Thursday, March 2, 2017

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 harmonic oscillating excitation 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 excitation.

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.

Lagrangian Field Theory formulates the relativistic quantum mechanical theory of interactions. It has dependent variables replaced by values of a field at a point in space-time  f (x,y,z,t). The equations of motion are obtained by the Action Principle where S is Action. Energy is conserved if the Lagrangian doesn’t change with time. 

The Euler-Lagrange Equation produces 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

Maxwell’s Equations
The L for Classical Electrodynamics:
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 Lagrangian is

3. The Quantum Electrodynamic U(1):

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

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

The Quantum Electrodynamics (QED) Lagrangian:
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) Lagrangian:
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) Lagrangian:

The first line represents the kinetic energy carried by W, Z, photon, and gluons. The second line is the interaction terms. The third line contains mass and the fourth line the left right parity interaction.

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.

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