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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3.1 Notes on the Representation Theory of Associative Algebras
To achieve our goal mentioned in Chap. 1 of disclosing the real secret geometrical meaning of Dirac spinors, we shall need to briefly recall some few results of the theory of representations of associative algebras. Propositions are presented without proofs and the interested reader may consult [3, 8, 12, 16, 20, 21] for details.
Let V be a finite dimensional linear space over \(\mathbb{K}\) (a division ring). Suppose that \(\dim _{\mathbb{K}}\mathbf{V} = n\), where \(n \in \mathbb{Z}\). We are interested in what follows in the cases where \(\mathbb{K} = \mathbb{R}, \mathbb{C}\) or \(\mathbb{H}\). In this case we also call V a vector space over \(\mathbb{K}\). When \(\mathbb{K} = \mathbb{H}\) it is necessary to distinguish between right or left \(\mathbb{H}\)-linear spaces and in this case V will be called a right or left \(\mathbb{H}\)-module. Recall that \(\mathbb{H}\) is a division ring (sometimes called a noncommutative field or a skew field) and since \(\mathbb{H}\) has a natural vector space structure over the real field, then \(\mathbb{H}\) is also a division algebra.
Definition 3.1
Let V be a vector space over \(\mathbb{R}\) and \(\dim _{\mathbb{R}}\mathbf{V} = 2m = n\). A linear mapping
such that
s called a complex structure mapping.
Definition 3.2
Let V be as in the previous definition. The pair \(\left (\mathbf{V,J}\right )\) is called a complex vector space structure and denote by \(\mathbf{V}_{\mathbb{C}}\) if the following product holds. Let \(\mathbb{C} \ni z = a +\mathrm{ i}b\) (\(\mathrm{i} = \sqrt{-1}\)) and let v∈V. Then
It is obvious that \(\dim _{\mathbb{C}} = \frac{m} {2}\).
Definition 3.3
Let V be a vector space over \(\mathbb{R}\). A complexification of V is a complex structure associated with the real vector space V ⊕V. The resulting complex vector space is denoted by \(\boldsymbol{V }^{\mathbb{C}}\). Let v,w ∈ V. Elements of \(\mathbf{V}^{\mathbb{C}}\) are usually denoted by c = v + i w, and if \(\mathbb{C} \ni z = a +\mathrm{ i}b\) we have
Of course, we have that \(\dim _{\mathbb{C}}\mathbf{V}^{\mathbb{C}} =\dim _{\mathbb{R}}\mathbf{V}\).
Definition 3.4
A \(\mathbb{H}\)-module is a real vector space S carrying three linear transformation, I, J and K each one of them satisfying
Exercise 3.5
Show that K 2 = −Id S
In what follows \(\mathcal{A}\) denotes an associative algebra on the commutative field \(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\) and \(\mathbb{F} \subseteq \mathcal{A}\).
Definition 3.6
Any subset \(I \subseteq \mathcal{A}\) such that
is called a left ideal of A.
Remark 3.7
An analogous definition holds for right ideals where Eq. ( 3.6) reads \(\psi a \in I,\forall a \in \mathcal{A},\forall \psi \in I\), for bilateral ideals where in this case Eq. ( 3.6) reads \(a\psi b \in I,\forall a,b \in \mathcal{A},\forall \psi \in I\).
Definition 3.8
An associative algebra \(\mathcal{A}\) is simple if the only bilateral ideals are the zero ideal and \(\mathcal{A}\) itself.
Not all algebras are simple and in particular semi-simple algebras are important for our considerations. A definition of semi-simple algebras requires the introduction of the concepts of nilpotent ideals and radicals. To define these concepts adequately would lead us to a long incursion on the theory of associative algebras, so we avoid to do that here. We only quote that semi-simple algebras are the direct sum of simple algebras and of course simple algebras are semi simple. Then, for our objectives in this chapter the study of semi-simple algebras is reduced to the study of simple algebras.
Definition 3.9
We say that e \(\mathbf{\in }\mathcal{A}\) is an idempotent element if e2 = e. An idempotent is said to be primitive if it cannot be written as the sum of two non zero annihilating (or orthogonal) idempotent, i.e., e≠ e1 + e2, with e1e2 = e2e1 = 0 and \(\mathrm{e}_{1}^{2} =\mathrm{ e}_{1},\mathrm{e}_{2}^{2} =\mathrm{ e}_{2}\).
We give without proofs the following theorems valid for semi-simple (and thus simple) algebras \(\mathcal{A}\):
Theorem 3.10
All minimal left (respectively right) ideals of semi-simple \(\mathcal{A}\) are of the form \(J = \mathcal{A}\mathrm{e}\) (respectively \(\mathrm{e}\mathcal{A}\) ), where e is a primitive idempotent of \(\mathcal{A}\) .
Theorem 3.11
Two minimal left ideals of a semi-simple algebra \(\mathcal{A}\) , \(J = \mathcal{A}\mathrm{e}\) and \(J = \mathcal{A}\mathrm{e}^{{\prime}}\) are isomorphic, if and only if, there exist a non null \(Y ^{{\prime}}\in J^{{\prime}}\) such that \(J^{{\prime}} = JY ^{{\prime}}\) .
Let \(\mathcal{A}\) be an associative and simple algebra on the field \(\ \mathbb{F}(\mathbb{R}\) or \(\mathbb{C})\), and let S be a finite dimensional linear space over a division ring \(\mathbb{K} \supseteq \mathbb{F}\) and let \(\mathbf{E} =\mathrm{ End}_{\mathbb{K}}\mathbf{S} =\mathrm{ Hom}_{\mathbb{K}}(\mathbf{S,S})\) be the endomorphism algebra of S.Footnote 1
Definition 3.12
A representation of \(\mathcal{A}\) in S is a \(\mathbb{K}\) algebra homomorphism Footnote 2 \(\rho: \mathcal{A}\rightarrow \) \(\mathbf{E} =\mathrm{ End}_{\mathbb{K}}\mathbf{S}\) which maps the unit element of \(\mathcal{A}\) to Id E . The dimension \(\mathbb{K}\) of S is called the degree of the representation.
Definition 3.13
The addition in S together with the mapping \(\mathcal{A}\times \mathbf{S \rightarrow S}\), (a, x) ↦ ρ(a)x turns S in a left \(\mathcal{A}\)-module,Footnote 3 called the left representation module.
Remark 3.14
It is important to recall that when \(\mathbb{K} = \mathbb{H}\) the usual recipe for \(\mathrm{Hom}_{\mathbb{H}}(\mathbf{S,S})\) to be a linear space over \(\mathbb{H}\) fails and in general \(\mathrm{Hom}_{\mathbb{H}}(\mathbf{S,S})\) is considered as a linear space over \(\mathbb{R}\), which is the centre of \(\mathbb{H}\).
Remark 3.15
We also have that if \(\mathcal{A}\) is an algebra on \(\mathbb{F}\) and S is an \(\mathcal{A}\)-module, then S can always be considered as a vector space over \(\mathbb{F}\) and if \(a \in \mathcal{A}\), the mapping \(\chi: a \rightarrow \chi _{a}\) with χ a (s) = a s, s∈S, is a homomorphism \(\mathcal{A}\rightarrow \mathrm{ End}_{\mathbb{F}}\mathbf{S}\), and so it is a representation of \(\mathcal{A}\) in S. The study of \(\mathcal{A}\) modules is then equivalent to the study of the \(\mathbb{F}\) representations of \(\mathcal{A}\).
Definition 3.16
A representation ρ is faithful if its kernel is zero, i.e., \(\rho (a)x = 0,\forall x \in \mathbf{S} \Rightarrow a = 0\). The kernel of ρ is also known as the annihilator of its module.
Definition 3.17
ρ is said to be simple or irreducible if the only invariant subspaces of \(\rho (a),\forall a \in \mathcal{A}\), are S and \(\left \{0\right \}\).
Then, the representation module is also simple. That means that it has no proper submodules.
Definition 3.18
ρ is said to be semi-simple, if it is the direct sum of simple modules, and in this case S is the direct sum of subspaces which are globally invariant under \(\rho (a),\forall a \in \mathcal{A}\).
When no confusion arises ρ(a)x may be denoted by a ∗ x or ax.
Definition 3.19
Two \(\mathcal{A}\)-modules S and \(\mathbf{S}^{{\prime}}\) (with the “exterior” multiplication being denoted respectively by ◊ and ∗) are isomorphicif there exists a bijection \(\varphi: \mathbf{S \rightarrow S}^{{\prime}}\) such that,
and we say that the representations ρ and \(\rho ^{{\prime}}\) of A are equivalent if their modules are isomorphic.
This implies the existence of a \(\mathbb{K}\)-linear isomorphism \(\varphi: \mathbf{S \rightarrow S}^{{\prime}}\) such that \(\varphi \circ \rho (a) =\rho ^{{\prime}}(a)\circ \varphi,\forall a \in \mathcal{A}\) or \(\rho ^{{\prime}}(a) =\varphi \circ \rho (a) \circ \varphi ^{-1}\). If \(\dim \mathbf{S} = n\), then \(\dim \mathbf{S}^{{\prime}} = n\).
Definition 3.20
A complex representation of \(\mathcal{A}\) is simply a real representation \(\rho: \mathcal{A}\rightarrow \mathrm{ Hom}_{\mathbb{R}}(\mathbf{S,S})\) for which
This means that the image of ρ commutes with the subalgebra generated by \(\left \{\mathrm{Id}_{\mathbf{S}},\mathbf{J}\right \} \sim \mathbb{C}\).
Definition 3.21
A quaternionic representation of \(\mathcal{A}\) is a representation \(\rho: \mathcal{A}\rightarrow \mathrm{ Hom}_{\mathbb{R}}(\mathbf{S,S})\) such that
This means that the representation ρ has a commuting subalgebra isomorphic to the quaternion ring.
The following theorem is crucial:
Theorem 3.22
(Wedderburn). If \(\mathcal{A}\) is simple algebra over \(\mathbb{F}\) then \(\mathcal{A}\) is isomorphic to \(\mathbb{D}(m)\) , where \(\mathbb{D}(m)\) is a matrix algebra with entries in \(\mathbb{D}\) (a division algebra), and m and \(\mathbb{D}\) are unique (modulo isomorphisms).
