No Arabic abstract
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.
Let (H,B) be an abstract Wiener space and let mu_{s} be the Gaussian measure on B with variance s. Let Delta be the Laplacian (*not* the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H. I will show that the heat operator exp(tDelta/2) is a contraction operator from L^2(B,mu_{s} to L^2(B,mu_{s-t}), for all t<s. More generally, the heat operator is a contraction from L^p(B,mu_{s}) to L^q(B,mu_{s-t}) for t<s, provided that p and q satisfy (p-1)/(q-1) leq s/(s-t). I give two proofs of this result, both very elementary.
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.
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.
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.
A special symplectic Lie group is a triple $(G,omega, abla)$ such that $G$ is a finite-dimensional real Lie group and $omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $ abla$. In this paper starting from a special symplectic Lie group we show how to ``deform the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $ abla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.