Abstract
We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } ⊂ C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on ∂ Ω × S 1 ). Several open problems are indicated throughout the survey.
Highlights
The present paper is a survey of known results on the mathematical analysis of weighted Bergman kernels and their applications to mathematical physics, such as the theory of quantization of states of mechanical systems, and back to complex analysis where some of the matters regarding weighted Bergman kernels arise
Schmidt’s bundle boundary constructions suggesting a connection that ought to be discovered among the mathematical analysis of weighted Bergman kernels and the physics of space-time singularities
The Forelli–Rudin–Ligocka–Peloso expansion is discussed in Section 6 which emphasizes the authors’ work [21] (itself relying on the treatment in [2] of weighted Bergman kernels as functions K : AW (Ω) → H A(Ω) on the Banach manifold of admissible weights) vis-a-vis to that by M
Summary
The present paper is a survey of known results on the mathematical analysis of weighted Bergman kernels and their applications to mathematical physics, such as the theory of quantization of states of mechanical systems, and back to complex analysis where some of the matters regarding weighted Bergman kernels arise. There is a fascinating formal resemblance between Naruki’s G-admissible metrics and B.G. Schmidt’s bundle boundary constructions (cf [24]) suggesting a connection that ought to be discovered among the mathematical analysis of weighted Bergman kernels and the physics of space-time singularities (cf e.g., C.J.S. Clarke [25]). Authors’ choice of mathematical physics results as related to the theory of weighted Bergman kernels occupies Section 4 and relies mainly on the scientific creation of K. The Forelli–Rudin–Ligocka–Peloso expansion is discussed in Section 6 which emphasizes the authors’ work [21] (itself relying on the treatment in [2] of weighted Bergman kernels as functions K : AW (Ω) → H A(Ω) on the Banach manifold of admissible weights) vis-a-vis to that by M.
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