Physlib.Relativity.Bispinors.Basic

Given a contravariant Lorentz vector $p^\mu$ (represented as `ℂT[.up]`), this function constructs its corresponding bispinor with upper spinor indices $p^{\alpha \dot{\beta}}$ (represented as `ℂT[.upL, .upR]`). The transformation is defined by contracting the Lorentz vector with the covariant Infeld-van der Waerden symbols $\sigma_{\mu}{}^{\alpha \dot{\beta}}$ (Pauli matrices with a Lorentz index): $$p^{\alpha \dot{\beta}} = \sigma_{\mu}{}^{\alpha \dot{\beta}} p^\mu$$ where $\alpha$ and $\dot{\beta}$ correspond to the upper left and upper right spinor indices respectively.

Given a contravariant Lorentz vector $p^\mu$ (represented as `ℂT[.up]`), this function constructs its corresponding bispinor with lower spinor indices $p_{\alpha \dot{\beta}}$ (represented as `ℂT[.downL, .downR]`). The transformation is defined by taking the upper bispinor $p^{\rho \dot{\sigma}}$ (constructed from $p^\mu$) and lowering its indices using the covariant spinor metric tensors (Levi-Civita tensors) $\varepsilon_{\alpha \rho}$ and $\varepsilon_{\dot{\beta} \dot{\sigma}}$: $$p_{\alpha \dot{\beta}} = \varepsilon_{\alpha \rho} \varepsilon_{\dot{\beta} \dot{\sigma}} p^{\rho \dot{\sigma}} = \varepsilon_{\alpha \rho} \varepsilon_{\dot{\beta} \dot{\sigma}} (\sigma_{\mu}{}^{\rho \dot{\sigma}} p^\mu)$$ where $\alpha$ and $\dot{\beta}$ correspond to the lower left and lower right spinor indices respectively.

Given a covariant Lorentz vector $p_\mu$, this function constructs its corresponding bispinor with upper spinor indices $p^{\alpha \dot{\beta}}$. This is achieved by contracting the vector with the Infeld-van der Waerden symbols $\sigma^{\mu \alpha \dot{\beta}}$ (represented as `σ^^^`), defined by the relation: $$p^{\alpha \dot{\beta}} = \sigma^{\mu \alpha \dot{\beta}} p_\mu$$ where $\alpha$ and $\dot{\beta}$ correspond to the left and right spinor indices (upL and upR) respectively.

Given a covariant Lorentz vector $p_\mu$, this function constructs the corresponding bispinor with lower spinor indices $p_{\alpha \dot{\beta}}$. This is defined by contracting the upper bispinor $p^{\alpha' \dot{\beta}'}$ (obtained from $p_\mu$ via `coBispinorUp`) with the covariant left and right spinor metrics $\epsilon_{\alpha \alpha'}$ and $\epsilon_{\dot{\beta} \dot{\beta}'}$ (represented as `εL'` and `εR'`): $$p_{\alpha \dot{\beta}} = \epsilon_{\alpha \alpha'} \epsilon_{\dot{\beta} \dot{\beta}'} p^{\alpha' \dot{\beta}'}$$ where $p^{\alpha' \dot{\beta}'} = \sigma^{\mu \alpha' \dot{\beta}'} p_\mu$. The resulting tensor has indices of type `downL` ($\alpha$) and `downR` ($\dot{\beta}$).

For a tensor \( p \), the contravariant bispinor with upper indices is equal to the contraction of the left and right metrics with the contravariant bispinor with lower indices: $$(\text{contrBispinorUp } p)^{\alpha \beta} = (\epsilon_L)^{\alpha \alpha'} (\epsilon_R)^{\beta \beta'} (\text{contrBispinorDown } p)_{\alpha' \beta'}$$ where \( \epsilon_L \) and \( \epsilon_R \) denote the left and right metrics, respectively.

For a covariant complex Lorentz tensor $p$, its up-type bispinor $p^{\alpha \beta}$ is equal to the contraction of its down-type bispinor $p_{\alpha' \beta'}$ with the left metric $\epsilon_L^{\alpha \alpha'}$ and the right metric $\epsilon_R^{\beta \beta'}$. This is expressed mathematically in index notation as $p^{\alpha \beta} = \epsilon_L^{\alpha \alpha'} \epsilon_R^{\beta \beta'} p_{\alpha' \beta'}$.