International Journal of Theoretical Physics

, Volume 56, Issue 1, pp 221–231 | Cite as

Extending Dualities to Trialities Deepens the Foundations of Dynamics

Article
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Abstract

Dualities are often supposed to be foundational, but they may come into conflict with a strong form of background independence, which is the principle that the dynamical equations of a theory not depend on arbitrary, fixed, non-dynamical structures. This is because a hidden fixed structures is needed to define the duality transformation. Examples include a fixed, absolute notion of time, a fixed non-dynamical background geometry, or the metric of Hilbert space. We show that this conflict can be eliminated by extending a duality to a triality. This renders that fixed structure dynamical, while unifying it with the dual variables. To illustrate this, we study matrix models with a cubic action, which have a natural triality symmetry. We show how breaking this triality symmetry by imposing different compactifications, which are expansions around fixed classical solutions, yields particle mechanics, string theory and Chern-Simons theory. These result from compactifying, respectively, one, two and three dimensions. This may explain the origin of Born’s duality between position and momenta operators in quantum theory, as well as some of the the dualities of string theory.

Keywords

Unification of physics Dualities Trialities 
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Notes

Acknowledgments

These ideas first occurred to me during a bus ride in Santa Barbara in the summer of 1986, following conversations with Louis Crane, Gary Horowitz and Andy Strominger about purely cubic string field theories [9]. They inspired my work on cubic matrix models [10, 11, 12]. But my interest in them was recently reinvigorated by several provocative conversations with Laurent Freidel, about his work on incorporating relative locality into string theory [7].

This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This research was also partly supported by grants from NSERC, FQXi and the John Templeton Foundation.

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Physics AstronomyUniversity of WaterlooWaterlooCanada
  3. 3.Department of PhilosophyUniversity of TorontoTorontoCanada

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