Spectral theory and quantum mechanics : with an introduction to the algebraic formulation

Cover......Page 1
Titlepage......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 9
1.1 On the book 1.1.1 Scope and structure......Page 17
1.1.3 General conventions......Page 20
1.2 On Quantum Mechanics 1.2.1 Quantum Mechanics as a mathematical theory......Page 21
1.2.2 QM in the panorama of contemporary Physics......Page 23
1.3.1 Open/closed sets and basic point-set topology......Page 26
1.3.2 Convergence and continuity......Page 28
1.3.3 Compactness......Page 30
1.3.4 Connectedness......Page 31
1.4.1 Measure spaces......Page 32
1.4.2 Positive s -additive measures......Page 35
1.4.3 Integration of measurable functions......Page 38
1.4.4 Riesz’s theorem for positive Borel measures......Page 41
1.4.6 Lebesgue’s measure on Rn......Page 43
1.4.7 The product measure......Page 47
1.4.8 Complex (and signed) measures......Page 48
1.4.9 Exchanging derivatives and integrals......Page 49
2 Normed and Banach spaces, examples and applications......Page 51
2.1.1 Normed spaces and essential topological properties......Page 52
2.1.2 Banach spaces......Page 56
Kn), the theorems of Dini and Arzelà–Ascoli......Page 58
2.1.4 Normed algebras, Banach algebras and examples......Page 61
2.2 Operators, spaces of operators, operator norms......Page 69
2.3.1 The Hahn–Banach theorem and its immediate consequences......Page 76
2.3.2 The Banach–Steinhaus theorem or uniform boundedness principle......Page 79
weak completeness of X......Page 81
2.3.4 Excursus: the theorem of Krein–Milman, locally convex metrisable spaces and Fréchet spaces......Page 85
2.3.5 Baire’s category theorem and its consequences: the open mapping theorem and the inverse operator theorem......Page 89
2.3.6 The closed graph theorem......Page 92
2.4 Projectors......Page 94
2.5 Equivalent norms......Page 96
2.6.1 The fixed-point theorem of Banach-Caccioppoli......Page 98
2.6.2 Application of the fixed-point theorem: local existence and uniqueness for systems of differential equations......Page 103
Exercises......Page 106
3.1 Elementary notions, Riesz’s theorem and reflexivity......Page 112
3.1.1 Inner product spaces and Hilbert spaces......Page 113
3.1.2 Riesz’s theorem and its consequences......Page 117
3.2 Hilbert bases......Page 121
3.3.1 Hermitian conjugation, or adjunction......Page 134
algebras and C*-algebras......Page 137
3.3.3 Normal, self-adjoint, isometric, unitary and positive operators......Page 142
3.4 Orthogonal projectors and partial isometries......Page 145
3.5 Square roots of positive operators and polar decomposition of bounded operators......Page 149
3.6 The Fourier-Plancherel transform......Page 157
Exercises......Page 168
4 Families of compact operators on Hilbert spaces and fundamental properties......Page 176
4.1.1 Compact sets in (infinite-dimensional) normed spaces......Page 177
4.1.2 Compact operators in normed spaces......Page 179
4.2 Compact operators in Hilbert spaces......Page 182
4.2.1 General properties and examples......Page 183
4.2.2 Spectral decomposition of compact operators on Hilbert spaces......Page 185
4.3.1 Main properties and examples......Page 191
4.3.2 Integral kernels and Mercer’s theorem......Page 199
4.4.1 General properties......Page 202
4.4.2 The notion of trace......Page 206
4.5 Introduction to the Fredholm theory of integral equations......Page 210
Exercises......Page 217
5.1 Unbounded operators with non-maximal domains......Page 223
Exercises......Page 249
5.1.1 Unbounded operators with non-maximal domains in normed spaces......Page 224
5.1.2 Closed and closable operators......Page 225
and the t operator......Page 226
5.1.4 General properties of the Hermitian adjoint operator......Page 227
5.2 Hermitian, symmetric, self-adjoint and essentially self-adjoint operators......Page 229
5.3.1 The position operator......Page 233
5.3.2 The momentum operator......Page 234
5.4.1 The Cayley transform and deficiency indices......Page 238
5.4.2 Von Neumann’s criterion......Page 242
5.4.3 Nelson’s criterion......Page 243
6.1 General principles of quantum systems......Page 253
6.2.1 The photoelectric effect......Page 255
6.2.2 The Compton effect......Page 256
6.3.1 De Broglie waves......Page 258
6.3.2 Schrödinger’s wavefunction and Born’s probabilistic interpretation......Page 259
6.4 Heisenberg’s uncertainty principle......Page 261
6.5 Compatible and incompatible quantities......Page 262
7 The first 4 axioms of QM: propositions, quantum states and observables......Page 264
