User:Toni 001/sandbox

=Supersymmetry=

Superfields
Superfields are functions of superspace. They have the form

$$ F(x, \theta, \bar{\theta}) = f(x) + \theta \phi(x) + \bar{\theta} \bar{\chi}(x) + \theta \theta m(x) + \bar{\theta} \bar{\theta} n(x) + \theta \sigma^\mu \bar{\theta} A_\mu(x) + \bar{\theta} \bar{\theta} \theta \lambda(x) + \theta \theta \bar{\theta} \psi(x) + \theta \theta \bar{\theta} \bar{\theta} d(x) $$.

Superpotential
A superpotential, denoted by $$ W(\Phi) $$, is a polynomial in chiral superfields. Chiral superfields satisfy $$ \bar{D}_\dot{\alpha} \Phi = 0 $$, where $$ \bar{D}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} - i \theta^\alpha \sigma^\mu_{\alpha \dot{\alpha}} \frac{\partial}{\partial x^\mu} $$ is the covariant derivative on superspace that commutes with the supersymmetry transformations. $$ \bar{D}_\dot{\alpha} $$ satisfies the product rule so $$ W(\Phi) $$ is a chiral superfield.

From superpotentials supersymmetry invariant interaction Lagrangians can be constructed:

$$ \mathcal{L}_1 = \int d^2 \theta W(\Phi) + \int d^2 \bar{\theta} \bar{W}(\Phi) $$

The integrals over the Grassmann numbers produce a term that transforms into spacetime derivatives under supersymmetry transformations. The spacetime derivatives do not change the action and therefore leave the equations of motion invariant.

=Algebras=

Harmonic oscillator
$$ [a, a^\dagger] = 1 $$

The basis vectors of the representation space are labeled by the eigenvalues of the number operator $$ N $$, which is defined as $$ N = a^\dagger a $$:

$$ \{| 0 \rangle, | 1 \rangle, ... \} $$

The elements of the one-parameter group generated by this algebra are given by $$ U(t) = e^{- \frac{i H t}{\hbar}} $$ where $$ H = \hbar \omega (N + 1 / 2) $$ and $$ t \in \mathbb{R} $$. $$ H $$ is called the generator of time translation.

Spin
$$ [S_i, S_j] = i \hbar \epsilon_{i j k} S_k $$

Angular momentum
$$ [L_i, L_i] = i \hbar \epsilon_{i j k} L_k $$

The basis vectors of the representation space can be labeled by the eigenvalues of of the total angular momentum operator $$ L^2 = {L_1}^2 + {L_2}^2 + {L_3}^2 $$ and $$ L_3 $$:

$$ \{| l, m \rangle| l = 0, 1, ... \land m = -l, ..., l \} = \{| 0, 0 \rangle, | 1, 1 \rangle, | 1, 0 \rangle, | 1, -1 \rangle, | 2, 2 \rangle, ... \} $$

The three-parameter group generated by this algebra consists of the elements $$ U(\mathbf{\theta}) = e^{-\frac{i}{\hbar} \theta_i J_i} $$. The $$ J_i $$ are called the generators of rotation.

Free particle on the real line
The algebra consists of a single element, the momentum operator $$ P $$. The basis vectors of the representation space are labeled by the eigenvalues of $$ P $$:

$$ \{| p \rangle| p \in \mathbb{R} \} $$

The elements of the one-parameter group are $$ T(x) = e^{\frac{i P x}{\hbar}} $$ for $$ x \in \mathbb{R} $$. $$ P $$ is called the generator of translations.

Two non-interacting particles
The algebra of two non-interacting free particles consists of two commuting single particle momentum operators $$ P \otimes 1 $$ and $$ 1 \otimes P $$. The representation space in a tensor product of two single particle representation spaces. The basis vectors of the representation space are:

$$ \{| p_1 \rangle \otimes | p_2 \rangle| p_1, p_2 \in \mathbb{R} \} $$

Lorentz algebra
$$ [M_{\mu \nu}, M_{\rho \sigma}] = i (\eta_{\mu \rho} M_{\nu \sigma} - \eta_{\mu \sigma} M_{\nu \rho} - \eta_{\nu \rho} M_{\mu \sigma} + \eta_{\nu \sigma} M_{\mu \rho}) $$

Poincaré algebra
$$ [P_\mu, P_\nu] = 0 $$

$$ [M_{\mu \nu}, P_\rho] = M_{\mu \rho} P_\nu - M_{\nu \rho} P_\mu $$

$$ [M_{\mu \nu}, M_{\rho \sigma}] = i (\eta_{\mu \rho} M_{\nu \sigma} - \eta_{\mu \sigma} M_{\nu \rho} - \eta_{\nu \rho} M_{\mu \sigma} + \eta_{\nu \sigma} M_{\mu \rho}) $$

Super Poincaré algebra
$$ [P_\mu, P_\nu] = 0 $$

$$ [P_\mu, {Q_\alpha}^A] = 0 $$

$$ [P_\mu, \bar{Q}_{\dot{\alpha} A}] = 0 $$

$$ \{{Q_\alpha}^A, {Q_\beta}^B \} = 0 $$

$$ \{\bar{Q}_{\dot{\alpha} A}, \bar{Q}_{\dot{\beta} B} \} = 0 $$

$$ [M_{\mu \nu}, P_\rho] = M_{\mu \rho} P_\nu - M_{\nu \rho} P_\mu $$

$$ [M_{\mu \nu}, M_{\rho \sigma}] = i (\eta_{\mu \rho} M_{\nu \sigma} - \eta_{\mu \sigma} M_{\nu \rho} - \eta_{\nu \rho} M_{\mu \sigma} + \eta_{\nu \sigma} M_{\mu \rho}) $$

$$ \{{Q_\alpha}^A, \bar{Q}_{\dot{\beta} B} \} = 2 \sigma_{\alpha \dot{\beta}}^\mu P_\mu {\delta^A}_B $$