2-Spinors, Dirac Spinors and Spacetime Geometry

2-spinors, Dirac spinors and spacetime geometry

(private notes - November 19, 2017)

D. Canarutto Dipartimento di Matematica e Informatica “U. Dini”, http://www.dma.uniﬁ.it/˜canarutto

November 19, 2017

Contents

1 Two-spinors 1 1.1 Complex spaces and conjugate spaces ...... 1 1.2 Hermitian tensors ...... 2 1.3 Two-spinor space ...... 4 1.4 2-spinors and Minkowski space ...... 6 1.5 The natural Dirac map ...... 8 1.6 Riemann sphere and celestial sphere ...... 10

2 2-spinors and 4-spinors 12 2.1 4-spinor bases ...... 13 2.2 Conjugation and Dirac adjoint ...... 13 2.3 The Dirac map in 4-spinor terms ...... 14

3 Additional structures 16 3.1 Observers and positive Hermitian metrics ...... 16 3.2 Charge conjugation ...... 18 3.3 Parity and time reversal ...... 19 3.4 Spin operators ...... 20 3.5 Particle momentum in 2-spinor form ...... 24 3.6 Helicity of Dirac Fermions ...... 27

4 Two-spinor bundles 31 4.1 Two-spinor connections ...... 31 4.2 Two-spinor tetrad ...... 36 4.3 Cotetrad ...... 37 4.4 Tetrad and connections ...... 38 4.5 The Dirac operator ...... 40 4.6 Fermi transport of spinors ...... 41 4.7 Flat spacetime ...... 47

I II CONTENTS

5 Optical geometry in curved spacetime 48 5.1 Real optical algebra ...... 48 5.2 Complexiﬁed optical algebra ...... 51 5.3 Optical spin ...... 53 5.4 Optical bundles ...... 54

6 2-spinor groups and 4-spinor groups 55 6.1 2-spinor groups ...... 55 6.2 2-spinor groups and Lorentz group ...... 60 6.3 Action of 2-spinor groups on 4-spinor space ...... 65 6.4 Dirac algebra in 2-spinor terms ...... 67 6.5 Cliﬀord group in 2-spinor terms ...... 76 6.6 4-spinor groups in 2-spinor terms ...... 83 6.7 Bundle structure of W ...... 86

A Spinors: further details and calculations 93 A.1 Complex spaces as real spaces ...... 93 A.2 Real volume forms on spinor spaces ...... 94 A.3 Two-spinor exchange formulas ...... 98 A.4 Pauli matrices ...... 99 A.5 Spinors as endomorphisms ...... 101 A.6 Spin groups ...... 102 A.7 4-momentum and spinors ...... 109 A.8 World-spinors ...... 115 A.9 Tensor algebra of world-spinors ...... 117 A.10 Spinor connection ...... 120 A.11 Spinor Laplacian ...... 123

B Clifford algebra 125 B.1 Clifford algebra ...... 125 B.2 Practical Clifford product calculations ...... 126 B.3 Clifford group ...... 129 B.4 Dirac algebra ...... 130 B.5 4-spinors ...... 131 B.6 Complexified spinor groups ...... 133 1

1 Two-spinors

1.1 Complex spaces and conjugate spaces

If A is a set and f : A → C is any map, then the conjugated map f¯ : A → C is deﬁned ¯ 1 ¯ 1 ¯ by f(a) ≡ f(a), a ∈ A. Moreover one sets tensor product and L(V , W ), A(V , W ) for the spaces of complex- linear and antilinear maps V → W , respectively (f : V → W is said to be antilinear if it is real-linear and f(iv) = −if(v) for all v ∈ V ). In particular one has the complex-dual and antidual spaces V F ≡ L(V , C) , V F ≡ A(V , C) , and the natural conjugation anti-isomorphism

V F → V F : λ 7→ λ¯ .

Next, consider bi(anti)dual spaces. Set V ≡ V F F, and call this the conjugate space of V . One obtains the isomorphisms

V =∼ V FF =∼ V F F , V ≡ V F F =∼ V FF .

In fact, any v ∈ V can be viewed as the linear map V F → C : λ 7→ hλ, vi or as the antilinear map V F → C : α 7→ hα,¯ vi; andv ¯ ∈ V can be viewed as the linear map V F → C : α 7→ hα, v¯i ≡ α(v). Summarizing, one has the four spaces

V ↔ V , V F ↔ V F , where the arrows indicate the conjugation anti-isomorphisms. Conjugated, dual and antidual spaces of these yield, up to natural isomorphisms, no new space: when the symbols F , F and are attached to V , in any number and in any order, an odd (even) number of ‘barred’ symbols yields a (non-)barred space, an odd (even) number of starred symbols yields a (non-)starred space. In particular, V F can be written as V F.

Remark. The symbols F , F and together with the identity constitute the Abelian group Z2 × Z2 (§??). Remark. According to another definition, the conjugate space of V is introduced as the complex vector space, on the same underlying set as V , where the product of a number c by a vector v is given byc ¯ v. This works also for infinite dimensional spaces; when the dimension is finite this definition is equivalent to the above one (the two spaces are isomorphic), but gives rise to notational complications, in particular with regard to coordinate expressions. Also, it seems preferable to maintain the standard use that, for any complex-valued function f and complex number c, the notation c f stands for the map a 7→ c f(a) , and this would conflict with the alternative definition. On the other hand, note that one could also introduce V as the vector space of all antilinear maps C → V . One has the usual identifications between tensor products and spaces of (multi)linear maps. In particular

L(V , W ) =∼ W ⊗ V F , A(V , W ) =∼ L(V , W ) =∼ W ⊗ V F . 2 1 TWO-SPINORS

The conjugation map can be extended to any tensor product by means of the universal property (namely it is applied to each factor). Contractions in tensor products are deﬁned when the contracted factors are either V and V F, or V and V F. A F Let (bA), 1 ≤ A ≤ dim V , be a basis of V , and (b ) its dual basis of V . The corresponding indices in the conjugate spaces are distinguished by a dot, namely one writes

¯ ¯A˙ A bA˙ ≡ bA , b ≡ b ,

¯ ¯A˙ F F so that {bA˙} is a basis of V and {b } its dual basis of V . For v ∈ V and λ ∈ V one has

A A˙ ¯ v = v bA , v¯ =v ¯ bA˙ , A ¯ ¯ ¯A˙ λ = λAb , λ = λA˙b ,

A˙ A ¯ wherev ¯ = v , λA˙ ≡ λA and Einstein summation convention is used. Remark. In some literature one ﬁnds also the convention of writing vA˙ instead ofv ¯A˙, and the like. This makes many formulas appear simpler, but in some cases may give rise to inconsistencies. On the other hand, one may question the utility of the ‘dotted indices’ con- ¯A ¯ vention: one might write simplyv ¯ = λ bA and the like, possibly avoiding misunderstandings, in complicate formulas, by explicitely carrying along the involved basis. However any index convention, including the standard one for Riemannian geometry (high and low indices), is never of a fundamental nature: its role is just that of making calculations easier and formulas more readable. If K is the matrix of an automorphism of V , then K¯ , K F (transposed matrix) and K¯ F (antitransposed matrix) are the matrices of the induced automorphisms of V , V F and V F, respectively. If τ is a tensor with indices of various types, then its components transform with the above matrices, each one for the appropriate type. The indices ofτ ¯ are all of dot-reversed type. Contractions in tensor products are indicated by the usual Einstein summation convention. Observe that dotted indices cannot be contracted with non-dotted indices.

1.2 Hermitian tensors Consider the space V ⊗ V ; this has a natural real linear (complex anti-linear) involution w 7→ w†, which on decomposable tensors reads

(u ⊗ v¯)† = v ⊗ u¯ .

Hence one has the natural decomposition of V ⊗ V into the direct sum of the real eigenspaces of the involution with eigenvalues ±1, respectively called the Hermitian and anti-Hermitian subspaces, namely V ⊗ V = (V ∨¯ V ) ⊕ i (V ∨¯ V ) . In other terms, the Hermitian subspace V ∨¯ V is constituted by all w ∈ V ⊗ V such that w† = w, while an arbitrary w is uniquely decomposed into the sum of an Hermitian and an anti-Hermitian tensor as 1 † 1 † w = 2 (w + w ) + 2 (w − w ) . 1 Note that w is Hermitian (resp. anti-Hermitian) iﬀ its real part

(V ∨¯ V )∗ =∼ V F ∨¯ V F , (i V ∨¯ V )∗ =∼ i V F ∨¯ V F , where ∗ denotes the real dual. A Hermitian 2-form is deﬁned to be a Hermitian tensor h ∈ V F ∨¯ V F. The associated quadratic form v 7→ h(v, v) is real-valued. The notions of signature and non-degeneracy of Hermitian 2-forms are introduced similarly to the case of real bilinear forms. If h is non- degenerate then it yields the isomorphism h[ : V → V F :v ¯ 7→ h(¯v, ); its conjugate map is an anti-isomorphism V → V F which, via composition with the canonical conjugation, can be seen as the isomorphism h¯[ : V → V F : v 7→ h( , v). The inverse isomorphisms h# and h¯# are similarly related to a Hermitian tensor h−1 ∈ V ∨¯ V . One has the coordinate expressions

¯A˙ A −1 AA˙ ¯ h = hAA˙b ⊗ b , h = h bA˙⊗ bA ,

[ A˙ ¯[ B ¯ B (h (¯v))B = hA˙B v¯ , (h (v))A˙ = hA˙B v = hBA˙v ,

# ¯ B A˙B ¯ ¯# A˙ A˙B ¯BA˙ (h (λ)) = h λA˙ , (h (λ)) = h λB = h λB , where C˙A A A˙C A˙ h hC˙B = δ B , h hB˙C = δ B˙ . Namely one has an index lowering and raising formalism similar to the pseudo-Euclidean case, but the operation changes the index’ kind from dotted to non-dotted and vice-versa. I’ll use the notations1

v† ≡ v¯[ ≡ v[ ∈ V F , λ† ≡ λ¯# ≡ λ† ∈ V .

The Hermitian conjugate of f ∈ End(V ) =∼ V ⊗ V F with respect to h is deﬁned to be

f † ≡ h# ◦ f¯F ◦ h¯[ ∈ End(V ) , where f¯F ∈ End(V F) is the anti-transpose of f. In coordinates

† A C˙A ¯D˙ (f ) B = h hD˙B f C˙ .

Equivalently, the deﬁnition of f † can be expressed as

h(f †u, v) = h(u, fv) , u, v ∈ V , h(f †u)†, vi = hu†, fvi , h(f †u)†, vi = hf Fu†, vi i.e. (f †u)† = f Fu† .

Obviously, one has (f †)† = f .

1Usually, particularly in the physics literature, t† and t denote objects which are related by their matrices being the Hermitian conjugate (i.e. transposed conjugate) of each other. For the Hermitian involution in V ⊗ V this is true in any basis. On the other hand, for Hermitian conjugation determined by a Hermitian metric, this relation between matrices of corresponding objects holds in an orthonormal basis. 4 1 TWO-SPINORS

The linear map f ∈ End(V ) is said to be unitary with respect to h if it fulﬁls

h ◦ (f × f) = h , i.e. h(fu)†, fvi = hu†, vi .

Unitarity of f can be equivalently expressed as

† † f ◦ f = f ◦ f = 11V , since2 hu†, vi = h(fu)†, fvi = hu†, f †fvi . Moreover, a unitary f fulﬁls

| det f|2 = 1 .

1.3 Two-spinor space Let S be a 2-dimensional complex vector space. Set Λ2 ≡ ∧2S, a 1-dimensional complex vector space, and identify Λ−2 ≡ Λ2F with ∧2SF through the rule3

0 1 0 2 F 0 ω(s ∧ s ) ≡ 2 ω(s, s ) , ∀ ω ∈ ∧ S , s, s ∈ S , 0 1 0 0 s ∧ s ≡ 2 (s⊗s −s ⊗s) . Then Λ−2 can be viewed as the space of ‘symplectic’ forms on S. Any ω ∈ Λ−2 \{0} has a unique ‘inverse’ or ‘dual’ element ω−1 ≡ ωF, i.e. ω(ω−1) = 1 . Denote by ω[ : S → SF the linear map given by

hω[(s), ti ≡ ω(s, t) , and by ω# : SF → S the linear map given by

hµ, ω#(λ)i ≡ ω−1(λ, µ) .

Then ω# = −(ω[)−1 . Namely the assignment of a non-zero element of Λ2 determines an index lowering and raising formalism, in which however one has to pay some attention to signs. The Hermitian subspace of Λ2 ⊗ Λ2 is a real vector space with a distinguished orientation, whose positively oriented semispace

2 2 2 + 2 L ≡ (Λ ∨¯ Λ ) ≡ {w ⊗ w,¯ w ∈ Λ } has the square root semi-space L, called the space of length units. Next, consider the complex 2-dimensional space

−1/2 U ≡ L ⊗ S . This is our 2-spinor space. Observe that the 1-dimensional space

2 −1 2 Q ≡ ∧ U = L ⊗ ∧ S

2 If V is a finite-dimensional vector space and A, B ∈ End(V ) are such that A ◦ B = 11V , then obviously A −1 −1 −1 −1 −1 and B are automorphisms of V and B ◦ A = (A ◦ B) = 11V . Then also B ◦ A = B ◦ (B ◦ A ) ◦ A = 11V . 3 This contraction, defined in such a way to respect the usual conventions in two-spinor literature, corresponds to half standard tensor-algebra contraction and 1/4 standard exterior-algebra contraction. 1.3 Two-spinor space 5 has a distinguished Hermitian metric, defined as the unit element in

2 F 2 F 2 2 2 ∼ (∧ U ) ∨¯ (∧ U ) = L ⊗ (Λ ∨¯ Λ ) = R . Hence there is the distinguished set of normalized symplectic forms on U, any two of them dif- fering by a phase factor. One says that elements of U and of its tensor algebra are ‘conformally r invariant’, while tensorializing by L one obtains ‘conformal densities’ of weight r. Consider an arbitrary basis

(ξA) of S and its dual basis (xA) of SF . This determines the mutually dual bases

AB 2 w ≡ ε ξA ∧ ξB = 2 ξ1 ∧ ξ2 of Λ ,

∗ A B −2 ∼ 2F w ≡ εAB x ∧ x = 2 x1 ∧ x2 of Λ = Λ .

AB where ε and εAB both denote the antisymmetric Ricci matrix, and the basis √ l ≡ w ⊗ w¯ of L . Then one also has the induced normalized basis

−1/2 (ζA) ≡ (l ⊗ ξA) of U , and its dual basis (zA) ≡ (l1/2 ⊗ xA) of U F ; the induced mutually dual bases

−1 A B −1 2 F ε ≡ l ⊗ w = εAB z ∧ z ∈ Q ≡ ∧ U and ∗ −1 −1 AB 2 ε ≡ ε ≡ l ⊗ w = ε ζA ∧ ζB ∈ Q ≡ ∧ U are then normalized too. The mutually inverse symplectic forms ε and ε∗ determine isomorphisms

ε[ : U → U F , ε# : U F → U by the rules hε[(u), vi ≡ ε(u, v) , hε#(λ), µi ≡ ε∗(λ, µ) , namely4 [ 1 # ∗ 1 ∗ ε (u) = ucε = 2 u|ε , ε (λ) = λcε = 2 λ|ε . These conventions can be checked by coordinate expressions. We have5

A B 1 A B B A A B ε = εAB z ∧ z = 2 εAB (z ⊗ z − z ⊗ z ) = εAB z ⊗ z = = 2 z1 ∧ z2 = z1 ⊗ z2 − z2 ⊗ z1 ,

[ A B 1 2 2 1 A B 1 2 2 1 ε (u) = εAB u z = u z − u z , ε(u, v) = εAB u v = u v − u v ,

4See also the above comment about duality and footnote 3. 5Remember (see [Go69], §I.6) that u|(λ ∧ µ) = (u|λ) ∧ µ + (−1)|λ| λ ∧ (u|µ). 6 1 TWO-SPINORS and similar expressions for ε#, with inverted index positions. In particular we have

[ 2 [ 1 # 1 # 2 ε (ζ1) = z , ε (ζ2) = −z , ε (z ) = ζ2 , ε (z ) = −ζ1 .

For u ∈ U and λ ∈ U F we write

u[ ≡ ε[(u) ∈ U F , λ# ≡ ε#(λ) ∈ U ,

[ A C # C BC uB ≡ (u )B = u εAB , λ ≡ (λ ) = λB ε .

Noting that6 BC C A BC A εAB ε = −δA ⇒ u εAB ε = −u we see that (u[)# = −u . Similarly (λ#)[ = −λ . Remark. In contrast to the usual 2-spinor formalism, the ‘symplectic’ 2-form ε is not a- AB priori fixed. Two-spinor indices may be lowered and raised by εAB and its inverse ε , but this operation depends from the chosen ε, hence on the chosen two-spinor basis (ξA). Moreover note that we use the convention that indices are moved by contraction with the first index in ε —indeed we’ll consistently use the same conventions also for other instances of index-moving operations determined by invertible 2-tensors. Then we find that ε[ and ε# are minus inverse isomorphisms. 4 Remark. An even more specialized space is the neutrino space, obtained by choosing a square root Q1/2 of Q as −1/2 −1 Uν ≡ Q ⊗ U = Λ ⊗ S .

On Uν , one has a distinguished symplectic 2-form. 4

1.4 2-spinors and Minkowski space

The fact that a normalized element ε ∈ QF is unique up to a phase factor, implies that there exists an intrinsic, well-deﬁned object ε ⊗ ε¯ ∈ QF ⊗ QF; this can be also seen as a natural bilinear formg ˜ on U ⊗ U, given for decomposable elements by

g˜(p ⊗ q,¯ r ⊗ s¯) = ε(p, r)ε ¯(¯q, s¯) .

The fact that any ε is non-degenerate implies thatg ˜ is non-degenerate too. In terms of index formalism, one can express this by saying that lowering and raising of couples (AA˙) of mutually conjugate indices, in a normalized basis of U, corresponds to an intrinsic operation.

In a normalized 2-spinor basis (ζA) , one writes

g˜AA˙BB˙ = εAB ε¯A˙B˙ .

AA˙ ¯ Namely, let w = w ζA ⊗ ζA˙ ∈ U ⊗ U ; then

AA˙ BB˙ AA˙ 11 22 12 21 g˜(w, w) = εAB ε¯A˙B˙w w = 2 det w = 2 (w w − w w ) .

1 Remark. The fact thatg ˜ (w) ≡ 2 g˜(w, w) can be written as the determinant of the matrix of w leads one to consider how is this determinant an intrinsecally deﬁned object. First, observe

6In the usual index moving formalism of (pseudo)Riemannian formalism one tends to regard tensors as objects defined independently of the positions of their indices. In other situations, like in the formalism under consideration, such identifications must be handled with greater care. 1.4 2-spinors and Minkowski space 7 that a linear map between different vector spaces has a well-defined determinant relatively to any fixed isomorphism between the two spaces. The case here is just slightly more involved: an element w ∈ U ⊗ U can be viewed as an anti-linear map U F → U, and these two spaces are also related by the anti-isomorphism induced by ε ; the latter is unique up to a phase factor, which is actually compensated by the anti-linearity. Next, consider the Hermitian subspace

H ≡ U ∨¯ U ⊂ U ⊗ U .

This is a 4-dimensional real vector space; for any given normalized basis (ζA) of U consider, in particular, the Pauli basis (τλ) of H associated with (ζA), namely

√1 τλ ≡ 2 σλ ≡ AA˙ ¯ √1 AA˙ ¯ ≡ τ λ ζA ⊗ ζA˙ ≡ 2 σλ ζA ⊗ ζA˙ ,

AA˙ where (σλ ) denotes the λ-th Pauli matrix (§A.4). The restriction ofg ˜ to the Hermitian subspace H (denoted still with the same symbol) turns out to be a Lorentz metric with signature (+, −, −, −); actually, a Pauli basis is readily seen to be orthonormal,7 that is

0 0 g˜λµ ≡ g˜(τλ , τµ) = ηλµ ≡ 2 δλδµ − δλµ .

Moreover the Lorentz metricg ˜ determines, up to sign, a normalized volume formη ˜ on H. AA˙ ¯ One might also say that (σλ ) is the matrix of 11H in the bases (σλ) and (ζA ⊗ ζA˙) ; sim- AA˙ ¯ ilarly, (τ λ ) is the matrix of 11H in the bases (τλ) and (ζA ⊗ ζA˙) . Then one can apply the above said ε-related index moving formalism to these components, and also move (“spacetime”) greek indices viag ˜λµ . λ Let (t ) be the (real) dual basis of (τλ) . Then one obtains

λ √1 λ A A˙ t ≡ 2 σ AA˙z ⊗ ¯z ,

λ √1 λ λ λ AA˙ λ namely t AA˙ = 2 σ AA˙ = τ AA˙, since σ AA˙σµ = 2δ µ as can be easily checked using the formulas reported in §A.4. One also writes

AA˙ BB˙ g˜λµ = τ λ τ µ εAB ε¯A˙B˙ ,

org ˜AA˙BB˙ = εAB ε¯A˙B˙ .

Proposition 1.1 An element w ∈ U ⊗ U = C ⊗ H is null, that is g˜(w, w) = 0 , iﬀ it is a decomposable tensor: w = u ⊗ s¯, u, s ∈ U . proof: A decomposable element u ⊗ s¯ ∈ U ⊗ U is immediately seen to be null (isotropic). In AA˙ ¯ order to show the converse, write w = w ζA ⊗ ζA˙ in a normalized 2-spinor basis. Then (see above) 0 =g ˜(w, w) = detwAA˙ , hence the two rows of the matrix wAA˙ are proportional, 1A˙ 2A˙ 1A˙ ¯ say w = c w , c ∈ C . One then gets w = (ζ1 + c ζ2) ⊗ (w ζA˙).

7 Note that τ0 is a Hermitian metric on U F; conversely, any ε-normalized positive Hermitian metric can be seen as a unit timelike future-pointing vector in H, namely an ‘observer’. 8 1 TWO-SPINORS

A null element in U ⊗ U is also in H iﬀ it is of the form ±u ⊗ u¯. Hence the null cone N ⊂ H is constituted exactly by such elements. Note how this fact yields a way of distinguish between time orientations: by convention, one chooses the future and past null-cones in H to be, respectively,

N + ≡ {u ⊗ u,¯ u ∈ U} , N − ≡ {−u ⊗ u,¯ u ∈ U} .

Proposition 1.2 For each orthonormal positively oriented basis (eλ) , such that e0 is future- oriented, there exists a normalized 2-spinor basis (ζA) whose associated Pauli basis (τλ) coincides with (eλ) . proof: The elements e0+e3 and e0−e3 are isotropic vectors, hence there exist u, v ∈ U such that √ √ e0 + e3 = 2 u ⊗ u¯ , e0 − e3 = 2 v ⊗ v¯ , and one gets

1 √1 e0 = 2 (e0 + e3) + (e0 − e3) = 2 (u ⊗ u¯ + v ⊗ v¯) ,

1 √1 e3 = 2 (e0 + e3) − (e0 − e3) = 2 (u ⊗ u¯ − v ⊗ v¯) .

2 Moreover, one has 1 =g ˜(e0 , e0) = |ε(u, v)| . By multiplying either u or v by an appropriate phase factor, one may redeﬁne them so that ε(u, v) = 1 . The elements

0 √1 0 √i e1 ≡ 2 (u ⊗ v¯ + v ⊗ u¯) , e2 ≡ 2 (−u ⊗ v¯ + v ⊗ u¯) are in the 2-dimensional space orthogonal to e0 and e3 . Now one can redeﬁne u and v , while iθ/2 −iθ/2 leaving all the above relations untouched, by multiplying u by e and v by e , θ ∈ R . One gets the transformation

0 √1 iθ −iθ 0 0 e1 7→ 2 (e u ⊗ v¯ + e v ⊗ u¯) = cos θ e1 − sin θ e2 ,

0 √i iθ −iθ 0 0 e2 7→ 2 (−e u ⊗ v¯ + e v ⊗ u¯) = sin θ e1 + cos θ e2 .

0 0 Since this is an arbitrary orientation preserving orthogonal transformation in the (e1 , e2)- plane, for an appropriate value of θ one gets (e1 , e2). One point worth stressing, related to the above proposition, is that any future-pointing timelike vector can√ be written as u ⊗ u¯ + v ⊗ v¯ , for suitable u, v ∈ U . Its Lorentz pseudo-norm is then given by 2 |ε(u, v)| .

