline{\xi}\right)} - e^{-f\left(\overline{\xi}, \overline{\xi}\right)}\right) \frac{\mathcal{G}\left(\overline{\xi}, \overline{\xi}\right)}{f\left(\overline{\xi}, \overline{\xi}\right)} q\_{a\prime} \tag{21}$$

with the iterated Poisson bracket defined as above. A direct comparison of the results in (20) and (21) to the solutions of the system of equations in (8) yields the dependencies of *f*(*ξ*, ˙ *ξ*) and *g*(*ξ*, ˙ *ξ*) on *ξ* and ˙ *ξ*, respectively:

$$f(\xi', \dot{\xi}) = \ln(\xi), \quad g(\xi', \dot{\xi}) = \frac{2\ln(\xi)\xi\xi}{1 - \xi^2}.\tag{22}$$

Finally, we are able to explicitly write down the generator of the extended canonical transformation Φ restricted to the constraint hypersurface *T*∗Σ × R, that is the physical sector. We call this restriction of Φ, which is a time-dependent canonical transformation on the reduced phase space, Γ*ξ* from now on. In a convenient notation, it has the following form:

$$\mathcal{G}(\mathbf{\tilde{\zeta}}, \dot{\mathbf{\xi}}, \mathbf{q}, \mathbf{p}) = \frac{1}{2} \ln(\xi) \left( \mathbf{q} \cdot \mathbf{p} + \mathbf{p} \cdot \mathbf{q} + h(\mathbf{\tilde{\zeta}}, \dot{\xi}) \mathbf{q}^2 \right), \quad h(\mathbf{\tilde{\zeta}}, \dot{\mathbf{\xi}}) := \frac{2 \mathbf{\tilde{\zeta}} \dot{\mathbf{\tilde{\zeta}}}}{1 - \xi^2}. \tag{23}$$

In fact, this classical generator precisely corresponds to the exponential operator found in Reference [17] for a quantized version of the time-dependent harmonic oscillator. It is worth noting that, regardless of the choice of coordinates, G takes the same form in either **q**, **p** or **Q**, **P**, that is it holds that G **<sup>q</sup>**(**Q**, **<sup>P</sup>**), **<sup>p</sup>**(**Q**, **P**) = G **Q**, **P** . Not surprisingly, we can switch between the autonomous Hamiltonian and the Lewis-Riesenfeld invariant in this framework, using the action of Γ*ξ* on (analytical) phase space functions, leading to:

$$H\_0\left(\Gamma\_{\tilde{\xi}}(\mathbf{q}), \Gamma\_{\tilde{\xi}}(\mathbf{p})\right) = H\_0\left(e^{X\_{\mathcal{Q}}}\mathbf{q}, e^{X\_{\mathcal{Q}}}\mathbf{p}\right) = \frac{1}{2}\left(\left(\xi(t)\mathbf{p} - \xi(t)\mathbf{q}\right)^2 + \frac{\omega\_0^2 \mathbf{q}^2}{\xi(t)^2}\right) =: I\_{LR} \tag{24}$$

*Universe* **2019**, *5*, 170

Of course this was how Γ*ξ* or rather Φ was constructed in the first place. However, relation (24) will be of importance in the quantum theory, where it is part of the time evolution operator (i.e., the Dyson series) associated to the time-dependent Hamiltonian. Furthermore, this will allow us to make contact to previous work and strictly derive the phase factor that was introduced by hand in Reference [13] in order to construct eigenfunctions of the time-dependent Schrödinger equation. Referring to the relational formalism outlined in for example, References [18,19], we reconsider the fact that the Lewis-Riesenfeld invariant strongly commutes with the constraint *C* as shown in (3). Hence, in this language *ILR* is a strong Dirac observable with respect to the constraint *<sup>C</sup>*(**<sup>q</sup>**, **p**, *t*, *pt*) if and only if *ξ*(*t*) satisfies the Ermakov Equation (5), connecting to the results presented in Reference [13] in the context of quantization. As a concluding remark, let us introduce *e*+ := 1 2**<sup>p</sup>**2, *e*− := −1 2**q**<sup>2</sup> and *h* := *qa pa*, which amount to the generators of the classical canonical transformation Γ*ξ* we derived in the preceding section. Then these three generators form a basis of the sl(2, R) algebra, which is evident due to the structure constants of their Poisson brackets. Hence, the exponential of these generators (or a subset thereof) constitutes a group element of SL(2, R) and consequently the classical canonical transformation Γ*ξ* is a real representation of SL(2, R) on the space of phase space polynomials or everywhere-analytic phase space functions, respectively.

