[Paper Review] Remarks on "Note about Hamiltonian formalism of healthy extended Hořava-Lifshitz gravity" by J. Klusoň

N. Kiriushcheva, S. V. Kuzmin+1 more|arXiv (Cornell University)|Dec 29, 2011

Black Holes and Theoretical Physics13 references3 citations

TL;DR

This paper challenges the claim that the healthy extended Hoýrava-Lifshitz gravity has a well-behaved Hamiltonian formulation. It demonstrates that the key assumption—solving a differential second-class constraint algebraically—fails in practice, leading to inconsistencies: the reduced Hamiltonian lacks quadratic momentum terms, preventing inverse Legendre transformation to recover the original Lagrangian. This indicates deep pathologies in the model, suggesting the 'cure' for earlier problems is worse than the original illness.

ABSTRACT

We reassess the conclusion by Klusoň (J. High Energy Phys. 1007 (2010) 038) that the Hamiltonian formulation of the healthy extended Hořava-Lifshitz gravity does not present any problem.

Motivation & Objective

• To re-express and critically reassess the Hamiltonian formulation of the healthy extended Hoýrava-Lifshitz gravity model as claimed in Klusoñ's work.
• To investigate whether the assumption of solving a differential second-class constraint algebraically is valid and physically consistent.
• To determine whether the resulting reduced Hamiltonian can be consistently linked back to the original Lagrangian via inverse Legendre transformation.
• To evaluate whether the model's 'healthiness' in the Hamiltonian framework is illusory due to overlooked dynamical inconsistencies.

Proposed method

• Re-derive the Hamiltonian formulation of the healthy extended Hoýrava-Lifshitz action using standard constrained dynamics techniques.
• Identify and analyze the time evolution of primary and secondary constraints, particularly focusing on the secondary constraint Θ₂ derived from p_N.
• Examine the solvability of the second-class constraint pair (p_N, Θ₂), which involves a non-algebraic differential equation for the lapse function N.
• Compare the Hamiltonian reduction process with standard field-theory examples (e.g. Maxwell, Yang-Mills, affine GR) where second-class constraints are algebraic and solvable.
• Perform inverse Legendre transformation on the reduced Hamiltonian to test consistency with the original Lagrangian action.
• Demonstrate that the absence of quadratic momentum terms in the reduced Hamiltonian prevents recovery of the original Lagrangian, indicating a fundamental inconsistency.

Experimental results

Research questions

• RQ1 Can the second-class constraint pair (p_N, Θ₂) in the healthy extended Hoýrava-Lifshitz model be consistently solved as an algebraic equation, as assumed in Klusoñ's analysis?
• RQ2 Does the reduced Hamiltonian derived from the constraint system allow for a consistent inverse Legendre transformation to recover the original Lagrangian?
• RQ3 Are the pathologies in the Hamiltonian formulation of the healthy extension a result of non-algebraic constraints, and how do they differ from standard field theories with algebraic second-class constraints?
• RQ4 Is the claim of 'healthiness' in the Hamiltonian formulation of the extended model justified, or does it mask deeper dynamical inconsistencies?
• RQ5 Does the failure to recover the Lagrangian from the reduced Hamiltonian indicate an intrinsic flaw in the model's canonical structure?

Key findings

• The secondary constraint Θ₂, derived from the time evolution of p_N, is a non-algebraic differential equation involving the lapse function N, not an algebraic constraint.
• The assumption that this differential constraint can be solved to eliminate N and p_N—treated as a second-class pair—fails to produce a consistent Hamiltonian reduction.
• The resulting reduced Hamiltonian is linear in constraints and lacks quadratic momentum terms, making it impossible to recover the original Lagrangian via inverse Legendre transformation.
• The absence of quadratic momentum terms in the reduced Hamiltonian contradicts the expected structure of a Lagrangian that is quadratic in velocities, indicating a fundamental inconsistency.
• The model's 'healthiness' in the Hamiltonian formulation is not supported by consistency checks, as the reduction process breaks the link between Lagrangian and Hamiltonian formulations.
• The paper concludes that the healthy extension introduces more severe pathologies than the original Hoýrava model, suggesting the cure is worse than the disease.