3.2 Real and Complex Clifford Algebras and Their Classification
Now, it is time to specialize the previous results to the Clifford algebras on the field \(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\). We are particularly interested in the case of real Clifford algebras. In what follows we take \(\mathbf{V} = \mathbb{R}^{n}\). We denote as in the previous chapter by \(\mathbb{R}^{p,q}\) (n = p + q) the real vector space \(\mathbb{R}^{n}\) endowed with a nondegenerate metric \(\mathbf{g}: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}\). Let \(\left \{E_{i}\right \},(i = 1,2,\ldots,n)\) be an orthonormal basis of \(\mathbb{R}^{p,q}\),
We recall (Definition 2.37 that the Clifford algebra \(\mathbb{R}_{p,q} = \mathcal{C}\ell(\mathbb{R}^{p,q})\) is the Clifford algebra over \(\mathbb{R}\), generated by 1 and the \(\left \{E_{i}\right \},(i = 1,2,\ldots,n)\) such that \(E_{i}^{2} = \mathbf{g}(E_{i},E_{i})\), \(E_{i}E_{j} = -E_{j}E_{i}(i\neq j)\), and E 1 E 2 … E n ≠ ± 1.
\(\mathbb{R}_{p,q}\) is obviously of dimension 2n and as a vector space it is the direct sum of vector spaces \(\bigwedge ^{k}\mathbb{R}^{n}\) of dimensions \(\binom{n}{k},0 \leq k \leq n\). The canonical basis of \(\bigwedge ^{k}\mathbb{R}^{n}\) is given by the elements \(e_{A} = E_{\alpha _{1}}\cdots E_{\alpha _{k}},1 \leq \alpha _{1} <\ldots <\alpha _{k} \leq n\). The element \(e_{J} = E_{1}\cdots E_{n} \in \bigwedge ^{n}\mathbb{R}^{n}\hookrightarrow \mathbb{R}_{p,q}\) commutes (n odd) or anticommutes (n even) with all vectors \(E_{1},\ldots,E_{n} \in \bigwedge ^{1}\mathbb{R}^{n} \equiv \mathbb{R}^{n}\). The center \(\mathbb{R}_{p,q}\) is \(\bigwedge ^{0}\mathbb{R}^{n} \equiv \mathbb{R}\) if n is even and it is the direct sum \(\bigwedge ^{0}\mathbb{R}^{n} \oplus \bigwedge ^{0}\mathbb{R}^{n}\) if n is odd.Footnote 4
All Clifford algebras are semi-simple. If p + q = n is even, \(\mathbb{R}_{p,q}\) is simple and if p + q = n is odd we have the following possibilities:
-
(a)
\(\mathbb{R}_{p,q}\) is simple \(\leftrightarrow e_{J}^{2} = -1 \leftrightarrow p - q\neq 1\) (mod 4) \(\leftrightarrow \) center of \(\mathbb{R}_{p,q}\) is isomorphic to \(\mathbb{C}\);
-
(b)
\(\mathbb{R}_{p,q}\) is not simple (but is a direct sum of two simple algebras) \(\leftrightarrow e_{J}^{2} = +1 \leftrightarrow p - q = 1\) (mod 4) \(\leftrightarrow \) center of \(\mathbb{R}_{p,q}\) is isomorphic to \(\mathbb{R} \oplus \mathbb{R}\).
Now, for \(\mathbb{R}_{p,q}\) the division algebras \(\mathbb{D}\) are the division rings \(\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{H}\). The explicit isomorphism can be discovered with some hard but not difficult work. It is possible to give a general classification of all real (and also the complex) Clifford algebras and a classification table can be found, e.g., in [20]. One convenient table is the following one (where μ = [n∕2] means the integer part of n∕2).
We denoted by \(\mathbb{R}_{p,q}^{0}\) the even subalgebra of \(\mathbb{R}_{p,q}\) and by \(\mathbb{R}_{p,q}^{1}\) the set of odd elements of \(\mathbb{R}_{p,q}\). The following very important result holds true
Proposition 3.23
\(\mathbb{R}_{p,q}^{0} \simeq \mathbb{R}_{p,q-1}\) and also \(\mathbb{R}_{p,q}^{0} \simeq \mathbb{R}_{q,p-1}.\)
Now, to complete the classification we need the following theorem:
Theorem 3.24 (Periodicity)
Footnote 5 We have
Remark 3.25
We emphasize here that since the general results concerning the representations of simple algebras over a field \(\mathbb{F}\) applies to the Clifford algebras \(\mathbb{R}_{p,q}\) we can talk about real, complex or quaternionic representation of a given Clifford algebra, even if the natural matrix identification is not a matrix algebra over one of these fields. A case that we shall need is that \(\mathbb{R}_{1,3} \simeq \mathbb{H}(2)\). But it is clear that \(\mathbb{R}_{1,3}\) has a complex representation, for any quaternionic representation of \(\mathbb{R}_{p,q}\) is automatically complex, once we restrict \(\mathbb{C} \subset \mathbb{H}\) and of course, the complex dimension of any \(\mathbb{H}\)-module must be even. Also, any complex representation of \(\mathbb{R}_{p,q}\) extends automatically to a representation of \(\mathbb{C} \otimes \mathbb{R}_{p,q}\).
Remark 3.26
\(\mathbb{C} \otimes \mathbb{R}_{p,q}\) is isomorphic to the complex Clifford algebra \(\mathcal{C}\ell_{p+q}\). The algebras \(\mathbb{C}\) and \(\mathbb{R}_{p,q}\) are subalgebras of \(\mathcal{C}\ell_{p+q}\)
3.2.1 Pauli, Spacetime, Majorana and Dirac Algebras
For the purposes of our book we shall need to have in mind that:
\(\mathbb{R}_{3,0}\) is called the Pauli algebra, \(\mathbb{R}_{1,3}\) is called the spacetime algebra, \(\mathbb{R}_{3,1}\) is called Majorana algebra and \(\mathbb{R}_{4,1}\) is called the Dirac algebra. Also, the following particular results, which can be easily proved, will be used many times in what follows:
In words: the even subalgebras of both the spacetime and Majorana algebras is the Pauli algebra. The even subalgebra of the Dirac algebra is the spacetime algebra and finally the Dirac algebra is the complexification of the spacetime algebra or of the Majorana algebra.
Equation ( 3.13) show moreover, in view of Remark 3.26 that the spacetime algebra has also a matrix representation in \(\mathbb{C}(4)\). Obtaining such a representation is very important for the introduction of the concept of a Dirac-Hestenes spinor, an important ingredient of the present work.
3.3 The Algebraic, Covariant and Dirac-Hestenes Spinors
3.3.1 Minimal Lateral Ideals of \(\mathbb{R}_{p,q}\)
We now give some results concerning the minimal lateral ideals of \(\mathbb{R}_{p,q}\).
Theorem 3.27
The maximum number of pairwise orthogonal idempotents in \(\mathbb{K}(m)\) (where \(\mathbb{K} = \mathbb{R}\) , \(\mathbb{C}\) or \(\mathbb{H}\) ) is m.
The decomposition of \(\mathbb{R}_{p,q}\) into minimal ideals is then characterized by a spectral set {e pq, j } of idempotents elements of \(\mathbb{R}_{p,q}\) such that:
-
(a)
\(\sum \limits _{i=1}^{n}\mathrm{e}_{pq,j} = 1\);
-
(b)
e pq, j e pq, k = δ jk e pq, j ;
-
(c)
the rank of e pq, j is minimal and non zero, i.e., is primitive.
By rank of e pq, j we mean the rank of the \(\bigwedge \mathbb{R}^{p,q}\) morphism, e pq, j : ϕ ↦ ϕe pq, j . Conversely, any ϕ ∈ I pq, j can be characterized by an idempotent e pq, j of minimal rank ≠ 0, with ϕ = ϕe pq, j .
We now need to know the following theorem [13]:
Theorem 3.28
A minimal left ideal of \(\mathbb{R}_{p,q}\) is of the type
where
is a primitive idempotent of R p,q and where \(e_{\alpha _{1}},\ldots,e_{\alpha _{k}}\) are commuting elements in the canonical basis of \(\mathbb{R}_{p,q}\) (generated in the standard way through the elements of a basis (E 1,… ,E p ,E p+1 ,…E p+q ) of \(\mathbb{R}^{p,q}\) ) such that \((e_{\alpha _{i}})^{2} = 1,\) (i = 1,2,…,k) generate a group of order 2 k , k = q − r q−p and r i are the Radon-Hurwitz numbers, defined by the recurrence formula r i+8 = r i + 4 and
Recall that \(\mathbb{R}_{p,q}\) is a ring and the minimal lateral ideals are modules over the ring \(\mathbb{R}_{p,q}\). They are representation modules of \(\mathbb{R}_{p,q}\), and indeed we have (recall the above table) the following theorem [13]:
Theorem 3.29
If p + q is even or odd with \(p - q\neq 1(\mathop{mod}\nolimits 4)\) , then
where (as we already know) \(\mathbb{K} = \mathbb{R}\) , \(\mathbb{C}\) or \(\mathbb{H}\) . Also,
and
where e is the representation of epq in \(\mathbb{K}(m)\) .
If p + q = n is odd, with p − q = 1 \((\mathop{mod}\nolimits 4)\) , then
with
and
With the above isomorphisms we can immediately identify the minimal left ideals of \(\mathbb{R}_{p,q}\) with the column matrices of \(\mathbb{K}(m)\).
