Bibliography. Apostol, T. M. (1974). Mathematical Analysis, 2nd edn. (Addison Wesley Publishing Company,

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1 Bibliography Apostol, T. M. (1974). Mathematical Analysis, 2nd edn. (Addison Wesley Publishing Company, Reading). Bargmann, V. (1954). On the Unitary Ray Representations of Continuous Groups (Annals of Mathematics 59), p Bargmann, V. (1964). Note on Wigner s Theorem on Symmetry Operations (Journal of Mathematical Physics 5), p Berberian, S. K. (1999). Fundamentals of Real Analysis (Springer, New York). Dirac, P. A. M. (1958, 1947, 1935, 1930). The Principles of Quantum Mechanics (Clarendon Press, Oxford). Greenberg, M. J. and Harper, J. R. (1981). Algebraic Topology: a First Course (Addison- Wesley Publishing Company, Redwood City, California). Heisenberg, W. (1925). Über Quantentheoretische Umdeutung Kinematischer und Mechanischer Beziehungen (Zeitschr. f. Phys. 33), p Hewitt, E. and Stromberg, K. (1965). Real and Abstract Analysis (Springer-Verlag, New York). Hilbert, D., Neumann, J. v., and Nordheim, L. (1927). Über die Grundlagen der Quantenmechanik (Mathematische Annalen 98(1)), p Holevo, A. S. (1982). Probabilistic and Statistical Aspects of Quantum Theory. (North- Holland Publishing Company, Amsterdam), second English edition published by Scuola Normale Superiore, Pisa, Horn, R. A. and Johnson, C. R. (2013). Matrix Analysis, 2nd edn. (Cambridge University Press). Jauch, J. M. (1968). Foundations of Quantum Mechanics (Addison-Wesley Publishing Company, Reading, Massachusetts). Jordan, P. (1926). Über Kanonische Transformationen in der Quantenmechanik (Zeitschr. f. Phys. 37), p Mackey, G. W.(1978). Unitary Group Representations in Physics, Probability, and Number Theory (The Benjamin/Cummings Publishing Company, Reading, Massachusetts). Munkres, J. R. (1991). Analysis on Manifolds (Addison-Wesley Publishing Company, Redwood City, California). Parthasarathy, K. R. (2005). Introduction to Probability and Measure (Hindustan Book Agency (India), New Delhi). Pauli, W. (1933). Die Allgemeinen Prinzipien der Wellenmechanik (Handbuch der Physik 24), p Reed, M. and Simon, B. (1980, 1972). Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, New York). 739

2 740 Hilbert Space and Quantum Mechanics Reeh, H. (1988). A Remark Concerning Canonical Commutation Relation (Journal of Mathematical Physics 29), p Riesz, F. and Sz.-Nagy, B. (1972). Leçons d Analyse Fonctionnelle, 6th edn. (Akadémiai Kiadó, Budapest), English translation of the 2nd edition: Functional Analysis, Dover Publications, New York, Royden, H. L. (1988). Real Analysis (Macmillan Publishing Company, New York). Rudin, W. (1976). Principles of Mathematical Analysis, 3rd edn. (McGraw-Hill Book Company, New York). Rudin, W. (1987). Real and Complex Analysis, 3rd edn. (McGraw-Hill Book Company, New York). Schrödinger, E. (1926). Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der Meinen (Annalen der Physik 79), p Shilov, G. E. (1973). Mathematical Analysis, Vol. 1 (MIT Press, Cambridge), (re-issued as Elementary Real and Complex Analysis by Dover Publications, Mineola, 1996). Shilov, G. E. (1974). Mathematical Analysis, Vol. 2 (MIT Press, Cambridge), (re-issued as Elementary Functional Analysis by Dover Publications, Mineola, 1996). Shilov, G. E. and Gurevich, B. L. (1966). Integral, Measure, and Derivative: a Unified Approach (Prentice Hall, Englewood Cliffs), (re-issued by Dover Publications, Mineola, 1977). Simmons, G. F. (1963). Introduction to Topology and Modern Analysis (McGraw-Hill Book Company, New York). Stone, M. H. (1930). Linear Transformations in Hilbert Space III: Operational Methods and Group Theory (Proc. Nat. Acad. Sci. U.S.A. 16), p Thaller, B. (1992). The Dirac Equation (Springer-Verlag, Berlin). von Neumann, J. (1931). Die Eindeutigkeit der Schrödingerschen Operatoren (Math. Ann. 104), p von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik (Springer- Verlag, Berlin), pages are quoted from the English translation, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, von Neumann, J. (1950). Functional Operators, Vol. 2 (Princeton University Press, Princeton). Weidmann, J. (1980). Linear Operators in Hilbert Spaces (Springer-Verlag, New York). Weyl, H. (1927). Quantenmechanik und Gruppentheorie (Zeitschr. f. Phys. 46), p Wichmann, E. H. (1971). Quantum Physics: Berkeley Physics Course, Vol. 4 (McGraw- Hill, New York).