1.5 The natural Dirac map

Next observe that an element of U ⊗ U can be seen as a linear map U F → U, while an element of U F ⊗ U F can be seen as a linear map U → U F. Then one deﬁnes the linear map √ γ˜ : U ⊗ U → End(U ⊕ U F): y 7→ γ˜(y) ≡ 2 y, y[F , √ i.e. γ˜(y)(u, χ) = 2ycχ , ucy[ ,

where y[ ≡ g˜[(y) . 1.5 The natural Dirac map 9

In particular for a decomposable y = p ⊗ q¯ one has √ γ˜(p ⊗ q¯)(u, χ) = 2hχ, q¯i p , hp[, ui q¯[ = √ = 2−ε¯(χ#, q¯) p , ε(p, u)ε ¯[(¯q) .

One obtains the coordinate expression √ AB˙ B A˙ γ˜(y)(u, χ) = 2 y χB˙ζA , y¯A˙B u ¯z = √ AB˙ B A˙ C˙D = 2 y χB˙ζA , yBA˙u ¯z , yBA˙ =y ¯A˙B ≡ ε¯C˙A˙ε¯DB y¯ .

Proposition 1.3 γ˜ is a Cliﬀord map; then, its restriction to H is a Dirac map. proof: From formula 2 in §A.3, for any p, q, r ∈ U one has

ε(p, q) r[ + ε(q, r) p[ + ε(r, p) q[ = 0 .

In order to prove thatγ ˜ is a Clifford map it suffices to show that the Clifford identity γ˜(y) ◦ γ˜(y0) +γ ˜(y0) ◦ γ˜(y) = 2g ˜(y, y0) 11holds for any two elements y, y0 of one basis of U ⊗ U ; since there exist bases constituted by null (isotropic) elements,8 it suffices to show that the Clifford identity holds for any two null elements in U ⊗ U ; such elements are just the decomposable ones, so set y = p ⊗ q¯, y0 = r ⊗ s¯, with p, q, r, s ∈ U . One gets (see also the following footnote 9) √ γ˜(p ⊗ q¯) ◦ γ˜(r ⊗ s¯)(u, χ) = 2γ ˜(p ⊗ q¯) −ε¯(χ#, s¯) r, ε(r, u)s ¯[ =

= 2 ε(r, u)ε ¯(¯s, q¯) p +ε ¯(¯s, χ#) ε(p, r)q ¯[ .

Hence

[˜γ(p ⊗ q¯) ◦ γ˜(r ⊗ s¯) +γ ˜(r ⊗ s¯) ◦ γ˜(p ⊗ q¯)](u + χ) = = 2 ε(r, u)ε ¯(¯s, q¯) p +ε ¯(¯s, χ#) ε(p, r)q ¯[ + ε(p, u)ε ¯(¯q, s¯) r +ε ¯(¯q, χ#) ε(r, p)s ¯[ =

= 2ε ¯(¯q, s¯)[ε(u, r) p + ε(p, u) r] + 2 ε(p, r) [¯ε(¯s, χ#)q ¯[ +ε ¯(χ#, q¯)s ¯[] =

= 2ε ¯(¯q, s¯)[−ε(r, p) u] + 2 ε(p, r)[−ε¯(¯q, s¯)(χ#)[] =

= 2 ε(p, r)ε ¯(¯q, s¯)(u, χ) = 2g ˜(p ⊗ q,¯ r ⊗ s¯)(u, χ) .

Recalling §B.1, we see that the condition thatγ ˜ is a Cliﬀord map can be equivalently stated as γ˜[w] ◦ γ˜[w] =g ˜(w, w) 11W , ∀ w ∈ U ⊗ U . The above relation can be easily checked in components as one gets

BA˙ A AA˙ B˙ γ˜[w]γ ˜[w](u, χ) = 2 w wAA˙u ζB , wAB˙w χA˙¯z =g ˜(w, w)(u, χ) .

8 For example (τ0+τ1 , τ0−τ1 , τ0+τ2 , τ0+τ3) , where (τλ) is a Pauli basis, is a null basis of H and hence also a null complex basis of U ⊗ U . 10 1 TWO-SPINORS

In fact, making index summation explicit we get

BA˙ BA˙ CC˙ 11 22 12 21 B w wAA˙ = w w ε¯C˙A˙εCA = (w w − w w ) δA =

1 B = 2 g˜(w, w) δA ,

AA˙ 1 A˙ wAB˙w = 2 g˜(w, w) δB˙ (similarly) . Thus, ﬁnally, one is led to regard

W ≡ U ⊕ U F as the space of Dirac spinors, decomposed into its Weyl subspaces.9 The ‘Dirac adjoint’ of the 4-spinor ψ = (u, χ) ∈ W is

ψ¯ = (¯χ, u¯) ∈ U F ⊕ U ≡ W F

−3/2 (actually, in order to obtain a conformally invariant Lagrangian, one takes ψ ∈ L ⊗ W ).

1.6 Riemann sphere and celestial sphere Let N ⊂ H be null cone. The celestial sphere is defined to be C ≡ PN , the (real) projective space of N ; namely C is the 2-sphere of all null directions in H . Moreover, I denote by R ≡ PS the Riemann sphere, the complex projective space of S . Finally, I denote by P: N → C and P : S → R the projections. The map ı : U → N : κ 7→ κ ⊗ κ¯ factorizes to a map R → C which turns out to be a diffeomorphism. Invertibility is proved at once because any element in C can be written, for some Pauli frame, as P(τ0 +τ3) = P(ζ1 ⊗ ζ¯1), which determines P(ζ1). Differentiability follows from the coordinate expression, which I am going to write down. Consider any fixed 2-spinor basis, and the Pauli basis determined by it. One obtains the λ 4 Cartesian coordinates (t, x, y, z) ≡ (x ): H → R and the associated spherical coordinates

+ (t, r, θ, φ): H˘ → R × R × (0, π) × (0, 2π) where H˘ ⊂ H is the open subset obtained by removing, at all times, the half plane characterized by y = 0, x ≥ 0. One has the relations

x = r sin θ cos φ , y = r sin θ sin φ , z = r cos θ .

Moreover one has λ ◦ √1 λ A A˙ x ı = 2 σ AA˙z ¯z , that is

0 ◦ ◦ √1 1 1 2 2 x ı ≡ t ı = 2 (z ¯z + z ¯z ) ,

1 ◦ ◦ √1 1 2 2 1 x ı ≡ x ı = 2 (z ¯z + z ¯z ) ,

2 ◦ ◦ √i 1 2 2 1 x ı ≡ y ı = 2 (z ¯z − z ¯z ) ,

3 ◦ ◦ √1 1 1 2 2 x ı ≡ z ı = 2 (z ¯z − z ¯z ) .

9 If there is no danger of confusion one could, by abuse of language, write u + χ instead of (u, χ). 1.6 Riemann sphere and celestial sphere 11

Then one obtains x − i y z1 ◦ ı = , z2 6= 0 , r − z z2

x + i y z2 ◦ ı = , z1 6= 0 , r + z z1 and also x − i y sin θ e−iφ = = cot( 1 θ) e−iφ , r − z 1 − cos θ 2

x + i y sin θ eiφ = = tan( 1 θ) eiφ . r + z 1 + cos θ 2 By abuse of language, I indicate still by (θ, φ) the induced coordinates on C. Let moreover w ≡ z1/z2 and w0 ≡ z2/z1 be the naturally induced projective coordinates on R. Then the correspondence C ↔ R is given in coordinates by

1 iφ w = cot( 2 θ) e , θ = 2 arctan(1/|w|) , φ = arg w , 0 1 −iφ 0 0 w = tan( 2 θ) e , θ = 2 arctan |w | , φ = − arg w . One has the canonical diﬀeomorphisms

R ≡ PS ≡ PU ≡ PSF ≡ PU F

(for the dual spaces, one observes that symplectic isomorphisms S ≡ SF and U ≡ U F are determined up to a complex factor). As a real manifold, R can be realized also as

R ≡ PS ≡ PU ≡ PSF ≡ PU F .

Note now that R is the quotient of S0 ≡ S \{0} by the free action (§??) of the group C× ≡ C \{0} , given by (a, s) 7→ a s . Because the quotient is a manifold and the projection 0 0 × × is smooth, also TR is the quotient of TS ≡ S × S by the induced action of TC ≡ C × C . Thus one has the commutative diagram

S0 × S −−−−→TP TR     y y S0 −−−−→ R P

× 0 Lemma 1.1 The induced action of TC on TS is given by

0 (a, b), (s, t) 7→ (a s, a t + b s) , a ∈ C \{0} , b ∈ C , s ∈ S , t ∈ S .

× 0 proof: Let I ⊂ R be an open neighbourhood of 0 , and g : I → C , s : I → S be smooth curves. By deﬁnition of tangent map one has

TL(∂g(0), ∂s(0)) = ∂L(g(t), s(t)) = ∂g(0) s(0) + g(0) ∂s(0) , t=0 projecting over L(g(0), s(0)) = g(0) s(0) . Then, setting g(0) ≡ a , ∂g(0) ≡ b , s ≡ s(0) and ∂s(0) ≡ (s, t) , one obtains the stated result. 12 2 2-SPINORS AND 4-SPINORS

Then one also has

Proposition 1.4 If (κ, λ), (κ0, λ0) ∈ S0 × S ≡ TS0 , then

( 0 0 0 κ = a κ TP(κ , λ ) = TP(κ, λ) ⇔ a ∈ C \{0} , b ∈ C . λ0 = a λ + b κ

Remark. The above proposition can also be proven directly without recalling the properties of the induced tangent Lie group action. Write (κ, λ) = dγ(0), (κ0, λ0) = dγ0(0), where 0 0 0 10 γ, γ : R → S are local diﬀerentiable curves. The condition TP(κ , λ ) = TP(κ, λ) holds iﬀ 0 0 one can choose these curves in such a way that P ◦ γ = P ◦ γ , namely γ = c γ with c : R → C . Thus dγ0(0) = c(0) γ(0) , c(0) dγ(0) + dc(0) γ(0) .

Setting a = c(0), b = dc(0), one obtains the stated result. −1 × It is easy to check that the inverse element (a, b) of (a, b) ∈ TC , and the product of 0 0 × the elements (a, b), (a , b ) ∈ TC are respectively given by 1 b (a, b)−1 = , − , a a (a, b) · (a0, b0) = (a a0 , a b0 + a0 b) .

2 2-spinors and 4-spinors

The precise equivalence between 4-spinor and 2-spinor settings for standard electrodynamics was exposed by Jadczyk and myself in [CJ97a, CJ97b] (see §B.5 for a review). In summary, one sees that a complex 2-dimensional vector space S yields, naturally and without any further assumptions, all the needed algebraic structures; conversely in a 4-spinor setting, provided one makes the minimum assumptions which are needed in order to formulate the physical theory, the 4-spinor space W naturally splits as U ⊕ U F (Weyl decomposition). Which setting one regards as fundamental is then mainly a matter of taste. The 2-spinor setting has the advantage that the fundamental objects have an immediate, almost trivial deﬁnition; the 4-spinor setting is perhaps closer to standard spacetime notation, and some formulas are more compact, while the relations among the various objects are more involved. I’ll regard the 2-spinor setting, as it was exposed in [C98, C00b], as the fundamental setting. In this section I’ll start from that setting in order to recover the Dirac representation in a detailed way. I will also expound the notational diﬀerences with the traditional formulation [IZ80]. Subsequently, I will be using the 4-spinor notation if convenient. The relation between the two settings in terms of structure groups will be examined in some detail in §6.

10 0 0 In general, consider a fibred manifold p : E M. If γ, γ : R → E are such that p ◦ γ = p ◦ γ , then obviously Tp ◦ dγ = Tp ◦ dγ0 . Conversely, if v, v0 ∈ TE are such that Tp(v) = Tp(v0) , then it is always possible to find two such p-equivalent curves fulfilling v = dγ(0) , v0 = dγ0(0) . In fact one can choose the local curves whose coordinate expression is given by

λ λ 0 λ i i i 0 0i x ◦ γ(t) = x ◦ γ (t) = v t , y ◦ γ(t) = v t , y ◦ γ (t) = v t ,

λ i λ 0i where v = v ∂λ + v ∂i , v = v ∂λ + v ∂i . In our present case, the ﬁbred manifold under consideration is S \{0} R . 2.1 4-spinor bases 13

2.1 4-spinor bases −1/2 1 2 Given a normalized basis (ζA) of U ≡ L ⊗ S, one obtains the basis (ζ1 , ζ2 , ¯z , ¯z ) of W ≡ U ⊕ U F. However the standard Weyl basis of W , giving rise to the usual Weyl representation, is deﬁned to be the basis (ζα), α = 1, 2, 3, 4, given by

1 2 (ζ1 , ζ2 , ζ3, ζ4) ≡ (ζ1 , ζ2 , −¯z , −¯z ) , 1 2 3 4 1 2 dual basis: (z , z , z , z ) = (z , z , −ζ¯1, −ζ¯2) .

Above, ζ1 is a simpliﬁed notation for (ζ1 , 0), and the like. Henceworth in this section, however, I shall use a somewhat more precise notation than usual, indicating an element of W as ψ = (u, χ) rather than u + χ. 0 The second important basis is the Dirac basis (ζα), α = 1, 2, 3, 4, where

0 √1 1 √1 ζ1 = 2 (ζ1 , ¯z ) = 2 (ζ1 − ζ3) , 0 √1 2 √1 ζ2 = 2 (ζ2 , ¯z ) = 2 (ζ2 − ζ4) , 0 √1 1 √1 ζ3 = 2 (ζ1 , −¯z ) = 2 (ζ1 + ζ3) , 0 √1 2 √1 ζ4 = 2 (ζ2 , −¯z ) = 2 (ζ2 + ζ4) . Dual basis:

01 √1 1 ¯ √1 1 3 z = 2 (z , ζ1) = 2 (z − z ) , 02 √1 2 ¯ √1 2 4 z = 2 (z , ζ2) = 2 (z − z ) , 03 √1 1 ¯ √1 1 3 z = 2 (z , −ζ1) = 2 (z + z ) , 04 √1 2 ¯ √1 2 4 z = 2 (z , −ζ2) = 2 (z + z ) .

0 β Write ζα = B α ζβ , where

1 0 1 0 1 0 −1 0 √1 0 1 0 1 −1 T √1 0 1 0 −1 (B) = 2 −1 0 1 0 , (B) = (B) = 2 1 0 1 0 . 0 −1 0 1 0 1 0 1 Note that B has unit determinant. One obtains

√1 0 0 √1 0 0 √1 0 0 √1 0 0 ζ1 = 2 (ζ1 + ζ3) , ζ2 = 2 (ζ2 + ζ4) , ζ3 = 2 (−ζ1 + ζ3) , ζ4 = 2 (−ζ2 + ζ4) ,

1 √1 01 03 2 √1 02 04 3 √1 01 03 4 √1 02 04 z = 2 (z + z ) , z = 2 (z + z ) , z = 2 (−z + z ) , z = 2 (−z + z ) .

2.2 Conjugation and Dirac adjoint Let ψ = (u, χ). According to my general complex space notation, I will write

ψ¯ = (¯u, χ¯) ∈ W = U ⊕ U F .

In the standard notation, this is denoted as ψ∗, while ψ¯ stands for the Dirac adjoint. Actually these two objects are strictly related. Consider the exchange map

s : W = U ⊕ U F → W F = U F ⊕ U :(u, χ) 7→ (χ, u) , 14 2 2-SPINORS AND 4-SPINORS

and denote by the same symbol the analogously deﬁned maps on W F, W and W F (then one may write s2 = 11). Then the Dirac adjoint is deﬁned to be simply

s(ψ¯) = (¯χ, u¯) ∈ W F = U F ⊕ U , which in the previous papers, for simplicity, I denoted simply by the usual notation ψ¯. Equivalently one may consider the Hermitian 2-form

0 0 0 0 k : W × W → C : (u, χ), (u , χ ) 7→ hχ,¯ u i + hχ , u¯i . and deﬁne the Dirac adjoint to be ψ[ ≡ k[(ψ) ≡ k(ψ, ) ∈ W F (by convention, index raising and lowering maps induced by 2-forms always work by contraction with the ﬁrst index of the 2-form; then, k¯[(ψ) = k( , ψ) ∈ W F). Note that k[ : W → W F is an antilinear map. Since it is obviously invertible, it turns out that k[ is an anti-isomorphism. Hence k is non-degenerate. In the natural 2-spinor basis and in the Weyl 4-spinor basis (ζα), respectively, one has 1 2 1 2 3 1 4 2 1 3 2 4 s(ψ) = χ1¯z + χ2¯z , u ζ1 + u ζ2 = −ψ ¯z − ψ ¯z − ψ ¯z − ψ ¯z , 1 2 1 2 1 2 3 4 ψ¯ = u¯ ζ¯1 +u ¯ ζ¯2 , χ¯1z +χ ¯2z = ψ¯ ζ¯1 + ψ¯ ζ¯2 + ψ¯ ζ¯3 + ψ¯ ζ¯4 , [ [ 1 2 1 2 3 1 4 2 1 3 2 4 ψ ≡ k (ψ) = χ¯1z +χ ¯2z , u¯ ζ¯1 +u ¯ ζ¯2 = −ψ¯ z − ψ¯ z − ψ¯ z − ψ¯ z . In the Weyl basis one can also write k = −¯z3 ⊗ z1 − ¯z4 ⊗ z2 − ¯z1 ⊗ z3 − ¯z2 ⊗ z4 . In the Dirac basis one obtains s(ψ) = ψ01¯z01 + ψ02¯z02 − ψ03¯z03 − ψ04¯z04 , k[(ψ) = ψ¯01z01 + ψ¯02z02 − ψ¯03z03 − ψ¯04z04 , k = ¯z01 ⊗ z01 + ¯z02 ⊗ z02 − ¯z03 ⊗ z03 − ¯z04 ⊗ z04 . Namely the Dirac basis diagonalizes k, which is seen to have signature (+, +, −, −).

2.3 The Dirac map in 4-spinor terms

Recall (1.5) that the natural Dirac mapγ ˜ : H ≡ U ∨¯ U F → End(W ) is given by √ √ [ AB˙ B A˙ γ˜[y](u, χ) = 2 ycχ , y¯ cu = 2 y χB˙ζA , yBA˙u ¯z . It is immediate to check that γ fulﬁls the relation11 k(γ[y]ψ, φ) = k(ψ, γ[y]φ) , y ∈ H, ψ, φ ∈ W .

Let (τλ) be the Pauli basis of H associated with a chosen basis (ζA) of U. Set

γλ ≡ γ˜[τλ] .

AA˙ Recalling (A.4) the expressions of the Pauli matrices (σλ ) and (σλAA˙) one ﬁnds AA˙ A A 1 1 2 2 γ0(ψ) ≡ γ0(u, χ) = σ0 χA˙ζA , σ0AA˙u ¯z = χ1 ζ1 + χ2 ζ2 , u ¯z + u ¯z = 3 4 1 2 = −ψ ζ1 − ψ ζ2 − ψ ζ3 − ψ ζ4 = 01 0 02 0 03 0 04 0 = ψ ζ1 + ψ ζ2 − ψ ζ3 − ψ ζ4 , 11In other terms, γ[y] is a k-Hermitian endomorphism. In matricial terms we have β˙ ¯α˙ β ¯α˙ α β β˙ α kβ˙β γ¯λ α˙ ψ φ = kα˙α ψ γλ β φ ⇒ kβ˙β γ¯λ α˙ = kα˙α γλ β . 2.3 The Dirac map in 4-spinor terms 15

AA˙ A A 2 1 1 2 γ1(ψ) ≡ γ1(u, χ) = σ1 χA˙ζA , σ1AA˙u ¯z = χ2 ζ1 + χ1 ζ2 , −u ¯z − u ¯z = 4 3 2 1 = −ψ ζ1 − ψ ζ2 + ψ ζ3 + ψ ζ4 = 04 0 03 0 02 0 01 0 = −ψ ζ1 − ψ ζ2 + ψ ζ3 + ψ ζ4 ,

AA˙ A A 2 1 1 2 γ2(ψ) ≡ γ2(u, χ) = σ2 χA˙ζA , σ2AA˙u ¯z = i −χ2 ζ1 + χ1 ζ2 , u ¯z − u ¯z = 4 3 2 1 = i (ψ ζ1 − ψ ζ2 − ψ ζ3 + ψ ζ4) = 04 0 03 0 02 0 01 0 = i (ψ ζ1 − ψ ζ2 − ψ ζ3 + ψ ζ4) ,

AA˙ A A 1 1 2 2 γ3(ψ) ≡ γ3(u, χ) = σ3 χA˙ζA , σ3AA˙u ¯z = χ1 ζ1 − χ2 ζ2 , −u ¯z + u ¯z = 3 4 1 2 = −ψ ζ1 + ψ ζ2 + ψ ζ3 − ψ ζ4 = 03 0 04 0 01 0 02 0 = −ψ ζ1 + ψ ζ2 + ψ ζ3 − ψ ζ4 .

0 Indicating by (γλ) and (γλ) the matrices of γλ in the Weyl basis (ζα) and in the Dirac basis 0 (ζα), respectively, one obtains 0 0 −1 0 0 0 0 −1 (0) (−11) (γ0) = −1 0 0 0 = , 0 −1 0 0 (−11) (0)

1 0 0 0 0 0 1 0 0 (11) (0) (γ0) = 0 0 −1 0 = , 0 0 0 −1 (0) (−11)

0 0 0 −1 0 0 0 −1 0 (0) (−σ1) (γ1) = (γ1) = 0 1 0 0 = , 1 0 0 0 (σ1) (0)

0 0 0 i ! 0 0 0 −i 0 (0) (−σ2) (γ2) = (γ2) = 0 −i 0 0 = , i 0 0 0 (σ2) (0)

0 0 −1 0 0 0 0 0 1 (0) (−σ3) (γ3) = (γ3) = 1 0 0 0 = . 0 −1 0 0 (σ3) (0) With regard to the correspondence between Cliﬀord algebra and exterior algebra of H,I indicate by γη the element corresponding to the volume form η. Then one has γη ≡ γ0γ1γ2γ3. One recovers the usual γ5 as −i γη. One obtains

γη(u, χ) ≡ γη(ψ) = i (u , −χ) = 1 2 3 4 = i (ψ ζ1 + ψ ζ2 − ψ ζ3 − ψ ζ4) = 03 0 04 0 01 0 02 0 = i (ψ ζ1 + ψ ζ2 + ψ ζ3 + ψ ζ4) , namely the matrices of γη in the Weyl and Dirac bases are

1 0 0 0 0 1 0 0 (11) (0) (γη) = i (γ5) = i 0 0 −1 0 = i , 0 0 0 −1 (0) (−11)

0 0 1 0 0 0 0 0 0 1 (0) (11) (γη) = i (γ5) = i 1 0 0 0 = i . 0 1 0 0 (11) (0) 16 3 ADDITIONAL STRUCTURES

Of course (§B.4) one has γλγη = −γηγλ .

