40 research outputs found
Mehler-Fock transforms and retarded radiative Green functions on hyperbolic and spherical spaces
We develop the theory of causal radiation Green functions on hyperbolic and
hyperspherical spaces using a constructive approach based on generalized
Mehler-Fock transforms. This approach focuses for on the kernel of the
transformation expressed in terms of hyperbolic angles with
. The kernel provides an explicit representation for the
generalized delta distribution which acts as the source term for the radiation,
and allows easy implementation of the causality or retardation condition and
determination of the Green function. We obtain the corresponding kernel
distribution on by analytic continuation of the kernel distribution of
the Helmholtz equation on , then show that this construction leads to the
proper retarded Green function for the wave equation. That result is then used
to establish the validity of a new generalized Mehler-Fock transformation for
. The present results clarify and extend those obtained
recently by Cohl, Dang, and Dunster.Comment: Important correction on left-hand side of Eq. (78), other minor
corrections; 17 pages, submitted to J. Math. Phy
Temporal breakdown and Borel resummation in the complex Langevin method
We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4
measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between
Euclidean and Lorentzian signature. Our main result is that the asymptotics for
short stochastic times t encapsulates information also about the equilibrium
aspects. The moments evaluated with the complex measure and with the real
measure defined by the stochastic Langevin equation have the same t -> 0
asymptotic expansion which is shown to be Borel summable. The Borel transform
correctly reproduces the time dependent moments of the complex measure for all
t, including their t -> infinity equilibrium values. On the other hand the
results of a direct numerical simulation of the Langevin moments are found to
disagree from the `correct' result for t larger than a finite t_c. The
breakdown time t_c increases powerlike for decreasing strength of the noise's
imaginary part but cannot be excluded to be finite for purely real noise. To
ascertain the discrepancy we also compute the real equilibrium distribution for
complex noise explicitly and verify that its moments differ from those obtained
with the complex measure.Comment: title changed, results on parameter dependence of t_c added,
exposition improved. 39 pages, 7 figure
Concentration estimates for band-limited spherical harmonics expansions via the large sieve principle
We study a concentration problem on the unit sphere for
band-limited spherical harmonics expansions using large sieve methods. We
derive upper bounds for concentration in terms of the maximum Nyquist density.
Our proof uses estimates of the spherical harmonics coefficients of certain
zonal filters. We also demonstrate an analogue of the classical large sieve
inequality for spherical harmonics expansions
Gauss Quadrature for Freud Weights, Modulation Spaces, and Marcinkiewicz-Zygmund Inequalities
We study Gauss quadrature for Freud weights and derive worst case error
estimates for functions in a family of associated Sobolev spaces. For the
Gaussian weight these spaces coincide with a class of modulation
spaces which are well-known in (time-frequency) analysis and also appear under
the name of Hermite spaces. Extensions are given to more general sets of nodes
that are derived from Marcinkiewicz-Zygmund inequalities. This generalization
can be interpreted as a stability result for Gauss quadrature
Group theoretical approach to quantum fields in de Sitter space I. The principal series
Using unitary irreducible representations of the de Sitter group, we
construct the Fock space of a massive free scalar field.
In this approach, the vacuum is the unique dS invariant state. The quantum
field is a posteriori defined by an operator subject to covariant
transformations under the dS isometry group. This insures that it obeys
canonical commutation relations, up to an overall factor which should not
vanish as it fixes the value of hbar. However, contrary to what is obtained for
the Poincare group, the covariance condition leaves an arbitrariness in the
definition of the field. This arbitrariness allows to recover the amplitudes
governing spontaneous pair creation processes, as well as the class of alpha
vacua obtained in the usual field theoretical approach. The two approaches can
be formally related by introducing a squeezing operator which acts on the state
in the field theoretical description and on the operator in the present
treatment. The choice of the different dS invariant schemes (different alpha
vacua) is here posed in very simple terms: it is related to a first order
differential equation which is singular on the horizon and whose general
solution is therefore characterized by the amplitude on either side of the
horizon. Our algebraic approach offers a new method to define quantum field
theory on some deformations of dS space.Comment: 35 pages, 2 figures ; Corrected typo, Changed referenc