[Paper Review] Faddeev-Jackiw quantization of spin-2 field

Michel Leclerc|ArXiv.org|Dec 20, 2006

Medical Imaging Techniques and Applications2 references3 citations

TL;DR

This paper applies the Faddeev-Jackiw constrained Hamiltonian formalism to the massless spin-2 field, demonstrating that unlike in electromagnetism, no gauge choice can eliminate all non-propagating components of the symmetric tensor field. The analysis reveals that the traceless-transverse tensor modes are the only physical degrees of freedom, but the constraints and gauge structure prevent a direct analogue of the Coulomb gauge, and the Hamiltonian remains dependent on non-propagating modes even after constraint reduction.

ABSTRACT

We apply the Faddeev-Jackiw method to the Hamiltonian analysis of the massless spin-two field. As expected, the reduced Hamiltonian contains only the traceless-transverse tensor, while some, but not all of the non-propagating components are determined by the constraints of the theory. In particular, it is concluded that no gauge choice can be imposed on the fields such that only the propagating modes remain in the theory, meaning that for the spin-2 field there is no direct analogue to the Coulomb gauge of electromagnetism. Implications for General Relativity are discussed.

Motivation & Objective

• To investigate whether a gauge choice exists for the spin-2 field that eliminates all non-propagating components, analogous to the Coulomb gauge in electromagnetism.
• To apply the Faddeev-Jackiw formalism to the Fierz-Pauli Lagrangian for linearized gravity and identify the physical degrees of freedom without prior gauge fixing.
• To clarify the structural differences between spin-1 (electromagnetism) and spin-2 (linearized gravity) theories in their Hamiltonian formulations, particularly regarding gauge invariance and constraint structure.
• To determine whether the Hamiltonian can be fully reduced to only the propagating modes without residual dependence on non-physical components.

Proposed method

• The Faddeev-Jackiw method is applied to the first-order form of the Fierz-Pauli Lagrangian for the spin-2 field, avoiding the introduction of unnecessary momentum variables.
• The independent fields are chosen as $ h_{0 heta}, h_{00}, h_{\mu\nu} $, with conjugate momenta derived directly from the Lagrangian, without treating them as constraints.
• The Hamiltonian is constructed by eliminating velocities using the momentum relations, resulting in a first-order action in terms of physical and auxiliary variables.
• The constraint surface is derived from the requirement that the symplectic structure is degenerate, leading to equations that determine the non-propagating components.
• The analysis shows that while the traceless-transverse part of $ h_{\mu\nu} $ is the only physical degree of freedom, the remaining components are constrained but not fully eliminated by gauge conditions.
• The gauge invariance of the Hamiltonian is examined, revealing that it is invariant only on the constraint surface, not independently of the gauge modes.

Experimental results

Research questions

• RQ1 Can a gauge condition be imposed on the spin-2 field such that only the traceless-transverse tensor modes remain, analogous to the Coulomb gauge in electromagnetism?
• RQ2 Why does the standard counting argument for propagating degrees of freedom fail to yield a complete gauge-fixing in linearized gravity?
• RQ3 How does the Faddeev-Jackiw formalism reveal the limitations of gauge fixing in spin-2 theories compared to spin-1 theories?
• RQ4 Is the Hamiltonian of the spin-2 field independent of non-propagating components after constraint reduction, or does it still depend on them?
• RQ5 What is the role of the constraint $ h^{\mu\nu}_{\ \ ,\mu,\nu} - \Delta h = 0 $ in determining the physical content of the theory?

Key findings

• The Faddeev-Jackiw method confirms that only the traceless-transverse part of the symmetric tensor $ h_{\mu\nu} $ corresponds to physical, propagating degrees of freedom.
• The reduced Hamiltonian contains only the traceless-transverse tensor, but the non-propagating components are not fully eliminated by any gauge choice.
• No gauge condition can be imposed that removes all 8 non-propagating components of the symmetric tensor field, unlike in the Maxwell theory where the Coulomb gauge eliminates the longitudinal and time-like modes.
• The constraint $ h^{\mu\nu}_{\ \ ,\mu,\nu} - \Delta h = 0 $ cannot be solved independently of the gauge degrees of freedom, preventing a strict traceless-transverse gauge.
• The Hamiltonian is not gauge-invariant in the same way as in electromagnetism; it depends on the gauge modes through the constraint surface, making it impossible to decouple them completely.
• An explicit counterexample is constructed where constraints and dynamical equations are satisfied, but the full field equations are not, due to violation of the missing equation $ \ddot{h} + \Delta h_{00} - 2\dot{h}^{0\mu}_{\ \ ,\mu} = 0 $, proving the system is underdetermined without this equation.