I'm delving into light-front quantization, with a focus on understanding the roles of good and bad fermions. Using Collins' formulation in Foundations of Perturbative QCD, we define the projectors as:
\begin{align} P_G = \frac{1}{2} \gamma^- \gamma^+ \qquad P_B = \frac{1}{2} \gamma^+ \gamma^- \end{align}
From which, the good and bad fermions can be expressed as:
\begin{align} \psi_G = P_G \psi \qquad \psi_B = P_B \psi \end{align}
Their Dirac conjugates are:
\begin{align} \overline{\psi_G} = \overline{\psi} P_B \qquad \overline{\psi_B} = \overline{\psi} P_G \end{align}
Given this setup, the light-front Hamiltonian, which is the Noether charge in the plus direction, is expressed as:
\begin{align} P_+ = \int dx^- d^2x_T \left[ \overline{\psi} ( -i \gamma^- \partial_- \gamma - i\gamma^k\nabla_k + M)\psi + \ldots \right] \end{align}
with the Heisenberg equation: \begin{align} i \frac{\partial\psi}{\partial x^+} = [\psi, P_+] \end{align}
The anticommutation relations for equal plus coordinates below are said to be compatible with the Heisenberg equation, in the sense that they together yield the Euler-Lagrange equations, \begin{align} \Big[ \psi_G(x), \overline{\psi_G} (w) \Big]_+ &= \frac{\gamma^-}{2} \delta(x^- - w^-) \delta^{(2)}(x_T - w_T) \\ \Big[ \psi_G(x), \psi_G (w) \Big]_+ &= 0 \\ \Big[ \overline{\psi_G} (x), \overline{\psi_G} (w) \Big]_+ &= 0 \end{align}
I'm facing difficulty in deriving the Euler-Lagrange equations using this setup and verifying this claim. While the process is straightforward for scalar fields, the introduction of the good and bad fermions seems to make the problem more complex.
Considering the mass term and using the defining properties of the projectors, I arrived at:
\begin{align} \overline{\psi} \psi &= \Big[ \overline{\psi} (P_G + P_B) \Big] \Big[ (P_G + P_B) \psi \Big] \\ &= \overline{\psi} [ P_G^2 + P_B^2 ] \psi \\ &= \overline{\psi_G} \psi_B + \overline{\psi_B} \psi_G \end{align}
From here I don't know how to continue, especially given the absence of commutators for the bad fermions. Solving directly for them using the constraint equation and then substituting into the Hamiltonian seems circular. Also, attempts to proceed without specifying the commutators proved unsuccesful.
I seek guidance on the following:
- How to derive the equations given this setup? Strategies or insights on addressing issues concerning the good and bad fermions.
- Clarification regarding the unindexed gamma in the Hamiltonian.
Any help would be greatly appreciated.
