[Paper Review] Covariant derivative of fermions and all that

Ilya L. Shapiro|arXiv (Cornell University)|Nov 7, 2016

Algebraic and Geometric Analysis4 citations

TL;DR

This paper provides a detailed pedagogical derivation of the covariant derivative for Dirac fermions in curved spacetime using the tetrad formalism, establishing the spin connection, commutator of covariant derivatives, and energy-momentum tensor. It derives the trace of the energy-momentum tensor for a massless Dirac field on a cosmological (FRW) background, showing it vanishes when the field depends only on time, consistent with conformal invariance.

ABSTRACT

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. On top of that, local conformal transformations for a Dirac fermion in curved spacetime are considered and we obtain the expression for the energy-momentum tensor on the cosmological background.

Motivation & Objective

• To provide a comprehensive, step-by-step derivation of the covariant derivative for Dirac fermions in curved spacetime for educational purposes.
• To clarify the role of the tetrad formalism and spin connection in enabling Lorentz-covariant derivatives for spinors.
• To derive the energy-momentum tensor for a free Dirac field and analyze its trace in relation to conformal symmetry.
• To compute the energy-momentum tensor on a Friedmann-Robertson-Walker (FRW) cosmological background, particularly in the massless limit.
• To highlight the physical interpretation of the field equations when the fermion depends only on time, showing a dust-like equation of state for massive fields.

Proposed method

• Uses the tetrad (vierbein) formalism to relate curved spacetime to local Minkowski space, enabling the definition of flat-space gamma matrices.
• Derives the spin connection from the requirement of metric compatibility and Lorentz invariance in the local tangent space.
• Constructs the covariant derivative of the Dirac spinor using the spin connection, ensuring local Lorentz covariance.
• Computes the commutator of covariant derivatives and relates it to the Riemann curvature tensor via the spin connection.
• Derives the energy-momentum tensor from the variation of the Dirac action with respect to the metric.
• Applies the formalism to the FRW metric, assuming time-dependent fermion fields only, and evaluates the resulting components of $T_{\mu\nu}$.

Experimental results

Research questions

• RQ1 How is the covariant derivative of a Dirac fermion constructed in curved spacetime using the tetrad formalism?
• RQ2 What is the explicit form of the spin connection and how does it ensure Lorentz covariance?
• RQ3 How does the commutator of covariant derivatives relate to the Riemann tensor in the context of spinor fields?
• RQ4 What is the trace of the energy-momentum tensor for a Dirac field in a cosmological (FRW) background, and what does it imply for conformal invariance?
• RQ5 Why does the energy-momentum tensor vanish for a massless Dirac field depending only on time in FRW spacetime?

Key findings

• The energy-momentum tensor for a massive Dirac field on a FRW background vanishes for spatial components ($T_{11} = T_{22} = T_{33} = 0$) and has a non-zero time-time component ($T_{00} \neq 0$), indicating a dust-like equation of state.
• For a massless Dirac field depending only on conformal time in FRW spacetime, the energy-momentum tensor vanishes on the shell ($T_{\mu\nu}|_{\text{on shell}} = 0$), consistent with conformal invariance.
• The trace of the energy-momentum tensor for the Dirac field is proportional to the mass and the field bilinear $\bar{\psi}\psi$, vanishing in the massless limit.
• The derivation confirms that the spin connection is a connection for the local Lorentz group, and its curvature yields the Riemann tensor via the commutator of covariant derivatives.
• The formalism is consistent with conformal symmetry: the trace of $T_{\mu\nu}$ vanishes when the mass is zero and the field is conformally coupled.
• The method applies to more general geometries, including non-Riemannian spaces with torsion, though such extensions require careful treatment of metricity.