Preface ........................................................ ix
1 Prologue ..................................................... 1
1.1 Reality and its description ............................. 1
1.2 A quantum education and evolution ....................... 8
References .................................................. 14
2 Quantum Algebra ............................................. 15
2.1 The quantum algebra of Dirac ........................... 15
2.2 The von Neumann perspective ............................ 19
2.3 The measurement algebra of Schwinger ................... 23
2.4 Weyl-Moyal algebra and the Moyal bracket ............... 35
2.5 Quantum algebras over phase space ...................... 49
2.6 Moshe Rato remembered .................................. 51
References .................................................. 55
3 Probability in the quantum world ............................ 57
3.1 The statistical interpretation of quantum theory ....... 57
3.2 The uncertainty principle of Heisenberg ................ 62
3.3 Hidden variables ....................................... 65
3.4 EPR .................................................... 74
3.5 Transition probabilities in the atom ................... 79
3.6 Feynman path integrals and the Feynman-Kac formula ..... 81
References .................................................. 90
4 Super geometry .............................................. 93
4.1 The evolution of classical geometry ..................... 93
4.2 Super geometry ........................................ 100
4.3 The theory of super manifolds ......................... 105
4.4 Super Lie groups ...................................... 116
References ................................................. 118
5 Unitary representations of super Lie groups ................ 121
5.1 Unitary representations of a super Lie group .......... 121
5.2 Super imprimitivity theorem for even homogeneous
spaces ................................................ 129
5.3 Super semidirect products and their unitary
irreducible representations ........................... 136
5.4 Super Poincare Lie algebras and Lie groups ............ 143
5.5 Unitary representations of super Poincare groups ...... 150
5.6 Super particles and their multiplet structure ......... 152
References ................................................. 153
6 Nonarchimedean physics ..................................... 155
6.1 Planck scale and the Volovich hypothesis .............. 155
6.2 Nonarchimedean symmetry groups and their multipliers .. 157
6.3 Elementary particles over nonarchimedean fields ....... 165
6.4 Nonarchimedean quantum field theory ................... 174
References ................................................. 176
7 Differential equations with irregular singularities ........ 179
7.1 Introduction .......................................... 180
7.2 Reduction theory ...................................... 184
7.3 Formal reduction at an irregular singularity and the
theory of linear algebraic groups ..................... 187
7.4 Analytic reduction theory: Stokes phenomenon .......... 193
7.5 The Stokes sheaf and the scheme structure on H1(St) ... 195
7.6 Reduction theory for families ......................... 197
References ................................................. 204
8 Mackey, Harish-Chandra, and representation theory .......... 207
8.1 George Mackey and his view of representation theory ... 207
8.2 Harish-Chandra as I knew him .......................... 212
8.3 Some reflections ...................................... 217
8.4 Fourier analysis on semisimple groups ................. 223
References ................................................. 235
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