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The Resource Born-Jordan Quantization : Theory and Applications

Born-Jordan Quantization : Theory and Applications

Label
Born-Jordan Quantization : Theory and Applications
Title
Born-Jordan Quantization
Title remainder
Theory and Applications
Creator
Subject
Language
eng
Member of
Cataloging source
MiAaPQ
Literary form
non fiction
Nature of contents
dictionaries
Series statement
Fundamental Theories of Physics Ser.
Series volume
v.182
Born-Jordan Quantization : Theory and Applications
Label
Born-Jordan Quantization : Theory and Applications
Link
http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=4333606
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Copyright
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Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Preface -- Contents -- 1 Introduction -- 1.1 From Quantum Theory to Quantum Mechanics -- 1.2 What We Do in This Book -- References -- Part IBorn-Jordan Quantization:Physical Motivation -- 2 On the Quantization Problem -- 2.1 Introduction -- 2.2 The Ordering Problem -- 2.3 What Is Quantization? -- 2.4 Motivation for Born--Jordan Quantization -- References -- 3 Quantization of Monomials -- 3.1 Polynomial Algebras -- 3.1.1 General Considerations and Notation -- 3.1.2 Commutation Relations -- 3.2 Some Common Orderings -- 3.2.1 Weyl Ordering -- 3.2.2 Born--Jordan Ordering -- 3.2.3 The Relation Between OpW (xrps) and OpBJ(xrps) -- 3.2.4 Shubin's x-Ordering -- 3.3 General Quantization Axioms for Monomials -- 3.3.1 The Domingo--Galapon Formula -- References -- 4 Basic Hamiltonian Mechanics -- 4.1 Hamiltonian Dynamics -- 4.2 Free Canonical Transformations -- 4.2.1 Free Symplectic Matrices -- 4.2.2 Hamilton's Two-Point Characteristic Function -- 4.3 The Action of a Dynamical System -- 4.3.1 The Poincaré--Cartan Invariant -- 4.3.2 The Action Functional -- 4.4 Short-Time Action -- 4.4.1 On M̀̀id-Point Rules'' -- 4.4.2 A Correct Short-Time Approximation -- 4.4.3 The Averaged Hamiltonian overlineH -- 4.4.4 Short-Time Approximations: The General Case -- References -- 5 Wave Mechanics and the Schrödinger Equation -- 5.1 Matter Waves -- 5.1.1 The Free Particle -- 5.1.2 The Free Propagator -- 5.2 The General Case -- 5.2.1 The Abstract Schrödinger Equation -- 5.2.2 The Approximate Wavefunction -- 5.2.3 Back to Schrödinger's Equation -- References -- Part IIMathematical Aspects of Born-JordanQuantization -- 6 The Weyl Correspondence -- 6.1 Definitions and Basics -- 6.1.1 Notation and Terminology -- 6.1.2 Traditional Definition of Weyl Operators (in Physics) -- 6.1.3 Traditional Definition of Weyl Operators (in Mathematics)
  • 6.2 Heisenberg and Grossmann--Royer Operators -- 6.2.1 Definition and Discussion -- 6.2.2 Symplectic Fourier Transform -- 6.2.3 Dynamical Interpretation of the Heisenberg Operator -- 6.3 Weyl Operators: Harmonic Analysis -- 6.3.1 Definition of a Weyl Operator -- 6.3.2 Harmonic Analysis -- 6.3.3 The Kernel of a Weyl Operator -- 6.4 Two Examples -- 6.4.1 The Weyl Quantization of Monomials -- 6.4.2 Physical Hamiltonians -- 6.5 L2-Boundedness of Weyl Operators -- 6.6 Adjoints and Products -- 6.6.1 The (Formal) Adjoint of a Weyl Operator -- 6.6.2 Composition Formulas -- References -- 7 The Cohen Class -- 7.1 The Wigner and Ambiguity Transforms -- 7.1.1 The Cross-Wigner Transform -- 7.1.2 The Cross-Ambiguity Function -- 7.1.3 A Reconstruction Formula -- 7.1.4 Relation with the Weyl Correspondence -- 7.2 The Cohen Class -- 7.2.1 Definition and General Properties -- 7.2.2 The Marginal Conditions -- 7.2.3 Generalization of Moyal's Identity -- 7.2.4 A Representation Result -- References -- 8 Born--Jordan Quantization -- 8.1 The Born--Jordan Kernel kBJ -- 8.1.1 Definition and First Properties -- 8.1.2 The Moyal Identity Is Not Satisfied -- 8.1.3 Some Properties of the sinc Function -- 8.2 Born--Jordan Operators -- 8.2.1 Definition and First Properties -- 8.2.2 The Relation Between Born--Jordan and Weyl Operators -- 8.3 On the Invertibility of Born--Jordan Quantization -- 8.3.1 A Non-injectivity Result -- 8.3.2 The Case of Monomials -- 8.3.3 The General Case -- References -- 9 Shubin's Pseudo-Differential Calculus -- 9.1 Definition and First Properties -- 9.1.1 The Kernel of a x-Operator -- 9.1.2 The Case of Monomials -- 9.2 The x-Wigner and Ambiguity Transforms -- 9.2.1 The Operators T"0362Tx(z0) and T"0362TGR,x(z0) -- 9.2.2 The Associated Transforms -- 9.2.3 Definition of Wigx and Ambx -- 9.2.4 Properties of x and -- 9.3 Back to Shubin's Operators
  • 9.3.1 Harmonic Decomposition of x-Operators -- 9.3.2 Products, Transposes, Adjoints -- References -- 10 Born--Jordan Pseudo-Differential Operators -- 10.1 Born--Jordan Pseudo-Differential Operators -- 10.1.1 Definition and Justification -- 10.1.2 Born--Jordan and Weyl Operators -- 10.2 The Born--Jordan Transform Revisited -- 10.2.1 Born--Jordan Versus Weyl -- 10.3 Tensor Products of Observables -- 10.3.1 Weyl Operators -- 10.3.2 The Born--Jordan Case -- 10.3.3 Illustration: The {u008D}Angular Momentum Dilemma{u008E} -- References -- 11 Weak Values and the Reconstruction Problem -- 11.1 The Notion of Weak Value -- 11.1.1 Motivation and Definition -- 11.1.2 Weak Values and the Cohen Class: The Phase Space Approach -- 11.2 A Complex Probability Density -- 11.2.1 A General Result Using the Cohen Class -- 11.2.2 The Born--Jordan Case -- 11.3 The Reconstruction Problem -- 11.3.1 A General Reconstruction Formula -- 11.3.2 The Born--Jordan Case: An Unsolved Problem -- References -- Part III Some Advanced Topics -- 12 Metaplectic Operators -- 12.1 The Symplectic Group -- 12.1.1 The Metaplectic Representation of -- 12.2 The Weyl Representation of Metaplectic Operators -- 12.2.1 The Symplectic Cayley Transform -- 12.2.2 A Factorization Result -- 12.2.3 Metaplectic Operators as Weyl Operators -- 12.3 The -Metaplectic Group -- 12.3.1 The Operators -- 12.3.2 Definition of -- References -- 13 Symplectic Covariance Properties -- 13.1 Symplectic Covariance of Weyl Operators -- 13.1.1 The Heisenberg and Grossmann--Royer Operators Revisited -- 13.1.2 Weyl Operators -- 13.1.3 Affine Covariance Properties -- 13.2 The Case of x-Operators -- 13.2.1 A Covariance Result -- 13.2.2 Application to the x-Wigner Function -- 13.3 The Case of Born--Jordan Operators -- 13.3.1 The Born--Jordan Metaplectic Group -- 13.3.2 The Born--Jordan Wigner Transform Revisited -- References
  • 14 Symbol Classes and Function Spaces -- 14.1 Shubin's Symbol Classes -- 14.1.1 The Shubin Symbol Classes Dum and Sum,m -- 14.1.2 Asymptotic Expansions of Symbols -- 14.1.3 A Reduction Result -- 14.1.4 First Continuity Results -- 14.2 Modulation Spaces -- 14.2.1 The Modulation Spaces Msq -- 14.2.2 The Generalized Sjöstrand Classes -- 14.3 Applications to Born--Jordan Operators -- 14.3.1 A Fundamental Result -- 14.3.2 Application: Some Boundedness Results -- References -- References -- Index
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online
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9783319279022
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rdamedia
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