CONSISTENT QUANTUM THEORY

By A. LALAN

CONTENTS INTRODUCTION SCOPE OF THIS BOOK QUANTUM STATES AND VARIABLES QUANTUM DYNAMICS MATHEMATICS I LINEAR ALGEBRA MATHEMATICS II CALCULUS, PROBABILITY THEORY QUANTUM REASONING QUANTUM MEASUREMENTS QUANTUM PARADOXES WAVE FUNCTIONS CLASSICAL AND QUANTUM PARTICLES PHYSICAL INTERPRETATION OF THE WAVE FUNCTION WAVE FUNCTIONS AND POSITION WAVE FUNCTIONS AND MOMENTUM TOY MODEL LINEAR ALGEBRA IN DIRAC NOTATION HILBERT SPACE AND INNER PRODUCT LINEAR FUNCTIONALS AND THE DUAL SPACE OPERATORS, DYADS PROJECTORS AND SUBSPACES ORTHOGONAL PROJECTORS AND ORTHONORMAL BASES COLUMN VECTORS, ROW VECTORS, AND MATRICES DIAGONALIZATION OF HERMITIAN OPERATORS TRACE POSITIVE OPERATORS AND DENSITY MATRICES FUNCTIONS OF OPERATORS PHYSICAL PROPERTIES CLASSICAL AND QUANTUM PROPERTIES TOY MODEL AND SPIN HALF CONTINUOUS QUANTUM SYSTEMS NEGATION OF PROPERTIES (NOT) CONJUNCTION AND DISJUNCTION (AND, OR) INCOMPATIBLE PROPERTIES PROBABILITIES AND PHYSICAL VARIABLES CLASSICAL SAMPLE SPACE AND EVENT ALGEBRA QUANTUM SAMPLE SPACE AND EVENT ALGEBRA RE?NEMENT, COARSENING, AND COMPATIBILITY PROBABILITIES AND ENSEMBLES RANDOM VARIABLES AND PHYSICAL VARIABLES AVERAGES COMPOSITE SYSTEMS AND TENSOR PRODUCTS DE?NITION OF TENSOR PRODUCTS EXAMPLES OF COMPOSITE QUANTUM SYSTEMS PRODUCT OPERATORS GENERAL OPERATORS, MATRIX ELEMENTS, PARTIAL TRACES PRODUCT PROPERTIES AND PRODUCT OF SAMPLE SPACES UNITARY DYNAMICS THE SCHR¨ODINGER EQUATION UNITARY OPERATORS TIME DEVELOPMENT OPERATORS TOY MODELS STOCHASTIC HISTORIES CLASSICAL HISTORIES QUANTUM HISTORIES EXTENSIONS AND LOGICAL OPERATIONS ON HISTORIES SAMPLE SPACES AND FAMILIES OF HISTORIES RE?NEMENTS OF HISTORIES UNITARY HISTORIES THE BORN RULE CLASSICAL RANDOM WALK SINGLE-TIME PROBABILITIES WAVE FUNCTION AS A PRE-PROBABILITY APPLICATION: ALPHA DECAY SCHR¨ODINGER’S CAT CONSISTENT HISTORIES CHAIN OPERATORS AND WEIGHTS CONSISTENCY CONDITIONS AND CONSISTENT FAMILIES EXAMPLES OF CONSISTENT AND INCONSISTENT FAMILIES RE?NEMENT AND COMPATIBILITY CHECKING CONSISTENCY SUPPORT OF A CONSISTENT FAMILY INITIAL AND ?NAL PROJECTORS HEISENBERG REPRESENTATION FIXED INITIAL STATE INITIAL PURE STATE CHAIN KETS UNITARY EXTENSIONS INTRINSICALLY INCONSISTENT HISTORIES EXAMPLES OF CONSISTENT FAMILIES TOY BEAM SPLITTER BEAM SPLITTER WITH DETECTOR TIME-ELAPSE DETECTOR TOY ALPHA DECAY QUANTUM INTERFERENCE TWO-SLIT AND MACH–ZEHNDER INTERFEROMETERS TOY MACH–ZEHNDER INTERFEROMETER DETECTOR IN OUTPUT OF INTERFEROMETER DETECTOR IN INTERNAL ARM OF INTERFEROMETER WEAK DETECTORS IN INTERNAL ARMS DEPENDENT (CONTEXTUAL) EVENTS AN EXAMPLE CLASSICAL ANALOGY CONTEXTUAL PROPERTIES AND CONDITIONAL PROBABILITIES DEPENDENT EVENTS IN HISTORIES DENSITY MATRICES DENSITY MATRIX AS A PRE-PROBABILITY REDUCED DENSITY MATRIX FOR SUBSYSTEM TIME DEPENDENCE OF REDUCED DENSITY MATRIX REDUCED DENSITY MATRIX AS INITIAL CONDITION DENSITY MATRIX FOR ISOLATED SYSTEM CONDITIONAL DENSITY MATRICES SOME GENERAL PRINCIPLES EXAMPLE: TOY BEAM SPLITTER INTERNAL CONSISTENCY OF QUANTUM REASONING INTERPRETATION OF MULTIPLE FRAMEWORKS MEASUREMENTS I MICROSCOPIC MEASUREMENT MACROSCOPIC MEASUREMENT, ?RST VERSION MACROSCOPIC MEASUREMENT, SECOND VERSION GENERAL DESTRUCTIVE MEASUREMENTS MEASUREMENTS II BEAM SPLITTER AND SUCCESSIVE MEASUREMENTS WAVE FUNCTION COLLAPSE NONDESTRUCTIVE STERN–GERLACH MEASUREMENTS MEASUREMENTS AND INCOMPATIBLE FAMILIES GENERAL NONDESTRUCTIVE MEASUREMENTS COINS AND COUNTERFACTUALS QUANTUM COINS STOCHASTIC COUNTERFACTUALS QUANTUM COUNTERFACTUALS DELAYED CHOICE PARADOX STATEMENT OF THE PARADOX SOME CONSISTENT FAMILIES QUANTUM COIN TOSS AND COUNTERFACTUAL PARADOX CONCLUSION INDIRECT MEASUREMENT PARADOX COMPARING MIN AND MOUT DELAYED CHOICE VERSION INTERACTION-FREE MEASUREMENT? INCOMPATIBILITY PARADOXES SIMULTANEOUS VALUES VALUE FUNCTIONALS PARADOX OF TWO SPINS TRUTH FUNCTIONALS PARADOX OF THREE BOXES TRUTH FUNCTIONALS FOR HISTORIES SINGLET STATE CORRELATIONS SPIN CORRELATIONS HISTORIES FOR THREE TIMES MEASUREMENTS OF ONE SPIN MEASUREMENTS OF TWO SPINS EPR PARADOX AND BELL INEQUALITIES BOHM VERSION OF THE EPR PARADOX COUNTERFACTUALS AND THE EPR PARADOX EPR AND HIDDEN VARIABLES BELL INEQUALITIES HARDY’S PARADOX THE ?RST PARADOX ANALYSIS OF THE ?RST PARADOX THE SECOND PARADOX ANALYSIS OF THE SECOND PARADOX DECOHERENCE AND THE CLASSICAL LIMIT PARTICLE IN AN INTERFEROMETER DENSITY MATRIX RANDOM ENVIRONMENT CONSISTENCY OF HISTORIES DECOHERENCE AND CLASSICAL PHYSICS QUANTUM THEORY AND REALITY QUANTUM VS CLASSICAL REALITY MULTIPLE INCOMPATIBLE DESCRIPTIONS THE MACROSCOPIC WORLD