Path integrals for higher derivative actions

Abstract

We consider Euclidean path integrals with higher derivative actions, including those that depend quadratically on acceleration, velocity and position. Such path integrals arise naturally in the study of stiff polymers, membranes with bending rigidity as well as a number of models for electrolytes. The approach used is based on the relation between quadratic path integrals and Gaussian fields and we also show how it can be extended to the evaluation of even higher order path integrals.

Brownian motion of a particle with higher-derivative dynamics

“…Different from the Langevin equation and Fokker-Planck equation, the path integral method offers another way to describe stochastic processes. We have noted that Kleinert [31] discussed the path integral for the Pais-Uhlenbeck oscillator, and that Dean et al [32] discussed the path integral of HDD in quadratic form. In both cases, they achieved analytical solutions of propagators.…”

Thermal Casimir interactions for higher derivative field Lagrangians: generalized Brazovskii models

“…and we have used the notation s i = sinh(ω i h) and c i = cosh(ω i h). This result was rederived, in a very different way to [20], in [22] by exploiting a link with the unconfined case.…”

“…Some special cases of these results were established even before the developments of the full theory [4]. Recently the authors have established an alternative derivation of Kleinert's results [22] based on a link between unconfined systems, which can be treated using Green's function methods, and confined systems (which correspond to the standard path integral). The aim of this paper is to exploit these results to explore the Casimir interaction arising in both unconfined and confined geometries For both confined and unconfined systems we examine the interaction between two parallel surfaces for the Brazovskii model field Hamiltonian [23,24] with various boundary conditions imposed at each surface.…”

Gel’fand–Yaglom type equations for calculating fluctuations around instantons in stochastic systems

“…Finally, we will derive the representation of the PDF prefactor without homogenization of the boundary conditions that has been stated at the beginning of this section and can more easily be computed for large system dimensions d. In the context of hydrodynamic shell models, Daumont, Dombre and Gilson [13] have derived a related expression for the influence of the quadratic fluctuations on the PDF prefactor of a A c c e p t e d M a n u s c r i p t one-dimensional observable by path integral calculations, but their derivation lead to a more complicated procedure, which they also did not discuss in the continuum limit. Furthermore, Dean, Miao and Podgornik have derived a similar expression involving algebraic Riccati equations in the case of constant coefficients in [29] via the Feynman-Kac formula. We adapt their derivation to our problem in remark 4.…”

“…Remark 4. It is also possible to derive (118) and (119) based solely on probabilistic methods, without explicit reference to the path integral computations that were utilized above, by adopting the techniques from [29]. Starting from (29), we note that, for suitable functions f , g : R d → R, the prefactor can be written as…”