This book addresses fundamental issues in mathematical physics, especially the structure of space-time. Quantum mechanics implicitly raises questions about the geometry of space-time: is it discrete, or is a continuous space-time consistent with quantum level measurements and the uncertainty principle?
The author begins with an exploration of the consequences of causality. For him, causality means that no signal can propagate faster than light; this is often called Einstein-Weyl causality. The author shows that causality can be defined as a partial order on any infinite set of points without any predefined mathematical structure. These “causally ordered spaces” can be densely embedded in continua in a unique way, and the causal order can be uniquely extended to the completed space. When these continua are finite-dimensional, they have unique local structures as differentiable manifolds. The first part of the book concentrates on a rigorous development of the embedding of a causally ordered space in a continuum.
The second part of the book addresses an argument raised by some physicists (such Eugene Wigner) that “there are no points” because points seem to be inconsistent with the uncertainty principle and some quantum measurement theories. The author’s constructions in the first part are contingent on the assumption that space-time is made up of geometrical points. Part 2 argues that one can make sense of the concept of point in quantum mechanics – and indeed that the basic issue is already present in classical mechanics. Much of the discussion involves theories of measurement, including von Neumann’s early theory as well as the theory due to Sewell that the author prefers.
The author suggests that this book is appropriate for advanced undergraduates and beginning graduate students in mathematics and physics. He does devote about a third of the book to mathematical appendices in order to provide the necessary background. Realistically this is a book for a devoted reader with special interests in this area and with graduate level training in mathematics and a decent physics background, or vice-versa. There are no exercises. This is really a research monograph.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
Prologue; Part I: Introduction to Part I; 1. Mathematical structures on sets of points; 2. Definition of causality on a structureless set; 3. The topology of ordered spaces; 4. Completions of ordered spaces; 5. Structures on order-complete spaces; Part II: Introduction to Part II; 6. Real numbers and classical measurements; 7. Special topics in quantum mechanics; 8. Von Neumann's theory of measurement; 9. Macroscopic observables in quantum physics; 10. Sewell's theory of measurement; 11. Summing-up; 12. Large quantum systems; Epilogue; Appendixes; References; Index.