3 Index µ-integrable function, 192 µ-measurable function, 191 σ-additivity, 152 σ-subadditivity, 152 σ-additivity of a p.v.m., 409 σ-algebra, 122 σ-algebra generated, 122 σ-algebra induced, 123 σ-finite additive function, 151 abelian algebra, 65 abelian group, 19 absolutely convergent, 71 absolutely precise preparation procedure, 636, 647 additive function, 151 additivity, 151 additivity of a p.v.a.m., 408 adjoint, 356 adjointable, 356 Alexandroff s theorem, 154 algebra, 65 algebra generated, 120 algebra of sets, 119 almost every, 158 almost everywhere, 158 antilinear operator, 274 antiunitarily equivalent, 275 antiunitary operator, 274 approximate point spectrum, 92 associative algebra, 65 automorphism of a group, 20 automorphism of a normed space, 95 automorphism of a projective Hilbert space, 316 automorphism of an algebra, 66 automorphism of an inner product space, 255 average of the results, 624 axiom of choice, 18 Banach algebra, 82 Banach space, 71 Bessel s inequality, 264 bijection, 12 bijective, 12 Borel σ-algebra, 124 Borel function, 150 Borel set, 124 bounded observable, 621 bounded operator, 74 bounded sesquilinear form, 285 bounded set, 23 C -algebra, 380 c.o.n.s., 289 c.o.p.u.g., 495 Carathéodory s theorem, 159 cartesian product, 5, 8 Cauchy sequence, 35 Cayley transform, 378, 385 change of variable theorem, 209 change of variable theorem for projection valued measures, 460 characteristic function, 10 closable operator, 90 closed ball, 26 closed graph theorem in Hilbert space, 362, 391 closed operator, 87 closed set, 25 closure of a set,

4 742 Hilbert Space and Quantum Mechanics closure of an operator, 90 coherent superposition of pure states, 640 commutator, 529 commuting operators, 529, 530 commuting projection valued measures, 415 commuting self-adjoint operators, 531 compact, 42 compatible observables, 675 compatible propositions, 668 complement, 4 complete measure, 157 complete metric space, 35 complete orthonormal system, 289 complete set of compatible observables, 682 completion, 36, 268 complex function, 8 composition of mappings, 13 computable self-adjoint operator, 605 connected, 47 conservative quantum system, 688 constant of motion, 694 continuous at a point, 31 continuous mapping, 31 continuous one-parameter group of automorphisms, 513 continuous one-parameter unitary group, 495 continuous spectrum, 370 convergent sequence, 22 convergent series of vectors, 70 copy of a physical system, 613 copy prepared in a state, 613 core, 366 countable set, 12 counterimage, 11 counting measure, 208 cover, 40 De Morgan s laws, 5 dense, 28 density matrix, 596 denumerable set, 12 derivative, 18 derivative of a mapping in a normed space, 497 determination of a proposition, 613 difference of sets, 4 differentiability of a mapping in a normed space, 496 differentiable, 18, 39 dilatation, 240 Dirac measure, 206 direct sum of Hilbert spaces, 269 discrete measure, 208 discrete observable, 622 disjoint, 4 distance, 21 domain, 7 dominating function, 195 e.s.a. operator, 365 eigenspace, 92 eigenvalue, 92 eigenvector, 92 ensemble, 613 epistemic probability, 630 equivalence class, 5 equivalence relation, 5 essential supremum, 430 essentially self-adjoint operator, 365 evaluable observable, 623 exact result, 621 expected result, 623 extended real line, 101 extension, 10 Fatou s lemma, 190 Fejér Riesz lemma, 465 filter for a proposition, 659 final set, 7 final subspace, 573 finite additive function, 151 finite dimensional Hilbert space, 303 finite set, 12 finite-dimensional linear space, 58 finite-dimensional spectral theorem for self-adjoint operators, 482 first kind determination of a proposition, 662 first kind implementation of a proposition, 662 first kind measurement of an observable, 663 Fourier expansion, 289 Fourier transform, 337 Fourier transform on L 2 (R), 343 Fubini s theorem, 220

