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In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks, for which parity is a symmetry, and is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It accounted for the fine details of the hydrogen spectrum in a completely rigorous way. 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; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, 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. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles. == Mathematical formulation == The Dirac equation in the form originally proposed by Dirac is: where is the wave function for the electron of rest mass with spacetime coordinates . The are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, is the speed of light, and is the Planck constant divided by . These fundamental physical constants reflect special relativity and quantum mechanics, respectively. Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have bearing on the problem of atomic spectra. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts 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 new elements in this equation are the 4 × 4 matrices and , and the four-component wave function . There are four components in because evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion). The 4 × 4 matrices and are all Hermitian and have squares equal to the identity matrix: : and they all mutually anticommute (if and are distinct): : : The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices, and the form of the wave function, have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th century work of the German mathematician Hermann Grassmann in his ''Lineale Ausdehnungslehre'' (''Theory of Linear Extensions''). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dirac equation」の詳細全文を読む スポンサード リンク
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