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Why is the Dirac operator considered so important - in both physics and (pure) mathematics?

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    $\begingroup$ define important - what would you say is the importance of the Dirac operator in pure mathematics? $\endgroup$ – lurscher Nov 14 '11 at 12:32
  • $\begingroup$ Let's be fair: it was invented within physics, for physicists, by a physicist. It may have ancillary relevance to maths, but not huge. @lurscher is also right; what's "important" here? $\endgroup$ – Noldorin Nov 14 '11 at 22:07
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It should be obvious why the Dirac operator is important in physics because of fermions. In mathematics, one could mention the following incomplete list.

  1. Atiyah-Singer Index Theorem for the (twisted) spin complex, see e.g., Nakahara, Geometry, Topology and Physics, 1990; or Berline, Getzler and Vergne, Heat Kernels and Dirac Operators, 2004.

  2. Connes's noncommutative differential geometry.

  3. Schrödinger–Lichnerowicz formula.

  4. Kostant's cubic Dirac operator.

For further information, see also nLab.

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Because it describes spinors (i.e. behavior of fermions). String Theory speaks about both bosons and fermions... when passing to a QFT, fermionic wavefunctions become spinor fields, and their dynamics require the Dirac operator. And this is where all the mathematical importance stems from: QFT and String Theory (topological field theories) is a pure math theory. In particular, I disagree completely with Noldorin's seemingly naive comment... Clifford algebras and spinors on manifolds is a huge field.

Sorry if this too brief of an explanation (for immediate purposes).

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  • $\begingroup$ Hi. Please feel free to elaborate whenever you're interested. :) I can then upvote your answer. $\endgroup$ – UGPhysics Nov 17 '11 at 18:09

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