Ask Question Asked 5 months ago. The text represents a part of the initial chapter of … The partial derivatives indeed commute unlike the covariant ones. The product is the number of cycles in the time period, independent of the units used (a scalar). We also have the curved-space version of Stokes's theorem using the covariant derivative and finally the exterior derivative and commutator, where Carroll seems to have made a very peculiar typo. Covariant derivatives (wrt some vector field; act on vector fields, or even on tensor fields). We present detailed pedagogical derivation of covariant derivative of fermions and some related expressions, including commutator of covariant derivatives and energy-momentum tensor of a free Dirac field. For a function the covariant derivative is a partial derivative so $\nabla_i f = \partial_i f$ but what you obtain is now a vector field, and the covariant derivative, when it acts on a vector field has an extra term: the Christoffel symbol: This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. The commutator acts on any tensor in any space of any dimensionality, so is foundational and general. I am wondering if there is a better formula for forms in particular.) The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). 1 Charged particles in an electromagnetic field 67 5. ) But this formula is the same for the divergence of arbitrary covariant tensors. The Commutator of Covariant Derivatives. The linear transformation ↦ (,) is also called the curvature transformation or endomorphism. I recently cam across a nice answer to that question, in a … ... Closely related to your question is what is the commutator of Lie derivative and Hodge dual *. The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. The commutator or Lie bracket is needed, in general, in order to "close up the quadrilateral"; this bracket vanishes if [itex]\vec{X}, \, \vec{Y}[/itex] are two of the coordinate vector fields in some chart. The structure equations define the torsion and curvature. Viewed 48 times 2. Commutator of covariant derivatives acting on a vector density. In differential geometry, the Lie derivative / ˈ l iː /, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. $\endgroup$ – Yuri Vyatkin Mar 14 '12 at 5:45 1 $\begingroup$ Let $\mathfrak n^\alpha$ be a vector density of weight 1. If they were partial derivatives they would commute, but they are not. a coordinate system, and you are talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference. QUANTUM FIELD THEORY II: NON-ABELIAN GAUGE INVARIANCE NOTES 3 Another way to deflne the fleld strength tensor F„” and to show its covariance in terms of the commutator of the covariant derivative. $\begingroup$ Partial derivatives are defined w.r.t. This is the notion of a connection or covariant derivative described in this article. 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