Remark. The Dirac basis diagonalizes γ0 (as well as k) , so it is adapted to the complex decomposition of W into the direct sum of the 2-dimensional eigenspaces of γ0 corresponding to eigenvalues ±1 . These are called (§??) the electron and positron eigenspaces, respectively. Commutators of Dirac matrices are relevant in several situation (in particular, in relation to the vector-space isomorphism between the Dirac algebra and the exterior algebra ∧H). For the Weyl representation one ﬁnds 1 (−σj) (0) 2 (γ0γj − γjγ0) = , (0) (σj) 1 k (σk) (0) 2 (γiγj − γjγi) = −i εij . (0) (σk)

Similarly, for the Dirac representation one ﬁnds 1 0 (0) (−σj) 2 (γ0γj − γjγ0) = , (−σj) (0) 1 0 k (σk) (0) 2 (γiγj − γjγi) = −i εij . (0) (σk)

Finally, in order to have the complete bases of the algebra, one considers the products γλγη , obtaining (0) (i σ0) (0) (i σj) (γ0γη) = , (γjγη) = , (−i σ0) (0) (i σj) (0) 0 (0) (i σ0) 0 (−i σj) (0) (γ0γη) = , (γjγη) = . (−i σ0) (0) (0) (i σj)

3 Additional structures

3.1 Observers and positive Hermitian metrics

A Hermitian 2-form h on U is an element in U F ∨¯ U F , hence it can be seen as an element in H∗ ; more precisely, h¯ ∈ H∗ . It will be said to be normalized (i.e. ε-normalized, or g˜-normalized) ifg ˜#(h) = h# , where h# is the contravariant inverse of h , characterized # ¯[ ¯ by h ◦ h = 11U . Note that this is diﬀerent form requiring that h , seen as an element in H∗ , has unit Lorentz pseudo-norm. For example, consider the Pauli basis determined 0 ¯0 by a normalized 2-spinor basis (ζA) ; then√ t has unit Lorentz pseudo-norm, but t is not ¯0 1 1 2 2 # √normalized in the above sense. Actually 2 t =z ¯ ⊗ z +z ¯ ⊗ z is normalized, and h = 0 2τ ¯0 = ζ¯1 ⊗ ζ1 + ζ¯2 ⊗ ζ2 ; instead, ¯t andτ ¯0 , while are obtained from one another through # [ g˜ andg ˜ , are not mutually inverse Hermitian metrics as the composition of their induced√ 1 # isomorphisms gives 2 11U . On the other hand, the Lorentz pseudo-norm of h and h is 2 . In general, given a positive-deﬁnite, normalized Hermitian metric h , there exist a normalized 2-spinor basis such that the components of h are hA˙A = δA˙A (this is an immediate consequence of proposition 1.2). Note that h¯ , seen as a world-covector, is always timelike and 3.1 Observers and positive Hermitian metrics 17

√ future-pointing, and its Lorentz pseudo-norm equals 2 . Similarly, negative-deﬁnite Her- mitian metrics correspond to past-pointing timelike covectors. Hermitian metrics of mixed signature (1, −1) correspond√ to spacelike covectors; actually, such metrics can always be written as proportional to 2¯t3 =z ¯1 ⊗ z1 − z¯2 ⊗ z2 , in an appropriate normalized 2-spinor basis. The basic observation resulting from the above discussion is that the assignments of an ‘observer’ in H and of a positive-deﬁnite Hermitian metric on U are equivalent, actually the two objects are nearly the same thing. In 4-spinor terms, the above equivalence is only slightly less obvious. If h is assigned, then it extends naturally to a Hermitian metric h on W , which can be characterized by

h(ψ, φ) = k(γ0ψ, φ) . For ψ ∈ W , one has [ [ h (ψ) = k (γ0ψ) ∈ W F . One has the basis expressions

[ [ 1 1 2 2 h (u, χ) ≡ h (ψ) ≡ h(ψ, ) = u¯ z +u ¯ z , χ¯1ζ¯1 +χ ¯2ζ¯2 = = ψ¯1z1 + ψ¯2z2 + ψ¯3z3 + ψ¯4z4 = = ψ¯01z01 + ψ¯02z02 + ψ¯03z03 + ψ¯04z04 . Namely, the Weyl and Dirac bases are both h-orthonormal. † In traditional texts one finds the relation ψ¯ = ψ γ0 , considered as the definition of the Dirac adjoint (but note that this amounts to define a canonical object in terms of two objects, [ [ both depending on the choice of an observer). I write this as k (ψ) = h (ψ) ◦ γ0 i.e. k(ψ, φ) = h(ψ, γ0φ), which follows from

k(ψ, φ) = k(γ0γ0ψ, φ) = k(γ0ψ, γ0φ) = h(ψ, γ0φ) , since (see §2.3) k(γλψ, φ) = k(ψ, γλφ) and γ0 ◦ γ0 = 11. The conjugate map h¯[ : W → W F is

[ [ 1 1 2 2 h¯ (u, χ) ≡ h¯ (ψ) ≡ h( , ψ) = u ¯z + u ¯z , χ1ζ1 + χ2ζ2 = = ψ1¯z1 + ψ2¯z2 + ψ3¯z3 + ψ4¯z4 = = ψ01¯z01 + ψ02¯z02 + ψ03¯z03 + ψ04¯z04 .

Traditionally, h¯[(ψ) is denoted as ψT, as it is a covariant object with the same components as ψ (in the special bases). One also ﬁnds the object denoted as ψ¯T, which corresponds in my notation to h¯# ◦ k[(ψ) ∈ W . One has

# [ 1 1 2 2 h¯ ◦ k (ψ) = γ0ψ = χ¯1ζ¯1 +χ ¯2ζ¯2 , u¯ z +u ¯ z = 3 4 1 2 = −ψ¯ ζ¯1 − ψ¯ ζ¯2 − ψ¯ ζ¯3 − ψ¯ ζ¯4 = ¯01 ¯0 ¯02 ¯0 ¯03 ¯0 ¯04 ¯0 = ψ ζ1 + ψ ζ2 − ψ ζ3 − ψ ζ4 . Recall that in general one has [CJ97a]

# [ # [ k ◦ h = h ◦ k = γ0 . Moreover for any y ∈ H one has † γy = γ0γyγ0 , † where γy ∈ End(W ) denotes the Hermitian transpose of γy with respect to h. 18 3 ADDITIONAL STRUCTURES

3.2 Charge conjugation

Charge conjugation, parity and time reversal are operators acting on solutions of the Dirac equation. In this and the next sections I deal with their algebraic forerunners, that are either linear or antilinear maps W → W . 2 F it A B Let ω ∈ ∧ U be a normalized 2-form, ω = e εAB z ∧ z . We deﬁne the charge conjugation anti-isomorphism Cω : W → W as

# [ −it # [ Cω(ψ) ≡ C(u, χ) = ω (¯χ), −ω¯ (¯u) = e ε (¯χ), −ε¯ (¯u) = −it AB A˙ B˙ = e ε χ¯A ζB , −ε¯A˙B˙u¯ ¯z = −it 2 1 1 2 = e −χ¯2ζ1 +χ ¯1ζ2 , u¯ ¯z − u¯ ¯z = −it 4 3 2 1 = e ψ¯ ζ1 − ψ¯ ζ2 − ψ¯ ζ3 + ψ¯ ζ4 = −it ¯04 0 ¯03 0 ¯02 0 ¯01 0 = e ψ ζ1 − ψ ζ2 − ψ ζ3 + ψ ζ4 , namely the matrix of Cω, seen as a linear map W → W , in the Weyl and Dirac bases and their conjugates, is 0 0 0 1 0 −it 0 0 −1 0 (Cω) = (Cω) = e 0 −1 0 0 . 1 0 0 0 One has

Cω ◦ Cω = 11W ,

γy ◦ Cω + Cω ◦ γy = 0 ⇔ Cω ◦ γy ◦ Cω = −γy , y ∈ H .

c T −it Traditionally, Cω(ψ) is denoted as ψ = ηc Cψ¯ , where ηc is a phase factor, here e ; one can write this as −it # [ −it Cω(ψ) = e C ◦ h¯ ◦ k (ψ) = e C γ0ψ , where C : W → W is a linear map. Denoting the (canonical) conjugation map by K : W → −it W , with inverse K¯ : W → W , one can write Cω = e C ◦ K ◦ γ0 , so that

it C = e Cω ◦ γ0 ◦ K¯ .

Note that this map C depends on the chosen 2-spinor basis, which determines the numerical factor eit for the given electromagnetic gauge ω.

Since Cω ◦ Cω = 11W one gets the decomposition of W into the direct sum of eigenspaces of Cω corresponding to eigenvalues ±1 , namely any ψ ≡ (u, χ) ∈ W can be uniquely decomposed as

1 1 ψ = 2 (ψ + Cωψ) + 2 (ψ − Cωψ) =

1 −it # −it [ 1 −it # −it [ = 2 (u + e χ¯ , χ − e u¯ ) + 2 (u − e χ¯ , χ + e u¯ ) .

Remark. An element ψ ∈ W is called a Majorana spinor if it fulﬁlls the condition Cω(ψ) = ψ , that is ( u = e−it χ¯# , (u, χ) = e−it (¯χ#, −u¯[) ⇒ χ = −e−it u¯[ . 3.3 Parity and time reversal 19

Using u[# = −u and λ#[ = −λ it easy to check that the two above conditions coincide, so that a Majorana spinor can be written either in the form12

ψ = (u, −ω¯[(¯u) = (u, −e−it u¯[) or in the form ψ = (ω#(¯χ), χ = (e−it χ¯#, χ) .

4 Recalling the Weyl and Dirac basis expressions

¯3 ¯ ¯4 ¯ ¯1 ¯ ¯2 ¯ ¯01 ¯0 ¯02 ¯0 ¯03 ¯0 ¯04 ¯0 γ0ψ = −ψ ζ1 − ψ ζ2 − ψ ζ3 − ψ ζ4 = ψ ζ1 + ψ ζ2 − ψ ζ3 − ψ ζ4 one obtains the matrix expressions

 ψ¯4  0 − 1 0 0  −ψ¯3 ¯3 ¯4 it −ψ   1 0 0 0  −ψ  e (Cωψ) =   =     , −ψ¯2  0 0 0 1  −ψ¯1 ψ¯1 0 0 − 1 0 −ψ¯2

 ψ¯04  0 0 0 − 1   ψ¯01 ¯03 ¯02 it 0 −ψ   0 0 1 0   ψ  e (Cωψ) =   =     , −ψ¯02  0 − 1 0 0  −ψ¯03 ψ¯01 1 0 0 0 −ψ¯04 namely the matrices of C, in the Weyl and Dirac bases and their conjugates, are

0 −1 0 0 0 0 0 −1 1 0 0 0 0 0 0 1 0 (C) = 0 0 0 1 , (C) = 0 −1 0 0 . 0 0 −1 0 1 0 0 0

02 2 ¯ Note that (C) = (C) = −(11), namely C ◦ C = −11W .

3.3 Parity and time reversal We saw how charge conjugation depends on the choice of an electromagnetic gauge ω (when no confusion arises we will write just C instead of Cω). The parity map P depends on the choice of an observer, while the time reversal map T depends on the choices of both an electromagnetic gauge and an observer. These are deﬁned by

P ≡ γ0 , T ≡ γηγ0C .

The basis expressions of C (§3.2) and of γ0 (§2.3) have already been written down. For time reversal one obtains

−it 2 1 1 2 T (u, χ) ≡ T (ψ) = i e (¯u ζ1 − u¯ ζ2 , χ¯2¯z − χ¯1¯z ) = −it 2 1 4 3 = i e (ψ¯ ζ1 − ψ¯ ζ2 + ψ¯ ζ3 − ψ¯ ζ4) = −it ¯02 0 ¯01 0 ¯04 0 ¯03 0 = i e (ψ ζ1 − ψ ζ2 + ψ ζ3 − ψ ζ4) ,

12In the recent literature one ﬁnds the notion of “Elko spinor”, presented as an original concept. As far as I can say, however, this is essentially the same as a Majorana spinor.(¿?) 20 3 ADDITIONAL STRUCTURES namely the matrix of T , seen as a linear map W → W , in the Weyl and Dirac bases and their conjugates, is 0 1 0 0 0 −it −1 0 0 0 (T ) = (T ) = i e 0 0 0 1 . 0 0 −1 0 The following identities are easily checked

P2 = 11 , T 2 = −11 ,

PT = TP = −γηC ,

PC = CP = γ0C ,

CT = −T C = γ0γη , (CT )2 = −(PC)2 = −(PT )2 = 11 ,

PCT = γη . The following adjoint actions will serve to determine the action of C, P and T on solutions of the Dirac equation:

−1 CγλC = −γλ , −1 −1 0 PγλP = T γλT = 2δ λγ0 − γλ , −1 PT γλ(PT ) = γλ , −1 −1 0 PCγλ(PC) = CT γλ(CT ) = −2δ λγ0 + γλ , −1 PCT γλ(PCT ) = −γλ .

Moreover observe that for the antilinear maps one has opposite adjoint action on iγλ; for example

−1 C(i γλ)C = i γλ , −1 0 T (i γλ)T = i (−2δ λγ0 + γλ) , and the like.

3.4 Spin operators Spin operators are algebraic operators, that is endomorphisms of the 4-spinor space W , which are generated by an angular momentum representation (§??). The construction depends on the choice of an observer τ0 .

3.4.1 Let H⊥ ⊂ H be the subspace of all vectors orthogonal to the chosen observer, and consider the map ⊥ i J : H → End(W ): y 7→ 2 γ0 γ[y] γη . If n ∈ H⊥ has unit length, i.e. g(n, n) = −1 , then

i Jn ≡ J[n] ≡ 2 γ0 γn γη ∈ End(W ) is called the spin operator (or spin map) in the n-direction. ~? In a Pauli basis one has j Jn = n Jj , where 1 Jj ≡ J[τj] = − 2 γ0 γj γ5 . 3.4 Spin operators 21

One ﬁnds the same expression for Jj in the Weyl and Dirac bases: 0 1 (σj) (0) (Jj) = (Jj) = 2 , (0) (σj) namely 1 2 1 1 2 J1(u, χ) ≡ J1(ψ) = 2 (u ζ1 + u ζ2 , χ2 ¯z + χ1 ¯z ) = 1 2 1 4 3 = 2 (ψ ζ1 + ψ ζ2 + ψ ζ3 + ψ ζ4) = 1 02 0 01 0 04 0 03 0 = 2 (ψ ζ1 + ψ ζ2 + ψ ζ3 + ψ ζ4) ,

i 2 1 1 2 J2(u, χ) ≡ J2(ψ) = 2 (−u ζ1 + u ζ2 , −χ2 ¯z + χ1 ¯z ) = i 2 1 4 3 = 2 (−ψ ζ1 + ψ ζ2 − ψ ζ3 + ψ ζ4) = i 02 0 01 0 04 0 03 0 = 2 (−ψ ζ1 + ψ ζ2 − ψ ζ3 + ψ ζ4) ,

1 1 2 1 2 J3(u, χ) ≡ J3(ψ) = 2 (u ζ1 − u ζ2 , χ1 ¯z − χ2 ¯z ) = 1 1 2 3 4 = 2 (ψ ζ1 − ψ ζ2 + ψ ζ3 − ψ ζ4) = 1 01 0 02 0 03 0 04 0 = 2 (ψ ζ1 − ψ ζ2 + ψ ζ3 − ψ ζ4) . 13 1 ◦ One has J[n]J[n] = 4 11, that is (2J[n]) (2J[n]) = 11. Then 2J[n] has eigenvalues ±1 , 1 ⊥ i.e. J[n] has eigenvalues ± 2 . Thus for each n ∈ H we have the decomposition of W into eigenspaces corresponding to these eigenvalues. The projections onto these eigenspaces are 1 1 Π[n] ≡ 2 (11 ± 2J[n]) = 2 11 ± J[n] , and have the matrix expressions j 1 σ0 ± n σj 0 Π[n] = 2 j . 0 σ0 ± n σj

In particular for n = τi , i = 1, 2, 3 , we get

J1 ) W = Span{ζ1 + ζ2 , ζ3 + ζ4} ⊕ Span{ζ1 − ζ2 , ζ3 − ζ4} = 0 0 0 0 0 0 0 0 = Span{ζ1 + ζ2 , ζ3 + ζ4} ⊕ Span{ζ1 − ζ2 , ζ3 − ζ4} ,

J2 ) W = Span{ζ1 + i ζ2 , ζ3 + i ζ4} ⊕ Span{ζ1 − i ζ2 , ζ3 − i ζ4} = 0 0 0 0 0 0 0 0 = Span{ζ1 + i ζ2 , ζ3 + i ζ4} ⊕ Span{ζ1 − i ζ2 , ζ3 − i ζ4} ,

J3 ) W = Span{ζ1 , ζ3} ⊕ Span{ζ2 , ζ4} = 0 0 0 0 = Span{ζ1 , ζ3} ⊕ Span{ζ2 , ζ4} . One also has the total spin map

jk ◦ 3 −g Jj Jk = 4 11W .

The maps J± ≡ J1 ± J2 are also important. One has 2 1 2 4 02 0 04 0 J+(u, χ) ≡ J+(ψ) = (u ζ1 , χ2 ¯z ) = ψ ζ1 + ψ ζ3 = ψ ζ1 + ψ ζ3 ,

13 Observing that γ0γ[n] + γ[n] γ0 = 2g(τ0 , n) = 0 , and remembering that ∀y ∈ H we have γηγη = −11, yγη = −γηy , we get 1 1 1 1 1 J[n]J[n] = − 4 γ0γ[n]γηγ0γ[n]γη = − 4 γ0γ[n]γ0γ[n]γηγη = 4 γ0γ[n]γ0γ[n] = − 4 γ0γ0γ[n]γ[n] = 4 11 . 22 3 ADDITIONAL STRUCTURES

1 2 1 3 01 0 03 0 J−(u, χ) ≡ J−(ψ) = (u ζ2 , χ1 ¯z ) = ψ ζ2 + ψ ζ4 = ψ ζ2 + ψ ζ4 .

3.4.2 The map J introduced above is essentially the angular momentum map (§??; see also §??) related to an angular momentum representation Σ by

Σ ≡ −i J : SU(U, τ0) → End(W ) , where SU(U, τ0) can be identified with SU(2) when an orthonormal basis is fixed. Actually on the 2-spinor space U , when an ‘observer’ τ0 is fixed , one has a natural such representation. In fact, consider the subspace iHˆ 0 ⊂ End(U) of all traceless anti-Hermitian endomorphisms of U (App. A.5). The tensor −i t0 ∈ U F ⊗ U F determines a real isomorphism

⊥ ξ : H → iHˆ 0 , which in a Pauli basis reads

√i i i A B ξ : τj 7→ ξj ≡ − 2 τˆj = − 2 σˆj = − 2 σj B ζA ⊗ z . ˆ Actually, it was already seen (§??) that this basis of iH0 (not to be confused with a basis (ξA) of S, see §1.3) fulﬁls the commutation rules k [ξi , ξj] = εij ξk , and so it is an angular momentum representation. In order to extend it to a representation on W one is lead to consider the conjugate transpose morphism

ξ¯∗ : H⊥ → End(U F) . However, remembering that a Pauli matrix coincides with its conjugate transpose, one has ¯∗ i matrices (ξj ) = + 2 (ˆσj) , so that ¯∗ ¯∗ k ¯∗ [ξi , ξj ] = −εij ξk .

This means that, in order to get the correct commutators, one has to take −ξ¯∗ in the U F component, namely Σ ≡ ξ ⊕ (−ξ¯∗): H ⊥ → End(W ) . There is some abuse of notation here, since the same symbol Σ is used for the angular momentum representation SU(2) → End(W ) and for the map H ⊥ → End(W ) ; this is common in physics texts. ⊥ Hence the basis (Σj) ≡ (Σ[τj]) of Σ(H ) fulﬁls the commutation rules k [Σi , Σj] = εij Σk . 1 Accordingly, an element in W represents a state of spin ± 2 in the direction characterized by ⊥ 1 a unit vector n ∈ H , namely it is an eigenvector of J[n] with eigenvalue ± 2 , iﬀ it is an i eigenvector of Σ[n] with eigenvalue ∓ 2 . Let now Σ∗ : H⊥ → End(W F) ∗ ∗ ∗ be the transpose representation. Since Σi Σj = (Σj Σi) one has ∗ ∗ ∗ k ∗ [Σi , Σj ] = [Σj, Σi] = −εij Σk . Thus the correct angular momentum representation on W F is ∗ ∗ Σ ≡ −Σ∗ i.e. J = −J ∗ . 3.4 Spin operators 23

3.4.3 It will be useful to have, ready at hand, formulas of the actions of spin maps on the elements of spinor bases. These formulas are actually particular cases of the above written ones.  ζ 7→ ζ , ζ 7→ ζ , ¯z1 7→ ¯z2 , ¯z2 7→ ¯z1 ,  1 2 2 1 J1 : ζ3 7→ ζ4 , ζ4 7→ ζ3 ,  0 0 0 0 0 0 0 0 ζ1 7→ ζ2 , ζ2 7→ ζ1 , ζ3 7→ ζ4 , ζ4 7→ ζ3 ,  ζ 7→ i ζ , ζ 7→ −i ζ , ¯z1 7→ i ¯z2 , ¯z2 7→ −i ¯z1 ,  1 2 2 1 J2 : ζ3 7→ i ζ4 , ζ4 7→ −i ζ3 ,  0 0 0 0 0 0 0 0 ζ1 7→ i ζ2 , ζ2 7→ −i ζ1 , ζ3 7→ i ζ4 , ζ4 7→ −i ζ3 ,  ζ 7→ ζ , ζ 7→ −ζ , ¯z1 7→ ¯z1 , ¯z2 7→ −¯z2 ,  1 1 2 2 J3 : ζ3 7→ ζ3 , ζ4 7→ −ζ4 ,  0 0 0 0 0 0 0 0 ζ1 7→ ζ1 , ζ2 7→ −ζ2 , ζ3 7→ ζ3 , ζ4 7→ −ζ4 .

 1 2 2 1 z 7→ −z , z 7→ −z , ζ¯1 7→ −ζ¯2 , ζ¯2 7→ −ζ¯1 , ∗  ∗ 3 4 4 3 J 1 ≡ −(J1) : z 7→ −z , z 7→ −z , z01 7→ −z02 , z02 7→ −z01 , z03 7→ −z04 , z04 7→ −z03 ,

 1 2 2 1 z 7→ i z , z 7→ −i z , ζ¯1 7→ i ζ¯2 , ζ¯2 7→ −i ζ¯1 , ∗  ∗ 3 4 4 3 J 2 ≡ −(J2) : z 7→ i z , z 7→ −i z , z01 7→ i z02 , z02 7→ −i z01 , z03 7→ i z04 , z04 7→ −i z03 ,

 1 1 2 2 z 7→ −z , z 7→ z , ζ¯1 7→ −ζ¯1 , ζ¯2 7→ ζ¯2 , ∗  ∗ 3 3 4 4 J 3 ≡ −(J3) : z 7→ −z , z 7→ z , z01 7→ −z01 , z02 7→ z02 , z03 7→ −z03 , z04 7→ z04 .

In fact the matrices of J1 and J3 are symmetric, while the matrix of J2 is anti-symmetric, and ∗ the matrix of (Jk) in the dual basis is the transpose matrix of (Jk) . So for example, one has ( hJ ∗z1 , ζ i = hz1 ,J (ζ )i = hz1, ζ i = 1 , 1 2 3 2 1 ⇒ J ∗z1 = z2 , ∗ 1 1 hJ1 z , ζαi = 0 , α 6= 2 , while ( hJ ∗z1 , ζ i = hz1 ,J (ζ )i = hz1, −i ζ i = −i , 2 2 2 2 1 ⇒ J ∗z1 = −i z2 . ∗ 1 2 hJ2 z , ζαi = 0 , α 6= 2 , s Remark. Taking the isomorphism W ↔ W F into account, J ∗ essentially coincides with J¯; more precisely, J ∗ = s ◦ J¯ ◦ s .

3.4.4 If n ∈ H⊥ is such that g(n, n) = −1 then, remembering

γη γη = −11 , y γη = −γη y ∀y ∈ H 24 3 ADDITIONAL STRUCTURES

(§2.3), and observing that γ0 γn + γn γ0 = 0 since g(τ0 , n) = 0 , we get

1 1 1 Jn Jn = − 4 γ0 γn γη γ0 γn γη = − 4 γ0 γn γ0 γn γη γη = 4 γ0 γn γ0 γn = 1 1 = − 4 γ0 γ0 γn γn = 4 11 .

3.5 Particle momentum in 2-spinor form 3.5.1 It has already been observed (§1.4) that any future-pointing non-spacelike element in H can be written in the form u ⊗ u¯ + v ⊗ v¯ , u, v ∈ U . If u and v are not proportional to each other, that is ε(u, v) 6= 0 , then the above expression is a timelike future-pointing vector; if ε(u, v) = 0 then it is a null vector. Correspondingly, any future-pointing non-spacelike element in H∗ can be written in the form (u ⊗ u¯ + v ⊗ v¯)[ , or

λ ⊗ λ¯ + ν ⊗ ν¯ , λ, ν ∈ U F .

In this section, this 2-spinor representation of such (co)vectors will be studied in some detail. Future-pointing elements in H (or H∗) represent a “conformally invariant” version of classical particle momenta. Translation to a scaled version, when needed, will be eﬀortless.

3.5.2 Let J + ⊂ H denote the set of all future-pointing non-spacelike elements, namely

J + = K+ ∪ N + ⊂ H where K+ and N + are deﬁned to be the subsets of H of all future-pointing timelike vectors and of all future-pointing null vectors, respectively; also, let

∗ ∗ ∗ J + = K+ ∪ N + ⊂ H∗ be the similarly deﬁned set of covectors. Remark. Note that each of these sets does not contain the zero element. When the analogous past sets are not considered, one simply writes J, K and N for J +, K+ and N +, and the like. Consider now the real quadratic maps

# √1 ˜p : U × U → J :(u, v) 7→ 2 (u ⊗ u¯ + v ⊗ v¯) ,

∗ √1 [ ˜p: U × U → J :(u, v) 7→ 2 (u ⊗ u¯ + v ⊗ v¯) ,

# ∼ F √1 # # p : W = U × U → J :(u, χ) 7→ 2 (u ⊗ u¯ +χ ¯ ⊗ χ ) ,

∗ ∼ F √1 [ [ p : W = U × U → J :(u, χ) 7→ 2 (u ⊗ u¯ +χ ¯ ⊗ χ) .

