31,066 research outputs found
Integration over connections in the discretized gravitational functional integrals
The result of performing integrations over connection type variables in the
path integral for the discrete field theory may be poorly defined in the case
of non-compact gauge group with the Haar measure exponentially growing in some
directions. This point is studied in the case of the discrete form of the first
order formulation of the Einstein gravity theory. Here the result of interest
can be defined as generalized function (of the rest of variables of the type of
tetrad or elementary areas) i. e. a functional on a set of probe functions. To
define this functional, we calculate its values on the products of components
of the area tensors, the so-called moments. The resulting distribution (in
fact, probability distribution) has singular (-function-like) part with
support in the nonphysical region of the complex plane of area tensors and
regular part (usual function) which decays exponentially at large areas. As we
discuss, this also provides suppression of large edge lengths which is
important for internal consistency, if one asks whether gravity on short
distances can be discrete. Some another features of the obtained probability
distribution including occurrence of the local maxima at a number of the
approximately equidistant values of area are also considered.Comment: 22 page
Attributing sense to some integrals in Regge calculus
Regge calculus minisuperspace action in the connection representation has the
form in which each term is linear over some field variable (scale of area-type
variable with sign). We are interested in the result of performing integration
over connections in the path integral (now usual multiple integral) as function
of area tensors even in larger region considered as independent variables. To
find this function (or distribution), we compute its moments, i. e. integrals
with monomials over area tensors. Calculation proceeds through intermediate
appearance of -functions and integrating them out. Up to a singular
part with support on some discrete set of physically unattainable points, the
function of interest has finite moments. This function in physical region
should therefore exponentially decay at large areas and it really does being
restored from moments. This gives for gravity a way of defining such
nonabsolutely convergent integral as path integral.Comment: 14 pages, presentation improve
Regge gravity from spinfoams
We consider spinfoam quantum gravity for general triangulations in the regime
, namely in the combined classical limit of large areas
and flipped limit of small Barbero-Immirzi parameter , where
is the Planck length. Under few working hypotheses we find that the flipped
limit enforces the constraints that turn the spinfoam theory into an effective
Regge-like quantum theory with lengths as variables, while the classical limit
selects among the possible geometries the ones satisfying the Einstein
equations. Two kinds of quantum corrections appear in terms of powers of
and . The result also suggests that the
Barbero-Immirzi parameter may run to smaller values under coarse-graining of
the triangulation.Comment: 18 pages, presentation substantially improve
Infrared spectroscopy of diatomic molecules - a fractional calculus approach
The eigenvalue spectrum of the fractional quantum harmonic oscillator is
calculated numerically solving the fractional Schr\"odinger equation based on
the Riemann and Caputo definition of a fractional derivative. The fractional
approach allows a smooth transition between vibrational and rotational type
spectra, which is shown to be an appropriate tool to analyze IR spectra of
diatomic molecules.Comment: revised + extended version, 9 pages, 6 figure
Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics
We review the Lagrangian formulation of Noether symmetries (as well as
"generalized Noether symmetries") in the framework of Calculus of Variations in
Jet Bundles, with a special attention to so-called "Natural Theories" and
"Gauge-Natural Theories", that include all relevant Field Theories and physical
applications (from Mechanics to General Relativity, to Gauge Theories,
Supersymmetric Theories, Spinors and so on). It is discussed how the use of
Poincare'-Cartan forms and decompositions of natural (or gauge-natural)
variational operators give rise to notions such as "generators of Noether
symmetries", energy and reduced energy flow, Bianchi identities, weak and
strong conservation laws, covariant conservation laws, Hamiltonian-like
conservation laws (such as, e.g., so-called ADM laws in General Relativity)
with emphasis on the physical interpretation of the quantities calculated in
specific cases (energy, angular momentum, entropy, etc.). A few substantially
new and very recent applications/examples are presented to better show the
power of the methods introduced: one in Classical Mechanics (definition of
strong conservation laws in a frame-independent setting and a discussion on the
way in which conserved quantities depend on the choice of an observer); one in
Classical Field Theories (energy and entropy in General Relativity, in its
standard formulation, in its spin-frame formulation, in its first order
formulation "`a la Palatini" and in its extensions to Non-Linear Gravity
Theories); one in Quantum Field Theories (applications to conservation laws in
Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).Comment: 27 page
Defining integrals over connections in the discretized gravitational functional integral
Integration over connection type variables in the path integral for the
discrete form of the first order formulation of general relativity theory is
studied. The result (a generalized function of the rest of variables of the
type of tetrad or elementary areas) can be defined through its moments, i. e.
integrals of it with the area tensor monomials. In our previous paper these
moments have been defined by deforming integration contours in the complex
plane as if we had passed to an Euclidean-like region. In the present paper we
define and evaluate the moments in the genuine Minkowsky region. The
distribution of interest resulting from these moments in this non-positively
defined region contains the divergences. We prove that the latter contribute
only to the singular (\dfun like) part of this distribution with support in the
non-physical region of the complex plane of area tensors while in the physical
region this distribution (usual function) confirms that defined in our previous
paper which decays exponentially at large areas. Besides that, we evaluate the
basic integrals over which the integral over connections in the general path
integral can be expanded.Comment: 18 page
Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
We study the fractional gravity for spacetimes with non-integer dimensions.
Our constructions are based on a geometric formalism with the fractional Caputo
derivative and integral calculus adapted to nonolonomic distributions. This
allows us to define a fractional spacetime geometry with fundamental
geometric/physical objects and a generalized tensor calculus all being similar
to respective integer dimension constructions. Such models of fractional
gravity mimic the Einstein gravity theory and various Lagrange-Finsler and
Hamilton-Cartan generalizations in nonholonomic variables. The approach
suggests a number of new implications for gravity and matter field theories
with singular, stochastic, kinetic, fractal, memory etc processes. We prove
that the fractional gravitational field equations can be integrated in very
general forms following the anholonomic deformation method for constructing
exact solutions. Finally, we study some examples of fractional black hole
solutions, fractional ellipsoid gravitational configurations and imbedding of
such objects in fractional solitonic backgrounds.Comment: latex2e, 11pt, 40 pages with table of conten
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