314 research outputs found
Initial Conditions for a Universe
In physical theories, boundary or initial conditions play the role of
selecting special situations which can be described by a theory with its
general laws. Cosmology has long been suspected to be different in that its
fundamental theory should explain the fact that we can observe only one
particular realization. This is not realized, however, in the classical
formulation and in its conventional quantization; the situation is even worse
due to the singularity problem. In recent years, a new formulation of quantum
cosmology has been developed which is based on quantum geometry, a candidate
for a theory of quantum gravity. Here, the dynamical law and initial conditions
turn out to be linked intimately, in combination with a solution of the
singularity problem.Comment: 7 pages, this essay was awarded First Prize in the Gravity Research
Foundation Essay Contest 200
Comments on the Sign and Other Aspects of Semiclassical Casimir Energies
The Casimir energy of a massless scalar field is semiclassically given by
contributions due to classical periodic rays. The required subtractions in the
spectral density are determined explicitly. The so defined semiclassical
Casimir energy coincides with that obtained using zeta function regularization
in the cases studied. Poles in the analytic continuation of zeta function
regularization are related to non-universal subtractions in the spectral
density. The sign of the Casimir energy of a scalar field on a smooth manifold
is estimated by the sign of the contribution due to the shortest periodic rays
only. Demanding continuity of the Casimir energy under small deformations of
the manifold, the method is extended to integrable systems. The Casimir energy
of a massless scalar field on a manifold with boundaries includes contributions
due to periodic rays that lie entirely within the boundaries. These
contributions in general depend on the boundary conditions. Although the
Casimir energy due to a massless scalar field may be sensitive to the physical
dimensions of manifolds with boundary, its sign can in favorable cases be
inferred without explicit calculation of the Casimir energy.Comment: 39 pages, no figures, references added, some correction
Absence of Singularity in Loop Quantum Cosmology
It is shown that the cosmological singularity in isotropic minisuperspaces is
naturally removed by quantum geometry. Already at the kinematical level, this
is indicated by the fact that the inverse scale factor is represented by a
bounded operator even though the classical quantity diverges at the initial
singularity. The full demonstation comes from an analysis of quantum dynamics.
Because of quantum geometry, the quantum evolution occurs in discrete time
steps and does not break down when the volume becomes zero. Instead, space-time
can be extended to a branch preceding the classical singularity independently
of the matter coupled to the model. For large volume the correct semiclassical
behavior is obtained.Comment: 4 pages, 1 figur
A Feynman-Kac Formula for Anticommuting Brownian Motion
Motivated by application to quantum physics, anticommuting analogues of
Wiener measure and Brownian motion are constructed. The corresponding Ito
integrals are defined and the existence and uniqueness of solutions to a class
of stochastic differential equations is established. This machinery is used to
provide a Feynman-Kac formula for a class of Hamiltonians. Several specific
examples are considered.Comment: 21 page
The Inverse Scale Factor in Isotropic Quantum Geometry
The inverse scale factor, which in classical cosmological models diverges at
the singularity, is quantized in isotropic models of loop quantum cosmology by
using techniques which have been developed in quantum geometry for a
quantization of general relativity. This procedure results in a bounded
operator which is diagonalizable simultaneously with the volume operator and
whose eigenvalues are determined explicitly. For large scale factors (in fact,
up to a scale factor slightly above the Planck length) the eigenvalues are
close to the classical expectation, whereas below the Planck length there are
large deviations leading to a non-diverging behavior of the inverse scale
factor even though the scale factor has vanishing eigenvalues. This is a first
indication that the classical singularity is better behaved in loop quantum
cosmology.Comment: 17 pages, 4 figure
Isotropic Loop Quantum Cosmology with Matter
A free massless scalar field is coupled to homogeneous and isotropic loop
quantum cosmology. The coupled model is investigated in the vicinity of the
classical singularity, where discreteness is essential and where the quantum
