83 research outputs found
Relational evolution of the degrees of freedom of generally covariant quantum theories
We study the classical and quantum dynamics of generally covariant theories
with vanishing a Hamiltonian and with a finite number of degrees of freedom. In
particular, the geometric meaning of the full solution of the relational
evolution of the degrees of freedom is displayed, which means the determination
of the total number of evolving constants of motion required. Also a method to
find evolving constants is proposed. The generalized Heinsenberg picture needs
M time variables, as opposed to the Heisenberg picture of standard quantum
mechanics where one time variable t is enough. As an application, we study the
parameterized harmonic oscillator and the SL(2,R) model with one physical
degree of freedom that mimics the constraint structure of general relativity
where a Schrodinger equation emerges in its quantum dynamics.Comment: 25 pages, no figures, Latex file. Revised versio
Bohmian arrival time without trajectories
The computation of detection probabilities and arrival time distributions
within Bohmian mechanics in general needs the explicit knowledge of a relevant
sample of trajectories. Here it is shown how for one-dimensional systems and
rigid inertial detectors these quantities can be computed without calculating
any trajectories. An expression in terms of the wave function and its spatial
derivative, both restricted to the boundary of the detector's spacetime volume,
is derived for the general case, where the probability current at the
detector's boundary may vary its sign.Comment: 20 pages, 12 figures; v2: reference added, extended introduction,
published versio
Relational time in generally covariant quantum systems: four models
We analize the relational quantum evolution of generally covariant systems in
terms of Rovelli's evolving constants of motion and the generalized Heisenberg
picture. In order to have a well defined evolution, and a consistent quantum
theory, evolving constants must be self-adjoint operators. We show that this
condition imposes strong restrictions to the choices of the clock variables. We
analize four cases. The first one is non- relativistic quantum mechanics in
parametrized form. We show that, for the free particle case, the standard
choice of time is the only one leading to self-adjoint evolving constants.
Secondly, we study the relativistic case. We show that the resulting quantum
theory is the free particle representation of the Klein Gordon equation in
which the position is a perfectly well defined quantum observable. The
admissible choices of clock variables are the ones leading to space-like
simultaneity surfaces. In order to mimic the structure of General Relativity we
study the SL(2R) model with two Hamiltonian constraints. The evolving constants
depend in this case on three independent variables. We show that it is possible
to find clock variables and inner products leading to a consistent quantum
theory. Finally, we discuss the quantization of a constrained model having a
compact constraint surface. All the models considered may be consistently
quantized, although some of them do not admit any time choice such that the
equal time surfaces are transversal to the orbits.Comment: 18 pages, revtex fil
The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity
The quantum behaviour of weak gravitational fields admits an adequate, albeit
approximate, description by those graviton states in which the expectation
values and fluctuations of the linearised gravitational field are small. Such
states must approximate corresponding states in full quantum gravity. We
analyse the nature of this approximation for the graviton vacuum state in the
context of kinematical Loop Quantum Gravity (LQG) wherein the constraints are
ignored. We identify the graviton vacuum state with kinematically
non-normalizable, distributional states in LQG by demanding that relations
between linearised operator actions on the former are mirrored by those of
their non-linear counterparts on the latter. We define a semi- norm on the
space of kinematical distributions and show that the identification is
approximate upto distributions which are small in this semi-norm. We argue that
our candidate states are annihilated by the linearised constraints (expressed
as operators in the full theory) to leading order in the parameter
characterising the approximation. This suggests the possibility, in a scheme
such as ours, of solving the full constraints order by order in this parameter.
The main drawback of our considerations is that they depend on certain
auxilliary constructions which, though mathematically well defined, do not
arise from physical insight. Our work is an attempt to implement an earlier
proposal of Iwasaki and Rovelli.Comment: 44 pages, no figure
Spin dependent observable effect for free particles using the arrival time distribution
The mean arrival time of free particles is computed using the quantum
probability current. This is uniquely determined in the non-relativistic limit
of Dirac equation, although the Schroedinger probability current has an
inherent non-uniqueness. Since the Dirac probability current involves a
spin-dependent term, an arrival time distribution based on the probability
current shows an observable spin-dependent effect, even for free particles.
This arises essentially from relativistic quantum dynamics, but persists even
in the non-relativistic regime.Comment: 5 Latex pages, 2.eps figures; discussions sharpened and references
added; accepted for publication in Physical Review
Loop Quantum Gravity: An Inside View
This is a (relatively) non -- technical summary of the status of the quantum
dynamics in Loop Quantum Gravity (LQG). We explain in detail the historical
evolution of the subject and why the results obtained so far are non --
trivial. The present text can be viewed in part as a response to an article by
Nicolai, Peeters and Zamaklar [hep-th/0501114]. We also explain why certain no
go conclusions drawn from a mathematically correct calculation in a recent
paper by Helling et al [hep-th/0409182] are physically incorrect.Comment: 58 pages, no figure
Loop Quantum Gravity
The problem of finding the quantum theory of the gravitational field, and
thus understanding what is quantum spacetime, is still open. One of the most
active of the current approaches is loop quantum gravity. Loop quantum gravity
is a mathematically well-defined, non-perturbative and background independent
quantization of general relativity, with its conventional matter couplings. The
research in loop quantum gravity forms today a vast area, ranging from
mathematical foundations to physical applications. Among the most significative
results obtained are: (i) The computation of the physical spectra of
geometrical quantities such as area and volume; which yields quantitative
predictions on Planck-scale physics. (ii) A derivation of the
Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical
picture of the microstructure of quantum physical space, characterized by a
polymer-like Planck scale discreteness. This discreteness emerges naturally
from the quantum theory and provides a mathematically well-defined realization
of Wheeler's intuition of a spacetime ``foam''. Long standing open problems
within the approach (lack of a scalar product, overcompleteness of the loop
basis, implementation of reality conditions) have been fully solved. The weak
part of the approach is the treatment of the dynamics: at present there exist
several proposals, which are intensely debated. Here, I provide a general
overview of ideas, techniques, results and open problems of this candidate
theory of quantum gravity, and a guide to the relevant literature.Comment: Review paper written for the electronic journal `Living Reviews'. 34
page
Trajectories for the Wave Function of the Universe from a Simple Detector Model
Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we
consider a simple model for quantum cosmology which includes, in the total
Hamiltonian, model detectors registering whether or not the system, at any
stage in its entire history, passes through a series of regions in
configuration space. We thus derive a variety of well-defined formulas for the
probabilities for trajectories associated with the solutions to the
Wheeler-DeWitt equation. The probability distribution is peaked about classical
trajectories in configuration space. The ``measured'' wave functions still
satisfy the Wheeler-DeWitt equation, except for small corrections due to the
disturbance of the measuring device. With modified boundary conditions, the
measurement amplitudes essentially agree with an earlier result of Hartle
derived on rather different grounds. In the special case where the system is a
collection of harmonic oscillators, the interpretation of the results is aided
by the introduction of ``timeless'' coherent states -- eigenstates of the
Hamiltonian which are concentrated about entire classical trajectories.Comment: 37 pages, plain Tex. Second draft. Substantial revision
Formation of dehydroalanine, lanthionine and lysinoalanine during heat treatment of?-Lactoglobuline A
- …