When a symplectic form ε ∈ ∧2U F is ﬁxed, ˜p# and ˜pare essentially the same objects as p# √1 and p . In such a situation one may represent a given element 2 (u ⊗ u¯ + v ⊗ v¯) of J by 3.5 Particle momentum in 2-spinor form 25

writing v ⊗ v¯ as (¯χ# ⊗ χ)# , or by writing u ⊗ u¯ as (λ ⊗ λ¯)# ; here, u, v ∈ U, χ ∈ U F, λ ∈ U F. ∗ Similar alternative representations can be used for an element in J. In such cases I’ll set

v ≡ −χ¯# ⇐⇒ χ =v ¯[ ,

u ≡ −λ# ⇐⇒ λ = u[ .

Then one also has

hχ,¯ ui = hv[, ui = ε(v, u) = −hλ, vi = hλ, χ¯#i = ε#(¯χ, λ) ,

hχ, u¯i = hv¯[, u¯i =ε ¯(¯v, u¯) = −hλ,¯ v¯i = hλ,¯ χ#i =ε ¯#(χ, λ¯) .

# iϕ [ Remark. More generally, requiring v ⊗ v¯ = (¯χ ⊗ χ¯) implies χ = e v¯ with ϕ ∈ R , or equivalently14 v¯ = −eiϕ χ# (a similar consideration holds about the relation between u and λ). The arbitrary phase factor ei ϕ won’t be considered; the essential point one has to remember is that the relation between p and ˜pdepends of the chosen ε .

3.5.3 If p = p(u, χ) ≡ ˜p(u, v) , i.e. p# ≡ g˜#(p) = p#(u, χ) , then for brevity I’ll write

µ2 ≡ g˜(p#, p#) = |ε(u, v)|2 = |hχ,¯ ui|2 .

Actually, it is easy to check (App.A.7.1) that h and h# are mutually inverse Hermitian metrics.

3.5.4 For each (s, σ) ∈ W one has (§1.5) γ[p#](s, σ) = hσ, u¯i u + hσ, v¯i v , ε(u, s)u ¯[ + ε(v, s)v ¯[ =

= hσ, u¯i u +ε ¯∗(χ, σ)χ ¯# , ε(u, s)u ¯[ + hχ,¯ si χ ,

In particular, γ[p#](u, v¯[) = ε¯(¯v, u¯) u , ε(v, u)v ¯[ = hχ, u¯i u , hχ,¯ ui χ .

14ε¯#(χ) =ε ¯#(ei ϕ ε¯[(¯v)) = e−i ϕ ε¯#(¯ε[(¯v)) = −e−i ϕ v¯ ⇒ v¯ = −ei ϕ ε¯#(χ). 26 3 ADDITIONAL STRUCTURES

1 # The observer τ determines (§3.3) a parity operator P ≡ γ[τ] = µ γ[p ] . This is given explicitely in §A.7.2, together with the associated time reversal morphism. The following expressions are easily obtained: 1 1 γ[τ](u, χ) = √ γ[p](u, χ) = hχ, u¯i u , hχ,¯ ui χ , 2 |hχ,¯ ui| |hχ,¯ ui|

1 hχ,¯ ui h¯[(u) = · uc(u[ ⊗ u¯[ +χ ¯ ⊗ χ) = χ , |hχ,¯ ui| |hχ,¯ ui|

iii) h¯[(u) = eiθ χ . iv) h#(χ) = e−iθ u . v) h(¯u, v) = 0 and |hχ,¯ ui| = h(¯u, u) . v’) h(¯u, v) = 0 and |hχ,¯ ui| = h(¯v, v) . proof: The formulas written just before the statement of this proposition show that condition i implies conditions ii, iii, iv, v and v’. Further implications will be proved according to the somewhat redundant15 scheme: ii ⇐⇒ iii ⇐⇒ iv ⇐⇒ v ⇐⇒ v0 ⇒ iv v ⇒ i

√1 ¯# # ¯[ ( ii ⇔ iii ) : It follows from γ[τ](u, χ) = 2 γ[h ](u, χ) = h (χ), h (u) .

15In practice, some implications are proved at once in both ways. Others are interesting though redundant. 3.6 Helicity of Dirac Fermions 27

( iii ⇔ iv ) : If h¯[(u) = eiθ χ then u = h#(h¯[(u)) = h#(eiθ χ) = eiθ h#(χ). Similarly, if h#(χ) = e−iθ u then χ = h¯[(h#(χ)) = h¯[(e−iθ u) = e−iθ h¯[(u). ( iv ⇒ v ) : h(¯u, v) = hh[(¯u), −χ¯#i = −he−iθ χ,¯ χ¯#i = e−iθ ε∗(¯χ, χ¯) = 0 . Moreover h(¯u, u) = hh¯[(u), u¯i = heiθ χ, u¯i = hχ,¯ ui hχ, u¯i/|hχ,¯ ui| = |hχ,¯ ui| . [ # ∗ [ [ ( v ⇒ iv ) : From 0 = h(¯u, v) = hh (¯u), −χ¯ i = −ε (¯χ, h (¯u)) one hasχ ¯ = c h (¯u), c ∈ C . Then hχ,¯ ui = c h(¯u, u) = c |hχ,¯ ui| ⇒ c = eiθ . ( v ⇒ v’ ) : Using also iv, already seen to be equivalent to v, one has h(¯v, v) = hh, χ# ⊗ χ¯#i = hh#, χ ⊗ χ¯i = hh#(χ), χ¯i = e−iθ hχ,¯ ui = |hχ,¯ ui| , hence also h(¯v, v) = |hχ,¯ ui| . [ 1 # ( v’ ⇒ iv ) : As in v ⇒ iv one hasχ ¯ = c h (u), c ∈ C , or u = c¯ h (χ) . Then, from 1 # 1 # 1 −iθ iθ hχ,¯ ui = hχ,¯ c¯ h (χ)i = c¯ h (χ, χ¯) = c¯ h(¯v, v) one hasc ¯ = e i.e. c = e . ( v ⇒ i ) : Using also v’, already seen to be equivalent to v, one sees that the couple (u, v) constitutes an h-orthogonal basis of U , whose elements have the same norm. Thus the couple (ζu , ζv) , deﬁned by 1 1 ζu ≡ √ u , ζv ≡ √ v , |hχ,u¯ i| |hχ,u¯ i| is an h-orthonormal basis16 of U ; hence 1 h# = ζ¯ ⊗ ζ + ζ¯ ⊗ ζ = u¯ ⊗ u +v ¯ ⊗ v . u u v v |hχ,¯ ui|

Condition i then follows from τ = √1 h¯# . 2 # # 1 Remark. In particular, τ0 = p (ζ1 , ζ2) ≡ p (ζ1 , z¯ ).

3.6 Helicity of Dirac Fermions (The helicity of photons is discussed in §5.3). ⊥ Recalling §3.4 we observe that in particular, for n ≡ τ3 ∈ H we have  ζ 7→ ζ , ζ 7→ −ζ , ¯z1 7→ ¯z1 , ¯z2 7→ −¯z2 ,  1 1 2 2 J3 : ζ3 7→ ζ3 , ζ4 7→ −ζ4 ,  0 0 0 0 0 0 0 0 ζ1 7→ ζ1 , ζ2 7→ −ζ2 , ζ3 7→ ζ3 , ζ4 7→ −ζ4 .

Hence  diag(1, 0, 1, 0) , 1 1 σ0 ± σ3 0  Π[n] = 2 11 ± J3 = 2 = 0 σ0 ± σ3 diag(0, 1, 0, 1) . We remark that for each observer and for each n ∈ H⊥ there exist a 2-spinor basis such that, in the induced Pauli basis, τ0 coincides with the observer and n = τ3 . If ψ ≡ u, λ¯ ∈ W then in such basis we have

1 1 2 ¯ 1 ¯ 2 J[n]ψ ≡ J3ψ = 2 (u ζ1 − u ζ2 , λ1 ¯z − λ2 ¯z ) .

1 Thus the condition J3(ψ) = ± 2 ψ reads 1 2 1 2 1 2 1 2 u ζ1 − u ζ2 , λ¯1 ¯z − λ¯2 ¯z = ± u ζ1 + u ζ2 , λ¯1 ¯z + λ¯2 ¯z ,

16 −iθ But not ε-normalized in general, since ε(ζu , ζv) = ε(u, v)/|ε(u, v)| =√ e . Also, note that (ζu , ζv) being h-orthonormal amounts to (u, v) being orthonormal relatively to h/µ = 2p/µ ¯ 2 . 28 3 ADDITIONAL STRUCTURES

2 1 which is fulﬁlled either for u = λ¯2 = 0 (positive eigenvalue) or for u = λ¯1 = 0 (negative eigenvalue), namely 1 1 2 2 either ψ = u ζ1 , λ¯1 ¯z or ψ = u ζ2 , λ¯2 ¯z . In order to examine the particle’s helicity we consider m p = √ u[ ⊗ u¯[ + λ ⊗ λ¯ , 2 |hλ, ui|

0 m 1 2 2 2 2 2 0 p⊥ = p − hp, τ0i t = p − 2 |hλ,ui| (|u | + |u | + |λ1| + |λ2| ) t = m # ¯ 0 = p − 2 |hλ,ui| h(¯u, u) + h (λ, λ) t ≡ p − pk , where h ≡ 2 t0. 0 # Observing that (t ) = τ0 we obtain

# i # # J[p⊥ ] = 2 γ0 γ[p − pk ] γη = i # i m # ¯ = 2 γ0 γ[p ] γη − 2 2 |hλ,ui| h(¯u, u) + h (λ, λ) γ0 γ0 γη = i # i m # ¯ = 2 γ0 γ[p ] γη − 4 |hλ,ui| h(¯u, u) + h (λ, λ) γη , Recalling

γη(u, λ¯) = i (u, −λ¯) , √ γu ⊗ u¯ + λ# ⊗ λ¯#(u, λ¯) = 2 hλ,¯ u¯i u , hλ, ui λ¯ , √ γu ⊗ u¯ + (−λ)# ⊗ (−λ¯)#(u, −λ¯) = 2 −hλ,¯ u¯i u , hλ, ui λ¯ ,

h(¯u, u) + h#(λ,¯ λ) = m hλ, ui h#(λ¯) , −hλ,¯ u¯i h[(u) + −u , λ¯ . 2 |hλ,ui| 2 Let us consider the case when ∃c ∈ C\{0} such that 1 h#(λ¯) ≡ λ¯ch# = c u ⇔ h[(u) ≡ hcu = λ¯ , c 1 i.e. h#(λ) ≡ h#cλ =c ¯u¯ ⇔ h[(¯u) ≡ u¯ch = λ . c¯ Then 1 # ¯ h(¯u, u) = c¯ hλ, ui , h (λ, λ) = c hλ, ui ⇒ 1 # ¯ ⇒ hλ, ui =c ¯ h(¯u, u) = c h (λ, λ) ⇒ ⇒ h#(λ,¯ λ) = |c|2 h(¯u, u) , 3.6 Helicity of Dirac Fermions 29 and, using the shorthand

|u|2 ≡ h(¯u, u) ⇒ hλ, ui =c ¯|u|2 , we get 1 + |c|2 J[p#](u, λ¯) = m c¯|u|2 c u , −c |u|2 1 λ¯ + |u|2 −u , λ¯ = ⊥ 2 |c| |u|2 c 2 1 + |c|2 = m |c|2 |u|2 u , −|u|2 λ¯ + |u|2 −u , λ¯ = 2 |c| |u|2 2 −1 + |c|2 −1 + |c|2 = m |u|2 u , |u|2 λ¯ = 2 |c| |u|2 2 2 m (|c|2−1) ¯ = 4 |c| (u , λ) .

On the other hand we have p⊥ = p − pk with m # ¯ 0 m 2 2 0 pk = 2 |hλ,ui| h(¯u, u) + h (λ, λ) t = 2 |c| |u|2 (1 + |c| ) |u| t =

m (1+|c|2) 0 0 = 2 |c| t ≡ p0 t , whence

g(p⊥, p⊥) = g(p, p) + g(pk , pk) − 2 g(p, pk) = g(p, p) − g(pk , pk) =

2 m2 (1+|c|2)2 m2 (|c|2−1)2 = m − 4 |c|2 = − 4 |c|2 .

For |c| = 1 we obtain g(p⊥, p⊥) = 0 ⇒ p⊥ = 0 . Indeed in this case

|hλ, ui| = h(¯u, u) = h#(λ,¯ λ) ,

17 # # which implies u ⊗ u¯ + λ ⊗ λ¯ ∝ τ0 . Assuming now |c|= 6 1 and setting

−1/2 2 |c| n ≡ |g(p⊥, p⊥)| p⊥ = m ||c|2−1| p⊥ we get ¯ 2 1 ¯ J[n](u, λ) = sgn(|c| − 1) · 2 (u, λ) . Proposition 3.2 Let ψ ≡ (u, λ¯) ∈ W be such that λ# 6∝ u . Then ψ is a helicity eigenstate, with respect to a given observer, if and only if there exists c ∈ C\{0} , |c|= 6 1 , such that λ¯ch# = c u .

1 2 More precisely, we get the helicity eigenvalue 2 sgn(|c| − 1) .

17 it # it 1 2 it 1 it 2 Letting c ≡ e we get λ¯ch = λ¯1 ζ1 + λ¯2 ζ2 = e (u ζ1 + u ζ2) that is λ¯1 = e u , λ¯2 = e u . Moreover we # −it 2 1 have λ = λ2 ζ1 − λ1 ζ2 = e (¯u ζ1 − u¯ ζ2), whence # # 2 2 λ ⊗ λ¯ = (|λ2| ζ1 ⊗ ζ¯1 + |λ1| ζ2 ⊗ ζ¯2 − λ2 λ¯1 ζ1 ⊗ ζ¯2 − λ1 λ¯2 ζ2 ⊗ ζ¯1) = 2 2 = |λ2| ζ1 ⊗ ζ¯1 + |λ1| ζ2 ⊗ ζ¯2 − λ2 λ¯1 ζ1 ⊗ ζ¯2 − λ1 λ¯2 ζ2 ⊗ ζ¯1 = 2 2 1 2 2 1 1 2 = |u | ζ1 ⊗ ζ¯1 + |u | ζ2 ⊗ ζ¯2 − u¯ u ζ1 ⊗ ζ¯2 − u¯ u ζ2 ⊗ ζ¯1 ,

1 2 2 2 1 2 2 1 u ⊗ u¯ = |u | ζ1 ⊗ ζ¯1 + |u | ζ2 ⊗ ζ¯2 + u u¯ ζ1 ⊗ ζ¯2 + u u¯ ζ2 ⊗ ζ¯1 ,

# ¯# 1 2 2 2 ¯ ¯ √1 ⇒ u ⊗ u¯ + λ ⊗ λ = (|u | + |u | )(ζ1 ⊗ ζ1 + ζ2 ⊗ ζ2) = 2 h(¯u, u) τ0 . 30 3 ADDITIONAL STRUCTURES proof: Let ψ be a helicity eigenstate. Then from a previous computation follows that, in a 2-spinor basis such that in the induced Pauli basis τ0 coincides with the observer and 1 ¯ 1 2 ¯ 2 ¯ # p⊥ = |p⊥| τ3 , one has either ψ = (u ζ1 , λ1 ¯z ) or ψ = (u ζ2 , λ2 ¯z ) , whence λch ∝ u . Con- versely, the above computation shows that the condition λ¯ch# = c u implies that (u, λ¯) is a helicity eigenstate with the stated value. Next we examine the helicity of states with isotropic 4-velocity. We start with ψ = (u, 0) . The 4-velocity k = u ⊗ u¯ is deﬁned up to a constant factor; this does not aﬀect our computation, that only depends from the versor of the spatial part of k (with respect to the chosen observer). We perform a coordinate computation, obtaining A A˙ ¯ 1 2 ¯ 2 2 ¯ 1 2 ¯ 2 1 ¯ k = u u¯ ζA ⊗ ζA˙ = |u | ζ1 ⊗ ζ1 + |u | ζ2 ⊗ ζ2 + u u¯ ζ1 ⊗ ζ2 + u u¯ ζ2 ⊗ ζ1 = λ = k τλ = kk + k⊥ ,

0 √1 1 2 2 2 kk = ht , ki τ0 = 2 (|u | + |u | ) τ0 ,

√1 1 2 2 1 √i 1 2 2 1 √1 1 2 2 2 k⊥ = 2 (u u¯ + u u¯ ) τ1 + 2 (u u¯ − u u¯ ) τ2 + 2 (|u | − |u | ) τ3 ,

1 1 2 2 2 2 g(k⊥ , k⊥) = − 2 (|u | + |u | ) . √ p 1 2 2 2 Setting n ≡ k⊥/ |g(k⊥ , k⊥)| = 2 k⊥/(|u | + |u | ) we obtain the matrix expression 1 2 3 J[n] = n J1 + n J2 + n J3 =

 1 1 2 2 2 1 2  2 (|u | −|u | ) u u¯ 0 0

 2 1 1 1 2 2 2  1  u u¯ (−|u | +|u | ) 0 0  =  2  1 2 2 2   |u | +|u |  0 0 1 (|u1|2−|u2|2) u1 u¯2   2  2 1 1 1 2 2 2 0 0 u u¯ 2 (−|u | +|u | ) whence u1 u1 2 2 u  1 u  1 J[n]   =   ⇒ J[n](u, 0) = (u, 0) .  0  2  0  2 0 0 # # # Now we consider ψ = (0, λ¯), k = λ ⊗ λ¯ . Since λ = λ2 ζ1 − λ1 ζ2 , we obtain J[n] p 1 with n ≡ k⊥/ |g(k⊥ , k⊥)| from that of the case k = u ⊗ u¯ via the replacements u → λ2 , 2 u → −λ1 , namely  1 2 2 ¯  2 (−|λ1| +|λ2| ) −λ2 λ1 0 0

 ¯ 1 2 2  1  −λ1 λ2 (|λ1| −|λ2| ) 0 0  J[n] =  2  |λ |2+|λ |2  1 2 2 ¯  1 2  0 0 (−|λ1| +|λ2| ) −λ2 λ1   2  ¯ 1 2 2 0 0 −λ1 λ2 2 (|λ1| −|λ2| ) yielding18  0   0   0  1  0  ¯ 1 ¯ J[n]  ¯  = − 2  ¯  ⇒ J[n](0, λ) = − 2 (0, λ) . −λ1 −λ1 −λ¯2 −λ¯2

18 The components of (0, λ¯) in the Weyl basis, in which the matrix J[n] is written, are (0, 0, −λ¯1, −λ¯2), however this conventional sign has no relevance in the present discussion. 31

# # If ψ ≡ (u, λ¯) ∈ W with ε (λ) ≡ λ = c u , c ∈ C\{0} , then 1 k = (1 + |c|2) u ⊗ u¯ = 1 + λ# ⊗ λ¯# . |c|2 Hence we can write J[n] as the block matrix ! Ju 0 J[n] = 0 Jλ¯ where

1 (|u1|2−|u2|2) u1 u¯2 ! 1 2 Ju = |u1|2+|u2|2 2 1 1 1 2 2 2 u u¯ 2 (−|u | +|u | )

1 (−|λ |2+|λ |2) −λ λ¯ ! 1 2 1 2 2 1 J¯ = . λ 2 2 |λ1| +|λ2| ¯ 1 2 2 −λ1 λ2 2 (|λ1| −|λ2| ) Then ﬁnally we obtain  u1  u1 2 2  u  1 u  ¯ 1 ¯ J[n]  ¯  = 2 ¯  ⇒ J[n](u, λ) = 2 (u, −λ) , −λ1 λ1 −λ¯2 λ¯2 showing that ψ ≡ (u, λ¯) ∈ W cannot be a helicity eigenstate if both u and λ are diﬀerent from zero.

Notes pasted from wikipedia: given massless particle appears to spin in the same direction along In physics, chirality may be found in the spin of a particle, its axis of motion regardless of point of view of the observer. where the handedness of the object is determined by the direc- For massive particles—such as electrons, quarks, and neutri- tion in which the particle spins. Not to be confused with helicity nos—chirality and helicity must be distinguished. In the case of (which is the projection of the spin along the linear momentum these particles, it is possible for an observer to change to a ref- of a subatomic particle), chirality is a purely quantum mechanical erence frame that overtakes the spinning particle, in which case phenomenon (like spin). Although both can have left-handed or the particle will then appear to move backwards, and its helicity right-handed properties, only in the massless case do they have (which may be thought of as ’apparent chirality’) will be reversed. a simple relation. In particular for a massless particle the helic- A massless particle moves with the speed of light, so a real ity is the same as the chirality while for an antiparticle they have observer (who must always travel at less than the speed of light) opposite sign. cannot be in any reference frame where the particle appears to re- The helicity of a particle is right-handed if the direction of verse its relative direction, meaning that all real observers see the its spin is the same as the direction of its motion. It is left-handed same chirality. Because of this, the direction of spin of massless if the directions of spin and motion are opposite. By convention particles is not affected by a Lorentz boost (change of viewpoint) for rotation, a standard clock, with its spin vector defined by the in the direction of motion of the particle, and the sign of the pro- rotation of its hands, tossed with its face directed forwards, has jection (helicity) is fixed for all reference frames: the helicity of left-handed helicity. Mathematically, helicity is the sign of the massless particles is a relativistic invariant (i.e. a quantity whose projection of the spin vector onto the momentum vector: left is value is the same in all inertial reference frames). negative, right is positive. With the discovery of neutrino oscillation, which implies that The chirality of a particle is more abstract. It is determined neutrinos have mass, the only observed massless particle is the by whether the particle transforms in a right- or left-handed rep- photon. The gluon is also expected to be massless, although the resentation of the Poincar group. (However, some representations, assumption that it is has not been conclusively tested. Hence, such as Dirac spinors, have both right- and left-handed compo- these are the only two particles now known for which helicity could nents. In cases like this, we can define projection operators that be identical to chirality, and only one of them has been confirmed project out either the right or left hand components and discuss by measurement. All other observed particles have mass and thus the right- and left-handed portions of the representation.) may have different helicities in different reference frames. It is still For massless particles—such as the photon, the gluon, and possible that as-yet unobserved particles, like the graviton, might the (hypothetical) graviton—chirality is the same as helicity; a be massless, and hence have invariant helicity like the photon.

4 Two-spinor bundles

4.1 Two-spinor connections 4.1.1 Consider any manifold M and a complex vector bundle S → M with 2-dimensional fibres. a Denote base coordinates as (x ); choose a local frame (ξA) of S, determining linear fibre 32 4 TWO-SPINOR BUNDLES coordinates (xA). According to the constructions of the previous sections, one now has the bundles Q, L, U, H over M, with smooth natural structures; the frame (ξA) yields the frames ε, l,(ζA) and (τλ) , respectively. Moreover for any rational number r ∈ Q one has the r semi-vector bundle L ; but observe that, for a given r, according to the bundle topology one may have no Qr, or one may have one or more (not isomorphic) such bundles; these roots always exist on a suitable restriction of the base manifold. Consider an arbitrary C-linear connection Γ on S → M, called a 2-spinor connection. In a A A the fibred coordinates (x , x ) Γ is expressed by the coefficients Γa B : M → C , namely the covariant derivative of a section s : M → S is expressed as

A A B a ∇s = (∂as − Γa B s ) dx ⊗ ξA . The rule ∇s¯ = ∇s yields a connection Γ¯ on S → M, whose coeﬃcients are given by Γ¯ A˙ = Γ A . a B˙ a B Of course one also has the induced connections on the dual bundles and on the various bundles constructed from S. I am going to show how Γ can be conveniently expressed in terms of the connections induced on Λ2 and H ; on turn, the connection induced on Λ2 can be expressed in terms of the connections induced on L and Q.

4.1.2

1/2 1/2 Denote by G and Y the connections induced on L and Q (the latter being deﬁned, possibly, only locally); as these bundles have 1-dimensional ﬁbres, it makes sense to denote 2 the connections induced on their squares L and Q ≡ ∧ U as 2 G and 2 Y , and the connection 2 2 induced on ∧ S ≡ L ⊗ ∧ U as 2(G + i Y ) ; actually one has

a ∇l = −2 Ga dx ⊗ l , a ∇ε = 2 i Ya dx ⊗ ε , −1 −1 a −1 ∇w ≡ ∇(l ⊗ ε) = 2(Ga + i Ya) dx ⊗ l ⊗ ε and the like.

Proposition 4.1 One has

G = <( 1 Γ A ) = 1 (Γ A + Γ¯ A˙ ) , a 2 a A 4 a A a A˙

Y = =( 1 Γ A ) = 1 (Γ A − Γ¯ A˙ ) . a 2 a A 4i a A a A˙

−1 A B A B 19 proof: Writing w = εAB x ∧ x = εAB x ⊗ x one has

−1 A C B B A C ∇aw = εAB ( Γa C x ⊗ x + Γa C x ⊗ x ) =

19 A The last passage uses the following observation. If (MB ) is a 2 × 2 matrix, then

C C MA εCB − MB εCA = µ εAB , µ ∈ C , since the left-hand side is antisymmetric in A ↔ B . By contracting both members with εAB one has

C A C B C MA δC − MB (−δC ) = 2 µ ⇒ µ = MC . 4.1 Two-spinor connections 33

C C A B = (εCB Γa A + εAC Γa B ) x ⊗ x =

C C A B = (εCB Γa A − εCA Γa B ) x ⊗ x =

C A B C −1 = εAB Γa C x ⊗ x ≡ Γa C w .

