Single-Valued Hamiltonian via Legendre-Fenchel Transformation and Time Translation Symmetry

Abstract

Under conventional Legendre transformation, systems with a non-convex Lagrangian will result in a multi-valued Hamiltonian as a function of conjugate momentum. This causes problems such as non-unitary time evolution of quantum state and non-determined motion of classical particles, and is physically unacceptable. In this work, we propose a new construction of single-valued Hamiltonian by applying Legendre-Fenchel transformation, which is a mathematically rigorous generalization of conventional Legendre transformation, valid for non-convex Lagrangian systems, but not yet widely known to the physics community. With the new single-valued Hamiltonian, we study spontaneous breaking of time translation symmetry and derive its vacuum state. Applications to theories of cosmology and gravitation are discussed.

Journal Version, 16pp. All results + conclusions un-changed, only minor refinements to clarify the importance of our new LFT method and its physics applications; references added

References in corpus (4)

• Generalized phase transitions in Lovelock gravity
• Models of Topology Change
• Hamiltonian description of singular Lagrangian systems with spontaneously broken time translation symmetry
• Canonical structure of higher derivative theories

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