The Method of Second QuantizationElsevier, 2 dic 2012 - 240 pagine The Method of Second Quantization deals with the method of second quantization and its use to solve problems of quantum mechanics involving an indefinite number of particles, mainly in field theory and quantum statistics. Topics covered include operations on generating functionals; linear canonical transformations; quadratic operators; and Thirring's four-fermion model. State spaces and the simplest operators are also described. This book is comprised of four chapters and begins with an overview of the method of second quantization and the relevant notations. The first chapter focuses on the connections between vectors and functionals and between operators and functionals, together with fundamental rules for operating on functionals. The reader is then introduced to the so-called quadratic operators and the linear canonical transformations closely connected with them. Quadratic operators reduced and not reduced to normal form are considered. The final chapter discusses the Thirring model, the simplest relativistically invariant model in quantum field theory, and explains why it includes infinities. This monograph will be of value to students and practitioners of mathematical physics. |
Sommario
| 1 | |
| 9 | |
Chapter II Linear Canonical Transformations | 87 |
Chapter III Quadratic Operators | 133 |
Chapter IV Thirrings Model in Quantum Field Theory | 181 |
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adjoint antisymmetric arbitrary bilinear form Bose bosons bounded operator calculation canonical trans commutation relations continual integral converges strongly defined defined by formula degrees of freedom denote dense set derivatives difficult to verify domain of definition dx dy elements equal equation Evidently exists expression Fermi fermions final find finite vectors finite-dimensional first fixed follows formal power series fulfilled functional A(a functional corresponding Grassmann algebra Hence Hermitian Hilbert space Hilbert-Schmidt operator infinite inner product invariant with respect inverse kernel Lemma limit linear canonical transformation matrix form norm normal form number of degrees operator H orthogonal orthonormal basis particles polynomial projection operator proof proper canonical transformation proved quadratic operator representation right member satisfies satisfying the condition scattering operator second quantization self-adjoint operator sequence space 9 square-integrable functions subsection subspace sufficient symmetric trace-class operator unitary operator vacuum vector variables verified