Physics Equations

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1 Electromagnetism
- 1.1 General
2 Special Relativity
- 2.1 Notation
- 2.2 Formula
3 General Relativity
4 Standard Model
- 4.1 Higgs Sector

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Electromagnetism

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General

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

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

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Special Relativity

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Notation

Formula

Mass
- $m (v μ) = m 0 γ$
- 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
- $E_{\mbox{rel}} = m(\vec v) c^2$
- $E tot = E rel + E pot$
- $E kin = E rel - m 0 c 2$
- $E^2 = (\vec p c)^2 + (m_0 c^2)^2$
- $E 0 = p 0 c$

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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.

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Standard Model

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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 $σ i / 2, Y / 2$ and $W^i_\mu, B_\mu$ are the $S U (2) L, U (1) Y$ 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 $m H$ . Technically, symmetry breaking is induced by a negative value: $m H = : i μ H$ and $μ H > 0$ .

The vacuum expectation value is

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