• DOI:10.1007/s10773-016-3168-7
• Corpus ID: 118550515

Extending Dualities to Trialities Deepens the Foundations of Dynamics

@article{Smolin2015ExtendingDT,
  title={Extending Dualities to Trialities Deepens the Foundations of Dynamics},
  author={Lee Smolin},
  journal={International Journal of Theoretical Physics},
  year={2015},
  volume={56},
  pages={221-231}
}

• L. Smolin
• Published 4 March 2015
• Mathematics
• International Journal of Theoretical Physics

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.