Protected SL(2,ℝ) symmetry in quantum cosmology

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

We show that the dynamics of Lorentzian geometries, associated with the semiclassical solutions of the Wheeler-DeWitt equation in the Hartle-Hawking minisuperspace model with no-boundary conditions, is chaotic due to homoclinic phenomena, and its basic features can be topologically characterized by the transversal crossings of homoclinic cylinders in the phase space of the model. The invariant manifolds of the dynamics of the classical Hamiltonian are the periodic orbits of the center manifold, a fractal set of homoclinic orbits which have an oscillatory approach to the periodic orbits of the center manifold, and an infinite set of unstable periodic orbits with arbitrarily large periods. The latter orbits have the homoclinic orbits as an accumulation set. These invariant manifolds may be considered to describe classical universes, with initial conditions determined by the Hartle-Hawking no-boundary proposal. We discuss an estimate for the amplitudes of semiclassical states (corresponding to distinct classical solutions) contributing to the quantum state of the early universe.

In this paper we consider the stability of some inflating brane-world models in quantum cosmology. It is shown that whereas the singular model based on the construction of inflating branes from Euclidean five-dimensional anti-de Sitter space is unstable to tensorial cosmological perturbations in the bulk, the nonsingular model which uses a five-dimensional asymptotically anti-de Sitter wormhole to construct the inflating branes is stable to these perturbations.

The flat, homogeneous, and isotropic universe with a massless scalar field is a paradigmatic model in loop quantum cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remain some aspects of its quantization which deserve a more detailed discussion. These aspects include the kinematical resolution of the cosmological singularity, the precise relation between the solutions of the densitized and nondensitized versions of the quantum Hamiltonian constraint, the possibility of identifying superselection sectors which are as simple as possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit associated with the theory in those sectors. We propose an alternative operator to represent the Hamiltonian constraint which is specially suitable to deal with all these issues in a detailed and satisfactory way. In particular, with our constraint operator, the singularity decouples in the kinematical Hilbert space and can be removed already at this level. Thanks to this fact, we can densitize the quantum Hamiltonian constraint in a well-controlled manner. Additionally, together with the physical observables, this constraint superselects simple sectors for the universe volume, with a discrete support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined WDW limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the classical singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes---focusing on states with a marked semiclassical behavior---or for a simplified model.

We study the polymer quantization of a homogeneous massive scalar field in the early Universe using a prescription inequivalent to those previously appearing in the literature. Specifically, we assume a Hilbert space for which the scalar field momentum is well defined but its amplitude is not. This is closer in spirit to the quantization scheme of loop quantum gravity, in which no unique configuration operator exists. We show that in the semiclassical approximation, the main effect of this polymer quantization scheme is to compactify the phase space of chaotic inflation in the field amplitude direction. This gives rise to an effective scalar potential closely resembling that of hybrid natural inflation. Unlike polymer schemes in which the scalar field amplitude is well defined, the semiclassical dynamics involves a past cosmological singularity; i.e., this approach does not mitigate the big bang.

The Einstein equations for an isotropic and homogeneous\nFriedmann--Robertson--Walker Universe in the presence of a quintessence scalar\nfield are shown to be described in a unified way, formally identical to the\ndynamics of a relativistic particle moving on a two--dimensional spacetime. The\ncorrect Lagrangian for the system is presented and used to construct a spinor\nquantum cosmology theory using Breit's prescription. The theory is\nsupersymmetric when written in Majorana representation. The spinor components\ninteract through a potential that unifies spacetime curvature and quintessence.\nAn exact supersymmetric solution for $k=0$ case is exhibited. This quantum\ncosmology model may be interpreted as a theory of interacting universes.\n

As a probe to explore the ability of invisibility cloaks to conceal objects in the quantum mechanics domain, we study the spontaneous emission rate of an excited two-level atom in the vicinity of an ideal invisibility cloaking. On this base, first, a canonical quantization scheme is presented for the electromagnetic field interacting with atomic systems in an anisotropic, inhomogeneous, and absorbing magnetodielectric medium which can suitably be used for studying the influence of arbitrary invisibility cloak on the atomic radiative properties. The time dependence of the atomic subsystem is obtained in the Schrodinger picture. By introducing a modified set of the spherical wave-vector functions, the Green tensor of the system is calculated via exact and discrete methods. In this formalism, the decay rate and as well the emission pattern of the aforementioned atom are computed analytically for both weak and strong coupling interaction, and then numerically calculations are done to demonstrate the performances of cloaking in the quantum mechanics domain. Special attention is paid to different possible orientations and locations of the atomic system near the spherical invisibility cloaking. Results in the presence and the absence of the invisibility cloak are compared. We find that the cloak works very well far from its resonance frequency to conceal a macroscopic object, whereas at near the resonance frequency the object is more visible than the situation where the object is not covered by the cloak.

