Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

Abstract

We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quaside Sitter spacetime. Our main interest lies in the question to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation and we show that its generalization to Fock space has to be chosen appropriately in order that the Shale-Stinespring condition is not violated, where we also compare our results to already existing ones in the literature. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution to the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.

Dirac’s Formalism for Time-Dependent Hamiltonian Systems in the Extended Phase Space

“…In Section 3, we studied the canonical transformation (35) which is a generalization of the Struckmeier transformation [19]. We showed that imposing the energy conservation law in the new variables gives rise to the equations (45). These equations admit solutions in which the mass of the system in the new variables can be considered as a function of the coordinate Q, that is, m 0 (Q).…”

“…In order to fixÃ(T), we use the third equation in (45) which, after inserting the former definitions, takes the form…”

“…The aim is to "transfer" the timedependence of the Hamilton equations to the coefficients of this canonical transformation, rendering the final Hamilton equations time-independent. As a result, we obtain three auxiliary equations given in (45). The first two of them can be easily solved, while the third gives rise to the Equation (50), which is a generalization of the auxiliary Equation (3).…”

“…This section considers the path integral formulation of this system using the action given in (4). It is worth mentioning that we can also implement Dirac's quantization procedure; using operators, this way of quantization was implemented in [45]. In our case, the extended phase space formalism allows us to consider the transformation (35) as a canonical transformation on each of the infinitesimal intervals in which the quantum-extended phase space is split.…”

Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass

“…For the three-body problem in the spherical universe, their perturbation theory analysis showed that the rate of precession of two small and nearly circular solutions of identical particles is proportional to the square root of their initial distance and inversely proportional to the square of the radius of the universe [5]. In addition, the three-body problem is also widely used in the evolution of binary systems [6] and the dynamic analysis of binary asteroids [7,8], as well as other fields in the universe such as dark matter, galaxies, GW170817 (GW is short for gravitational wave), and Mukhanov-Sasaki Hamiltonian dynamics, and so forth (see References [9][10][11][12][13] for more information).…”