Let us briefly summarize what we have established in the previous section. Starting from an explicitly time-dependent Hamiltonian and its associated Lewis-Riesenfeld invariant, we systematically constructed a time-dependent canonical transformation on an extended phase space, which removes the time dependence of the original Hamiltonian. Let us stress at this point that *<sup>H</sup>*(*t*), *ILR*(*t*) and *H*0 are in fact the *same object* in *different coordinates* on the extended phase space. Consequently, we were able to construct the associated infinitesimal generator of this symplectic map and established the notion of a reduced phase space with the prospect of a corresponding unitary transformation in the one-particle quantum theory. The construction of the latter will be the content of the next section.

### **3. Quantization: One-Particle Hilbert Space**

In this section we will present the quantization of the time-dependent canonical transformation derived in the last section on the one-particle Hilbert space. This allows to transform each mode of the single-mode Mukhanov-Sasaki Hamiltonian into a harmonic oscillator with constant frequency. In Section 4 we will discuss in which sense the results obtained in this section can be generalized to field theories. The unitary implementation of the symplectic transformation we considered can be used for constructing an analytic solution to the time-dependent Schrödinger equation in the form of a unitary time evolution operator.

### *3.1. Canonical Quantization of the Time-Dependent Canonical Transformation*

From the classical theory, the relevant algebra is P = *C* <sup>∞</sup>(*T*<sup>∗</sup>R*<sup>d</sup>*), {., .}, · equipped with the Poisson bracket and pointwise multiplication, which is the algebra of elementary variables of a classical point-particle in d-dimensional Euclidean space. This algebra can be further extended by an involution operation leading to the Poisson \*-algebra that will be our starting point for the canonical quantization. In the following we can restrict our discussion to the case d=1 which is sufficient for the quantization of the single mode Mukhanov-Sasaki system. As a first step we define a quantization map Q that maps elements of P into an abstract operator algebra Q(P). Given any two smooth phase space functions *f* , *g* ∈ P we have

$$\mathcal{Q}: \mathcal{P} \to \mathcal{Q}(\mathcal{P}), \qquad \{f, \emptyset\} \mapsto \mathcal{Q}(\{f, \emptyset\}) = -i[\mathcal{Q}(f), \mathcal{Q}(\emptyset)] \in \mathcal{Q}(\mathcal{P}), \tag{25}$$

where we have set *h*¯ = 1. Requiring that Q is function-preserving, that is Q *<sup>F</sup>*(*q*, *p*) = *F* Q(*q*), Q(*p*) for any real function *F* as usually required for any quantization map, we can now directly write down the quantum version of the generator for the one-parameter family (i.e., time-dependent) of canonical transformations Γ*ξ* on *T*∗Σ and its exponential:

$$\mathcal{Q}(\mathcal{G}) := \mathcal{G} = \frac{1}{2} \ln(\mathfrak{J}) \left( \mathfrak{H} \mathfrak{P} + \mathfrak{H} \mathfrak{H} + h(\mathfrak{J}, \mathfrak{J}) \mathfrak{q}^2 \right), \tag{26}$$

where *q*ˆ, *p*ˆ denote elements of the abstract operator algebra Q(P). For later convenience we quantize the inverse of Γ*ξ* and hence the inverse map, that is due to the minus sign in the quantization prescription and to be closer to existing results in the literature, since the mapping to the autonomous Hamiltonian is classically achieved by the inverse of Φ:

$$\mathcal{Q}(\Gamma\_{\vec{\xi}}^{-1}) = \exp\left\{i\left[\mathcal{Q}(\mathcal{G})\_{\prime}.\right]\right\} = \exp\left\{i\left[\mathcal{G}\_{\prime}.\right]\right\} = \exp\left\{i\operatorname{ad}\_{\mathcal{G}}\right\} =: \operatorname{Ad}\_{\mathbb{F}\_{\vec{\xi}}^{-1}}$$

with 'ad' and 'Ad' denoting the adjoint representation of the Lie algebra and the corresponding Lie group, respectively. Now since we want to define the action of Γ ˆ *ξ* on some Hilbert space we need a representation that maps the abstract operators into the set of linear operators on a Hilbert space respecting the commutator relations of the abstract algebra, that is *π* : Q(P) → L(H) such that *π*(Q( P)) = H, *<sup>π</sup>*(Q({*q*, *p*})) = <sup>−</sup>*<sup>i</sup>*[*π*(Q(*q*)), *π*(Q(*p*))] as well as *<sup>π</sup>*(Q({*q*, *q*})) = <sup>−</sup>*<sup>i</sup>*[*π*(Q(*q*)), *π*(Q(*q*))] and *<sup>π</sup>*(Q({*p*, *p*})) = <sup>−</sup>*<sup>i</sup>*[*π*(Q(*p*)), *<sup>π</sup>*(Q(*p*))]. If not otherwise stated we will work with the standard Schrödinger position representation given by (*<sup>π</sup>*, H = *<sup>L</sup>*2(<sup>R</sup>, *dx*)) with

$$\begin{aligned} \pi\_{\boldsymbol{q}}(\mathcal{Q}(\boldsymbol{q})) &= \quad \pi\_{\boldsymbol{q}}(\boldsymbol{\dot{q}}) : \mathcal{S}(\mathbb{R}) \to \mathcal{S}(\mathbb{R}), \quad (\pi\_{\boldsymbol{q}}(\boldsymbol{\dot{q}})\Psi)(\boldsymbol{q}) = \boldsymbol{q}\Psi(\boldsymbol{q}),\\ \pi\_{\boldsymbol{q}}(\mathcal{Q}(\boldsymbol{p})) &= \quad \pi\_{\boldsymbol{q}}(\boldsymbol{\dot{p}}) : \mathcal{S}(\mathbb{R}) \to \mathcal{S}(\mathbb{R}), \quad (\pi\_{\boldsymbol{q}}(\boldsymbol{\dot{p}})\Psi)(\boldsymbol{q}) = -\boldsymbol{i}\frac{d\Psi}{d\boldsymbol{q}}(\boldsymbol{q}). \end{aligned}$$

Here S(R) denotes the space of Schwartz functions on R. Given the representation we can define the action of Γ ˆ *ξ* on both operators and elements Ψ in S(R) lying dense in *<sup>L</sup>*2(<sup>R</sup>, *dq*) according to the prescription:

$$\pi\_q(\mathcal{O}) \mapsto \mathrm{Ad}\_{\mathbb{F}\_{\tilde{\mathbb{F}}}}(\pi\_q(\mathcal{O})) := \mathbb{T}\_{\tilde{\mathbb{F}}} \pi\_q(\mathcal{O}) \, \mathbb{T}\_{\tilde{\mathbb{F}}}^{\mathfrak{t}} \qquad \Psi \mapsto \mathbb{T}\_{\tilde{\mathbb{F}}} \Psi := \sum\_{n=0}^{\infty} \frac{\left(i \pi\_q(\hat{\mathcal{G}})\right)^n}{n!} \Psi,\tag{27}$$

where we used the abbreviation Γ ˆ *ξ* := *<sup>π</sup>q*(Γ<sup>ˆ</sup> *ξ* ) to keep our notation compact. Let us briefly check that the the adjoint action of Γ ˆ *ξ* on *<sup>π</sup>q*(*q*<sup>ˆ</sup>) and *<sup>π</sup>q*(*p*<sup>ˆ</sup>) is consistent. We have:

$$\operatorname{Ad}\_{\mathbb{T}\_{\xi}}(\pi\_{\mathfrak{q}}(\vec{\eta})) = \sum\_{n=0}^{\infty} \frac{(-i^2)^n}{n!} \left( \ln(\xi) \right)^n \pi\_{\mathfrak{q}}(\vec{\eta}) = \xi \pi\_{\mathfrak{q}}(\vec{\eta}),\tag{28}$$

where the iterated commutator [*<sup>π</sup>q*(*A*<sup>ˆ</sup>), *<sup>π</sup>q*(*B*<sup>ˆ</sup>)](*n*) is defined similarly to the iterated Poisson bracket with an identity at the zeroth order. For *<sup>π</sup>q*(*p*<sup>ˆ</sup>) we ge<sup>t</sup> as expected:

$$\mathrm{Ad}\_{\mathbb{F}\_{\xi}}(\pi\_{\mathfrak{q}}(\mathfrak{p})) = \sum\_{n=0}^{\infty} \frac{2^{n}}{n!} \left( \ln(\mathfrak{f}) \right)^{n} \pi\_{\mathfrak{q}}(\mathfrak{p}) + \sum\_{n=0}^{\infty} \frac{(\mathfrak{f}^{2})^{2n+1}}{(2n+1)!} \left( \ln(\mathfrak{f}) \right)^{2n+1} h(\mathfrak{f}, \dot{\mathfrak{f}}) \pi\_{\mathfrak{q}}(\mathfrak{q}) = \frac{\pi\_{\mathfrak{q}}(\mathfrak{p})}{\mathfrak{f}} + \dot{\xi} \pi\_{\mathfrak{q}}(\mathfrak{q}), \quad \text{(29)}$$

