Saturday, July 1, 2017

Standard Model Lagrangian

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 producing forces (Bosons) from integer spin interactions and matter (Fermions) from half-integer spin.

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 using S as Action.

The Euler-Lagrange Equation minimizes S and 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, L

Starting with 1863 Maxwell’s Equations

The L for Classical Electrodynamics:

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

Introducing a mass term:

Introducing a source term produces J(x)
For the case of a field with mass and spin (1/2 and 1) interaction:

The Klein Gordon EOM for Spin ( 0 ) (Higgs field):

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

f(x)  = Ce-i(p.x-Et)

The EOM for Spin = 1/2 Dirac Eq.:

The EOM for Spin = 1 (Boson) Proca Eq.:

3. Quantum Electrodynamic U(1):

Tomonaga, Feynman, Schwinger (1945-58) developed Quantum Electrodynamics: a precise description of electromagnetic interactions.

Feynman Diagram:

Feynman transition probabilities are calculated from a Feynman diagram where (for example) Fermions (spin 1/2 with charge) are destroyed to create a virtual Boson (spin 1 without charge) that is then destroyed to create new Fermions.

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

A photon is a spin 1 massless Boson interference packet in the electromagnetic field that has no rest mass, but has quanta E = hv and always travels at speed c.

An electron is a spin 1/2 Fermion interference packet in the electromagnetic field that has a rest mass.

A quark (Gell-Mann, Zweig 1960) is a spin 1/2 Fermion interference packet that interacts with electromagnetic, weak, and strong fields and has a rest mass.

The Quantum Electrodynamics (QED) Lagrangian:
4. Quantum ElectroWeak  SU(2):

Weinberg and Salam (1967) developed a gauge theory requiring three gauge bosons (W+-,Z). The Quantum ElectroWeak (QEW) Lagrangian:

5. Quantum Chromodynamics SU(3):

Han, Nambu, Greenburg (1970) 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 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.

Guth (1981) Inflation Theory.

7. The Big Bang
The Higgs field is unstable to symmetry breaking. After 10^-12 seconds, the SU(2)xU(1) symmetry breaks and the electron acquires mass, the neutrino stays massless, the W+-, Z acquire mass and the massless photon emerges.

A simple calculation of the Higgs' mass has suggested new science.

References:

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

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

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

 Schwichtenberg, J., Springer, London, 2015.