3.3.2 Algorithm for Finding Primitive Idempotents of \(\mathbb{R}_{p,q}\)
With the ideas introduced above it is now a simple exercise to find primitive idempotents of \(\mathbb{R}_{p,q}\). First we look at Table 3.1 and find the matrix algebra to which our particular Clifford algebra \(\mathbb{R}_{p,q}\) is isomorphic. Suppose \(\mathbb{R}_{p,q}\) is simple.Footnote 6 Let \(\mathbb{R}_{p,q} \simeq \mathbb{K}(m)\) for a particular \(\mathbb{K}\) and m. Next we take an element \(\mathbf{e}_{\alpha _{1}} \in \left \{\mathbf{e}_{A}\right \}\) from the canonical basis \(\left \{\mathbf{e}_{A}\right \}\) of \(\mathbb{R}_{p,q}\) such that
Next we construct the idempotent \(\mathrm{e}_{pq} = (1 + \mathbf{e}_{\alpha _{1}})/2\) and the ideal \(I_{pq} = \mathbb{R}_{p,q}\mathrm{e}_{pq}\) and calculate \(\mathrm{\dim }_{\mathbb{K}}(I_{pq})\). If \(\mathrm{\dim }_{\mathbb{K}}(I_{pq}) = m\), then e pq is primitive. If \(\mathrm{\dim }_{\mathbb{K}}(I_{pq})\neq m\), we choose \(\mathbf{e}_{\alpha _{ 2}} \in \left \{\mathbf{e}_{A}\right \}\) such that \(\mathbf{e}_{\alpha _{ 2}}\) commutes with \(\mathbf{e}_{\alpha _{1}}\) and \(\mathbf{e}_{\alpha _{ 2}}^{2} = 1\) and construct the idempotent \(\mathrm{e}_{pq}^{{\prime}} = (1 + \mathbf{e}_{\alpha _{1}})(1 + \mathbf{e}_{\alpha _{2}})/4\). If \(\mathrm{\dim }_{\mathbb{K}}(I_{pq}^{{\prime}}) = m\), then \(\mathrm{e}_{pq}^{{\prime}}\) is primitive. Otherwise we repeat the procedure. According to Theorem 3.28 the procedure is finite.
3.3.3 \(\mathbb{R}_{p,q}^{\star }\), Clifford, Pinor and Spinor Groups
The set of the invertible elements of \(\mathbb{R}_{p,q}\) constitutes a non-abelian group which we denote by \(\mathbb{R}_{p,q}^{\star }\). It acts naturally on \(\mathbb{R}_{p,q}\) as an algebra homomorphism through its twisted adjoint representation (\(\mathrm{\hat{A} d}\)) or adjoint representation (Ad)
Definition 3.30
The Clifford-Lipschitz group is the set
Note in Eq. ( 3.26b) the restriction to the even (\(\mathbb{R}_{p,q}^{\star (0)}\)) and odd \((\mathbb{R}_{p,q}^{\star (1)}\)) parts of \(\mathbb{R}_{p,q}^{\star }\).
Definition 3.31
The set \(\Gamma _{p,q}^{0} = \Gamma _{p,q} \cap \mathbb{R}_{p,q}^{0}\) is called special Clifford-Lipschitz group.
Definition 3.32
The Pinor group Pin p. q is the subgroup of \(\Gamma _{p,q}\) such that
where
Definition 3.33
The Spin group Spin p, q is the set
It is easy to see that Spin p, q is not connected.
Definition 3.34
The Special Spin Group \(\mathrm{Spin}_{p,q}^{e}\) is the set
The superscript e, means that \(\mathrm{Spin}_{p,q}^{e}\) is the connected component to the identity. We can prove that \(\mathrm{Spin}_{p,q}^{e}\) is connected for all pairs (p, q) with the exception of \(\mathrm{Spin}^{e}(1,0) \simeq \mathrm{ Spin}^{e}(0,1)\).
We recall now some classical results [17] associated with the pseudo-orthogonal groups O p, q of a vector space \(\mathbb{R}^{p,q}\) (n = p + q) and its subgroups. Let G be a diagonal n × n matrix whose elements are G ij
with p positive and q negative numbers.
Definition 3.35
O p, q is the set of n × n real matrices L such that
Equation ( 3.32) shows that O p, q is not connected.
Definition 3.36
SO p, q , the special (proper) pseudo orthogonal group is the set of n × n real matrices L such that
When p = 0 (q = 0) SO p, q is connected. However, SO p, q (for, p, q ≠ 0) is not connected and has two connected components for p, q ≥ 1.
Definition 3.37
The group \(\mathrm{SO}_{p,q}^{e}\), the connected component to the identity of SO p, q will be called the special orthochronous pseudo-orthogonal group.Footnote 7
Theorem 3.38
Ad \(_{\left \vert \mathrm{Pin}_{p,q}\right.}:\mathrm{ Pin}_{p,q} \rightarrow \mathrm{ O}_{p,q}\) is onto with kernel \(\mathbb{Z}_{2}\) .
Ad \(_{\left \vert \mathrm{Spin}_{p,q}\right.}:\mathrm{ Spin}_{p,q} \rightarrow \mathrm{ SO}_{p,q}\) is onto with kernel \(\mathbb{Z}_{2}\) . Ad \(_{\left \vert \mathrm{Spin}_{p,q}^{e}\right.}:\mathrm{ Spin}_{p,q}^{e} \rightarrow \mathrm{ SO}_{p,q}^{e}\) is onto with kernel \(\mathbb{Z}_{2}\) .
We have,
The group homomorphism between \(\mathrm{Spin}_{p,q}^{e}\) and SOe(p, q) will be denoted by
The following theorem that first appears in [20] is very important.
Exercise 3.39
(Porteous). Show that for p + q ≤ 4, \(\mathrm{Spin}^{e}(p,q) =\{ u \in \mathbb{R}_{p,q}\vert u\tilde{u} = 1\}\).
Solution
We must show that for any \(u \in \mathbb{R}_{p,q}^{0}\), N(u) = ±1 and \( \mathbf{\mathit{x}} \in \mathbb{R}^{p,q}\) we have that \(\mathrm{\hat{A} d}_{u}( \mathbf{\mathit{x}}) \in \mathbb{R}^{p,q}\). But when \(u \in \mathbb{R}_{p,q}^{0}\), \(\mathrm{\hat{A}d}_{u}( \mathbf{\mathit{x}}) = u \mathbf{\mathit{x}}u^{-1}\). We must then show that
Since \(u \in \mathbb{R}_{p,q}^{0}\) we have that \( \mathbf{\mathit{y}} \in \mathbb{R}_{p,q}^{1}\). Let e
i
, i = 1, 2, 3, 4 an orthonormal basis of \(\mathbb{R}_{p,q}\), p + q = 4. Now, 
. Writing
we get
from which follows that \( \mathbf{\mathit{y}} \in \mathbb{R}^{p,q}\).
3.3.4 Lie Algebra of Spin \(_{1,3}^{e}\)
It can be shown [14, 16, 23] that for each \(u \in \mathrm{ Spin}_{1,3}^{e}\) it holds \(u = \pm e^{F},F \in \bigwedge ^{2}\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) and F can be chosen in such a way to have a positive sign in Eq. ( 3.33), except in the particular case F 2 = 0 when u = −e F. From Eq. ( 3.33) it follows immediately that the Lie algebra of \(\mathrm{Spin}_{1,3}^{e}\) is generated by the bivectors \(F \in \bigwedge ^{2}\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) through the commutator product.
Exercise 3.40
Show that when F 2 = 0 we must have u = −e F.
3.4 Spinor Representations of \(\mathbb{R}_{4,1}, \mathbb{R}_{4,1}^{0}\) and \(\mathbb{R}_{1,3}\)
We investigate now some spinor representations of \(\mathbb{R}_{4,1}, \mathbb{R}_{4,1}^{0}\) and \(\mathbb{R}_{1,3}\) which will permit us to introduce the concepts algebraic, Dirac and Dirac-Hestenes spinors in the next section.
Let \(b_{0} = \left \{\mathbf{e}_{0},\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right \}\) be an orthogonal basis of \(\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\), such that \(\mathbf{e}_{\mu }\mathbf{e}_{\nu } + \mathbf{e}_{\nu }\mathbf{e}_{\mu } = 2\eta _{\mu \nu }\), with \(\eta _{\upsilon \mu } =\mathrm{ diag}(+1,-1,-1,-1)\). Now, with the results of the previous section we can verify without difficulties that the elements e, \(\mathrm{e}^{{\prime}},\mathrm{e}^{{\prime\prime}}\in \mathbb{R}_{1,3}\)
are primitive idempotents of \(\mathbb{R}_{1,3}\). The minimal left ideals,Footnote 8 \(I = \mathbb{R}_{1,3}\mathrm{e}\), \(I^{{\prime}} = \mathbb{R}_{1,3}\mathrm{e}^{{\prime}}\), \(I^{{\prime\prime}} = \mathbb{R}_{1,3}\mathrm{e}^{{\prime\prime}}\) are right two dimension linear spaces over the quaternion field (\(\mathbb{H}\mathrm{e} =\mathrm{ e}\mathbb{H} =\mathrm{ e}\mathbb{R}_{1,3}\mathrm{e}\)).
An elements \(\Phi \in \mathbb{R}_{1,3}\frac{1} {2}(1 + \mathbf{e}_{0})\) has been called by Lounesto [15] a mother spinor. Footnote 9 Let us see the justice of this denomination. First recall from the general result of the previous section that \(\frac{\mathrm{Pin}_{1,3}} {\mathbb{Z}_{2}} \simeq \mathrm{ O}_{1,3}\), \(\frac{\mathrm{Spin}_{1,3}} {\mathbb{Z}_{2}} \simeq \mathrm{ SO}_{1,3}\), \(\frac{\mathrm{Spin}_{1,3}^{e}} {\mathbb{Z}_{2}} \simeq \mathrm{ SO}_{1,3}^{e}\), and \(\mathrm{Spin}_{1,3}^{e} \simeq \mathrm{ Sl}(2, \mathbb{C})\) is the universal covering group of \(\mathcal{L}_{+}^{\uparrow }\equiv \mathrm{ SO}_{1,3}^{e}\), the special (proper) orthochronous Lorentz group. We can show [10, 11] that the ideal \(I = \mathbb{R}_{1,3}\mathrm{e}\) carries the D (1∕2, 0) ⊕ D (0, 1∕2) representation of \(\mathrm{Sl}(2, \mathbb{C})\). Here we need to know [10, 11] that each \(\Phi \) can be written as
and where the ψ i are formally complex numbers, i.e., each \(\psi _{i} = (a_{i} + b_{i}\mathbf{e}_{2}\mathbf{e}_{1})\) with \(a_{i},b_{i} \in \mathbb{R}\) and the set \(\left \{s_{i},i = 1,2,3,4\right \}\) is a basis in the mother spinors space.
