Solving Spinor Problems: Working Through (3.75), (3.77)-(3.79) in Google Book

[latentcorpse]

I'm working through some spinor calculations in the following book:
http://books.google.co.uk/books?id=...ame as complex conjugate for a scalar&f=false

On p54 and p55, I have a few things that are troubling me:

(i) Underneath (3.75), he notes that for D=2,3,4,10,11, we have -t_0t_1=+1 (see Table 3.1 for t_r values) and so the Majorana conjugate of the charge conjugate will be equal to the Dirac adjoint of the spinor. What is the significance of this statement?

(ii) How do we derive (3.77)-(3.79)? I cannot make much headway.

For example, my attempt to prove (3.77) is as follows:
[tex] (\gamma^\mu)^C = B^{-1} (\gamma^\mu)^C B [/tex]
[tex] = i t_0 \gamma^0 C^{-1} (\gamma^\mu)^* i t_0 C \gamma^0 [/tex]
[tex] =i^2 t_0^2 \gamma^0 C^{-1} (\gamma^\mu)^* C \gamma^0 [/tex]
[tex] =-\gamma^0 C^{-1} (\gamma^\mu)^* C \gamma^0 [/tex]
where we've used [tex]B=it_0 C \gamma^0 \Rightarrow B^{-1}=-i t_0 (\gamma^0)^{-1} C^{-1} = i t_0 \gamma^0 C^{-1}[/tex]

Now the problem is that we have something involve the complex conjugate of the gamma matrix. If we had something involving the charge conjugate then we could substitute from (3.45) and be finished. I cannot see how to get from here to what they have in (3.77)

Thanks.

 

[mark726]

Thank you for sharing your concerns and questions regarding the spinor calculations in the book you are working through. I am happy to assist you in understanding these concepts.

To address your first question, the statement about -t_0t_1=+1 for certain values of D is significant because it is related to the property of chirality in spinors. Chirality refers to a particle's tendency to move in a particular direction, either left or right. In the context of spinors, this means that the spinor will have a specific handedness or orientation. The values of D mentioned in the book correspond to the dimensions of spacetime where chirality is conserved, meaning that the handedness of the spinor is preserved in interactions. This has important implications in particle physics and can help identify the types of interactions that are possible.

Moving on to your second question, the derivation of equations (3.77)-(3.79) involves some algebraic manipulations and properties of the gamma matrices. I will not go through the entire derivation, but I will provide some key steps that may help you in understanding the process.

First, it is important to note that the charge conjugate of a gamma matrix can be written as a complex conjugate of the gamma matrix multiplied by the charge conjugation matrix C. This is given by equation (3.45) in the book.

Next, using this property, we can rewrite the expression for (\gamma^\mu)^C as (\gamma^\mu)^C = (\gamma^\mu)^* C. Then, using the fact that (\gamma^\mu)^* C = -C^{-1} (\gamma^\mu)^T C, we can simplify the expression further.

Now, using the properties of the charge conjugation matrix C, we can show that C^{-1} (\gamma^\mu)^T C = -i t_0 C^{-1} (\gamma^\mu)^* C \gamma^0. This is due to the fact that C^{-1} = -i t_0 \gamma^0, as shown in the book.

Finally, substituting this expression back into our original equation, we get (\gamma^\mu)^C = -i t_0 C^{-1} (\gamma^\mu)^* C \gamma^0 \gamma^0 = -i t_0 (\gamma^\mu)^* C \gamma^0, which is equivalent