Book / The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose

This monumental overview of mathematical physics could more adequately be handled by 4-6 separate book reviews. For anyone who has ever read popular physics and had a deep desire to grasp the real math, here it is! Boiled down to a convenient 1,085 pages. It took Penrose 6 years to write, and sales have been so brisk (four printings so far) that he's had to focus solely on finding and fixing errors, leaving no time to publish answers to the exercises.

The first 16 chapters (378 pages) are solely mathematics, giving a quick brush up on number theory, complex numbers, Riemann surfaces, Fourier series, hypercomplex numbers, Cantor's infinities, manifolds, fiber bundles, guage connections, exterior calculus, and so on. Then we head into spacetime, analyzed in the Aristotelian, Galelian, Newtonian, Minkowski, and Einstein pictures. Then relativity, basic quantum mechanics, entanglement, the "Standard Model," quantum field theory, etc., etc. -- explained in detail using the math from the first part.

There are two ways to approach this book. As Penrose suggests, and many of us noticed long ago, (a) you can learn a lot by reading the easy parts and skimming the rest. Or else (b) there is enough material for at least 4 semesters of class work. In the latter scenario, budget (say) 10-15 hours a week over a 2 year period, download the 47 page errata sheet from Penrose's website, and work through the 1,000+ exercises and problems in the text. (Also consider purchasing a copy of 'Mathematica' which was designed to perform physics calculations, and perhaps take a class to get proficient with it.)

I'm about 2/3 of the way through, and have learned many things. For example, the infinite dimensional "Hilbert space" is not a physical spacetime, but a state space of all the variables in some system. An "operator" is a differential function, such as d/dx + (d/dx)^2. Einstein owed a huge debt to Riemann, Minkowski, Poincare, and others. Tensors are mandatory, central to general relativity, and grasping their cryptic notation would be an achievement. However, when Maxwell's Equations are presented, and this seemed rather gratuitous to me, they are given solely in the form of hieroglyphic tensor diagrams!

On a more humorous level, it seems Penrose (a British citizen at Oxford Univ.) is not above national rivalries. For example, his account of quantum entanglement covers the work of John S. Bell, a "Northern Irish" physicist, and several British-sounding scientists, but does not mention the famed "Aspect Experiment" nor any Continental scientists, and his bibliography lists no papers by Alain Aspect, a Frenchman!

I may extend this review after I complete the book, but there is so much material that it creates a definite level of fatigue -- concepts going in one ear and out the other! I am reading through in mode "A", trying to get a feel for what is there, but it remains to be seen whether I will go back through in mode "B" and spend two years working the exercises. As if this were a "Dummies" book, all exercises are graded into three levels of difficulty: easy, ponder this, and hard, signified by little icons showing the face of a pupil. But, trust me, this is NOT a "Dummies" book!