Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator in the setting of pure mathematics:


* Which underlying structure (metrics, bundles, etc.) a manifold have in order to be able to define a Dirac?
* What are the domain a range of the Dirac operator (perhaps the set of sections of a certain bundle?)
* Is there an intuitive explanation of what the Dirac operator *does*? (say, along the lines of "the Laplacian represents diffusion, as in the heat equation $\partial\_tu=\Delta u$).


I will be grateful for any suggestions.



Exact Definition of Dirac Operator
I would think that these [these notes](http://math.uchicago.edu/~amathew/dirac.pdf) by Akhil Mathew provide the "exact definition" you are asking for:


![](https://ilorentz.org/beenakker/MO/DiracOperator.png)




---


In response to the follow-up question "which is the first Clifford module used in the physics context":


Two different representations ([modules](http://en.wikipedia.org/wiki/Clifford_module)) of the Clifford algebra were studied in early work on the Dirac equation. Paul Dirac himself used a quaternionic representation, while Ettore Majorana used a real representation. 


The physics implications were entirely different: Dirac had a complex field equation, and concluded that the field and its complex conjugate described two different particles. One was the electron, the complex conjugate was unknown at the time. Dirac hypothesised that it described a positively charged "antiparticle", with the same mass as the electron. This "positron" was discovered shortly afterwards, a triumph of mathematical physics.


Majorana, in contrast, had a real wave equation and hypothesised that it would describe charge-neutral particles that were their own antiparticle. We still do not know whether such particles exist in nature (the neutrino may or may not be of this type).


*Historical note:* the real representation of the Dirac equation is called the [Majorana equation,](http://en.wikipedia.org/wiki/Majorana_equation) but this was actually a rediscovery: [Eddington](http://rspa.royalsocietypublishing.org/content/121/788/524) had published it a decade before Majorana, so "Eddington-Majorana equation" would be a more appropriate name.


