Table of Contents
Preface ix
1 Forms of quantum mechanics 1-1
1.1 The Schrodinger and Heisenberg pictures 1-2
1.2 Interaction picture 1-4
1.3 Density-matrix formulation of quantum mechanics 1-7
1.3.1 Mixed states 1-9
1.3.2 Density matrix and ensemble average 1-12
1.3.3 Time dependence of the density matrix 1-15
1.4 Further contents of Heisenberg's commutation relations 1-19
1.4.1 Rotation group and its extension to the Lorentz group 1-19
1.4.2 Harmonic oscillators and Fock space 1-20
1.4.3 Dirac's two-oscillator system 1-21
References 1-22
2 Lorentz group and its representations 2-1
2.1 Lie algebra of the Lorentz group 2-2
2.2 Two-by-two representation of the Lorentz group 2-5
2.3 Four-vectors in the two-by-two representation 2-6
2.4 Transformation properties in the two-by-two representation 2-9
2.5 Subgroups of the Lorentz group 2-10
2.6 Decompositions of the Sp(2) matrices 2-11
2.6.1 Bargmann decomposition 2-11
2.6.2 Iwasawa decomposition 2-13
2.7 Bilinear conformal representation of the Lorentz group 2-13
References 2-14
3 Internal space-time symmetries 3-1
3.1 Wigner's little groups 3-3
3.1.1 O(3)-like little group for massive particles 3-4
3.1.2 E(2)-like little group for massless particles 3-5
3.1.3 O(2,1)-like little group for imaginary-mass particles 3-7
3.2 Little groups in the light-cone coordinate system 3-9
3.3 Two-by-two representation of the little groups 3-13
3.4 One expression with three branches 3-15
3.5 Classical damped oscillators 3-18
References 3-20
4 Photons and neutrinos in the relativistic world of Maxwell and Wigner 4-1
4.1 The Lorentz group and Wigner's little groups 4-2
4.2 Massive and massless particles 4-7
4.3 Polarization of massless neutrinos 4-9
4.3.1 Dirac spinors and massless particles 4-10
4.4 Scalars, vectors, tensors, and the polarization of photons 4-11
4.4.1 Four-vectors 4-13
4.4.2 Second-rank tensor 4-14
4.4.3 Higher spins 4-17
References 4-18
5 Wigner functions 5-1
5.1 Basic properties of the Wigner phase-space distribution function 5-2
5.2 Time dependence of the Wigner function 5-4
5.3 Wave packet spread 5-6
5.4 Harmonic oscillators 5-8
5.5 Minimum uncertainty in phase space 5-10
5.6 Density matrix 5-13
5.7 Measurable quantities 5-15
References 5-19
6 Coherent states of light 6-1
6.1 Phase-number uncertainty relation 6-3
6.2 Baker-Campbell-Hausdorff relation 6-4
6.3 Coherent states 6-7
6.4 Symmetry of coherent states 6-10
6.5 Coherent states in phase space 6-13
6.6 Single-mode squeezed states 6-16
References 6-18
7 Squeezed states and their symmetries 7-1
7.1 Two-mode states 7-2
7.2 Unitary transformations 7-3
7.3 Symmetries of two-mode states 7-6
7.4 Dirac matrices and O(3,3) symmetry 7-8
7.5 Symmetries in phase space 7-10
7.6 Two coupled oscillators 7-15
References 7-20
8 Entanglement and entropy 8-1
8.1 Density matrix and entropy 8-2
8.2 Two-by-two density matrices 8-4
8.3 Density matrix for two-oscillator states 8-5
8.4 Entropy for the two-mode state 8-7
8.5 Entangled excited states 8-9
8.6 Wigner functions and uncertainty relations 8-12
References 8-15
9 Ray optics and optical activities 9-1
9.1 Ray optics using the group of ABCD matrices 9-2
9.1.1 Diagonalization properties of the ABCD matrices 9-3
9.1.2 Decompositions of the ABCD matrices 9-5
9.1.3 Recomposition of the ABCD matrices 9-7
9.2 Physical examples using ABCD matrices 9-9
9.2.1 Optics using multilayers 9-12
9.2.2 Ray optics applied to cameras 9-15
9.3 Optical activities 9-17
9.3.1 Computation of the transformation matrix U 9-19
9.3.2 Correspondence to space-time symmetries 9-22
References 9-24
10 Polarization optics 10-1
10.1 Jones vector, phase shifters, and attenuators 10-2
10.1.1 Squeeze and phase shift 10-4
10.1.2 Rotation of the polarization axes and combined effects 10-6
10.1.3 The SL(2, c) content of polarization optics 10-9
10.2 New filters and possible applications 10-10
10.3 Non-orthogonal coordinate systems 10-12
References 10-14
11 Stokes parameters and Poincaré sphere 11-1
11.1 Polarization optics and decoherence 11-2
11.2 Coherency matrix and Stokes parameters 11-3
11.3 Poincaré sphere 11-5
11.3.1 Two concentric Poincaré spheres 11-5
11.3.2 O(3, 2) symmetry of the Poincaré sphere 11-7
11.3.3 The Poincaré circle 11-9
11.3.4 Diagonalization of the coherency matrix 11-10
11.4 The entropy problem 11-11
11.5 Further symmetries from the Poincaré sphere 11-11
11.5.1 Momentum four-vector and the Poincaré sphere 11-11
11.5.2 Mass variation within O(3, 2) symmetry 11-13
References 11-14
Appendix A Covariant harmonic oscillators and the quark-parton puzzle A-1
A.1 The covariant harmonic oscillator A-2
A.1.1 Differential equations of the covariant harmonic oscillator A-3
A.1.2 Normalizable solutions of the relativistic oscillator equations A-4
A.1.3 Lorentz transformations of harmonic oscillator wave functions A-9
A.1.4 Covariant phase-space picture of harmonic oscillators A-11
A.2 Quark-parton puzzle A-14
A.2.1 Lorentz-covariant quark model A-15
A.2.2 Feynman's parton picture A-18
A.2.3 Proton structure function and form factor A-20
A.2.4 Coherence in momentum-energy space A-28
A.2.5 Hadronic temperature A-29
References A-30
Index I-1
