[Paper Review] Dual descriptions of spin two massive particles in $D=2+1$

D. Dalmazi, E. L. Mendonça|arXiv (Cornell University)|Nov 30, 2008

Cosmology and Gravitation Theories1 citations

TL;DR

This paper presents a dual formulation of massive spin-2 particles in 2+1 dimensions using a master action that interpolates between first-, second-, and third-order self-dual models. It establishes explicit duality maps and proves the absence of ghosts in the third-order model due to the triviality of the Einstein-Hilbert action in 2+1D, while also demonstrating quantum equivalence between opposite-helicity models and a generalized self-dual model with quadratic Einstein-Hilbert, Chern-Simons, and Fierz-Pauli terms, avoiding sign conflicts via a first-order Chern-Simons term.

ABSTRACT

In the first part of this work we show the decoupling (up to contact terms) of redundant degrees of freedom which appear in the covariant description of spin two massive particles in $D=2+1$. We make use of a master action which interpolates, without solving any constraints, between a first, second and third order (in derivatives) self-dual model. An explicit dual map between those models is derived. In our approach the absence of ghosts in the third order self-dual model, which corresponds to a quadratic truncation of topologically massive gravity, is due to the triviality (no particle content) of the Einstein-Hilbert action in $D=2+1$. In the second part of the work, also in $D=2+1$, we prove the quantum equivalence of the gauge invariant sector of a couple of self-dual models of opposite helicities (+2 and -2) and masses $m_+$ and $m_-$ to a generalized self-dual model which contains a quadratic Einstein-Hilbert action, a Chern-Simons term of first order and a Fierz-Pauli mass term. The use of a first order Chern-Simons term instead of a third order one avoids conflicts with the sign of the Einstein-Hilbert action.

Motivation & Objective

• To demonstrate the decoupling of redundant degrees of freedom in a covariant description of massive spin-2 particles in 2+1 dimensions.
• To construct a master action that interpolates between first-, second-, and third-order self-dual models without solving constraints.
• To establish an explicit dual map between these models and prove the absence of ghosts in the third-order model.
• To prove quantum equivalence between self-dual models of opposite helicities (+2 and -2) and a generalized self-dual model containing Einstein-Hilbert, Chern-Simons, and Fierz-Pauli terms.
• To resolve sign conflicts in the Einstein-Hilbert action by employing a first-order Chern-Simons term instead of a third-order one.

Proposed method

• Use of a master action that smoothly interpolates between first-, second-, and third-order derivative self-dual models in 2+1 dimensions.
• Derivation of an explicit dual map between the first-, second-, and third-order models via the master action.
• Leveraging the trivial particle content of the Einstein-Hilbert action in 2+1D to explain the absence of ghosts in the third-order model.
• Construction of a generalized self-dual model combining quadratic Einstein-Hilbert, first-order Chern-Simons, and Fierz-Pauli mass terms.
• Proof of quantum equivalence between the gauge-invariant sectors of opposite-helicity self-dual models and the generalized model.
• Use of a first-order Chern-Simons term to avoid inconsistencies arising from the negative sign of the Einstein-Hilbert action in 2+1D.

Experimental results

Research questions

• RQ1 How can redundant degrees of freedom in the covariant description of massive spin-2 particles in 2+1D be decoupled without solving constraints?
• RQ2 What is the explicit dual map between first-, second-, and third-order self-dual models in 2+1 dimensions?
• RQ3 Why is the third-order self-dual model ghost-free, and how does the triviality of the Einstein-Hilbert action in 2+1D contribute to this?
• RQ4 Can the gauge-invariant sectors of self-dual models with opposite helicities (+2 and -2) be quantum mechanically equivalent to a generalized self-dual model?
• RQ5 How does using a first-order Chern-Simons term resolve sign conflicts with the Einstein-Hilbert action in the context of massive spin-2 theories?

Key findings

• The master action successfully interpolates between first-, second-, and third-order self-dual models in 2+1 dimensions without requiring constraint solving.
• An explicit dual map is derived between the first-, second-, and third-order models, establishing their physical equivalence.
• The absence of ghosts in the third-order model is attributed to the trivial particle content of the Einstein-Hilbert action in 2+1 dimensions.
• The gauge-invariant sectors of self-dual models with opposite helicities (+2 and -2) are quantum equivalent to a generalized self-dual model containing quadratic Einstein-Hilbert, first-order Chern-Simons, and Fierz-Pauli mass terms.
• Using a first-order Chern-Simons term instead of a third-order one avoids sign conflicts with the Einstein-Hilbert action, ensuring consistency in the quantum formulation.
• The generalized self-dual model provides a consistent framework for massive spin-2 particles in 2+1 dimensions, unifying different derivative-order formulations.