AG Mathematische Physik, Michael Heins (Würzburg): A Holomorphic Perspective of Strict Deformation Quantization

Michael Heins (Würzburg)

A Holomorphic Perspective of Strict Deformation Quantization

Abstract:
Classical mechanical systems may be modelled by means of Poisson algebras, which physically arise as the observables corresponding to the system. The quantization scheme called Deformation Quantization establishes a second product on the observable algebra, a so-called star product. Imposing a classical and semiclassical limit condition with respect to a positive parameter ℏ then intertwines the classical with the quantum multiplication. To facilitate the construction, it is both customary and advisable to take a step back and pass to formal power series with coefficients in the classical algebra and indeterminant ℏ. One thus speaks of a formal deformation quantization.

This talk is about overcoming the formal character of the resulting theory, which is a
necessity for physically meaningful statements, and accounts for the fact that ℏ is a physical constant and not a free parameter. As the formal star product is, in particular, itself given by a power series, this naturally leads into the realm of holomorphic functions, both in finite and infinite dimensions. After briefly reviewing the formal situation, we discuss a recently proposed notion of strict deformation quantization and investigate how one can use established results from complex analysis to think about these objects. Along the way, we shall illustrate the abstract ideas in the setting of standard ordered quantization on the cotangent bundle of the real line.