Abstract

In the conventional formalism of physics, with one time, systems with different Hamiltonians or Lagrangians have different physical interpretations and are considered to be independent systems unrelated to each other. However, in this paper we construct explicitly canonical maps in one-time (1T) phase space (including timelike components, specifically the Hamiltonian) to show that it is appropriate to regard various 1T physics systems, with different Lagrangians or Hamiltonians, as being duals of each other. This concept is similar in spirit to dualities discovered in more complicated examples in field theory or string theory. Our approach makes it evident that such generalized dualities are widespread. This suggests that, as a general phenomenon, there are hidden relations and hidden symmetries that conventional 1T physics does not capture, implying the existence of a more unified formulation of physics that naturally supplies the hidden information. In fact, we show that two-time (2T) physics in (d + 2) dimensions is the generator of these dualities in 1T physics in d dimensions by providing a holographic perspective that unifies all the dual 1T systems into one. The unifying ingredient is a gauge symmetry in phase space. Via such dualities it is then possible to gain new insights toward new physical predictions not suspected before, and suggest new methods of computation that yield results not obtained before. As an illustration, we will provide concrete examples of 1T systems in classical mechanics that are solved analytically for the first time via our dualities. These dualities in classical mechanics have counterparts in quantum mechanics and field theory, and in some simpler cases they have already been constructed in field theory. We comment on the impact of our approach on the meaning of space-time and on the development of new computational methods based on dualities.