Abstract
Let us consider the four-dimensional pseudo-Euclidean Minkowski space with the metric signature (+, +, +, ā) referred to an Cartesian coordinate system with the variables x
i and holonomic vector basis 
(iā=ā1, 2, 3, 4). The contravariant and covariant components of the metric tensor of the Minkowski space calculated in the coordinate system x
i are defined by the matrix.
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Notes
- 1.
- 2.
For the spinor equations of type (5.18), describing fields with half-integral spin (the Dirac equations, the Heisenberg equation, etc.), function \(\varLambda = \overset {\circ }\varLambda +f (\psi , \psi ^+) \) with the corresponding choice of an algebraic function f, is the Lagrangian. Formula (5.61) shows that the Lagrangian, describing fields of the half-integer spin, within the framework of known classical theories is represented in the form of sum Īā=āĪ 1ā+āĪ 2ā+āĪ 3ā+āĪ int, where
$$\displaystyle \begin{gathered} \varLambda _1=-\alpha S^i\partial _i\eta + \frac 12\alpha \big(-S_iS^i+m_1^2\eta ^2\big), \\ \varLambda _2=-\frac 12\alpha \mu ^{ ij}\partial _iu_j + \frac 14\alpha \bigg (-\frac 12\mu _{ ij}\mu ^{ij} +m_2^2u_iu^i\bigg), \\ \varLambda _3=-\frac 14\alpha \big(\dot Z^{ ij}\partial _iZ_j + Z^{ ij}\partial _i\dot Z_j\big) + \frac 14\alpha \bigg (-\frac 12\dot Z_{ij} Z^{ij} +m_3^2\dot Z_iZ^i\bigg). \end{gathered} $$Here m 1, m 2, m 3 are constants. Functions Ī 1, Ī 2, Ī 3 with arbitrary quantities S i, u s, Ī¼ js, Ī·, Z s, Z js are the Lagrangians for the Proca equations describing, respectively, a neutral field of the spin 0, a neutral field of the spin 1 and a charged field of the spin 1. Function Ī int does not depend on derivatives of tensor fields. A possible physical interpretation of this equality is considered in [76, 80].
- 3.
- 4.
Tensor equations in the components of the complex tensor C ij, equivalent to the Weyl equations, are obtained in [74, 82], see also [50, p. 221]. In order to avoid misunderstanding we recall that the spinor in the Weyl and Dirac equations is considered as the geometric object in the Minkowski space and its components are defined up to a common sign.
- 5.
- 6.
For the Dirac equations the components T ij define the Einstein energy-momentum tensor.
- 7.
- 8.
The field of the spinor Ī¾ is used here only to determine the tetrad \(\breve {\boldsymbol e}^{\circ }_a\) and is not related to the Weyl equation (5.129).
- 9.
The spinor equations of another form from which the Frenet-Serret equations also follow, were considered in [68].
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Zhelnorovich, V.A. (2019). Tensor Forms of Differential Spinor Equations. In: Theory of Spinors and Its Application in Physics and Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-27836-6_5
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