Resource recommendation for Vector/Tensor/Operator and Spinor geometric understanding

Question

I am a Bachelor's Physics student,currently on my 5th Semester (3rd Year 1st semester). We are at a point in Physics where the mathematics is getting seriously hard. Our Bachelor program in itself is badly constructed. We had math only for 3 semesters and most of the things that were taught during this time were stuff we already knew from High school but in the university they were explained more in details. In this semester we are dealing the Tensors, even though up until this point nothing was explained regarding what a tensor is, its geometrical interpretations etc.

What I want is for a book suggestion that has a detailed (maybe even illustrated to a certain extend) geometrical interpretation of the above concepts (i know what a vector is,but how it's different from a spinor etc i don't know). Or a book that explains the mathematics that one ought to know in order to understand QFT.

Today for example the concept of a spinor was explained, in the following way :

Equations which:

transform $Ψ(x^μ)= (S(Λ))^{-1} Ψ'(x'^μ)$ (we are trying to show the lorentz covariance of the Dirac Eq.)

and

$Λ^ν_μ γ^μ=(S(Λ))^{-1}γ^νS(Λ)$ (this equation was derived by comparing the Dirac eq.in two inertial systems).

Are called 4 component Lorentz-Spinor.

And that was it about the spinor. But this type explanation tells me literally nothing (since my mathematical knowledge are not sufficient).

So is there a book one can recommend?

Answer 1

While I’m not aware of any book covering all of “vector/tensor/operator and spinor geometric understanding” that’s suitable at an undergraduate level, I can offer a few recommendations regarding spinors mainly.

Possibly the best, and specifically undergraduate-targeted introduction to spinors – available free online – is Andrew Steane’s pedagogical article, which you can download from https://arxiv.org/abs/1312.3824 . (He’s a professor of physics at Oxford U., and sometimes contributes here.) You’ll need little more than vector analysis and some idea of what a mathematical group is (think of an abstract algebraic representation of the completed turns of a Rubik’s cube – every 90° rotation and combination thereof (i.e. associativity) results in another configuration (i.e. closure) of the cube, and each operation has an inverse as well as there being an identity operation (0° turn).) The article also includes a number of diagrams, which are helpful in showing some of the ways in which spinors differ from vectors. I strongly recommend starting with Steane.

A longer and more advanced text is J M Cole’s high-quality translation of Jean Hladik’s (sic) “Spinors in Physics”, (Springer, 1999; ISBN-13: 978-0387986470 ISBN-10: 0387986472 ) available from Amazon but try to get it via a university library or look for a seond-hand copy. Although it’s listed as a graduate text, Hladik develops the ideas carefully enough that I think an undergraduate would get a lot from the earlier sections. He starts with spinors in 3-space before moving on, in the 2nd half of the book, to spinors in spacetime. He begins with Elie Cartan’s original definition of a spinor, which he then explains a lot more clearly than Cartan does in his classic Theory of Spinors (which latter is more suited to professional mathematicians, IMO). But still in Chapter 1, Hladik also considers alternative definitions of spinors, partly to highlight spinors’ geometric properties. As you may have sensed, spinors are widely regarded as mysterious, even by some leading mathematicians.

It’s worth taking a look at the first half of https://en.wikipedia.org/wiki/Spinor for n overview, although it rather runs away with itself in the latter half.

Also https://en.wikipedia.org/wiki/Spinors_in_three_dimensions outlines Cartan’s definition, more approachably that Cartan, but in rather less detail than Hladik’s careful presentation.

Finally, and perhaps a little self-indulgently, I would also recommend a look at parts of Simon L. Altmann’s Rotations, Quaternions and Double Groups (Dover, 1986). It’s overall too advanced for undergraduate study, but has some good introductory chapters, including much of the essential maths and a brief mention of tensors in relation to spinors. It’s main virtue, IMO, is that he analyses the ideas underlying the representation of rotations in careful detail while highlighting some of the pitfalls in less careful presentations. I found it helpful for getting some sense of the physical geometries involved. (Incidentally, I attended some of his lectures on group theory, in Oxford, many years ago.)