SL(2,C) to Lorentz in Carmeli's Theory of Spinors

[vanhees71]

I think you should forget these books by Carmeli and use another one. As already said, a great book on the representation theory of the Lorentz and Poincare groups and their Lie algebras is

Sexl, Urbandtke, Relativity, Groups, Particles, Springer

My own attempt to explain these issues, you find in my QFT manuscript (Appendix B):

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

There I avoid the introduction of matrices and work with the SL(2,C) tensor formalism with undotted and dotted indices, which is much safer to avoid all kinds of mistakes, as indicated by samalkhaiat in his last posting.

 

[arkajad]

There I avoid the introduction of matrices

There is nothing wrong with working with matrices - provided you know what you are doing. And working with matrices is not that difficult. In fact, it is straightforward and elementary. There are no reasons to complicate things that can be explained and calculated in an elementary way. Easier, quicker, less prone to errors.
After all this is all about SL(2,C) and SL(2,C) is a group of matrices.

And if for some reasons you have aversion to the books by Carmeli, here is p. 305 from "Differential Geometry and Lie groups for Physicists" by Marian Fecko, Cambridge 2006.

Here is some little complication as Fecko first introduces sigma with tilde, and then gets rid of this tilde by using eta! But it is all the same.

 

[arkajad]

Essentially the same reasoning about the matrix group SL(2,C) is also in the online book by Jean Gallier "http://www.seas.upenn.edu/~jean/diffgeom.pdf", p. 227 in July 10, 2013 edition. Gallier book is also valuable because on p. 102 he warns against serious errors concerning SL(2,C) in the monograph "Matrix groups. An introduction to Lie Groups" by Andrew Baker, Springer 2002

 

[vanhees71]

I'd be utmost suspicious against books that write $\Lambda^{\mu}_{\nu}$ instead of ${\Lambda^{\mu}}_{\nu}$, but I don't know the book by Fecko. So I don't want to say anything against it only from this one quote.

I have nothing against matrices. You only have to define them properly. Using the SL(2,C)-tensor index formalism, there is, however no such trouble reflected in this thread with a lot of misunderstandings and trivial mistakes. If you write
$${\sigma^{\mu}}_{A \dot{B}}$$
it's clear from the very beginning, what transforms how. Then you only have to take into account that index raising and lowering in the SL(2,C) business is done with the skew-symmetric bilinear form
$$\epsilon_{AB}=\epsilon_{\dot{A} \dot{B}}=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad \epsilon_{A \dot{B}}=0.$$
Then all the discussions in this thread most probably would have been avoided. That's all I'm saying.

Of course, many calculations can be abbreviated by using the matrix-vector notation, but one must make sure to define properly, which matrix denotes which SL(2,C)-spinor or SL(2,C)-transformation matrix elements!

 

[arkajad]

Of course, many calculations can be abbreviated by using the matrix-vector notation, but one must make sure to define properly, which matrix denotes which SL(2,C)-spinor or SL(2,C)-transformation matrix elements!

But the discussion is not about spinors. The discussion is about matrix groups: $SL(2,C)$ and $SO^+(3,1)$. Spinors is a different subject. It comes later after you first understand simple matrix groups. To understand matrix groups you have to operate on matrices. How to multiply them, conjugate them, take determinants, traces, exponentials etc. This is all elementary algebra and group theory. Spinors come later, when we learn differential geometry, frame bundles, spin bundles etc. Of course one can talk about tensors and spinors even without differential geometry, in a flat space, or as a part of the theory of groups representations (or even "Clifford modules"), but, as I mentioned above, first you need to understand groups that are at work. And these groups are simple matrix groups that can be studied with matrix algebra methods. Nothing more is needed (well, also some topology, but that can be also made simple).

 

[arkajad]

For those who have aversion to matrices, to ease their sufferings, here is a piece from "Matrix groups: An Introduction to Lie Group Theory" by Andrew Baker, Springer 2002

He has $S\mapsto ASA^*$ instead of $Q\mapsto gQg^\dagger$. Notation is different, but the content is always the same: it is all about properties of the matrix group SL(2,C).

 

[vanhees71]

The matrices live on a vector space, which in the case of SU(2) or SL(2,C) is called spinor space. Also the various finite-dimensional (irreducible) representations define spinors spaces. In the case of SU(2) or SL(2,C) one can show that all finite-dimensional representations can be found by reduction of representations induced by the fundamental representations on tensor products.

 

[arkajad]

The matrices live on a vector space

The matrices are just matrices - tables of numbers, with definite rules of multiplication etc.
To calculate determinant of a matrix you do not have to use vector space (though you can, it becomes more complicated). To calculate trace of a matrix you do not have to use vector space, though you can, it becomes more complicated.

Matrices can be interpreted as operating on vector space, but such an interpretation is not always needed. It is of course good to know, but you do not have always use all that you know. You use that what is needed. Killing a mosquito with a gun is of course possible, but is it always wise?

 

[arkajad]

Some comments of more general nature:

I think much of the confusion comes from the fact that some physicists fail to distinguish between two different mathematical objects. For instance "Lorentz group". How we define it? There are two different ways of defining it. One way: it is the group of 4x4 real matrices with the property that ... etc.
Second way: Suppose we have a four-dimensional real vector space V endowed with a scalar product of signature (+---). The Lorentz group is the group of isometries of this vector space.
The point is that these two definitions define objects in two different categories. Then comes a theorem:
Given an orthonormal basis in V there is an isomorphism from one mathematical object to another. Lorentz groups of two definitions becomes isomorphic. But one has to remember that if we chose a different orthonormal basis in V, the identification of the two objects will change! So, they are isomorphic, but there are many different isomorphisms and no one is better than other. The same, with appropriate changes, can be applied to SL(2,C).

I am talking here about the first kind. Others want to talk about the second. Once we understand the difference between these two categories, there will be no place for confusion.

 

[samalkhaiat]

The matrices are just matrices - tables of numbers, with definite rules of multiplication etc.
To calculate determinant of a matrix you do not have to use vector space (though you can, it becomes more complicated). To calculate trace of a matrix you do not have to use vector space, though you can, it becomes more complicated.

Matrices can be interpreted as operating on vector space, but such an interpretation is not always needed. It is of course good to know, but you do not have always use all that you know. You use that what is needed. Killing a mosquito with a gun is of course possible, but is it always wise?

You shouldn't read Carmeli's texts then. Find a book that suit your "understanding". Believe me, there are a plenty of them.

 

[arkajad]

You shouldn't read Carmeli's texts then. Find a book that suit your "understanding". Believe me, there are a plenty of them.

Indeed, there are good books. Carmeli's book are god. Also Frankel "The Geometry of Physics", Cambridge 1997. Also Naimark "Linear Representations of the Lorentz Group", Pergamon Press, 1964. Penrose and Rindler "Spinors and Space-Time" is not so good, as there are no clear definitions and theorems - just one long talk. It confuses those physicists who do not have deep enough mathematical background, so that they may not know how to make vague things mathematically precise. I think Penrose and Rindler are partly responsible for not understanding the difference between groups of matrices and groups of transformations.

Also Huggett and Tod, "An Introduction to Twistors Theory", Cambridge 1994 (2nd ed) is not bad:

Though it is not free of errors.