that these conditions could be met if , , and are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here. Given the factorization in terms of these matrices, one can now write down immediately an equation with to be determined. Applying again the matrix operator on both sides yields Taking shows that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is Setting and because , the Dirac equation is produced as written above. Covariant form and relativistic invariance To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows: and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that ) where there is an implied summation over the values of the twice-repeated index , and is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is The complete system is summarized using the Minkowski metric on spacetime in the form where the bracket expression denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature . The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory. The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum