Coherent States and Anti-Wick Quantization

Abstract

The theory of coherent states plays an important role in various aspects of representation theory, and of, course, in quantum mechanics from which it originates. Historically, the notion of coherent state goes back to Schrödinger’s 1926 work [143] on non-dispersing wavepackets for a harmonic oscillator. In 1932 von Neumann [130] considered sets of coherent states associated with a division of phase space into quantum cells. The modern theory was initiated by Glauber’s 1963 work [62] in quantum optics (Glauber was awarded the 2005 Nobel Prize in Physics for his contributions; we mention that there was a controversy about priorities involving the physicist Sudarshan, also famous for his work on coherent states). The theory of coherent states has since then been applied to a variety of problems in mathematics and mathematical physics, and has been extended in various directions, for instance within the framework of anti-Wick (also called Toeplitz, or Berezin) quantization (Berezin [13]) which we will study later in this chapter.