5 Index 743 function, 8 function of a self-adjoint operator, 483 function of an X-valued observable, 620 function of an observable in quantum mechanics, 643 function of two observables in quantum mechanics, 677 function of two self-adjoint operators, 544 function preserving, 643, 678 g.l.b., 6 Galilean relativity, 715 Galilei-covariance, 719 Galilei-invariance, 719 generator of a c.o.p.u.g., 507 Gram Schmidt orthonormalization, 259 graph, 9 greatest lower bound, 6 group, 18 Hahn s theorem, 165 Hamiltonian, 691 Heine Borel theorem, 43 Heisenberg canonical commutation relation, 382 Heisenberg picture, 696 Hellinger Toeplitz theorem, 365 Hermite c.o.n.s., 333 Hermite function, 262 Hermite polynomial, 262 Hilbert space, 268 homomorphism of algebras, 66 homomorphism of groups, 20 ideal determination of a proposition, 662 ideal filter, 660 ideal implementation of a proposition, 662 ideal measurement of an observable, 663 identity mapping, 10 identity of an algebra, 65 image, 11 implementation of a filter, 660 implementation of a proposition, 619 implementation of a state, 619 implementation of an isomorphism of projective Hilbert spaces, 317, 318 impossible result, 621 improper eigenfunction, 647 indexed family, 8 initial set, 7 initial subspace, 573 injection, 12 injective, 12 inner product, 248 inner product space, 248 integrable function, 192 integral, 178, 180, 185 integral over a subset, 202 integral with respect to a projection valued measure, 429, 443, 446 interference term, 640 interior, 24 intersection, 4 invariant subspace, 550 inverse, 12 inverse Fourier transform, 337 involution, 380 irreducible pair of c.o.p.u.g. s, 700 irreducible pair of self-adjoint operators, 700 irreducible representation of WCR, 700 irreducible set of operators, 567 isometry, 572 isomorphism of algebras, 66 isomorphism of groups, 20 isomorphism of inner product spaces, 255 isomorphism of metric spaces, 22 isomorphism of normed spaces, 94 isomorphism of projective Hilbert spaces, 316 kinematic Galilei group, 716, 720 l.u.b., 6 Lüder s reduction axiom, 662 least upper bound, 6 Lebesgue integrable functions, 192 Lebesgue integral, 192 Lebesgue measure on R, 235 Lebesgue measure on R n, 237 Lebesgue measure on bounded interval, 243 Lebesgue s dominated convergence theorem, 195 Lebesgue Stieltjes measure, 233 limit of a mapping in a normed space, 496 limit of a sequence, 22 linear basis, 56 linear combination, 56 linear dimension, 58

6 744 Hilbert Space and Quantum Mechanics linear functional, 59 linear manifold, 52 linear manifold generated, 53 linear operator, 59 linear operator in, 59 linear operator on, 59 linear space, 51 linearly dependent, 56 linearly independent, 56 lower bound, 6 Lusin s theorem, 172 mapping, 7 measurable function, 137 measurable mapping, 133 measurable sets, 122 measurable space, 122 measurable subspace, 123 measure, 157 measure space, 157 measurement of an observable, 618 metric space, 21 metric subspace, 22 microparticle, 653 microstate, 629 mixed state, 639 mixture of states, 639 monotone class, 132 monotone convergence theorem, 180, 188 monotonicity, 151 multiplication operators, 458 negation of a proposition, 615, 616 non-relativistic quantum particle, 721 norm, 69 norm of a bounded operator, 75 normalized vector, 258 normed algebra, 82 normed space, 69 null measure, 157 null space, 59 o.n.s., 258 o.n.s. complete in a subspace, 289 observable, 616 one-dimensional projection, 392 open ball, 23 open set, 23 operator, 59 orthogonal complement, 265 orthogonal decomposition, 276 orthogonal decomposition mapping, 387 orthogonal decomposition theorem, 276 orthogonal dimension of a Hilbert space, 298 orthogonal dimension of a subspace, 299 orthogonal projection, 387 orthogonal subset, 257 orthogonal sum of subspaces, 402 orthogonal vectors, 257 orthogonality between subsets, 267 orthonormal system, 258 outer measure, 158 P-measurable function, 430 p.v.a.m., 408 p.v.m., 409 parallelogram law, 253 Parseval s identities, 289 partial isometry, 572 partial ordering, 6 partial sum, 70 partially isometric operator, 572 partition, 6 phase space, 630 physical system, 613 point spectrum, 92 polar decomposition, 575 polarization identity, 247 positive linear functional, 227 positive operator, 571 possible result, 621 premeasure, 152 principle of relativity, 715 probability function, 613, 615 probability measure, 157 probability of a proposition in a state, 613 product σ-algebra, 128 product distance, 37 product measure, 213 product measure space, 213 product of operators, 60 product of projection valued measures, 416 projection, 391 projection mappings, 10 projection postulate, 662 projection theorem, 276 projection valued additive mapping, 408 projection valued measure, 409

