Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer

Weyl-Underhill-Emmrich (WUE) quantization and its generalization are considered. It is shown that an axiomatic definition of the Stratonovich-Weyl (SW) quantizer leads to severe difficulties. Quantization on the cylinder within the WUE formalism is discussed.Comment: 15+1 pages, no figure

Deformation Quantization of Bosonic Strings

Deformation quantization of bosonic strings is considered. We show that the light-cone gauge is the most convenient classical description to perform the quantization of bosonic strings in the deformation quantization formalism. Similar to the field theory case, the oscillator variables greatly facilitates the analysis. The mass spectrum, propagators and the Virasoro algebra are finally described within this deformation quantization scheme.Comment: 33+1 pages, harvmac file, no figure

Deformation Quantization of Geometric Quantum Mechanics

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$.Comment: 27+1 pages, harvmac file, no figure

Deformation quantization of cosmological models

The Weyl-Wigner-Groenewold-Moyal formalism of deformation quantization is applied to cosmological models in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in a flat minisuperspace. The de Sitter cosmological model is worked out in detail and the computation of the Wigner functions for the Hartle-Hawking, Vilenkin and Linde wave functions are done numerically. The Wigner function is analytically calculated for the Kantowski-Sachs model in (non)commutative quantum cosmology and for string cosmology with dilaton exponential potential. Finally, baby universes solutions are described in this context and the Wigner function is obtained.Comment: 37 pages, 16 figure

Uncertainty Relations in Deformation Quantization

Robertson and Hadamard-Robertson theorems on non-negative definite hermitian forms are generalized to an arbitrary ordered field. These results are then applied to the case of formal power series fields, and the Heisenberg-Robertson, Robertson-Schr\"odinger and trace uncertainty relations in deformation quantization are found. Some conditions under which the uncertainty relations are minimized are also given.Comment: 28+1 pages, harvmac file, no figures, typos correcte