#
Standard
Model Lagranian

The Standard Model of Particle Physics has helped unravel the hidden symmetries within the design of the Universe. Here we examine the steps
in building the Standard Model.

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

The concepts of quantum field
theory are profound and explain why electrons are identical because each
electron is an excitation of the electron quantum field and therefore they have
identical properties. Quantum fields include classes of particles for the
Fermi-Dirac statistics and Bose-Einstein statistics.

A field is represented by
amplitude (scalar, vector, complex number, spinor, or tensor) at some
space-time point. Gravity and electromagnetism are fields. Fields can oscillate
in space-time and produce wave-like excitations. The matter itself is an
excitation in the field.

The Lagrangian, L,
formulation is used to describe quantum fields in order to find Hamilton’s
Principle of Least Action which yields the Euler-Lagrange Equation.

**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:

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

The Lagrangian density function for a massless field f is

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

f(x) = e

^{-+i(p.x-Et)}For the case of a field with mass and interaction:

**3**. We are now ready to present the Quantum Electrodynamic (QED: 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.

The

The

**Quantum Electrodynamics (QED)**Lagrangian:

**4**. Next the Quantum ElectroWeak Theory (SU(2); Salam, Weinberg) is a gauge theory requiring three gauge bosons (W+-,Z).

The

**Quantum ElectroWeak (QEW)**Lagranian:

**5**. Quantum Chromodynamics (QCD: SU(3); Han, Nambu, Greenburg) describes the strong force mediated by gauge bosons called gluons carrying a unique kind of charge called color.

The

**Quantum Chromodynamics (QCD)**Lagranian:We are now ready to present the Standard Model.

**6**. The current structure of elementary particle physics is called the Standard Model [SU

_{C}(3) x SU

_{L}(2) x U

_{Y}(1) gauge theory].

Where
U

_{Y}(1) = local gauge invariance-Abelian
SU

_{L}(2) =local gauge invariance-Non-Abelian
SU

_{C}(3) = local gauge invariance-Non-Abelian
The

**Standard Model****(SM)**Lagranian:The first line represents the kinetic energy carried by W, Z, photon, and gluon. The second line is the interaction terms. The third line contains mass and the fourth line the left right parity interaction.

**References:**

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[1] Cox, B. and Forshaw, J.,

*Why does E=mc*, Da Capo Press, Cambridge, MA 2009.^{2}^{}
[2] Lancaster, T., and
Blundell, S. j.,

*Quantum Field Theory*, Oxford, UK, 2014.
[3] Robinson, M.,

*Symmetry and the Standard Model*, Springer, London, 2011.