[Paper Review] Landau meets Newton: time translation symmetry breaking in classical mechanics

Liu Zhao, Wei Xu+1 more|arXiv (Cornell University)|Aug 29, 2012

Advanced Thermodynamics and Statistical Mechanics6 references3 citations

TL;DR

This paper proposes a nonstandard Hamiltonian formulation in classical mechanics where the Hamiltonian is the square of the canonical Hamiltonian, enabling spontaneous time translation symmetry breaking when the potential becomes negative. It establishes a direct analogy to Landau's theory of second-order phase transitions and reformulates the ΛCDM cosmological model as a ground state with broken time translation symmetry.

ABSTRACT

Every classical Newtonian mechanical system can be equipped with a nonstandard Hamiltonian structure, in which the Hamiltonian is the square of the canonical Hamiltonian up to a constant shift, and the Poisson bracket is nonlinear. In such a formalism, time translation symmetry can be spontaneously broken, provided the potential function becomes negative. A nice analogy between time translation symmetry breaking and the Landau theory of second order phase transitions is established, together with several example cases illustrating time translation breaking ground states. In particular, the $Λ$CDM model of FRW cosmology is reformulated as the time translation symmetry breaking ground states.

Motivation & Objective

• To demonstrate that time translation symmetry breaking can occur in classical Newtonian systems through a nonstandard Hamiltonian structure.
• To establish a formal analogy between time translation symmetry breaking and Landau's theory of second-order phase transitions.
• To reformulate the ΛCDM model of cosmology as a ground state with broken time translation symmetry using an upside-down harmonic potential.
• To explore the dynamical emergence of time translation symmetry breaking in classical systems with negative potentials.
• To provide a classical mechanical framework that may inform quantum gravity and cosmological models, particularly in relation to the Wheeler-DeWitt equation.

Proposed method

• Introduces a nonstandard Hamiltonian $ H_2 = (H_1)^2 + E_0 $, where $ H_1 $ is the canonical Hamiltonian, to describe the same Newtonian dynamics as the standard formulation.
• Employs a nonlinear Poisson bracket $ \{x,v\}_2 = \frac{1}{v^2 + 2U(x)} $, distinct from the canonical $ \{x,v\}_1 = 1 $, to maintain consistency with the modified Hamiltonian.
• Applies Landau-type free energy minimization to the squared Hamiltonian $ H_2 $, treating the potential's sign as an order parameter analogous to temperature in phase transitions.
• Analyzes the ground state condition $ H_2 = 0 $ for systems with negative potentials, showing that this corresponds to a time-translation-symmetry-breaking phase.
• Reformulates the Friedmann-Robertson-Walker (FRW) cosmological model with $ \Lambda $ and spatial curvature $ k $ as a mechanical system with an upside-down harmonic potential.
• Demonstrates that the $ \Lambda $CDM model emerges as a ground state with $ H_2 = 0 $, where $ k \leq 0 $ is required for consistency.

Experimental results

Research questions

• RQ1 Can time translation symmetry be spontaneously broken in classical Newtonian systems through a nonstandard Hamiltonian formulation?
• RQ2 How does the analogy between time translation symmetry breaking and Landau's second-order phase transition theory manifest in classical mechanics?
• RQ3 What conditions allow for the dynamical emergence of time translation symmetry breaking in classical systems?
• RQ4 Can the $ \Lambda $CDM cosmological model be consistently described as a ground state with broken time translation symmetry in a classical mechanical framework?
• RQ5 What is the role of the nonlinear Poisson bracket and squared Hamiltonian in enabling time translation symmetry breaking?

Key findings

• Time translation symmetry breaking occurs in classical systems when the potential function becomes negative, provided the system is described by the nonstandard Hamiltonian $ H_2 = (H_1)^2 + E_0 $ and nonlinear Poisson bracket.
• The ground state of the system corresponds to $ H_2 = 0 $, which is achieved when the canonical Hamiltonian $ H_1 $ vanishes, leading to a time-translation-symmetry-breaking phase.
• The $ \Lambda $CDM model of FRW cosmology is reformulated as a mechanical system with an upside-down harmonic potential, where the ground state condition $ H_2 = 0 $ corresponds to the de Sitter universe with $ \Lambda > 0 $.
• The spatial curvature $ k $ must be non-positive (preferably zero) for the ground state to exist, ensuring consistency with the $ H_2 = 0 $ condition.
• The nonstandard Hamiltonian formulation is more natural than the canonical one for cosmological models, as it aligns with the Wheeler-DeWitt equation and general relativistic Hamiltonian constraints.
• The mechanism of time translation symmetry breaking is dynamically accessible: initial conditions may preserve symmetry, but the system can evolve into a symmetry-broken phase.