# Lie derivative for spinor fields According to Kosmann's definition, for a spinor field, $\psi$, its Lie derivative with respect to a vector field, $X$, is given by$\mathcal{L}_X \psi:=X^a \nabla_a \psi-\frac{1}{8} \nabla_{[a} X_{b]}\left[\gamma^a, \gamma^b\right] \psi,$where $\gamma^a$ are Dirac matrices. When $X^a$ is a Killing vector field, it reduces to $\mathcal{L}_X \psi:=X^a \nabla_a \psi-\frac{1}{4} \nabla_a X_b \gamma^a \gamma^b \psi.$ ## Refs - [Wikipedia](https://en.wikipedia.org/wiki/Lie_derivative#The_Lie_derivative_of_a_spinor_field) - reviews - [[2008#Sharipov (Notes)]] - [[2003#Matteucci (Thesis)]] - [[2025#Giotopoulos (Notes)]] - derivation - see sec.2.1 of [[1999#Figueroa-O'Farrill (Notes)]] ## History - [[Lichnerowicz1963]] - first correct definition of Lie derivative for spinor fields but only with respective to infinitesimal isometries - [[1971#Kosmann]] - generalises Lichnerowicz's definition to include all infinitesimal transformations - it is natural but still *ad hoc*: it is not necessarily the unique way to define a Lie derivative for spinors - [[BourguignonGauduchon1992]] - gives a definition that provides Kosmann's definition a geometric justification - bad: - it only coincides with Kosmann's definition on spinor fields - it automatically preserves the metrics -> not always what we want - [[1996#Fatibene, Ferraris, Francaviglia, Godina (Aug)]] - first correct geometric explanation of Kosmann's definition - [[GodinaMatteucci2002]] - introduces a reductive $G$-structure, which generalises earlier definitions - [[2005#Godina, Matteucci]] - expanded version of their 2002 paper, with applications