Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation

194751 Print indd Spectral Theory and Quantum Mechanics Valter Moretti Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation Second Edition UNITEXT 110.

UNITEXT 110 Valter Moretti Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation Second Edition UNITEXT - La Matematica per il 3+2 Volume 110 Editor-in-chief A Quarteroni Series editors L Ambrosio P Biscari C Ciliberto C De Lellis M Ledoux V Panaretos W.J Runggaldier More information about this series at http://www.springer.com/series/5418 Valter Moretti Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation Second Edition 123 Valter Moretti Department of Mathematics University of Trento Povo, Trento Italy Translated by: Simon G Chiossi, Departamento de Matemática Aplicada (GMA-IME), Universidade Federal Fluminense ISSN 2038-5714 ISSN 2532-3318 (electronic) UNITEXT - La Matematica per il 3+2 ISSN 2038-5722 ISSN 2038-5757 (electronic) ISBN 978-3-319-70705-1 ISBN 978-3-319-70706-8 (eBook) https://doi.org/10.1007/978-3-319-70706-8 Library of Congress Control Number: 2017958726 Translated and extended version of the original Italian edition: V Moretti, Teoria Spettrale e Meccanica Quantistica, © Springer-Verlag Italia 2010 1st edition: © Springer-Verlag Italia 2013 2nd edition: © Springer International Publishing AG 2017 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Bianca Preface to the Second Edition In this second English edition (third, if one includes the first Italian one), a large number of typos and errors of various kinds have been amended I have added more than 100 pages of fresh material, both mathematical and physical, in particular regarding the notion of superselection rules—addressed from several different angles—the machinery of von Neumann algebras and the abstract algebraic formulation I have considerably expanded the lattice approach to Quantum Mechanics in Chap 7, which now contains precise statements leading up to Solèr’s theorem on the characterization of quantum lattices, as well as generalised versions of Gleason’s theorem As a matter of fact, Chap and the related Chap 11 have been completely reorganised I have incorporated a variety of results on the theory of von Neumann algebras and a broader discussion on the mathematical formulation of superselection rules, also in relation to the von Neumann algebra of observables The corresponding preparatory material has been fitted into Chap Chapter 12 has been developed further, in order to include technical facts concerning groups of quantum symmetries and their strongly continuous unitary representations I have examined in detail the relationship between Nelson domains and Gårding domains Each chapter has been enriched by many new exercises, remarks, examples and references I would like once again to thank my colleague Simon Chiossi for revising and improving my writing For having pointed out typos and other errors and for useful discussions, I am grateful to Gabriele Anzellotti, Alejandro Ascárate, Nicolò Cangiotti, Simon G Chiossi, Claudio Dappiaggi, Nicolò Drago, Alan Garbarz, Riccardo Ghiloni, Igor Khavkine, Bruno Hideki F Kimura, Sonia Mazzucchi, Simone Murro, Giuseppe Nardelli, Marco Oppio, Alessandro Perotti and Nicola Pinamonti Povo, Trento, Italy September 2017 Valter Moretti vii Preface to the First Edition I must have been or when my father, a man of letters but well-read in every discipline and with a curious mind, told me this story: “A great scientist named Albert Einstein discovered that any object with a mass can't travel faster than the speed of light” To my bewilderment I replied, boldly: “This can't be true, if I run almost at that speed and then accelerate a little, surely I will run faster than light, right?” My father was adamant: “No, it's impossible to what you say, it's a known physics fact” After a while I added: “That bloke, Einstein, must've checked this thing many times … how you say, he did many experiments?” The answer I got was utterly unexpected: “Not even one I believe He used maths!” What did numbers and geometrical figures have to with the existence of an upper limit to speed? How could one stand by such an apparently nonsensical statement as the existence of a maximum speed, although certainly true (I trusted my father), just based on maths? How could mathematics have such big a control on the real world? And Physics ? What on earth was it, and what did it have to with maths? This was one of the most beguiling and irresistible things I had ever heard till that moment… I had to find out more about it This is an extended and enhanced version of an existing textbook written in Italian (and published by Springer-Verlag) That edition and this one are based on a common part that originated, in preliminary form, when I was a Physics undergraduate at the University of Genova The third-year compulsory lecture course called Theoretical Physics was the second exam that had us pupils seriously climbing the walls (the first being the famous Physics II, covering thermodynamics and classical electrodynamics) Quantum Mechanics, taught in Institutions, elicited a novel and involved way of thinking, a true challenge for craving students: for months we hesitantly faltered on a hazy and uncertain terrain, not understanding what was really key among the notions we were trying—struggling, I should say—to learn, together with a completely new formalism: linear operators on Hilbert spaces At that time, actually, we did not realise we were using this mathematical theory, and for many mates of mine, the matter would have been, rightly perhaps, completely futile; Dirac's bra vectors were what they were, and that’s it! They were certainly not elements in the topological dual of the Hilbert space The notions of Hilbert space and dual topological space had no right of abode in the mathematical toolbox of the