Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgement -- To the reader -- Prologue -- Part I Causality and differentiable structure -- Introduction to Part I -- 1 Mathematical structures on sets of points -- 1.1 Mathematical structures on finite sets -- 1.2 Mathematical structures on countably infinite sets -- 1.3 Mathematical structures on uncountable sets -- 1.4 Global geometrical structures on Rn -- 1.5 Local geometrical structures on n-manifolds -- 1.5.1 Riemannian and Lorentz structures -- 1.5.2 The conformal structure -- 1.5.3 The Weyl projective structure -- 2 Definition of causality on a structureless set -- 2.1 Light rays -- 2.2 The order axiom -- 2.3 The identification axiom -- 2.4 Light cones -- 2.4.1 Timelike points -- 2.4.2 Extension of order -- 2.5 The cone axiom -- 2.6 D-sets -- 2.7 Properties of D-sets -- 2.7.1 D-sets and timelike order -- 2.7.2 Light rays from T-interiors of D-sets -- 2.7.3 Incidence of light rays on cone boundaries -- 2.7.4 A D-set is an S-space -- 2.7.5 New D-sets from old -- 2.7.6 An incidence theorem for β-boundaries -- 2.7.7 Spacelike separation in D-sets -- 2.7.8 Timelike order and D-subsets -- 2.8 The local structure axiom -- 2.9 Ordered spaces -- 3 The topology of ordered spaces -- 3.1 The order topology -- 3.1.1 The Tychonoff property -- 3.1.2 Order equivalence -- 3.2 Homogeneity properties -- 3.2.1 The standard maps -- 3.2.2 Three or more dimensions -- 4 Completion of ordered spaces -- 4.1 Spaces in which light rays are complete -- 4.2 Metric and uniform completions -- 4.2.1 Uniformizability of ordered spaces -- 4.2.2 Complete uniformizability -- 4.3 The concept of order completion -- 4.3.1 The order uniformity on D-sets -- 4.4 Extension of order: notations and definitions -- 4.5 Extension of order: topological results.

4.5.1 Symmetry properties -- 4.5.2 Separation theorems -- 4.5.3 Density lemmas -- 4.6 Extending the order to M -- 4.6.1 The segment l[a, η] -- 4.6.1.1 Properties of the segment l[a, η] -- 4.6.2 The segment l[epsilon, η] -- 4.6.2.1 Properties of the segment l[ξ, η] -- 4.6.3 Extending light-ray segments -- 4.7 Verification of the axioms -- 4.7.1 The order axiom -- 4.7.2 The identification axiom -- 4.7.3 The cone axiom -- 4.7.4 D-sets in -- 4.7.5 The local structure axiom -- 5 Structures on order-complete spaces -- 5.1 Timelike curves in M -- 5.2 Parametrization of D-intervals -- 5.2.1 2 -cells in D-intervals -- 5.2.2 Cylindrical coordinates on D-intervals -- 5.3 Final results -- 5.3.1 Contractibility of order intervals -- 5.3.2 The local differentiable structure -- Part II Geometrical points and measurement theory -- Introduction to Part II -- 6 Real numbers and classical measurements -- 6.1 The impact of quantum theory -- 6.2 Precise measurements in classical mechanics -- 6.2.1 Space, time and measurement in classical mechanics -- 6.2.2 Perfect rulers -- 6.3 Discussion -- 6.4 The role of the experimentalist -- 6.5 Born's probability interpretation of classical mechanics -- 7 Special topics in quantum mechanics -- 7.1 The formalism of quantum mechanics -- 7.1.1 Pure states -- 7.1.2 Mixed states -- 7.1.3 Partial traces -- 7.1.4 Dynamics -- 7.1.5 Observables -- 7.2 The probability interpretation of quantum mechanics -- 7.2.1 Quantum theory of measurement -- 7.2.2 Remarks on notations -- 7.3 Superselection rules -- 7.4 The Galilei group -- 7.4.1 Projective representations of the Galilei group -- 7.4.1.1 Factor systems and group exponents -- 7.4.1.2 The extended Galilei group -- 7.4.1.3 Mass in nonrelativistic quantum mechanics -- 7.4.2 Bargmann's superselection rule -- conservation of mass -- 7.4.3 The Galilei group in two dimensions.

7.5 Theorems of von Neumann and Stone -- Reeh's example -- 7.5.1 The Stone-von Neumann uniqueness theorem -- 7.5.2 Reeh's example -- 7.5.3 Stone's theorem -- 8 Von Neumann's theory of measurement -- 8.1 Overview -- 8.2 Von Neumann's initial considerations -- 8.2.1 The inference from observation -- 8.2.2 Measurement of operators with continuous spectra -- 8.2.3 The quantum measurement problem -- 8.3 Von Neumann's Chapter VI -- 8.3.1 Formulation of the problem -- 8.3.2 Composite systems -- 8.3.3 Discussion of the measuring process -- 8.4 Wigner's reservations -- 8.5 Reconsideration of von Neumann's theory -- 8.5.1 Entanglement -- 8.5.2 Description of the measuring apparatus -- 9 Macroscopic observables in quantum physics -- 9.1 Commuting self-adjoint operators -- 9.1.1 Commuting observables with discrete spectra -- 9.2 The uncertainty principle and commuting observables -- 9.2.1 The uncertainty principle -- 9.2.2 Minimum uncertainty product states -- 9.2.3 Determination of… -- 9.2.4 Von Neumann's commuting operators… -- 9.2.5 The von Neumann-Wigner characterization of macroscopic measurements -- 9.3 Answers to Wigner, I -- 10 Sewell's theory of measurement -- 10.1 Preliminary discussion -- 10.2 The object-apparatus interaction -- 10.3 The macroscopic observables of A -- 10.4 Expectations and conditional expectations of observables -- 10.4.1 Explicit expressions -- 10.4.2 Properties of F r,s -- alpha -- 10.5 Ideal measurements -- 10.5.1 Interpretation -- 10.5.2 The probability interpretation in Sewell's theory -- 10.6 Consistency and robustness of Sewell's generic model -- 10.6.1 The question of consistency -- 10.6.2 The question of robustness -- 10.7 Answers to Wigner, II -- 11 Summing-up -- 11.1 Bell's example -- 11.2 Extension of Sewell's scheme -- 11.2.1 Degenerate eigenvalues -- 11.2.2 The case of dim… -- 11.3 The Araki-Yanase theorem.

