How Do You Quantize Spinor Theory in Quantum Field Theory?

[Jan Paniev]

Homework Statement

Problem 3.4e of Peskin & Schroeder Introduction to Quantum Field Theory. Quantize the spinor theory of item (a) of this exercise, where the spinor $$\chi$$ is the first two components of the Dirac spinor ($$\psi_L$$). Find a Hermitean Hamiltonian and the correct creation/annihilation operators that diagonalize it.

Homework Equations

The Majorana mass equation for these components $$\[i\bar{\sigma}\cdot\partial\chi=im\sigma^2\chi^*\]$$
(item a), the anticommutation relations for the components of the spinor
$$\[\{\chi_a(x),\chi^\dagger_b(y) \}=\delta_{ab}\delta(\vec{x}-\vec{y})\]$$.

The Attempt at a Solution

Many. The problem is that I cannot find the correct expansions of the fields and the correct normalizations to diagonalize the Hamiltonian. The fact that the equations mix $$\chi$$ with $$\chi^\dagger$$ doesn't allow to use the usual methods and I could not find this done in any book. Pointing to a book or a partial/full solution in the internet would already be a great help. General suggestions and tips are also very, very welcome.

 

[lmwpearce]

You're presumably not still working on this, but see Giunti & Kim's Fundamentals of Neutrino Physics & Astrophysics, section 6.2.5. The key is that after writing the Hamiltonian as chi^dagger partial_0 chi, the partial_0 acts both to the left and to the right (which is often skipped over in P&S, which is usually but not always valid). Here, it's important to cancelling the aa and a^dagger a^dagger terms. (Strictly speaking, the Hamiltonian with the partial_0 acting in both directions differs from the Hamiltonian with it acting in one direction by a surface term.)