Exercise 3.41
Prove Eq. ( 3.39).
Now we determine an explicit relation between representations of \(\mathbb{R}_{4,1}\) and \(\mathbb{R}_{3,1}\). Let \(\left \{\mathbf{f}_{0},\mathbf{f}_{1},\mathbf{f}_{2},\mathbf{f}_{3},\mathbf{f}_{4}\right \}\) be an orthonormal basis of \(\mathbb{R}_{4,1}\) with
Define the pseudo-scalar
Put
we can immediately verify that
Taking into account that \(\mathbb{R}_{1,3} \simeq \mathbb{R}_{4,1}^{0}\) we can explicitly exhibit here this isomorphism by considering the map j \(: \mathbb{R}_{1,3} \rightarrow \mathbb{R}_{4,1}^{0}\) generated by the linear extension of the map j \(^{\#}: \mathbb{R}^{1,3} \rightarrow \mathbb{R}_{4,1}^{0}\), \(\mathtt{j}^{\#}(\mathbf{e}_{\mu }) = \mathcal{E}_{\mu } = \mathbf{f}_{\mu }\mathbf{f}_{4}\), where \(\mathcal{E}_{\mu }\), (μ = 0, 1, 2, 3) is an orthogonal basis of \(\mathbb{R}^{1,3}\). Note that j \((1_{\mathbb{R}_{1,3}}) = 1_{\mathbb{R}_{4,1}^{0}}\), where \(1_{\mathbb{R}_{1,3}}\) and \(1_{\mathbb{R}_{4,1}^{0}}\) (usually denoted simply by 1) are the identity elements in \(\mathbb{R}_{1,3}\) and \(\mathbb{R}_{4,1}^{0}\). Now consider the primitive idempotent of \(\mathbb{R}_{1,3} \simeq \mathbb{R}_{4,1}^{0}\),
and the minimal left ideal \(I_{4,1}^{0} = \mathbb{R}_{4,1}^{0}\mathrm{e}_{41}^{0}\).
The elements \(Z \in I_{4,1}^{0}\) can be written analogously to \(\Phi \in \mathbb{R}_{1,3}\frac{1} {2}(1+\) e 0) as,
where
and where
are formally complex numbers, \(a_{i},b_{i} \in \mathbb{R}\).
Consider now the element \(f \in \mathbb{R}_{4,1}\)
with \(\mathfrak{i}\) defined as in Eq. ( 3.41).
Since \(f\mathbb{R}_{4,1}\,f = \mathbb{C}f = f\mathbb{C}\) it follows that f is a primitive idempotent of \(\mathbb{R}_{4,1}\). We can easily show that each \(\Phi \in I = \mathbb{R}_{4,1}f\) can be written
With the methods described in [10, 11] we find the following representation in \(\mathbb{C}(4)\) for the generators \(\mathcal{E}_{\mu }\) of \(\mathbb{R}_{4,1} \simeq \mathbb{R}_{1,3}\)
where 1 2 is the unit 2 × 2 matrix and \(\sigma _{i}\), (i = 1, 2, 3) are the standard Pauli matrices. We immediately recognize the γ-matrices in Eq. ( 3.49) as the standard ones appearing, e.g., in [4].
The matrix representation of \(\Psi \in I\) will be denoted by the same letter in boldface, i.e., \(\Psi \mapsto \boldsymbol{\Psi } \in \mathbb{C}(4)f\), where
We have
Equations ( 3.49)–( 3.51) are sufficient to prove that there are bijections between the elements of the ideals \(\mathbb{R}_{1,3}\frac{1} {2}(1+\) e 0), \(\mathbb{R}_{4,1}^{0}\frac{1} {2}(1 + \mathcal{E}_{0})\) and \(\mathbb{R}_{4,1}\frac{1} {2}(1 + \mathcal{E}_{0})\frac{1} {2}(1 + \mathfrak{i}\mathcal{E}_{1}\mathcal{E}_{2})\).
We can easily find that the following relation exist between \(\Psi \in \mathbb{R}_{4,1}f\) and \(Z \in \mathbb{R}_{4,1}^{0}\frac{1} {2}(1 + \mathcal{E}_{0}),\)
Decomposing Z into even and odd parts relative to the Z 2-graduation of \(\mathbb{R}_{4,1}^{0} \simeq \mathbb{R}_{1,3}\), Z = Z 0 + Z 1 we obtain \(Z^{0} = Z^{1}\mathcal{E}_{0}\) which clearly shows that all information of Z is contained in Z 0. Then,
Now, if we take into account that \(\mathbb{R}_{4,1}^{0}\frac{1} {2}(1 + \mathcal{E}_{0}) = \mathbb{R}_{4,1}^{00}\frac{1} {2}(1 + \mathcal{E}_{0})\) where the symbol \(\mathbb{R}_{4,1}^{00}\) means \(\mathbb{R}_{4,1}^{00} \simeq \mathbb{R}_{1,3}^{0} \simeq \mathbb{R}_{3,0}\) we see that each \(Z \in \mathbb{R}_{4,1}^{00}\frac{1} {2}(1 + \mathcal{E}_{0})\) can be written
Then putting Z 0 = ψ∕2, Eq. ( 3.54) can be written
The matrix representation of ψ and Z in \(\mathbb{C}(4)\) (denoted by the same letter in boldface) in the matrix representation generated by the spin basis given by Eq. ( 3.48) are
3.5 Algebraic Spin Frames and Spinors
We introduce now the fundamental concept of algebraic spin frames.Footnote 10 This is the concept that will permit us to define spinors (steps (i)–(vii)).Footnote 11
-
(i)
In this section \((\mathbf{V},\boldsymbol{\eta })\) refers always to Minkowski vector space.
-
(ii)
Let \(\mathrm{SO}(\mathbf{V},\boldsymbol{\eta })\) be the group of endomorphisms of V that preserves \(\boldsymbol{\eta }\) and the space orientation. This group is isomorphic to SO1, 3 but there is no natural isomorphism. We write \(\mathrm{SO}(\mathbf{V},\boldsymbol{\eta }) \simeq \mathrm{ SO}_{1,3}\). Also, the connected component to the identity is denoted by \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) and \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta }) \simeq \mathrm{ SO}_{1,3}^{e}\). Note that \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) preserves besides orientation also the time orientation.
-
(iii)
We denote by \(\mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\) the Clifford algebraFootnote 12 of \((\mathbf{V},\boldsymbol{\eta })\) and by \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta }) \simeq \mathrm{ Spin}_{1,3}^{e}\) the connected component of the spin group \(\mathrm{Spin}(\mathbf{V},\boldsymbol{\eta }) \simeq \mathrm{ Spin}_{1,3}\). Consider the 2: 1 homomorphism \(\mathbf{L}:\mathrm{ Spin}^{e}(\mathbf{V},\boldsymbol{\eta }) \rightarrow \mathrm{ SO}^{e}(\mathbf{V},\boldsymbol{\eta })\), \(u\mapsto \mathbf{L}(u) \equiv \mathbf{L}_{u}\). \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\) acts on V identified as the space of 1-vectors of \(\mathcal{C}\ell(\mathbf{V},\eta ) \simeq \mathbb{R}_{1,3}\) through its adjoint representation in the Clifford algebra \(\mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\) which is related with the vector representation of \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) as followsFootnote 13:
$$\displaystyle\begin{array}{rcl} & & \mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\text{ } \ni \text{}u\mapsto \mathrm{Ad}_{ u} \in \mathrm{ Aut}(\mathcal{C}\ell(\mathbf{V},\eta )) \\ & & \phantom{\mathrm{Spin}^{e}(\mathbf{V},1}\left.\mathrm{Ad}_{ u}\right \vert _{\mathbf{V}}: \mathbf{V} \rightarrow \mathbf{V},\text{ }\mathbf{v}\mapsto u\mathbf{v}u^{-1}\text{ } = \mathbf{L}_{ u} \odot \mathbf{v}. {}\end{array}$$(3.57)In Eq. ( 3.57) L u ⊙v denotes the standard action L u on v and where we identified \(\mathbf{L}_{u} \in \mathrm{ SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) with L u ∈ V⊗V ∗ and
$$\displaystyle{ \boldsymbol{\eta }\left (\mathbf{L}_{u} \odot \mathbf{v,L}_{u} \odot \mathbf{v}\right ) =\boldsymbol{\eta } \left (\mathbf{v,v}\right ). }$$(3.58) -
(iv)
Let \(\mathcal{B}\) be the set of all oriented and time oriented orthonormal basisFootnote 14 of V. Choose among the elements of \(\mathcal{B}\) a basis b 0 = {b 0 ,….,b 3}, hereafter called the fiducial frame of V. With this choice, we define a 1 − 1 mapping
$$\displaystyle{ \boldsymbol{\Sigma }:\mathrm{ SO}^{e}(\mathbf{V},\boldsymbol{\eta }) \rightarrow \mathcal{B}, }$$(3.59)given by
$$\displaystyle{ \mathbf{L}_{u}\boldsymbol{\mapsto \Sigma }(\mathbf{L}_{u}):=\boldsymbol{ \Sigma }_{\mathbf{L}_{u}} = \mathbf{L}\mathit{b}_{0} }$$(3.60)where \(\boldsymbol{\Sigma }_{\mathbf{Lu}} = \mathbf{L}_{u}\mathit{b}_{0}\) is a short for \(\{\mathbf{e}_{1},\ldots.,\mathbf{e}_{3,}\} \in \mathcal{B}\), such that denoting the action of L u onb i ∈ b 0 by L u ⊙b i we have
$$\displaystyle{ \mathbf{e}_{i} = \mathbf{L}_{u} \odot \mathbf{b}_{i}:= L_{\cdot i}^{j\cdot }\mathbf{b}_{ j},\ \ \ i,j = 0,\ldots,3. }$$(3.61)In this way, we can identify a given vector basis b of V with the isometry L u that takes the fiducial basis b 0 to b. The fiducial basis b 0 will be also denoted by \(\boldsymbol{\Sigma }_{\mathbf{L}_{0}}\), where L 0 = e, is the identity element of \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\).