(Alternatively, one may write w−1 = x1 ∧ x2 and use the standard calculation of the covariant −1 1 2 derivative of a volume form, directly finding ∇aw = ( Γa 1 + Γa 2) ). Next, l−2 ≡ w−1 ⊗ w¯ −1 (which can be identified with the naturally induced conformal metric in the fibres of S ∨¯ S = L ⊗ H), and one has

−2 −2 −1 −1 4 Ga l = ∇al = ∇a(w ⊗ w¯ ) =

= (Γ C + Γ¯ C˙ ) w¯ −1 , a C a C˙

⇒ G = 1 (Γ C + Γ¯ C˙ ) = <( 1 Γ C ) . a 4 a C a C˙ 2 a C

Finally

−1 −1 C −1 ∇aε = ∇a(l ⊗ w ) = −2 Ga l ⊗ w + Γa C l ⊗ w =

C C C = ( Γa C − 2 Ga)ε = [ Γa C − <( Γa C )] ε =

C = i =( Γa C ) ε .

Corollary 4.1 Since Ga and Ya are real, one sees that Γ actually yields a semi-linear connection on L, and that the induced linear connection on Q is Hermitian (preserves its natural Hermitian structure).

4.1.3 The coeﬃcients of the connection Γ˜ induced on U are given by

˜ A A A Γa B = Γa B − Ga δ B ,

−1/2 A as one sees from ∇aζB = ∇a(l ⊗ ξB ) = Ga ζB − Γa B ζA . Let Γ˜ be the connection induced on U ⊗ U, and Γ0 the connection induced on S ⊗ S. ˜ AA˙ ˜ λ ˜ ¯ Denote by Γa BB˙ and Γa µ the coeﬃcients of Γ in the bases ζA ⊗ ζA˙ and τλ , namely

˜ AA˙ ˜ λ AA˙ µ Γa BB˙ = Γa µ τ λ τ BB˙ .

0 AA˙ 0 λ 0 ¯ ¯ Moreover, denote by Γ a BB˙ and Γ a µ the coeﬃcients of Γ in the bases ξA ⊗ ξA˙ ≡ l ⊗ ζA ⊗ ζA˙ and l ⊗ τλ (actually, since l is a 1-dimensional factor, one could just say that these are the 0 components of Γ in the bases ζA and τλ), fulﬁlling a similar relation. From an immediate 34 4 TWO-SPINOR BUNDLES calculation20 one has

Γ0 AA˙ = Γ A δA˙ + δA Γ¯ A˙ , a BB˙ a B B˙ B a B˙

Γ˜ AA˙ = Γ˜ A δA˙ + δA Γ˜¯ A˙ = a BB˙ a B B˙ B a B˙

0 AA˙ A A˙ = Γ a BB˙ − 2 Ga δ B δ B˙ =

= Γ A δA˙ + δA Γ¯ A˙ − 2 G δA δA˙ . a B B˙ B a B˙ a B B˙

Proposition 4.2 The connections Γ0 and Γ˜ are reducible to real connections on S ∨¯ S and H ≡ U ∨¯ U, respectively. proof: This follows from the fact that their coeﬃcients are real.

Proposition 4.3 The connection Γ˜ induced on H by any 2-spinor connection is metric, namely ∇[Γ]˜˜ g = 0 .

−2 proof: The metricg ˜ can be identiﬁed with the identity of the bundle L , namely it is the −1 −1 −2 2 + canonical section 1 ≡ w ⊗ w ≡ ε ⊗ ε : M → L ⊗ L ≡ M × R , which obviously has vanishing covariant derivative. ˜ λ 21 Because of the metricity, the coeﬃcients Γa µ are antisymmetric, that is

˜ λµ ˜ µλ ˜ λµ ˜ λ νµ Γa + Γa = 0 , Γa ≡ Γa ν η , and traceless, that is ˜ λ Γa λ = 0 .

The latter formula expresses ∇aη˜ = 0 . On the other hand, one has

Γ0 λ = 2 (Γ A + Γ¯ A˙ ) = 8 G , ⇒ ∇ η0 = 8 G η0 , a λ a A a A˙ a a a where η0 is the volume form associated with the conformal metric l2 g˜ of S ∨¯ S.

4.1.4 The above relations between Γ and the induced connections can be inverted as follows.

Proposition 4.4 One has

Γ A = (−G + i Y ) δA + 1 Γ0 AA˙ = a B a a B 2 a BA˙

A 1 ˜ AA˙ = (Ga + i Ya) δ B + 2 Γa BA˙ . 20∇ (ξ ⊗ ξ¯ ) = −(Γ A ξ ⊗ ξ¯ + Γ¯ A˙ ξ ⊗ ξ¯ ) = −(Γ A δA˙ + δA Γ¯ A˙ ) ξ ⊗ ξ¯ . By the way, this corre- a B B˙ a B A B˙ a B˙ B A˙ a B B˙ B a B˙ B B˙ 0 AA˙ sponds to the Penrose decomposition of the symbol Γ a BB˙ (§A.9). 21 # λµ # λµ ˜ ν ˜ ν Fromg ˜ = η Θλ ⊗ Θµ one ﬁnds 0 = ∇ag˜ = −η (Γa λ Θν ⊗ Θµ + Γa µ Θλ ⊗ Θν ) = ˜ λµ ˜ µλ = −(Γa + Γa )Θλ ⊗ Θµ . 4.1 Two-spinor connections 35 proof: Taking the trace of the expression of Γ0 in terms of Γ one ﬁnds Γ0 AA˙ = 2 Γ A + δA Γ¯ A˙ = a BA˙ a B B a A˙

A A = 2 Γa B + 2 δ B (Ga − i Ya) . ˜ AA˙ 0 AA˙ A A˙ Moreover, from Γa BB˙ = Γ a BB˙ − 2 Ga δ B δ B˙ one has ˜ AA˙ 0 AA˙ A Γa BA˙ = Γ a BA˙ − 4 Ga δ B =

A A = 2 Γa B − 2 δ B (Ga + i Ya) .

In 4-spinor formalism this reads (§A.10) α α 1 ˜ λµ α Γa β = (Ga + i Ya) δ β + 4 Γa (˜γλ ∧ γ˜µ) β = α 1 ˜ λµ α = (Ga + i Ya) δ β + 8 Γa (˜γλ γ˜µ − γ˜µ γ˜λ) β = α 1 ˜ λµ α = (Ga + i Ya) δ β + 4 Γa (˜γλ γ˜µ) β , α ¯F where now Γa β stands for the coeﬃcients of the naturally induced connection ( Γ , Γ ) on F W ≡ U ⊕M U and α, β = 1, .., 4. A similar relation holds among the curvature tensors, namely22

A A 1 0 AA˙ Rab B = 2 (dG − i dY )ab δ B + 2 R ab BA˙ =

A 1 ˜ AA˙ = −2 (dG + i dY )ab δ B + 2 Rab BA˙ , where R, R0 and R˜ are the curvature tensors of Γ, Γ0 and Γ,˜ respectively. In 4-spinor formalism one has α α 1 ˜ λµ α Rab β = −2 (dG + i dY )ab δ β + 4 Rab (˜γλ ∧ γ˜µ) β . Of course, the structure of the bundles is reﬂected in the way the connection coeﬃcients transform under a local gauge transformation K : M → Gl(2, C). One obtains 0 A −1 A D C −1 A C Γ a B = (K )C KB Γa D − (K )C ∂aKB , 0 1 Ga = Ga − 2 ∂a log |det K| , 0 1 Ya = Ya − 2 ∂a arg det K , ˜0 λ ˜ −1 λ ˜ ρ ˜ ν ˜ −1 λ ˜ ν Γ a µ = (K )ν Kµ Γa ρ − (K )ν ∂aKµ .

4.1.5 A different way of writing the coefficients of Γ is Γ A = Γλ σ A . a B a λ B The use of this notation is examined in App.A.10.2 (see also §??); in general, this kind of notation is suitable for describing any linear connection of a complex vector bundle with 2- dimensional fibres, and turns out to be particularly meaningful when the connection under consideration preserves some given fibred Hermitian metric; in that case, the components of i the connection relatively to an orthonormal frame have the property that the coefficients Γa , i i = 1, 2, 3 , are immaginary. In particular, if Γ fulfils ∇[ Γ]τ0 = 0 then Γa : M → i R . 22 a b a b b a a b (∂aYb − ∂bYa) dx ⊗ dx = ∂aYb (dx ⊗ dx − dx ⊗ dx ) = 2 ∂aYb dx ∧ dx = 2 dY and the like. 36 4 TWO-SPINOR BUNDLES

4.2 Two-spinor tetrad

From now on I assume that M is a real 4-dimensional manifold. Consider a linear morphism

Θ:TM → S ⊗ S = C ⊗ L ⊗ H , i.e. a section ∗ Θ: M → C ⊗ L ⊗ H ⊗ T M

(all tensor products are over M). Its coordinate expression is

λ a AA˙ ¯ a Θ = Θa τλ ⊗ dx = Θa ζA ⊗ ζA˙⊗ dx ,

λ AA˙ Θa, Θa : M → C ⊗ L .

Usually one considers the particular case when Θ is real, namely valued in the Hermitian subspace L ⊗ H ⊂ S ⊗ S , and non-degenerate; this means that Θ is a ﬁbred isomorphism TM → L ⊗ H over M, whose inverse is denoted by

← ← Θ ≡ Θ: L ⊗ H → TM or, as a section, ← −1 ∗ Θ: M → L ⊗ TM ⊗ H .

In the real non-degenerate case Θ can be viewed as a ‘scaled’ tetrad (or soldering form, or λ vierbein). In that case the coeﬃcients Θa are real-valued (more exactly, valued in R ⊗ L) AA˙ ¯ A˙A AA˙ while the coeﬃcients Θa are Hermitian, i.e. Θa = Θa . By abuse of language, in general, Θ will be called a possibly degenerate tetrad. From Θ one obtains the following objects23

∗ 2 ∗ ∗ g ≡ Θ g˜ : M → C ⊗ L ⊗ T M ⊗ T M ,

∗ 4 4 ∗ η ≡ Θ η˜ : M → C ⊗ L ⊗ ∧ T M ,

γ ≡ γ˜ ◦ Θ:TM → L ⊗ End(W ) , with the coordinate expressions

λ µ a b AA˙ BB˙ a b g = ηλµ Θa Θb dx ⊗ dx = εAB ε¯A˙B˙Θa Θb dx ⊗ dx , η = det(Θ) dx0 ∧ dx1 ∧ dx2 ∧ dx3 , √ AA˙ ¯ B˙ B a γ = 2 Θa (ζA ⊗ ζA˙ + εAB ε¯A˙B˙¯z ⊗ z ) ⊗ dx .

23Here, and whenever a distinction is needed, an object ‘living’ in H is marked by a tilda, while the same unmarked symbol denotes the spacetime object corresponding to it via Θ . 4.3 Cotetrad 37

Namely

a b g = gab dx ⊗ dx ,

λ µ AA˙ BB˙ 2 gab = ηλµ Θa Θb = εAB ε¯A˙B˙Θa Θb : M → C ⊗ L ;

η =η ˘d4x ,

4 η˘ = det Θ : M → C ⊗ L ;

a γ = γa ⊗ dx , √ AA˙ ¯ B˙ B γa = 2 Θa (ζA ⊗ ζA˙ + εAB ε¯A˙B˙¯z ⊗ z ): M → L ⊗ End(W ) .

If and only if Θ is real and non-degenerate, the above objects turn out to be a conformal Lorentz metric, the corresponding (up to sign, unique) conformal volume form and a conformal Cliﬀord map. Moreover, in the non-degenerate case, one has

b λ ab ← b −1 ab −2 Θµ ≡ Θa ηλµ g = Θµ : M → C ⊗ L , g : M → C ⊗ L .

A non-degenerate tetrad, together with a two-spinor frame, yields mutually dual orthonor- ∗ −1 λ ∗ mal frames (Θλ) of L ⊗ TM and (Θ ) of L ⊗ T M , given by

← a Θλ ≡ Θ(τλ) = Θλ ∂xa , ∗ λ ∗ λ λ a Θ ≡ Θ (t ) = Θa dx .

One also writes

∗ λ a γ = γλ ⊗ Θ = γa ⊗ dx ,

γλ ≡ γ(Θλ): M → End(W ) , λ γa ≡ γ(∂xa) = Θa γλ : M → L ⊗ End(W ) .

4.3 Cotetrad The following constructions, which hold for any (possibly degenerate) Θ, are needed in order to formulate a field theory which is never singular. ∗ One can define a natural ‘exterior’ product of elements in any given fibre of H ⊗M T M by requiring that, for decomposable tensors, it is given by24

∗ (y1 ⊗ α1) ∧ (y2 ⊗ α2) = (y1 ∧ y2) ⊗ (α1 ∧ α2) , α1 , α2 ∈ T M , u1 , u2 ∈ H ; equivalently, this product is characterized by

1 ∗ θ1 ∧ θ2 | u1 ∧ u2 = 2 (θ1|u1) ∧ (θ2|u2) − (θ1|u2) ∧ (θ2|u1) , θ1 , θ2 ∈ H ⊗ T M (all the above algebraic formulas hold ﬁbrewise). Consider now the exterior products

q q q q ∗ ∧ Θ: M → C ⊗ L ⊗ ∧ H ⊗ ∧ T M , q = 1, 2, 3, 4 . 24Of course, this construction holds for linear maps between any two vector spaces. 38 4 TWO-SPINOR BUNDLES

In particular, one has ∧2Θ ≡ Θ ∧ Θ , that is

∧2Θ(u ∧ v) = Θ(u) ∧ Θ(v) , with coordinate expression

2 λ µ a b ∧ Θ = ΘaΘb (τλ ∧ τµ) ⊗ (dx ∧ dx ) . Next, consider the linear map over M

∗ 4 4 ∗ Θ:(˘ S ⊗ S) ⊗ T M → C ⊗ L ⊗ ∧ T M deﬁned by ˘ 1 ◦ 1 ◦ 3 Θ(ξ) ≡ 3! ∗˜ (ξ ∧ Θ ∧ Θ ∧ Θ) = 3! ∗˜ [ξ ∧ (∧ Θ)] , where ∗˜ is the Hodge operator, in the exterior algebra ∧ (H), determined byη ˜. One ﬁnds the coordinate expression

˘ ˘ a λ 4 1 abcd µ ν ρ λ 4 Θ(ξ) = Θλ ξa d x ≡ 3! ε ελµνρ Θb Θc Θd ξa d x , λ a λ ξ = ξa τλ ⊗ dx , ξa : M → C ⊗ L .

∗ 4 4 ∗ Now Θ˘ can be seen as a bilinear map (S ⊗ S) × T M → C ⊗ L ⊗ ∧ T M over M, or also as a linear map 4 4 ∗ S ⊗ S → C ⊗ L ⊗ TM ⊗ ∧ T M over M. Using the latter point of view, in the non-degenerate case one writes

← Θ˘ = Θ ⊗ η .

Then, in general one may regard Θ˘ as a kind of ‘pseudo-inverse’ of Θ, deﬁned even if Θ is degenerate. I call it the co-tetrad. The above construction can be immediately generalized, for p = 0, 1, 2, 3, 4, to a map

(p) p p ∗ 4 4 ∗ Θ˘ : ∧ (S ⊗ S) ⊗ (∧ T M) → C ⊗ L ⊗ ∧ T M by ˘ (p) 1 ◦ 4−p Θ (ξ) ≡ (4−p)! ∗˜[ξ ∧ (∧ Θ)] ; I will be concerned with Θ˘ (1) = Θ˘ and Θ˘ (2). Note that Θ˘ (0) = η.

4.4 Tetrad and connections A further point about the non-degenerate case deals with connections, and will turn out to be essential for ﬁeld theories: if Γ is a complex-linear connection on S, and G and Γ˜ are the induced connections on L and H, then a non-degenerate tetrad Θ : TM → L ⊗ H yields a unique connection Γ on TM, characterized by the condition

∇[Γ ⊗ Γ]Θ˜ = 0 .

Moreover Γ is metric, i.e. ∇[Γ]g = 0. Denoting by Γ λ the coeﬃcients of Γ in the frame ← a µ 0 Θλ ≡ Θ(l ⊗ τλ) one obtains λ ˜ λ λ Γa µ = Γa µ + 2 Ga δ µ . 4.4 Tetrad and connections 39

˜ λ ˜ λ The curvature tensors of Γ and Γ are related by Rab µ = Rab µ , or

c ˜ λ c µ Rabd = Rab µ Θλ Θd .

Hence the Ricci tensor and the scalar curvature are given by

b ˜ λ b µ Rad = Rabd = Rab µ Θλ Θd , a ˜ λµ b a Ra = Rab Θλ Θµ .

Remark. In the degenerate case there is no induced spacetime connection Γ, but one still has the two-spinor connection Γ and its decomposition. In general, the connection Γ will have non-vanishing torsion. This is the tensor ﬁeld T : M → TM ⊗ ∧2T∗M deﬁned by

T (u, v) = ∇uv − ∇vu − [u, v] , where u, v : M → TM are any two vector ﬁelds, and has the coordinate expression

c c c T ab = −Γab + Γba .

˜ λ The torsion can be expressed through the components of Θ and the coeﬃcients Γa µ as follows. First, note that the condition ∇Θ = 0 reads

λ λ ˜ λ µ ˜ c λ 0 = ∇aΘb = ∂aΘb − Γa µ Θb + Γab Θc , c λ λ ˜ λ µ i.e. Γab Θc = −∂aΘb + Γa µ Θb , c ← c λ ˜ λ µ or Γab = Θλ −∂aΘb + Γa µ Θb , ˜ λ c λ λ ← b Γa µ = (Γab Θc + ∂aΘb ) Θµ .

Then one gets

λ c λ c c Θc T ab = Θc (−Γab + Γba) = λ λ µ λ λ µ = ∂aΘb − Γa µ Θb − ∂bΘa + Γb µ Θa = λ ˜ λ µ λ λ ˜ λ µ λ = ∂aΘb − Γa µ Θb − Ga Θb − ∂bΘa + Γb µ Θa + Gb Θa λ µ ˜ λ λ = ∂[aΘb] + Θ[a Γb] µ + 2 Θ[a Gb] .

We remark that the torsion can be seen as the Fr¨olicher-Nijenhuis bracket

T˜ ≡ T cΘ = [Γ0, Θ] : M → ∧2T∗M ⊗ H0 , M

0 where H = L ⊗ H, Γ0 : H0 → T∗M ⊗ TH0 H0 0 is the induced connection on H M, and Θ is seen as a vertical-valued form

Θ: H0 → T∗M ⊗ VH0 . H0 40 4 TWO-SPINOR BUNDLES

4.5 The Dirac operator Given a tetrad and a two-spinor connection, one introduces the Dirac operator acting on sections M → W . In view of actual ﬁeld theories, one takes sections

−3/2 ψ ≡ (u, χ): M → L ⊗ W . The components of ψ are taken to be scaled functions, namely

α A A˙ α A −3/2 ψ = ψ ζα ≡ (u ζA , χA˙¯z ) , ψ , u , χA˙ : M → C ⊗ L . One has

γ˜# ≡ γ˜ ◦ g˜# : H∗ → End(W ) , orγ ˜# : M → H ⊗ End(W ) , −3/2 ∗ ∇ψ : M → L ⊗ T M ⊗ W , M so that

D # E D # E −3/2 ∗ γ˜ ∇ψ ≡ γ˜ ⊗ ∇ψ : M → L ⊗ H ⊗ T M ⊗ W ,

D # E λα β a or γ˜ ∇ψ =γ ˜ β∇aψ τλ ⊗ dx ⊗ ζα =

a λAA˙ λ A A˙ = τλ ⊗ dx ⊗ σ ∇aχA˙ζA , σ AA˙∇au ¯z .

If Θ is non-degenerate, then one gets the Dirac operator essentially by direct translation of the usual 4-spinor setting. Namely, taking into account

← −1 ∗ Θ: M → L ⊗ H ⊗ TM one sets

D← D # EE −5/2 ∇/ ψ ≡ Θ γ˜ ∇ψ : M → L ⊗ W ,

← a λ α β a α β or ∇/ ψ = Θλ γ˜ β∇aψ ζα = γ β∇aψ ζα = ← a λAA˙ λ A A˙ = Θλ σ ∇aχA˙ζA , σ AA˙∇au ¯z .

The above construction can be modiﬁed in order to obtain a setting which remains valid if Θ is degenerate (and even non-Hermitian). Since

3 ∗ 4 ∗ Θ:˘ M → C ⊗ L ⊗ H ⊗ TM ⊗ ∧ T M , one introduces

˘ D D # EE 3/2 4 ∗ ∇/ ψ ≡ Θ˘ γ˜ ∇ψ : M → L ⊗ W ⊗ ∧ T M , /˘ ˘ a λAA˙ λ A A˙ 4 or ∇ψ = Θλ σ ∇aχA˙ζA , σ AA˙∇au ¯z ⊗ d x , with d4x ≡ dx1 ∧ dx2 ∧ dx3 ∧ dx4 . In the non-degenerate case one simply has ∇/˘ ψ = ∇/ ψ ⊗ η , while the two coordinate expressions diﬀer by a factor det Θ . 4.6 Fermi transport of spinors 41

4.6 Fermi transport of spinors A 1-dimensional timelike submanifold L ⊂ M can be seen as a ‘pointlike observer’, or as the world-line of a ‘detector’. Note that there is a natural inclusion TL ⊂ TM. The restriction of the spacetime time metric is a Riemann metric on L, which yields the detecor’s ‘proper time’. Throughout this §4.6 we’ll assume a tetrad Θ and a spinor connection Γ to be fixed, namely we’ll work in a given gravitational field background. Moreover, for simplicity, we’ll assume Ga = 0 as it is in the standard fied theories (§??). ∼ Since Θ is fixed, it will be convenient to make the identification TM = L ⊗ H (and the like) in order to simplify our notations. Remember that the scalar product of elements in TM 2 is valued into R ⊗ L , while the tensor product of elements in H is real valued; we express 2 this fact by saying that the spacetime metric g is L -scaled, while the metric on H is unscaled (or ‘conformally invariant’).

4.6.1 Rivisitation of the standard Fermi transport

Denote as TLM L and HL L the restrictions of the bundles TM M and H M ∼ −1 to the base L (the ﬁbres over elements in L are the same). Then HL = L ⊗ TLM has one distinguished section, namely the unit future-pointing scaled vector ﬁeld

−1 −1 ∼ τ : L → L ⊗ TL ⊂ L ⊗ TLM = HL . We now consider the linear morphism over L

2 Φ:TL → ∧ HL : v 7→ Φv ≡ 2 (∇vτ) ∧ τ .

a 1 2 3 4 4 Choose base coordinates (x ) ≡ (x , x , x , x ) adapted to L, namely such that ∂x4 ≡ ∂/∂x is tangent to L at the points of L; let moreover (τλ) be any orthonormal frame of HL such that τ0 ≡ τ ; then one gets the coordinate expression ˜ 0j 4 Φ = 2 Γ4 dx ⊗ τj ∧ τ0 . By ‘lowering the second index’ of Φ through the metric one gets a linear morphism

[ ∗ Φ : L → HL ⊗ HL ≡ End(HL) , namely [ 0 0 Φv = ∇vτ0 ⊗ t − τ0 ⊗ ∇vt , with coordinate expression

[ 4 ˜ j 0 ˜ 0 j Φ = −dx ⊗ (Γ4 0 τj ⊗ t + Γ4 j τ0 ⊗ t ) ,

λ where the dual frame of (τλ) was denoted as (t ). Note that [ 0 [ j [ λ (Φ )4 0 = (Φ )4 j = (Φ )4 λ = 0 . ˜ The bundle HL L has of course the connection naturally induced by Γ : the covariant derivative of any section X : L → HL is deﬁned to be the map v 7→ ∇vX : L → HL , namely ˜ ˜ 0 0 0 ∇vX ≡ ∇v[Γ]X is the restriction of ∇v0 [Γ]X for any local extensions v and X of v and X . [ ∗ But now we observe that Φ can be viewed as a section L → T L ⊗L End(HL) , according to 42 4 TWO-SPINOR BUNDLES

[ [ Φv ≡ vcΦ . Hence we are able to introduce a new connection of HL L , namely the Fermi connection25 ˜ [ ΓF ≡ Γ + Φ .

