Holomorphic Methods in Mathematical Physics

We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this setting using the Caputo and Liouville operators. Finally, we give an extension of these results to the case of quaternions, in particular in the slice hyperholomorphic setting.

A review of the Hodge and Hopf-algebraic approach to QFT.

The purpose of this contribution is to give a very brief introduction to Quantum Mechanics for an audience of mathematicians. I will follow Segal's approach to Quantum Mechanics paying special attention to algebraic issues. The usual representation of Quantum Mechanics on Hilbert spaces is also discussed.

In this work, we present some elementary properties of Segal-Bargmann space and some properties of unitary Segal Bargmann transform with applications to differential operators arising out of diffusion problem or of reggeon field theory.

This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from readers, I've decided to make a presentable half-finished version available here. Further chapters will be added in future updates.

Introductory Notes in Bosonic String Theory and its Canonical Quantization.

These notes were written following lectures I had the pleasure of giving on this subject at Keio University, during November and December 2004. The first part is about new applications of Jordan algebras to the geometry of Hermitian symmetric spaces and to causal semi-simple symmetric spaces of Cayley type. The second part will present new contributions for studing (non commutative) Hardy spaces of holomorphic functions on Lie semi-groups which is a part of the so called Gelf...

This paper provides a pedagogical introduction to the quantum mechanical path integral and its use in proving index theorems in geometry, specifically the Gauss-Bonnet-Chern theorem and Lefschetz fixed point theorem. It also touches on some other important concepts in mathematical physics, such as that of stationary phase, supersymmetry and localization. It is aimed at advanced undergraduates and beginning graduates, with no previous knowledge beyond undergraduate quantum mec...

In this paper we develop a quantization method for flat compact manifolds based on path integrals. In this method the Hilbert space of holomorphic functions in the complexification of the manifold is used. This space is a reproducing kernel Hilbert space. A definition of the Feynman propagator, based on the reproducing property of this space, is proposed. In the $\mathbb{R}^n$ case the obtained results coincide with the known expressions.

We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on $\mathbb{C}$ and $\mathbb{C}^2$ and the slice hyperholomorphic Bargmann-Fock space on $\mathbb{H}$. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two di...