Since the group \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) is not simple connected their elements cannot distinguish between frames whose spatial axes are rotated in relation to the fiducial vector frame \(\boldsymbol{\Sigma }_{\mathbf{L}_{0}}\) by multiples of 2π or by multiples of 4π. For what follows it is crucial to make such a distinction. This is done by introduction of the concept of algebraic spin frames.
Definition 3.42
Let \(\mathit{b}_{0} \in \mathcal{B}\) be a fiducial frame and choose an arbitrary \(u_{0} \in \mathrm{ Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\). Fix once and for all the pair (u 0, b 0) with u 0 = 1 and call it the fiducial algebraic spin frame.
Definition 3.43
The space \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta }) \times \mathcal{B} =\{ (u,\mathit{b}),u\mathit{b}u^{-1} = u_{0}\mathit{b}_{0}u_{0}^{-1}\}\) will be called the space of algebraic spin frames and denoted by \(\mathcal{S}\).
Remark 3.44
It is crucial for what follows to observe here that Definition 3.43 implies that a given \(b \in \mathcal{B}\) determines two, and only two, algebraic spin frames, namely (u, b) and (−u, b), since \(\pm u\mathit{b}(\pm u^{-1}) = u_{0}\mathit{b}_{0}u_{0}^{-1}\).
-
(v)
We now parallel the construction in (iv) but replacing \(\mathrm{SO}^{e}(\mathbf{V},\boldsymbol{\eta })\) by its universal covering group \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\) and \(\mathcal{B}\) by S. Thus, we define the 1 − 1 mapping
$$\displaystyle{ \begin{array}{cccc} \boldsymbol{\Xi }:&\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })& \rightarrow & \mathcal{S}, \\ & u & \mapsto &\boldsymbol{\Xi }(u)\mathbf{:} =\boldsymbol{ \Xi }_{u} = (u,\mathit{b})\text{,}\end{array} }$$(3.62)where ubu −1 = b 0.
The fiducial algebraic spin frame will be denoted in what follows by \(\boldsymbol{\Xi }_{0}\). It is obvious from Eq. ( 3.62) that \(\boldsymbol{\Xi }(-u) \equiv \boldsymbol{ \Xi }_{\mathbf{-}u} = (\mathbf{-}u,\mathit{b})\neq \boldsymbol{\Xi }_{u}\).
Definition 3.45
The natural right action of \(a \in \mathrm{ Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\) denoted by ⊙ on \(\mathcal{S}\) is given by
Observe that if \(\boldsymbol{\Xi }_{u^{{\prime}}} = (u^{{\prime}},\mathit{b}^{{\prime}}) = u^{{\prime}}\odot \boldsymbol{ \Xi }_{0}\) and \(\boldsymbol{\Xi }_{u} = (u,\mathit{b}) = u \odot \boldsymbol{ \Xi }_{0}\) then,
Note that there is a natural 2 − 1 mapping
such that
Indeed,
This means that the natural right actions of \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\), respectively on \(\mathcal{S}\) and \(\mathcal{B}\), commute. In particular, this implies that the algebraic spin frames \(\boldsymbol{\Xi }_{u},\boldsymbol{\Xi }_{-u} \in \mathcal{S}\), which are, of course distinct, determine the same vector frame \(\Sigma _{\mathbf{L}_{u}} = \mathbf{s}(\boldsymbol{\Xi }_{u}) = \mathbf{s}(\boldsymbol{\Xi }_{-u}) = \Sigma _{\mathbf{L}_{-u}}\). We have,
Also, from Eq. ( 3.65), we can write explicitly
where the meaning of Eq. ( 3.68) of course, is that if \(\boldsymbol{\Sigma }_{\mathbf{L}_{u}} =\boldsymbol{ \Sigma }_{\mathbf{L}_{-u}} = \mathit{b} =\{ \mathbf{e}_{0}\mathbf{,\ldots.,e}_{3,}\} \in \mathcal{B}\) and \(\boldsymbol{\Sigma }_{\mathbf{L}_{u_{ 0}}} = \mathit{b}_{0} \in \mathcal{B}\) is the fiducial frame, then
In resume, we can say that the space \(\mathcal{S}\) of algebraic spin frames can be thought as an extension of the space \(\mathcal{B}\) of vector frames, where even if two vector frames have the same ordered vectors, they are considered distinct if the spatial axes of one vector frame is rotated by an odd number of 2π rotations relative to the other vector frame and are considered the same if the spatial axes of one vector frame is rotated by an even number of 2π rotations relative to the other frame. Even if the possibility of such a distinction seems to be impossible at first sight, Aharonov and Susskind [1] claim that it can be implemented physically in a spacetime where the concept of algebraic spin frame is enlarged to the concept of spin frame used for the definition of spinor fields. See Chap. 7 for details.
-
(vi)
Before we proceed an important digression on the notation used below is necessary. We recalled above how to construct a minimum left (or right) ideal for a given real Clifford algebra once a vector basis \(\mathit{b} \in \mathcal{B}\) for \(\mathbf{V}\hookrightarrow \mathcal{C}\ell(\mathbf{V},\mathbf{g})\) is given. That construction suggests to label a given primitive idempotent and its corresponding ideal with the subindex b. However, taking into account the above discussion of vector and algebraic spin frames and their relationship we find useful for what follows (specially in view of the Definition 3.46 and the definitions of algebraic and Dirac-Hestenes spinors (see Definitions 3.48 and 3.50 below) to label a given primitive idempotent and its corresponding ideal with a subindex \(\boldsymbol{\Xi }_{u}\). This notation is also justified by the fact that a given idempotent is according to definition 3.48 representative of a particular spinor in a given algebraic spin frame \(\boldsymbol{\Xi }_{u}\).
-
(vii)
Next we recall Theorem 3.28 which says that a minimal left ideal of \(\mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\) is of the type
$$\displaystyle{ I_{\boldsymbol{\Xi }_{u}} = \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\mathrm{e}_{\boldsymbol{\Xi }_{u}} }$$(3.70)where \(\mathrm{e}_{\boldsymbol{\Xi }_{u}}\) is a primitive idempotent of \(\mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\).
It is easy to see that all ideals \(I_{\boldsymbol{\Xi }_{u}} = \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\mathrm{e}_{\boldsymbol{\Xi }_{u}}\) and \(I_{\boldsymbol{\Xi }_{u^{{\prime}}}} = \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\mathrm{e}_{\boldsymbol{\Xi }_{u^{{\prime}}}}\) such that
\(u,u^{{\prime}}\in \mathrm{ Spin}^{e}(\mathbf{V},\eta )\) are isomorphic. We have the
Definition 3.46
Any two ideals \(I_{\boldsymbol{\Xi }_{u}} = \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\mathrm{e}_{\boldsymbol{\Xi }_{u}}\) and \(I_{\boldsymbol{\Xi }_{u^{{\prime}}}} = \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\mathrm{e}_{\boldsymbol{\Xi }_{u^{{\prime}}}}\) such that their generator idempotents are related by Eq. ( 3.71) are said geometrically equivalent.
Remark 3.47
If u is simply an element of the Clifford group, then the ideals are said to be algebraically equivalent.
But take care, no equivalence relation has been defined until now. We observe moreover that we can write
an equation that will play a key role in what follows.
3.6 Algebraic Dirac Spinors of Type \(I_{\boldsymbol{\Xi }_{u}}\)
Let \(\{I_{\boldsymbol{\Xi }_{u}}\}\) be the set of all ideals geometrically equivalent to a given minimal \(I_{\boldsymbol{\Xi }_{u_{o}}}\) as defined by Eq. ( 3.72). Let be
Let \(\boldsymbol{\Xi }_{u},\boldsymbol{\Xi }_{u^{{\prime}}} \in S\), \(\Psi _{\boldsymbol{\Xi }_{u}} \in I_{\boldsymbol{\Xi }_{u}}\), \(\Psi _{\boldsymbol{\Xi }_{u^{{\prime}}}} \in I_{\boldsymbol{\Xi }_{u^{{\prime}}}}\). We define an equivalence relation \(\mathcal{E}\) on \(\mathfrak{T}\) by setting
if and only if and
Definition 3.48
An equivalence class
is called an algebraic spinor of type \(I_{\boldsymbol{\Xi }_{u}}\) for \(\mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\). \(\psi _{\boldsymbol{\Xi }_{u}} \in I_{\boldsymbol{\Xi }_{u}}\) is said to be a representative of the algebraic spinor \(\boldsymbol{\Psi }_{\boldsymbol{\Xi }_{u}}\) in the algebraic spin frame \(\Xi _{u}\).
We observe that the pairs \((\boldsymbol{\Xi }_{u},\Psi _{\boldsymbol{\Xi }_{u}})\) and \((\boldsymbol{\Xi }_{-u},\Psi _{\boldsymbol{\Xi }_{-u}}) = (\boldsymbol{\Xi }_{-u},-\Psi _{\boldsymbol{\Xi }_{u}})\) are equivalent, but the pairs \((\boldsymbol{\Xi }_{u},\Psi _{\boldsymbol{\Xi }_{u}})\) and \((\boldsymbol{\Xi }_{-u},-\Psi _{\boldsymbol{\Xi }_{-u}}) = (\boldsymbol{\Xi }_{-u},\Psi _{\boldsymbol{\Xi }_{u}})\) are not. This distinction is essential in order to give a structure of linear space (over the real field) to the set \(\mathfrak{T}\). Indeed, a natural linear structure on \(\mathfrak{T}\) is given by
Remark 3.49
The definition just given is not a standard one in the literature [5, 8]. However, the fact is that the standard definition (licit as it is from the mathematical point of view) is not adequate for a comprehensive formulation of the Dirac equation using algebraic spinor fields or Dirac-Hestenes spinor fields which will be introduced in Chap. 7
We end this section recalling that as observed above a given Clifford algebra \(\mathbb{R}_{p,q}\) may have minimal ideals that are not geometrically equivalent since they may be generated by primitive idempotents that are related by elements of the group \(\mathbb{R}_{p,q}^{\star }\) which are not elements of \(\mathrm{Spin}^{e}(\mathbf{V},\boldsymbol{\eta })\) [see Eqs. ( 3.36)–( 3.38)] where different, non geometrically equivalent primitive ideals for \(\mathbb{R}_{1,3}\) are shown). These ideals may be said to be of different types. However, from the point of view of the representation theory of the real Clifford algebras all these primitive ideals carry equivalent (i.e., isomorphic) modular representations of the Clifford algebra and no preference may be given to any one.Footnote 15 In what follows, when no confusion arises and the ideal \(I_{\boldsymbol{\Xi }_{u}}\) is clear from the context, we use the wording algebraic Dirac spinor for any one of the possible types of ideals.
