26 Classical Theory of Fields II

Ashok Goyal

1. Introduction

Historically the concept of Action at a distance was developed to explain the force experienced by a body in the presence of gravitation and electro–magnetic forces. The concept of fields was later developed by Faraday and Maxwell to account for action at a distance experienced by a charge particle in the presence of other charged and currents. Maxwell developed the theory of electro–magnetic fields through Maxwell equations which accounted for all electro–magnetic interactions including light. In Maxwell’s theory fields are physical entities. The Lagrangian and Hamiltonian formulation of electro–magnetic fields was later developed. The quantization of electro–magnetic fields was achieved by Werner Heisenberg, Wolfgang Pauli and Paul Dirac among others to give the theory of Quantum Electro–dynamics. The theory turned out to be the most accurate theory for the prediction of electro–magnetic phenomenon and predicted the existence of photons as light particles mediating electro–magnetic interactions between charged particles. The theory of Quantum Electro–dynamics has been tested to unprecedented accuracy to as much a nine decimal points agreement between theory and experimental observation. It remains as the crowning glory of the success of quantum mechanics.

2. Electro–Magnetic Fields

Maxwell equations relate electro–magnetic field to charge and current densities. The equations are :

There are six components of electric and Magnetic fields, but they are related by Maxwell equations

For the electromagnetic Lagrangian in free space

The Lagrangian is not unique and neither is it gauge invariant. We can always add the four divergence of a vector without changing the action and hence the equations of motion remain unchanged. Thus adding a term  to the energy momentum tensor (28.28) we get

In General Theory of Relativity, the matrix  is no longer constant and depends on the space-time coordinates. Thus

we obtain the equation of motion in a gravitational field as

5. Summary

• The electro – magnetic field Lagrangian requires the gauge fixing terms to reduce the redundant four degrees of freedom to the physical two degrees of freedom.
• The electro – magnetic field equation of motion is o = 0 appropriate for a massless vector field.
• The motion of a particle in the gravitational field is determined by the Christaffel Symbol