Perturbative improvement of the WKB approximation (original) (raw)
Related papers
Cosmological particle production and the precision of the WKB approximation
Physical Review D, 2005
Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.
Semiclassical Approximations to Cosmological Perturbations
2007
We apply several methods related to the WKB approximation to study cosmological perturbations during inflation, obtaining the full power spectra of scalar and tensor perturbations to first and to second order in the slow-roll parameters. We compare our results with those derived by means of other methods, in particular the Green's function method, and find agreement for the slow-roll structure.
P o S ( I S F T G ) 0 1 5 P o S ( I S F T G ) 0 1 5 New approximation methods in General Relativity
2009
We show how approximate solutions of the two-body problem in General Relativity, and the approximate solutions of Einstein’s equations in vacuo can be constructed using small deformations of geodesics and of Einstein space-times embedded into a pseudo-Euclidean flat space of higher dimension. The method consists in using expansions of equations around a given simple solution (a circular orbit in the case of geodesics, and Minkowskian or Schwarzschild space in the case of Einstein’s equations) in a series of powers of small deformation parameter, and then solving by iteration the corresponding linear systems of differential equations.
Analytic approximations, perturbation methods, and their applications
The paper summarizes the parallel session B3 Analytic approximations, perturbation methods, and their applications of the GR18 conference. The talks in the session reported notably recent advances in black hole perturbations and post-Newtonian approximations as applied to sources of gravitational waves.
The WKB approximation in the deformed space with the minimal length and minimal momentum
Journal of Physics A: Mathematical and Theoretical, 2008
A Bohr-Sommerfeld quantization rule is generalized for the case of the deformed commutation relation leading to minimal uncertainties in both coordinate and momentum operators. The correctness of the rule is verified by comparing obtained results with exact expressions for corresponding spectra.
Improved WKB analysis of cosmological perturbations
Physical Review D - Particles, Fields, Gravitation and Cosmology, 2005
Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from sub-horizon to super-horizon scales, and include the region near the turning point. We employ both a perturbative Green's function technique and an adiabatic (or "semiclassical") expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green's function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful.
Annals of Physics, 2004
We present a divergence-free WKB theory, which is a new semiclassical theory modified by nonperturbative quantum corrections. Conventionally, the WKB theory is constructed upon a trajectory that obeys the bare classical dynamics expressed by a quadratic equation in momentum space. Contrary to this, the divergence-free WKB theory is based on a higher-order algebraic equation in momentum space, which represents a dressed classical dynamics. More precisely, this higher-order algebraic equation is obtained by including quantum corrections to the quadratic equation, which is the bare classical limit. An additional solution of the higher-order algebraic equation enables us to construct a uniformly converging perturbative expansion of the wavefunction. Namely, our theory removes the notorious divergence of wavefunction at a turning point from the WKB theory. Moreover, our theory is able to produce wavefunctions and eigenenergies more accurate than those given by the traditional WKB method. In addition, the divergence-free WKB theory that is based on the cubic equation allows us to construct a uniformly valid wavefunction for the nonlinear Schr€ odinger equation (NLSE). A recent short letter [T. Hyouguchi, S. Adachi, M. Ueda, Phys. Rev. Lett. 88 (2002) 170404] is the opening of the divergencefree WKB theory. This paper presents full formalism of this theory and its several applications concerning wavefunction and eigenenergy to show that our theory is a natural extension of the traditional WKB theory that incorporates nonperturbative quantum corrections.
The CWKB Method of Particle Production Near the Chronology Horizon
2002
In this paper we investigate the phenomenon of particle production of massles scalar field, in a model of spacetime where the chronology horizon could be formrd, using the method of complex time WKB approximation (CWKB). For the purpose, we take two examp les in a model of spacetime, one already discussed by Sushkov, to show that the mode of particle production near chronology horizon possesses the similar characteristic features as are found while discussing particle production in time dependent curved ba ckground. We get identical results as that obtained by Sushkov in this direction. We find, in both the examples studied, that the total number of particles remain finite at the moment of the formation of the chronology horizon.