The covariant derivative associated with ΓF turns out to have the expression

[ DvX ≡ ∇v[ΓF]X = ∇vX − Φv(X) =

= ∇vX + g(∇vτ , X) τ − g(τ , X) ∇vτ : L → HL ,

for any sections v : L → TL and X : L → HL . The usual Fermi derivative is deﬁned as a derivation with respect to the detector’s proper time, that is −1 DX ≡ Dτ X : L → L ⊗ HL ,X : L → HL .

Proposition 4.5 For any v : L → TL and X,Y : L → HL one has

v.(X · Y ) = (DvX) · Y + X · DvY. proof: It follows from the fact that Γ˜ is metric and Φ is anti-symmetric, so that Φ[ is valued into the Lorentz group (see §4.6.4 for more details about that). We can directly verify our statement by observing that the Lie derivative along v of the scalar ﬁeld X · Y is well-deﬁned on L independently of extensions. We then have

[ [ v.(X · Y ) = (∇vX) · Y + X · ∇vY = [DvX + Φv(X)] · Y + X · [DvY + Φv(Y )] =

[ [ [ [ = (DvX) · Y + X · DvY + Φv(X ,Y ) + Φv(Y ,X ) = (DvX) · Y + X · DvY, since Φv is antisymmetric.

A section X : L → HL which is covariantly constant relatively to ΓF (namely DX = 0 , or DvX = 0 for all v : L → TL) is said to be Fermi-transported along L; a Fermi-transported section is uniquely determined26 by the value it takes at any point of L. A few points are worth stressing:

• The scalar product of Fermi-transported vectors is constant along L (this follows at once from the above proposition).

• τ itself is Fermi-transported; if f : L → R then D(f τ) = (τ.f) τ .

⊥ ⊥ • If X : L → HL (the subbundle of HL orthogonal to τ) then also DvX : L → HL , ⊥ coinciding with the orthogonal projection onto HL of the ordinary covariant derivative ∇vX .

Thus ΓF preserves the splitting of HL into the direct sum of its subbundles parallel and orthogonal to τ . Moreover one also has Fermi-transported orthonormal frames (τλ) of HL such that τ0 ≡ τ (one only has to ﬁx the frame at some point of L and ‘Fermi-transport’ it).

25 We recall that the difference between any two connections on a vector bundle E B is a tensor field ∗ B T B ⊗B End(E). 26 This follows from a well-known result about general connections, since ΓF is a true connection on the restricted bundle HL L for fixed L. 4.6 Fermi transport of spinors 43

In any orthonormal frame (not necessarily Fermi-transported) one has the coordinate expressions

4 λ k ˜ j 0 j 4 k ˜ j DvX = v ∂4X τλ − X Γ4 k τj ≡ v.X τ0 + v.X − v X Γ4 k τj , 4 λ k ˜ j DX = Θ0 ∂4X τλ − X Γ4 k τj , which are independent of any extensions of v and X . Remark. The definition of the Fermi derivative could be extended to the case when L is spacelike, but cannot be immediately extended to a derivative along a null 1-dimensional submanifold,27 since in the latter case there exists no normalized tangent vector (τ). Moreover, the Fermi transport along an arbitrary timelike curve cannot be seen as parallel transport relatively to some connection on H M. However, a different kind of extension can be devised. For this purpose, we first note that [ ∗ the section Φ : L → T L ⊗ End(HL) can be extended via the spacetime metric to a section

[ ∗ Φ : L → TLM ⊗ End(HL) .

Namely we set [ [ vcΦ ≡ g(v, τ)τcΦ , v ∈ TLM . Suppose that M is ﬁlled with congruence of timelike 1-dimensional submanifolds, with normalized tangent vector ﬁeld τ : M → H. Then considering the above said extension for all said submanifolds we obtain a section

Φ[ : M → T∗M ⊗ End(H) =∼ T∗M ⊗ End(TM) . M M

Consider now the new spacetime connection Γ˜ + Φ[. This has the property that the parallel transport along lines of the chosen congruence coincides with Fermi transport there; the same is not true, however, for lines which do not belong to the chosen congruence. Also, note that the transport along spacelike lines orthogonal to the lines of the congruence coincides with ordinary parallel transport.

4.6.2 Fermi transport of 2-spinors

Let UL L be the restriction of the bundle U M to the base manifold L . Introducing an appropriate Fermi transport for spinors amounts essentially to deﬁning a 28 modiﬁcation ΓF of the connection Γ on UL L, in such a way that the induced connection ¯ ΓF ⊗ ΓF on HL L coincides with ΓF . The solution to the problem of determining ΓF is not unique, as we’ll see, so one has got to describe the family of all solutions (in the next section, the results obtained here will be extended to 4-spinors). There is a natural procedure we can follow: writing down an analogous of the relation between Γ˜ and Γ(§4.1). We start from the two-spinor form of Φ[, namely [ AA˙ 4 ¯ B B˙ ∗ Φ = Φ4 BB˙dx ⊗ ζA ⊗ ζA˙⊗ z ⊗ ¯z : M → T L ⊗ End(H) ,

AA˙ λ AA˙ µ 1 λ AA˙ µ with Φ4 BB˙ = Φ4 µ τ λ t BB˙ = 2 Φ4 µ σλ σ BB˙ .

27Samuel and Nityananda [SN00] have introduced a somewhat diﬀerent transport law for polarization vectors along non-geodesic null curves. 28For simplicity, we denote the restriction of Γ by the same symbol. 44 4 TWO-SPINOR BUNDLES

By taking half the trace of Φ[ relatively to its conjugate 2-spinor indices we get the section ∗ φ : L → T L ⊗ End(UL) L

A 4 B which has the coordinate expression φ = φ4 B dx ⊗ ζA ⊗ z with A 1 AA˙ φ4 B = 2 Φ4 BA˙ . A simple calculation, using the properties of the Pauli matrices, gives then A 1 ˜ 0j A φ4 B = 2 Γ4 σj B . Note that A 4 Tr(φ) = φ4 A dx = 0 λ AA˙ (in agreement with Φ4 λ = Φ4 AA˙ = 0). Conversely, it’s not diﬃcult to show—by standard 2-spinor algebra—that AA˙ A A˙ A ¯ A˙ Φ4 BB˙ = φ4 B δ B˙ + δ B φ4 B˙ .

Next, we introduce the spinor Fermi connection on UL L ,

ΓF ≡ Γ + φ , which has the coordinate expression (Γ ) A = Γ A + φ A = (G + i Y ) δA + 1 Γ˜ AA˙ + 1 Φ AA˙ F 4 B 4 B 4 B 4 4 B 2 4 BA˙ 2 4 BA˙

A 1 AA˙ = (G4 + i Y4) δ B + 2 (ΓF)4 BA˙ .

If v : L → TL and u : L → UL are sections, then ∇ [Γ ]uA = ∇ [Γ]uA − v4 φ A uB = v4 (∂ uA − Γ A uB − 1 Γ˜ 0j σ A uB ) . v F v 4 B 4 4 B 2 4 j B ¯ Proposition 4.6 The connection ΓF ⊗ ΓF induced by ΓF on HL L coincides with the the 0 Fermi connection ΓF . Moreover, any other linear connection ΓF of UL L yielding ΓF diﬀers ∗ from ΓF by a term of the type i α ⊗ 11 with α : L → T L , namely 0 A A A ( ΓF )4 B = ( ΓF)4 B + i α4 δ B . ¯ proof: The coeﬃcients of ΓF ⊗ ΓF are (Γ ⊗ Γ¯ ) AA˙ = Γ A δA˙ + δA Γ¯ A˙ = (Γ A + φ A ) δA˙ + δA (Γ¯ A˙ + φ¯ A˙ ) = F F 4 BB˙ F4 B B˙ B F4 B˙ 4 B 4 B B˙ B 4 B˙ 4 B˙ = (Γ A δA˙ + δA Γ¯ A˙ ) + (φ A δA˙ + δA φ¯ A˙ ) = 4 B B˙ B 4 B˙ 4 B B˙ B 4 B˙ ˜ AA˙ AA˙ = Γ4 BB˙ + Φ4 BB˙ . 0 Now we observe that any other connection ΓF of UL L can be written as ΓF + Ξ , with ∗ 0 Ξ: L → T L ⊗ End(UL) . The condition that ΓF yields ΓF can be written as Γ A δA˙ + δA Γ¯ A˙ = (Γ A + Ξ A ) δA˙ + δA (Γ¯ A˙ + Ξ¯ A˙ ) , F4 B B˙ B F4 B˙ F4 B 4 B B˙ B F4 B˙ 4 B˙ A A˙ A ¯ A˙ ⇒ Ξ4 B δ B˙ + δ B Ξ4 B˙ = 0 .

A A A short discussion then shows that Ξ4 B = ξ δ B with ξ : L → i R .

Conclusion: we obtained a family of connections of the restricted bundle UL L . Each element of the family yields the standard Fermi transport, and is characterized by the arbitrary choice of an imaginary function on L . ΓF is a distinguished element of the family, so we see it as the natural generalization of Fermi transport to 2-spinors. 4.6 Fermi transport of spinors 45

4.6.3 Fermi transport of 4-spinors

The coeﬃcients of the connection induced by Γ on U F M are ∗ Γ¯ A˙ = −Γ¯ A˙ . aB˙ a B˙ ∗ The couple ( Γ, Γ)¯ then constitutes the induced (4-spinor) connection on the bundle U ⊕U F ≡ W M. Its coeﬃcients can be expressed [C00b] in the form α α 1 ˜ λµ α Γa β = i Ya δ β + 4 Γa (γλ γµ) β , α, β = 1, 2, 3, 4 .

Its restriction to WL L can then be modiﬁed in order to obtain a 4-spinor Fermi connection, that is the connection ∗ ∗ ¯ ¯ ¯∗ ( ΓF , ΓF) = ( Γ+φ , Γ−φ ) ∗ ¯ obtained from ΓF by a similar procedure. Namely, this new connection diﬀers from ( Γ, Γ) by the section ¯∗ ∗ (φ, −φ ): L → T L ⊗ End(WL) , where the transpose conjugate

¯∗ ∗ F ∗ F φ : L → T L ⊗ UL ⊗ UL ≡ T L ⊗ End(UL ) , has the coordinate expression

¯∗ A˙ 1 ¯ A˙A 1 ˜ 0j A˙ (φ )4B˙ = 2 Φ4 B˙A = 2 Γ4 σ¯j B˙ . After some calculations we also ﬁnd ¯∗ 1 1 ˜ 0j 4 (φ, −φ ) = 4 γˆ(Φ) = 4 Γ4 dx ⊗ (γ0 γj − γj γ0) , whereγ ˆ : ∧H → End W is the natural extension of the Dirac map. For simplicity, let us indicate a connection on UL L and the induced connection on WL L by the same symbol; then the induced 4-spinor Fermi connection of WL L can be written as ¯∗ 1 ΓF = Γ + (φ, −φ ) = Γ + 4 γˆ(Φ) . 0 Any other member ΓF of the family of 4-spinor Fermi connections are obtained from the above expression via the replacement φ → φ + i α ⊗ 11U , namely 0 ΓF = ΓF + i α ⊗ 11W . ± ± In §3.5 it was shown that ψ ≡ (u, χ) ∈ W is in W iﬀ hχ, u¯i ∈ R . More precisely, if that is the case then

# # √1 # 2 # # 2 γ[p ]ψ = µ ψ , p ≡ 2 u ⊗ u¯ + (¯χ ⊗ χ) , µ ≡ g˜(p , p ) = |hχ, u¯i| . Hence one proves: ± Proposition 4.7 Fermi transport preserves the subbundles W L. More precisely, a ± ± Fermi transported section ψ ≡ (u, χ): L → W is valued into W if and only if ψ(τ) ∈ Wτ for any one τ ∈ L. Furthermore, ψ projects over the Fermi constant section

# √1 # p ≡ 2 u ⊗ u¯ + (¯χ ⊗ χ) , 2 # # which is such that |hχ, u¯i| ≡ g˜(p , p ) = constant. The above result will be used in the deﬁnition of free-particle states for electrons and positrons (§??). 46 4 TWO-SPINOR BUNDLES

4.6.4 Group considerations A detailed study of the relations between 2-spinor groups, 4-spinor groups and the Lorentz group was exposed in [C07]. In this section I’ll just recall a few results which are relevant in the present discussion. Consider the group Sl(U) ≡ {K ∈ Aut(U) : det K = 1} , which preserves the two-spinor structure. Its relations with the special orthochronous Lorentz group and the orthochronous Spin group are described by the commutative diagram

- ↑ - ¯ Sl(U) Lor+(H) K K ⊗ K 3 > : ? ? ↑ −1 Spin (W ) K, (K¯ F)

One has the isomorphic Lie algebras LLor ≡ LLor(H), LSpin ≡ LSpin(W ) and

LSl ≡ LSl(U) =∼ {φ ∈ End(U) : Tr φ = 0} .

Furthermore LLor(H) is isomorphic, as a vector space, to ∧2H. Thus one has the diagram of isomorphisms

LSl(U) - LLor(H) φ - Φ[ 6 6 6 6 :

? ? ? ? LSpin(W ) - ∧2H (φ, −φ¯∗)Φ - where the relations among the above objects are as follows. a)Φ[ ∈ LLor(H) ⊂ End(H) = H ⊗ H∗ is obtained from Φ ∈ ∧2H ⊂ H ⊗ H through the isomorphism g[ : H → H∗ determined by the Lorentz metric. b) φ ∈ LSl(U) ⊂ U ⊗ U F is one-half the trace of Φ[ relatively to the conjugate factors. [ ¯ Conversely, Φ = φ ⊗ 11U + 11U ⊗ φ . Hence Tr φ = 0 .

1 ¯∗ F c) 4 γˆ(Φ) = (φ, −φ ) ∈ End U ⊕ End U ⊂ End W , whereγ ˆ : ∧H → End W is the natural extension of the Dirac map to the exterior algebra of H. Furthermore, it should be observed that the biggest group which preserves the two-spinor structure is not Sl(U) but rather the ‘complexiﬁed’ group

c Sl (U) ≡ {K ∈ Aut(U): | det K| = 1} = U(1) × Sl(U) /Z2 , which leaves any symplectic form of U invariant up to a phase factor. Its Lie algebra is

c ∼ LSl (U) = {A ∈ End(U): < Tr A = 0} = i R ⊕ LSl(U) .

Accordingly, any θ ∈ LSlc(U) can be uniquely decomposed as 4.7 Flat spacetime 47

1 1 d) θ = 2 (Tr θ) 11+ θ − 2 (Tr θ) 11 ≡ i α 11+ φ , α ∈ R , φ ∈ LSl(U) , with 11 ≡ 11U , and one has θ ⊗ 11+¯ 11 ⊗ θ¯ = φ ⊗ 11+¯ 11 ⊗ φ¯ .

In other words, θ ∈ LSlc(U) determines an element Φ[ ∈ LLor(H) via its traceless part φ . In the bundle setting of §?? the above spaces and groups become vector bundles and group bundles over M. Consider sections

φ : M → T∗M ⊗ LSl(U) , M

∗ c θ ≡ i α ⊗ 11U + φ : M → T M ⊗ LSl (U) , M

Φ[ : M → T∗M ⊗ LLor(H) , M

1 ¯F ∗ 4 γˆ(Φ) = (φ, −φ ): M → T M ⊗ LSpin(W ) , M fulfilling the same mutual relations a, b, c and d as the previously considered algebraic objects with the same names (α : M → T∗M is now a real 1-form). Such Lie-algebra-bundle valued 1-forms can be seen as differences between linear connections preserving the respective vector bundle structures (while the curvature tensors are 2-forms valued in the same Lie-algebra-bundles). More precisely, it’s not difficult to prove: Proposition 4.8 Let Γ and Γ0 be 2-spinor connections, and Γ˜, Γ˜0 the respectively induced connections of H → M. Then

θ ≡ i α ⊗ 11+ φ ≡ Γ−Γ0 : M → T∗M ⊗ LSlc(U) M and Φ[ ≡ Γ˜−Γ˜0 : M → T∗M ⊗ LLor(H) M fulﬁl the above relations a, b, c, d. In particular, Φ[ only depends on the traceless part φ of θ .

4.7 Flat spacetime In agreement with the 2-spinor setting, and in view of electrodynamics, one may describe ﬂat spacetime as a triple (M, S, Θ); here, M is a 4-dimensional real aﬃne space, with ‘derived’ ∗ vector space DM ; S is a complex 2-dimensional vector space; Θ ∈ L ⊗ H ⊗ D M is a constant non-degenerate tetrad. This setting determines the isomorphism ∼ ∼ TM = M × DM = M × (L ⊗ H) .

A time+space splitting (t, x): H → T × X determines a splitting (denoted with the same symbols) (t, x):DM → (L ⊗ T ) × (L ⊗ X) . Here, T is a Euclidean 1-dimensional (future-)oriented vector space and can thus be identiﬁed with R; X is a Euclidean 3-dimensional oriented vector space. If a Pauli frame (τλ) ⊂ H is adapted to this splitting, then one has the corresponding −1 scaled frame (Θλ) ⊂ L ⊗ DM and the ‘scaled’ coordinates λ i 4 (x ) ≡ (t, x ): M → L ⊗ R . 48 5 OPTICAL GEOMETRY IN CURVED SPACETIME

Note that the associated polar coordinates are scaled in part, as

2 (t, r, θ, φ):DM \{0} → (R ⊗ L) × L × R .

By ﬁxing a reference point o, one also obtains ‘scaled’ coordinates

λ i 4 (xo ) ≡ (to, xo): M → L ⊗ R and the like. Similarly, one has the dual splitting

∗ −1 ∗ −1 ∗ (τ, y):D M → (L ⊗ T ) × (L ⊗ X ) .

The associated coordinates on D∗M have inverse scaling, with respect to the analogous coordinates on DM. λ If needed for clarity, induced (scaled) coordinates on M will be denoted as xo and the like.

5 Optical geometry in curved spacetime

The term ‘optical geometry’ was introduced [Nu96, Tr97] for certain generalizations of the geometric structure determined by a conguence of null lines. I shall only consider the case of physical spacetime with fixed gravitational background, that is fixed soldering form Θ ; this ∼ −1 determines an identification TM = L ⊗ H . Basically I’ll expose an unscaled approach (then of course one can attach unit spaces at will to the considered bundles).

5.1 Real optical algebra Consider ﬁrst an algebraic setting, based on the assignment of a 2-dimensional vector space S and on the spaces associated with it (§1.3, 1.4). Let L ⊂ N ⊂ H be a 1-dimensional isotropic subspace of H , and

L⊥ ≡ {y ∈ H : g(y, k) = 0 , k ∈ L} ⊂ H its orthogonal space in H . Then

dim L⊥ = 3 , L ⊂ L⊥ .

Now set B ≡ L⊥/L , so that dim B = 2 and one has the exact sequence

0 → L → L⊥ → B → 0 .

Remark. The assignment of L is equivalent to that of an element [k] ∈ C ≡ PK , the celestial sphere in H (§1.6). Actually k ∈ L means [k] ≡ L . 5.1 Real optical algebra 49

Because L is isotropic, the above sequence does not split by virtue of the Lorentz metric. There is no distinguished complementary subspace to L in L⊥ ; an element in B is an equivalence class in L⊥ , two elements being equivalent if they diﬀer by an element in L , and there is no distinguished representative. The most natural way to ﬁx a splitting is through + ∼ ⊥ ⊥ the choice of an ‘observer’ τ ∈ K1 ; then one writes B = L ∩ hτi , namely b ∈ B can be 29 ⊥ characterized by the unique bτ ∈ L such that g(τ, bτ ) = 0 . There exists a Pauli basis (τλ) ⊂ H such that τ0 + τ3 ∈ L ; moreover, given an observer τ one can always choose this basis in such a way that τ0 ≡ τ , so that the only freedom left is an Euclidean rotation in the plane generated by τ1 and τ2 . Thus (τ1 , τ2) can be seen as a basis of B , and as such will be denoted by the same symbols if no confusion arises. Next, observe that the Lorentz metric g of H (I’ll drop the tilde for the rest of this section) determines a (negative) Euclidean metric gB on B by the rule

gB ([v], [w]) ≡ g(v, w) , which actually does not depend on the chosen representatives of the equivalence classes [v], [w] ∈ B . Namely

(gB )ij = gij , i, j = 1, 2 , so that (τ1 , τ2) is an orthonormal basis of B .

Up to sign, gB determines a volume form ηB on B . Note that the positive Euclidean metric −gB yields the same volume form, and that choosing between gB and −gB does not give rise to a distinguished orientation. However, for any chosen orientation the two cases 1 2 determine opposite Hodge isomorphisms ∗B : B → B , since by taking ηB = t ∧ t one has ∗ τ = ±g#(τ |η ) = ±g#(t2) = ∓τ , ∗ τ = ±g#(τ |η ) = ±g#(−t1) = ±τ . B 1 B 1 B B 2 B 2 B 2 B B 1 In order to choose an orientation on B , note that for k ∈ L and b ∈ B the tensor k ∧ b ∈ ∧2H is well deﬁned. Then one ﬁnds Proposition 5.1 There is a unique choice of an orientation in B such that

−∗(k ∧ b) = k ∧ (∗Bb) . proof: It can be seen by a coordinate calculation in a Pauli basis as above. Set k = τ0+τ3 (the component k0 = k3 can be taken to have any non-vanishing value). Since η = t0 ∧ t1 ∧ t2 ∧ t3 one has

# 0 1 2 3 # 0 1 2 3 ∗(k ∧ τ1) = g (τ0 + τ3) ∧ τ1 |t ∧ t ∧ t ∧ t = g (τ0 ∧ τ1 + τ3 ∧ τ1)|t ∧ t ∧ t ∧ t =

# 2 3 0 2 = g t ∧ t + t ∧ t = τ2 ∧ τ3 − τ0 ∧ τ2 = (τ0 + τ3) ∧ (−τ2) =

= k ∧ (−τ2) ,

# 0 1 2 3 # 0 1 2 3 ∗(k ∧ τ2) = g (τ0 + τ3) ∧ τ2 |t ∧ t ∧ t ∧ t = g (τ0 ∧ τ2 + τ3 ∧ τ2)|t ∧ t ∧ t ∧ t =

# 1 3 0 1 = g −t ∧ t − t ∧ t = −τ1 ∧ τ3 + τ0 ∧ τ1 = (τ0 + τ3) ∧ τ1 =

= k ∧ τ1 .

Comparing these with the above written expression for ∗B one gets the stated result. 29In the case of the electromagnetic vector (§??), this is essentially the so-called radiation gauge. 50 5 OPTICAL GEOMETRY IN CURVED SPACETIME

Henceforth, the Hodge isomorphism ∗B will be ﬁxed by requiring −∗(k ∧ b) = k ∧ (∗Bb), 1 2 2 1 namely as determined by the positive Euclidean metric −gB = t ⊗ t + t ⊗ t and by the 1 2 choice of the volume form ηB = t ∧ t , so that ∗Bτ1 = τ2 and ∗Bτ2 = −τ1 . Note that 2 (∗B) = 11B , so that B turns out to be a 1-dimensional complex vector space. If one identiﬁes

τ1 with the real unit and τ2 with the imaginary unit, then ∗B can be identiﬁed with the multiplication by i ≡ τ2 . Passing now to consider dual spaces, one has the exact sequence

0 → B∗ → L⊥∗ → L∗ → 0 , which also splits when an observer is given. In that case, taking an adapted basis as above one has bases (t1, t2) of B∗ ,(t1, t2, t0 +t3) of L⊥∗ and (t0 +t3) of L∗ . Furthermore, the metric g yields the vector spaces

[ [ [ ∗ ∗ L ≡ g (L) = {c k , c ∈ C, k ∈ L} ⊂ N ⊂ H ,

L⊥[ ≡ g[(L⊥) = {y[ ∈ H∗ : hy[, ki = 0, k ∈ L} ⊂ H∗ ,

(L[)⊥ ≡ {y[ ∈ H∗ : g#(y[, k[) = 0, k ∈ L} = {y[ ∈ H∗ : hy[, ki = 0, k ∈ L} = = (L⊥)[ ,

[ 0 3 [ 1 2 0 3 ⊥[ and one has the bases g (τ0 + τ3) = (t − t ) of L and (t , t , t − t ) of L . One notes that L[ is distinct from L∗ (actually if k ∈ L then hk[, ki = 0), and L⊥[ is distinct from L⊥∗ . However: Proposition 5.2 There is a natural isomorphism

B∗ =∼ L[⊥/L[ given by h[θ], [y]i ≡ hθ, yi , θ ∈ L[⊥, y ∈ L⊥ . The Hodge isomorphism of ∧H∗ is determined by the contravariant metric g# and the contravariant volume form

# # −1 η = g (η) = −η = −τ0 ∧ τ1 ∧ τ2 ∧ τ3 = τ1 ∧ τ2 ∧ τ3 ∧ τ0 (or, equivalently, by ∗t ≡ t#|η), while one has

(−g )#(η ) = g#(η ) = η−1 = τ ∧ τ . B B B B B 1 2 Thus one ﬁnds Proposition 5.3 [ [ ∗ −∗(k ∧ β) = k ∧ (∗Bβ) , β ∈ B . proof: It follows from the analogous contravariant case and the properties of the Hodge isomorphism; in order to check it by direct calculations one sets k[ = t0 − t3 , whence

[ 1 # 0 3 1 # 0 1 3 1 ∗(k ∧ t ) = g (t − t ) ∧ t |τ1 ∧ τ2 ∧ τ3 ∧ τ0 = g (t ∧ t − t ∧ t )|τ1 ∧ τ2 ∧ τ3 ∧ τ0 =

# 2 3 0 2 0 3 2 = g −τ2 ∧ τ3 + τ0 ∧ τ2 = −t ∧ t − t ∧ t = (t − t ) ∧ (−t ) =

= k[ ∧ (−t2) , 5.2 Complexiﬁed optical algebra 51

2 # 0 3 2 # 0 2 3 2 ∗(k ∧ t ) = g (t − t ) ∧ t |τ1 ∧ τ2 ∧ τ3 ∧ τ0 = g (t ∧ t − t ∧ t )|τ1 ∧ τ2 ∧ τ3 ∧ τ0 =

# 1 3 0 1 0 3 1 = g τ1 ∧ τ3 − τ0 ∧ τ1 = t ∧ t + t ∧ t = (t − t ) ∧ t =

= k[ ∧ t1 .

