Hamiltonian diagonalization in hybrid quantum cosmology

Abstract

We explore the possibility of selecting a natural vacuum state for scalar and tensor gauge-invariant cosmological perturbations in the context of hybrid quantum cosmology, by identifying those variables for the description of the perturbations that display a dynamical behavior adapted in a specific way to the evolution of the entire cosmology. We make use of a canonical formulation of the whole of the cosmological system (background geometry and perturbations) in which the perturbative gauge-invariant degrees of freedom are identified as canonical variables. Introducing background-dependent linear canonical transformations that respect the spatial symmetries of the background on these perturbations and completing those canonical transformations for the entire system, we are able to characterize a generic collection of annihilation and creationlike variables that obey the dynamics dictated by a respective collection of Hamiltonians. We then impose that such Hamiltonians possess no self-interaction terms so that, in a Fock representation with normal ordering, they act diagonally on the basis of n-particle states. This leads to a semilinear first-order partial differential equation with respect to the background for the coefficients that define the annihilation and creationlike variables for all Fourier modes, as well as to a very precise ultraviolet characterization of them. Such first-order equation contains, in the imaginary part of its complex solutions, the complicated second-order field equation that typically arises for the time-dependent frequency of the perturbations in the context of quantum field theory in curved spacetimes. We check that the asymptotic knowledge acquired allows one to select the standard vacua in Minkowski and de Sitter spacetimes. Finally, we discuss the relation of our vacuum and the standard

Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology

“…In particular, it displays the ultraviolet behavior expected to guarantee Hadamard-like properties. This result supports the potential robustness of our criterion to select a privileged fermionic vacuum state in FLRW cosmologies, specially considering that the very same criterion of asymptotic diagonalization has succeeded in predicting the BD vacuum for the well-known cases of scalar and tensor perturbations [42]. On the other hand, our work shows that, at least in certain cases, the criterion employed in hybrid quantum cosmology to (i) determine a dynamical splitting between the homogeneous and inhomogeneous sectors in phase space, and (ii) select a Fock representation for the inhomogeneities, can indeed result into a complete removal of both types of ambiguities, even when this criterion was initially based solely on ultraviolet considerations.…”

“…Such a criterion turns out to completely fix, in an asymptotic expansion, the dynamical definition of those operators, expansion from which one may hope to uniquely determine them globally. In the restricted context of QFT in a de Sitter background cosmology, this hope was actually realized for the scalar perturbations, resulting in the specification of the well-known BD vacuum [42]. In this paper, we show in detail how the criterion of asymptotic diagonalization also serves to uniquely fix the choice of vacuum in de Sitter for fermionic perturbations of the Dirac type.…”

“…This criterion of choice has been further restricted in the context of fermionic perturbations in hybrid quantum cosmology, in such a way that certain effective backreaction to the Hamiltonian constraint does not develop divergences, without the need of introducing any regularization procedure [41]. Finally, and motivated by these previous conditions, the most recent works on the theoretical formulation of the hybrid quantization of both fermionic and scalar cosmological perturbations have proposed an approach that aims to remove all the ambiguities (up to irrelevant phases) in the choice of the canonical algebra associated with the perturbations, that is going to be quantizedà la Fock [22,42]. The approach tries and constructs a quantum description of the inhomogeneities such that the local structure of the Hamiltonian contains no self-interaction contributions in terms of the annihilation and creation operators, and thus it is asymptotically diagonalized in the ultraviolet regime of short scales.…”

Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology

“…In this sense, one can interpret that this vacuum and the corresponding splitting of degrees of freedom in the hybrid quantization approach are those that get best adapted to the cosmological evolution. Finally, it is remarkable that this criterion of asymptotic diagonalization reproduces the standard choices of vacuum state in well-understood situations, within the scheme of QFT in a curved classical background [39,40]. This happens e.g., in the case of a flat spacetime, as well as for a de Sitter cosmology, scenario where the Bunch-Davies state is a natural vacuum [41].…”

Fock quantization of the Dirac field in hybrid quantum cosmology: Relation with adiabatic states

“…Indeed, this was first done for fermionic perturbations in inflationary cosmologies [30] in an attempt to find a representation such that some kind of quantum backreaction on the homogeneous cosmological sector remains finite without the need of a regularization scheme, and that one gets a Hamiltonian constraint that is properly defined on the dense set of the Fock space spanned by the n-particle/antiparticle states [31]. Additionally, it is possible to further refine the description of the inflationary cosmology and arrive at a recurrence relation by which the dynamics of the annihilation and creation operators that describe the fermionic, scalar, and tensor perturbations become diagonal in the asymptotic limit of infinitely large particle/antiparticle wave numbers [32,33].…”