Fabio Bagarello - Pseudo-Bosons and Their Coherent States (Mathematical Physics Studies)

The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathematics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geometry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory.

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Doing research is fun! I like when I find something unclear, and I like more when I understand what is going on. The point is that, quoting George Bernard Shaw, science never solves a problem without creating ten more. For this reason, from time to time, I force myself to stop considering new problems and I try to spend some time to summarize what I have done so far. And the best way to clarify the state of the art is putting all results together, giving them the form of a self-contained book. However, on the way, sometimes I meet crossroads leading me in different directions with respect to those I had in mind at the beginning of my adventure. This is particularly true for these notes, which, in my original idea, should have been a short paper on a very specific topic, the Baker-Campbell-Hausdorff formula for pseudo-bosonic operators. While preparing this paper I realized that, in connection with this formula, there was much more to say, and from several points of view: mathematical details are relevant, but also physical applications are of interest, of course. Then I started to add material to the original manuscript, and the new material required more and more discussions, lemmas, theorems, and convergence tests. And, at a certain stage, it became evident that I was no more writing a (now, long) paper, but something different. For this reason, I also added some introductory material in Chaps. , in the attempt to make the final result sufficiently self-contained. The result is a book which is essentially about several aspects of ladder operators of a special kind. These are deformed versions of the lowering and raising bosonic operators all people working in quantum mechanics know very well. I will discuss the sets of vectors on which they act as ladders, some appearances in quantum mechanics, and the coherent states connected to these deformed annihilation operators. I will also discuss extensions to distributions of this framework, and the use of my particular coherent states in the computation of transition probabilities. I really hope that reading these notes will be interesting, and maybe inspiring, to you. In the meantime, enjoy the reading!

Acknowledgments

Writing a book is a hard task! One needs results to describe, time to put them in the right perspective, more time to write, and still other time to finalize the project. In the meantime, life goes on (and most probably doesnt care at all about your book and any other project you might have!).

But time is not the only ingredient! More than time, I need friends to help me during my work, and a friendly environment around me to work without distractionsor with pleasant distractions. And my friends deserve many thanks for their continuous support. So, thank you Jean Pierre (Gazeau), Camillo (Trapani), Franco (Oliveri), Francesco (Gargano), Miloslav (Znojil), Sergii (Kuel), Hiroshi and Atsushi (Inoue), Andreas (Fring), and Joshua (Feinberg). You have been precious, and very patient, all these years! And it is not over yet! I also like to thank Federico Roccati for his precious reading of the manuscript. And let me also thank my wonderful magnificent six, always on my side, in one way or another: Angelo, Cosimo, Fino, Graziella, Ina, Vincenzo. Thank you very much

As always, I dedicate this book to my beloved parents, Giovanna and Benedetto, and to Federico, Giovanna, and Grazyna, with all my love. I really cannot imagine something better than sharing my days with you!

Palermo, Italy Fabio Bagarello