The Enigma of Probability and Physics

Lazar Mayants is a recent Russian emigre noted for his work in theoretical physics. He was previously a professor at several universities of the Soviet Union and a distinguished member of the Academy of Sciences of the U.S.S.R, where he worked for about 30 years. In this book he presents a unique, extremely detailed, and embracive version of a subject that has suffered for a long time from numerous internal imperfections. His approach is new and original, the material covered features not only the foundations of the science of probability but also most of its applications, including statistical and quantum mechanics. The key methodolOgical principle underlying the book is of extraordinary significance and deserves special attention. The treatment excels in thoroughness of presentation, in its fulness of mathe­ matical detail and the abundance of physical examples. The book is intended for a wide range of people interested in probability and its connection with modern science. It is written as a text for advanced students, and I predict that a reader who masters all its contents will become an expert in the subject of both prob­ ability and its physical implications, while enjoying its understanding and use. HENRY MARGENAU Veritas nihil veretur nisi abscondi (truth 'What tremendously easy riddles you ask!' Humpty Dumpty growled out. fears nothing except being hidden). Latin proverb Lewis Carroll, Through the Looking Glass, Chap. 6. Preface The history of producing this book is rather complicated and not quite usual.

I. Fundamentals of Probabilistics.- 1 / Principal Concepts.- 1.1. Concrete Objects.- 1.2. Abstract Objects.- 1.3. More about Concrete and Abstract Objects.- 1.4. Measure of a Set of Concrete Objects.- 1.5. Experimental Determination of the Measures of Sets of Concrete Objects.- 1.6. The Statistical Method.- 1.7. Probability. Preliminary Consideration.- 1.8. Mutually Adequate Sets.- 1.9. Probability. General Definition.- Problems.- Suggested References.- 2 / Main Theorems.- 2.1. Sum and Product of Events.- 2.2. Addition Theorem.- 2.3. Multiplication Theorem.- 2.4. Sequence of Random Tests.- 2.5. Law of Large Numbers.- 2.6. Limiting Cases of Binomial Law.- Problems.- Suggested References.- 3 / Random Variables.- 3.1. Definition.- 3.2. Probability Distribution.- 3.3. Joint Probability Distribution for Functions of Random Variables.- 3.4. Mathematical Expectation. Moments.- 3.5. Characteristic Function.- Problems.- Suggested References.- 4 / Some Aspects of Statistics.- 4.1. Preliminary Considerations.- 4.2. Statistical Experiment.- 4.3. Numerical Statistical Experiment.- 4.4. Concluding Remarks.- Suggested References.- 5 / States of Abstract Objects.- 5.1. Introductory Remarks.- 5.2. Description of States.- 5.3. Necessary Mathematics.- 5.4. Two Specific Modes of Description of a State.- 5.5. Additional Mathematics.- Problems.- Suggested References.- 6 / Hamiltonian Random Variables.- 6.1. Lagrangian Equations and Hamiltonian Equations.- 6.2. Hamiltonian Random Variables.- 6.3. Canonically Conjugate Operators.- 6.4. Quantum Approach.- 6.5. Standard Deviations of Canonically Conjugate Random Variables.- Problems.- Suggested References.- 7 / Random Fields.- 7.1. Definition.- 7.2. Two Types of Related Finite-Dimensional Random Variables.- 7.1. Lagrangian and Hamiltonian Partial Equations.- 7.2. Hamiltonian Random Fields.- Suggested References.- II. Fundamentals of Probabilistic Physics.- 8 / General Considerations.- 8.1. Preliminaries.- 8.2. Classification of Physical Systems.- 8.3. Two Possible Types of Problems.- 8.4. Conservation Laws.- Suggested References.- 9 / Equilibrium Classical Statistical Mechanics.- 9.1. Microcanonical Distribution.- 9.2. Canonical Distribution.- 9.3. A Separate Atom in a Thermostat.- 9.4. Thermodynamic Functions.- 9.5. The Ideal (Perfect) Gas.- 9.6. Important Notes.- Suggested Reference.- 10 / Quantum Mechanics.- 10.1. Principal Propositions.- 10.2. Operators for Physical Quantities.- 10.2.1. Coordinates and Momenta.- 10.2.2. Time and Energy.- 10.2.3. Hamiltonian.- 10.2.4. Angular Momentum.- 10.2.5. Derivatives.- 10.2.6. Quantities Having a Finite Number of Values.- 10.3. Schrödinger Equation.- 10.4. A Free Particle.- 10.5. A Particle in a One-Dimensional Potential Box.- 10.6. One-Dimensional Harmonic Oscillator.- 10.7. A Particle in a Central Field.- 10.8. Equilibrium Quantum Statistical Mechanics.- Problems.- Suggested References.- 11 / Kinetics of Physical Transformations.- 11.1. Processes in Concrete Physical Systems.- 11.2. Probabilistic Treatment of Transformations.- 11.3. Basic Principles of the Transitional Configuration Theory.- 11.4. Leaving a Potential Well Through a Potential Barrier.- 11.5. Nonexponential Decay Law.- 11.6. Intramolecular Rearrangements.- 11.7. A Criterion of the Possibility of an Intramolecular Rearrangement.- Problems.- Suggested References.- 12 / Electromagnetic Field and Photons.- 12.1. Preliminaries.- 12.2. The Four-Dimensional Space Treatment.- 12.3. Physical Consideration.- 12.4. The Emon.- 12.5. Relativistic Aberration and Doppler Effect.- Problems.- Suggested References.- III. Methodological Problems.- 13 / Problems Related to Probability.- 13.1. Basic Phenomenon of Probabilistics.- 13.2. Mises’ Definition of Probability.- 13.3. Kolmogorov’s Probability Theory.- 13.4. Bertrand Paradox 285 Bibliography.- 14 / Problems Related to Physics.- 14.1. Gibbs’ Paradox and Indistinguishability of Particles.- 14.2. Classical Limit of Quantum Mechanics.- 14.3 Energy and Time.- 14.4. One-Particle Relativistic Equations.- 14.5. Possible States of a Conservative System.- 14.6. Conventional Decay Theories.- 14.7.Time—Energy Uncertainty Relation.- 14.8. Measurement and Related Problems.- 14.1. Wave-Corpuscle Duality and Reality of Motion.- 14.2. Reality of Motion and Potential Energy.- 14.3. Einstein—Podolsky—Rosen’s Paradox.- 14.4. Bell’s Theorem.- 14.5. Second Quantization.- Appendix 1. Proof of Equations (6.16”) and (6.16”).- Appendix 2. Derivation of Equations of Section 6.3.- Appendix 3. A General Rotation-Vibration Hamiltonian.- Answers to Problems.