Weyl Spinors, SO(1,3) algebra and calculations

[earth2]

Hey guys,

something that puzzles me everytime I stumble across spinors is the following:

I know that i can express Dirac spinors in terms of2-component Weyl spinors (dotted/undotted spinors).
Now, if i do that, i can reexpress for instance the Lorentz or conformal algebra in terms of Weyl spinors.

In the literature (http://arxiv.org/abs/1001.3871) one finds for the momentum generator and the conformal generator:

$$p_{\alpha \dot{\alpha}}=\lambda_\alpha \lambda_\dot{\alpha}$$
$$D=[\lambda_\alpha \partial_{\lambda_\alpha}+\alpha<-> \dot{\alpha}]$$.

(i don't care about normalization for my question).
So, if i read the notation correctly D is just a number but $$p_{\alpha \dot{\alpha}}$$ is a 2x2 matrix.

The commutator of these two reads $$[D,p_{\alpha\dot{\alpha}}]=p_{\alpha\dot{\alpha}}$$.

If I spell that out explicitely I get $$\lambda_\alpha\delta_\beta^\alpha\lambda_{\dot{\beta}}+\lambda_\alpha\lambda_\beta\lambda_{\dot{\beta}}\partial_\alpha-\lambda_\beta \lambda_{\dot{\beta}}\lambda_\alpha\partial_\alpha + dotted part$$

Now, obviously the second and third term are cancelling each other to make the commutation relation work, i.e. we have to treat lambda_x and lambda_{\dot{x}} as numbers. Here is where my problem is:

We said in the beginning that the lambdas are spinors spinors (.i.e 2x1 vectors) so how can we look at them as just numbers in the above equations?


So, in my understanding the second term reads as (2x1 vector) times 2x2 matrix and the third term as 2x2 matrix times 2x1 vector. But i can't just commute an expression like that in order to make the 2nd and 3rd term cancel...
I just don't get as to when to regard lambda as a vector, when as a number.. it is SO confusing...

Help is appreciated!
Thanks

 

[fzero]

Now, obviously the second and third term are cancelling each other to make the commutation relation work, i.e. we have to treat lambda_x and lambda_{\dot{x}} as numbers.

Spinor components are actually Grassmann (i.e. anticommuting) numbers.

Here is where my problem is:

We said in the beginning that the lambdas are spinors spinors (.i.e 2x1 vectors) so how can we look at them as just numbers in the above equations?


So, in my understanding the second term reads as (2x1 vector) times 2x2 matrix and the third term as 2x2 matrix times 2x1 vector. But i can't just commute an expression like that in order to make the 2nd and 3rd term cancel...
I just don't get as to when to regard lambda as a vector, when as a number.. it is SO confusing...

Help is appreciated!
Thanks

If you write tensors out in terms of indexed components, then you are dealing with sums of products of numbers. For example

$$r_j m_{ij} = (\mathbf{m} \mathbf{r})_i$$

is just the product of a matrix and a vector. From the pattern with which the indices were summed, we were able to deduce the order of the matrix and vector in the product.

 

[earth2]

Thanks for your reply!

So does that mean that if i write everything in terms of indices (like I did in my first post) I am in fact looking at the individual entries of the matrix, i.e. I'll deal with plain and simple numbers (may they be Grassmann or not)?

 

[fzero]

Thanks for your reply!

So does that mean that if i write everything in terms of indices (like I did in my first post) I am in fact looking at the individual entries of the matrix, i.e. I'll deal with plain and simple numbers (may they be Grassmann or not)?

Yes, this is one of the advantages of using index notation for many calculations. As long as you keep track of the order of indices, you will preserve all of the matrix properties of the expression. Of course, higher rank tensors don't even have a simple matrix notation, so index notation is often the simplest representation.