Corinne A. Manogue


Dimensional Reduction and Octonions

My long-term goal in this research is to describe the fundamental symmetries of physics by exploiting the symmetries inherent in exceptional mathematical structures involving the octonions, a non-commutative, non-associative extension of the complex numbers.

My results have included:

  • a demonstration that the non-linear Virasoro constraints of bosonic string theory and the superstring equations in 10-dimensions are both simple restatements of the octonionic multiplication rule;
  • an explicit alternative to exponentiation for the description of finite Lorentz transformations which evades the twin hazards of non-commutativity and non-associativity;
  • an exploration of the eigenvectors and eigenvalues of octonionic matrices;
  • a generalization of Mobius transformations to ten-dimensions, laying the groundwork for a possible relationship between twistor theory and superstring theory;
  • a proposal to use the octonionic formalism to reduce ten spacetime dimensions to four, without the usual compactification required by superstring theory;
  • the application of this symmetry breaking to the Dirac equation resulting in a particle spectrum with the correct number of generations and spin/helicity properties to describe precisely three generations of leptons.
  • Field Theory in Curved Spacetime

    What different physical phenomena can create particles out of the vacuum? Beginning with Klein's suggestion (soon after the Dirac equation was postulated) that strong electric fields should spontaneously create particles, and revivified with Hawking's proposal that black holes should also do so, this question has always been central to the interpretation of quantum field theory in background spacetimes. Surprisingly, the key is usually a precise, unambiguous definition of the vacuum itself.

    My own contributions have included:

    While these research problems involve deep issues regarding the interpretation of quantum field theory, the calculations themselves involve simple models, carefully chosen to illuminate the important physics while minimizing mathematical complications. Research on these problems is certainly accessible to graduate students and often to undergraduates.

    Much of the work discussed here has been completed in collaboration with other scientists. Please see my publication list for details.

    If you have comments or suggestions, email me at

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    Last Update 6/17/09

    © Corinne Manogue, 2005.