Problem with anticommutation of spinors

[ansgar]

In e.g. Burgess and Moore - standard model a primer

it is stated that for two spinors (majorana)

$$\bar{\psi_1}\psi_2 = (\bar{\psi_1}\psi_2)^T = - \psi_2^T \bar{\psi_1}^T$$

since the spinors are anticommuting objects, thus ordering reversion gives -1

but they also state that

$$(\psi_1^\dagger\psi_2 )^* = (\psi_1^\dagger \psi_2)^\dagger = \psi_2^\dagger \psi_1$$

i.e. "without a minus sign" (explicity written in text)

I am SOOO confused, how how how can this be consistent??

best regards

37 views so far, come on I know that you can help me :)

 

[azntoon]

The minus sign comes from the fact that the Majorana spinors are anticommuting objects. That is, for any two Majorana spinors, a and b, we have that a * b = -b * a. This means that when we take the transpose of a product of two Majorana spinors, we get the same result as taking the reverse order of the product, but with a minus sign. So, the minus sign in the first equation comes from the fact that \bar{\psi_1}\psi_2 is the transpose of (\bar{\psi_1}\psi_2)^T. In the second equation, however, there is no minus sign because we are not taking the transpose of the product. We are simply taking the complex conjugate of it, which does not change the order of the product.