26 research outputs found
Space of spaces as a metric space
In spacetime physics, we frequently need to consider a set of all spaces
(`universes') as a whole. In particular, the concept of `closeness' between
spaces is essential. However, there has been no established mathematical theory
so far which deals with a space of spaces in a suitable manner for spacetime
physics.
Based on the scheme of the spectral representation of geometry, we construct
a space of all compact Riemannian manifolds equipped with the spectral measure
of closeness. We show that this space of all spaces can be regarded as a metric
space. We also show other desirable properties of this space, such as the
partition of unity, locally-compactness and the second countability. These
facts show that this space of all spaces can be a basic arena for spacetime
physics.Comment: To appear in Communications in Mathematical Physics. 20 page
Spectral Evolution of the Universe
We derive the evolution equations for the spectra of the Universe.
Here "spectra" means the eigenvalues of the Laplacian defined on a space,
which contain the geometrical information on the space.
These equations are expected to be useful to analyze the evolution of the
geometrical structures of the Universe.
As an application, we investigate the time evolution of the spectral distance
between two Universes that are very close to each other; it is the first
necessary step for the detailed analysis of the model-fitting problem in
cosmology with the spectral scheme.
We find out a universal formula for the spectral distance between two very
close Universes, which turns out to be independent of the detailed form of the
distance nor the gravity theory. Then we investigate its time evolution with
the help of the evolution equations we derive.
We also formulate the criteria for a good cosmological model in terms of the
spectral distance.Comment: To appear in Phys. Rev.
Asymptotic Principal Values and Regularization Methods for Correlation Functions with Reflective Boundary Conditions
Smearing effect due to the spread of a probe-particle on the Brownian motion near a perfectly reflecting boundary
Quantum fluctuations of electromagnetic vacuum are investigated in a
half-space bounded by a perfectly reflecting plate by introducing a probe
described by a charged wave-packet distribution in time-direction. The
wave-packet distribution of the probe enables one to investigate the smearing
effect upon the measured vacuum fluctuations caused by the quantum nature of
the probe particle. It is shown that the wave-packet spread of the probe
particle significantly influences the measured velocity dispersion of the
probe. In particular, the asymptotic late-time behavior of its -component, , for the wave-packet case is quite different from the test
point-particle case ( is the coordinate normal to the plate). The result for
the wave-packet is \sim 1/\t^2 in the late time (\t is the
measuring time), in stead of the reported late-time behavior for a point-particle probe. This result can be quite significant
for further investigations on the measurement of vacuum fluctuations.Comment: 8 page
Partition Function for (2+1)-Dimensional Einstein Gravity
Taking (2+1)-dimensional pure Einstein gravity for arbitrary genus as a
model, we investigate the relation between the partition function formally
defined on the entire phase space and the one written in terms of the reduced
phase space. In particular the case of is analyzed in detail.
By a suitable gauge-fixing, the partition function basically reduces to
the partition function defined for the reduced system, whose dynamical
variables are . [The 's are the Teichm\"uller
parameters, and the 's are their conjugate momenta.]
As for the case of , we find out that is also related with another
reduced form, whose dynamical variables are and .
[Here is a conjugate momentum to 2-volume .] A nontrivial factor
appears in the measure in terms of this type of reduced form. The factor turns
out to be a Faddeev-Popov determinant coming from the time-reparameterization
invariance inherent in this type of formulation. Thus the relation between two
reduced forms becomes transparent even in the context of quantum theory.
Furthermore for , a factor coming from the zero-modes of a differential
operator can appear in the path-integral measure in the reduced
representation of . It depends on the path-integral domain for the shift
vector in : If it is defined to include , the nontrivial factor
does not appear. On the other hand, if the integral domain is defined to
exclude , the factor appears in the measure. This factor can depend
on the dynamical variables, typically as a function of , and can influence
the semiclassical dynamics of the (2+1)-dimensional spacetime.
These results shall be significant from the viewpoint of quantum gravity.Comment: 21 pages. To appear in Physical Review D. The discussion on the
path-integral domain for the shift vector has been adde
Switching effect upon the quantum Brownian motion near a reflecting boundary
The quantum Brownian motion of a charged particle in the electromagnetic
vacuum fluctuations is investigated near a perfectly reflecting flat boundary,
taking into account the smooth switching process in the measurement.
Constructing a smooth switching function by gluing together a plateau and the
Lorentzian switching tails, it is shown that the switching tails have a great
influence on the measurement of the Brownian motion in the quantum vacuum.
Indeed, it turns out that the result with a smooth switching function and the
one with a sudden switching function are qualitatively quite different. It is
also shown that anti-correlations between the switching tails and the main
measuring part plays an essential role in this switching effect. The switching
function can also be interpreted as a prototype of an non-equilibrium process
in a realistic measurement, so that the switching effect found here is expected
to be significant in actual applications in vacuum physics.Comment: 12 pages, 2 figures This version is just to correct the author-lis
Evolution of the discrepancy between a universe and its model
We study a fundamental issue in cosmology: Whether we can rely on a
cosmological model to understand the real history of the Universe. This
fundamental, still unresolved issue is often called the ``model-fitting problem
(or averaging problem) in cosmology''. Here we analyze this issue with the help
of the spectral scheme prepared in the preceding studies.
Choosing two specific spatial geometries that are very close to each other,
we investigate explicitly the time evolution of the spectral distance between
them; as two spatial geometries, we choose a flat 3-torus and a perturbed
geometry around it, mimicking the relation of a ``model universe'' and the
``real Universe''. Then we estimate the spectral distance between them and
investigate its time evolution explicitly. This analysis is done efficiently by
making use of the basic results of the standard linear structure-formation
theory.
We observe that, as far as the linear perturbation of geometry is valid, the
spectral distance does not increase with time prominently,rather it shows the
tendency to decrease. This result is compatible with the general belief in the
reliability of describing the Universe by means of a model, and calls for more
detailed studies along the same line including the investigation of wider class
of spacetimes and the analysis beyond the linear regime.Comment: To be published in Classical and Quantum Gravit