defining anti-commutation relations: where is the Minkowski metric element, and the indices run over 0,1,2 and 3. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation where are the Pauli matrices, and the chiral representation: the are the same, but The slash notation is a compact notation for where is a four-vector (often it is the four-vector differential operator ). The summation over the index is implied. Dirac adjoint and the adjoint equation The Dirac adjoint of the spinor field is defined as Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the ) that one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by : where the partial derivative acts from the right on : written in the usual way in terms of a left action of the derivative, we have Klein–Gordon equation Applying to the Dirac equation gives That is, each component of the Dirac spinor field satisfies the Klein–Gordon equation. Conserved current A conserved current of the theory is Another approach to derive this expression is by variational methods, applying Noether's theorem for the global symmetry to derive the conserved current Solutions Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected. Plane-wave solutions Plane-wave solutions are those arising from an ansatz which models a particle with definite 4-momentum where For this ansatz, the Dirac equation becomes an equation for : After picking a representation for the gamma matrices , solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see here). For example, in the chiral representation for , the solution space is parametrised by a vector , with where and is the Hermitian matrix square-root. These plane-wave solutions provide a starting point for canonical quantization. Lagrangian formulation Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by: If one varies this with respect to one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to one gets the Dirac equation.