A bstract In this note, we develop a prescription for describing BPS black hole microstates in terms of a holographic conformal quantum mechanics (CQM) model dual to the near-horizon AdS 2 geometry of the black hole. We use 1/2 BPS small black holes in a 4D $$ \mathcal{N} $$ N = 4 toroidal heterotic compactification as well as 1/8 BPS large black holes in a 4D $$ \mathcal{N} $$ N = 8 Type II toroidal compactification as test cases for our approach. In each case, the SL(2, ℤ ) modular symmetries of the known generating function of the exact microstate degeneracies enables the latter to be expressed as a Rademacher series expansion, with each summand consisting of phases and a modified Bessel function of the first kind. We make a motivated ansatz that the de Alfaro-Fubini-Furlan model (DFF) coupled to a simple harmonic oscillator is a universal sector of the holographic CQM dual to the BPS black hole’s near-horizon AdS 2 geometry, and demonstrate how in both cases, the two parameters of this putative CQM, the DFF coupling as well as the oscillator frequency, exactly encode both the index and the argument of the Bessel function. Consequently, we extract the leading, logarithmic and all sub-leading power law black hole entropy contributions from calculations in the CQM. In the $$ \mathcal{N} $$ N = 4 case, the DFF ansatz is sufficient to successfully reproduce the exact microscopic generating function from the CQM.

In principio erat verbum. It seems appropriate to add the first few words of St. John's gospel to the Latin quotations that were bandied about at the beginning of the recent Workshop on Cosmology, which was held at the Fermi National Accelerator Laboratory in Batavia, Ill. This outburst of Latinity and classical scholarship, which even included a dispute over whether the word apparatus belongs to the second or the fifth declension, seemed an appropriate beginning for a conference on a subject the origins of the universe that has fascinated scholars ever since classical times. In the beginning was the word, en arche hen ho logos. One of the meanings of logos in this context is a word descriptive of the basic structure of the universe and, more than descriptive, a dynamical word that outlines and perhaps even determines the course of the history that follows the beginning. From the philosophers of ancient Hellas to the professors of physics in modern universities, most have expected that this word would be mathematical. It is the last generation or two that has expected it to be quantum mechanical. Quantum mechanics has to describe everything, said Murray Gell-Mann of Caltech in Pasadena in the opening talk of the workshop. And it became clear that by everything he meant the work of historians and crime detectives and the songs of birds as well as the motions of galaxies. But he added: questions are all murky and border on the philosophical. The main reason for the dominance of quantum mechanics was cited by James Hartle of the University of California at Santa Barbara: laws of physics are quantum mechanical; quantum cosmology is the proper framework. Nevertheless, the large objects that dominate the universe as we now see it operate according to classical mechanics. The serious problem for all of physics, and for quantum cosmology in particular, is to find some kind of linkage by which classical physics can be generated out of quantum mechanics. A similar question occurs historically: If we trace the expansion of the universe backwards we eventually come to a point in history when the universe was so small that it had to behave as a whole as a quantum mechanical system. Figuring out how the present came out of the past is the problem.

Coherent state functional integral for the minisuperspace model of loop quantum cosmology is studied. By the well-established canonical theory, the transition amplitude in the path integral representation of loop quantum cosmology with alternative dynamics can be formulated through group averaging. The effective action and Hamiltonian with higher-order quantum corrections are thus obtained. It turns out that for a nonsymmetric Hamiltonian constraint operator, the Moyal (star)-product emerges naturally in the effective Hamiltonian. For the corresponding symmetric Hamiltonian operator, the resulted effective theory implies a possible quantum cosmological effect in large scale limit in the alternative dynamical scenario, which coincides with the result in canonical approach. Moreover, the first-order modified Friedmann equation still contains the particular information of alternative dynamics and hence admits the possible phenomenological distinction between the different proposals of quantum dynamics.

Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. The lectures covered quantum mechanics for closed systems like the universe, generalized quantum mechanics, time in quantum mechanics, the quantum mechanics spacetime, and practical quantum cosmology. References have not been updated.

