Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method
58
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The Picard-Fuchs equation is a powerful mathematical tool which has numerous applications in physics, for it allows to evaluate integrals without resorting to direct integration techniques. We use this equation to calculate both the classical action and the higher-order WKB corrections to it, for the sextic double-well potential and the Lame potential. Our development rests on the fact that the Picard-Fuchs method links an integral to solutions of a differential equation with the energy as a parameter. Employing the same argument we show that each higher-order correction in the WKB series for the quantum action is a combination of the classical action and its derivatives. From this, we obtain a computationally simple method of calculating higher-order quantum-mechanical corrections to the classical action, and demonstrate this by calculating the second-order correction for the sextic and the Lame potential. This paper also serves as a self-consistent guide to the use of the Picard-Fuchs equation.
rate research
Read More
We give an explicit differential equation which is expected to determine the instanton partition function in the presence of the full surface operator in N=2 SU(N) gauge theory. The differential equation arises as a quantization of a certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and Tsuda.
We calculate the statistical entropy of a quantum field with an arbitrary spin propagating on the spherical symmetric black hole background by using the brick wall formalism at higher orders in the WKB approximation. For general spins, we find that the correction to the standard Bekenstein-Hawking entropy depends logarithmically on the area of the horizon. Furthermore, we apply this analysis to the Schwarzschild and Schwarzschild-AdS black holes and discuss our results.
We extend topological string methods in order to perform WKB approximations for quantum mechanical problems with higher order potentials efficiently. This requires techniques for the evaluation of the relevant quantum periods for Riemann surfaces beyond genus one. The basis of these quantum periods is fixed using the leading behaviour of the classical periods. The full expansion of the quantum periods is obtained using a system of Picard-Fuchs like operators for a sequence of integrals of meromorphic forms of the second kind. Discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve. In this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of SL$(2,mathbb{Z})$ as we check for a symmetric sextic potential.
Two known 2-dim SUSY quantum mechanical constructions - the direct generalization of SUSY with first-order supercharges and Higher order SUSY with second order supercharges - are combined for a class of 2-dim quantum models, which {it are not amenable} to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained - the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related flipped potentials are established.
A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