which precisely corresponds to the inverse of the transformation of *q* and *p* generated by the classical Hamiltonian vector field *X*G. As discussed in Section 2.3, the dynamics of the classical theory is generated by the physical Hamiltonian *<sup>H</sup>*(**<sup>q</sup>**, **p**, *t*). Thus, we can directly consider the corresponding Schrödinger equation in the one dimensional case that is given by

$$i\frac{\partial}{\partial t}\Psi(q,t) = \frac{1}{2}\left(\pi\_{\emptyset}(\not p)^2 + \omega^2(t)\pi\_{\emptyset}(\not q)^2\right)\Psi(q,t).$$

and unitarily equivalent to the corresponding Heisenberg equations for *<sup>π</sup>q*(*q*) and *<sup>π</sup>q*(*p*). If we apply the transformation induced by Γ*ξ* on the Hamiltonian and Ψ, which is the natural choice since classically, the replacement of **q**, **p** in terms of **Q**, **P** (the inverse extended map Φ) achieved our aim of mapping *H*(*t*) to *H*0, we end up with:

Γ ˆ *<sup>ξ</sup>*12*<sup>π</sup>q*(*p*<sup>ˆ</sup>)<sup>2</sup> + *<sup>ω</sup>*<sup>2</sup>(*t*)*<sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup> − *i ∂∂t*Γˆ †*ξ*Γˆ *<sup>ξ</sup>*<sup>Ψ</sup>(*q*, *t*) = 0 ⇐⇒ 12Γ<sup>ˆ</sup> *<sup>ξ</sup><sup>π</sup>q*(*p*<sup>ˆ</sup>)<sup>2</sup> + *<sup>ω</sup>*<sup>2</sup>(*t*)*<sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup>Γ<sup>ˆ</sup> †*ξ* − *i*Γˆ *ξ ∂*Γ<sup>ˆ</sup> †*ξ ∂t* − *i ∂∂t* !Γˆ *<sup>ξ</sup>*<sup>Ψ</sup>(*q*, *t*) = 0 ⇐⇒ 12*<sup>π</sup>q*(*p*<sup>ˆ</sup>)<sup>2</sup> *ξ*2 + *<sup>ξ</sup>ω*<sup>2</sup>(*t*)*<sup>ξ</sup>* + ¨*<sup>ξ</sup><sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup> − *i ∂∂t* !Γˆ *<sup>ξ</sup>*<sup>Ψ</sup>(*q*, *t*) = 0 ⇐⇒ 12*ξ*<sup>2</sup> *<sup>π</sup>q*(*p*<sup>ˆ</sup>)<sup>2</sup> + *<sup>ω</sup>*20*<sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup> − *i ∂∂t* !Γˆ *<sup>ξ</sup>*<sup>Ψ</sup>(*q*, *t*) = 0 (30)

$$\iff \qquad \left[\frac{1}{\xi^2}\hat{H}\_0 - i\frac{\partial}{\partial t}\right]\hat{\Gamma}\_{\xi}\Psi(q,t) = 0,\tag{31}$$

with *H* ˆ 0 := 12 *<sup>π</sup>q*(*p*<sup>ˆ</sup>)<sup>2</sup> + *<sup>ω</sup>*20*<sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup>. In the second step, we used the *t*-derivative of the one-parameter family of transformations Γ ˆ † *ξ* , which has already been derived in References [14,17]. We can rediscover their result by using the explicit form of the generator *<sup>π</sup>q*(G<sup>ˆ</sup>) using (26) and a Baker-Campbell-Hausdorff decomposition of Γˆ *ξ* in the position representation. Later a similar but slightly generalized procedure for the occupation number representation will be discussed in Section 3.2. We realize that *H* ˆ 0 in Equation (31) does *not* carry any explicit time dependence, hence we can construct a solution of the Schrödinger equation in (31) by integration. Further note that the inverse square of the time scaling function *ξ*(*t*) precisely corresponds to the Lagrange multiplier *<sup>λ</sup>*(*s*) = d*t*/d*s* that is involved in the extended classical Hamiltonian (14). Given this result we can now give an explicit solution of the Schrödinger equation as was already shown in Reference [17]:

