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 ${\bar {D}}_{\dot {\alpha }}=-{\frac {\partial }{\partial {\bar {\theta }}^{\dot {\alpha }}}}-i\theta ^{\alpha }\sigma _{\alpha {\dot {\alpha }}}^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ 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 $U(t)=e^{-{\frac {iHt}{\hbar }}}$ where $H=\hbar \omega (N+1/2)$ and $t\in \mathbb {R}$. $H$ is called the generator of time translation.

Spin

$[S_{i},S_{j}]=i\hbar \epsilon _{ijk}S_{k}$

Angular momentum

$[L_{i},L_{i}]=i\hbar \epsilon _{ijk}L_{k}$

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 $T(x)=e^{\frac {iPx}{\hbar }}$ 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

$[P_{\mu },P_{\nu }]=0$

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

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

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

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

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

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

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