It is something of a truism that modern physics concerns itself with space-time. We generally presuppose, gratia not just Einstein but Minkowski, Lorentz, and even Riemann himself, that the universe “is” a Riemannian manifold with a metric whose signature is something like (+,+,+,-), and wonderful discoveries follow in this context: space time is indeed the locale for twentieth and even twenty first century physics and cosmology, with the problem of unifying the enterprise so as also to take in quantum mechanics being the ever-present siren’s song.
But what about time-space, the phrase in the title of the book under review? Why is Laudal concerned with this entity and its geometry? Well, in the first place, time-space is not space-time: in point of fact a time-space is a moduli space “where the space-time of classical physics bec[omes] a section of the universal fiber space … defined on [this moduli space] … of the physical systems we … consider.” Under this regime “[m]easurable time … turn[s] out to be a metric on the time-space, measuring all possible infinitesimal changes of the state of the objects in the family we are studying.” This is very exciting, of course, and indeed, Laudal notes that we get “a physics where there are no infinite velocities, and … the principle of relativity comes for free.”
This is only the tip of the iceberg. The backdrop, or rather the mathematical framework for these considerations, includes nothing less than non-commutative (algebraic) geometry, and, to be sure, and as the Introduction makes clear (in 17 pages), an awful lot of physics enters in early: quantum theory, quantum field theory, and cosmology provide the launch pad for the next chapter on phase spaces and the Dirac derivation.
The next chapter engenders a real infusion of non-commutative geometrical themes, finishing with a discussion of non-commutative schemes and a section tantalizingly titled, “Morphisms, Hilbert schemes, fields, and strings.” The compact book’s last two chapters concern, respectively, the geometry of time-spaces and “the General Dynamical Law,” and the non-commutative algebraic geometric angle on (physical) interactions.
Laudal introduces the book, in his Preface, with the engaging comment that “[t]his book is the result of the author’s struggle to understand modern physics … but is really just a study of the mathematical notion of moduli, based upon my version of non-commutative algebraic geometry.” Then the Preface ends with the following quote:
The fact that the introduction of a non-commutative deformation theory, the basic ingredient in my version of non-commutative algebraic geometry, might lead to a better understanding of the part of modern physics that I had never understood before, occurred to me during a memorable stay at the university of Catania, Italy, in 1992. To check this out, has since then been my main interest, and hobby.
Thus, Geometry of Time-Spaces is a labor of love on the part of a scholar who has a deep idea to promote: it’s a very interesting idea indeed, and relates to some very important avant garde topics. It is indeed a fascinating piece of work, well worth reading.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.