Big bounce as scattering of wave function at big crunch

A gauge-invariant quantum theory of the Friedmann-Robertson-Walker (FRW) universe with dust is studied in terms of the Ashtekar variables. We use the reduced phase space quantization which has following advantages: (i) fundamental variables are all gauge invariant, (ii) there exists a physical time evolution of gauge-invariant quantities, so that the problem of time is absent and (iii) the reduced phase space can be quantized in the same manner as in ordinary quantum mechanics. In the FRW model, the dynamical components of the Ashtekar variables are given by a single quantity $p$ and its conjugate momentum, where $p$ is related to the scale factor $a$ as $a\propto \sqrt{|p|}$ and its sign gives the orientation of triads. We solve a scattering problem in terms of ingoing and outgoing energy eigenstates. We show that the incident wave is reflected in rate $1/4$ and transmitted in rate $3/4$ at the classical singularity $p=0$. Analyzing the dynamics of a wave packet, we show that the classical initial singularity is replaced by a big bounce in quantum theory. A possible interpretation of the result is that the wave function of the universe has been in a superposition of states representing right-handed and left-handed systems before the big bounce.Comment: 9 pages, 4 figure

Black hole and baby universe in a thin film of 3He-A

Condensed matter black hole analogues may provide guidance in grappling with difficult questions about the role of short distance physics in the Hawking effect. These questions bear on the very existence of Hawking radiation, the correlations it may or may not carry, the nature of black hole entropy, and the possible loss of information when a black hole evaporates. We describe a model of black hole formation and evaporation and the loss of information to a disconnected universe in a thin film of 3He-A, and we explain why the existence of Hawking radiation has not yet been demonstrated in this model. [We would like this article to be accessible to researchers in both condensed matter and gravitational physics, hence we include more than the usual amount of introductory material.]Comment: 21 pages, Chapter for book, "Artificial Black Holes", eds. M. Novello, M. Visser, and G. Volovik (World Scientific, 2002), based on a talk by TJ at the Workshop on Analog Models of General Relativity, held at CBPF in Rio de Janeiro, October 16-20, 200

Disturbance by optimal discrimination

We discuss the disturbance by measurements which unambiguously discriminate between given candidate states. We prove that such an optimal measurement necessarily changes distinguishable states indistinguishable when the inconclusive outcome is obtained. The result was previously shown by Chefles~[Phys. Lett. A 239, 339 (1998)] under restrictions on the class of quantum measurements and on the definition of optimality. Our theorems remove these restrictions and are also applicable to infinitely many candidate states. Combining with our previous results, one can obtain concrete mathematical conditions for the resulting states. The method may have a wide variety of applications in contexts other than state discrimination

Gauge-invariant construction of quantum cosmology

We present and analyze a gauge-invariant quantum theory of the Friedmann-Robertson-Walker universe with dust. We construct the reduced phase space spanned by gauge-invariant quantities by using the so-called relational formalism at the classical level. The reduced phase space thereby obtained can be quantized in the same manner as an ordinary mechanical system. We carry out the quantization and obtain the Schr\"{o}dinger equation. This quantization procedure realizes a possible resolution to the problem of time and observables in canonical quantum gravity. We analyze the classical initial singularity of the theory by evolving a wave packet backward in time and evaluating the expectation value of the scale factor. It is shown that the initial singularity of the Universe is avoided by the quantum gravitational effects.Comment: 8 pages, 7 figures, accepted for publication in Phys. Rev.

