Abstract
This chapter reviews the classification of the real and complex Clifford algebras and analyze the relationship between some particular algebras that are important in physical applications, namely the quaternion algebra \((\mathbb{R}_{0,2})\), Pauli algebra \((\mathbb{R}_{3,0})\), the spacetime algebra \((\mathbb{R}_{1,3})\), the Majorana algebra \((\mathbb{R}_{3,1})\) and the Dirac algebra \((\mathbb{R}_{4,1})\). A detailed and original theory disclosing the hidden geometrical meaning of spinors is given through the introduction of the concepts of algebraic, covariant and Dirac-Hestenes spinors. The relationship between these kinds of spinors (that carry the same mathematical information) is elucidated with special emphasis for cases of physical interest. We investigate also how to reconstruct a spinor from their so-called bilinear invariants and present Lounesto’s classification of spinors. Also, Majorana, Weyl spinors, the dotted and undotted algebraic spinors are discussed with the Clifford algebra formalism.
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Notes
- 1.
Recall that \(\mathrm{Hom}_{\mathbb{K}}(\mathbf{V,W})\) is the algebra of linear transformations of a finite dimensional vector space V over \(\mathbb{K}\) into a finite vector space W over \(\mathbb{K}\). When V=W the set End\(_{\mathbb{K}}\mathbf{V =}\mathrm{Hom}_{\mathbb{K}}(\mathbf{V,V})\) is called the set of endomorphisms of V.
- 2.
We recall that a \(\mathbb{K}\)-algebra homomorphism is a \(\mathbb{K}\)-linear map ρ such that \(\forall X,Y \in \mathcal{A},\rho (XY ) =\rho (X)\rho (Y )\).
- 3.
We recall that there are left and right modules , so we can also define right modular representations of \(\mathcal{A}\) by defining the mapping \(\mathbf{S} \times \mathcal{A}\rightarrow \mathbf{S}\), (x, a) ↦ xρ(a). This turns S in a right \(\mathcal{A}\)-module, called the right representation module.
- 4.
For a proof see [20].
- 5.
See [20].
- 6.
Once we know the algorithm for a simple Clifford algebra it is straightforward to devise an algorithm for the semi-simple Clifford algebras.
- 7.
This nomenclature comes from the fact that SO\(^{e}(1, 3) = \mathcal{L}_{+}^{\uparrow }\) is the special (proper) orthochronous Lorentz group. In this case the set is easily defined by the condition \(L_{0}^{0} \geq +1\). For the general case see [17].
- 8.
According to Definition 3.47 these ideals are algebraically equivalent. For example, \(\mathrm{e}^{{\prime}} = u\mathrm{e}u^{-1}\), with \(u = (1 + \mathbf{e}_{3})\notin \Gamma _{1,3}\).
- 9.
Elements of \(I^{{\prime}}\) are sometimes called Hestenes ideal spinors.
- 10.
The name spin frame will be reserved for a section of the spinor bundle structure \(\mathbf{P}_{\mathrm{Spin}_{1,3}^{e}}(M)\) which will be introduced in Chap. 7
- 11.
This section follows the developments given in [22].
- 12.
We reserve the notation \(\mathbb{R}_{p,q}\) for the Clifford algebra of the vector space \(\mathbb{R}^{n}\) equipped with a metric of signature (p, q), p + q = n. \(\mathcal{C}\ell(\mathbf{V},\mathbf{g})\) and \(\mathbb{R}_{p,q}\) are isomorphic, but there is no canonical isomorphism. Indeed, an isomorphism can be exhibit only after we fix an orthonormal basis of V.
- 13.
\(\mathrm{Aut}(\mathcal{C}\ell(\mathbf{V},g))\) denotes the (inner) automorphisms of \(\mathcal{C}\ell(\mathbf{V},g)\).
- 14.
We will call the elements of \(\mathcal{B}\) (in what follows) simply by orthonormal basis.
- 15.
- 16.
In Physics literature the components of J, S and K when written in terms of covariant Dirac spinors are called bilinear covariants.
- 17.
Sometimes they are also called ‘complex quaternions’. This last terminology will become obvious in a while.
- 18.
We omit in the following the term representative and call the elements of I simply by algebraic contravariant undotted spinors. However, the reader must always keep in mind that any algebraic spinor is an equivalence class, as defined and discussed in Sect. 4.6.
- 19.
The matrix representation of the elements of the ideals \(\mathbf{I,\dot{I}}\), are of course, 2 × 2 complex matrices (see, [10], for details). It happens that both columns of that matrices have the same information and the representation by column matrices is enough here for our purposes.
- 20.
The symbol adiag means the antidiagonal matrix.
- 21.
Recall that γ μ are the Dirac matrices defined by Eq. ( 3.49).
- 22.
As a suggestion for solving the above exercise the reader may consult [10].
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Rodrigues, W.A., Capelas de Oliveira, E. (2016). The Hidden Geometrical Nature of Spinors. In: The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics, vol 922. Springer, Cham. https://doi.org/10.1007/978-3-319-27637-3_3
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