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Massive Spin-2: Field-equations, Propagators, Massless-limit, and Perihelion Precessions

Th. A. Rijken, J. W. Wagenaar

TL;DR

The paper develops a massive spin-2 framework using an auxiliary-field (AF) formulation with an imaginary scalar ghost to achieve a smooth massless limit and circumvent the vDVZ discontinuity. Through Dirac constrained quantization and careful treatment of the auxiliary fields, it derives the spin-2 propagator and shows that, in the $b \to \infty$ limit, the theory yields a consistent massless spin-2 content while preserving correct Mercury perihelion precession. The analysis decomposes contributions from spin-2, scalar, and ghost sectors to the gravitational potential, showing that scalar-ghost cancellations restore Einsteinian predictions in the massless limit, and that finite-mass corrections are negligible within solar-system bounds. The work connects graviton mass effects to cosmological terms and provides a quantitative assessment against observational constraints, arguing that the imaginary ghost is essential for achieving the desired limits. Overall, it presents a coherent, quantized, AF-based approach to massive gravity with a physically viable massless limit and measurable—but currently tiny—deviations in planetary motion.

Abstract

This paper presents the quantization of massive and massless spin-2 particles, using the auxiliary field method. The issue, the so-called vDVZ-discontinuity, whether the perihelion precessions for a massive graviton are in agreement with the data, is studied in the context of this spin-2 theory in tree-approximation. In the context of this setting, it is found, that a massive gravitation model with an imaginary scalar ghost, for a small graviton mass is compatible with the perihelion-precession of Mercury, etc..

Massive Spin-2: Field-equations, Propagators, Massless-limit, and Perihelion Precessions

TL;DR

The paper develops a massive spin-2 framework using an auxiliary-field (AF) formulation with an imaginary scalar ghost to achieve a smooth massless limit and circumvent the vDVZ discontinuity. Through Dirac constrained quantization and careful treatment of the auxiliary fields, it derives the spin-2 propagator and shows that, in the limit, the theory yields a consistent massless spin-2 content while preserving correct Mercury perihelion precession. The analysis decomposes contributions from spin-2, scalar, and ghost sectors to the gravitational potential, showing that scalar-ghost cancellations restore Einsteinian predictions in the massless limit, and that finite-mass corrections are negligible within solar-system bounds. The work connects graviton mass effects to cosmological terms and provides a quantitative assessment against observational constraints, arguing that the imaginary ghost is essential for achieving the desired limits. Overall, it presents a coherent, quantized, AF-based approach to massive gravity with a physically viable massless limit and measurable—but currently tiny—deviations in planetary motion.

Abstract

This paper presents the quantization of massive and massless spin-2 particles, using the auxiliary field method. The issue, the so-called vDVZ-discontinuity, whether the perihelion precessions for a massive graviton are in agreement with the data, is studied in the context of this spin-2 theory in tree-approximation. In the context of this setting, it is found, that a massive gravitation model with an imaginary scalar ghost, for a small graviton mass is compatible with the perihelion-precession of Mercury, etc..

Paper Structure

This paper contains 35 sections, 295 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Massive Gravitation Field, Euler-Lagrange Equations
  3. Quantization
  4. Dirac-theory: The Hamiltonian and Constraints
  5. Field-Commutators and Spin-2 Field Propagator
  6. Massless Limit: Scalar-tensor or Imaginary-ghost Theory
  7. Elimination $\rho(x)$-field, New field Commutators
  8. Elimination $\rho(x)$-field
  9. Commutators and Propagator spin-2 for $b \rightarrow \infty$
  10. Asymptotic fields. Physical Contents $h_{\mu\nu}$-field
  11. Massive $M_2 \ne 0$ case
  12. Massless $M=M_2=0$ case
  13. Momentum-space asymptotic states
  14. Graviton-mass and the Perihelion-precession of Planets
  15. Interaction Spin-2 Particles with a Scalar-field
  16. ...and 20 more sections

Figures (7)

  • Figure 1: Three regions in $(b,\lambda)$-space. Region I, III: spin-2 and spin-0 imaginary ghost. Region II: (massive) spin-2 and spin-0.
  • Figure 2: Graviton-exchange between mass M and mass m.
  • Figure 3: Complex $p_0$-plane contour $\Gamma = R + \delta(-\omega_p) -\delta(\omega_p)$. Here, $\delta$-operator is Leray's coboundary operator.
  • Figure 4: BS-Integral Equation
  • Figure 5: One-meson and planar two-meson exchange, etc. Feynman graphs. The solid lines denote scalar heavy particles, e.g. the sun and the planet. The dashed lines refer to the scalar mesons.
  • ...and 2 more figures