Spinors in the Four-Dimensional Pseudo-Euclidean Space

Zhelnorovich, Vladimir A.

doi:10.1007/978-3-030-27836-6_3

Vladimir A. Zhelnorovich²

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Abstract

Let us consider the four-dimensional pseudo-Euclidean vector space $ E_4^1 $ of index 1 referred to an orthonormal basis , i = 1, 2, 3, 4.

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Notes

1.
By virtue of definition the components of the Levi-Civita pseudotensor satisfy the following relations
$$\displaystyle \begin{gathered} \varepsilon _{pqmn}\varepsilon ^{ijks}=-\begin{vmatrix} \delta _p^i & \delta _q^i & \delta _m^i & \delta _n^i \\ \delta _p^j & \delta _q^j & \delta _m^j & \delta _n^j \\ \delta _p^k & \delta _q^k & \delta _m^k & \delta _n^k \\ \delta _p^s & \delta _q^s & \delta _m^s & \delta _n^s \end{vmatrix} ,\quad \varepsilon _{pqmn}\varepsilon ^{ijkn}=-\begin{vmatrix} \delta _p^i & \delta _q^i & \delta _m^i \\ \delta _p^j & \delta _q^j & \delta _m^j \\ \delta _p^k & \delta _q^k & \delta _m^k \end{vmatrix} , \\ \varepsilon _{pqmn}\varepsilon ^{ijmn}=-2\big( \delta _p^i\delta _q^j- \delta _p^j\delta _q^i\big) ,\quad \varepsilon _{pjmn}\varepsilon ^{ijmn}=-6\delta ^i_p,\quad \\ \varepsilon _{pqmn}\varepsilon ^{pqmn}=-24. \end{gathered} $$
2.
In physical literature the matrix of the components of the second rank spinor E ⁻¹ = ∥e ^BA∥ is usually denoted by symbol C (in the Dirac theory the matrix C is operator of the charge conjugation).
3.
Real representations of the γ-matrices exist in four-dimensional pseudo-Euclidean spaces $E_4^1$, $E_4^2$. In the space $E_4^3$ (and in $E_4^2$) there are purely imaginary representations γ-matrices, obtained from real γ-matrices in spaces $E_4^1$, $E_4^2$ by multiplication by imaginary unit $\mathrm{i} =\sqrt {-1}$. Therefore in the spaces $E_4^1$, $E_4^2$ and $E_4^3$ there are the real spinor representations of corresponding pseudo-orthogonal groups. In four-dimensional pseudo-Euclidean space $E_4^2$ with the metric signature (+, +, −, −) there are also real semispinors. In spaces $E_4^0$, $E_4^4$ do not exist real and imaginary representations for γ _i and therefore in these spaces there are no real spinor representations (see in connection with it also Chap. 1).
4.
See Appendix C.
5.
A connection between the first-rank spinors and complex tensors C in the space $E_4^1$ for the first time was considered, apparently, by Whittaker [73]. The Whittaker formulas define each component of spinor ψ ^A in terms of the tensor components C up to the sign and therefore do not carry out a one-to-one connection between the spinors and the tensors C. In subsequent papers of various authors some particular cases have been considered (real and two-components spinors in the four-dimensional and three-dimensional spaces), but the explicit formulas, that realize one-to-one connection between spinors and tensors, have not been obtained. After a number of attempts to establish such connection it was appeared opinion on not reducibility of the first-rank spinor to tensors and “elementary nature” of the spinor. The one-to-one invariant connection between spinors and various systems of tensors has been established in [74, 75] in the spaces of any dimension. Various aspects of the connection between tensors and spinors were considered, for example, in [15, 35, 57, 67–70]. A geometric illustration of two-component spinors in the four-dimensional space is given in [50].
6.
Sometimes (especially in physical literature) a four-component spinor in the space $E_4^1$ is called a bispinor, while ξ, η are called, respectively, undotted and dotted spinors in the space $E_4^1$.
7.
For two-component spinors one uses also the following connection between contravariant and covariant components of a spinor ξ ₂ = ξ ¹, ξ ₁ = −ξ ² that is related to different definition of covariant and contravariant components ε _AB, ε ^AB of the metric spinor. With such definition of the connection between ξ _A and ξ ^A in some formulas the sign changes.
8.
We use here the Pauli matrices (3.93), which differ by the factor $1/\sqrt {2} $ from the matrices used in [50]. Therefore formulas (3.124) and (3.125) differ from the corresponding Newman–Penrose formulas by the numerical factor.
9.
The orthonormal vector tetrads defined by the spinor field of the first-rank in the space $E_4^1$, were introduced by Gürsey [32], see also Takabayasi [65, 66].
10.
The quantities p ⁱ = ρπ ⁱ, q ⁱ = ρξ ⁱ, S ⁱ = ρσ ⁱ, j ⁱ = ρu ⁱ are defined by relations(3.126) and for ρ = 0. However, one-to-one connection between ψ and quantities Ω, N, p ⁱ, q ⁱ, S ⁱ, j ⁱ does not exist either. For example, for semispinors Ω = N = 0, p ⁱ = q ⁱ = 0. One nonzero isotropic vector with components S ⁱ = j ⁱ or S ⁱ = −j ⁱ remaining in this case from tetrad ρ ĕ_a, determines the semispinor components up to a factor $\exp (\mathrm{i} \varphi )$ only.
11.
The null tetrad $\breve {\boldsymbol e}^{\circ } _a$, determined by the equations analogous to Eqs. (3.156) and (3.154) for the spinor fields with given invariants ρ = 2, η = 0, was used in [31].
12.
As it was already noted (see p. 18), specifying only the tensor components C ^ij determines two spinors with components ψ and iγ ⁵ ψ. Under transformation ψ →iγ ⁵ ψ the components M ^ij pass into − M ^ij. Therefore specifying C ^ij and M ^ij completely determines the spinor ψ.

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Institute of Mechanics, Moscow State University, Moscow, Russia
Vladimir A. Zhelnorovich

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Zhelnorovich, V.A. (2019). Spinors in the Four-Dimensional Pseudo-Euclidean Space. In: Theory of Spinors and Its Application in Physics and Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-27836-6_3

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DOI: https://doi.org/10.1007/978-3-030-27836-6_3
Published: 25 October 2019
Publisher Name: Springer, Cham
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