Talk:Dirac operator

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This article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 09:48, 10 November 2007 (UTC)

generalized Dirac Operators

How about mentioning generalized Dirac Operators. Unfortunately I'm not 100% sure about the definition right now. It might be D is called generalized Dirac operator iff D^2 - \nabla ^* \nabla is differential operator of first order. But I'm not sure, whether \nabla has to meet requirements regarding the compatibility with the metric or the Bilinearform used to to define the Cliffordmultiplikation, used to define the Diracoperator. JanCK (talk) 01:30, 19 November 2007 (UTC)

Clifford algebra -- a better way to introduce this subject ?

At the moment the 5 example cases given seem a bit unmotivated, arbitrary and haphazard.

I'm wondering whether it might be good to present the n-dimensional Euclidean (or Pseudo-Euclidean) Clifford algebra case first,

$D=\sum e_{j}\;{\partial \over \partial x_{j}}$

so (tacitly assuming that the bases e_i and the partial derivatives commute)

$D^{2}=\sum \sum e_{i}e_{j}\;{\partial \over \partial x_{i}}{\partial \over \partial x_{j}}$

Then to say that this equates to the Laplacian (or its generalisation the d'Alembertian) if

1. the squared terms e_i² correspond to the metric
2. all the cross-terms (e_ie_j + e_je_i) identically equal zero.

These are the conditions that characterise Clifford algebra -- a family of noncommutative algebras ((explain a bit more, in particular grading: 1 scalar, vectors, bivectors etc.)

Then one could go on to present examples in four dimensions (Minkowski), three dimensions and two dimensions, showing how these relate to original operator of Dirac, the ordinary vector calculus operator del, and the Cauchy-Riemann operator; before introducing other generalisations, and Clifford analysis.

I think that might make an easier entry-point, to build up the understanding of someone who might currently be put off that we currently initially go straight in talking about a tangent bundle over a line.

Any thoughts? Jheald (talk) 18:03, 26 July 2011 (UTC)

Yes, please try to do it if you are able to. I am also a bit confused of the following issues:

• The sign convention should be fixed or at least mentioned: sometimes d^2=-\laplace, for example in the first example $-i\partial _{x}$. And one could ask: why i?
• In the second example, does it have a specific meaning that $\psi :R^{2}\to C$ and not $R^{2}\to R^{2}$ or $C\to C$?
• In the third example, it should be explained what is $\gamma ^{i}$.
• In general, the word "signature" should be somewhere in the article; concerning Clifford algebras, the most standard Dirac operator (coinciding with the physical one) is that for $\mathbb {R} ^{n,1}$. However, the square is then not laplacian, but probably the D'alembert operator -- or more general, it should be said what is "laplacian".
• In the fourth example, it should be mentioned on which space D acts. The remark about "special case of a spinor bundle" is a bit confusing. Probably the space on which it acts is either the space of Clifford-valued smooth functions on R^n, or spinor-valued smooth functions on R^n.

I hope I will have some time to improve it, but if you try to reorganize it first, Jheald, it would be good. Franp9am (talk) 16:29, 27 August 2011 (UTC)

Yup. The entire article is haphazard and sloppy. Oh well. 67.198.37.16 (talk) 22:38, 18 November 2020 (UTC)

Free Dirac operator

Maybe someone wants to mention the free dirac operator $D_{0}$

$D_{0}:=c\alpha \cdot (-i\hbar \nabla _{x})+mc^{2}\beta$

Where $\alpha =(\alpha _{1},\alpha _{2},\alpha _{3})$ are the off-diagonal Dirac matrices and $m$ the mass of an electron a. It acts on a four-component wave function $\psi (x,t)\in L_{x}^{2}(\mathbb {R} ^{3},\mathbb {C} ^{4})$. It can also be extended to a self-adjoint operator on that domain. — Preceding unsigned comment added by 129.187.111.157 (talk) 14:30, 9 September 2011 (UTC)

Done. 67.198.37.16 (talk) 22:01, 18 November 2020 (UTC)