[Paper Review] Canonical formulation of Pais-Ulhenbeck action and resolving the issue of branched Hamiltonian

Kaushik Sarkar, Nayem Sk+2 more|arXiv (Cornell University)|Jul 13, 2015

Black Holes and Theoretical Physics7 references4 citations

TL;DR

This paper proposes fixing the Ricci scalar (R) at the boundary in higher-order gravity theories to resolve inconsistencies in canonical formulations, particularly the loss of the Gibbons-Hawking-York (GHY) boundary term and the issue of branched Hamiltonians. By modifying Horowitz’s formalism with an auxiliary variable and enforcing R-fixing, the authors establish a one-to-one correspondence between higher-order oscillator and gravity actions, yielding a unique, non-canonical-transformable Hamiltonian that regulates unitarity issues via higher-order terms.

ABSTRACT

Shortcomings of Dirac's constrained analysis in the context of fourth order Pais-Uhlenbeck oscillator action and the appearance of badly affected phase-space Hamiltonian for a generalized fourth order oscillator action, following Ostrogradski, Dirac and Horowitz's formalism, require a viable canonical formulation. This is achieved only after fixing appropriate variables at the end points and taking care of the counter surface terms obtained from variational principle. In the process a one-to-one correspondence between different higher order theories has been established. On the other hand the issue of branched Hamiltonian appearing in the presence of velocities with degree higher than two in the Lagrangian, has not been resolved uniquely as yet. However, often such terms appear with higher order theory, gravity in particular. Here we show that canonical formulation of higher order theory takes care of the issue elegantly.

Motivation & Objective

• To resolve the inconsistency in canonical formulations of higher-order gravity, where Ostrogradski’s and Dirac’s methods fix velocity but lose the GHY boundary term.
• To address the mismatch in boundary conditions across different higher-order theories, especially regarding the role of acceleration and curvature.
• To establish a unified framework for higher-order theories by proposing that the Ricci scalar (R) — as a proxy for acceleration — be fixed at the boundary.
• To demonstrate that fixing R at the boundary is consistent with scalar-tensor equivalence and Noether symmetry, which require R to be constant.
• To develop a modified Horowitz formalism that preserves the GHY term and enables a unique, non-canonical Hamiltonian formulation.

Proposed method

• Propose fixing the Ricci scalar (R) at the boundary instead of velocity, to maintain consistency with scalar-tensor equivalence and Noether symmetry.
• Modify Horowitz’s auxiliary variable technique to accommodate R-fixing, ensuring compatibility with higher-order actions.
• Use the auxiliary variable method to construct a canonical formulation that establishes a one-to-one correspondence between higher-order oscillator and gravity actions.
• Derive the Hamiltonian using the modified formalism, showing it is not related to standard Ostrogradski or Dirac Hamiltonians via canonical transformation.
• Demonstrate that higher-order terms (e.g., R²) can regulate branched Hamiltonians by suppressing multivaluedness and restoring unitarity.
• Apply the formalism to isotropic and homogeneous cosmology, showing it leads to a well-posed Schrödinger-like equation with probabilistic interpretation.

Experimental results

Research questions

• RQ1 Why does fixing velocity at the boundary in Ostrogradski’s or Dirac’s formalism lead to the loss of the Gibbons-Hawking-York (GHY) boundary term in higher-order gravity?
• RQ2 How can a consistent canonical formulation be established for higher-order gravity that preserves the GHY term and respects scalar-tensor equivalence?
• RQ3 What is the role of the Ricci scalar (R) as a boundary condition, and why is it more physically consistent than fixing velocity or extrinsic curvature?
• RQ4 Can a modified Horowitz formalism with auxiliary variables yield a unique Hamiltonian not related by canonical transformation to standard formulations?
• RQ5 How do higher-order curvature terms (e.g., R²) regulate the issue of branched Hamiltonians and restore unitarity in quantum time evolution?

Key findings

• Fixing the Ricci scalar (R) at the boundary is necessary to preserve the Gibbons-Hawking-York (GHY) boundary term, which is physically linked to black hole entropy.
• The proposed method of fixing R at the boundary is consistent with scalar-tensor equivalence, where R must be constant in Jordan and Einstein frames.
• Noether symmetry in higher-order gravity demands R to be constant, providing a theoretical justification for fixing R at the boundary.
• The modified Horowitz formalism establishes a one-to-one correspondence between higher-order oscillator and gravity actions, yielding a unique Hamiltonian not related to standard formulations via canonical transformation.
• Higher-order terms such as R² regulate branched Hamiltonians by suppressing multivaluedness, enabling a well-defined probabilistic interpretation in the Schrödinger equation.
• The resulting canonical formulation leads to a semiclassical approximation strongly peaked around classical inflationary solutions, supporting its physical viability.