**1. The Universe**was created out of interacting quantum fields of harmonic oscillating excitations with particle-like eigenfunctions.

A field is represented at some space-time point by an amplitude (scalar, vector, complex number, spinor, or tensor)that oscillates in space-time to produce wave-like excitations.

Quantum field theory explains why electrons are identical because each electron is an excitation of the electron quantum field with identical properties using Fermi-Dirac statistics.

Lagranian Field Theory formulates the relativistic quantum mechanical theory of interactions. It has dependent variables replaced by values of a field at a point in spacetime f (x,y,z,t) so that the equations of motion are obtained by the Action Principle. S is called Action which is a set of n independent variables.

Energy is conserved if the Lagrangian doesn’t change with time. The result is the Euler-Langrange Equation:

Therefore, the Lagrangian describes quantum fields through Hamilton’s Principle of Least Action to satisfy the Euler-Lagrange Equation producing the model's equation of motion.

The steps to construct the Standard Model of Quantum Field Theory start with the classical Lagrangian, L.

**2. The Lagrangian density function,**

L, for Classical Electrodynamics:

Maxwell’s Equations

Then

Next consider, the Lagrangian density function for a massless field:

Adding mass creates the scalar (spin j=0) Klein Gordon

The solutions to the Klein Gordon Equation are simple plane waves subject to relativistic constraint:

f(x) = e

^{-+i(p.x-Et)}Introducing a source term produces J(x)

For the case of a field with mass and interaction:

The relativistic quantum mechanical Dirac Lagranian is

**3. The Quantum Electrodynamic U(1):**

Tomonaga, Schwinger, Feynman) is a precise description of electromagnetic interactions.

Feynmn Diagram:

A loop in a Feynman diagram indicts a divergence (infinite integral) that must be renormalized for calculations.

**4. Next the Quantum ElectroWeak Theory SU(2):**

Salam and Weinberg developed a gauge theory requiring three gauge bosons (W+-,Z). The Quantum ElectroWeak (QEW) Lagranian:

**5. Quantum Chromodynamics SU(3):**

Han, Nambu, Greenburg described the strong force mediated by gauge bosons, called gluons, carrying a unique kind of charge called color. The Quantum Chromodynamics (QCD) Lagranian:

**6. The Standard Model SU(3) x SU(2) x U(1):**

The current structure of elementary particle physics is called the Standard Model. The Standard Model (SM) Lagranian:

**References:**

[1] Cox, B. and Forshaw, J., Why does E=mc2, Da Capo Press, Cambridge, MA 2009.

[2] Lancaster, T., and Blundell, S. j., Quantum Field Theory, Oxford, UK, 2014.

[3] Robinson, M., Symmetry and the Standard Model, Springer, London, 2011.