Lagrangians of physics and the game of Fisher-information transfer

Abstract

The Lagrangians of physics arise out of a mathematical game between a ‘‘smart’’ measurer and nature (personified by a demon). Each contestant wants to maximize his level of Fisher information I. The game is zero sum, by conservation of information in the closed system. The payoff of the game introduces a variational principle—extreme physical information (EPI)—which fixes both the Lagrangian and the physical constant of each scenario. The EPI approach provides an understanding of the relationship between measurement and physical law. EPI also defines a prescription for constructing Lagrangians. The prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which I remains invariant. As an example, when applied to the smart measurement of the space-time coordinate of a particle, the symmetry used is that between position-time space and momentum-energy space. Then the unitary transformation is the Fourier one, and EPI derives the following: the equivalence of energy, momentum, and mass; the constancy of Planck’s parameter h; and the Lagrangian that implies both the Klein-Gordon equation and the Dirac equation of quantum mechanics.