Lagrangian is basic formulation of Quantum Field Theory

Quantum field theory is a unification of special relativity and quantum mechanics. This theory formed the framework of the standard model in particle physics. Mathematical foundation in quantum field theory is the formulation of Lagrangian. We can observe a system by looking from its Lagrangian. Then by using Euler-Lagrange equation, the equation of motion of the system can be obtained. And many more works can be done from the Lagrangian.
Let us consider properties of the Lagrangian further. If the field has a kinetic energy T and the potential  V, then Lagrangian is

In the continuous case we actually work with the density of the Lagrangian

If we integrate the Lagrangian over time, we get a new quantity called the action S

Action is functional because the action always takes functions as arguments and produces a number. Particles always take the path with the smallest action. To find the path, then the variation of the action should be minimized. This is done by describing the action as the minimum mean value and a mean variation.

The action is minimum if satisfied

In quantum field theory, since that used is a lagrangian density, then the equation for the action will be

For the sake of abbreviation, the Lagrangian density is often called just Lagrangian.
This is an example of a Lagrangian for a free scalar particle

The first term is the kinetic energy (containing ) and the second term is the rate of mass (containing ).

Traveling Solutions of Massless Nonlinear Klein-Gordon Equation

Let us consider the mass-less nonlinear Klein-Gordon (NKG) equation as following,

Assume D'Alembertian stands for one time and one dimensional space. Then the equation becomes,

Solution of the equation can be obtained more simple by reducing the two variables (x,t) become one new variable z that containing mixed x and t with certain velocity. This assumptions that will be called the traveling solution. The NKG equation that of course is an partial differential equation will be formed an ordinary differential equation.

Using this assumption, then the equation becomes

Assume in a case of abbreviation notation , then times the equation with ,

Integration the equation in respect of z. Assume , then we can take integration constants are equal zero.

Therefore we have traveling solution of massless NKG. Because of the integral form is very simple, we can solve it straightforwardly using ordinary method. But for many other case in solving traveling solution problem, we often need elliptic integral methods to solve the integrations.

Integration of the 4-Dimensional Green’s Function

Let us assume a field must be satisfied to the equation of motion.

           (1)

Where and are D' Alembertian operator and an external source respectively.

Solution of the wave equation can be obtain by using Fourier's representation,




Where is a Green's function that must obey



Sometimes is called Feynman propagator.

The Fourier's representation solution can be proved by substitute it into the wave equation in Eq. (1).



As can be seen, it was proved easily. Now, the problem is how to determine the value of .
To calculate it, let's get back to the equation of Green's function and involving the Fourier transform
into and . Suppose the Fourier transform is



then the equation becomes,



From the relation, we have a conclusion that is



Remembering that stands for . Therefore, the Green's function becomes



Assume is an angle between and . In order to calculate volume integral can be more simple by using spherical coordinates. Then the Green's function has value,



Integration both parts and , then we have



Therefore we have a result that for the massless field the green function can be obtained as