Student Friendly Introduction to QFT · Problem 27

Spinors: Spin ½ Fields

Physics — QFT · dirac-3-momentum-operator-derivation

Problem

Derive the 3-momentum operator (4-107),

P^i = \int \not{p}^i d^3 x = \sum_{r, p^i} p^i\left(N_r(p^i) + \bar{N}_r(p^i)\right) \quad \to \quad \mathbf{P} = \sum_{r,\mathbf{p}} \mathbf{p}\left(N_r(\mathbf{p}) + \bar{N}_r(\mathbf{p})\right), \tag{4-107}

using the relation between 3-momentum density and conjugate momentum of the field from Box 2-2, with

n=1, \phi^1 = \psi

;

n=2, \phi^2 = \bar{\psi}