Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Square-Preserving and Symplectic Structure and Scheme for Quantum System

Abstract: The time-dependent Schrodinger equation is a square-preserving and symplectic (SPS) transformation. The canonical equations of quantum systems are deduced by using eigenfunction expansion. The normal-square of wavefunction of the quantum systems is an invariant integral of the canonical equations and then the symplectic schemes that based on both Cayley transformation and diagonal Pade approximation to exp(x) are also square-preserving. The evaluated example show that the SPS approach is reasonable and effective for solving time-evolution of quantum system.

A symplectic structure for connections on surfaces with boundary

Abstract: For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

Mass-Weighted Symplectic Forms for the N-Body Problem

Abstract: Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in the N-body problem. These forms are introduced, compared to previous approaches, and their properties are discussed. Applications to symplectic mappings, the definition of action-angle variables for the Kepler problem, and Hamiltonian perturbation theory are outlined

The k-local Pauli Commuting Hamiltonians Problem is in P

Abstract: Given a Hamiltonian that is a sum of commuting few-body terms, the commuting Hamiltonian problem is to determine if there exists a quantum state that is the simultaneous eigenstate of all of these terms that minimizes each term individually. This problem is known to be in the complexity class quantum Merlin-Arthur, but is widely thought to not be complete for this class. Here we show that a limited form of this problem when the individual terms are all made up of tensor products of Pauli matrices is efficiently solvable on a classical computer and thus in the complexity class P. The problem can be thought of as the classical XOR-SAT problem over a symplectic vector space. This class of problems includes instance Hamiltonians whose ground states possess topological entanglement, thus showing that such entanglement is not always a barrier for the more general problem.

A Class of Integrable Geodesic Flows on the Symplectic Group and the Symmetric Matrices

Abstract: This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian This analysis is extended to cover flows on symmetric matrices when an isomorphism with the symplectic Lie algebra does not hold The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting