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

+ 4 like - 0 dislike
2060 views

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
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

+ 3 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user Qmechanic
answered Nov 14, 2011 by (3,120 points)
+ 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 (590 points)
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification