2021/05/18 by Micho Durdevich, Durdevich, Micho, Stephen Bruce Sontz +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Operator Algebras (math.OA) #Quantum Physics (quant-ph) #math.OA #quant-ph
paper · pdf · doi:10.48550/arxiv.2105.08400
46 Classical Pages
arxiv created 2021/05/18 · arxiv updated 2021/05/19
The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associated to the symbols from another quadratic non-commutative algebra, which is interpretable as polynomials over a plane to be quantized. Such a conceptual framework promotes interesting non-trivial conditions on the initial scalar product. These are analyzed in detail. Various illustrative examples are computed.