[2] The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later.
It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin.
Dirac did not fully appreciate the importance of his results; however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics.
[citation needed] Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, thus allowing the atom to be treated in a manner consistent with relativity.
Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity—which were based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus—had failed, and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem.
Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics.
The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford.
The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy.
which says that the length of this four-vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects,
Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion.
As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator (see also half derivative) thus:
Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B, C and D are matrices, with the implication that the wave function has multiple components.
The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.
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.
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.
Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector.
For the operator γμ∂μ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index.
This matrix is given the special symbol γ5, owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors.
It takes a leading role when questions of parity arise because the volume element as a directed magnitude changes sign under a space-time reflection.
Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector Aμ can be defined as
On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form:
This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra.
The separation, done here, of the Dirac spinor into large and small components depends on the low-energy approximation being valid.
The components that were neglected above, to show that the Pauli theory can be recovered by a low-velocity approximation of Dirac's equation, are necessary to produce new phenomena observed in the relativistic regime – among them antimatter, and the creation and annihilation of particles.
According to the postulates of quantum mechanics, such quantities are defined by self-adjoint operators that act on the Hilbert space of possible states of a system.
Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory.
To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied.
Alternatively the four coupled linear first-order partial differential equations for the four quantities that make up the wave function can be written as a vector.
Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.
which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-helicity spinors, where the coupling strength is proportional to the mass: