Physlib.Relativity.PauliMatrices.Relations

In the context of complex Lorentz tensors for $SL(2, \mathbb{C})$, the contraction of two Pauli matrices over their common Lorentz vector index $\nu$ satisfies the following identity: $$\sigma_{\nu}{}^{\alpha \dot{\beta}} \sigma^{\nu \alpha' \dot{\beta}'} = 2 \epsilon^{\alpha \alpha'} \epsilon^{\dot{\beta} \dot{\beta}'}$$ where: - $\sigma_{\nu}{}^{\alpha \dot{\beta}}$ and $\sigma^{\nu \alpha' \dot{\beta}'}$ are the Pauli matrices (intertwining Lorentz vectors and Weyl spinors). - $\alpha, \alpha'$ are left-handed (undotted) spinor indices. - $\dot{\beta}, \dot{\beta}'$ are right-handed (dotted) spinor indices. - $\epsilon^{\alpha \alpha'}$ is the left-handed spinor metric tensor ($\epsilon_L$). - $\epsilon^{\dot{\beta} \dot{\beta}'}$ is the right-handed spinor metric tensor ($\epsilon_R$).

In the context of complex Lorentz tensors for $SL(2, \mathbb{C})$, the contraction (trace) over the spinor indices of two Pauli matrices, one with covariant spinor indices and the other with contravariant spinor indices, is proportional to the Minkowski metric: $$\sigma_{\mu \dot{\beta} \alpha} \sigma_\nu{}^{\alpha \dot{\beta}} = 2 \eta_{\mu\nu}$$ where: - $\sigma_{\mu \dot{\beta} \alpha}$ and $\sigma_\nu{}^{\alpha \dot{\beta}}$ are Pauli matrices acting as intertwining operators between Lorentz vectors and Weyl spinors. - $\mu, \nu$ are covariant Lorentz vector indices. - $\alpha$ is a left-handed (undotted) spinor index and $\dot{\beta}$ is a right-handed (dotted) spinor index. - $\eta_{\mu\nu}$ is the covariant Minkowski metric.

In the theory of complex Lorentz tensors for the group $SL(2, \mathbb{C})$, the contraction of two Pauli matrices over their spinor indices is equal to twice the covariant Minkowski metric: $$\sigma_{\mu}{}^{\alpha \dot{\beta}} \sigma_{\nu \dot{\beta} \alpha} = 2 \eta_{\mu \nu}$$ where: - $\sigma_{\mu}{}^{\alpha \dot{\beta}}$ is a Pauli matrix with a covariant Lorentz index $\mu$ and contravariant spinor indices $\alpha$ and $\dot{\beta}$. - $\sigma_{\nu \dot{\beta} \alpha}$ is a Pauli matrix with three covariant indices (Lorentz index $\nu$ and spinor indices $\dot{\beta}$ and $\alpha$). - $\eta_{\mu \nu}$ is the covariant Minkowski metric.

In the framework of complex Lorentz tensors for $SL(2, \mathbb{C})$, the Pauli matrices $\sigma$ satisfy a symmetric anticommutation-like relation. For contravariant Lorentz indices $\mu, \nu$, left-handed spinor indices $\alpha, \alpha'$, and a right-handed spinor index $\dot{\beta}$, it holds that: $$\sigma^{\mu \alpha \dot{\beta}} \sigma_{\nu \dot{\beta} \alpha'} + \sigma^{\nu \alpha \dot{\beta}} \sigma_{\mu \dot{\beta} \alpha'} = 2 \eta^{\mu \nu} \delta^{\alpha}_{\alpha'}$$ where $\eta^{\mu \nu}$ is the contravariant Minkowski metric and $\delta^\alpha_{\alpha'}$ is the unit tensor (Kronecker delta) for left-handed Weyl spinors.

In the framework of complex Lorentz tensors for $SL(2, \mathbb{C})$, the Pauli matrices $\sigma$ satisfy the following anticommutation identity involving right-handed (dotted) spinor indices: $$\sigma_{\mu \dot{\beta} \alpha} \sigma^{\nu \alpha \dot{\beta}'} + \sigma_{\nu \dot{\beta} \alpha} \sigma^{\mu \alpha \dot{\beta}'} = 2 \eta_{\mu}^{\phantom{\mu}\nu} \delta_{\dot{\beta}}^{\phantom{\dot{\beta}}\dot{\beta}'}$$ where: - $\mu, \nu$ are indices representing Lorentz vectors (covariant and contravariant respectively). - $\alpha$ is a left-handed (undotted) spinor index, over which the terms are contracted. - $\dot{\beta}, \dot{\beta}'$ are right-handed (dotted) spinor indices (covariant and contravariant respectively). - $\eta_{\mu}^{\phantom{\mu}\nu}$ is the mixed Minkowski metric, acting as a Kronecker delta $\delta_{\mu}^{\nu}$ for vector indices. - $\delta_{\dot{\beta}}^{\phantom{\dot{\beta}}\dot{\beta}'}$ is the Kronecker delta (unit tensor) for right-handed Weyl spinors.