Space and Time in the Microworld

A way of understanding the laws which govem the worId of elementary particIes has not been found yet. Present-day theoretical physicists have to be satisfied with compromises which, at the best, promise some success at the expense of generality and unity. U nder these circumstances a critical analysis of the basic concepts of modem quantum theory may be timely and usefuI. It is hoped that the value of such an analysis may be preserved even if, in the near future, new ways of understanding the basis of elementary particIe physics are discovered. In this monograph one specific aspect of this analysis is treated, namely the problems of geometry in the microworld. An out­ line of geometrical measurements in the macroworld was given pre­ viously. These measurements seem to be c1ear enough for at least a certain set of problems to be considered as a starting point for discussing the situation in the microworld. The concepts and methods which are useful in the macroworld may only indirectly be carried over into the microworld and they require a high degree of abstraction. In comprehending the physical content of dynamic variables which have geometric meaning, for example, the space-time partic1e coor­ dinates x, y, z, t it is of ten necessary to have recourse to gedanken experiments which, although not feasible in practice, can nevertheless be compatible with the basic principles of geometry and quantum mechanics.

I. Geometrical Measurements in the Macroworld.- 1. The Arithmetization of Space-Time.- 2. The Physical Methods of Arithmetization of Space-Time.- 3. On Dividing the Manifold of Events into Space and Time.- 4. The Aftine Manifold.- 5. The Riemann Manifold.- 6. The Physics of Arithmetization of the Space-Time Manifold.- 7. Arithmetization of Events in the Case of the Non-Linear Theory of Fields.- 8. The General Theory of Relativity and the Arithmetization of Space-Time.- 9. Chronogeometry.- II. Geometrical Measurements in the Microworld.- 10. Some Remarks on Measurements in the Microworld.- 11. The Measurement of Coordinates of the Microparticles.- 12. The Mechanics of Measuring Coordinates of Microparticles.- 13. Indirect Measurement of a Microparticle Coordinates at a Given Instant in Time.- III. Geometrical Measurements in the Microworld in the Relativistic Case.- 14. The Fermion Field.- 15. The Uncertainty Relation for Fermions.- 16. The Boson Field.- 17. The Localization of Photons.- 18. The Diffusion of Relativistic Packets.- 19. The Coordinates of Newton and Wigner.- 20. The Measurement of a Microparticle's Coordinates in the Relativistic Case.- IV. The Role of Finite Dimensions of Elementary Particles.- 21. The Polarization of Vacuum. The Dimensions of an Electron.- 22. The Electromagnetic Structure of Nucleons.- 23. The Meson Structure of Nucleons.- 24. The Structure of Particles in Quantized Field Theory.- V. Causality in Quantum Theory.- 25. A Few Remarks on Causality in the Classical Theory of Fields.- 26. Causality in Quantum Field Theory.- 27. The Propagation of a Signal "Inside" a Microparticle.- 28. Microcausality in the Quantum Field Theory.- 29. Microcausality in the Theory of Scattering Matrices.- 30. Causality and the Analytical Properties of theScattering Matrix.- VI. Macroscopic Causality.- 31. Formal ?-matrix Theory.- 32. Space-Time Descriptions Using the ?-matrix.- 33. The Scale for the Asymptotic Time T.- 34. Unstable Particles (Resonances).- 35. Conditions of Macroscopic Causality for the S-matrix.- 36. Examples of Acausal Influence Functions.- 37. An Example of Constructing an Acausal Scattering Matrix.- 38. The Dispersion Relation for the Acausal ?a-Matrix.- VII. A Generalization of Causal Relationships and Geometry.- 39. Two Possible Generalizations.- 40. Euclidean Geometry in the Microworld.- 41. Stochastic Geometry.- 42. Discrete Space-Time.- 43. Quasi-Particlcs in Quantized Space.- 44. Fluctuations of the Metric.- 45. Nonlinear Fields and the Quantization of Space-Time.- VIII. Experimental Questions.- 46. Concluding Remarks on the Theory.- 47. Experimental Consequences of Local Acausality.- 48. Experimental Results of Models with the "External" Vector.- Appendices.