Search results for "Integral form"

Erratum to: Classical and Quantum Dynamics: From Classical Paths to Path Integrals

2017

Propagators for Particles in an External Magnetic Field

2001

In order to describe the propagation of a scalar particle in an external potential, we begin again with the path integral $$ K(r',t';r,0) = \int_{r,(0)}^{r',(t')} {[dr(t)]} \exp \left\{ {\frac{{\text{i}}} {\hbar }S[r(t)]} \right\} $$ (1) with $$ S[r(t)] = \int_0^{t'} {dt} L(r,\dot r). $$

Quantum Spin-Tunneling:A Path Integral Approach

1995

We investigate the quantum tunneling of a large spin in a crystal field and an external magnetic field. The twofold degeneracy of the corresponding classical ground state is removed due to tunneling. The tunnel splitting ΔE o of the ground state is calculated by use of a path integral formalism. It is shown that coherent spin state path integrals do not yield a reasonable result. However a “bosonlzation” of the spin system yields excellent results in the semiclassical limit. This result follows from the coherent spin state approach from replacing the spin quantum number s by s + 1/2 which causes a renormalization of the preexponential factor of ΔE o .

Note on the super-extended Moyal formalism and its BBGKY hierarchy

2017

We consider the path integral associated to the Moyal formalism for quantum mechanics extended to contain higher differential forms by means of Grassmann odd fields. After revisiting some properties of the functional integral associated to the (super-extended) Moyal formalism, we give a convenient functional derivation of the BBGKY hierarchy in this framework. In this case the distribution functions depend also on the Grassmann odd fields.

Solving the Cut-Off Wave Numbers in Partially filled Rectangular Waveguides with Ferrite by the Cauchy Integral Method

2005

The modal analysis of the off-centered rectangular waveguide loaded with a vertical slab of ferrite material, biased in the y-direction by a DC magnetic field, leads to the resolution of a transcendent equation whose infinite solutions are the TE/sub m0/ cutoff wave numbers in the guide. The method based on the Cauchy integral (Delvest L.M. and Lyness, J.N., 1967) is becoming very popular for solving such equations. This powerful method is described for solving the propagation constant in a partially ferrite filled waveguide. The method is used to calculate the propagation constant of the fundamental TE mode for some configurations used in the literature about ferrites. Results obtained in …

Magnetoelectric effects in superconductors due to spin-orbit scattering : Nonlinear σ-model description

2021

We suggest a generalization of the nonlinear σ model for diffusive superconducting systems to account for magnetoelectric effects due to spin-orbit scattering. In the leading orders of spin-orbit strength and gradient expansion, it includes two additional terms responsible for the spin-Hall effect and the spin-current swapping. First, assuming a delta-correlated disorder, we derive the terms from the Keldysh path integral representation of the generating functional. Then we argue phenomenologically that they exhaust all invariants allowed in the effective action to the leading order in the spin-orbit coupling (SOC). Finally, the results are confirmed by a direct derivation of the saddle-poi…

Path Integral Formulation of Quantum Electrodynamics

2020

Let us consider a pure Abelian gauge theory given by the Lagrangian $$\displaystyle\begin{array}{rcl} \mathcal{L}_{\text{photon}}& =& -\frac{1} {4}F_{\mu \nu }F^{\mu \nu } \\ & =& -\frac{1} {4}\left (\partial _{\mu }A_{\nu } - \partial _{\nu }A_{\mu }\right )\left (\partial ^{\mu }A^{\nu } - \partial ^{\nu }A^{\mu }\right ){}\end{array}$$ (36.1) or, after integration by parts, $$\displaystyle\begin{array}{rcl} \mathcal{L}_{\text{photon}}& =& -\frac{1} {2}\left [-\left (\partial _{\mu }\partial ^{\mu }A_{\nu }\right )A^{\nu } + \left (\partial ^{\mu }\partial ^{\nu }A_{\mu }\right )A_{\nu }\right ] \\ & =& \frac{1} {2}A_{\mu }\left [g^{\mu \nu }\square - \partial ^{\mu }\partial ^{\nu }\righ…

Quantitative approximation of certain stochastic integrals

2002

We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.

Ship Roll Motion under Stochastic Agencies Using Path Integral Method

2009

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamica…

Path integral solution for nonlinear systems under parametric Poissonian white noise input

2016

Abstract In this paper the problem of the response evaluation in terms of probability density function of nonlinear systems under parametric Poisson white noise is addressed. Specifically, extension of the Path Integral method to this kind of systems is introduced. Such systems exhibit a jump at each impulse occurrence, whose value is obtained in closed form considering two general classes of nonlinear multiplicative functions. Relying on the obtained closed form relation liking the impulses amplitude distribution and the corresponding jump response of the system, the Path Integral method is extended to deal with systems driven by parametric Poissonian white noise. Several numerical applica…