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Gauge fields, knots and gravity. (English) Zbl 0843.57001

Series on Knots and Everything. 4. Singapore: World Scientific. xii, 465 p., $ 81.00; £56.00/hbk (1994).
Written jointly by a mathematician and a physicist this book aims to explain everything about gauge fields, knots, and gravity ‘from scratch’ in a self-contained manner and is meant to be an elementary introduction to these topics. With great candour the authors admit that these subjects are full of mysteries. Well, in a market dominated world, everything is valued in terms of money and subjects which attract megabucks are naturally more important than those that do not – the quality of truth is less important than the grants won. Astrophysics with obvious interrelations with space science and ballistics and modern quantum physics with similar interrelations with the technology of nuclear and laser weapons are the subjects in which governments have poured large sums of money: the results are spectacular. Quite a few distinguished topologists have found the inspiration for developing the geometry of low-dimensional manifolds in these highly lucrative fields and it cannot be disputed that the mathematics thus developed is good and valid. On the other hand on the physics side, much of the developments has been speculative, hasty and mysterious, though a few of the ideas thus formulated have captured rare glimpses of the truth that has led to the developments in topology mentioned above. Though the aim was to provide a theory of everything, no completely consistent theory of anything in physics has emerged from these deliberations.
This book aims to provide a systematic account of the more important developments thus made for both mathematicians and physicists. The account is presented in an interesting informal and conversational style with a great sense of humour, though many readers will have difficulty in seeing the funny side of a mere difference in sign in a pair of equations when they are described as such. The style adopted has many advantages but one drawback is that the authors have found it often necessary to use twenty words where five will do, this might not have been such a bad thing after all but the poor reader has to pay for all the unnecessary extra words. Again judging by style, the book is meant more for the physicist than for the mathematician as the informality has made the definitions and proofs somewhat lacking in a precision that one expects in works on mathematics. As for the mysteries, they arise because physicists often try to sell ad hoc explanations which explain a particular phenomenon but are inconsistent with the existing mathematical model as a valid theory without even trying to construct a different or even modified self-consistent mathematical model. A good example is the notion of phase of a state vector in quantum mechanics: in a rigorously formulated model of quantum mechanics the phase plays no part at all but many physical phenomena are explained by assigning the phase some significance on an ad hoc basis. This indicates that the rigorous quantum theory as a model for what actually happens is probably inadequate and needs modification and of course attempts have been made to formulate just such a modified theory without success. Thus the situation is that some regularities in the physical system have been identified but the correct explanations and the correct model are yet to be discovered.
The authors deal with this matter in the following way: they say that they are going to do something sloppy, then they use the phase to explain something and then they point out that this violates the rules they have laid down for phase in quantum theory and then they say somewhat vaguely that the phase is indeed needed for certain purposes leaving the reader a little bewildered about what is going on. While the book describes the mathematics reasonably effectively without being formal as is done in most mathematics books meant for physicists, it does little to describe either the methodology of modern physics or the physical phenomena for those mathematicians who have not studied physics to a significant level.
There are a certain number of bad printing mistakes such as on page 27 where the Section on Tangent Bundles begins with “Often it is nice to think…”. Notwithstanding these defects, this is a most useful and timely book, albeit a bit expensive considering that its size is probably twice as much as is really necessary for imparting the same amount of information.

MSC:

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory
83C45 Quantization of the gravitational field
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