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.

In the framework of complex Lorentz tensors for $SL(2, \mathbb{C})$, the contraction of the Pauli matrix tensor $\sigma^{\mu \alpha \dot{\beta}}$ (with contravariant Lorentz index $\mu$, left-handed spinor index $\alpha$, and right-handed spinor index $\dot{\beta}$) and the Pauli matrix tensor with dualized spinor indices $\sigma^{\nu \tau(\alpha') \tau(\dot{\beta})}$ over the right-handed spinor index $\dot{\beta}$ is equivalent to the tensor constructed from the sum of the products of their rational components. Specifically, for a multi-index $b = (\mu, \alpha, \nu, \alpha')$, the component is given by: $$\sum_{x=0}^1 \text{pauliContrComponent}(\mu, \alpha, x) \cdot \text{pauliContrDownComponent}(\nu, x, \alpha')$$ where $\tau$ denotes the index duality involution and `ofRat` converts the rational-complex component calculation into a complex Lorentz tensor.

In the framework of complex Lorentz tensors for $SL(2, \mathbb{C})$, the contraction of a Weyl-dualized Pauli matrix $\sigma^\mu_{\alpha \dot{\beta}}$ with a Pauli matrix $\sigma^{\nu \alpha \dot{\beta}'}$ over the shared left-handed spinor index $\alpha$ is equal to the tensor constructed from rational complex components. Specifically, for Lorentz indices $\mu, \nu$ and right-handed spinor indices $\dot{\beta}, \dot{\beta}'$, the result is given by: $$\sigma^\mu_{\alpha \dot{\beta}} \sigma^{\nu \alpha \dot{\beta}'} = \text{ofRat} \left( \sum_{x \in \{0, 1\}} (\sigma_\mu)_{x \dot{\beta}} (\sigma^\nu)^{x \dot{\beta}'} \right)$$ where $(\sigma_\mu)_{x \dot{\beta}}$ and $(\sigma^\nu)^{x \dot{\beta}'}$ represent the rational-complex components of the respective Pauli matrices, and `ofRat` is the map embedding these rational components into the complex tensor space.

The contraction of the complex four-dimensional Levi-Civita tensor $\epsilon_{\mu\nu\rho\kappa}$ with the contravariant Pauli tensor $\sigma^{\kappa \alpha \dot{\beta}}$ is equal to its contraction with the covariant Pauli tensor $\sigma_{\kappa}{}^{\alpha \dot{\beta}}$. Mathematically, $$\epsilon_{\mu\nu\rho\kappa} \sigma^{\kappa \alpha \dot{\beta}} = \epsilon_{\mu\nu\rho\kappa} \sigma_{\kappa}{}^{\alpha \dot{\beta}}$$ where the left-hand side involves a Lorentz-dualized index contraction and the right-hand side uses the covariant form of the Pauli tensor.

In the framework of complex Lorentz tensors for $SL(2, \mathbb{C})$, let $\sigma^\mu$ denote the Pauli matrices (the soldering form) with indices in the contravariant vector, left-handed spinor, and right-handed spinor representations. Let $\bar{\sigma}^\nu$ denote the dual Pauli matrices (obtained by index dualization $\tau$), $\eta^{\mu\nu}$ denote the Minkowski metric, and $\epsilon^{\mu\nu\rho\kappa}$ denote the four-dimensional Levi-Civita tensor. The product of three Pauli matrices satisfies the following identity: $$\sigma^\mu \bar{\sigma}^\nu \sigma^\rho = \eta^{\mu\nu} \sigma^\rho - \eta^{\mu\rho} \sigma^\nu + \eta^{\nu\rho} \sigma^\mu + i \epsilon^{\mu\nu\rho\kappa} \sigma_\kappa$$ In index notation, using $\alpha, \beta, \gamma, \dots$ for spinor indices and $\mu, \nu, \rho, \dots$ for Lorentz vector indices, this is expressed as: $$(\sigma^\mu)_{\alpha \dot{\gamma}} (\bar{\sigma}^\nu)^{\dot{\gamma} \delta} (\sigma^\rho)_{\delta \dot{\beta}} = \eta^{\mu\nu} (\sigma^\rho)_{\alpha \dot{\beta}} - \eta^{\mu\rho} (\sigma^\nu)_{\alpha \dot{\beta}} + \eta^{\nu\rho} (\sigma^\mu)_{\alpha \dot{\beta}} + i \epsilon^{\mu\nu\rho\kappa} (\sigma_\kappa)_{\alpha \dot{\beta}}$$ where $i$ is the imaginary unit.

The product of three Pauli matrices in the configuration $\bar{\sigma}^\mu \sigma^\nu \bar{\sigma}^\rho$ satisfies the conjugate three-Pauli identity: $$\bar{\sigma}^\mu \sigma^\nu \bar{\sigma}^\rho = \eta^{\mu\nu} \bar{\sigma}^\rho - \eta^{\mu\rho} \bar{\sigma}^\nu + \eta^{\nu\rho} \bar{\sigma}^\mu - i \epsilon^{\mu\nu\rho\kappa} \bar{\sigma}_\kappa$$ where: - $\sigma^\mu$ are the Pauli matrices and $\bar{\sigma}^\mu$ are their conjugate (dualized) forms. - $\eta^{\mu\nu}$ is the Minkowski metric. - $\epsilon^{\mu\nu\rho\kappa}$ is the four-dimensional Levi-Civita tensor. - $i$ is the imaginary unit. - The indices on the left-hand side are contracted to represent the matrix product $(\bar{\sigma}^\mu \sigma^\nu \bar{\sigma}^\rho)_{\dot{\beta} \alpha'}$.