7.1 The pillars of the standard interpretation of quantum phenomenology......Page 265
7.2.1 States as probability measures......Page 267
7.2.2 Propositions as sets, states as measures on them......Page 269
7.2.3 Set-theoretical interpretation of the logical connectives......Page 270
7.2.4 “Infinite” propositions and physical quantities......Page 271
7.2.5 Intermezzo: basics on the theory of lattices......Page 273
7.2.6 The distributive lattice of elementary propositions for classical systems......Page 275
7.3 Propositions on quantum systems as orthogonal projectors......Page 276
7.3.1 The non-distributive lattice of orthogonal projectors on a Hilbert space......Page 277
7.3.2 Recovering the Hilbert space from the lattice......Page 284
7.4 Propositions and states on quantum systems......Page 286
7.4.1 Axioms A1 and A2: propositions, states of a quantum system and Gleason’s theorem......Page 287
7.4.2 The Kochen–Specker theorem......Page 294
7.4.3 Pure states, mixed states, transition amplitudes......Page 295
7.4.4 Axiom A3: post-measurement states and preparation of states......Page 300
7.4.5 Superselection rules and coherent sectors......Page 302
7.4.6 Algebraic characterisation of a state as a noncommutative Riesz theorem......Page 305
7.5.1 Axiom A4: the notion of observable......Page 309
7.5.2 Self-adjoint operators associated to observables: physical motivation and basic examples......Page 312
7.5.3 Probability measures associated to state/observable couples......Page 317
Exercises......Page 319
8 Spectral Theory I: generalities, abstract C*-algebras and operators in B(......Page 321
8.1.1 Basic notions in normed spaces......Page 323
8.1 Spectrum, resolvent set and resolvent operator......Page 322
8.1.2 The spectrum of special classes of normal operators in Hilbert spaces......Page 326
8.1.3 Abstract C*-algebras: Gelfand-Mazur theorem, spectral radius, Gelfand’s formula, Gelfand–Najmark theorem......Page 328
8.2.1 Abstract C*-algebras: functional calculus for continuous maps and self-adjoint elements......Page 334
homomorphisms of C*-algebras, spectra and positive elements......Page 337
8.2.3 Commutative Banach algebras and the Gelfand transform......Page 341
8.2.4 Abstract C*-algebras: functional calculus for continuous maps and normal elements......Page 346
functional calculus for bounded measurable functions......Page 348
8.3.1 Spectral measures, or PVMs......Page 356
8.3.2 Integrating bounded measurable functions in a PVM......Page 358
8.3.3 Properties of operators obtained integrating bounded maps with respect to PVMs......Page 364
8.4.1 Spectral decomposition of normal operators in B(......Page 371
8.4.2 Spectral representation of normal operators in B(......Page 376
8.5 Fuglede’s theorem and consequences......Page 383
8.5.1 Fuglede’s theorem......Page 384
8.5.2 Consequences to Fuglede’s theorem......Page 386
Exercises......Page 387
9.1 Spectral theorem for unbounded self-adjoint operators......Page 391
9.1.1 Integrating unbounded functions with respect to spectral measures......Page 392
9.1.2 Von Neumann algebra of a bounded normal operator......Page 404
9.1.3 Spectral decomposition of unbounded self-adjoint operators......Page 405
9.1.4 Example with pure point spectrum: the Hamiltonian of the harmonic oscillator......Page 413
9.1.5 Examples with pure continuous spectrum: the operators position and momentum......Page 417
9.1.6 Spectral representation of unbounded self-adjoint operators......Page 418
9.1.7 Joint spectral measures......Page 419
9.2 Exponential of unbounded operators: analytic vectors......Page 421
9.3.1 Strongly continuous one-parameter unitary groups, von Neumann’s theorem......Page 425
9.3.2 One-parameter unitary groups generated by self-adjoint operators and Stone’s theorem......Page 429
9.3.3 Commuting operators and spectral measures......Page 436
Exercises......Page 440
10.1 Abstract differential equations in Hilbert spaces......Page 443
10.1.1 The abstract Schrödinger equation (with source)......Page 445
10.1.2 The abstract Klein–Gordon/d’Alembert equation (with source and dissipative term)......Page 451
10.1.3 The abstract heat equation......Page 459
10.2.1 Tensor product of Hilbert spaces and spectral properties......Page 462
10.2.2 Tensor product of operators (typically unbounded) and spectral properties......Page 467
10.2.3 An example: the orbital angular momentum......Page 470
10.3 Polar decomposition theorem for unbounded operators......Page 473
