25 Classical Theory of Fields I

Ashok Goyal

1. Introduction

In classical physics, matter is considered discrete made up of discrete particles (atoms, molecules, etc). Space and time are considered continuous entities. So far we developed techniques like the Lagrangian and Hamiltonian formalism to deal with discrete systems with a finite number of degrees of freedom. In practice however, there are a large number of dynamical systems which have an infinite number of degrees of freedom and can be better described by a variable   ( ) where takes continuous values. A simple example is that of a vibrating string. Thus in the case of continuous media such as macroscopic bodies of solid, liquid or gases we need to deal with a continuous variable. Another example is that of non-material objects. We are familiar with the interaction of material bodies with gravitational and electromagnetic forces. These forces are represented by ‘fields’, the electric, magnetic and gravitation field. The classical field theory is of great interest and in this and next unit, we will develop the Lagrangian and Hamiltonian formalism for the fields. The study of Relativistic Quantum Fields has found to be of paramount importance in modern theory of particles and their interactions, in condensed matter physics in string theories and in theories of gravitation.

In the Standard Model of Particle Physics which is the most successful theory of nature, both the material particles and their interactions are represented by relativistic field. The interactions between them is governed by specific symmetry principles under which the theory is invariant. The material particles as well as the particles that mediate interactions manifest themselves as quantas of the corresponding fields. For example, the electrons are the quantuas of certain electron-field and the photons are the quanta of electro-magnetic field.

2. Transition from Discrete to the Continuous System

Consider an infinitely long one-dimensional elastic solid which can undergo longitudinal vibrations. We can represent this by an infinite chain of interacting particles. For simplicity, we assume the infinite chain to consist of an infinite number of particles of mass connected by springs of unstretched length a and spring constant .

we replace→ In the continuous limit, the sum over can be replaced by an integral over . The Lagrangian in the continuous limit can thus be written as

At this stage let us clarify the role of the variable . It is not the generalized coordinate as in a discrete system. It serves the same role as played by the discrete index . In the discrete case each value of corresponds to a different generalized coordinate, here for each continuous value of , there is a field   ( ) which plays the role of generalized coordinate. In addition it also depends on the time variable .

i.e. the action is given by a four–dimensional volume integral of the Lagrangian density . Classical field equations can be obtained by minimizing the action under an infinitesimal displacement of the field

3. Hamiltonian Formalism

The momenta conjugate to the field( ) is defined as

Obtain the Euler–Lagrangian equation of motion and the Hamiltonian density.The Euler–Lagrangian equations are

4. Noether Theorem

In classical mechanics if the Hamiltonian is time independent, then the energy is conserved. Likewise if the Hamiltonian is invariant under three dimensional space translation, the momentum is conserved. Invariance under three dimensional rotation leads to the conservation of angular momentum. There is thus a deep relationship between symmetry and conservation laws. A precise mathematical correspondence between symmetry and conservation laws is given by Noether’s theorem.

and we have a conservation principle.

Space–time Symmetry :

Let us now consider

an infinite system and the symmetry associated with space–time translation.

Define the Energy-Momentum Tensor as

Which is conserved i.e.= 0(27.27) Integration of the energy momentum tensor, will give the conserved charges which in this case are the energy and momentum.Consider the component

and we get the conservation of energy and momentum under space–time translational symmetry.

5. Summary

•  A field is specified by a set of numbers at each space – time point. When we go from discrete to continuous media with infinite number of degrees of freedom, the field ( ) plays the role of the generalized coordinates and becomes a continuous variable.
• If the action is invariant under some space time symmetry or under some internal symmetry operation, there is an associated conservation law given by Noether’s theorem.