# Electromagnetism

### General

• $k = \frac{2 \pi}{\lambda} = \frac{\omega}{c} = \frac{2 \pi \nu}{c}$
• $\lambda = \frac{2 \pi}{k} = \frac{c}{\nu}= \frac{c 2 \pi}{\omega}= \frac{h}{p}$
• $\nu = \frac{1}{T} = \frac{c}{\lambda} = \frac{\omega}{2 \pi} = \frac{c k }{2 \pi} = \frac{E}{h}$
• $\omega = \frac{2 \pi }{T} = 2 \pi \nu = \frac{c 2 \pi}{\lambda} = k c$

• $E = \hbar \omega = h \nu$
• $\vec p = \hbar \vec k = \frac{h}{\lambda}$
• $c = \frac{\omega}{k} = \nu \lambda$

# Special Relativity

### Notation

• $x^\mu := \left(ct, \vec x \right)$
• $\partial^{\mu} = \frac{\partial}{\partial x_\mu} = \left( \frac{\partial}{c \partial t}, - \nabla \right)^T$ and $\partial_{\mu} = \frac{\partial}{\partial x^\mu} = \left( \frac{\partial}{c \partial t}, \nabla \right)^T$
• d'Alembertian: $\square = \partial_\mu \partial^\mu = \frac{\partial^2}{c^2 \partial t^2} - \Delta$

### Formula

• Mass
• m(vμ) = m0γ
• Invariant mass: $m = \sqrt{\frac{(p^\mu)^2}{c^2}}$
• Momentum
• $p^\mu = m_0 v^\mu = (E/c, \vec p)^T$
• $\vec p = m_0 \gamma \vec v$
• Energy
• $E_{\mbox{rel}} = m(\vec v) c^2$
• Etot = Erel + Epot
• Ekin = Erelm0c2
• $E^2 = (\vec p c)^2 + (m_0 c^2)^2$
• E0 = p0c

# General Relativity

• Gravitational field equations: $G^{\mu \nu} = - \frac{8 \pi G_N}{c^4} T^{\mu \nu}$
• Einstein tensor: $G^{\mu \nu} = R^{\mu \nu} - \frac{1}{2} g^{\mu \nu} R$
• Ricci tensor: $R_{\mu \nu} = R^\lambda_{\; \; \mu \nu \lambda}$
• Riemann tensor: $R^\sigma_{\; \; \mu \nu \lambda}$
• Curvature scalar: $R=R^\lambda_{\; \; \lambda} = g^{\mu \nu} R_{\mu \nu}$

For the role of the cosmological constant, see the 5th force.

# Standard Model

### Higgs Sector

The $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$ invariant Lagrangian of the Standard Model describes massless gauge bosons interacting with massless fermions. In order for mass terms to be generated, one introduces fundamental complex scalar (Higgs) fields φ, which couple gauge-invariantly to the gauge bosons via the covariant derivatives

$\mathcal D_\mu \phi \mathcal D^\mu \phi^\dagger, \qquad \mathcal D_\mu := \partial_\mu + ig \sum_{i=1}^3 W^i_\mu \frac{\sigma^i}{2} +i g^\prime \frac{Y}{2} B_\mu,$

where $g, g^\prime$ and σi / 2,Y / 2 and $W^i_\mu, B_\mu$ are the SU(2)L,U(1)Y couplings and group generators and gauge fields, respectively and couple gauge-invariantly to the fermions through the Yukawa coupling of the form

$-\lambda_Y \left[ (\bar \psi_L \phi ) \psi_R + \bar \psi_R (\phi^\dagger \psi_L) \right].$

A natural and general way to construct a Higgs Lagrangian compatible with the Standard Model is to require gauge invariance and renormalizability:

$\mathcal L_H = (\mathcal D_\mu \phi)^\dagger \mathcal D^\mu \phi - m_H^2 \phi^\dagger \phi -\lambda_H (\phi^\dagger \phi )^2$
$=: \mathcal L^{kin}_H - \mathcal V_H (\phi) ,$

where $\mathcal V_H := m_H^2 (\phi^\dagger \phi) + \lambda_H (\phi^\dagger \phi)^2$ is the scalar potential.

The spontaneous symmetry breaking is driven by the 'mass' term mH. Technically, symmetry breaking is induced by a negative value: mH = :iμH and μH > 0.

The vacuum expectation value is

$\langle \phi \rangle = v / \sqrt{2}$,

where $v = \frac{\mu}{\sqrt{\lambda_H}}$.

The mass terms for the fermions and bosons originate from $\mathcal L^{kin}_H$.

So the Higgs mechanism refers to the spontaneous breaking of local (gauge) symmetries by the vacuum.

For a further analysis, go here.