Mathematics – Logic
Scientific paper
May 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.3655d&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 10, pp. 3655-3657 (2006).
Mathematics
Logic
Scientific paper
Bryce Seligman DeWitt (1923 2004), a friend and mentor to many, was a towering figure in the development of the quantum theories of gravity and gauge fields. To appreciate his uniqueness, one must recall the history through which he lived. From DeWitt's birth date through 1965, general relativity (GR) was considered to have so few empirically testable predictions that its practitioners in English-speaking countries were largely banished to mathematics departments. When the discoveries of cosmological background radiation, quasars and pulsars made it clear that GR does model important science at astronomical scales, the theory still appeared remote from the microcosmic concerns of most research physicists. Every course on elementary-particle physics began by listing the four fundamental forces of nature; then the lecturer would cross off the line 'gravitational' and say 'we will have no more to say about that one'. As recently as 1980, high-energy theorists lecturing on phase transitions in the early universe often started their timelines with a cartoon of a dragon to represent the unknowable initial subnanosecond.
The isolation of GR from the rest of theoretical physics was intensified by the special nature of its mathematical tools. Particle physicists could recognize that condensed-matter people were doing quantum field theory; nuclear and molecular physicists used the same quantum mechanics. In the early days, the heavily indexed tensors of GR betokened a kinship with continuum mechanics (similarly exiled to engineering departments), but when relativists fell under the spell of index-free differential forms and algebraic topology, their isolation became complete. But only briefly: by around 1976 non-Abelian gauge theories had become the core of the standard model even as they were increasingly explicitly expounded in terms of the geometry and topology of Lie groups and fibre bundles. By 1985 the superstring revolution was under way, and soon the professors were saying that gravity was not only an honourable member of the forces of nature, but ultimately the source of them all. It has become a cliche that the unification of relativity with quantum theory is the central problem of contemporary physics.
In the 1950s, before strings, non-Abelian monopoles, or even quasars, Bryce DeWitt was applying the quantum-field-theoretical methods and conceptual framework of his mentor, Julian Schwinger, to gravity. His central insight was that the conceptual and technical problems of quantum gravity were closely analogous to those of gauge theories. He developed a unified, relentlessly abstract, and highly personal vision of the fundamentals of physical theory. It was, and is, expressed in idiosyncratic and condensed notation often different from the languages of mainstream field theorists, traditional relativists, and mathematicians alike. In short, he has never been easy reading. His ideas were systematically presented in famous lecture series at the Les Houches summer schools on Relativity, Groups and Topology in 1963 [1] and 1983 [2], the book Supermanifolds [3], and a number of (relatively) shorter expositions that have been widely read. By the middle 1970s the methods that he had developed mainly for gravity were widely recognized as very useful for Yang Mills gauge theories, and his work had become part of the mainstream.
Now, another 20 years after the second Les Houches, we have this final testament of Bryce DeWitt's ideas. At over 1000 pages in two volumes in a fabric-covered slipcase, it is the sort of work usually described as 'magisterial' (meaning, perhaps, 'no one has yet succeeded in reading it all the way through'). Over the years, of course, DeWitt learned many new things and thought of ways to say the old things better. Accordingly, the new books consist of reworkings of the most important parts of the older writings together with some new material. Oxford University Press is to be thanked and congratulated for the care it has lavished on this set, which is an important monument and reference but presumably not a moneymaker. Every university library must buy it, but probably few individuals will, at least not in the near term when most of those with the seniority (i.e., cash) to make the purchase already own, or have read, the Les Houches lectures. What remains to be seen is to what extent a later generation will discover it as a valuable resource.
Parts I and II develop the basic classical and quantum kinematics of fields and other dynamical systems. The presentation is conducted in the utmost generality, allowing for dynamical quantities that may be anticommuting (supernumbers) and theories subject to the most general possible gauge symmetry— in fact, such symmetries are called 'invariant flows' to emphasize that they may not form a gauge group in the conventional sense. The basic ingredients are action functionals and the Peierls bracket, a manifestly covariant replacement for the Poisson bracket and equal-time commutation relations. Nothing was more central to the DeWitt gospel than these, and the book begins with a paragraph of condemnation of Hamiltonians and canonical formulations with constraints as dysfunctional nineteenth-century baggage. For DeWitt the logical progression is
Peierls bracket → Schwinger action principle → Feynman functional integral
although he points out that the historical development was in the opposite order. The word 'global' in the title, presumably chosen to avoid duplicating the second Les Houches lectures, refers to this overall framework.The word is somewhat misleading, since in many respects DeWitt's work entails a concentration on local over global quantities. For example, chapter 2 states forcefully that local gauge symmetries are both more fundamental and more ubiquitous than global symmetries.
It must be pointed out that the Peierls Schwinger DeWitt approach, despite some advantages over initial-value formulations, has some troubles of its own. In particular, it has never completely escaped from the arena of scattering theory, the paradigm of conventional particle physics. One is naturally led to study matrix elements between an 'in-vacuum' and an 'out-vacuum' though such concepts are murky in situations, such as big bangs and black holes, where the ambient geometry is not asymptotically static in the far past and future.
The newest material in the treatise appears in two chapters in part II devoted to the interpretation of quantum theory, incorporating some unpublished work of David Deutsch on the meaning of probability in physics. Such discussions are unavoidably polemical; DeWitt takes a firm stand ('Everett's ['many worlds'] interpretation has been adopted by the author out of practical necessity: he knows of no other [acceptable one]'), but he acknowledges that 'each physicist has his own manner of understanding quantum mechanics', and the philosophical differences have little import for how theories are applied in practice. In the end DeWitt's many-world theory comes out very similar to the more recent 'decoherent histories' approach, which (in some versions, at least) attributes physical reality to quantities whose measurements can be predicted with certainty—thereby having as much kinship with hidden-variable theories as with the extreme Everett view. I recall a conversation with Bryce in which he said (in paraphrase) 'there is a deeper reality underneath the quantum reality. [In that sense Einstein, Podolsky and Rosen were right.] But it is not the classical reality. [That is, a naive hidden-variable picture does not apply at the microscopic level.]' This occurred probably in the late 1970s, long after DeWitt became a public partisan of the Everett interpretation and long before this book, so I do not believe that it represents a wavering of his faith in many worlds, just a nuance in what that meant to him.
Parts III through V apply the formalism in depth to successively more difficult classes of systems: quantum mechanics, linear (free) fields, and interacting fields. DeWitt's characteristic tools of effective actions, h
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