This has been a really refreshing book for me to read. I studied mathematics in college but haven't really exercised that part of my brain since graduating and working as a programmer. Although this book is probably inaccessible to someone without some formal mathematical training, it's still one-of-a-kind. Nobody else of Penrose's stature has ever attempted to go from zero to string theory in a single volume, with all the physics and mathematics explained and very little left out.
For me, it's really been nice to finally satisfy a lifetime of curiosity that had built up about quantum theory. My fascination with it has never been enough to drive me to be a physicist, but it was enough for me to feel uneasy about not really knowing the underlying mathematics.
I finished it but certainly can't claim to have understood everything in it. I read it on and off for several weeks so it definitely took some effort for me.
It's an incredibly dense book. There's nothing else out there quite like it. Penrose also has a pretty idiosyncratic method of presenting ideas. This means that sometimes you can't cross-reference his explanations even if you want to. He covers some exotic and nonstandard topics as well. I haven't personally seen his diagrammatic notation for tensor algebra anywhere else, for instance. And the book was my first and only exposure to hyperfunctions.
The "trick", honestly, is to already have some prior exposure to most of the tricky mathematical ideas, like differential forms and fiber bundles. I was by no means an expert on differential geometry but I'd been exposed to it a few times as an undergraduate.
The other "trick" is simply not to skip around. I was tempted at first to browse and read whatever topics interested me; this book is not structured that way. He constantly references backwards in the book, and even if you know the mathematical material some of his references can be confusing. This is probably the greatest weakness of the book. For instance, gauge connections aren't really covered fully in any single section of the book. They're discussed briefly in the chapter on fiber bundles, and then again in the chapter on electromagnetism, and then again I believe in the chapter on quantum mechanics, and none of these discussions is complete, but they are if taken together.
One trick is to stop worring about completely understanding the maths on your first pass through the material. Instead make note of what you were confused about and continue on.
If for instance you find that you need a refresher on complex integration, don't stop reading and do quarter's worth of review of complex analysis. Don't expect to complete a year of differential geometry in the few chapters that lead to Gravity.
After you finish through the a first pass, definitely go back to sections that you want to understand better and do the necessary work to improve your understanding.
http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...
This has been a really refreshing book for me to read. I studied mathematics in college but haven't really exercised that part of my brain since graduating and working as a programmer. Although this book is probably inaccessible to someone without some formal mathematical training, it's still one-of-a-kind. Nobody else of Penrose's stature has ever attempted to go from zero to string theory in a single volume, with all the physics and mathematics explained and very little left out.
For me, it's really been nice to finally satisfy a lifetime of curiosity that had built up about quantum theory. My fascination with it has never been enough to drive me to be a physicist, but it was enough for me to feel uneasy about not really knowing the underlying mathematics.