Υ μ ∈ Λ 1 only corresponds to the geometrical calculations, but Ω μ ∈ Λ 3 leads to dynamical effects. Only in this representation, we can clearly define classical concepts such as coordinate, speed, momentum and spin for a spinor, and then derive the classical mechanics in detail. This form of connection is determined by metric, independent of Dirac matrices. In this paper, by means of Clifford algebra, we split the spinor connection into geometrical and dynamical parts ( Υ μ, Ω μ ), respectively. In the previous works, we usually used spinor covariant derivative directly, in which the spinor connection takes a compact form and its physical meaning becomes ambiguous. The spinor field is used to explain the accelerating expansion of the universe and dark matter and dark energy. The spinor connection has been constructed and researched in many works. The interaction between spinors and gravity is the most complicated and subtle interaction in the universe, which involves the basic problem of a unified quantum theory and general relativity. The Dirac equation for spinor is a magic equation, which includes many secrets of nature. From this paper we find that Clifford algebra has irreplaceable advantages in the study of geometry and physics. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table S a b μ ν, which plays an important role in the interaction between spinor and gravity. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. This uncertainty increases the difficulty of deriving rigorous expression. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields however, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton’s second law of particles. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. To split the spinor connection into the Keller connection Υ μ ∈ Λ 1 and the pseudo-vector potential Ω μ ∈ Λ 3 not only makes the calculation simpler, but also highlights their different physical meanings. Presented is a derivation of the the Einstein equation and then the DiracĮquation in curved space.By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of the Dirac equation, the energy-momentum tensor of spinors and so on. Vierbein field theory is the most natural way to represent a relativistic Toĭate, one of the most important applications of the vierbein representation isįor the derivation of the correction to a 4-spinor quantum field transported inĬurved space, yielding the correct form of the covariant derivative. Vierbein fields to take the place of the conventional affine connection. Einstein discovered the spin connection in terms of the Proportional to the vierbein field as it would be if gravity were strictly a Theory-the correction to the derivative (the covariant derivative) is not Gauge field but not exactly like the vector potential field does in Yang-Mills Einstein's vierbein theory isĪ gauge field theory for gravity the vierbein field playing the role of a It is based on the vierbein field takenĪs the "square root" of the metric tensor field. Download a PDF of the paper titled Einstein's vierbein field theory of curved space, by Jeffrey Yepez Download PDF Abstract: General Relativity theory is reviewed following the vierbein field theoryĪpproach proposed in 1928 by Einstein.
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