QED

Gauge Invariance

  1. Gauge Invariance in Classical Electrodynamics
    Maxwell's equation
    maxwell1
    suggests that there is a vector potential fulfilling
    vectorpotential
    The magnetic field is unchanged if one adds a gradient of an arbitrary scalar field Λ:
    gaugetrafo1
    Similar in line, the Maxwell equation
    maxwell2
    suggests that there is a scalar potential V fulfilling
    scalarpotential
    In this case one can add a time derivative of an arbitrary scalar field Λ to the scalar potential V
    gaugetrafo2
    without changing the electric field.

    To summarize this in a covariant notation: The field-strength tensor
    fieldstrengthtensor
    with
    fourpotential
    is unchanged under a 'gauge transformation'
    gaugetrafo3
    with Λ(x) being an arbitrary function.
    The same electrodynamics can be described by many different four-vector potentials. This is what is meant by GAUGE INVARIANCE of classical electrodynamics.

    The two Maxwell equations from above are then rewritten as
    maxwell12
    The two remaining Maxwell equations
    maxwell3
    and
    maxwell4
    can be written in the compact form
    maxwell34
    with the electromagnetic current being
    emcurrent


    CONSEQUENCES
    1) The electromagnetic current is conserved:
    emcurrentconservation
    2) The time derivative of the electric field in the fourth Maxwell equation guaranteeing local charge conservation leads also to the prediction of electromagnetic waves:
    emwaveequation
    In the absence of external electromagnetic currents and using the Lorentz gauge
    lorentzgauge
    one obtains for each compoenent of the four-potential (identified with the photon field) a Klein-Gordon equation for a massless particle:
    emwaveequationlorentzgauge
  2. Phase invariance of the Dirac field
  3. The Aharanov-Bohm effect