The most important property concerning algebraic Dirac spinors is a coincidence given by Eq. ( 3.78) below. It permit us to define a new kind of spinors.
3.7 Dirac-Hestenes Spinors (DHS)
Let \(\boldsymbol{\Xi }_{u} \in \mathcal{S}\) be an algebraic spin frame for \((\mathbf{V},\boldsymbol{\eta })\) such that
Then, it follows from Eq. ( 3.54) that
when
Then, each \(\Psi _{\boldsymbol{\Xi }_{u}} \in I_{\boldsymbol{\Xi }_{u}}\) can be written as
From Eq. ( 3.75) we get
A possible solution for Eq. ( 3.81) is
Let \(\mathcal{S}\times \mathcal{C}\ell(\mathbf{V},\boldsymbol{\eta })\) and consider an equivalence relation \(\mathcal{E}\) such that
if and only if \(\phi _{\boldsymbol{\Xi }_{u^{{\prime}}}}\) and \(\phi _{\boldsymbol{\Xi }_{u}}\) are related by
This suggests the following
Definition 3.50
The equivalence classes \([(\boldsymbol{\Xi }_{u},\phi _{\boldsymbol{\Xi }_{u}})] \in \mathcal{S}\times C\ell(\mathbf{V},\boldsymbol{\eta })/\mathcal{E}\) are the Hestenes spinors.
Among the Hestenes spinors, an important subset is the one consisted of Dirac-Hestenes spinors where \([(\boldsymbol{\Xi }_{u},\psi _{\boldsymbol{\Xi }_{u}})] \in (\mathcal{S}\times \mathcal{C}\ell^{0}(\mathbf{V},\boldsymbol{\eta }))/\mathcal{E}\).
We say that \(\phi _{\boldsymbol{\Xi }_{u}}\) \((\psi _{\boldsymbol{\Xi }_{u}})\) is a representative of a Hestenes (Dirac-Hestenes) spinor in the algebraic spin frame \(\boldsymbol{\Xi }_{u}\).
3.7.1 What is a Covariant Dirac Spinor (CDS)
Let \(\mathbf{L}^{{\prime}}: \mathcal{S}\rightarrow \mathcal{B}\) and let \(\mathbf{L}^{{\prime}}(\boldsymbol{\Xi }_{u}) =\{ \mathcal{E}_{0},\mathcal{E}_{1},\mathcal{E}_{2},\mathcal{E}_{3}\}\) and \(\mathbf{L}^{{\prime}}(\boldsymbol{\Xi }_{u^{{\prime}}}) =\{ \mathcal{E}_{0}^{{\prime}},\mathcal{E}_{1}^{{\prime}},\mathcal{E}_{2}^{{\prime}},\mathcal{E}_{3}^{{\prime}}\}\) with \(\mathbf{L}^{{\prime}}(\boldsymbol{\Xi }_{u}) = u\mathbf{L}^{{\prime}}(\boldsymbol{\Xi }_{o})u^{-1}\), \(\mathbf{L}(\boldsymbol{\Xi }_{u^{{\prime}}}) = u^{{\prime}}\mathbf{L}^{{\prime}}(\boldsymbol{\Xi }_{o})u^{{\prime}-1}\) be two arbitrary basis for \(\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{4,1}\).
As we already know \(f_{\boldsymbol{\Xi }_{0}} = \frac{1} {2}(1 + \mathcal{E}_{0})\frac{1} {2}(1 + \mathfrak{i}\mathcal{E}_{1}\mathcal{E}_{2})\) [Eq. ( 3.48)] is a primitive idempotent of \(\mathbb{R}_{4,1} \simeq \mathbb{C}(4)\). If \(u \in \mathrm{ Spin}\left (1,3\right ) \subset \mathrm{ Spin}(4,1)\) then all ideals \(I_{\boldsymbol{\Xi }_{u}} = I_{\boldsymbol{\Xi }_{0}}u^{-1}\) are geometrically equivalent to \(I_{\boldsymbol{\Xi }_{0}}\). From Eq. ( 3.49) we can write
where
and
Since \(\Psi _{\boldsymbol{\Xi }_{u^{{\prime}}}} = \Psi _{\boldsymbol{\Xi }_{u}}(u^{{\prime}-1}u)^{-1}\), we get
Then
where \(S_{ik}(u^{-1}u^{{\prime}})\) are the matrix components of the representation in \(\mathbb{C}(4)\) of (\(u^{-1}u^{{\prime}}\)) \(\in \mathrm{ Spin}_{1,3}^{e}\). As proved in [10, 11] the matrices S(u) correspond to the representation D (1∕2, 0) ⊕ D (0, 1∕2) of \(\mathrm{Sl}(2, \mathbb{C}) \simeq \mathrm{ Spin}_{1,3}^{e}\).
Remark 3.51
We remark that all the elements of the set \(\{I_{\boldsymbol{\Xi }_{u}}\}\) of the ideals geometrically equivalent to \(I_{\boldsymbol{\Xi }_{0}}\) under the action of \(u \in \mathrm{ Spin}_{1,3}^{e} \subset \mathrm{ Spin}_{4,1}^{e}\) have the same image \(I = \mathbb{C}(4)f\) where f is given by Eq. ( 3.47), i.e.,
where γ μ , μ = 0, 1, 2, 3 are the Dirac matrices given by Eq. ( 3.50). Then, if
it follows that
for all \(\{\mathcal{E}_{\mu }\}\), \(\{\mathcal{E}_{\mu }^{{\prime}}\}\) such that \(\mathcal{E}_{\mu }^{{\prime}} = (u^{{\prime}-1}u)\mathcal{E}_{\mu }(u^{{\prime}-1}u)^{-1}\). Observe that all information concerning the geometrical images of the algebraic spin frames \(\boldsymbol{\Xi }_{u}\), \(\boldsymbol{\Xi }_{u^{{\prime}}},\ldots,\) under \(\mathbf{L}^{{\prime}}\) disappear in the matrix representation of the ideals \(I_{\boldsymbol{\Xi }_{u}},I_{\boldsymbol{\Xi }_{u^{{\prime}}}},\ldots.,\) in \(\mathbb{C}(4)\) since all these ideals are mapped in the same ideal \(I = \mathbb{C}(4)f\).
Taking into account Remark 3.51 and taking into account the definition of algebraic spinors given above and Eq. ( 3.86) we are lead to the following
Definition 3.52
A covariant Dirac spinor for \(\mathbb{R}^{1,3}\) is an equivalence class of pairs \((\Xi _{u}^{m},\boldsymbol{\Psi })\), where \(\Xi _{u}^{m}\) is a matrix algebraic spin frame associated to the algebraic spin frame \(\boldsymbol{\Xi }_{u}\) through the \(S(u^{-1}) \in D^{(\frac{1} {2},0)} \oplus D^{(0,\frac{1} {2} )}\) representation of \(\mathrm{Spin}_{1,3}^{e},u \in \mathrm{ Spin}_{1,3}^{e}\). We say that \(\boldsymbol{\Psi,\Psi }^{{\prime}}\in \mathbb{C}(4)f\) are equivalent and write
if and only if,
Remark 3.53
The definition of CDS just given agrees with that given in [6] except for the irrelevant fact that there, as well as in the majority of Physics textbook’s, authors use as the space of representatives of a CDS a complex four-dimensional space \(\mathbb{C}^{4}\) instead of \(I = \mathbb{C}(4)f\).
3.7.2 Canonical Form of a Dirac-Hestenes Spinor
Let \(\mathbf{v} \in \mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) be a non lightlike vector, i.e., v 2 = v ⋅ v ≠ 0 and consider a linear mapping
with \(\psi \in \mathbb{R}_{1,3}\) and \(\rho \in \mathbb{R}^{+}\). Now, recall that if \(R \in \mathrm{ Spin}_{1,3}^{e}\) then \(\mathbf{w} = R\mathbf{v}\tilde{R}\) is such that w 2 = v 2. It follows that the most general solution of Eq. ( 3.91) is
where \(\beta \in \mathbb{R}\) is called the Takabayasi angle and \(\mathbf{e}_{5} = \mathbf{e}_{0}\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3} \in \bigwedge \nolimits ^{4}\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) is the pseudoscalar of the algebra. Now, Eq. ( 3.92) shows that \(\psi \in \mathbb{R}_{1,3}^{0} \simeq \mathbb{R}_{3,0}.\) Moreover, we have that \(\psi \tilde{\psi }\neq 0\) since
3.7.2.1 The Secret
Now, let \(\psi _{\boldsymbol{\Xi }_{u}}\) be a representative of a Dirac-Hestenes spinor (Definition 3.50) in a given spin frame \(\boldsymbol{\Xi }_{u}\). Since \(\psi _{\boldsymbol{\Xi }_{u}} \in \mathbb{R}_{1,3}^{0} \simeq \mathbb{R}_{3,0}\) we have disclosed the real geometrical meaning of a Dirac-Hestenes spinor. Indeed, a Dirac-Hestenes spinor such that \(\psi _{\boldsymbol{\Xi }_{u}}\tilde{\psi }_{\boldsymbol{\Xi }_{u}}\neq 0\) induces the linear mapping given by Eq. ( 3.91), which rotates a vector and dilate it. Observe, that even if we started our considerations with \(\mathbf{v} \in \mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) and v 2 ≠ 0, the linear mapping ( 3.91) also rotates and ‘dilate’ a light vector.
