A differential Galois approach to path integrals

@article{MoralesRuiz2019ADG,
  title={A differential Galois approach to path integrals},
  author={Juan Jos'e Morales-Ruiz},
  journal={arXiv: Mathematical Physics},
  year={2019}
}
  • J. Morales-Ruiz
  • Published 14 October 2019
  • Mathematics, Physics
  • arXiv: Mathematical Physics
We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself : if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational… Expand
1 Citations
Differential Galois Theory and Integration
In this chapter, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methodsExpand

References

SHOWING 1-10 OF 64 REFERENCES
Integrability of hamiltonian systems and differential Galois groups of higher variational equations
Abstract Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the GaloisExpand
Galois Theory of Linear Differential Equations
Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential GaloisExpand
Galoisian obstructions to non-Hamiltonian integrability
Abstract We show that the main theorem of Morales, Ramis and Simo (2007) [6] about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to theExpand
GALOISIAN OBSTRUCTIONS TO INTEGRABILITY OF HAMILTONIAN SYSTEMS: STATEMENTS AND EXAMPLES
An inconvenience of all the Galoisian formulations of Ziglin non-integrability Theory, is the regularity assumption of the singular points in the variational equation (the variational equation is ofExpand
Galosian Obstructions to Integrability of Hamiltonian Systems II
An inconvenience of all the known galoisian formulations of Ziglin's non-integrability theory is the Fhchsian condition at the singular points of the variational equations. We avoid this restriction.Expand
The hamiltonian path integrals and the uniform semiclassical approximations for the propagator
Abstract The generalized path expansion scheme is defined for path integration in phase-space. Within this framework we study the semiclassical limits to the propagator, both in the momentum and theExpand
GALOISIAN OBSTRUCTIONS TO INTEGRABILITY OF HAMILTONIAN SYSTEMS II ∗
By applying the results of our previous paper [19], we obtain non–integrability results for the following four Hamiltonian systems: the Bianchi IX Cosmological Model, the Sitnikov system of celestialExpand
Liouvillian propagators, Riccati equation and differential Galois theory
In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schr¨Expand
Functional determinants in quantum field theory
Functional determinants of differential operators play a prominent role in theoretical and mathematical physics, and in particular in quantum field theory. They are, however, difficult to compute inExpand
Integrability of dynamical systems through differential Galois theory : a practical guide
We survey recent advances in the non-integrability criteria for Hamiltonian Systems which involve the differential Galois group of variational equations along particular solutions. The emphasis is onExpand
...
1
2
3
4
5
...