Causal Feature of Central Singularity and Gravitational Mass

Mass of singularity is defined, and its relation to whether the singularity is spacelike, timelike or null is discussed for spherically symmetric spacetimes. It is shown that if the mass of singularity is positive (negative) the singularity is non-timelike (non-spacelike). The connection between the sign of the mass and the force on a particle is also discussed.Comment: 10 pages REVTeX 3.0, TIT/HEP-246/COSMO-4

Hamiltonian structures for compact homogeneous universes

Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the \Teich parameters, parameterizing the purely global geometry. One of the key ingredients of our arguments is a suitable mathematical expression for quotient manifolds, where the universal cover metric carries all the degrees of freedom of geometrical variations, i.e., the covering group is fixed. We discuss general problems concerned with the use of this expression in the context of general relativity, and demonstrate the reduction of the Hamiltonians for some examples. For our models, all the dynamical degrees of freedom in Hamiltonian view are unambiguously interpretable as geometrical deformations, in contrast to the conventional open models.Comment: 27 pages including 2 figures. REVTeX and epsf package

Renormalization group and critical behaviour in gravitational collapse

We present a general framework for understanding and analyzing critical behaviour in gravitational collapse. We adopt the method of renormalization group, which has the following advantages. (1) It provides a natural explanation for various types of universality and scaling observed in numerical studies. In particular, universality in initial data space and universality for different models are understood in a unified way. (2) It enables us to perform a detailed analysis of time evolution beyond linear perturbation, by providing rigorous controls on nonlinear terms. Under physically reasonable assumptions we prove: (1) Uniqueness of the relevant mode around a fixed point implies universality in initial data space. (2) The critical exponent $\beta_{BH}$ and the unique positive eigenvalue $\kappa$ of the relevant mode is exactly related by $\beta_{BH} = \beta /\kappa$, where $\beta$ is a scaling exponent. (3) The above (1) and (2) hold also for discretely self-similar case (replacing ``fixed point'' with ``limit cycle''). (4) Universality for diffent models holds under a certain condition. According to the framework, we carry out a rather complete (though not mathematically rigorous) analysis for perfect fluids with pressure proportional to density, in a wide range of the adiabatic index $\gamma$. The uniqueness of the relevant mode around a fixed point is established by Lyapunov analyses. This shows that the critical phenomena occurs not only for the radiation fluid but also for perfect fluids with $1 < \gamma \lesssim 1.88$. The accurate values of critical exponents are calculated for the models.Comment: ReVTeX, 42 pages with 8 embedded PS figures using "boxedeps.tex." This is a replacement, which (1) presents more straightforward presentation by sending some detailed "proofs" into the appendices, (2) corrects some minor errors in the first versio

Hamiltonians for Compact Homogeneous Universes

We briefly show how we can obtain Hamiltonians for spatially compact locally homogeneous vacuum spacetimes. The dynamical variables are categorized into the curvature parameters and the Teichm\"{u}ller parameters. While the Teichm\"{u}ller parameters usually parameterise the covering group of the spatial sections, we utilise another suitable parameterization where the universal cover metric carries all the dynamical variables and with this we reduce the Hamiltonians. For our models, all dynamical variables possess their clear geometrical meaning, in contrast to the conventional open models.}Comment: 3 pages, LaTeX, uses mprocl.sty. Talk given at the Eighth Marcel Grossmann Meeting on General Relativity, 22-27 June 1997, Jerusale

Killing Tensors and Conserved Quantities of a Relativistic Particle in External Fields

We generalize Killing equations to a test particle system which is subjected to external force. We relax the conservation condition by virtue of reparametrization invariance of a particle orbit. As a result, we obtain generalized Killing equations which have hierarchical structure on the top of which a conformal Killing equation exists.Comment: Proceedings of the 12th Marcel Grossmann Meeting on General Relativity (MG 12), Paris, France, 12-18 Jul 200

State protection by quantum control before and after noise

We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. We extend our previous result on nonexistence of "truly quantum" protocols that protect an unknown qubit state against the depolarizing noise better than "classical" ones [Phys. Rev. A, 95, 022321 (2017)] in two directions. First, we show that the statement is also true in any finite-dimensional Hilbert spaces, which was previously conjectured, the optimal protocol is either the do nothing protocol or the discriminate and reprepare protocol, depending on the strength of the noise. Second, in the case of a qubit, we show that essentially the same conclusion holds for any unital noise. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory.Comment: 8 pages, 2 figure