9 - Vector Fields and Gauge Invariance

Summary

Presenting the canonical quantization of a massive vector field, the existence of a massless limit when this field is coupled to a conserved current, and the gauge invariance of this massless limit; a preliminary understanding of QED.

Introduction

Now that we have found fermionic fields, the need is for polarized fields that can represent the photon. Experimentally, photons have only the two transverse polarizations and no possibility of polarization along the direction of motion. Since these polarizations are vectors in Minkowski space, we shall begin our investigation using a vector field, a set of four hermitian fields A_μ that transform as a vector under the Lorentz-group action on the fields.

The most general Lagrangian density for a free vector field contains two free parameters, a and b. After careful analysis of this Lagrangian density, we find in Section 9.1 that the classical theory is only free from off-shell modes and modes with longitudinal polarizations when a = -1 and b ≠ 0, in which case the vector field is massive. We also observe that when a = -1 and b = 0, the vector field is massless and has a gauge symmetry which makes the longitudinal modes non-physical.

Rather than set up the functional integral machinery necessary for quantizing the gauge field theory, in this chapter we treat the gauge theory as the massless limit of a massive vector field theory.