model is non-singular, as well as in the regime of large volumes, where it
displays the expected semiclassical features. The particular matter content
(massless, free scalar) is chosen to illustrate how the discrete structure
regulates pathological behavior caused by kinetic terms of matter Hamiltonians
(which in standard quantum cosmology lead to wave functions with an infinite
number of oscillations near the classical singularity). Due to this
modification of the small volume behavior the dynamical initial conditions of
loop quantum cosmology are seen to provide a meaningful generalization of
DeWitt's initial condition.Comment: 18 pages, 4 figure
Noise Kernel in Stochastic Gravity and Stress Energy Bi-Tensor of Quantum Fields in Curved Spacetimes
The noise kernel is the vacuum expectation value of the (operator-valued)
stress-energy bi-tensor which describes the fluctuations of a quantum field in
curved spacetimes. It plays the role in stochastic semiclassical gravity based
on the Einstein-Langevin equation similar to the expectation value of the
stress-energy tensor in semiclassical gravity based on the semiclassical
Einstein equation. According to the stochastic gravity program, this two point
function (and by extension the higher order correlations in a hierarchy) of the
stress energy tensor possesses precious statistical mechanical information of
quantum fields in curved spacetime and, by the self-consistency required of
Einstein's equation, provides a probe into the coherence properties of the
gravity sector (as measured by the higher order correlation functions of
gravitons) and the quantum nature of spacetime. It reflects the low and medium
energy (referring to Planck energy as high energy) behavior of any viable
theory of quantum gravity, including string theory. It is also useful for
calculating quantum fluctuations of fields in modern theories of structure
formation and for backreaction problems in cosmological and black holes
spacetimes.
We discuss the properties of this bi-tensor with the method of
point-separation, and derive a regularized expression of the noise-kernel for a
scalar field in general curved spacetimes. One collorary of our finding is that
for a massless conformal field the trace of the noise kernel identically
vanishes. We outline how the general framework and results derived here can be
used for the calculation of noise kernels for Robertson-Walker and
Schwarzschild spacetimes.Comment: 22 Pages, RevTeX; version accepted for publication in PR
Dynamical Initial Conditions in Quantum Cosmology
Loop quantum cosmology is shown to provide both the dynamical law and initial
conditions for the wave function of a universe by one discrete evolution
equation. Accompanied by the condition that semiclassical behavior is obtained
at large volume, a unique wave function is predicted.Comment: 4 pages, 1 figur
Loop Quantum Cosmology II: Volume Operators
Volume operators measuring the total volume of space in a loop quantum theory
of cosmological models are constructed. In the case of models with rotational
symmetry an investigation of the Higgs constraint imposed on the reduced
connection variables is necessary, a complete solution of which is given for
isotropic models; in this case the volume spectrum can be calculated
explicitly. It is observed that the stronger the symmetry conditions are the
smaller is the volume spectrum, which can be interpreted as level splitting due
to broken symmetries. Some implications for quantum cosmology are presented.Comment: 21 page
A Massive Renormalizable Abelian Gauge Theory in 2+1 Dimensions
The standard formulation of a massive Abelian vector field in
dimensions involves a Maxwell kinetic term plus a Chern-Simons mass term; in
its place we consider a Chern-Simons kinetic term plus a Stuekelberg mass term.
In this latter model, we still have a massive vector field, but now the
interaction with a charged spinor field is renormalizable (as opposed to super
renormalizable). By choosing an appropriate gauge fixing term, the Stuekelberg
auxiliary scalar field decouples from the vector field. The one-loop spinor
self energy is computed using operator regularization, a technique which
respects the three dimensional character of the antisymmetric tensor
. This method is used to evaluate the vector self
energy to two-loop order; it is found to vanish showing that the beta function
is zero to two-loop order. The canonical structure of the model is examined
using the Dirac constraint formalism.Comment: LaTeX, 17 pages, expanded reference list and discussion of
relationship to previous wor
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