11.4 Impossibility theorems of Shimony and Busch -- 11.5 The Heisenberg cut -- 11.6 Measurement of continuous spectra -- 11.7 Adequacy of Sewell's scheme -- 11.8 Answers to Wigner, III -- 11.8.1 The tension with the theory of relativity -- 12 Large quantum systems -- 12.1 Elementary excitations in superfluid helium -- 12.2 Hilbert bundles -- 12.3 Bundle representations -- 12.3.1 Landau excitations -- 12.3.1.1 Unitary representations of E3 and H -- 12.3.1.2 Landau excitations as bundle representations -- 12.3.1.3 Unstable excitations -- 12.4 Dynamics on Banach bundles -- 12.5 Operator algebras and states -- 12.5.1 C-algebras -- 12.5.2 States on C-algebras -- 12.6 Representations of C-algebras -- 12.6.1 The Gelfand-Naimark-Segal theorem -- 12.6.2 The weak operator topology on a representation -- 12.7 Algebraic description of infinite systems -- 12.7.1 C-dynamical systems -- 12.7.2 Observables at infinity -- 12.8 Temporal evolution of reduced descriptions -- Epilogue -- Causal automorphisms of an ordered space -- Narrowing the von Neumann-Heisenberg gap in measurement theory -- Sets, real numbers and physics -- Interaction of quantum and classical systems -- Superseparability and noninterferometry -- Is nature law abiding? -- Mathematical appendices -- A1 Sets and mappings -- A1.1 Sets -- A1.2 New sets from old -- A1.2.1 Union, intersection and difference -- A1.2.2 Subsets -- A1.2.3 Boolean algebras -- A1.3 Maps -- A1.4 Finite and infinite sets -- A1.5 Power sets -- A1.5.1 Cardinal numbers -- A1.6 Products of sets -- A1.7 The axiom of choice and the well-ordering theorem -- A1.8 Russell's paradox -- A2 The real number system -- A2.1 Irrational numbers -- A2.2 The real line -- A2.2.1 Dedekind's construction -- A2.2.2 The least upper bound property -- A2.2.3 Cantor's procedure -- A2.2.4 Well-ordering and real numbers -- A3 Point-set topology.

A3.1 Topological spaces -- A3.1.1 Closed sets, interior, boundary -- A3.1.2 Comparison of topologies -- A3.2 Continuous functions -- A3.2.1 Homeomorphisms -- A3.3 New spaces from old -- A3.3.1 Subspaces -- A3.3.2 Topological products -- A3.3.3 Quotient spaces -- A3.4 Countability and separation axioms -- A3.4.1 Second countability and separability -- A3.4.2 Separation properties -- A3.4.2.1 The T0-separation property -- A3.4.3 Separation by continuous functions -- A3.5 Metric spaces and the metric topology -- A3.5.1 Metrizability -- A3.6 Compactness -- A3.6.1 Some properties of compact spaces -- A3.6.2 Local compactness -- A3.6.3 One-point compactification -- A3.7 Connectedness and path-connectedness -- A3.8 The theory of convergence -- A3.8.1 Filters -- A3.8.1.1 Filters on topological spaces -- A3.9 Topological groups -- A4 Completions -- A4.1 Metric completion -- A4.2 Uniformities -- A4.2.1 Definition and general properties -- A4.2.2 Uniformizability of a topological space -- A4.2.3 Metrizability of a uniform space -- A4.3 Complete uniform spaces -- A4.3.1 Uniform completion -- A4.3.2 Miscellaneous results -- A4.3.3 Complete uniformizability -- A4.3.4 Total boundedness -- A5 Measure and integral -- A5.1 Normed spaces -- A5.2 Function spaces -- A5.2.1 The sup norm -- A5.2.2 The 1-norm. Incompleteness of the Riemann integral -- A5.2.3 The p-norms -- the 2-norm -- A5.3 The Riemann-Stieltjes integral -- A5.3.1 Step functions as integrators -- A5.4 alpha-algebras, measures and integrals -- A5.4.1 Measurable sets -- A5.4.2 Measures -- A5.4.3 Integrals -- A5.5 Haar measure -- A5.6 The Lebesgue integral and the Lp-spaces -- A5.6.1 Construction of measures -- A5.6.2 The Lebesgue measure on -- A5.6.3 The space L1([a, b], dx) -- A5.6.4 The spaces Lp -- A5.6.5 The space Lp -- A5.7 Noninvariant measures on R -- A5.7.1 Signed measures.

the Radon-Nikodym theorem.