Essential contents on quantization of fields and second quantization, which are necessary for undergraduate and graduate level, are presented in this chapter. An innovative approach is used for the presentations and the discussions on field quantization and second quantization for systems of Bosons and Fermions. The chapter covers field operators, the procedure of second quantization and its use for the Schrödinger equation for both Bosons and Fermions, matrix representation of Fermionic operators, number operators for both Bosons and Fermions, and systems of weakly interacting Bosons and Fermions. A good number of solved examples and unsolved exercises clarify the basic concepts behind quantization of fields and second quantization.Essential contents on quantization of fields and second quantization, which are necessary for undergraduate and graduate level, are presented in this chapter. An innovative approach is used for the presentations and the discussions on field quantization and second quantization for systems of Bosons and Fermions. The chapter covers field operators, the procedure of second quantization and its use for the Schrödinger equation for both Bosons and Fermions, matrix representation of Fermionic operators, number operators for both Bosons and Fermions, and systems of weakly interacting Bosons and Fermions. A good number of solved examples and unsolved exercises clarify the basic concepts behind quantization of fields and second quantization.

CRC Press eBooksJun 18, 20210 citations

Deformation quantization of geometric quantum mechanics

H García-Compeán, J F Plebanski, M Przanowski +1 more

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrödinger spinless field is considered. Under the assumption that the phase space of the Schrödinger field is ∞ , 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 P ∞ endowed with the Fubini-Study Kählerian 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/ℏ.Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrödinger spinless field is considered. Under the assumption that the phase space of the Schrödinger field is ∞ , 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 P ∞ endowed with the Fubini-Study Kählerian 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/ℏ.

arXiv:hep-th/0112049Journal of Physics A Mathematical and GeneralMay 3, 20026 citations

Deformation Quantization of Geometric Quantum Mechanics

Hugo García‐Compeán, Jerzy F Plebanski, Maciej Przanowski +1 more

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 ${1 \\over \\hbar}$ .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 ${1 \\over \\hbar}$.

Dec 7, 20014 citations

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}