Quantum Field Theory can be considered as "field + quantization". Followings is a mathematical formulation (at 2nd year undergraduate level) on the construction of quantum field and its application to elementary particle physics.

The dynamic of the field is determined by the field equation. The field equation for the neutral scalar meson field is a very simple kind of Klein-Gordon Equation as shown below:

where m is a parameter related to the mass, is the field, which is a complex function (with real and imaginary parts) of x, y, z, and t (simply represented by x in the equation),

are the Laplacians in four dimensional space-time and three dimensional space respectively. The wave equations and many systems in physics and engineering are constructed with these Laplacian operators. .

The field can be expressed in a series expansion in terms of the harmonic functions and the coefficients ck's, where k is a four dimensional vector related to the momentum and energy of the particle:

which is just a Fourier Series where the coefficients are to be determined by the field: