Introduction to Supersymmetry
This article is one of Lie Group & Representation contents.
Supersymmetry is a symmetry that encompasses fermions and bosons, in other words, combines light and matter.
Although essential for high-energy theoretical physics, the minimal supersymmetry model(MSSM) has been experimentally disproved.
Nevertheless, we must learn supersymmetry theory.
The first reason is to understand BRST quantization. Familiarized with supersymmetry is needed to understand the process of assuming ghost fermions to quantize bosons in QFT.
The second reason is that supersymmetry is the only way to combine space-time symmetry and internal symmetry. Unique scalability should not be discussed in a purely mathematical or aesthetic realm, which tells us that we are missing the big picture.
Lorentz group
generator:
$[J_i, J_j] = i\epsilon_{ijk} J_k$, $[K_i, K_j] = -i\epsilon_{ijk} J_k$, $[J_i, K_j] = i\epsilon_{ijk} K_j$
Change $J^{\pm}_j = \frac{1}{2}(J_j \pm i K_j)$, then $[J^{\pm}_i, J^{\pm}_j] = i\epsilon_{ijk} J^{\pm}_k$, $[J^{\pm}_i, J^{\pm}_j] = 0$.
Lie group $SU(2) \times SU(2)$ have same Lie algebra with $SL(2,\mathbb{C})$.
casimir operator:
eigenvalue:
Poincare group
$[P_\mu, P_\nu] = 0$, $[J_i, P_j] = i\epsilon_{ijk} P_k$, $[J_i, P_0]=0$, $[K_i, P_j]=-iP_0$, $[K_i, P_0] = -iP_j$
With $M_{ij} = \epsilon_{ijk} J_k$, $[P_\mu, P_\nu] = 0$,
$[M_{\mu\nu}, M_{\rho\sigma}] = ig_{\nu\rho} M_{\mu\sigma} - ig_{\mu\rho} M_{\nu\sigma} + ig_{\mu\sigma} M_{\nu\rho}$, $[M_{\mu\nu}, P_\rho] = -ig_{\rho\mu} P_\nu + ig_{\rho\nu} P_\mu$.
Casimir operator:
Vector and Tensor
This is for boson, like photon.
Weyl Spinor
Representation of $SL(2,\mathbb{C})$.
Let two complex component object $\psi = \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix}$. Then $\mathcal{M} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in SL(2, \mathbb{C})$ as
$\psi_\alpha \rightarrow \psi'_\alpha = \mathcal{M}_\alpha^\beta \psi_\beta$, with $\alpha, \beta = 1, 2$ labling the components.
Left
Dirac spinor
Two Weyl Spinor
This is for fermion, like electron.
Supersymmetry Algebra
New generator added on Poincare algebra.
Undotted spinor $Q^I_\alpha$ and dotted spinor $\bar{Q}^I_\dot{\alpha}$ with index $I=1, \cdots N$.
$[P_\mu, Q_\alpha^I] = 0$,
$[P_\mu, \bar{Q}_\dot{\alpha}^I] = 0$,
$[M_{\mu\nu}, Q_\alpha^I] = i(\sigma_{\mu\nu})_\alpha^\beta Q_\beta^I$,
$[M_{\mu\nu}, \bar{Q}^{I \dot{\alpha}}] = i(\bar{\sigma}_{\mu\nu})^\dot{\alpha}_\dot{\beta} \bar{Q}^{I\dot{\beta}}$
$\{Q^I_\alpha, \bar{Q}^J_\dot{\beta}\} = 2\sigma^\mu_{\alpha \dot{\beta}} P_\mu \delta^{IJ}$, $\{Q^I_\alpha, Q^J_\beta\} = \epsilon_{\alpha \beta} Z^{IJ}$, $\{\bar{Q}^I_\dot{\alpha}, \bar{Q}^J_\dot{\beta}\} = \epsilon_{\dot{\alpha} \dot{\beta}} (Z^{IJ})^*$
Superspace
Superfield
Susy Gauge Theory
Broken Susy
Propagetor
grassmann variable
https://physics.stackexchange.com/questions/530370/what-is-the-meaning-of-a-grassmann-variable
Reference
오선근 - 초대칭성 물리학 입문
https://arxiv.org/pdf/hep-th/0101055.pdf
Wess,Bagger - supersymmetry and supergravity