Banach Gelfand Triples and Born-Jordan Quantization

The aim of this project is to open a new research line at the crossroad of time-frequency analysis (TFA) and quantum mechanics (QM) by studying jointly the theory of Banach Gelfand triples and Born-Jordan pseudo-differential calculus, with a strong focus on their applications. TFA is basically trying to analyze signals by representing them over phase space; QM explains the nature and behavior of matter and energy on the atomic and subatomic level. While both disciplines are using similar mathematical formalisms (and sometimes even language), there has been little interpenetration because of conceptual differences. Motivated by a concrete example arising in TFA (the Segal algebra, a particular modulation space) Hans Feichtinger has recently introduced the notion of Banach Gelfand triple (BGT), and given applications of this notion to the theory Gabor frames and classical Fourier analysis. BGTs both simplifies and extends the scope of the classical Hilbert Gelfand triples which have played an important role in quantum mechanics (it has justified Diracs bra-ket Dirac formalism and allowed the study of generalized eigenvalues and vectors). On the other hand, one of the most fruitful formulations of QM uses the theory of pseudo- differential operators, which are closely related to the notion of quantization. It turns out that mathematicians working in TFA have recently rediscovered one of the oldest quantization schemes, going back to the physicists Born and Jordan (1925), and showed that the corresponding pseudo-differential formalism yields much better numerical results than the usual Kohn-Nirenberg or Weyl formalism; these theoretical results have been fully corroborated by numerical experiments (Boggiatto and his collaborators, Turunen). There are also strong indications that the Born-Jordan correspondence is physically the correct quantization scheme (as opposed to the usual Weyl quantization). The study of this new pseudo-differential calculus is still in its infancy, and deserves to be studied and developed, not only because it has obvious applications to TFA and QM, but also because it exhibits unexpected difficulties (for instance, the Born-Jordan correspondence is neither injective nor surjective, as opposed to the usual pseudo-differential correspondences!) Moreover, nothing is known about the spectral properties of Born-Jordan pseudo-differential operators (spectral properties strongly depend on the chosen quantization, as well-known examples show); the natural framework for the study of these properties seems to the theory of BGTs as soon as the involved symbols belong to non-standard classes (e.g. the Shubin or Sjöstrand classes). Thus, a complete theory for the action of Born-Jordan operators on functional spaces (e.g. modulation spaces or Schatten classes) is necessary. Parts of this project are applications and consequences of the results obtained during the previous FWF Projects FWF P23902 und P20442, which have produced a large number of publications and new results.

The aim of the current project was to expand and analyze from a mathematical perspective the quantization scheme proposed in 1925 by Born and Jordan (and Heisenberg). This research has led to a large number of published papers, and to a book published by Springer-Verlag. We have expanded the original approach using mathematical tools such as the Cohen classes, Gabor frame theory, and harmonic analysis. Quite surprisingly we have found that the Born Jordan quantization rule is actually the correct rule, as opposed to the more commonly used Weyl quantization. This fact is corroborated by recent joint work with a team of Max-Planck-Institut quantum chemists in Garching. The project is therefore certainly not ended, but has rather led to new and exciting perspectives, with potential applications to applied Sciences (quantum computing and optics, chemistry, cryptography).