Space and Time in the Microworld – Blokhintsev

In this post, we will see the book Space and Time in the Microworld by D. I. Blokhintsev .

About the book

In comprehending the physical content of dynamic variables which have geometric meaning, for example, the space-time particle coordinates x, y, z, t it is often 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. In a desert sea of abstract constructions there is a still larger distance between macroscopic concepts of space-time and the way of employing the coordinates x, y, z and t in relativistic quantum field theory.

It is shown in this monograph that if elementary particles have a structure it is doubtful whether the coordinates of elementary particles, x, y, z, t, can even be defined exactly, let alone the coordinates of the elements which make up these particles (if they do not exist only in our imagination). This important fact is revealed in even the most favourable gedanken experiments.

From this fact, doubt arises about the logical validity of using the symbols x, y, z, t as the space-time coordinates to describe phenomena inside elementary particles. This allows theoreticians a certain freedom of choice of space-time and causal relationships within elementary particles; in other words, an arbitrariness of choice of the geometry in
the small.

The last chapters of this book describe some models used to illustrate the situation described above. In concluding, experimental data and experimental possibilities relating to geometric and causal problems in the microworld are discussed.

The book was translated from Russian by Zdenka Smith and was published in 1970.

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Contents

INTRODUCTION

I. GEOMETRICAL MEASUREMENTS IN THE MACROWORLD 1

1. The Arithmetization of Space-Time 1
2. The Physical Methods of Arithmetization of Space-Time 3
3. On Dividing the Manifold of Events into Space and Time 10
4. The Affine Manifold 19
5. The Riemann Manifold 23
6. The Physics of Arithmetization of the Space-Time Manifold 28
7. Arithmetization of Events in the Case of the Non-Linear Theory of Fields 33
8. The General Theory of Relativity and the Arithmetization of Space-Time 37
9. Chronogeometry 42

II. GEOMETRICAL MEASUREMENTS IN THE MICROWORLD 44

10. Some Remarks on Measurements in the Microworld 44
11. The Measurement of Coordinates of the Microparticles 46
12. The Mechanics of Measuring Coordinates of Microparticles 52
13. Indirect Measurement of a Microparticle Coordinates at a Given Instant in Time 64

III. GEOMETRICAL MEASUREMENTS IN THE MICROWORLD IN THE RELATIVISTIC CASE 69

14. The Fermion Field 69
15. The Uncertainty Relation for Fermions 74
16. The Boson Field 77
17. The Localization of Photons 82
18. The Diffusion of Relativistic Packets 86
19. The Coordinates of Newton and Wigner 89
20. The Measurement of a Microparticle’s Coordinates in the Relativistic Case

IV. THE ROLE OF FINITE DIMENSIONS OF ELEMENTARY PARTICLES 95

21. The Polarization of Vacuum. The Dimensions of an Electron 95
22. The Electromagnetic Structure of Nucleons 99
23. The Meson Structure of Nucleons 108
24. The Structure of Particles in Quantized Field Theory 114

V. CAUSALITY IN QUANTUM THEORY 124

25. A Few Remarks on Causality in the Classical Theory of Fields 124
26. Causality in Quantum Field Theory 132
27. The Propagation of a Signal “Inside” a Microparticle 141
28. Microcausality in the Quantum Field Theory 147
29. Microcausality in the Theory of Scattering Matrices 153
30. Causality and the Analytical Properties of the Scattering Matrix 159

VI. MACROSCOPIC CAUSALITY 173

31. Formal S-matrix Theory 173
32. Space-Time Descriptions Using the S-matrix 182
33. The Scale for the Asymptotic Time T 187
34. Unstable Particles (Resonances) 191
35. Conditions of Macroscopic Causality for the S-matrix 200
36. Examples of Acausal Influence Functions 207
37. An Example of Constructing an Acausal Scattering Matrix 211
38. The Dispersion Relation for the Acausal S_{a},-Matrix 219

VII. A GENERALIZATION OF CAUSAL RELATIONSHIPS AND GEOMETRY 226

39. Two Possible Generalizations 226
40. Euclidean Geometry in the Microworld 232
41. Stochastic Geometry 237
42. Discrete Space-Time 243
43. Quasi-Particles in Quantized Space 250
44. Fluctuations of the Metric 255
45. Nonlinear Fields and the Quantization of Space-Time 261

VIII. EXPERIMENTAL QUESTIONS 269

46. Concluding Remarks on the Theory 269
47. Experimental Consequences of Local Acausality 270
48. Experimental Results of Models with the “External” Vector 278

APPENDICES 282

BIBLIOGRAPHY 326

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