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Mathematical Foundations of Quantum Mechanics
K. R. Parthasarathy
Revised with the assistance of M. Krishna
J51OI H I N D U S T A N lEULl BOOK AGENCY
Contents
Preface Chapter 1.PROBABILITY THEORY ON THE LATTICE OF PROJECTIONS IN A HILBERT SPACE 1.1. Gleason's theorem 1.2. Observables and expectation 1.3. Wigner's theorem 1.4. Covariant description of a quantum mechanical system 1.5.Observables arising from a covariant description 1.6. Hahn-Hellinger theorem Chapter 2. SYSTEMS WITH A CONFIGURATION UNDER A GROUP ACTION 2.1. Position as an observable 2.2. Examples of imprimitivitysystems 2.3. The imprimitivity theorem of Mackey for transitive group actions 2.4. Equivalence of canonical imprimitivity systems 2.5. Irreducibility of canonical imprimitivity systems 2.6. Existenceof p.u.a representations 2.7. An alternative description of the induced representation 2.8. Inducing in stages vii 1 1 15 21 30 32 33 43 43 45 49 59 66 67 71 74
Chapter 3. MULTIPLIERS ON LOCALLYCOMPACT GROUPS 85 3.1. Borel multipliers on general locally compact groups 85 3.2. Multipliers on compact groups 90 3.3. Symmetric multipliers on abelian groups 91 3.4. Multipliers on semidirectproducts 92 3.5. Multipliers on Lie groups 94 3.6. Multipliers on W1 x W1 and T n x T n 98 3.7. Multipliers on semisimple Lie groups 101 3.8. Multipliers on a semidirect product of a vector space and alinear Lie group 105 3.9. The Galilean group and its multipliers 109
vi
CONTENTS
3.10.
The inhomogeneous Lorentz group and its multipliers
115
Chapter 4. THE BASIC OBSERVABLES OF AQUANTUM MECHANICAL SYSTEM 125 4.1. Projective unitary antiunitary representations of semidirect products when the normal component is abelian 125 4.2. Projective unitary representations of the coveringgroup of the Galilean group 131 4.3. A description of observables of a quantum mechanical system covariant under the action of the Galilean group 139 4.4. The Hamiltonian of an N-particle system 142...
K. R. Parthasarathy
Revised with the assistance of M. Krishna
J51OI H I N D U S T A N lEULl BOOK AGENCY
Contents
Preface Chapter 1.PROBABILITY THEORY ON THE LATTICE OF PROJECTIONS IN A HILBERT SPACE 1.1. Gleason's theorem 1.2. Observables and expectation 1.3. Wigner's theorem 1.4. Covariant description of a quantum mechanical system 1.5.Observables arising from a covariant description 1.6. Hahn-Hellinger theorem Chapter 2. SYSTEMS WITH A CONFIGURATION UNDER A GROUP ACTION 2.1. Position as an observable 2.2. Examples of imprimitivitysystems 2.3. The imprimitivity theorem of Mackey for transitive group actions 2.4. Equivalence of canonical imprimitivity systems 2.5. Irreducibility of canonical imprimitivity systems 2.6. Existenceof p.u.a representations 2.7. An alternative description of the induced representation 2.8. Inducing in stages vii 1 1 15 21 30 32 33 43 43 45 49 59 66 67 71 74
Chapter 3. MULTIPLIERS ON LOCALLYCOMPACT GROUPS 85 3.1. Borel multipliers on general locally compact groups 85 3.2. Multipliers on compact groups 90 3.3. Symmetric multipliers on abelian groups 91 3.4. Multipliers on semidirectproducts 92 3.5. Multipliers on Lie groups 94 3.6. Multipliers on W1 x W1 and T n x T n 98 3.7. Multipliers on semisimple Lie groups 101 3.8. Multipliers on a semidirect product of a vector space and alinear Lie group 105 3.9. The Galilean group and its multipliers 109
vi
CONTENTS
3.10.
The inhomogeneous Lorentz group and its multipliers
115
Chapter 4. THE BASIC OBSERVABLES OF AQUANTUM MECHANICAL SYSTEM 125 4.1. Projective unitary antiunitary representations of semidirect products when the normal component is abelian 125 4.2. Projective unitary representations of the coveringgroup of the Galilean group 131 4.3. A description of observables of a quantum mechanical system covariant under the action of the Galilean group 139 4.4. The Hamiltonian of an N-particle system 142...
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