Totally Anti-symmetric Spinor Tensors in Minkowski Space

TL;DR

The paper addresses proving the total anti-symmetry of the volume spinor tensor in four-dimensional Minkowski space within the spinor formalism, focusing on the spinor representation $e_{AA'BB'CC'DD'}=i\bigl(\epsilon_{AB}\epsilon_{CD}\bar{\epsilon}_{A'C'}\bar{\epsilon}_{B'D'}-\epsilon_{AC}\epsilon_{BD}\bar{\epsilon}_{A'B'}\bar{\epsilon}_{C'D'}\bigr)$. It achieves this via an index-based algebraic treatment of spinor tensors, exploiting the fundamental antisymmetric form $ε_{AB}$ and the group structure of $SL(2,\mathbb{C})$. The work also derives explicit bases for the totally antisymmetric $(0,2)$ and $(0,3)$ tensor spaces and presents a concise proof for the $(0,4)$ volume form. These results bolster the use of spinor methods in formulating and analyzing Maxwell's equations, duality transformations, and knotted electromagnetic configurations, with potential extensions to quantum information and topological invariants. The findings emphasize computational efficiency and conceptual clarity afforded by spinor representations, suggesting further exploration of spinor-based invariants and applications.

Abstract

The spinor tensor $ε_{AB}$ has a special property that its elements can be formulated into an algebraic expression of the indices. All the totally anti-symmetric tensors in Minkowski space are expressed by $ε_{AB}$. By using the property, we give a simple proof of the total anti-symmetry for the volume spinor tensor.