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arXiv · 1310.3769

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

Huan-Hang Chi·Hong-Jian He
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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.

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Huan-Hang Chi, Hong-Jian He. 2014-06-02. Single-Valued Hamiltonian via Legendre-Fenchel Transformation and Time Translation Symmetry. https://doi.org/10.1016/j.nuclphysb.2014.05.017

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