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The landscape of theoretical physics: a global view. From point particles to the brane world and beyond, in search of a unifying principle. (English) Zbl 1059.81004

Fundamental Theories of Physics 119. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-7006-6). xx, 367 p. (2001).
This book is a research monograph of an unusual kind. In it the author has collected his rather personal views and those of a comparatively small research community on what the basic ingredients of a fundamental theory of space, time and matter should be. The unifying theme is the extension and application of the “fifth parameter formalism” (originally introduced for the relativistic point particle by Fock and Stueckelberg) and of the so-called “geometric calculus” based on Clifford algebras (first formulated by Hestenes) to the dynamics of membranes, which are considered to be the fundamental constituents of matter and space-time itself.
The first part of the book deals with the relativistic point particle and its quantization. Here the author makes clear his preference for the so-called unconstrained action, according to which the particle moves in space-time rather than in space with respect to an independent evolution parameter \(\tau\), entailing a continuous mass spectrum in the quantum theory. This formulation has proved useful as a technical tool in quantum field theory, where, however, it appears at most in intermediary calculations and a projection onto the mass shell is always performed at the end. By contrast, the author attributes to \( \tau \) a genuine physical meaning as representing the flow of time. He next proposes that every physical quantity is a “polyvector” in the sense of the “geometrical calculus”, i.e. a direct sum of multivectors of all degrees, where “vectors” are elements of the linear space spanned by a certain preferred set of generators of the Clifford algebra, identified with the “space-time basis”. Rewriting the conventional constrained point-particle action in terms of polyvectors and keeping only a minimum of nontrivial polyvector components yields the unconstrained action. This result is central to the author’s philosophy, because an analogous relation also holds for membranes.
The rest of Part I is devoted to the first and second quantization of the unconstrained action (which will be commented on below), a discussion of the realization of spinors and Grassmann numbers as Clifford algebra elements, and the relativity of signature and dimension of the space(-time) metric arising from different choices of basic generators in the Clifford algebra (suggesting the interpretation of the space of polyvectors as a “pandimensional continuum”). The relativity of signature also motivates the discussion of the pseudo-Euclidean harmonic oscillator as a model for the cancellation of the cosmological constant (also commented on below) in the final chapter of Part I.
The subject of Part II is the generalization of the concepts developed in Part I to an unconstrained theory of membranes. An unconstrained membrane (of any dimension) is represented by a point in an infinite-dimensional “membrane space” that evolves with respect to a parameter \(\tau\). Only for a special choice of metric in membrane space the standard membrane action is recovered after an appropriate projection, implying that unconstrained membranes are not identified under active diffeomorphisms or, physically speaking, that the points on the membrane possess individuality. Polyvectors are employed not only to obtain the unconstrained action from the constrained one, but also as wavefunctionals of first quantization, and it is proposed to identify their multivector components with the definite number states of Fock space. Consideration of the external field problem for membranes then leads the author to argue that there is no background space independent of a membrane configuration but that space-time is nothing but a (dense) membrane configuration (containing also branes of lower dimension), an effective space-time metric being determined by the Einstein equations with the membranes as sources via an action that is originally defined in membrane space. This is interpreted as a realization of the concept of a reference fluid introduced by DeWitt and Rovelli.
Part III adopts the point of view that our space-time is just one 4-dimensional member of such a membrane configuration. In this picture 4-dimensional gravity is “induced” (in the sense of Sakharov) on our 4-brane by quantum fluctuations of its embedding functions, whereas matter arises from its self-intersections or intersections with other branes. The spirit of this model is similar to the “brane-world” scenario of Randall and Sundrum to which it is compared in a dedicated section. In the last formal chapter the author proposes a solution to the “problem of time” that is often attributed to quantum gravity: Observing that wavefunctionals may be concentrated on spacelike hypersurfaces (and on material sources contained within them) that move in space-time with respect to \(\tau\), he considers \( \tau\) as “true” time.
The final part of the book is entitled “Beyond the Horizon” and presents mainly the author’s own philosophical view on the nature of quantum mechanics as well as a non-technical discussion of some peculiar issues of unconstrained membrane dynamics.
It is clear from the text as well as from the comprehensive list of references at the end of the book that the author has worked on the Stueckelberg formulation and classical membrane actions for decades and is an expert on the pertinent literature, including the application of “geometric calculus”. As regards mathematics, the presentation is consistent (with the possible exception of the use of non-commuting partial derivatives in the definition of curvature), but never goes beyond the formal level. This leaves much to be desired, in particular the most important issue of the consistency of the quantum theory of membranes (constrained or unconstrained) is left open. Most physicists will not follow the author when he approves of the sacrifice of (micro)causality and positivity of the energy in order to adhere to the Stueckelberg particle (which admits tachyons) and its second quantization or to the pseudo-Euclidean oscillator. (The main objection to unbounded negative energy is not gravitational repulsion but the loss of stability of our world!) The expectation that an ultrahyperbolic metric signature will make string theory well-defined in any dimension is also unfounded. On the philosophical side there will be objections to the identification of the wavefunction with consciousness, and based on this, the suggestion of the immortality of human beings in Part IV (which is otherwise quite readable). A comparatively minor deficiency of the book is its poor editing, manifested in many (mostly trivial) misprints and linguistic errors.
Summarizing: What separates the theory exposed in this book from mainstream approaches like string/M theory or loop quantum gravity is the level of sophistication with respect to both internal consistency and phenomenology. A bold approach it is, yes. But, to use a famous phrase of Bohr, is it crazy enough?

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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