Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymplectic Microgeometry IV: Quantization
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schrodinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.
rate research
Read More
We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.
Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $gamma$ topology defined by the author, to $bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${bar H}(y,q,p)$ and thus yields an effective Hamiltonian. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $bar H$ coincides with Mathers $alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard. A previous version of this paper relied on the former On the capacity of Lagrangians in $T^*T^n$ which has been withdrawn. The present version of Symplectic Homogenization does not rely on it anymore.
We say that a subset of a symplectic manifold is symplectically (neighbourhood) excisable if its complement is symplectomorphic to the ambient manifold, (through a symplectomorphism that can be chosen to be the identity outside an arbitrarily small neighbourhood of the subset). We use time-independent Hamiltonian flows, and their iterations, to show that certain properly embedded subsets of noncompact symplectic manifolds are symplectically neighbourhood excisable: a ray, a Cantor brush, a box with a tail, and -- more generally -- epigraphs of lower semi-continuous functions; as well as a ray with two horns, and -- more generally -- open-rooted finite trees.
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $mathbb{C}P^2# Noverline{mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.
We construct a new family of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. We prove various structural properties of the capacities and discuss the connections with the equivariant L-infinity structure on symplectic cohomology and curve counts with tangency conditions. We also give some preliminary computations in basic examples and show that they give new state of the art symplectic embedding obstructions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