Moreover from η# = τ ∧ τ one has B 1 2

∗ t1 = −g[ (t1|η#) = −g[ (τ ) = t2 , ∗ t2 = −g[ (t2|η#) = −g[ (−τ ) = −t1 . B B B B 2 B B B B 1

5.2 Complexiﬁed optical algebra

Consider the complexiﬁed vector space BC ≡ C ⊗ B . One has natural identiﬁcations ⊗ L⊥ B =∼ C , C C ⊗ L BF =∼ ⊗ B∗ . C C

The metric gB and the Hodge map ∗B can be naturally extended to operations on BC . Then one has the decomposition of BC into the direct sum of the two eigenspaces of i ∗B , namely B = B+ ⊕ B− . C C C An element of B+ (resp. B−) can be written as C C β = b ± i∗b , b ∈ B .

Since30 b · ∗b = 0 and b2 = (∗b)2 one sees that

β2 = (b ± i∗b)2 = 0 , (b + i∗b) · (b − i∗b) = 2b2 .

Conversely, the vanishing square property characterizes those elements of BC which are either in B+ or in B− . C C Similarly one has

BF = (BF)+ ⊕ (BF)− =∼ (B+)F ⊕ (B−)F . C C C C C

+ Now for any vector space V (real or complex) denote by V /R the manifold constituted by all real half-lines in V starting from 0. Let

F ≡ B∗/ + , F ≡ BF/ + , F ± ≡ (BF)±/ + . R C C R C C R Through g one can identify the above manifolds with spheres. Namely, if b ∈ B∗, then one B √ can identify [b] ∈ F with b/ −b2, which belongs to the unit sphere in B∗ , and [b ± i∗b] ∈ F ± √ C 2 with (b ± i∗b)/ −b ≡ [b] ± i∗[b]. The Hodge isomorphism ∗B passes to the quotient, so that

F ± ≡ {ϕ ∈ F : i∗ϕ = ±ϕ} . C C 30When no confusion arises, I will use the shorthands v · w ≡ g(v, w) and v2 ≡ g(v, v). Furthermore I’ll write simply ∗ for ∗B . 52 5 OPTICAL GEOMETRY IN CURVED SPACETIME

The 2-spinor formulation of the above complex geometry is particularly expressive. First, remember that an element in U ⊗ U = C ⊗ H is isotropic iﬀ it is decomposable. Then k ∈ L can be written as k = κ ⊗ κ¯ , κ ∈ U , where κ is characterized by k up to a phase factor. Equivalently, [ [ [ k = κ ⊗ κ¯ , κ = κ ≡ ε (κ) ∈ U F . Proposition 5.4

B+ = {[ ⊗ u¯]: ⊗ ¯ ∈ L , u ∈ U } , C κ κ κ B− = {[u ⊗ ¯]: ⊗ ¯ ∈ L , u ∈ U } . C κ κ κ

Similarly,

(BF)+ = {[κ ⊗ λ¯]: κ ⊗ κ¯ ∈ L[ , λ ∈ U F } , C

(BF)− = {[λ ⊗ κ¯]: κ ⊗ κ¯ ∈ L[ , λ ∈ U F } . C

⊥ proof: An element in BC is an equivalence class of isotropic elements in C ⊗ L , two elements being equivalent if their diﬀerence is proportional to k ∈ L . An isotropic element in C ⊗ H = U ⊗ U can be written in 2-spinor form as u ⊗ v¯ , and the condition that it be in C ⊗ L⊥ , that is orthogonal to k = κ ⊗ κ¯ ∈ L , yields

0 = ε(u, κ)ε ¯(¯v, κ¯) ⇒ either u = κ or v = κ . The quickest way to see how the two cases above describe B± is by an adapted Pauli basis as C introduced in §5.1 (clearly it can also be seen as a basis of the complexiﬁed space). One has

i∗(τ1 + i τ2) = i τ2 + τ1 , i∗(τ1 − i τ2) = i τ2 − τ1 , hence √1 (τ +i τ ) = ζ ⊗ ζ¯ ∈ B+ when identiﬁed with an element in B via the observer τ . 2 1 2 1 2 C C 0 √1 ¯ Now from k ∝ 2 (τ0 + τ3) = ζ1 ⊗ ζ1 one has κ ≡ ζ1 , so that the general expression [κ ⊗ u¯] for elements in B+ follows. Note that changing u by a term proportional to changes ⊗ u¯ C κ κ by a term proportional to k , namely does not change [κ ⊗ u¯] . The other cases are similarly checked. A summary of coordinate expressions related to the above proposition, in the adapted Pauli basis introduced in §5.1, will be useful. One writes

√1 ¯ k = 2 (τ0 + τ3) = ζ1 ⊗ ζ1 ⇒ k = κ ⊗ κ¯ with κ = ζ1 ,

[ √1 0 3 2 2 [ 2 [ k = 2 (t − t ) = z ⊗ ¯z ⇒ k = κ ⊗ κ¯ with κ = z = −κ , hence one ﬁnds31

√1 √i √1 i∗ 2 (τ1 + i τ2) = 2 (τ2 − i τ1) = 2 (τ1 + i τ2)

⇒ B+ 3 √1 (τ + i τ ) = ζ ⊗ ζ¯ ≡ ⊗ u¯ , C 2 1 2 1 2 κ

√ 0 −i √ 0 i 31Remember (§A.4) that 2 (τ ) = and 2 (t2) = . 2 i 0 −i 0 5.3 Optical spin 53

√1 √i √1 i∗ 2 (τ1 − i τ2) = 2 (τ2 + i τ1) = − 2 (τ1 − i τ2)

⇒ B− 3 √1 (τ − i τ ) = ζ ⊗ ζ¯ ≡ u ⊗ ¯ , C 2 1 2 2 1 κ

√1 1 2 √i 2 1 √1 1 2 i∗ 2 (t + i t ) = 2 (t − i t ) = 2 (t + i t )

⇒ (BF)+ 3 √1 (t + i t2) = z2 ⊗ ¯z1 ≡ κ ⊗ λ¯ , C 2 1

√1 1 2 √i 2 1 √1 1 2 i∗ 2 (t − i t ) = 2 (t + i t ) = − 2 (t − i t )

⇒ (BF)− 3 √1 (t1 − i t2) = z1 ⊗ ¯z2 ≡ λ ⊗ κ¯ . C 2

Also note that

2 1 [ ¯ √1 [ √1 1 2 − z ⊗ ¯z = g (ζ1 ⊗ ζ2) = 2 g (τ1 + i τ2) = − 2 (t + i t ) ,

1 2 [ ¯ √1 [ √1 1 2 − z ⊗ ¯z = g (ζ2 ⊗ ζ1) = 2 g (τ1 − i τ2) = − 2 (t − i t ) , so that ( B+ → (BF)+ , g[ : C C B− → (BF)− , C C and the like for the inverse isomorphism g# . Remark. The dual basis of √1 √1 ¯ ¯ 2 (τ1 + i τ2) , 2 (τ1 − i τ2) = ζ1 ⊗ ζ2 , ζ2 ⊗ ζ1 is √1 1 2 √1 1 2 1 2 2 1 2 (t − i t ) , 2 (t + i t ) = z ⊗ ¯z , z ⊗ ¯z .

Thus one sees that

g[B+ = (BF)+ = (B−)F , C C C

g[B− = (BF)− = (B+)F . C C C

This oddity is related to the fact that the subspaces B± are g-isotropic. C

5.3 Optical spin

∗ The angular momentum maps J : H⊥ → End(W ) and J ≡ −J ∗ : H⊥ → End(W F) studied in §3.4 determine angular momentum maps

∗ J ⊗ 11+ 11 ⊗ J : H⊥ → End(W ⊗ W F) ,

∗ J ⊗ 11+ 11 ⊗ J : H⊥ → End(W F ⊗ W ) . 54 5 OPTICAL GEOMETRY IN CURVED SPACETIME

When no confusion arises, these will be denoted again by the symbol J . In particular, one is interested to their restrictions to the subspaces U ⊗ U ⊂ W ⊗ W F and U F ⊗ U F ⊂ W F ⊗ W F . Recalling the formulas listed at the end of §3.4 one immediately ﬁnds

J3(ζ1 ⊗ ζ¯1) = J3(ζ2 ⊗ ζ¯2) = 0 ,

1 1 2 2 J3(z ⊗ ¯z ) = J3(z ⊗ ¯z ) = 0 ,

J3(ζ1 ⊗ ζ¯2) = ζ1 ⊗ ζ¯2 ,J3(ζ2 ⊗ ζ¯1) = −ζ2 ⊗ ζ¯1 ,

2 1 2 1 1 2 1 2 J3(z ⊗ ¯z ) = z ⊗ ¯z ,J3(z ⊗ ¯z ) = −z ⊗ ¯z .

The above formulas imply that J3 , the spin map in the direction of the spatial direction corresponding to k ∈ L relatively to the observer τ0 , naturally give rise to spin maps JB3 ∗ on B and J on BF . Moreover the fact that J (k) = J (ζ ⊗ ζ¯ ) = 0 means that the C B3 C 3 3 1 1 construction is actually independent of the observer, namely the definition of JB3 is intrinsic. On the contrary, J and J can be defined on B and BF only as restrictions to subspaces 1 2 C C orthogonal to K and to the chosen observer. Thus, one finds that B+ and (BF)+ are the eigenspaces of J corresponding to the C C B3 eigenvalue (helicity) +1 , while B− and (BF)− are the eigenspaces corresponding to the C C eigenvalue (helicity) −1 . Observe that there is no subspace corresponding to helicity 0 . Recalling §?? one sees that JB3 , though arising from an angular momentum map, is not a component of an angular momentum map JB .

5.4 Optical bundles Let Z ⊂ M be a submanifold of dimension ≥ 1. An optical bundle is deﬁned to be a vector −1 subbundle L ⊂ L ⊗ TZ over Z whose ﬁbres are 1-dimensional null (i.e. isotropic) spaces. Then Z turns out to be a disjoint union of null lines, G Z = Zα . α∈A

If dim Z = 1 then Z is either a null line, or a discrete collection of isotropic lines. If dim Z = n > 1 then Z is a congruence of null lines, locally described by n−1 real parameters (namely, the ‘index set’ A is an n−1-dimensional manifold). Note that, since the soldering form Θ is ﬁxed, giving an optical bundle L M is equivalent to giving a section of the bundle R ≡ PS M of Riemann spheres. Hence, at any point x ∈ M one has the subspace Lx ⊂ Nx ⊂ Hx and all the constructions of §5.1 and §5.2. Henceforth in this section I’ll consider an optical bundle L → Z such that the null curves 2 generating Z are geodesics, and denote by k : Z → L ⊗ L a section such that ∇kk = 0 . 0 ⊥ 0 −2 0 If v, v : Z → L are such that v = v + σk, σ : Z → R ⊗ L , then k · ∇kv = k · ∇kv = 0 0 and ∇kv = ∇kv + (k.σ)k. Namely the operator ∇k passes to the quotient: if b : Z → B is a section, then ∇ b : Z → B. The same holds for sections from M to B∗, B , B±. In other k C C terms: if Zα is any one of the collection of null geodesics, then ∇k determines connections of these bundles over Zα . One has

∇kgB = 0 . 55

2 [ [ 1 [ Using k = 0 and dk = A∇k + 2 T ck , where T is the torsion (§??), one ﬁnds

∇k[ck = 0 , [ [ [ c b k | dk + T ckck = 0 , i.e. (k | dk )a + T ab k kc = 0 .

[ [ [ Since ∇kk = 0 one has ∇k (rk + v, sk + w) = ∇k (v, w), r, s ∈ R. Then the tensor ﬁeld ∇k[ : Z → T∗M ⊗ T∗M passes to the quotient, namely it gives rise to a bilinear form

Dk[ : Z → B∗ ⊗ B∗ .

Proposition 5.5 One has [ [ gB · Dk = div k . proof: Consider an orthonormal frame (eλ). One has

[ [ [ [ [ [ div k ≡ g · ∇k = ∇k (e0 , e0) − ∇k (e1 , e1) − ∇k (e2 , e2) − ∇k (e3 , e3) .

0 [ [ Replacing k = k (e0 + e1) in the equalities h∇kk , e0i = 0 and h∇e1 k , ki = 0, and subtracting [ [ them, one ﬁnds ∇k (e0 , e0) − ∇k (e1 , e1) = 0, which yields the stated result.

The geodesic equation ∇kk = 0 can be rewritten ∇kκ ⊗ κ¯ + κ ⊗ ∇kκ¯ = 0, which means

∇kκ = i r κ , r : M → R .

One can rescale κ so that ∇kκ = 0 . (1) 0 R In fact, let s : M → R and κ ≡ i sκ; if k.s + r = 0, namely s(t) = r(t) dt along each 0 0 2 0 geodesic, then ∇kκ = 0 ⇒ ∇k0 κ = s ∇kκ = 0 .

6 2-spinor groups and 4-spinor groups

6.1 2-spinor groups 6.1.1

As a set, the group Aut(S) is the subset of End(S) ≡ S ⊗ SF constituted by all invertible endomorphisms: Aut(S) ≡ {K ∈ End(S) : det K 6= 0} .

Thus Aut(S) is a complex open submanifold of End(S) , dimC Aut(S) = 4 . When an arbitrary ∼ basis (ξA) is fixed, then one gets an isomorphism Aut(S) = Gl(2, C) , the latter being the group of all complex 2×2 matrices with non-vanishing determinant. The Lie algebra of Aut(S) can be identified with T11Aut(S) , the tangent space at the identity, and as a vector space coincides with the whole vector space End(S) ; when a basis 2 2 is fixed, this Lie algebra can be identified with the Lie algebra C ⊗ C of all compex 2×2 matrices. The set B(S) of all bases of S is a group-affine space with derived group Gl(2, C) : if 0 0 (ξA), (ξA) ∈ B(S) are any two bases then there is a unique K ∈ Aut(S) such that KξA = ξA , A A = 1, 2 ; on turn, this determines the matrix (K) ≡ (K B ) , that is the matrix of K in the 0 A basis (ξA) , the first of the couple. One has ξB = K B ξA , namely the components of the 0 elements of the “new” basis (ξA) in the “old” basis (ξA) are written in the columns of (K). 56 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

Similarly, the set BF(S) of all bases of SF is a group-aﬃne space with derived group Gl(2, ) : if (xA), (x0A) ∈ BF(S) then there is a unique K0 ∈ Aut(S) such that xAK0 = x0A , C ← ← 0 0 0A A B A = 1, 2 ; if these are the dual bases of (ξA) and (ξA) above then K = K and x = K B x , namely the components of of the elements of the “new” dual basis (x0A) in the “old” basis ← (xA) are written in the rows of (K).

6.1.2

• The multiplicative group C ≡ C \{0} can be identiﬁed with the subgroup of Aut(S) constituted by all automorphism of the type c 11S , c ∈ C \{0} ; its Lie algebra is just C (that is, more precisely, C11S ). • + Then one has the subgroups R ≡ R \{0} and R , with Lie algebra R , and the subgroup U(1) ≡ {exp(i φ) , φ ∈ R} with Lie algebra i R . • C ≡ C \{0} and its subgroups are normal subgroups (§??) of Aut(S). Moreover, note that these subgroups are naturally “numerical”, without any need to ﬁx a basis.

6.1.3 Next, consider the two following subgroups of Aut(S):

c c Sl (S) ≡ {K ∈ Aut(S): | det K| = 1} , dimR Sl (S) = 7 , c Sl(S) ≡ {K ∈ Aut(S) : det K = 1} ⊂ Sl (S) , dimC Sl(S) = 3 .

Note that, as well as C \{0} an its subgroups, these subgroups are intrinsically deﬁned independently of any further structure possibly given on S ; actually, the groups Slc(V ) and Sl(V ) are intrinsically deﬁned on any complex vector space V . Slc(S) can be seen as the group of all automorphisms of S which preserves any complex volume form ω up to a phase factor , namely

Slc(S) = {K ∈ Aut(S): K Fω = ei φω , ∀ ω ∈ ∧2SF} ; note that the phase factor exp(i φ) ∈ U(1) is independent of ω (it only depends only on K). Sl(S) can be seen as the group of all automorphisms of S which preserve any complex volume form ω , namely Sl(S) = {K ∈ Aut(S): K Fω = ω , ∀ ω ∈ ∧2SF} . Furthermore, one has32

c ∼ Sl (S) = U(1)×˜ Sl(S) ≡ U(1) × Sl(S) /Z2 .

c ∼ c ∼ For each fixed basis of S one gets isomorphisms Sl (S) = Sl (2, C) and Sl(S) = Sl(2, C), in which every K ∈ Slc(S) is identified with a matrix (K) such that | det(K)| = 1 , and every K ∈ Sl(S) is identified with a matrix (K) such that det(K) = 1 . 2 Let w ∈ ∧ S, and Be(S, w) the set of all bases (ξA) of S such that ξ1 ∧ ξ2 = c w , c ∈ U(1) . c Then Be(S, w) is a group-affine space with derived group Sl (2, C) . Similarly, let B(S, w) be

32The map U(1)×Sl(S) → Slc(S):(c, K) 7→ c K is a two-to-one epimorphism, so that Slc(S) is the quotient 0 0 0 0 of U(1) × Sl(S) by the equivalence relation (c, K) ∼ (c ,K ) ⇐⇒ c K = c K . In other terms, here Z2 is identiﬁed with the normal subgroup of U(1) × Sl(S) generated by (−1, −11). 6.1 2-spinor groups 57

the set of all bases (ξA) of S such that ξ1 ∧ ξ2 = w . Then B(S, w) is a group-aﬃne space with derived group Sl(2, C). One has the Lie algebras (§??, proposition ?? on page ??) LSlc(S) =∼ {A ∈ End(S): < Tr A = 0} ,

LSl(S) =∼ {A ∈ End(S) : Tr A = 0} . Hence c LSl (S) = i R ⊕ LSl(S) .

6.1.4

Let now k ∈ SF ∨¯ SF be a positive Hermitian metric, and denote by33 ← U(S, k) ≡ {K ∈ Aut(S): K† = K} the group of all endomorphisms which preserve k . These fulﬁl | det K| = 1 , thus one has U(S, k) ⊂ Slc(S) . Moreover set ← SU(S, k) ≡ U(S, k) ∩ Sl(S) = {K ∈ Aut(S): K† = K, det K = 1} . The one gets ∼ U(S, k) = U(1)×˜ SU(S, k) ≡ U(1) × SU(S, k) /Z2 . One gets the Lie algebras LU(S, k) = {A ∈ End(S): A + A† = 0} ,

LSU(S, k) = {A ∈ End(S): A + A† = 0 , Tr A = 0} . Thus one has

dimR LU(S, k) = dimR U(S, k) = 4 ,

dimR LSU(S, k) = dimR SU(S, k) = 3 ,

LU(S, k) = i R ⊕ LSU(S, k) . The set O(S, k) of all k-orthonormal bases of S is a group-aﬃne space with derived group † −1 U(2) ≡ {M ∈ Gl(2, C): M = M } ; 0 if (ξA), (ξA) ∈ O(S) are any two k-orthonormal bases then there is a unique K ∈ U(S, k) such 0 A that KξA = ξA , A = 1, 2 ; on turn, this determines the matrix (K) ≡ (K B ) ∈ U(2) . Let moreover O+(S, k) ⊂ O(S, k) be the subset constituted of all k-orthonormal bases of S which are positively oriented according to any given orientation:34 this is a group-aﬃne space with derived group † −1 SU(2) ≡ {M ∈ Gl(2, C): M = M , det(M) = 1} ; 0 + if (ξA), (ξA) ∈ O (S, k) are any two k-orthonormal bases with the same orientation then there 0 is a unique K ∈ SU(S, k) such that KξA = ξA , A = 1, 2 ; on turn, this determines the matrix A (K) ≡ (K B ) ∈ SU(2) .

33Remember (§1.2) that, for any K ∈ Aut(S), K† ≡ k# ◦ K¯ F ◦ k[ ≡ k# ◦ KF ≡ k[ ∈ Aut(S). 34 0 + 0 0 + In a complex space this condition means (ξA), (ξA) ∈ O (S) iﬀ ξ1 ∧ ξ2 = r ξ1 ∧ ξ2 with r ∈ R . 58 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

6.1.5 The group SU(2) is constituted by all complex 2 × 2 matrices of the form

ei α cos γ ei β sin γ ! a b ! 2 2 = , |a| + |b| = 1 , α, β, γ ∈ R . −e−i β sin γ e−i α cos γ −¯b a¯

From the second expression one sees that SU(2) is can be identiﬁed, as a real manifold, with 4 the unit sphere in R . Similarly, the group U(2) is constituted by all complex 2 × 2 matrices of the form

ei α cos γ ei β sin γ ! a b ! ei φ = c , |a|2 + |b|2 = |c|2 = 1 . −e−i β sin γ e−i α cos γ −¯b a¯

i 35 The Lie algebra LSU(2) has the basis (− 2 σi), i = 1, 2, 3 . The assignment of a k- orthonormal basis of S determines the basis of LSU(S, k) constituted by the endomorphisms i whose matrices in the chosen basis are exactly (− 2 σi) . Every element in SU(2) can be expressed as the exponentiation of some element in LSU(2) as

 |θ| θ3 |θ| −i θ1−θ2 |θ|  cos 2 − i |θ| sin 2 |θ| sin 2 i j   i j 1/2 exp(− 2 θ σj) = , |θ| ≡ (δij θ θ ) .  −i θ1+θ2 |θ| |θ| θ3 |θ|  |θ| sin 2 cos 2 + i |θ| sin 2

i The Lie algebra LU(2) has the basis (− 2 σλ), λ = 0, 1, 2, 3 . Every element in U(2) can be expressed as the exponentiation of some element in LU(2) as one ﬁnds simply

i 0 i λ − 2 θ i j exp(− 2 θ σλ) = e exp(− 2 θ σj) .

6.1.6 When a (possibly scaled) positive Hermian metric k on S is given then the space of all endomorphisms of S splits into the direct sum

End(S) = H(S, k) ⊕ i H(S, k) of the subspaces of all k-Hermitian and anti-Hermitian endomorphisms. Since LSl(S) ∩ H(S, k) = LSU(S, k) , one also get the splittings

LSl(S) = LSU(S, k) ⊕ i LSU(S, k) ,

c LSl (S) = LSU(S, k) ⊕ i LSU(S, k) ⊕ i R .

