[NOTES/QFT-01001] Examples of Classical Fields

Analytical dynamics

The analytical dynamics will here mean the Lagrangian or Hamiltonian form of dynamics. In the Lagrangian formulation the states of physical system in classical mechanics are described by {\it generalized coordinates} \(q_k\) and {\it generalized velocities} \(\dot{q}_k\). The dynamical variables of a physical system are functions of generalized coordinates and generalized velocities
The law of dynamics is formulated as a variational principle known as   action principle. The equations of motion are Euler Lagrange equations.
The information about the interactions of a system is coded in the form of Lagrangian of the system as a function of generalized coordinates and generalized momenta.

The classical systems of interest in mechanics are those with finite degrees of freedom.

Examples of Classical Fields

Vibrating string
Not all systems can be described by finite number of generalized coordinates. An example is a vibrating string with end points fixed. The state of a vibrating string is completely specified by giving the displacement of the string and its velocity for each point along the length of the spring. The displacement \(y(x,t)\), for each \(x\), serves the purpose of a generalized coordinate. Thus this system has infinite number of degrees of freedom. The time variation of  displacement of the  are described by the wave equation.

Electromagnetic fields
The electromagnetic waves are described by specifying the electric and magnetic fields. These fields obey Maxwell's equations.
The Maxwell's equations provide the equations of motion. The interaction of a charged particle and electromagnetic fields is  described by Maxwell's equations and the Lorentz force equation.

Can one think of them as Euler Lagrange equations of motion?  Obtain them from an action principle? If yes what is the action?

For this purpose we need to describe the electromagnetic fields in terms of scalar and vector potentials. Thus, in the new scheme of things,  the potentials  \(\phi(\vec{r},t), \vec{A}(\vec{r},t)\) at each point \(\vec{r}\) become the generalized coordinates of the electromagnetic field.

Relativistic wave equations
Attempts to combine relativity and quantum mechanics led to relativistic quantum mechanics. For example Klein Gordon equation \begin{equation} \frac{1}{c^2}\frac{ \partial^2\phi}{\partial t^2}- \nabla^2 \phi(x) - m^2 \phi(x)=0 \end{equation} was proposed to describe relativistic quantum mechanics of a free spin zero particle. Very soon it was realized that these theories, relativistic quantum mechanics, have insurmountable problems and a consistent formulation of interacting particles is not possible.

The above description of a relativistic particle by wave function satisfying Klein Gordon equation  was given up as a quantum description.

The Klein Gordon equation and Dirac equation were  reinterpreted as classical  field equations. Thus we have spin zero particles described by Klein Gordon field and spin half particles described by fields obeying Dirac equation.

Schrodinger equation as classical field equation
In the picture just described, the Klein Gordon equation, Dirac equation are not regarded as quantum equations of motion. They are taken  as classical equations of motion and a consistent interpretation emerged after quantization of these fields. As we will see that it is possible to give an action principle such that for these equations, Klein Gordon and Dirac  equations, are arise as Euler Lagrange equations. This is achieved by reinterpreting the Klein Gordon wave function and Dirac wave function as a classical fields with field values at different space points as generalized coordinates.

What has been said above for the Klein Gordon and Dirac equations, can be repeated  for non relativistic  Schrodinger equation for a point particle. The time dependent Schrodinger equation for a point particle is \begin{equation} i\hbar \frac{\partial \psi(\vec{r},t)}{\partial t} = - \frac{\hbar^2}{2m} \nabla^2 \psi (\vec{r},t)+ V(\vec{r})\psi(\vec{r},t). \end{equation} The quantum point particle is fully described by its wave function; and the set of all values of wave function at different  space points  can be thought of as being the set of generalized coordinates required for a complete description of the quantum particle. In this way of viewing the quantum description, the Schrodinger equation comes out as (details later) Euler Lagrange equation corresponding to the action \begin{equation} \int dt \iiint d^3x \Big( i\hbar \psi^*(\vec{r},t) \dd[\psi(\vec{r},t)]{t} -\frac{\hbar^2}{2m} \big|\nabla \psi(\vec{r},t)\big|^2 - \psi^*(\vec{r},t) V(\vec{r}) \psi(\vec{r},t)\Big)\end{equation}

Analytical dynamics
In the above examples, we have described several systems of interest with infinite degrees of freedom. In each example we know the field equation. The analytical formulation of classical field theory starts with an action principle followed by Lagrangian and Hamiltonian forms of dynamics. Having formulated the dynamics for classical fields,  one needs the form of Lagrangian, or the Hamiltonian, which will give the correct equations of motion from the action principle.