3.7.3 Bilinear Invariants and Fierz Identities
Definition 3.54
Given a representative \(\psi _{\boldsymbol{\Xi }_{u}}\) of a DHS in the algebraic spin frame field \(\Xi _{u}\) the bilinear invariantsFootnote 16 associated with it are the objects: \(\sigma -\star \omega \in (\bigwedge \nolimits ^{0}\mathbb{R}^{1,3} +\bigwedge \nolimits ^{4}\mathbb{R}^{1,3})\hookrightarrow \mathbb{R}_{1,3}\), \(J = J_{\mu }\mathbf{e}^{\mu } \in \mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\), \(S = \frac{1} {2}S_{\mu \nu }\mathbf{e}^{\mu }\mathbf{e}^{\nu } \in \bigwedge \nolimits ^{2}\mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{ 1,3}\), \(K = K_{\mu }\mathbf{e}^{\mu } \in \mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\) such that
and where \(\star \omega = -\mathbf{e}_{5}\rho \sin \beta\)
The bilinear invariants satisfy the so called Fierz identities, which are
Exercise 3.55
Prove the Fierz identities.
3.7.4 Reconstruction of a Spinor
The importance of the bilinear invariants is that once we know ω, \(\sigma\), J, K and F we can recover from them the associate covariant Dirac spinor (and thus the DHS) except for a phase. This can be done with an algorithm due to Crawford [7] and presented in a very pedagogical way in [14–16]. Here we only give the result for the case where \(\sigma\) and/or ω are non null. Define the object \(\mathfrak{B} \in \mathbb{C} \otimes \mathbb{R}_{1,3} \simeq \mathbb{R}_{4,1}\) called boomerang and given by (\(\mathrm{i} = \sqrt{-1}\))
Then, we can construct \(\boldsymbol{\Psi } = \mathfrak{B}f \in \mathbb{R}_{4,1}f\), where f is the idempotent given by Eq. ( 3.47) which has the following matrix representation in \(\mathbb{C}(4)\) (once the standard representation of the Dirac gamma matrices are used)
Now, it can be easily verified that \(\boldsymbol{\Psi } = \mathfrak{B}f\) determines the same bilinear covariants as the ones determined by \(\psi _{\boldsymbol{\Xi }_{u}}\). Note however that this spinor is not unique. In fact, \(\mathfrak{B}\) determines a class of elements \(\mathfrak{B}\xi\) where \(\xi\) is an arbitrary element of \(\mathbb{R}_{4,1}f\) which differs one from the other by a complex phase factor.
3.7.5 Lounesto Classification of Spinors
A very interesting classification of spinors have been devised by Lounesto [14–16] based on the values of the bilinear invariants. He identified six different classes and proved that there are no other classes based on distinctions between bilinear covariants. Lounesto classes are:
-
1.
\(\sigma \neq 0,\;\;\;\omega \neq 0\).
-
2.
\(\sigma \neq 0,\;\;\;\omega = 0\).
-
3.
\(\sigma = 0,\;\;\;\omega \neq 0\).
-
4.
\(\sigma = 0 =\omega,\;\;\;K\neq 0,\;\;\;S\neq 0\).
-
5.
\(\sigma = 0 =\omega,\;\;\;K = 0,\;\;\;S\neq 0\).
-
6.
\(\sigma = 0 =\omega,\;\;\;K\neq 0,\;\;\;S = 0\).
The current density J is always non-zero. Type 1, 2 and 3 spinor are denominated Dirac spinor for spin-1/2 particles and type 4, 5, and 6 are singular spinors respectively called flag-dipole, flagpole and Weyl spinor. Majorana spinor is a particular case of a type 5 spinor. It is worthwhile to point out a peculiar feature of types 4, 5 and 6 spinor: although J is always non-zero, we have due to Fierz identities that J 2 = −K 2 = 0.
Spinors belonging to class 4 have not previously been identified in the literature. For the applications we have in mind we are interested (besides Dirac spinors which belong to classes 1 or 2 or 3) in spinors belonging to classes 5 and 6, respectively the Majorana and Weyl spinors.
Remark 3.56
In [2] Ahluwalia-Khalilova and Grumiller introduced from physical considerations a supposedly new kind of spinors representing dark matter that they dubbed ELKO spinors. The acronym stands for the German word Eigenspinoren des Ladungskonjugationsoperators. It has been proved in [9] that from the algebraic point of view ELKO spinors are simply class 5 spinors. In [2] it is claimed that differently from the case of Dirac, Majorana and Weyl spinor fields which have mass dimension 3∕2, ELKO spinor fields must have mass dimension 1 and thus instead of satisfying Dirac equation satisfy a Klein-Gordon equation. A thoughtful analysis of this claim is given in Chap. 16.
3.8 Majorana and Weyl Spinors
Recall that for Majorana spinors \(\sigma = 0\), ω = 0, K = 0, S ≠ 0, J, ≠ 0
Given a representative ψ of an arbitrary Dirac-Hestenes spinor we may construct the Majorana spinors
Note that defining an operator C : ψ ↦ ψ e 01 (charge conjugation) we have
i.e., Majorana spinors are eigenvectors of the charge conjugation operator. Majorana spinors satisfy
For Weyl spinors \(\sigma =\omega = S = 0\) and K ≠ 0, J ≠ 0.
Given a representative ψ of an arbitrary Dirac-Hestenes spinor we may construct the Weyl spinors
Weyl spinors are ‘eigenvectors’ of the chirality operator e 5 = e 0 e 1 e 2 e 3, i.e.,
We have also,
For future reference we introduce the parity operator acting on the space of Dirac- Hestenes spinors. The parity operator P in this formalism [13] is represented in such a way that for \(\psi \in \mathbb{R}_{1,3}^{0}\)
The following Dirac-Hestenes spinors are eigenstates of the parity operator with eigenvalues ± 1:
where \(\psi _{\pm }:=\psi _{ W}^{\pm }\)
3.9 Dotted and Undotted Algebraic Spinors
Dotted and undotted covariant spinor fields are very popular subjects in General Relativity. Dotted and undotted algebraic spinor fields may be introduced using the methods of Chap. 7 and are briefly discussed in Exercise 7.63. A preliminary to that job is a deep understanding of the algebraic aspects of those concepts, i.e., the dotted and undotted algebraic spinors which we now discuss. Their relation with Weyl spinors will become apparent in a while.
Recall that the spacetime algebra \(\mathbb{R}_{1,3}\) is the real Clifford algebra associated with Minkowski vector space \(\mathbb{R}^{1,3}\), which is a four dimensional real vector space, equipped with a Lorentzian bilinear form
Let {e 0, e 1, e 2, e 3} be an arbitrary orthonormal basis of \(\mathbb{R}^{1,3}\), i.e.,
where the matrix with entries η μ ν is the diagonal matrix diag(1, −1, −1, −1). Also, {e 0, e 1, e 2, e 3} is the reciprocal basis of {e 0, e 1, e 2, e 3}, i.e., \(\boldsymbol{\eta }(\mathbf{e}^{\mu },\mathbf{e}_{\nu }) =\delta _{ \nu }^{\mu }\). We have in obvious notation
where the matrix with entries η μ ν is the diagonal matrix diag(1, −1, −1, −1).
The spacetime algebra \(\mathbb{R}_{1,3}\) is generate by the following algebraic fundamental relation
As we already know (Sect. 3.7.1) the spacetime algebra \(\mathbb{R}_{1,3}\) as a vector space over the real field is isomorphic to the exterior algebra \(\bigwedge \mathbb{R}^{1,3} = \bigoplus \nolimits _{j=0}^{4}\bigwedge \nolimits ^{j}\mathbb{R}^{1,3}\) of \(\mathbb{R}^{1,3}\). We code that information writing \(\bigwedge \mathbb{R}^{1,3}\hookrightarrow \mathbb{R}_{1,3}\). Also, we make the following identifications: \(\bigwedge \nolimits ^{0}\mathbb{R}^{1,3} \equiv \mathbb{R}\) and \(\bigwedge \nolimits ^{1}\mathbb{R}^{1,3} \equiv \mathbb{R}^{1,3}\). Moreover, we identify the exterior product of vectors by
and also, we identify the scalar product of vectors by
Then we can write
Now, an arbitrary element \(\mathbf{C} \in \mathbb{R}_{1,3}\) can be written as sum of nonhomogeneous multivectors, i.e.,
where \(s,c_{\mu },c_{\mu \nu },c_{\mu \nu \rho },p \in \mathbb{R}\) and c μ ν , c μ ν ρ are completely antisymmetric in all indices. Also e 5 = e 0 e 1 e 2 e 3 is the generator of the pseudoscalars. Recall also that as a matrix algebra we have that \(\mathbb{R}_{1,3} \simeq \mathbb{H}(2)\), the algebra of the 2 × 2 quaternionic matrices.