10.3.1 Properties of operators A*A, square roots of unbounded positive self-adjoint operators......Page 474
10.3.2 Polar decomposition theorem for closed and densely-defined operators......Page 478
10.4.1 The Kato-Rellich theorem......Page 480
10.4.2 An example: the operator -. +V and Kato’s theorem......Page 482
Exercises......Page 488
11.1 Round-up and remarks on axioms A1, A2, A3, A4 and superselection rules......Page 490
11.2 Axiom A5: non-relativistic elementary systems......Page 497
11.2.1 The canonical commutation relations (CCRs)......Page 499
11.2.2 Heisenberg’s uncertainty principle as a theorem......Page 500
11.3.1 Families of operators acting irreducibly and Schur’s lemma......Page 501
11.3.2 Weyl’s relations from the CCRs......Page 503
11.3.3 The theorems of Stone–von Neumann and Mackey......Page 510
11.3.4 The Weyl......Page 513
11.3.5 Proof of the theorems of Stone–von Neumann and Mackey......Page 517
11.3.6 More on “Heisenberg’s principle”: weakening the assumptions and extension to mixed states......Page 523
11.3.7 The Stone–von Neumann theorem revisited, via the Heisenberg group......Page 524
11.3.8 Dirac’s correspondence principle and Weyl’s calculus......Page 526
Exercises......Page 529
12.1 Definition and characterisation of quantum symmetries......Page 531
12.1.1 Examples......Page 533
12.1.2 Symmetries in presence of superselection rules......Page 534
12.1.3 Kadison symmetries......Page 535
12.1.4 Wigner symmetries......Page 537
12.1.5 The theorems of Wigner and Kadison......Page 539
12.1.6 The dual action of symmetries on observables......Page 549
12.2.1 Projective and projective unitary representations......Page 554
12.2.3 Central extensions and quantum group associated to a symmetry group......Page 559
12.2.4 Topological symmetry groups......Page 562
12.2.5 Strongly continuous projective unitary representations......Page 567
12.2.6 A special case: the topological group R......Page 569
12.2.7 Round-up on Lie groups and algebras......Page 574
12.2.8 Symmetry Lie groups, theorems of Bargmann, Gårding, Nelson, FS3......Page 582
12.2.9 The Peter–Weyl theorem......Page 593
12.3.1 The symmetry group SO(3) and the spin......Page 598
12.3.3 The Galilean group and its projective unitary representations......Page 602
12.3.4 Bargmann’s rule of superselection of the mass......Page 609
Exercises......Page 612
13 Selected advanced topics in Quantum Mechanics......Page 617
13.1.2 Dynamical symmetries......Page 620
13.1.3 Schrödinger’s equation and stationary states......Page 623
13.1.4 The action of the Galilean group in position representation......Page 631
13.1.5 Basic notions of scattering processes......Page 633
13.1.6 The evolution operator in absence of time homogeneity and Dyson’s series......Page 639
13.1.1 Axiom A6: time evolution......Page 618
13.1.7 Antiunitary time reversal......Page 643
13.2.1 Pauli’s theorem......Page 645
13.2.2 Generalised observables as POVMs......Page 646
13.3.1 Heisenberg’s picture and constants of motion......Page 648
13.3.2 A short detour on Ehrenfest’s theorem and related mathematical issues......Page 653
13.3.3 Constants of motion associated to symmetry Lie groups and the case of the Galilean group......Page 655
13.4.1 Axiom A7: compound systems......Page 660
13.4.2 Entangled states and the so-called “EPR paradox”......Page 661
13.4.3 Bell’s inequalities and their experimental violation......Page 663
13.4.4 EPR correlations cannot transfer information......Page 666
13.4.5 The phenomenon of decoherence as a manifestation of the macroscopic world......Page 669
13.4.6 Axiom A8: compounds of identical systems......Page 670
13.4.7 Bosons and Fermions......Page 672
Exercises......Page 674
14.1 Introduction to the algebraic formulation of quantum theories......Page 676
14.1.1 Algebraic formulation and the GNS theorem......Page 677
14.1.2 Pure states and irreducible representations......Page 683
14.1.3 Hilbert space formulation vs algebraic formulation......Page 686
14.1.4 Superselection rules and Fell’s theorem......Page 689
14.1.5 Proof of the Gelfand–Najmark theorem, universal representations and quasi-equivalent representations......Page 692
14.2.1 Further properties of Weyl......Page 695
14.2.2 The Weyl C*-algebra CW (......Page 699
14.3 Introduction to Quantum Symmetries within the algebraic formulation......Page 700
14.3.1 The algebraic formulation’s viewpoint on quantum symmetries......Page 701
14.3.2 (Topological) symmetry groups in the algebraic formalism......Page 703
Appendix......Page 705
References......Page 717
Index......Page 722