Path Integrals and Coherent States of SU(2) and SU(1,1)

By: A Inomata, Hiroshi Kuratsuji, C C Gerry

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Hardcover | 1 December 1991

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328 Pages

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22.86 x 17.15 x 2.54

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The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals of SU(2) and SU(1,1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problems in curved space and others. The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent states, formulation of coherent path integrals for spin and unitary spin, and semiclassical quantization are presented. Applications are made to the study of quantum fluctuation, the nonlinear field model and phase holonomy. The final chapters present the theory of the SU(1,1) coherent states and their applications. The radial Coulomb problem, the Morse oscillator, and the large-N approximation are discussed. Applications to problems in quantum optics such as squeezed states, interaction with the squeezed vacuum states, and phase operator formalism are also included. This book is intended as an introduction to the subject as well as a work of reference.

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Preface
Path Integrals for SU(2) and SU(1, 1)	p. 1
Introduction	p. 3
Techniques for Path Integration	p. 8
Rules of Thumb in Path Integral Calculus	p. 8
Change of Variables in a Path Integral	p. 12
Local Time Rescaling Techniques	p. 16
Path Integrals in Polar Coordinates	p. 22
Asymptotic Recombination Techniques	p. 28
Dimensional Extension Techniques	p. 32
Path Integrals on SU(2)	p. 40
The Group SU(2)	p. 40
Path Integrals on SU(2)	p. 56
The Poschl-Teller Oscillator	p. 61
The Hulthen Potential	p. 68
Path Integrals for SU(1, 1)	p. 76
The Group SU(1, 1)	p. 76
Continuous Basis for Representations of SU(1, 1)	p. 85
The Modified Poschl-Teller System	p. 92
Harmonic Oscillators Revisited	p. 98
The Coulomb Problem	p. 106
The Morse Oscillator	p. 112
Exactly Path-Integrable Examples	p. 115
Basic Soluble Examples	p. 115
Other Soluble Examples	p. 117
Appendix. Representations of SU(2) and SU(1, 1)	p. 121
Path Integrals in the SU(2) Coherent State Representation and Related Topics	p. 139
Introduction	p. 141
Grauber States Revisited	p. 145
SU(2) Coherent States and their Generalization	p. 149
Elementary Construction by Spin Algebra	p. 149
Lie Group Construction of the GCS	p. 157
Path Integrals for Spin and Unitary Spin	p. 164
Path Integrals for Spin	p. 164
Path Integrals for Unitary Spin	p. 169
Semiclassical Quantization Theory	p. 174
Formula without Quantum Fluctuations	p. 174
Formula with Quantum Fluctuations	p. 178
Topological Aspect of Semiclassical Quantization	p. 184
Applications	p. 188
Topological Invariants in Many-Particle Systems	p. 188
Geometric Characteristics of the Canonical Term	p. 191
Nonlinear Field Model	p. 194
General Formulation	p. 194
P[subscript 1](C) Model (Continuous Spin)	p. 197
Phase Holonomy	p. 202
General Remarks	p. 202
Coherent State Path Integrals for Adiabatic Motion	p. 205
Appendix A. The Kernel Function	p. 210
Appendix B. Invariant Measure of the Grassmann Manifold	p. 211
Appendix C. The Poisson Bracket in Generalized Phase Space	p. 213
SU(1, 1) Coherent States and Path Integrals for SU(1, 1)	p. 219
Introduction	p. 221
SU(1, 1) and the Perelemov Coherent States	p. 224
SU(1, 1) and Its UIRs	p. 224
Examples of SU(1, 1) as an SGA	p. 227
SU(1, 1) Coherent States	p. 229
Path Integral and Classical Dynamics	p. 233
Path Integrals	p. 234
Conserved Noether Currents for Dynamical Groups	p. 238
Most General Coherent Preserving Hamiltonian	p. 239
Time Dependent Invariants	p. 244
Direct Path Integral Approach to Time Evolution	p. 246
Applications in Quantum Mechanics	p. 251
Radial Isotropic Oscillator	p. 251
Radial Coulomb Problem	p. 257
The Morse Oscillator	p. 266
Application to the Large-N Approximation	p. 271
Application to Quantum Optics	p. 281
Introduction	p. 281
Squeezed States and SU(1, 1) CS	p. 284
Interactions of the Squeezed Vacuum States	p. 289
Phase Operator Formalism for SU(1, 1)	p. 296
Berry's Phase in the Degenerate Parametric Amplifier	p. 301
Related Matters	p. 308
Addendum - Further Developments	p. 309
Table of Contents provided by Blackwell. All Rights Reserved.

Path Integrals and Coherent States of SU(2) and SU(1,1)

At a Glance

Hardcover

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