I’ve just finished reading Roger Penrose’s very thick and challenging book (more than 1000 pages !) called "The Road to Reality : A Complete Guide to the Laws of the Universe". Penrose seems to hope that even people having difficulty with fractions would turn around to understand what he’s trying to say. The chance for that is a mite puny. With all sorts of mathematical ideas and formulations, Penrose explains and discusses his views on the development of fundamental particle physics and cosmology with unavoidably (special and general) relativity and quantum mechanics (and QFT) in its core. The last chapter (34th) is probably the easiest to read, in which Penrose was contemplating about the future trends and possibilities. It ends with a short and indeterministic fairy tale epilogue.
I sometimes think of Penrose as Oxford’s counterpart of Stephen Hawking in Cambridge and he’s probably more of a mathematician than a physicist. He did care more about the mathematical rigor and criticize physicists of not being so. I have to confess that I’m not familiar with a lot of the mathematical concepts. The book offers the illusion of opportunities to learn about them but in my opinion it doesn’t. The author seems to indulge in talking about the subjects from his vantage point of being an expert. It’s rather like a wine connoisseur talking to other wine connoisseurs. Sometimes, I’ve felt that I was reading just a list of statements of the theories. When I came across more familiar topics, I got the ability to understand and judge what I was reading; but if not, it’s sometimes probably difficult to really learn from the book unless (I suppose) you read all his references quoted. At the beginning, I did try a few of his “exercises” in the footnotes and they’re non-trivial. Some of them ask you to derive some formulae such as Doppler effect but Doppler effect was probably not obvious before Doppler, right ?
This is only the 2nd time that I read about the “Twistor Theory” (last time was in Lee Smolin’s “The Trouble with Physics"), in which Penrose has worked for ~40 years (on and off). From Penrose, the Twistor Theory seems to contribute as many mathematical ideas as the String Theory. But obviously, Penrose doesn’t quite believe in supersymmetry and higher than 4 dimensions in spacetime and the Twistor Theory resides in 4 dimension spacetime and normally doesn’t have supersymmetry. Both of these theories at the moment are kind of driven by the beauty and some of their miracles … Needless to say, Penrose is definitely not a String Theory enthusiast and shares some of the worries that I’ve read about from L. Smolin and P. Woit. Penrose does mention quite a bit about Hawking’s and others’ (and his) contributions in applying general relativity to understand the evolution and various features of our universe. I’ve read about quantum loop gravity theory elsewhere but it’s probably the 1st time that I read about the “spin networks” as another sort of spacetime quantization and quantum gravity theory.
Somehow I’ve learnt from Penrose that the complex numbers may play a fundamental role in physics rather than just a tool (in my mind previously). The advanced mathematics occupies almost the 1st 400 pages of this book. I’ve easily forgot what I’ve read and sometimes needed to go back to look them up even for the simplest thing. Eg. at the last few pages, when I saw “simple group”, I almost had no idea about. Going back to the earlier chapters, you’d get to know that simple group is ~”non-trivial normal group” and then you had to know what “normal group” is and so on … There are tons of other mathematical concepts including all sorts of tensors, groups and topologies which I’ve sometimes felt that they don’t really help explain the subjects. For the same topic of quantum entanglement, I believe Brian Green has done a significantly better job in explaining its awkwardness. But I do enjoy reading Penrose’s introduction of separated U and R operations in quantum mechanics and his gravitational OR (objective reduction) of the quantum state. Penrose seems also like to promote the usage of his (?) diagrammatic notions which he claimed that he’d been using for 20 years but I was probably too stupid to understand their usefulness.
Nevertheless, I have to say that I was indeed exposed to various modern mathematical concepts. A student who has recently asked me to write recommendation letter(s) for graduate school applications told me that she was trying to solve “Riemann hypothesis” — not sure whether she’s really digging very deep or just trying to impress me. At that time I didn’t actually know exactly what it was. Now, I have a slightly better idea as Penrose has mentioned about it in this book.
Penrose of course has his own biases or beliefs (eg. towards Twistor Theory). He strongly believes that there must be fundamental change(s) to understand quantum mechanics (that we can’t really rely on the stopgap Copenhagen interpretation) before we can make further progress into quantum gravity and other fundamental areas. It’s also a typical westerner of him that the reader is kind of given the impression that the beginning of all knowledge came from the Greek. I’ve often been annoyed by this impression (due to my own bias and culture). There was occasional humor such as why quantum leap would mean big improvement when it should really mean miniscule development