In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Hamilton and in 1928 by Paul Dirac. The question which concerned Dirac was to factorise formally the Laplace operator of the Minkowski space, to get an equation for the wave function which would be compatible with special relativity.

In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If

where ∆ is the (positive, or geometric) Laplacian of V, then D is called a Dirac operator.

Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian, which could be characterized in ${\displaystyle \mathbb {R} ^{n}}$ as ${\displaystyle \Delta =\nabla ^{2}=\sum _{j=1}^{n}{\Big (}{\frac {\partial }{\partial x_{j}}}{\Big )}^{2}}$ (which is negative-definite, in the sense that ${\displaystyle \int _{\mathbb {R} ^{n}}{\overline {\varphi (x)}}\Delta \varphi (x)\,dx=-\int _{\mathbb {R} ^{n}}|\nabla \varphi (x)|^{2}\,dx<0}$ for any smooth compactly supported function ${\displaystyle \varphi (x)}$ which is not identically zero), and the "geometric", positive-definite Laplacian defined by ${\displaystyle \Delta =-\nabla ^{2}=-\sum _{j=1}^{n}{\Big (}{\frac {\partial }{\partial x_{j}}}{\Big )}^{2}}$.

W.R. Hamilton defined "the square root of the Laplacian" in 1847 in his series of articles about quaternions:

<...> if we introduce a new characteristic of operation, ${\displaystyle \triangleleft }$, defined with relation to these three symbols ${\displaystyle ijk,}$ and to the known operation of partial differentiation, performed with respect to three independent but real variables ${\displaystyle xyz,}$ as follows: $${\displaystyle \triangleleft ={\frac {i\mathrm {d} }{\mathrm {d} x}}+{\frac {j\mathrm {d} }{\mathrm {d} y}}+{\frac {k\mathrm {d} }{\mathrm {d} z}};}$$ this new characteristic ${\displaystyle \triangleleft }$ will have the negative of its symbolic square expressed by the following formula : $${\displaystyle -\triangleleft ^{2}={\Big (}{\frac {\mathrm {d} }{\mathrm {d} x}}{\Big )}^{2}+{\Big (}{\frac {\mathrm {d} }{\mathrm {d} y}}{\Big )}^{2}+{\Big (}{\frac {\mathrm {d} }{\mathrm {d} z}}{\Big )}^{2};}$$ of which it is clear that the applications to analytical physics must be extensive in a high degree.

D = −i ∂_x is a Dirac operator on the tangent bundle over a line.

Consider a simple bundle of notable importance in physics: the configuration space of a particle with spin ⁠1/2⁠ confined to a plane, which is also the base manifold. It is represented by a wavefunction ψ  : R² → C²