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A Dirac operator is a differential operator acting on a vector bundle $V$ over a Riemannian manifold $M$:

$$ \tag{1} D^{2}=\Delta $$

Where $\Delta$ in the Laplacian (in the Euclidean space).

An example of a Dirac operator is Feynman's Dirac operator. In covariant form:

$$ \tag{2} D=\gamma^{\mu} \partial_{\mu} $$

Now, this is the operator that appears in the Dirac Equation (in its covariant form):

$$ \tag{3} (i\gamma^{\mu}\partial_{\mu} - m)\psi = 0 $$

whose solutions $\psi$ are spinors, describing relativistic particles with spin $1/2$.

There are other Dirac operator's, for example (more examples in Wikipedia), the Laplace-Beltrami operator, which is a generalization of the Laplace operator (that is described in the Euclidean space) in the Riemannian space. Have

Question: Can I rewrite the Dirac equation in terms of a Dirac operator like this?

$$ \tag{4} (iD - m)\psi = 0 $$

Question: In that case, what kind of relativistic-quantum system describes the Dirac equation with the Laplace-Beltrami operator?

As far as I know, the Laplace-Beltrami operator can be used to rewrite the wave equation.

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The Dirac operator on an $N$-dimensional Euclidean-signature Riemann manifold $(M,g)$ is a bit more compicated than you make out. It takes the form $$ {\mathcal D}= \gamma^a e^\mu_a \left(\partial_\mu + \textstyle{\frac 12} \sigma^{bc}\,\omega_{bc\mu}\right)= \gamma^a D_a. $$ Here the ${\bf e}_a\equiv e_a^\mu\partial_\mu$ compose an orthonormal vielbein on $M$, the $\gamma^a$ are Hermitian matrices obeying $$ \{\gamma^a,\gamma^b\}= 2\delta^{ab}, $$ the $$ \sigma^{ab} = \textstyle{\frac 14} [\gamma^a,\gamma^b] $$ are the skew-Hermitian spinor generators of $\mathfrak {so}(N)$ which obey $$ [\sigma^{ab},\sigma^{cd}]= \delta^{bc} \sigma^{ad}- \delta^{ac} \sigma^{bd}+\delta^{ad} \sigma^{bc}-\delta^{bd} \sigma^{ac}, $$ and $$ D_a \equiv D_{{\bf e}_a}=e^\mu_a \left(\partial_\mu + \textstyle{\frac 12} \sigma_{bc}\,{ \omega^{bc}}_\mu\right) $$ is the covariant derivative acting on the components of a Dirac spinor. The ${ \omega^{bc}}_\mu$ are sometimes called the "spin connection" although they are just the components Levi-Civita connection in the vielbein frame.

The square of the Dirac operator is given by Lichnerowicz' identity $$ ({\mathcal D})^2= \nabla^2 - \frac 14 R, $$ where $$ \nabla^2\equiv\frac 1{\sqrt{g}} D_\mu \sqrt{g}g^{\mu\nu} D_\nu $$ is the "rough'', or "connection'' Laplacian acting on spinors so $$ D_\mu= \left(\partial_\mu + \textstyle{\frac 12} \sigma_{bc}\,{ \omega^{bc}}_\mu\right). $$ There are (not short) proofs of Lichnerowicz' result elsewhere on Stack Exchange, but I have no time to find them for you.

I think for other Laplace equation you may like is the Hodge laplacian $$ (\delta+d)^2=-\nabla^2_{\rm Hodge} $$ which involves the exterior derivative $d$ and its adjoint $\delta$ which act on differential forms. It is not a simple Laplacian but is related to it via the Bochner-Weitzenboeck identity. Like the Lichnerowicz LaplacIan, the Hodge Laplacian has a square root $$ D=d+\delta $$ whose action can be made to look like Dirac's.

Write $$ \psi_{\alpha\beta}= \sum_{p=0}^d \frac 1 {p!} A_{\mu_1\mu_2\cdots \mu_p}(x)(\gamma^{\mu_1} \ldots \gamma^{mu_p})_{\alpha\beta} $$ If we let $d+\delta$ act on the Euclidean space forms $$ \frac 1 {p!} A_{\mu_1\mu_2\cdots \mu_p}(x)dx^{\mu_1}\ldots dx^{\mu_p} $$ in the usual way, then the action of the two-spinor $\psi_{\alpha\beta}$ is $$ \psi_{\alpha\beta}\mapsto [(d+\delta)\psi]_{\alpha\beta}=(\gamma^\mu)_{\alpha\beta'} \partial_\mu \psi_{\beta'\beta}. $$ Here, in flat space only, $\beta$ acts rather as a spectator. These fermions are called Kahler-Dirac fermions, or in a natural lattice version, Kogut-Susskind staggered fermions. In the lattice version the $\beta$ indices are called "tastes" are are regarded as "fermion-doubling" lattice artifacts. I believe that Kahler-Dirac fermions were introduced by Edington --- although I have no reference for this. I do not know of anywhere that they exist in nature, but they have a large footprint in the theory literature.

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"Kahler-Dirac" fermions were first introduced by Landau and Ivanenko: Ivanenko, D., and Landau, L. (1928). Zeitschriftfiir Physik, 48, 340. See also https://lib-extopc.kek.jp/preprints/PDF/1985/8508/8508107.pdf for some history and the relation to the Dirac equation

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