3.9.1 Pauli Algebra
Next, we recall (again) that the Pauli algebra \(\mathbb{R}_{3,0}\) is the real Clifford algebra associated with the Euclidean vector space \(\mathbb{R}^{3,0}\), equipped as usual, with a positive definite bilinear form. As a matrix algebra we have that \(\mathbb{R}_{3,0} \simeq \mathbb{C}\left (2\right )\), the algebra of 2 × 2 complex matrices. Moreover, we recall that \(\mathbb{R}_{3,0}\) is isomorphic to the even subalgebra of the spacetime algebra, i.e., writing \(\mathbb{R}_{1,3} = \mathbb{R}_{1,3}^{(0)} \oplus \mathbb{R}_{1,3}^{(1)}\) we have,
The isomorphism is easily exhibited by putting \(\boldsymbol{\sigma }^{i} = \mathbf{e}^{i}\mathbf{e}^{0}\), i = 1, 2, 3. Indeed, with δ ij = diag(1, 1, 1), we have
which is the fundamental relation defining the algebra \(\mathbb{R}_{3,0}\). Elements of the Pauli algebra will be called Pauli numbers.Footnote 17 As a vector space over the real field, we have that \(\mathbb{R}_{3,0}\) is isomorphic to \(\bigwedge \mathbb{R}^{3,0}\hookrightarrow \mathbb{R}_{3,0} \subset \mathbb{R}_{1,3}\). So, any Pauli number can be written as
where \(s,p_{i},p_{ij},p \in \mathbb{R}\) and p ij = −p ji and also
Note that i 2 = −1 and that i commutes with any Pauli number. We can trivially verify
In that way, writing \(\mathbb{R}_{3,0} = \mathbb{R}_{3,0}^{(0)} + \mathbb{R}_{3,0}^{(1)}\), any Pauli number can be written as
with
3.9.2 Quaternion Algebra
Equation ( 3.121) show that the quaternion algebra \(\mathbb{R}_{0,2} = \mathbb{H}\) can be identified as the even subalgebra of \(\mathbb{R}_{3,0}\), i.e.,
The statement is obvious once we identify the basis \(\{1,\mathit{\hat{\imath}},\mathit{\hat{\jmath}},\mathbf{\mathit{\hat{k}}}\}\) of \(\mathbb{H}\) with
which are the generators of \(\mathbb{R}_{3,0}^{(0)}\). We observe moreover that the even subalgebra of the quaternions can be identified (in an obvious way) with the complex field, i.e., \(\mathbb{R}_{0,2}^{(0)} \simeq \mathbb{C}\). Returning to Eq. ( 3.117) we see that any \(\mathbf{P} \in \mathbb{R}_{3,0}\) can also be written as
where
with \(l_{k} = -\varepsilon _{k}^{i\text{ }j}p_{ij} \in \mathbb{R}\). The important fact that we want to emphasize here is that the subspaces \((\mathbb{R}\oplus \bigwedge \nolimits ^{1}\mathbb{R}^{3,0})\) and \((\bigwedge \nolimits ^{2}\mathbb{R}^{3,0} \oplus \bigwedge \nolimits ^{3}\mathbb{R}^{3,0})\) do not close separately any algebra. In general, if \(\mathbf{A,C} \in (\mathbb{R}\oplus \bigwedge \nolimits ^{1}\mathbb{R}^{3,0})\) then
To continue, we introduce
Then, i \(= -\boldsymbol{\sigma }_{1}\boldsymbol{\sigma }_{2}\boldsymbol{\sigma }_{3}\) and the basis \(\{1,\hat{\imath}\) \(,\hat{\jmath},\hat{k}\}\) of \(\mathbb{H}\) can be identified with {1, − i \(\boldsymbol{\sigma }_{1},\mathbf{-}\) i \(\boldsymbol{\sigma }_{2}\), \(\mathbf{-}\text{I}\boldsymbol{\sigma }_{3}\}\).
Now, we know that \(\mathbb{R}_{3,0} \simeq \mathbb{C}\left (2\right )\). This permit us to represent the Pauli numbers by 2 × 2 complex matrices, in the usual way (\(\mathrm{i} = \sqrt{-1}\)). We write \(\mathbb{R}_{3,0} \ni \mathbf{P}\mapsto P \in \mathbb{C}(2)\), with
3.9.3 Minimal Left and Right Ideals in the Pauli Algebra and Spinors
The elements \(\mathrm{e}_{\pm } = \frac{1} {2}(1 +\boldsymbol{\sigma } _{3}) = \frac{1} {2}(1 + \mathbf{e}_{3}\mathbf{e}_{0}) \in \mathbb{R}_{1,3}^{(0)} \simeq \mathbb{R}_{ 3,0}\), \(\mathrm{e}_{\pm }^{2} =\mathrm{ e}_{\pm }\) are minimal idempotents of \(\mathbb{R}_{3,0}\). They generate the minimal left and right ideals
From now on we write e = e+. It can be easily shown (see below) that, e.g., I=I + has the structure of a 2-dimensional vector space over the complex field [10, 13], i.e., \(\mathbf{I} \simeq \mathbb{C}^{2}\). The elements of the vector space I are called representatives of algebraic contravariant undotted spinors Footnote 18 and the elements of \(\mathbb{C}^{2}\) are the usual contravariant undotted spinors used in physics textbooks. They carry the \(D^{(\frac{1} {2},0)\text{ }}\) representation of \(\mathrm{Sl}(2, \mathbb{C})\) [17]. If \(\boldsymbol{\varphi }\mathbf{\in I}\) we denote by \(\varphi \in \mathbb{C}^{2}\) the usual matrix representativeFootnote 19 of \(\boldsymbol{\varphi }\) is
Denoting by \(\mathbf{\dot{I}} =\mathrm{ e}\mathbb{R}_{1,3}^{(0)}\) the space of the algebraic covariant dotted spinors, we have the isomorphism, \(\mathbf{\dot{I} \simeq (}\mathbb{C}^{2})^{\dag }\simeq \mathbb{C}_{2}\), where † denotes Hermitian conjugation. The elements of \((\mathbb{C}^{2})^{\dag }\) are the usual contravariant spinor fields used in physics textbooks. They carry the \(D^{(0,\frac{1} {2} )\text{ }}\) representation of \(\mathrm{Sl}(2, \mathbb{C})\) [17]. If \(\mathop{\boldsymbol{\xi }}\limits^{\cdot }\in \mathbf{\dot{I}}\), then its matrix representation in \((\mathbb{C}^{2})^{\dag }\) is a row matrix usually denoted by
The following representation of \(\mathop{\boldsymbol{\xi }}\limits^{\cdot }\in \mathbf{\dot{I}}\) in \((\mathbb{C}^{2})^{\dag }\) is extremely convenient. We say that to a covariant undotted spinor \(\xi\) there corresponds a covariant dotted spinor \(\dot{\xi }\) given by
with
We can easily find a basis for I and \(\mathbf{\dot{I}}\). Indeed, since \(\mathbf{I} = \mathbb{R}_{1,3}^{(0)}\mathrm{e}\) we have that any \(\boldsymbol{\varphi }\mathbf{\in I}\) can be written as
where
Analogously we find that any \(\mathop{\boldsymbol{\xi }}\limits^{\cdot }\in \mathbf{\dot{I}}\) can be written as
Defining the mapping
we have
From this it follows the identification
and then, each Pauli number can be written as an appropriate sum of Clifford products of algebraic contravariant undotted spinors and algebraic covariant dotted spinors. And, of course, a representative of a Pauli number in \(\mathbb{C}^{2}\) can be written as an appropriate Kronecker product of a complex column vector by a complex row vector.
Take an arbitrary \(\mathbf{P \in }\mathbb{R}_{3,0}\) such that
where \(p^{_{\mathbf{k}_{1}\mathbf{k}_{2}\ldots \mathbf{k}_{j}}} \in \mathbb{R}\) and
With the identification \(\mathbb{R}_{3,0} \simeq \mathbb{R}_{1,3}^{(0)} \simeq \mathbf{I} \otimes _{\mathbb{C}}\mathbf{\dot{I}}\), we can also write
where the \(\mathbf{P}_{\ \dot{B}}^{A} = \mathbf{X}_{\ \dot{B}}^{A} + \mathbf{iY}_{\ \dot{B}}^{A}\), \(\mathbf{X}_{\ \dot{B}}^{A},\mathbf{Y}_{\ \dot{B}}^{A} \in \mathbb{R}\).
Finally, the matrix representative of the Pauli number \(\mathbf{P} \in \mathbb{R}_{3,0}\) is \(P \in \mathbb{C}(2)\) given by
with \(P_{\ \dot{B}}^{A} \in \mathbb{C}\) and
It is convenient for our purposes to introduce also covariant undotted spinors and contravariant dotted spinors. Let \(\varphi \in \mathbb{C}^{2}\) be given as in Eq. ( 3.130). We define the covariant version of undotted spinor \(\varphi \in \mathbb{C}^{2}\) as \(\varphi ^{{\ast}} \in (\mathbb{C}^{2})^{t} \simeq \mathbb{C}_{2}\) such that
whereFootnote 20 \(\varepsilon _{AB} =\varepsilon ^{AB} =\mathrm{ adiag}(1,-1)\). We can write due to the above identifications that there exists \(\varepsilon \in \mathbb{C}(2)\) given by Eq. ( 3.133) which can be written also as
where \(\boxtimes \) denotes here the Kronecker product of matrices. We have, e.g.,
We now introduce the contravariant version of the dotted spinor
as being \(\dot{\xi }^{{\ast}}\in \mathbb{C}^{2}\) such that
where \(\varepsilon _{\dot{A}\dot{B}} =\varepsilon ^{\dot{A}\dot{B}} =\mathrm{ adiag}(1,-1)\). Then, due to the above identifications we see that there exists \(\dot{\varepsilon }\in \mathbb{C}(2)\) such that
Also, recall that even if {s A },\(\{\mathbf{s}_{\dot{A}}\}\) and \(\{s^{\dot{A}}\}\),{s A} are bases of distinct spaces, we can identify their matrix representations, as it is obvious from the above formulas. So, we have \(s_{A} \equiv s_{\dot{A}}\) and also \(s^{\dot{A}} = s^{A}\). This is the reason for the representation of a dotted covariant spinor as in Eq. ( 3.132). Moreover, the above identifications permit us to write the matrix representation of a Pauli number \(\mathbf{P} \in \mathbb{R}_{3,0}\) as, e.g.,
besides the representation given by Eq. ( 3.142).
Exercise 3.57
Consider the ideal \(I = \mathbb{R}_{1,3}\frac{1} {2}(1 -\mathbf{e}_{0}\mathbf{e}_{3})\). Show that ϕ ∈ I is a representative in a spin frame \(\boldsymbol{\Xi }_{u}\) of a covariant Dirac spinorFootnote 21 \(\Psi \in \mathbb{C}(4)\frac{1} {2}(1 +\underline{\gamma } _{0})(1 +\mathrm{ i}\underline{\gamma }_{1}\underline{\gamma }_{2})\). Let ψ be the representative (in the same spin frame \(\boldsymbol{\Xi }_{u}\)) of a Dirac-Hestenes spinor, associated to a mother spinor \(\Phi \in \mathbb{R}_{1,3}\frac{1} {2}(1 + \mathbf{e}_{0})\) by \(\Phi =\psi \frac{1} {2}(1 + \mathbf{e}_{0})\). Show that ϕ ∈ I can be written as \(\phi =\psi \frac{1} {2}(1 + \mathbf{e}_{0})\frac{1} {2}(1 -\mathbf{e}_{0}\mathbf{e}_{3})\).
-
(a)
Show that ϕ e 5 = ϕ e 21.
-
(b)
Weyl spinors are defined as eigenspinors of the chirality operator, i.e., \(\underline{\gamma }_{5}\Psi _{\pm } = \pm \mathrm{i}\Psi _{\pm }\). Show that Weyl spinors corresponds to the even and odd parts of ϕ.
-
(c)
Relate the even and odd parts of ϕ to the algebraic dotted and undotted spinors.Footnote 22
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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