[Paper Review] Particle with spin S=3/2 in Riemannian space-time

В. М. Редьков|arXiv (Cornell University)|Sep 18, 2011

Relativity and Gravitational Theory24 references3 citations

TL;DR

This paper investigates the Rarita-Schwinger field for spin-3/2 particles in Riemannian space-time, deriving a generalized wave equation with curvature-coupled covariant derivatives and spinor connections. It identifies a gauge-like symmetry for massless spin-3/2 fields when the Ricci tensor satisfies $ R_{\alpha\beta} - \frac{1}{2}Rg_{\alpha\beta} = 0 $, which ensures consistency and removes unphysical degrees of freedom in such regions.

ABSTRACT

Equations for 16-component vector-bispinor field, originated from Rarita-Schwinger Lagrangian for spin 3/2 field extended to Riemannian space-time are investigated. Additional general covariant constrains for the field are produced, which for some space-time models greatly simplify original wave equation. Peculiarities in description of the massless spin 3/2 field are specified. In the flat Minkowski space for massless case there exist gauge invariance of the main wave equation, which reduces to possibility to produce a whole class of trivial solutions in the the form of 4-gradient of arbitrary (gauge) bispinor function, Ψ^{0}_{c} = \partial_{c} ψ. Generalization of that property for Riemannian model is performed; it is shown that in general covariant case solutions of the gradient type Ψ^{0}_β = ( abla_β + Γ_β)Ψexist in space-time regions where the Ricci tensor obeys an identity R_{αβ} - {1 \over 2} R g_{αβ} = 0.

Motivation & Objective

• To extend the Rarita-Schwinger formalism for spin-3/2 fields to general Riemannian space-time with curvature and torsion.
• To derive a consistent set of field equations for massive and massless spin-3/2 particles in a generally covariant framework.
• To identify conditions under which the massless spin-3/2 field exhibits gauge invariance, ensuring physical consistency.
• To simplify the original 16-component field equations using generalized constraints in specific space-time models.

Proposed method

• Formulates a generally covariant Lagrangian for a 16-component vector-bispinor field using curved-space Dirac matrices and spin connections.
• Derives the field equations from the Rarita-Schwinger Lagrangian via variational principles, incorporating covariant derivatives $ D_{\alpha} = \nabla_{\alpha} + \Gamma_{\alpha} - ieA_{\alpha} $.
• Applies a spinor representation of Dirac matrices and uses identities involving $ \gamma^\alpha(x) $, $ \Gamma_\alpha(x) $, and the spin connection to simplify the equations.
• Performs a similarity transformation on the field using $ S^\beta_\alpha = \delta^\beta_\alpha - \frac{1}{3}\gamma_\alpha(x)\gamma^\beta(x) $ to diagonalize the equation structure.
• Transforms the wave equation into a form involving the Levi-Civita tensor and the dual gamma matrix $ \gamma^5 $, yielding $ \gamma^5 \epsilon^{\nu\sigma\mu}_\rho \gamma_\mu (iD_\nu - \frac{mc}{2\hbar}\gamma_\nu) \tilde{\Psi}_\sigma = 0 $.
• Analyzes the existence of gradient-type solutions $ \tilde{\Psi}^0_\beta = (\nabla_\beta + \Gamma_\beta)\Psi $ in the massless limit, leading to a curvature condition on the Ricci tensor.

Experimental results

Research questions

• RQ1 Under what conditions does the massless spin-3/2 field in Riemannian space-time admit gauge-like solutions of the form $ \tilde{\Psi}^0_\beta = (\nabla_\beta + \Gamma_\beta)\Psi $?
• RQ2 How does the Rarita-Schwinger Lagrangian generalize to curved space-time while preserving consistency and eliminating unphysical degrees of freedom?
• RQ3 What role does the Ricci tensor play in determining the existence of physical, gauge-invariant solutions for the massless spin-3/2 field?
• RQ4 Can the 16-component vector-bispinor field be simplified via a similarity transformation to reveal a more transparent dynamical structure?
• RQ5 What is the geometric condition on the space-time curvature that ensures the massless spin-3/2 field is free from pathological degrees of freedom?

Key findings

• The massless spin-3/2 field admits solutions of the form $ \tilde{\Psi}^0_\beta = (\nabla_\beta + \Gamma_\beta)\Psi $ if the Ricci tensor satisfies $ R_{\alpha\beta} - \frac{1}{2}Rg_{\alpha\beta} = 0 $.
• This condition ensures that the commutator $ [D_\nu, D_\sigma] $ vanishes when acting on scalar bispinors, which is necessary for the gradient solution to satisfy the wave equation.
• The resulting condition $ R_{\alpha\beta} - \frac{1}{2}Rg_{\alpha\beta} = 0 $ is equivalent to the vanishing of the Einstein tensor, indicating that such solutions exist in vacuum-like regions of space-time.
• In the massive case, the field equations are derived from a generalized Rarita-Schwinger Lagrangian with curvature and gauge coupling, leading to a 16-component system with nontrivial coupling between spinor and vector indices.
• The transformation $ S^\beta_\alpha = \delta^\beta_\alpha - \frac{1}{3}\gamma_\alpha(x)\gamma^\beta(x) $ simplifies the field equations into a form involving the Levi-Civita tensor and dual gamma matrices.
• The final form of the wave equation for the massless case is $ i\gamma^5 \epsilon^{\nu\sigma\mu}_\rho \gamma_\mu (\nabla_\nu + \Gamma_\nu) \tilde{\Psi}_\sigma = 0 $, which exhibits manifest gauge-like structure under the derived condition.