$$\Psi(q,t) = \hat{\Gamma}\_{\xi}^{\dagger} \exp\left\{-i\pi\_{\emptyset}(\hat{H}\_{0}) \int\_{t\_{0}}^{t} \frac{\mathbf{d}\tau}{\mathcal{J}^{2}(\tau)}\right\} \hat{\Gamma}\_{\xi,0} \Psi(q,t\_{0}), \quad \Psi(q,t\_{0}) \in \mathcal{S}(\mathbb{R})\_{t\_{0}} \tag{32}$$

with <sup>S</sup>(R)*<sup>t</sup>*0 denoting a one-parameter family of Schwarz spaces, each corresponding to a different initial time *t*0. In a cosmological context, this behavior is a very natural one, as the *instantaneous vacuum* on cosmological backgrounds shows an analogous behavior. Using that ˆ *ILR*(*t*) = Γˆ †*ξ H*ˆ 0Γˆ *ξ* , the time evolution in Equation (32) can also be rewritten as:

$$\mathcal{U}(t\_0, t) = \mathbb{I}\_{\xi}^{\mathsf{t}} \exp \left\{ -i\pi\_q(\boldsymbol{\varOmega}\_0) \int\_{t\_0}^{t} \frac{\mathrm{d}\tau}{\xi^2(\tau)} \right\} \mathbb{I}\_{\xi, 0} = \exp \left\{ -i\pi\_q(\boldsymbol{I}\_{LR}) \int\_{t\_0}^{t} \frac{\mathrm{d}\tau}{\xi^2(\tau)} \right\} \mathbb{I}\_{\xi}^{\mathsf{t}} \mathbb{I}\_{\xi, 0} \tag{33}$$

At this point let us further discuss the result in the quantum theory: Firstly, the integrand in the exponential corresponds exactly to our time-rescaling transformation in (11) that we naturally obtained in the extended phase space approach of the classical theory. Secondly, if we compare the result here to that in Reference [13], they use the eigenstates of the Lewis-Riesenfeld invariant multiplied by a phase factor to construct the solutions of the time-dependent Schrödinger equation. Now, if Γ ˆ † *ξ*Γ ˆ *<sup>ξ</sup>*,0<sup>Ψ</sup>(*q*, *<sup>t</sup>*0) corresponds to an eigenstate of the Lewis-Riesenfeld invariant, then this reproduces precisely the phase factor that was introduced in Reference [13] in a rather ad hoc manner. In fact, it can be easily shown that Γ ˆ † *<sup>ξ</sup>*Ψ0(*q*, *<sup>t</sup>*0) for the time-independent vacuum Ψ0 corresponds to the time-dependent vacuum state of the Lewis-Riesenfeld invariant as we will see later. The expression in Equation (33) corresponds to the unique time evolution operator, that is the Dyson series associated to the time-dependent Hamiltonian *H* ˆ (*t*), since it satisfies identical initial conditions. Moreover, *U*ˆ (*<sup>t</sup>*0, *t*) is closely related to

the unitary operator found in Reference [14] (see the equation above (3.15) in that reference). In our framework, it is very natural to find the time-independent Hamiltonian in the central exponential operator on the left-hand-side of (33). The reason for this is twofold: Firstly, the extended canonical transformation Φ maps the Hamiltonian *H*ˆ (*t*) into the time-independent one *H*ˆ 0 by transforming the Schrödinger equation via Γˆ *ξ* . Secondly, the time-rescaling that is used in the extended phase space appears as a Lagrange multiplier in the extended Hamiltonian constraint and consequently as the integrand in the time-evolution operator. Lastly, let us mention that compared to Reference [14] we use a slightly different Ermakov equation here because the prefactor of *ξ*−<sup>3</sup>(*t*) in the Ermakov equation in (5) corresponds to the squared frequency *ω*<sup>2</sup> 0 of the time-independent oscillator. In the prospects of a field theoretical treatment of this transformation, it is rather unnatural to map every time-dependent mode *<sup>ω</sup>***k**(*t*) onto the Minkowski case *ω*(0) **k** = 1 for all **k** as done in Reference [14]. As we will discuss later on, our choice of mapping *<sup>ω</sup>***k**(*t*) onto *ω*(0) **k** = *k* is of advantage when we analyze the implementation of the unitary map on the bosonic Fock space in Section 4.
