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Why is the Dirac operator so important - in both physics and mathematics?

+ 4 like - 0 dislike
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Why is the Dirac operator considered so important - in both physics and (pure) mathematics?

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user UGPhysics
asked Nov 11, 2011 in Theoretical Physics by UGPhysics (155 points) [ no revision ]
define important - what would you say is the importance of the Dirac operator in pure mathematics?

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user lurscher
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?

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user Noldorin

2 Answers

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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.

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user Qmechanic
answered Nov 15, 2011 by Qmechanic (2,580 points) [ no revision ]
+ 2 like - 0 dislike

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).

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user Chris Gerig
answered Nov 14, 2011 by Chris Gerig (540 points) [ no revision ]
Hi. Please feel free to elaborate whenever you're interested. :) I can then upvote your answer.

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user UGPhysics

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