Information, Geometry, and Physics Seminar

In Bargmann–Fock quantization, an observable g acts by multiplication followed by projection onto holomorphic quantum states. Under the Bargmann transform, this compressed multiplication operator is closely related to Weyl quantization, with the heat evolution g^(1/4) appearing at a distinguished endpoint time. Berger and Coburn conjectured that boundedness of the resulting quantum operator should be equivalent to boundedness of this endpoint heat transform.

I will discuss recent work showing that this conjectural picture breaks down in a rather dramatic way. In every dimension, one can construct symbols g for which the quantized operator is bounded, even Hilbert–Schmidt, while g^(1/4) is unbounded. I will explain the phase-space mechanism behind the construction, emphasizing the roles of oscillation, heat flow, and localization in the Bargmann–Fock setting.