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Electroweak Theory

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Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
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Countdown to the Standard Model
From a framework to a model Yang-Mills Theory
Quantum Electrodynamics
Quantum Chromodynamics
Electroweak Theory
Higgs Mechanism
Standard Model
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Problems with the Standard Model
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
Related De Donder-Weyl Theory

Electroweak Theory is a Quantum Field Theory which unifies Electromagnetism and the Weak Nuclear Force in a single Gauge Field with gauge group SU(2)_L \times U(1)_Y, where SU(2)_L acts only on Left-Handed Fermions, and U(1)_Y is "hypercharge". The Higgs Mechanism gives mass to the W and Z components of this gauge field, and breaks the symmetry to U(1)_{electromagnetism}, as described by Quantum Electrodynamics.

Lagrangian DensityEdit this section

It can be easily shown like as for other Quantum Field Theoryies, that the Dirac Equation for Electroweak Theory is: \left( i\hbar \not{\partial }-{{m}_{Weak}}{{c}_{0}}-\frac{{{g}_{W}}\alpha \not{W}+{{g}_{F}}Y\not{F}}{2} \right)\psi =0

The Lagrangian Density excluding field strength is given by:

{{\mathsf{\mathcal{L}}}_{EW}}={{c}_{0}}\bar{\psi }\left( i\hbar \not{\partial }-{{m}_{Weak}}{{c}_{0}}-\frac{{{g}_{W}}\alpha \not{W}+{{g}_{F}}Y\not{F}}{2} \right)\psi

Here, {{m}_{\text{Weak}}} is the mass of the W/Z boson, W is the Weak Field, F is the Photon Field, and Y is the Weak Hypercharge.

If one lets  \mathcal{Q}=\left[ \begin{matrix}
   {{g}_{W}}\alpha   \\
   {{g}_{F}}Y  \\
\end{matrix} \right]


\mathsf{\mathcal{V}}=\left[ \begin{matrix}
   {\not{W}}  \\
   {\not{F}}  \\
\end{matrix} \right]

then the Lagrangian Density (with a field strength I

{{\mathsf{\mathcal{L}}}_{EW}}=-\frac{{{I}^{\mu \nu \rho }}{{I}_{\mu \nu \rho }}}{4}+{{c}_{0}}\bar{\psi }\left( i\hbar \left( \not{\partial }+\frac{i}{\hbar }\frac{\cdot \mathsf{\mathcal{V}}}{2} \right)-{{m}_{\mathrm{Weak}}}{{c}_{0}} \right)\psi =-\frac{1}{4}{{I}^{\mu \nu \rho }}{{I}_{\mu \nu \rho }}+i\hbar {{c}_{0}}\bar{\psi }{{\not{\nabla }}_{EW}}\psi -\bar{\psi }{{m}_{EW}}c_{0}^{2}\psi

Dirac EquationEdit this section

From the result of the last section, we see that

i\hbar \not\nabla_{EW}\psi -m_\mathrm{Weak}c_0\psi = 0

Electroweak Symmetry BreakingEdit this section

Main Article: Higgs Mechanism

As per Experimental Results, Photons are massless, whereas Weak Force mediating W and Z Bosons are massive. If these W and Z Bosons are massive, then this means that Electroweak Symmetry is broken. This Electroweak Symmetry Breaking is the result of the Higgs Mechanism.

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