We sketch a quantum mechanical framework for the universe as a whole. Within that framework we propose a program for describing the ultimate origin in quantum cosmology of the quasiclassical domain of familiar experience and for characterizing the process of measurement. Predictions in quantum mechanics are made from probabilities for sets of alternative histories. Probabilities can be assigned only to sets of histories that approximately decohere. Decoherence is defined and the mechanism of decoherence is reviewed. Decoherence requires a sufficiently coarse-grained description of alternative histories of the universe. A quasiclassical domain consists of a branching set of alternative decohering histories, described by a coarse graining that is maximally refined consistent with decoherence, with individual branches that exhibit a high level of classical correlation in time. A quasiclassical domain is emergent in the universe as a consequence of the initial condition and the action function of the elementary particles. It is an important question whether all the quasiclassical domains are roughly equivalent or whether there are various essentially inequivalent ones. A measurement is a correlation with variables in a quasiclassical domain. An observer (or information gathering and utilizing system) is a complex adaptive system that has evolved to exploit the relative predictability of a quasiclassical domain. We suggest that resolution of many of the problems of interpretation presented by quantum mechanics is to be accomplished, not by further scrutiny of the subject as it applies to reproducible laboratory situations, but rather by an examination of alternative histories of the universe, stemming from its initial condition, and a study of the problem of quasiclassical domains.

The theory of a neutral vector field is studied by introducing an auxiliary scalar field interacting with it. The vector field propagator of the theory is the same as that obtained by Yang and Lee using the $\ensuremath{\xi}$-limiting process. The parameter $\ensuremath{\xi}$ now enters as an interaction parameter. We show, further, that it is a gauge parameter and the physical content of the theory is independent of its value. Using a canonical quantization scheme, it is found that the contribution of the scalar field to the energy of the system is negative, and one is forced to introduce an indefinite metric. To have a positive norm of the states for arbitrary interactions involving production of these hypothetical scalar particles, the values of $\ensuremath{\xi}$ have to be restricted so that the energy of the system is less than ${m}^{2}{\ensuremath{\xi}}^{\ensuremath{-}1}$, $m$ being the mass of the vector meson. Most of the formalism can be easily extended to the theory of charged vector mesons. The main purpose of the theory is to investigate the possibility of renormalizing the interactions of the vector field with currents whose divergences do not vanish. It is found that an extra term in the interaction Lagrangian makes the auxiliary-field equation independent of the current and hence of the additional interaction. It is shown that the $S$ matrix cannot produce these particles and that there need be no restrictions on the parameter $\ensuremath{\xi}$ for this case so long as one deals with initial states having no scalar mesons.

We study the behavior of a massless, quantized, scalar field on a two-dimensional cylinder spacetime as it responds to the time-dependent evolution of a Mamev-Trunov potential of the form $V(x,t) = 2 \xi \delta(x) \theta(-t)$. We begin by constructing mode solutions to the classical Klein-Gordon-Fock equation with potential on the whole spacetime. For a given eigen-mode solution of the IN region of the spacetime ($t<0$), we determine its evolution into the OUT region ($t>0$) through the use of a Fourier decomposition in terms of the OUT region eigen-modes. The classical system is then second quantized in the canonical quantization scheme. On the OUT region, there is a unitarily equivalent representation of the quantized field in terms of the OUT region eigen-modes, including zero-frequency modes which we also quantize in a manner which allows for their interpretation as particles in the typical sense. After determining the Bogolubov coefficients between the two representations, we study the production of quanta out of the vacuum when the potential turns off. We find that the number of "particles" created on the OUT region is finite for the standard modes, and with the usual ambiguity in the number of particles created in the zero frequency modes. We then look at the renormalized expectation value of the stress-energy-tensor on the IN and OUT regions for the IN vacuum state. We find that the resulting stress-tensor can violate the null, weak, strong, and dominant energy conditions because the standard Casimir energy-density of the cylinder spacetime is negative. Finally, we show that the same stress-tensor satisfies a quantum inequality on the OUT region.

Using recent developments in expressing one-loop partition functions in Euclidean AdS2 space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in AdS2 (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global AdS2 boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the CFT1 (CQM), each living near one of the two boundaries of global AdS2, in a state at global time τ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean AdS2 disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the AdS2/CFT1 version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in AdS2. Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian AdS2 and show that it precisely matches the boundary entanglement entropy.