1,721,048 research outputs found
New perspectives on the irregular singular point of the wave equation for a massive scalar field in Schwarzschild space-time
No full textFor a massive scalar field in a fixed Schwarzschild background, the radial wave
equation obeyed by Fourier modes is first studied. After reducing such a radial wave
equation to its normal form, we first study approximate solutions in the neighborhood
of the origin, horizon and point at infinity, and then we relate the radial with the Heun
equation, obtaining local solutions at the regular singular points. Moreover, we obtain the
full asymptotic expansion of the local solution in the neighborhood of the irregular singular
point at infinity. We also obtain and study the associated integral representation of the
massive scalar field. Eventually, the technique developed for the irregular singular point is
applied to the homogeneous equation associated with the inhomogeneous Zerilli equation
for gravitational perturbations in a Schwarzschild background
On the physical and mathematical foundations of quantum physics via functional integrals
No full textIn order to preserve the leading role of the action principle in formulating all field theories
one needs quantum field theory, with the associated BRST symmetry, and Feynman
DeWitt Faddeev Popov ghost fields. Such fields result from the fiber-bundle structure of
the space of histories, but the physics-oriented literature used them formally because a rigorous
theory of measure and integration was lacking. Motivated by this framework, this paper exploits
the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral
was proposed. The Henstock-Kurzweil integral is first introduced, because it makes it possible
to integrate functions like exp(ix**2). The Lebesgue measure on R(infinity) is then built and used to
define the measure on every separable Hilbert space. The subsequent step is the construction of
a new Hilbert space KS**2[R**n], which contains L**2[R*n] as a continuous dense embedding, and
contains both the test functions D[R**n] and their dual D*[R**n], the Schwartz space of distributions,
as continuous embeddings. This space allows us to construct the Feynman path integral in
a manner that maintains its intuitive and computational advantages. We also extend this space
to KS**2[H], where H is any separable Hilbert space. Last, the existence of a unique universal
definition of time, tau(h), that we call historical time, is proven. We use tau(h) as the order parameter
for our construction of Feynman's time ordered operator calculus, which in turn is used to
extend the path integral in order to include all time-dependent groups and semigroups with a
reproducing kernel representation
Fondamenti matematici della teoria classica dei campi
Full text linkVengono studiati i fondamenti matematici e fisici della teoria classica dei campi
DeWitt Boundary Condition in One-Loop Quantum Cosmology
No full textDeWitt’s suggestion that the wave function of the universe should vanish at the classical Big Bang singularity is considered here within the framework of one-loop quantum cosmology. For pure gravity at one loop about a flat four-dimensional background bounded by a 3-sphere, three choices of boundary conditions are considered: vanishing of the linearized magnetic curvature when only transverse-traceless gravitational modes are quantized; a one-parameter family of mixed boundary conditions for gravitational and ghost modes; and diffeomorphism-invariant boundary conditions for metric perturbations and ghost modes. A positive ζ(0) value in these cases ensures that, when the three-sphere boundary approaches zero, the resulting one-loop wave function approaches zero. This property may be interpreted by saying that, in the limit of small three-geometry, the resulting one-loop wave function describes a singularity-free universe. This property holds for one-loop functional integrals, which are not necessarily equivalent to solutions of the quantum constraint equations
Numerov and phase-integral methods for charmonium
Full text linkThis paper applies the Numerov and phase-integral methods to the stationary
Schrodinger equation that studies bound states of charm anti-charm quarks. The
former is a numerical method well suited for a matrix form of second-order
ordinary di erential equations, and can be applied whenever the stationary
states admit a Taylor-series expansion. The latter is an analytic method that
provides, in principle, even exact solutions of the stationary Schrodinger
equation, and well suited for applying matched asymptotic expansions and higher
order quantization conditions. The Numerov method is found to be always in
agreement with the early results of Eichten et al., whereas an original
evaluation of the phase-integral quantization condition clarifies under which
conditions the previous results in the literature on higher-order terms can be
obtained.Comment: 23 pages, 2 tables and 2 figures. In the final version, the
presentation and the numerical analysis have been improved, and a misprint in
Eq. (2.3) has been amende
Highlights of symmetry groups
No full textThe concepts of symmetry and symmetry groups are
at the heart of several developments in modern theoretical and
mathematical physics. The present paper is devoted to a number of
selected topics within this framework: Euclidean and rotation
groups; the properties of fullerenes in physical chemistry; Galilei,
Lorentz and Poincar'e groups; conformal transformations and
the Laplace equation; quantum groups and Sklyanin algebras.
