Ion takes a compact type and its physical which means becomes ambiguous. In this paper, by implies of Clifford algebra, we split the spinor connection into geometrical and dynamical parts (, ), respectively . This kind of connection is determined by metric, independent of Dirac matrices. Only in this representation, we are able to clearly define classical ideas including coordinate, speed, momentum and spin for a spinor, and after that derive the classical mechanics in detail. 1 only corresponds to the geometrical calculations, but 3 results in dynamical effects. couples with the spin Sof a spinor, which gives location and navigation functions for any spinor with small energy. This term is also connected together with the origin with the magnetic field of a celestial physique . So this form of connection is beneficial in understanding the subtle relation involving spinor and space-time. The classical theory for any spinor moving in gravitational field is firstly studied by Mathisson , then developed by Goralatide web Papapetrou  and Dixon . A detailed deriva-Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Copyright: 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is definitely an open access short article distributed below the terms and situations of your Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ four.0/).Symmetry 2021, 13, 1931. https://doi.org/10.3390/symhttps://www.mdpi.com/journal/symmetrySymmetry 2021, 13,two oftion is often identified in . By the PK 11195 Inhibitor commutator with the covariant derivative with the spinor [ , ], we acquire an additional approximate acceleration on the spinor as follows a ( x ) = – h R ( x )u ( x )S ( x ), 4m (1)where R may be the Riemann curvature, u 4-vector speed and S the half commutator in the Dirac matrices. Equation (1) results in the violation of Einstein’s equivalence principle. This challenge was discussed by quite a few authors . In , the precise Cini ouschek transformation and the ultra-relativistic limit in the fermion theory were derived, however the FoldyWouthuysen transformation just isn’t uniquely defined. The following calculations also show that the usual covariant derivative involves cross terms, which is not parallel towards the speed uof the spinor. To study the coupling impact of spinor and space-time, we have to have the energy-momentum tensor (EMT) of spinor in curved space-time. The interaction of spinor and gravity is thought of by H. Weyl as early as in 1929 . You can find some approaches for the general expression of EMT of spinors in curved space-time [4,8,25,26]; on the other hand, the formalisms are often fairly difficult for practical calculation and unique from each other. In [6,11], the space-time is generally Friedmann emaitre obertson alker type with diagonal metric. The energy-momentum tensor Tof spinors might be straight derived from Lagrangian on the spinor field in this case. In [4,25], in line with the Pauli’s theorem = 1 g [ , M ], 2 (two)where M is a traceless matrix related to the frame transformation, the EMT for Dirac spinor was derived as follows, T = 1 two (i i) ,(three)where = is the Dirac conjugation, will be the usual covariant derivatives for spinor. A detailed calculation for variation of action was performed in , along with the final results were a bit unique from (two) and (3). The following calculation shows that, M is still related with g, and delivers nonzero contribution to T normally situations. The precise kind of EMT is a lot more.