majority ix x Preface to the First Edition of my fellows, even if they would soon come back in through the back door, with the course Mathematical Methods of Physics taught by Prof G Cassinelli Mathematics, and the mathematical formalisation of physics, had always been my flagship to overcome the difficulties that studying physics presented me with, to the point that eventually (after a Ph.D in Theoretical Physics) I officially became a mathematician Armed with a maths’ background—learnt in an extracurricular course of study that I cultivated over the years, in parallel to academic physics—and eager to broaden my knowledge, I tried to formalise every notion I met in that new and riveting lecture course At the same time, I was carrying along a similar project for the mathematical formalisation of General Relativity, unaware that the work put into Quantum Mechanics would have been incommensurably bigger The formulation of the spectral theorem as it is discussed in x 8, is the same I learnt when taking the Theoretical Physics exam, which for this reason was a dialogue of the deaf Later my interest turned to Quantum Field Theory, a subject I still work on today, though in the slightly more general framework of QFT in curved spacetime Notwithstanding, my fascination with the elementary formulation of Quantum Mechanics never faded over the years, and time and again chunks were added to the opus I begun writing as a student Teaching this material to master’s and doctoral students in mathematics and physics, thereby inflicting on them the result of my efforts to simplify the matter, has proved to be crucial for improving the text It forced me to typeset in LaTeX the pile of loose notes and correct several sections, incorporating many people’s remarks Concerning this, I would like to thank my colleagues, the friends from the newsgroups it.scienza.fisica, it.scienza.matematica and free.it.scienza.fisica, and the many students—some of which are now fellows of mine—who contributed to improve the preparatory material of the treatise, whether directly or not, in the course of time: S Albeverio, G Anzellotti, P Armani, G Bramanti, S Bonaccorsi, A Cassa, B Cocciaro, G Collini, M Dalla Brida, S Doplicher, L Di Persio, E Fabri, C Fontanari, A Franceschetti, R Ghiloni, A Giacomini, V Marini, S Mazzucchi, E Pagani, E Pelizzari, G Tessaro, M Toller, L Tubaro, D Pastorello, A Pugliese, F Serra Cassano, G Ziglio and S Zerbini I am indebted, for various reasons also unrelated to the book, to my late colleague Alberto Tognoli My greatest appreciation goes to R Aramini, D Cadamuro and C Dappiaggi, who read various versions of the manuscript and pointed out a number of mistakes I am grateful to my friends and collaborators R Brunetti, C Dappiaggi and N Pinamonti for lasting technical discussions, for suggestions on many topics covered in the book and for pointing out primary references At last, I would like to thank E Gregorio for the invaluable and on-the-spot technical help with the LaTeX package In the transition from the original Italian to the expanded English version, a massive number of (uncountably many!) typos and errors of various kinds have been corrected I owe to E Annigoni, M Caffini, G Collini, R Ghiloni, A Iacopetti, M Oppio and D Pastorello in this respect Fresh material was added, Preface to the First Edition xi both mathematical and physical, including a chapter, at the end, on the so-called algebraic formulation In particular, Chap contains the proof of Mercer’s theorem for positive Hilbert–Schmidt operators The analysis of the first two axioms of Quantum Mechanics in Chap has been deepened and now comprises the algebraic characterisation of quantum states in terms of positive functionals with unit norm on the C à -algebra of compact operators General properties of Cà -algebras and à -morphisms are introduced in Chap As a consequence, the statements of the spectral theorem and several results on functional calculus underwent a minor but necessary reshaping in Chaps and I incorporated in Chap 10 (Chap in the first edition) a brief discussion on abstract differential equations in Hilbert spaces An important example concerning Bargmann’s theorem was added in Chap 12 (formerly Chap 11) In the same chapter, after introducing the Haar measure, the Peter–Weyl theorem on unitary representations of compact groups is stated and partially proved This is then applied to the theory of the angular momentum I also thoroughly examined the superselection rule for the angular momentum The discussion on POVMs in Chap.13 (ex Chap 12) is enriched with further material, and I included a primer on the fundamental ideas of non-relativistic scattering theory Bell’s inequalities (Wigner’s version) are given considerably more space At the end of the first chapter, basic point-set topology is recalled together with abstract measure theory The overall effort has been to create a text as self-contained as possible I am aware that the material presented has clear limitations and gaps Ironically—my own research activity is devoted to relativistic theories—the entire treatise unfolds at a non-relativistic level, and the quantum approach to Poincaré’s symmetry is left behind I thank my colleagues F Serra Cassano, R Ghiloni, G Greco, S Mazzucchi, A Perotti and L Vanzo for useful technical conversations on this second version For the same reason, and also for translating this elaborate opus into English, I would like to thank my colleague S G Chiossi Trento, Italy October 2012 Valter Moretti References [Str05b] [StWi00] [Tak00] [Tau61] [Tes09] [Var84] [Var07] [Vla02] [Wal94] [War75] [Wei99] [Wei40] [Wes78] [Wig59] [Wie80] [Zan91] [ZFC05] [Zeh00] 935 Strocchi, F.: Symmetry Breaking Lecture Notes in Physics Springer, Berlin (2005) Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and all that, Revised edn Princeton University Press, Princeton, Revised edn (2000) Takesaki, M.: Theory of Operator Algebra, vols I, II, III, 2nd print for the first (1979) edn Springer, Berlin (2000) Taub, H.A (ed.): J von Neumann, Collected Works, Volume III: Rings of Operators, Pergamon Press, Oxford (1961) Teschl, G.: Mathematical Methods in Quantum Mechanics with Applications to Schrdinger Operators Graduate Studies in Mathematics, vol 99 American Mathematical Society, Rhode Island (2009) Varadarajan, V.S.: Lie Groups Lie Algebras and their Representations Springer, Berlin (1984) Varadarajan, V.S.: Geometry of Quantum Theory, 2nd edn Springer, Berlin (2007) Valdimirov, V.S.: Methods of the Theory of Generalized Functions (Analytical Methods and Special Functions), 1st edn CRC Press, Boca Raton (2002) Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics Chicago University Press, Chicago (1994) Warner F.W.: Foundation of Differentiable Manifolds and Lie Groups Springer, New York (1975) Weinberg, S.: The Quantum Theory of Fields, vols 1, 2, Cambridge University Press, Cambridge (2005) A Weil.: L’integration dans le groupes topologiques et ses applications Herman, Paris, A.S.I (1940) von Westenholtz, C.: Differential Forms in Mathematical Physics North-Holland, Amsterdam (1978) Wigner, E.: Group Theory and its Applications to the Quantum Theory of Atomic Spectra Academic Press, Boston (1959) Wiedmann, J.: Linear Operators in Hilbert Spaces Springer, Berlin (1980) Zanziger, S.: Coherence and quantum mechanics In: Gans, W (ed.) Large-Scale Molecular Systems, pp 327–332 Plenum Press, New York (1991) Zachos, C., Fairlie, D., Curtright, T.: Quantum Mechanics in Phase Space World Scientific, Singapore (2005) Zeh, H.D.: The meaning of decoherence In: [BGJKS00] Index Symbols A H t (t), 832 i∈I with I of arbitrary cardinality, 117 ∨ j∈J a j , 316 ∧ j∈J a j , 316 (A|B)2 with A, B Hilbert–Schmidt operators, 216 ( | ), 5, 111 (a, +∞], 19 (σ ) (H⊗n )λ , 860 ) ⊗n (H )(σ ± , 862 :=, < K >, 113 < X , · · · , X n >, 84 A B, A, B sets, 461 A, A , N , 504, 632 AηR, 603 A ⊂ B with A, B operators, 253 A∗∗···∗ , 261 Aut (G), 917 Br (x0 ), 41 C(J ; S), 541 C(K), 48 C(K; Kn ), 47 C(X), 54 C W (X, σ ), 904 C ∗ -algebra, 135, 394, 401 C ∞ (A) with A operator, 277, 513 C ∞ (Rn ), 80 C k map, 919 C k (J ; H), 541 C k (Ω), 919 C k (Ω; Rm ), 919 C0 (X), 54 Cb (X), 54 Cc (X), 28, 54 D(T ), 252 Fol(ω), 878 G(T ), 86, 252 G L(V ), 917 G L(n, C), 705 G L(n, R), 705, 727 G A, 52 H (V, R), 740 I O(3), 688 I O(n), 706 I nt (S), 81 J , 772 Jk , 773 K ⊥ , 108 K i , 781 K er (T ), 132 L(X), 54 L , 571 L (Rn , d x), 264 L (X, μ), 112 L αn (x), 805 L p (X, μ), 56 M(n, C), 727 M(n, R), 727 M(X), 19, 352 M α (x), 171 Mb (X), 54, 352, 424 MR (X), 19 O(3), 682 O(n), 706, 728 (A) PE , 348 P (T ) , 494 P (T ) with T bounded normal, 449 (f) PE , 311 Pi , 507, 571 P j , 266 © Springer International Publishing AG 2017 V Moretti, Spectral Theory and Quantum Mechanics, UNITEXT - La Matematica per il 3+2 110, https://doi.org/10.1007/978-3-319-70706-8 937 938 Pk , 625 Rλ (A), 395 Ran(T ), 132 S matrix, 813 S(A), 898 S(X), 352, 434 S L(n, R), 706 S O(3), 729, 769 S O(n), 706, 729 SU (2), 729, 769 SU (n), 706 S G , 775 T ≥ U , T, U bounded on a Hilbert space, 141 T ∗ , 132 T ∗ with T unbounded, 256 T2 topology, 12, 42, 54, 307 TK , 206 U (1)-central extension of a group by a multiplier function, 703 U (n), 706 U (t, t ), 818 W ((t, u)), 631 W (z), 640 W ∗ -algebra, 164 X i , 264, 507, 571, 624 Ylm , 572 [−∞, a), 19 [−∞, a], 19 [−∞, ∞], 19 [a, +∞], 19 Laplace operator, 584 Aρ , 358, 619 f , 476 Ω± , 811 T , 423 a , 409, 421 ⇒ (logical implication), 310 (logical ‘not’ or logical complement), 309 B30DT B30D, 60 B30DμB30D with μ complex or signed measure, 65 B30D B30D, 41 B30D B30D∞ , 47 δi j , 626 p (N), 58 p (X), 57 γ ∗ for γ symmetry (Kadison or Wigner automorphism), 687 , 5, X s(x) d P(x) with s simple, 434 X f dμ with μ complex measure, 36 Index f (x)d P(x) with f bounded, measurable, 436 n Rn f (t)Vt dt with {Vt }t∈R strongly continuous operators at t, 521 X f (x)d P(x) with f measurable, not necessarily bounded, 477, 489 λ-eigenspace, 142 λ-eigenvector, 142 A ρ , 358, 619 A (A), 513 B (X), 18 G , 774 μ-integrable function, 25 ) μ(T ψ , 499 μψ , 441 μρA , 619 μψ,φ , 441 μac , μ pa , μsing , 34 ¬, 313 ν ≺ μ, 30, 36 ⊕ j∈J A j with A j operators, 171 ⊕ j∈J ψ j and ψ1 ⊕ · · · ⊕ ψn with ψ j vectors, 148 ⊥, 255 φa , 408 π π , 139 π1 ≈ π2 , 899 in LR (H), 367 exp of a Lie group, 725 ess ran( f ), 534 ess sup| f |, 58 S with S a set, 12 c, R, 19 lim inf, 15 lim √ sup, 15 A, 155, 578 {A} with A operator, 260 { f, g}, 659 ψ(φ| ) and (φ| )ψ, ρ(A), 395 σ , 18 σ -additive measure over a lattice, 373 σ -additive positive measure, 20, 55 σ -complete lattice, 313 σ -strong topology, 166 σ -weak topology, 166 σ (A), 395 σ (A), 415 σ X (A), 166 σc (A), 395 σd (T ), 499 σi , i = 1, 2, 3, 332 X Index σ p (A), 395 σ p (T ), 208 σr (A), 395 σ X Y (A), 166 σA (a), 405 σac (T ), 500 σap (T ), 500 σess (T ), 499 σ pc (T ), 500 σ pr (T ), 500 σsing (T ), 500 ∼ in LR (H), 367 ⊂, ⊃, τ , 255 ∨n∈N an , 314 |A|, 158, 579 |A| with A operator, 158 |μ| with μ complex measure, 35 ||A||1 with A operator of trace class, 228 ||A||2 with A Hilbert–Schmidt operator, 214 || ||, 111 (P) || ||∞ , 434 || ||2 , 112 || ||∞ , 65 ∧n∈N an , 314 ∗ -algebra, 135 ∗ -algebra or C ∗ -algebra generated by a subset, 135 ∗ -anti-automorphism, 906 ∗ -automorphism, 135, 906 ∗ -homomorphism, 135, 409, 421 ∗ -isomorphism, 135 ∗ -weak topology, 75 a.e., 23 a ⊥ b in an orthocomplemented lattice, 313 co(E), 73 d± (A), 273 dzdz, 130 f (T ) with T self-adjoint and f measurable, neither necessarily bounded, 499 f (T, T ∗ ) with T bounded normal and f bounded measurable, 427, 445 f (T, T ∗ ) with T bounded normal and f continuous, 423 f (a), 409 f (a, a ∗ ), 421 h, 2, p( ), 41 r (a), 402 s- lim, 76 sing(A), 212 supp(P), 432 supp( f ), 16 939 tr A with A of trace class, 231 tr E , 857 v1 ⊗ · · · ⊗ with vi vectors, 561 w-∂ α , 265 w- lim, 76 w∗ - lim, 76 H (n), 655 D (Rn ), 171 DG , 744 E (‘and’, logical conjunction), 309 F f , 172 Ft , 307 F− g, 172 G Gelfand transform, 416 Hn+1 , 307 Jk , 772 L (X, μ), 25 L , 571 L p (X, μ), 55 Li , 571 LR S (H S ), 599 O (‘or’, logical disjunction), 309 Pn , 858 S (Rn ), 80, 171 C, N, R+ , B (X), 307 W (X, σ ), 642 A , 512 A+ , 412 B(X), 59 B(X, Y), 59 B1 (H), 228 B2 (H), 214 B∞ (X), 201 B∞ (X, Y), 201 J(H), 694 L(X), 59 L(X, Y), 59 M , 162 R1 ⊗ R2 with R1 , R2 von Neumann algebras, 570 RS , 599 RS von Neumann algebra of observables of a physical system, 599 S(H)adm , 609, 891 Sp (H)adm , 891 G1 ⊗ψ G2 , 917 G1 ⊗ G2 with G , G groups, 917 H(C), 130 H⊗n , 858 H1 ⊗ · · · ⊗ Hn with Hi Hilbert spaces, 563 940 H S , 596 Hψ , 277 H p , 499, 813 X , 59 X∗ , 59 X1 ⊕ · · · ⊕ Xn , 84 A with A operator, 254 inf, 915 sup, 915 F , phase space, 312 L (H), 319 LR (H), 364 RS von Neumann algebra of observables of a physical system, 326 S(H), 333 S(H)adm , 381 Sp (H), 340 Sp (H)adm , 380 H S , 325 ess ran P ( f ), 501 σ -additivity, 20 σ -algebra, 18, 55 σ -finite, 21 A Abelian algebra, 51 Abelian group, 916 Abelian projector, 369 Abelian superselection rules, 611 Absolutely continuous function, 32, 283 Absolutely continuous measure with respect to another, 30 Absolutely continuous spectrum, 500 Absorbing set, 73 Abstract differential equations, 540 AC lattice, 359, 361 Active transformation, 667 Adjoint operator (general case), 256 Adjoint operator or Hermitian conjugate operator, 132 Algebra, 50 Algebra automorphism, 51 Algebra homomorphism, 51 Algebraic dual, 59 Algebraic formulation of quantum theories, 374, 615, 867 Algebraic multiplicity, 235 Algebraic state invariant under a quantum symmetry, 907 Algebraic state on B∞ (H), 375 Algebra isomorphism, 51 Algebra of sets, 18 Index Algebra with unit, 51 Almost everywhere, 23 Analytic function with values in a Banach space, 394 Analytic vector, 277, 513 Analytic vector of a representation of a Lie group, 747 Annihilation operator, 504, 632 Anti-isomorphic Hilbert spaces, 117 Anti-unitary operator, 276, 671 Approximate point spectrum, 500 Arzelà–Ascoli theorem, 49 Asymptotic completeness, 814 Atlas, 920 Atomic lattice, 359 Atomic proposition, 326, 343 Atomistic lattice, 359 Attractive Coulomb potential, 589, 804 Axiom of choice, 915 B Baire’s category theorem, 81 Baker-Campbell-Hausdorff formula, 631, 727 Balanced set, 73 Banach algebra, 51, 135, 394, 401 Banach lemma, 203 Banach’s inverse operator theorem, 84 Banach space, 45 Bargmann–Fock–Hilbert space, 129 Bargmann’s superselection rule for the mass, 784 Bargmann’s theorem, 736 Basis of a Hilbert space, 117, 119 Basis of a topology, 11, 78 Bell’s inequalities, 850 Beppo Levi’s monotone convergence theorem, 26 Bessel’s inequality, 119 Bi-invariant Haar measure, 707 Bijective map, Boolean σ -algebra, 313 Boolean algebra, 313 Boost along the ith axis, 781 Borel σ -algebra, 18, 307 Borel-measurable function, 19, 307 Borel measure, 306, 307 Borel set, 307 Bosonic Fock space, 864 Bosons, 862 Bounded functional, 60 Bounded lattice, 313 Index Bounded operator, 60 Bounded projector-valued measure, 432 Bounded Rn or Cn -valued function, 16 Bounded set, 41, 200 Bounded set in Rn or Cn , 15 Busch’s theorem, 830 C Calkin’s theorem, 205 Canonical commutation relations, 626 Canonical injection of a central extension, 703 Canonical projection, 85 Canonical projection of a central extension, 703 Canonical symplectic form, 640 Cartan’s theorem, 725 Cauchy property, 44 Cauchy–Schwarz inequality, 108 Cauchy sequence, 44 Cayley transform, 271 CCR, 626 Central carrier of an operator, 369 Central charge, 736, 781 Central projector, 367 Centre of a group, 916 Centre of an orthocomplemented lattice, 313 Characteristic function, 30, 129 Character of a Banach algebra with unit, 415 Character of a finite-dimensional group representation, 768 Character of an Abelian topological group, 752 Chronological reordering operator, 819 Closable operator, 254 Closed map, 14 Closed operator, 253 Closed set, 11 Closure of an operator, 254 Closure of a set, 12 Closure operator associated to a Galois connection, 390 Coherent sectors (of superselection), 378, 381, 609, 669 Coherent superposition, 340 Collapse of the wavefunction, 343 Commutant, 162 Commutant of an (generally unbounded) operator, 260 Commutant of a set of (generally unbounded) self-adjoint operators, 512 Commutative algebra, 51 941 Commutative Gelfand–Najmark theorem, 418 Commutative group, 916 Commutator of a Lie algebra, 722 Commuting elements in an orthocomplemented lattice, 313 Commuting operators, 260, 604 Commuting orthogonal projections, 146 Commuting spectral measures, 529 Compact, 15, 198 Compact-open topology, 752 Compact operator, 200 Compatible and incompatible propositions, 317 Compatible and incompatible quantities, 300, 304 Compatible observables, 604 Complete lattice, 313 Completely additive algebraic state, 879 Completely additive measure over a lattice, 374 Completely continuous operator, 200 Complete measure, 23 Complete measure space, 23 Complete metric space, 79, 81 Complete normed space, 45 Complete orthonormal system, 119 Complete set of commuting observables, 605 Complex measure, 35, 36 Complex spectral measure associated to two vectors, 441 Complex-valued simple function, 352 Compound quantum system, 844 Compton effect, 292 Conjugate observables, 626 Conjugate or adjoint operator (in a normed space), 64 Conjugation of the charge, 692 Conjugation operator, 276, 671 Connected components, 17 Connected set, 16 Connected space, 16 Constant of motion, 832 Continuous Borel measure on R, 33 Continuous function, 14 Continuous functional, 61 Continuous functional calculus, 407 Continuous map, 42, 79 Continuous operator, 61 Continuous projective representation, 708 Continuous spectrum of an operator, 395 Continuous superselection rules, 383 942 Contravariant vector, 924 Convergent sequence, 14, 42, 79 Convex hull of a set, 73 Convex (linear) combination, 335 Convex set, 72, 73, 113, 335 Coordinatisation problem, 361 Copenhagen interpretation, 298, 305 Core of an operator, 263 Cotangent space, 923 Countable set, 117 Counting measure, 57 Covariant vector, 924 Covering map, 723 Covering property, 359 Covering space of a topological space, 723 Creation operator, 504, 632 Cyclic vector for a ∗ -algebra representation, 139 D Darboux’s theorem, 640 De Broglie wavelength, 296 Decoherence, 857 Deficiency indices, 273 Deformation quantisation, 873 Degenerate operator, 223, 249 De Morgan’s law, 314 Dense set, 12 Density matrix, 341, 597 Diffeomorphism, 922 Differentiation inside an integral, 37 Dimension function, 368 Dini’s theorem on uniform convergence, 48 Dirac measure, 306 Dirac’s correspondence principle, 657 Direct decomposition into factors of R, 371 Direct-integral decomposition into factors, 370 Direct product of groups, 917 Direct sum, 84 Direct sum of C ∗ -algebras, 137 Direct sum of von Neumann algebras, 171 Discrete spectrum, 499 Discrete subgroup, 724 Distance, 78 Distributive lattice, 313 Division algebra, 138 Division ring, 138 Dixmier-Malliavin’s theorem, 747 Domain of an operator, 252 Dual action of a symmetry on observables, 687, 700, 910 Index Dual space of an Abelian topological group, 752 Dual vector space, 924 Du Bois-Reymond lemma, 265 Dye’s theorem, 694 Dynamical flow, 795 Dynamical symmetry, 797 Dyson series, 819 E Ehrenfest theorem, 836 Eigenspace, 141 Eigenvalue, 141 Eigenvector, 141 Embedded submanifold, 922 Entangled states, 848 Entire function, 130 EPR paradox, 848 Equicontinuous family of operators, 71 Equicontinuous sequence of functions, 49 Equivalent norms, 89 Equivalent projective unitary representations, 699 Essentially bounded map for a PVM, 434 Essentially self-adjoint operator, 259 Essential norm with respect to a PVM, 434 Essential rank, 534 Essential rank of a measurable function with respect to a PVM, 501 Essential spectrum, 499 Essential supremum, 58 Euclidean, or standard, distance, 79 Expansion of a compact operator with respect to its singular values, 212 Exponential mapping of a Lie group, 725 Extension of an operator, 253 Extremal element of a convex set, 72 Extreme element in a convex set, 335 F Factor (von Neumann algebra), 164, 366 Factor of type In , 368 Factor of type I I1 , 366, 368 Factor of type I I I , 368 Factor of type I I I∞ , 368 Faithful algebraic state, 877 Faithful representation, 696, 918 Faithful representation of a ∗ -algebra, 139 Fatou’s lemma, 27 Fermionic Fock space, 864 Fermions, 862 Final space of a partial isometry, 151 Index Finite measure, 21 Finite projector, 367 First integral, 832 Fischer–Riesz theorem, 56 Fischer–Riesz theorem, L ∞ case, 58 Fixed-point theorem, 91 Fock space, 566, 863 Folium of an algebraic state, 878 Fourier–Plancherel transform, 178 Fourier transform, 172 Fréchet space, 80 Fredholm equation of the first kind, 237 Fredholm equation of the second kind, 239 Fredholm equation of the second kind with Hermitian kernel, 238 Fredholm’s alternative, 240 Free representation, 918 Frobenius theorem, 138 FS3 theorem (Flato, Simon, Snellman, Sternheimer) on the existence of unitary representations of Lie groups, 750 Fubini–Tonelli theorem, 34 Fuglede-Putnam-Rosenblum theorem, 466 Fuglede’s theorem, 464 Functional calculus, 407 Function of bounded variation, 32 G Gårding space, 525, 744 Gårding’s theorem, 745 Galilean group, 706, 774, 808, 839 Galois connection on a complete lattice, 390 Gauge algebra, 616 Gauge group, 616 Gauge symmetry, 616, 667 Gauge transformation, 616, 667 Gaussian or quasi-free algebraic states, 905 Gelfand ideal, 877 Gelfand–Mazur Theorem, 402 Gelfand-Najmark theorem, 407, 895 Gelfand’s formula for the spectral radius, 404 Gelfand transform, 416 General linear group, 705 Generator of a unitary representation of a Lie group, 743 Generator (self-adjoint) of a strongly continuous one-parameter group of unitary operators, 527 Generators of a Weyl ∗ -algebra, 642 Gleason-Montgomery-Zippin theorem, 720 Gleason’s theorem, 331 943 Gleason’s theorem for general von Neumann algebras, 373 GNS representation, 646 GNS theorem, 874 GNS theorem for ∗ -algebras with unit, 885 Gram–Schmidt orthonormalisation process, 126 Graph of an operator, 86, 252 Greatest lower bound, 915 Groenewold–van Hove Theorem, 660 Ground state, 911 Ground state of the hydrogen atom, 806 Group, 916 Group automorphism, 917 Group homomorphism, 696, 916 Group isomorphism, 917 Gyro-magnetic ratio of the electron, H Haag theorem, 814 Haar measure, 707, 755 Hadamard’s theorem, 404 Hahn–Banach theorem, 67 Hamiltonian formulation of classical mechanics, 307 Hamiltonian of the harmonic oscillator, 503 Hamiltonian operator, 795 Hamilton’s equations, 307 Hausdorff (or T2 ) space, 12, 42, 54, 307 Heine–Borel theorem, 15 Heisenberg’s picture, 831 Heisenberg’s relations, 626, 751 Heisenberg’s uncertainty principle, 298 Hellinger-Toeplitz theorem, 260 Hermite functions, 128, 505, 632 Hermite polynomial, 129 Hermite polynomial Hn , 129 Hermitian inner product, 108 Hermitian operator, 259 Hermitian or self-adjoint element, 135 Hermitian semi-inner product, 108 Hilbert basis, 117, 119 Hilbert-Schmidt operator, 214 Hilbert space, 112 Hilbert space associated to a physical system, 325 Hilbert space of a non-relativistic particle of mass m > and spin 0, 624 Hilbert sum of Hilbert spaces, 147, 163, 255, 566 Hilbert’s theorem on compact operators, 208 Hilbert’s theorem on the spectral expansion of a compact operator, 209 944 Hilbert tensor product, 563 Hille–Yosida theorem, 528, 529 Hölder’s inequality, 55 Homeomorphism, 14 Homogeneous Volterra equation C([a, b]), 94 Homotopy, 17 Hydrogen atom, 805 Index on I Ideal and ∗ -ideal, 205 Idempotent operator, 87 Identical particles, 858 Imprimitivity condition, 689 Imprimitivity system, 690 Imprimitivity theorem of Mackey, 690 Incoherent superposition, 340 Incompatible observables, 604 Incompatible propositions, 318 Indirect or first-kind measurement, 343 Induced topology, 11 Inertial frame system, 624 Infimum, 915 Infinite projector, 367 Infinite tensor product of Hilbert spaces, 566 Initial space of a partial isometry, 151 Inner continuity, 20 Inner product space, 108 Inner regular measure, 21 Integral of a bounded measurable map with respect to a PVM, 436 Integral of a function with respect to a complex measure, 36 Integral of a function with respect to a measure, 25 Integral of a measurable, not necessarily bounded, function for a PVM, 489 Integral of a simple function with respect to a PVM, 434 Interior of a set, 81 Internal point of a set, 11 Invariant subspace, 59, 629, 918 Invariant subspace under a ∗ -algebra representation, 139 Inverse dual action of a symmetry on observables, 688, 701, 832, 907 Inverse Fourier transform, 172 Inverse operator theorem of Banach, 84 Invertible map, Involution, 135 Irreducible family of operators, 629 Irreducible lattice, 359 Irreducible orthocomplemented lattice, 313 Irreducible projective unitary representation, 698 Irreducible representation, 918 Irreducible representation of a ∗ -algebra, 139 Irreducible space for a family of operators, 629 Irreducible subspace for a family of operators on H, 629 Irreducible unitary representation, 699 Isometric element, 135 Isometric operator, 141 Isometry, 42, 111, 270 Isometry group of R3 , 688 Isomorphic algebras, 51 Isomorphically isomorphic algebras, 51 Isomorphism of Hilbert spaces, 112, 141 Isomorphism of inner product spaces, 111 Isomorphism of normed spaces, 42 Isotopic spin, 612 J Jointly continuous map, 44 Joint spectral measure, 509, 510, 604, 753 Joint spectrum, 510 Jordan algebra, 624, 694, 870, 872 Jordan product, 623, 694, 871 K Kadison automorphism, 670 Kadison symmetry, 670 Kadison’s theorem, 682 Kato–Rellich theorem, 582 Kato’s theorem, 586, 801 Kernel of a group homomorphism, 916 Kernel of an operator, 132 Klein–Gordon/d’Alembert equation, 548 Klein-Gordon equation, 905 Kochen–Specker theorem, 334 Krein–Milman theorem, 78 L Laguerre function, 129 Laguerre polynomial, 129, 805 Lattice, 312 Lattice automorphism, 315 Lattice homomorphism, 315 Lattice isomorphism, 315 Least upper bound, 915 Lebesgue-measurable function, 31 Lebesgue measure on Rn , 31 Index Lebesgue measure on a subset, 31 Lebesgue’s decomposition theorem for measures on R, 34 Lebesgue’s dominated convergence theorem, 27 Left and right orbit of a subset by a group, 707 Left-invariant and right-invariant measure, 707 Legendre polynomials, 127 Lidskii’s theorem, 234 Lie algebra, 659, 722 Lie algebra homomorphism, 722 Lie algebra isomorphism, 722 Lie algebra representation, 723 Lie group, 720 Lie group homomorphism, 720 Lie group isomorphism, 720 Lie-Jordan algebra, 872 Lie subgroup, 724 Lie theorem, 724 Limit of a sequence, 42, 79 Limit point, 14 Lindelöf’s lemma, 13 Linear (left) representation of a group, 918 Linear representation of a group, 918 Liouville’s equation, 308 Liouville’s theorem, 308 Lipschitz function, 33 Local chart, 920 Local existence and uniqueness for firstorder ODEs, 97 Local homomorphism of Lie groups, 720 Local isomorphism of Lie groups, 720 Locally compact space, 15, 54, 198, 307 Locally convex space, 73 Locally integrable map, 265 Locally Lipschitz function, 96 Locally path-connected, 17 Locally square-integrable function, 588 Logical conjunction, ‘and’, 309 Logical disjunction, ‘or’, 309 Logical implication, 310 Logic of admissible elementary propositions in presence of superselection rules, 382 Logic of elementary propositions of a quantum system, 325, 596, 599 Logic of the von Neumann algebra R, 364 Loomis–Sikorski theorem, 316 Lorentz group, 706 Lower bound, 915 Lower bounded set, 915 945 LSZ formalism, 814 Lüders-von Neumann collapse postulate, 342, 598 Luzin’s theorem, 29 M Mackey’s theorem, 640 Maximal Abelian von Neumann subalgebra, 606 Maximal ideal, 415 Meagre set, 81 Mean value, 619 Measurable function, 19 Measurable space, 18 Measure absolutely continuous with respect to another measure, 36 Measure concentrated on a set, 22 Measure di Borel, 21 Measure dominated by another, 30, 36 Measure space, 20, 55, 306 Measuring operators, 829 Mercer’s theorem, 224 Metric, 78 Metric space, 78 Metrisable topological space, 80 Minkowski’s inequality, 55 Mixed algebraic state, 873 Mixed state, 340, 597 Mixture, 340 Modulus of an operator, 158 Momentum operator, 266, 507, 571, 624 Momentum representation, 691 Monotonicity, 20, 432 Multi-index, 80 Multiplicity of a singular value, 212 Multiplier of a projective unitary representation, 698 N Nelson’s theorem on commuting spectral measures, 751 Nelson’s theorem on essential selfadjointness (Nelson’s criterion), 279 Nelson’s theorem on the existence of unitary representations of Lie groups, 750 Neumark’s theorem, 831 Non-destructive or indirect measurements, 598 Non-destructive testing, 343 Non-meagre set, 81 Nonpure state, 340 Norm, 40 946 Normal coordinate system, 726 Normal element, 135 Normal operator, 140 Normal operator (general case), 259 Normal state of a von Neumann algebra, 374, 879 Normal states of a C ∗ -algebra, 878 Normal subgroup, 916 Normal vector, 108 Normed space, 40 Normed unital algebra, 51 Norm topology of a normed space, 41, 78 Nowhere dense set, 81 Nuclear operator, 228 Number operator, 504, 632 Nussbaum lemma, 278 O Observable, 348, 598 Observable function of another observable, 351 One-parameter group of operators, 517 One-parameter group of unitary operators, 517 Open ball, 41 Open map, 14, 82 Open mapping theorem (of Banach– Schauder)), 82 Open metric ball, 81 Open neighbourhood of a point, 11 Open set, 11, 41, 78 Operator affiliated to a von Neumann algebra, 603 Operator norm, 60 Operators of spin, 770 Orbital angular momentum, 571 Orbital angular momentum operator, 571 Ordered set, 915 Ortho-automorphism, 693 Orthochronous Lorentz group, 706 Orthocomplemented lattice, 313 Orthogonal complement, 313 Orthogonal elements in an orthocomplemented lattice, 313 Orthogonal group, 706 Orthogonal projector, 144 Orthogonal space, 108 Orthogonal system, 119 Orthogonal vectors, 108 Orthomodular lattice, 313, 359 Orthonormal system, 119 Outer continuity, 21 Index Outer regular measure, 22 P Paley-Wiener theorem, 181 Parallelogram rule, 109 Parastatistics, 863 Parity inversion, 668, 690, 729 Partial isometry, 151, 160, 579 Partially ordered set, 915 Partial order, 915 Partial trace, 857 Passive transformation, 668 Path-connected, 17 Pauli matrices, 138, 332, 680, 770, 825 Pauli’s theorem, 827 Permutation group on n elements, 858 Peter-Weyl theorem, 755, 763, 767 Phase spacetime, 307 Photoelectric effect, 291 Plancherel theorem, 178 Planck’s constant, 2, Poincaré group, 706 Poincaré sphere, 680 Point spectrum of an operator, 395 Poisson bracket, 659 Polar decomposition of bounded operators, 159, 160 Polar decomposition of closed denselydefined operators., 579 Polar decomposition of normal operators, 161 Polarisation formula, 109 Poset, 915 Position operator, 264, 507, 571, 624 Positive element in a C ∗ -algebra with unit, 412 Positive operator, 141 Positive-operator valued measure (POVM), 829 Positive square root, 155 Preparation of system in a pure state, 343, 606 Probabilistic state, 308 Probability amplitude, 340 Probability measure, 21, 306 Product measure, 34 Product structure, 922 Product topology, 13, 43, 85, 253 Projection, 87 Projection space, 87 Projective representation of a symmetry group, 696 Index Projective space, 336 Projective unitary representation of a group, 698 Projective unitary representations of the Galilean group, 778 Projector-valued measure, 348, 431 Projector-valued measure su R, 350 Properly infinite projector, 367 Pullback, 926 Pure algebraic state, 873 Purely atomic Borel measure on R, 33 Purely continuous spectrum, 500 Purely residual spectrum, 500 Pure state, 340, 597 Pushforward, 926 PVM, 348 PVM P concentrated on supp(P), 432 PVM on R, 350 PVM on X, 431 Q Quantum group associated to a group, 705 Quantum logic, 346 Quantum Noether theorem, 834 Quantum state, 327, 333, 597 Quantum symmetry, 666 Quantum symmetry in the algebraic formulation, 907 Quasi-equivalent representations of a ∗ algebra, 899 Quaternions, 138 R Radon measure, 128 Radon–Nikodym derivative, 30, 36 Radon–Nikodym theorem, 30, 36 Range of an operator, 132 Real-analytic manifold, 920 Realisation of a Weyl ∗ -algebra, 645 Reflexive space, 70, 116 Regular complex Borel measure, 65 Regular measure, 22 Relatively compact set, 15, 198 Representation of a ∗ -algebra, 139 Representation of a Weyl ∗ -algebra, 643 Residual spectrum of an operator, 395 Resolvent, 395 Resolvent identity, 396 Resolvent set, 395 Resonance, 556 Restricted Galilean group, 775 Riesz’s theorem for complex measures, 65 947 Riesz’s theorem for complex measures on Rn , 66 Riesz’s theorem for positive Borel measures, 28, 376 Riesz’s theorem on Hilbert spaces, 115 Right-invariant Haar measure, 707 Right regular representation, 762 Right representation of a group, 701 S Scattering, 811 Scattering operator, 813 Schröder–Bernstein theorem, 124 Schrödinger’s equation, 297, 542 Schrödinger’s picture, 832 Schrödinger’s wavefunction, 296 Schur’s lemma, 629 Schwartz distribution, 81 Schwartz space on Rn , 80, 171 Second-countable space, 13, 78 Second real cohomology group, 740 Segal–Bargmann transformation, 664 Segal symmetry, 693, 695 Self-adjoint operator, 141 Self-adjoint operator (general case), 259 Semidirect product, 688 Semidirect product of groups, 917 Seminorm, 40 Semisimple Lie algebra, 723 Semisimple Lie group, 723 Separable Borel measures and L p spaces, 128 Separable Hilbert space, 124 Separable lattice, 359 Separable L p measures and spaces, 128 Separable measure, 127 Separable topological space, 12 Separating elements, 69 Sequentially compact, 198 Sequentially continuous map, 43 Set of admissible pure states in presence of superselection rules, 380, 609 Set of admissible states in presence of superselection rules, 381, 609 Set of atoms of a Borel measure on R, 33 Set of the first category, 81 Set of the second category, 81 Sharp state, 308 Signed measure, 35 Simple C ∗ -algebra, 885 Simple function, 24, 434 Simple group, 730, 916 948 Simple Lie algebra, 723 Simple Lie group, 724 Simply connected space, 17 Singular measure with respect to another, 30 Singular spectrum, 500 Singular values of a compact operator, 212 Smooth manifold, 920 Smooth map, 919 Smooth structure, 920 Smooth vector of a representation of a Lie group, 747 SNAG theorem, 753 Solèr–Holland’s theorem, 362 Space of analytic vectors of a unitary representation of a Lie group, 747 Space of effects, 829 Spatial isomorphism of von Neumann algebras, 167 Special Jordan algebra, 624 Special orthochronous Lorentz group, 706, 730 Special orthogonal group, 706 Special unitary group, 706 Spectral decomposition for normal operators, 449 Spectral decomposition of unbounded selfadjoint operators, 494 Spectral measure associated to a vector, 441 Spectral multiplicity, 462 Spectral radius, 402 Spectral representation of normal operators in B(H), 455 Spectral representation of unbounded selfadjoint operators, 508 Spectral measure on R, 350 Spectral measure on X, 431 Spectrum of a commutative Banach algebra with unit, 415 Spectrum of an operator, 395 Spectrum of the Hamiltonian of the hydrogen atom, 805 Spherical harmonics, 572 Spin, 300, 769 Spin statistical correlation, 862 Spontaneous symmetry breaking, 908 Square root of an operator, 153 Standard deviation, 619 Standard domain, 252 Standard symplectic basis of a symplectic vector space, 640 Standard topology, 12 Statistical operator, 341, 597, 878 Stone representation theorem, 316 Index Stone’s formula, 523, 536 Stone’s theorem, 523, 743 Stone–von Neumann theorem, 640 Stone–von Neumann theorem, alternative version, 641 Stone-Weierstrass theorem, 55, 127 Strongly continuous one-parameter group of operators, 517 Strongly continuous projective unitary representation, 711 Strongly continuous semigroup of operators, 528, 560 Strong Segal automorphism, 695 Strong topology, 74 Structure constants of a Lie algebra, 725 Structure constants of the Galilean group, 776 Sub-additivity, 20, 41, 432 Subalgebra, 51 Subgroup, 916 Subgroup of pure Galilean transformations, 776 Subgroup of space rotations, 776 Subgroup of space translations, 776 Subgroup of time displacements, 795 Subgroup of time translations, 776 Subrepresentation of a ∗ -algebra, 898 Superposition principle of states, 341 Superselection charge, 611 Superselection observables, 611 Superselection rule of the angular momentum, 379, 773, 826 Superselection rule of the electric charge, 379 Superselection rules, 607, 669, 889 Support of a complex measure, 36 Support of a function, 16 Support of a measure, 22 Support of a measure over a lattice, 374 Support of a projector-valued measure, 432 Supremum, 915 Symmetric operator, 259 Symmetry group, 696 Symplectic coordinates, 307 Symplectic form, 639 Symplectic linear map, 639 Symplectic vector space, 639 Symplectomorphism, 639 T Tangent space, 923 Tensor product of Hilbert spaces, 563 Index Tensor product of vectors, 561 Tensor product of von Neumann algebras, 570 Theorem corresponding to Heisenberg’s Uncertainty Principle, 627 Theorem corresponding to Heisenberg’s Uncertainty Principle for mixed states, 654 Theorem corresponding to Heisenberg’s Uncertainty Principle, strong version, 654 Theorem of Arzelà–Ascoli, 49 Theorem of Banach–Alaoglu, 77 Theorem of Banach–Mazur, 50 Theorem of Banach–Steinhaus, 71 Theorem of characterisation of pure algebraic states, 880 Theorem on absolutely convergent series, 35 Theorem on Hilbert-space completion, 112 Theorem on ∗ -homomorphisms of unital C ∗ algebras, 411 Theorem on positive elements in a C ∗ algebra with unit, 413 Theorem on regular values, 923 Theorem on solutions to Fredholm equations of the second kind with Hermitian kernels, 238 Theorem on the commutant of tensor products of von Neumann algebras, 570 Theorem on the continuity of positive functionals over C ∗ -algebras with unit, 877 Theorem on the eigenvalues of compact operators on normed spaces, 202 Theorem on the invariance of the spectrum, 412 Theorem on the representability of algebraic quantum symmetries, 907 Time displacement symmetry, 776, 795, 832, 911 Time-dependent dynamical symmetry, 797 Time-dependent Schrödinger equation, 800 Time-evolution operator, 795 Time-evolution operator in absence of time homogeneity, 818 Time homogeneity, 794, 818 Time reversal, 691, 822–824 Topological dual, 59, 117 Topological group, 690, 705 Topological irreducibility, 629, 918 Topological space, 11 Topological vector space, 74 Topology, 11 949 Topology of a metric space, 78 Total angular momentum of a particle with spin, 771 Total order relation, 915 Total variation of a measure, 35 Trace-class operator, 228 Trace of an operator of trace class, 232 Trace’s invariance under cyclic permutations, 233 Transition amplitude, 340, 597 Transition probability, 340, 672 Transitive representation, 918 Triangle inequality, 41, 78 Trotter formula, 529 Tychonoff’s theorem, 16 U Ultrastrong topology, 166 Ultraweak topology, 166 Uniform boundedness principle, 71 Uniform topology, 74 Unital algebra, 51 Unitarily equivalent irreducible representations of the CCRs, 867 Unitarily equivalent representations of a group, 699 Unitarily equivalent representations of ∗ algebras, 139 Unitary element, 135 Unitary group, 706 Unitary operator, 112, 141 Unitary representation of a group, 699 Unitary transformation, 112 Universal covering of a topological space, 723 Universal representation of a C ∗ -algebra, 898 Upper bound, 915 Upper bounded set, 915 Urysohn’s Lemma, 16, 377 V Vector of uniqueness, 277 Vector subspace, Volterra equation, 221 Volterra equation of the second kind, 243 Volterra operator, 222 Von Neumann algebra, 162, 164, 363 Von Neumann algebra generated by a bounded normal operator and its adjoint, 492 950 Index Von Neumann algebra generated by an operator, 260 Von Neumann algebra generated by a subset of B(H), 165, 364 Von Neumann algebra of observables, 599 Von Neumann’s double commutant theorem, 162 Von Neumann’s theorem on iterated projectors, 347 Von Neumann’s theorem on the continuity of one-parameter groups of unitary operators, 519 Von Neumann’s theorem on the existence self-adjoint extensions (von Neumann’s criterion), 276 Weak Segal automorphism, 695 Weak topology, 74 Weak topology on a normed space, 74 Well-ordering axiom, 915 Weyl ∗ -algebra, 642 Weyl C ∗ -algebra, 904 Weyl C∗ -algebra of a symplectic vector space, 646 Weyl calculus, 657, 660 Weyl–Heisenberg group, 655 Weyl’s (commutation) relations, 632, 642 Wigner automorphism, 672 Wigner symmetry, 672 Wigner’s theorem, 675 W Wavefunction, 341, 625 Wave operators, also known as Møller operators, 811 Weak derivative, 265 Weakly continuous one-parameter group of operators, 517 Weakly non-degenerate bilinear form, 639 Y Yukawa potential, 589, 806 Z Zero-measure set, 23 Zero representation of an algebra, 139 Zorn’s lemma, 123, 915 ... Moretti Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation Second Edition 123 Valter Moretti Department of. .. roughly speaking, the physical theory of the atomic and sub-atomic world, while GSR is the physical theory of gravity, the macroscopic world and cosmology (as recently Introduction and Mathematical. .. axioms of QM, and more advanced topics like quantum symmetries and the algebraic formulation of quantum theories Quantum symmetries and symmetry groups (both according to Wigner and to Kadison)