For example, graphite can be vaporized by laser irradiation,
producing a remarkably stable cluster consisting of 60 carbon
atoms. The corresponding theoretical model considers a truncated
icosahedron, i.e. a polygon with 60 vertices and 32 faces,
12 of which are pentagonal and 20 hexagonal. The
C_{60} molecule obtained when a carbon atom is placed at each
vertex of this structure has all valences satisfied by two single
bonds and one double bond. In other words, a structure in which a
pentagon is completely surrounded by hexagons is stable. Thus,
a ``cage'' in which all 12 pentagons are completely surrounded
by hexagons has optimum stability. On a more formal side, the
exactly solvable models of quantum and statistical physics can
be studied with the help of the quantum inverse problem method.
The problem of enumerating the discrete quantum systems which can
be solved by the quantum inverse problem method reduces to the
problem of enumerating the operator-valued functions that satisfy
an equation involving a fixed solution of the quantum
Yang-Baxter equation. Two basic equations exist which provide a
systematic procedure for obtaining completely integrable lattice
approximations to various continuous completely integrable
systems. This analysis leads in turn to the discovery of
Sklyanin algebras
Nariai spacetime: orbits, scalar self force and Poynting-Robertson-like external force
No full textAfter studying properties of the Nariai solution, including its geodesics, in spherical and de Sitter
coordinates, two kinds of accelerated motion are investigated in detail: either observers at rest with respect
to the coordinates, or observers in radial motion. Next, massless scalar perturbations of Nariai spacetime in
absence of sources are worked out, and an explicit example out of the black hole context of analytic selfforce
calculation is obtained. Last, self-force effects are studied as well, together with some variant of the
type of Poynting-Robertson external force, and also building a test electromagnetic field and a test
gravitational field in Nariai spacetime geometry
Lack of strong ellipticity in Euclidean quantum gravity
No full textRecent work in Euclidean quantum gravity has studied boundary conditions which
are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using
the de Donder gauge-averaging functional, this scheme leads to both normal and tangential
derivatives in the boundary conditions. In the present paper, it is proved that the corresponding
boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues
for Euclidean quantum gravity on manifolds with boundary
Projective infinity with spherical symmetry in space-time geometry
No full textThis paper points out that, in a four-dimensional spherically symmetric spacetime
manifold, one can consider coordinate transformations expressed by fractional linear
maps which give rise to isometries and make it possible to bring infinity down
to a finite distance. Schwarzschild and Nariai spacetimes are then described in
projectively transformed coordinates
Fractional linear maps in general relativity and quantum mechanics
No full textThis paper studies the nature of fractional linear transformations in a general relativ-
ity context as well as in a quantum theoretical framework. Two features are found to
deserve special attention: the first is the possibility of separating the limit-point condition
at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful
in a context in which one wants to look for a correspondence between essentially
self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of
Bondi-Metzner-Sachs transformations in general relativity. The analogy therefore arising,
suggests that further investigations might be performed for a theory in which the
role of fractional linear maps is viewed as a bridge between the quantum theory and
general relativity. The second aspect to point out is the possibility of interpreting the
limit-point condition at both ends of the positive real line, for a second-order singular
differential operator, which occurs frequently in applied quantum mechanics, as the limiting
procedure arising from a very particular Kleinian group which is the hyperbolic
cyclic group. In this framework, this work finds that a consistent system of equations can
be derived and studied. Hence one is led to consider the entire transcendental functions,
from which it is possible to construct a fundamental system of solutions of a second-order
differential equation with singular behavior at both ends of the positive real line, which
in turn satisfy the limit-point conditions. Further developments in this direction might
also be obtained by constructing a fundamental system of solutions and then deriving
the differential equation whose solutions are the independent system first obtained. This
guarantees two important properties at the same time: the essential self-adjointness of
a second-order differential operator and the existence of a conserved quantity which is
an automorphic function for the cyclic group chosen
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