Author: Nadir Jeevanjee
Audience: Physics and Maths Graduates
Reviewer: Mike James
Both tensors and groups are at the foundation of modern physics and many aspects of computing.
This is a book for graduate students of math and physics - but mostly physics. If you have tried to master tensors and group theory as they apply to modern physics you may well have been horrified by the brutal abstract math approach of the typical book. Both subjects lend themselves well to the theorem proof with no kind word to help you understand how it all fits together or why you are even doing all of it.
This book is a lot better, but not perfect. It tries hard to get you to understand what is going on by talking around the math, but it doesn't quite go far enough. It is inevitable that you will occasionally be left wondering what is going on. In plain terms there is still room for a book on both of these topics that explains what it is all about first and then presents the math to back it all up.
The book divides into two parts - first we have a long introduction to linear algebra and tensors and then the second main topic is groups.
Before we get to the details there is a short "this is a tensor" chapter. This sets the scene and motivates much of what follows and basically says a tensor is a multilinear function - from which everything else follows. Of course if you don't know about dual spaces and other ideas it is difficult to work out what follows for yourself. So chapter 2 starts working through the standard ideas - span, basis, dual space etc. The approach is for the physicist because spaces of matrices are introduced along with specific examples such as the Pauli matrices. Theoretical difficulties with infinite dimensional spaces are glossed over, thankfully and examples of function spaces are included.
If you aren't prepared to put some work in you might be left behind by many of the examples. The angular momentum operators are introduced very early and of course there is a very real sense in which all of this theory is motivated by angular momentum and spin. By the end of the chapter we have covered real and complex inner product spaces.
Next we have tensors. Given that dual spaces have already been introduced we have definitions of covariance and contravariance, but without relating the ideas to the physical reasons why some things are best treated as covariance and other as contravariant. The ideas go deeper than the mathematics presented here.
On to cover the tensor product and symmetric and antisymmetric spaces. This is not only useful for general physics but it is what you have to master to fully understand quantum computing which goes on in the product space of qubits.
This brings the first part of the book to a close. It is short and not too theorem-proof bound to be impossible to understand. If you do read it then be prepared to think quite hard about some of the examples which tend to be more advanced than the main text.
The second part of the book dives into groups without too much motivation. The whole aim is to get on to Lie groups as quickly as possible. I can imagine that if this is your first introduction to groups it might be too fast. The second problem is that the important groups of classical and quantum physics are introduced equally quickly and the problem is that there is no complete list of symbols. What this means is that you inevitably end up having to go back and discover what SL or M is. Not too much of a problem, unless this is your first exposure to the nomenclature and you read the book in short bursts.
From here we move on to Lie algebra and its key ideas, such as the exponential map and so on. If you are prepared for it then the book will get you to understand that the Lie algebra is the infinitesimal form of the group, the tangent space of the group as a manifold and that the connection between the infinitesimal and the macro is the exponential map. However it is still possible to get lost in the detail.
Once we have the basics over the next topic is representation theory. This is introduced with very little motivation. It basically doesn't tell you why we are interested in representations - from the physics point of view. It doesn't say we seek representations of continuous symmetry groups because we need them to build symmetries into quantum mechanical theories. A physical system may have rotational symmetry but its quantum mechanical model needs something like a unitary representation of the same symmetry group to work. The point being that the we have to have the same symmetries in the spaces that our models occupy as the real world. Of course, today this has been turned around and we suppose that any valid symmetry of the model is also a symmetry of the real world. It is in this sense that representation theory has become quantum mechanics. It is so powerful that it even pushes us to quantum field theory because there is no finite dimensional representation of the Poincare group.
This is the least satisfactory part of the book, but still better than many others - it just needs more lower level motivation.
Overall, this is a good read but mainly if you already know the motivations behind what is going on. There is a long term trend in physics textbooks to move in the direction of mathematical textbooks. The whole point of physics is to say something about the real world and expressing everything in abstract symbols in theorem-proof form may say it but only in a whisper. We need to be much clearer about what the actual physics is the math is expressing. While this book only succumbs to math envy for part of its length, it could still add more physics to the math.
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