Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userClassical KMS Functionals and Phase Transitions in Poisson Geometry
79
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinsteins seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide a complete characterization of the convex cone of KMS measures.
rate research
Read More
We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson, [LM], is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators.
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to Quantum Mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the Physics literature.
We prove the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings, and explicitly construct the unstable mode of the inner catenoid, by studying the spectrum of an exactly solvable one-dimensional Schrodinger operator with an asymmetric Darboux-Poschl-Teller potential.
A Poisson realization of the simple real Lie algebra $mathfrak {so}^*(4n)$ on the phase space of each $mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $mathfrak {so}^*(4n)$ on the Poisson manifold $T^*mathbb H_*^n/mathrm{Sp}(1)$. (Here $mathbb H_*^n:=mathbb H^nbackslash {0}$ and the Hamiltonian action of $mathrm{Sp}(1)$ on $T^*mathbb H_*^n$ is induced from the natural right action of $mathrm{Sp}(1)$ on $mathbb H_*^n$. )
These are lecture notes for the course Poisson geometry and deformation quantization given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes theorem, de Rhams theorem and Frobenius theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinsteins tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introduction to Poisson geometry where we also cover Courant structures, Dirac structures, the local splitting theorem, symplectic foliations and Poisson maps. The fourth chapter is about deformation quantization where we cover the Moyal product, $L_infty$-algebras, Kontsevichs formality theorem, Kontsevichs star product construction through graphs, the globalization approach to Kontsevichs star product and the operadic approach to formality. The fifth chapter is about the quantum field theoretic approach to Kontsevichs deformation quantization where we cover functional integral methods, the Moyal product as a path integral quantization, the Faddeev-Popov and BRST method for gauge theories, infinite-dimensional extensions, the Poisson sigma model, the construction of Kontsevichs star product through a perturbative expansion of the functional integral quantization for the Poisson sigma model for affine Poisson structures and the general construction.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