7 Index 745 projection valued measure of a self-adjoint operator, 479 projective Hilbert space, 316 proof by contradiction, 2 proof by contraposition, 2 proof by induction, 2 proposition, 612, 615 pure state, 639 purely quantum state, 639 Pythagorean theorem, 258 quantized observable, 646 quantized result, 645 quantum particle model, 721 quantum theory, 636 quotient set, 6 range, 7 ray, 316 real function, 8 reducing subspace, 553 reduction of an operator, 553 regular measure, 170 relation, 5 representation of the Weyl commutation relation, 698 representative, 5 resolution of the identity, 420 resolvent set, 91 restriction of a function, 10 restriction of an operator, 60 reversible quantum system, 688 Riemann integrable function, 244 Riemann integral, 244 Riesz representation theorem, 284 Riesz Fisher theorem, 281 Riesz Fréchet theorem, 284 Riesz Markov theorem, 227 s.a. operator, 365 scalar, 51 scalar multiplication, 52 scalar product, 248 Schrödinger equation, 692 Schrödinger picture, 696 Schrödinger representation of WCR, 700 Schwartz space of functions of rapid decrease, 55, 249, 336 Schwarz inequality, 251, 252 Schwarz inequality in C N, 272 Schwarz inequality in l 2, 273 section of a function, 210 section of a set, 210 self-adjoint operator, 365 semialgebra, 117 separable, 29 sequence, 8 series, 23, 70, 110 series of projections, 405 sesquilinear form in, 247 sesquilinear form on, 247 Shur s lemma, 570 simple function, 142 singleton set, 4 space translation, 720 spectral family, 419 spectral theorem for self-adjoint operators, 475 spectral theorem for unitary operators, 469 spectrum of an observable, 621 spectrum of an operator, 91 square integrable function, 319 standard deviation of the results, 624 standard extension, 185 state, 612, 615 state preparation, 612 state reduction, 660, 666 stationary state, 693 statistical operator, 596 statistical theory, 613 Stern Gerlach device, 658 Stone s theorem, 504 Stone s theorem in one dimension, 511 Stone von Neumann uniqueness theorem, 710 Stone Weierstrass approximation theorem, 85 subadditivity, 151 subalgebra, 65 subgroup, 19 subsequence, 13 subspace, 72 subspace generated, 72 sum of a series, 23, 70, 110 sum of linear spaces, 54 sum of normed spaces, 73 sum of operators, 61 sum of sets, 53 superposition principle, 640

8 746 Hilbert Space and Quantum Mechanics superselection rules, 637 support, 34 surjection, 12 surjective, 12 symmetric operator, 364 system, 613 tight measure, 170 Tonelli s theorem, 218 total ordering, 6 trace class, 581 trace class operators, 581 trace of a positive operator, 578 trace of a trace class operator, 588 trajectory of a state, 692 transition probability, 662 translation, 239 triangle inequality in C N, 272 triangle inequality in l 2, 273 trigonometric polynomial, 84 trivial observable, 720 two-valued observable, 628 ubp closed, 147 ubp limit, 147 uncertainty, 623 uncertainty relations, 686 uncountable set, 12 uniformly continuous, 31 union, 4 unitarily equivalent, 273 unitarily-antiunitarily equivalent, 276 unitarily-antiunitarily equivalent representations of a quantum system, 653 unitary irreducible representation of a group, 570 unitary operator, 273 unitary representation of a group, 557 upper bound, 6 vector, 51 vector space, 52 vector sum, 51 velocity transformation, 720 von Neumann s reduction axiom, 662 WCR, 698 weight, 639 Wigner s theorem, 304 X-valued observable, 616, 619 yes-no observable, 628 zero Hilbert space, 271 zero linear space, 54