The analogous splittings

LSl(2, C) = LSU(2) ⊕ i LSU(2) , c LSl (2, C) = LSU(2) ⊕ i LSU(2) ⊕ i R

35 Here the σi’s are the Pauli matrices. This basis is orthogonal but not orthonormal with respect to the i i k i Killing metric, and fulﬁlls the relation [(− 2 σi), (− 2 σj )] = εij (− 2 σk). 6.1 2-spinor groups 59

c of the corresponding matrix algebras always determine splittings of LSF(S) and LSl (S) when a basis of S is chosen; if the basis is orthonormal relatively to a Hermitian metric k , then the two splitting procedures give the same result. A basis of LSl(2, C) which is adapted to the i above said splitting is constituted by the three matrices (− 2 σi) and by the three matrices 1 i c ( 2 σi), i = 1, 2, 3 ; add − 2 σ0 for a basis of LSl (2, C). So LSl(2, C) is the complexiﬁed space of LSU(2) . Accordingly, any element of Sl(2, C) can be written as ei α cos γ ei β sin γ ! , α, β, γ ∈ C . −e−i β sin γ e−i α cos γ

The subspace i LSU(2) generates, via exponentiation, those elements of Sl(2, C) which are related to k-boosts, namely

 3 |θ| 1 2 |θ|  |θ| θ sinh 2 (θ −i θ ) sinh 2 cosh 2 + |θ| |θ| 1 j   exp( 2 θ σj) =   =  1 2 |θ| 3 |θ|  (θ +i θ ) sinh 2 |θ| θ sinh 2 |θ| cosh 2 − |θ|

 θ3 sinh |θ| (θ1−i θ2) sinh |θ|  q 1+ |θ|(1+cosh |θ|) |θ|(1+cosh |θ|) = cosh |θ|+1   . 2  (θ1+i θ2) sinh |θ| θ3 sinh |θ|  |θ|(1+cosh |θ|) 1− |θ|(1+cosh |θ|)

6.1.7

+ The group R × SU(2) can be identiﬁed with the multiplicative group H\{0} , where H is the 36 ∼ quaternion algebra. Its Lie algebra coincides with H = R ⊕ LSU(2) . This is the structure group of S with an assigned scaled positive hermitian metric. For −1 example, let h ∈ U F ∨¯ U F = L ⊗ SF ∨¯ SF be a positive Hermitian metric on U , thus −1 + L -scaled on S . Then the set O (S, h) of all h-orthogonal bases with a given orientation + (see footnote 34) is a group-aﬃne space, with derived group R × SU(2) . + Since SU(U, h) also acts on S, one can set SU(S, h) ≡ R × SU(U, h) ; the assignment + of an orthogonal basis determines a group isomorphism SU(S, h) → R × SU(2) . Also, note that SU(S, h) is not canonically isomorphic to H\{0} (see footnote 36). + Similarly, R × Sl(S) can be seen as the subgroup of Aut(S) which preserves any given scaled symplectic form.

6.1.8 Each automorphism of S determines an automorphism of the associated spaces, in such a way to preserve the composition rule. Hence one has group homomorphisms from Aut(S) onto the Aut groups of all those spaces. Note that some of these homomorphisms are not epimorphisms, as each image groups preserves the naturally induced geometric structure.

Starting from an arbitrary K ∈ Aut(S) and an arbitrary basis (ξA) of S one gets

36 2 2 Here H is identiﬁed (§??) with the subspace of C ⊗ C generated by the elements

11 , ıi ≡ −i σ1 , j ≡ −i σ2 , kk ≡ −i σ3 , which fulﬁl ıi2 = j2 = kk2 = ıijkk= −11, ıij = −jıi = kk, jkk= −kkj = ıi, kkıi = −ıikk= j. It is straightforward to check that if R ∈ SO(3) then the triple (ıi0 ≡ R ıi, j0 ≡ R j, kk0 ≡ R kk)still fulﬁls the same axioms. 60 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

transformation induced matrix space group transformation induced basis group

S Aut(S) K (ξA) Gl(2, C)

2 . . ∧ S C \{0} .K. w ≡ ξ1 ∧ ξ2 C \{0}

+ . . √ + L R |.K.| l ≡ w ⊗ w¯ R . . . . Q ≡ ∧2U U(1) |.K.|−1 .K. ε∗ ≡ l−1 ⊗ w U(1)

. . − 1 − 1 c ∼ c . . 2 2 c U Sl (U) = Sl (S) |.K.| K (ζA) ≡ (l ξA) Sl (2, C)

. . . . 1 √ − 1 √ Q1/2 U(1) ±(.K./|.K.|) 2 ε∗=l 2 ⊗ w U(1)

− 1 . .− 1 √ 2 ∼ . . 2 Uν ≡ Q ⊗ U Sl(Uν) = Sl(U) ±.K. K (± ε ⊗ ζA) Sl(2, C) . . ↑ . . −1 ¯ ¯ ↑ H Lor+(H) |.K.| K ⊗ K (ζA ⊗ ζA˙) , (τλ) SO (1, 3)

. .− 1 . . 1 ↑ . . 2 . . 2 ‡ 1 2 ↑ W Spin (W ) (.K. K × .K. K ) (ζ1, ζ2, −¯z , −¯z ) Spin (1, 3)

. . where .K. is a shorthand for det K. The two last lines will be examined in the next sections.

6.2 2-spinor groups and Lorentz group 6.2.1 Up to an obvious transposition one can make the identiﬁcation

End(U) ⊗ End(U) =∼ (U ⊗ U F) ⊗ (U ⊗ U F) =∼ U ⊗ U ⊗ U F ⊗ U F =∼ End(U ⊗ U) .

If no confusion arises one then writes

¯ AA˙ A ¯ A˙ (K ⊗ K) BB˙ = K B K B˙ ,K ∈ End(U) ,

¯ λ A ¯ A˙ λ BB˙ (K ⊗ K) µ = K B K B˙τ AA˙τ µ .

The open submanifold Aut(U ⊗ U) is constituted by all elements in End(U ⊗ U) with non- vanishing determinant; the product group Aut(U) × Aut(U) can be identiﬁed with the subgroup of Aut(U ⊗ U) constituted by all decomposable elements,37 i.e. elements which can be written as K ⊗ H¯ with K,H ∈ Aut U . The Lie algebra of this subgroup is the subalgebra of End(U ⊗ U) =∼ End(U) ⊗ End(U) constituted by all elements which can be written in the

37In order to see that an automorphism of U ⊗ U can be non-decomposable, consider

1 2 ¯ K ≡ (2 ζ1 ⊗ z + ζ2 ⊗ z ) ∈ Aut U ,A ≡ 11U ⊗ K + K ⊗ 11U ∈ End(U) ⊗ End(U) .

Then A can be written, when seen as an element in U ⊗ U ⊗ U F ⊗ U F, as

1 1 2 1 1 2 2 2 A = 4 ζ1 ⊗ ζ¯1 ⊗ z ⊗ ¯z + 3 ζ2 ⊗ ζ¯1 ⊗ z ⊗ ¯z + 3 ζ1 ⊗ ζ¯2 ⊗ z ⊗ ¯z + 2 ζ2 ⊗ ζ¯2 ⊗ z ⊗ ¯z . ¯ Thus A ∈ Aut(U ⊗ U) , since the matrix of A in the basis (ζA ⊗ ζA˙) is diagonal, the diagonal being (4, 3, 3, 2) . On the other hand, if one tries to write A as B ⊗ C¯ , B,C ∈ Aut U, then one gets the system of equations 1 ¯1 2 ¯1 1 ¯2 2 ¯2 {B 1 C 1 = 4,B 2 C 1 = 3,B 1 C 2 = 3,B 2 C 2 = 2} , which is easily seen to have no solution. 6.2 2-spinor groups and Lorentz group 61 form38 ¯ B ⊗ 11U + 11U ⊗ C,B,C ∈ End(U) . By abuse of notation, this subgroup of Aut(U ⊗ U) identiﬁed with Aut(U) × Aut(U) is sometimes written as Aut(U) ⊗ Aut(U) ; of course, this must not be intended as a true tensor product.39 Next consider the group morphism

Aut(U) → Aut(U) ⊗ Aut(U): K 7→ K ⊗ K,¯ whose image is a proper subgroup Aut(U) ∨¯ Aut(U) ⊂ Aut(U) ⊗ Aut(U).

Proposition 6.1 Aut(U) ∨¯ Aut(U) ⊂ Aut(U ⊗ U) preserves the splitting U ⊗ U = H ⊕i H and the causal structure of H . proof: There exist bases of H composed of isotropic elements; these are also complex bases of isotropic elements of U ⊗ U. Then A ∈ Aut(U ⊗ U) preserves the splitting and the causal structure iﬀ it sends any element of the form u ⊗ u¯ in an element of the form v ⊗ v¯ . This is ¯ obviously true for every K ⊗ K ∈ Aut(U) ∨¯ Aut(U). Remark. Aut(U) ∨¯ Aut(U) is not the subgroup of all automorphisms of U ⊗ U which preserve the splitting U ⊗ U = H ⊕ i H and the causal structure of H . For example if F ∈ U ⊗ U F then the mapping u ⊗ v¯ 7→ hF ⊗ F,¯ v¯ ⊗ ui shares the same properties (see §6.6, 6.5). The Lie algebra of Aut(U) ∨¯ Aut(U) is the subalgebra of End(U) ⊗ End(U) constituted by all elements which can be written in the form

¯ Λ[X] ≡ X ⊗ 11U + 11U ⊗ X,X ∈ End(U) . Writing i i 1 A B i X = − 2 X σˆi , σˆi ≡ 2 σi B ζA ⊗ z ,X ∈ C , i = 1, 2, 3 , one has the coordinate expressions

AA˙ i i A A˙ ¯ i A A˙ (Λ[X]) BB˙ = 2 −X σi B δ B˙ + X δ B σ¯i B˙ ,

λ 1 AA˙ λ BB˙ (Λ[X]) µ = 2 (Λ[X]) BB˙σ AA˙σµ . Then one ﬁnds  0 −i (X1 − X¯ 1) −i (X2 − X¯ 2) −i (X3 − X¯ 3)   −i (X1 − X¯ 1) 0 −X3 − X¯ 3 X2 + X¯ 2  λ 1   (Λ[X]) µ =   2  2 ¯ 2 3 ¯ 3 1 ¯ 1  −i (X − X ) X + X 0 −X − X    −i (X3 − X¯ 3) −X2 − X¯ 2 X1 + X¯ 1 0

One then sees that elements in the Lie algebra of Aut(U) ∨¯ Aut(U) restrict to endomorphisms of H, actually they constitute the vector space of all g-antisymmetric endomorphisms of H namely the Lie algebra LLor(H, g).

38This is obviously a proper subalgebra of End(U ⊗ U) , as one sees from the fact that the matrix of any of ¯ its elements, in the basis (ζA ⊗ ζA˙) , is a block matrix with two (2×2)-blocks. 39In particular, note that a linear combination of automorphisms may have vanishing determinant. 62 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

Let a normalized 2-spinor basis be ﬁxed; then the isomorphism LSl(U) ↔ LLor(H, g), taking into account the isomorphism LLor(H, g) ↔ ∧2H∗ induced by the Lorentz metric g , 40 associates the basis (νi ;ν ˇi) with the basis (ρi ;ρ ˇi), i = 1, 2, 3 , where

1 1 A B νi ≡ −iν ˇi νˇi ≡ 2 σˆi ≡ 2 σi B ζA ⊗ z ,, 0 i ρi ≡ −∗ρˇi , ρˇi ≡ 2 t ∧ t .

6.2.2

Accordingly, one may denote as Slc(U) ∨¯ Slc(U) the subgroup of of Aut(U) ∨¯ Aut(U) constituted of all automorphisms of U ⊗ U which are of the form K ⊗ K¯ , K ∈ Slc(U) ; but then Slc(U) ∨¯ Slc(U) = Sl(U) ∨¯ Sl(U) ≡ {K ⊗ K¯ : K ∈ Sl(U) } .

Since K preserves ε up to a phase factor, K ⊗ K¯ preserves ε ⊗ ε¯ ≡ g ; moreover it is easily checked that any Pauli basis is transformed into a Pauli basis, so that Sl(U) ∨¯ Sl(U) restricted ↑ ↑ to H is a subgroup of the special ortochronous Lorentz group Lor+(H) ≡ Lor+(H, g) . Ac- ↑ tually Sl(U) ∨¯ Sl(U)|H coincides with Lor+(H) ; this follows from proposition 1.2 on page 8: for each orthonormal positively oriented basis (eλ) of H, such that e0 is future-oriented, there exists a normalized 2-spinor basis (ζA) whose associated Pauli basis (τλ) coincides with (eλ) . c ↑ It will be shown that the epimorphism Sl (U) Lor+(H) turns out to be a principal bundle ↑ with ﬁbre U(1) , while its restriction Sl(U) Lor+(H) is 2-to-1 .

6.2.3

When a Hermitian metric h on U is given then one has the splittings

LSl(U) = LSU(U, h) ⊕ i LSU(U, h) ,

c LSl (U) = LSU(U, h) ⊕ i LSU(U, h) ⊕ i R .

√1 ¯ Moreover h also determines an “observer” τ0 ≡ 2 h , which on turn determines (proposi- ↑ 41 tion ?? on page ??) the splitting of the Lie algebra LLor(H) ≡ LLor+(H) as

LLor(H) = LLorR(H, τ0) ⊕ LLorB(H, τ0) .

If one chooses an ε-normalized 2-spinor basis (ζA) of U, such that the element τ0 of the corresponding Pauli basis of H coincides with the given observer, then one considers the bases (νi ;ν ˇi) of LSl(U) and (ϑi ; ϑˇi) of LLor(H) , where the latter is constituted by the elements 2 ∗ of End H corresponding via g to the basis (ρi ;ρ ˇi) of ∧ H . These bases are onstituted by the endomorphisms (respectively of U and H) whose matrices are

i 1 (νi) = − 2 (σi) , (ˇνi) = i νi = 2 (σi) .

40Note that the Hodge isomorphism restricts to a complex structure on ∧2H∗. 41Remember that LLor(H) is constituted by all g-antisymmetric endomorphisms of H, namely it is isomor- 2 phic (as a vector space) to ∧ H , via g . Elements of LLorR(H, τ0) , which generate pure rotations, correspond 2 ⊥ to elements in ∧ H , while elements of LLorB(H, τ0) , which generate boosts, correspond to elements of the ⊥ type τ0 ∧ y with y ∈ H . 6.2 2-spinor groups and Lorentz group 63

 0 0 0 0   0 0 0 0   0 0 0 0  (ϑ ) = 0 0 0 0 , (ϑ ) = 0 0 0 1 , (ϑ ) = 0 0 −1 0 , 1  0 0 0 −1  2  0 0 0 0  3  0 1 0 0  0 0 1 0 0 −1 0 0 0 0 0 0

 0 1 0 0   0 0 1 0   0 0 0 1  (ϑˇ ) = 1 0 0 0 , (ϑˇ ) = 0 0 0 0 , (ϑˇ ) = 0 0 0 0 . 1  0 0 0 0  2  1 0 0 0  3  0 0 0 0  0 0 0 0 0 0 0 0 1 0 0 0 These bases are adapted to the respective splittings.

6.2.4 Consider the real symmetric 2-form

◦ KLSl : LSl(U) × LSl(U) → R :(A, B) 7→ 2 < Tr(A B) , and the real symmetric 2-form

1 ◦ KLLor : LLor(H) × LLor(H) → R :(Y,Z) 7→ 2 Tr(Y Z) .

Then one checks that the bases (νi ;ν ˇi) and (ϑi ; ϑˇi) are both orthonormal; the signature of the above metrics turns out to be

(− , − , − , + , + , +) .

All this means that the choice of the underlying 2-spinor frame determines an isometry ( ν ↔ ϑ , LSl(U) ↔ LLor(H): i i νˇi ↔ ϑˇi .

One also sees that the splittings of the two algebras, determined by the choice of an “observer”, can’t be into arbitrary subspaces: instead, the two components must be mutually orthogonal subspaces of opposite signature.

6.2.5 It is easily checked that the endomorphisms constituting the two above bases also fulﬁl the same commutation relations

k k k [νi , νj] = εij νk , [ˇνi , νˇj] = −εij νk , [ˇνi , νh] = εij νˇk , k ˇ ˇ k ˇ k ˇ [ϑi , ϑj] = εij ϑk , [ϑi , ϑj] = −εij ϑk , [ϑi , ϑj] = εij ϑk .

Thus the above said isometry LSl(U) ↔ LLor(H) , determined by the choice of a 2-spinor frame, also turns out to be an isomorphism of Lie algebras; as such, it yields a local iso- ↑ 42 morphism Sl(U) ↔ Lor+(H) between neighbourhoods of the respective identities. So, for

42At this point one may recall that LLor(H) is the vector space of all elements in H ⊗ H∗ which are antisymmetric relatively to the Lorentz metric g ; in other terms, the index lowering isomorphism [g : H ⊗ H∗ → H∗ ⊗ H∗ restricts to a vector space isomorphism LLor(H) → ∧2H∗. Actually, the basis [ (ϑi ; ϑˇi) corresponds via g to the basis (ρi ;ρ ˇi) deﬁned by

2 3 3 1 1 2 0 1 0 2 0 3 ρ1 ≡ 2 e ∧ e , ρ2 ≡ 2 e ∧ e , ρ3 ≡ 2 e ∧ e , ρˇ1 ≡ 2 e ∧ e , ρˇ2 ≡ 2 e ∧ e , ρˇ3 ≡ 2 e ∧ e . 64 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS example, one has the correspondence

−i (s+i t)/2 ! e 0 = exp (s + i t) ν3) ←→ 0 ei (s+i t)/2 cosh t 0 0 sinh t ˇ  0 cos s − sin s 0  ←→ exp s ϑ3 + t ϑ3 =   .  0 sin s cos s 0  sinh t 0 0 cosh t

The 2-to-1 epimorphism shows itself in the factor 1/2 which is found in the exponents of the left-hand side. The right-hand side describes a boost along the x3-direction together with a spatial rotation around the same axis. Note that exp s ϑ3 + t ϑˇ3 = exp(s ϑ3) ◦ exp(t ϑˇ3) , since [ϑ3 , ϑˇ3] = 0 . Also, it can be checked that if the matrix of K ∈ Aut U in the basis

(ζA) is the above 2 × 2-matrix, then the above 4 × 4-matrix is the matrix of the corresponding element K ⊗ K¯ ∈ Aut H in the induced Pauli basis (τλ) . The same relation holds among the various corresponding automorphisms expressed via exponentiation in §??; in particular

r r  r z sinh (x−i y) sinh  cosh + 2 2 2 r r 1 1 1  r r  = exp( 2 x σ1 + 2 y σ2 + 2 z σ3) ←→ (x+i y) sinh 2 r z sinh 2 r cosh 2 − r  x sinh r y sinh r z sinh r  cosh r r r r  x sinh r x2(cosh r−1) xy(cosh r−1) xz(cosh r−1)   1+  ˇ ˇ ˇ  r r2 r2 r2  ←→ exp(x ϑ1 + y ϑ2 + z ϑ3) =    y sinh r xy(cosh r−1) y2(cosh r−1) yz(cosh r−1)   r 2 1+ 2 2   r r r  z sinh r xz(cosh r−1) yz(cosh r−1) z2(cosh r−1) 1+ r r2 r2 r2 where r ≡ px2 + y2 + z2 . Remark. The above relation shows that one can ﬁx a consistent smooth way of choosing, ↑ ¯ for each boost L ∈ Lor+ , one B ∈ Sl(U) such that B ⊗ B = L . In fact, let the matrix of L in any given Pauli basis (induced by the assignment of a 2-spinor basis) be the above 4 × 4-matrix; then (x, y, z) is uniquely determined, and so is B ∈ Aut U whose matrix is the above 2 × 2-matrix. Of course one also has (−B) ⊗ (−B¯) = B ⊗ B¯ = L , nevertheless the said choice of a unique B is well-deﬁned (though somewhat arbitrary), and this correspondence is smooth.

[ 2 ∗ 1 Moreover, this restriction of g turns out to be an isometry, where the metric in ∧ H is the restriction of − 2 ∗ ∗ times the metric induced by g on H ⊗ H . Actually in this metric the basis (ρi ;ρ ˇi) turns out to be orthonormal with signature (− , − , − , + , + , +) . Also, the Hodge isomorphism ∗ : ∧2H∗ → ∧2H∗ determines a complex 2 ∗ # [ −1 structure on ∧ H , whose basis expression turns out to be ∗ρˇi = ρi ; through g ≡ ( g) one gets a complex structure on LLor(H) , and the isomorphism LSl(U) ↔ LLor(H) determined by any chosen 2-spinor basis turns out to be a complex isomorphism. Furthermore, the isomorphism [g : LLor(H) → ∧2H∗ determines a Lie algebra structure on ∧2H∗, the Lie product of two decomposable 2-forms turning out to be

1 [a ∧ b, c ∧ d] = 2 g(b, c) a ∧ d − g(a, c) b ∧ d − g(b, d) a ∧ c + g(a, d) b ∧ c . 6.3 Action of 2-spinor groups on 4-spinor space 65

6.2.6

AA˙ ¯ In the Pauli basis τλ = τ λ ζA ⊗ ζA˙ one has

˜ µ −1 µ AA˙ B ¯ B˙ K λ ≡ |det K| τ BB˙τ λ K A K A˙ , which can be obtained by D E D E ˜ µ µ ˜ µ B B˙ AA˙ 0 ¯ 0 ¯ K λ = t , K(τλ) = τ BB˙z ⊗ ¯z , τ λ K (ζA) ⊗ K (ζA˙) = D E −1 µ B B˙ AA˙ B ¯ B˙ ¯ = |det K| τ BB˙z ⊗ ¯z , τ λ K A K A˙ζB ⊗ ζB˙ =

−1 µ AA˙ B ¯ B˙ = |det K| τ BB˙τ λ K A K A˙ , where K0 ≡ |det K|−1/2 K . Remark. Since Aut(S) is connected one sees that K˜ preserves orientation, so that one has a way of selecting a ‘positive’ orientation on H : those orientations for which the ‘Pauli basis’ (τλ) induced by any basis of S is positively oriented. Of course, this choice of an orientation depends on the Pauli matrices, whose choice is a mere convention, but not on the considered basis of S . A similar argument holds about time-orientation. ˜ 1 ¯ Remark. Thus K is the restriction to H of |det K| K ⊗ K ∈ Aut(U ⊗ U) . The restriction to i H is a special orthochronous Lorentz transformation, too. Moreover, K ⊗ K¯ preserves the decomposition U ⊗ U = H ⊕i H and also the conformal structure of S ⊗ S = L ⊗ (H ⊕i H). The 2-to-1 epimorphisms shows itself in K ⊗ K¯ = (−K) ⊗ (−K¯ ).

6.3 Action of 2-spinor groups on 4-spinor space 6.3.1 There are two natural ways by which an automorphism A ∈ Aut U yields an element in Aut U F , namely

AF : U F → U F : λ 7→ λ ◦ A ≡ λA ,

← ← ← AF : U F → U F : λ 7→ λ ◦ A ≡ λA.

However, only the second way determines a group morphism (actually an isomorphism) Aut U ,→ Aut U F, since one gets

←− ← ← ← ← (AB)F = (B A)F = (A)F ◦ (B)F , A, B ∈ Aut U .

Taking the complex conjugate construction one also gets a group monomorphism ← Aut U ,→ Aut W : A 7→ A• ≡ A, A¯F .

This is a proper monomorphism, since

← A, A¯F) ∈ (U ⊗ U F) ⊕ (U F ⊗ U) ⊂

⊂ (U ⊗ U F) ⊕ (U ⊗ U) ⊕ (U F ⊗ U F) ⊕ (U F ⊗ U) =

= (U ⊕ U F) ⊗ (U F ⊗ U) = W ⊗ W F . 66 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

6.3.2

Remember that the space W = U ⊕ U F is naturally endowed with the following geometric structures:

• the Hermitian metric k : W × W → C ;

• the complex symplectic form (ε, ε¯F) ∈ ∧2U F ⊕ ∧2U ⊂ ∧2W , which is actually deﬁned up to an arbitrary phase factor;

• the Dirac mapγ ˜ : H → End W .

One then looks at how the above objects are aﬀected by the action of the group Aut U.

6.3.3 ← • The Hermitian metric k is preserved by any A ≡ A, A¯F , A ∈ Aut U. In fact, setting φ ≡ (s, σ) ∈ W , ψ ≡ (u, χ) ∈ W one has

← ← ← ← kA•φ, A•ψ = k(As, σA¯), (Au, χA¯) = hσ¯A, Aui + hχA,¯ A¯s¯i =

= hσ,¯ ui + hχ, s¯i = k(φ, ψ) .

6.3.4 The symplectic form is transformed according to

(A•)F(ε, ε¯F) = det A ε, det A¯−1ε¯F .

In particular, if K ∈ Slc(U) then det K = det K¯ −1, and so

(K•)F(ε, ε¯F) = det K ε, ε¯F ,K ∈ Slc(U) i.e. | det K| = 1 ;

c Thus K• ∈ Sp (W , ε +ε ¯F) , the subgroup of Aut(W ) constituted by all automorphisms which preserve ε +ε ¯F ≡ (ε, ε¯F) up to an arbitrary phase factor. If K ∈ Sl(U) then K• is in the “symplectic group” Sp(W , ε +ε ¯F) , the subgroup of Aut(W ) constituted by all automorphisms which preserve (ε, ε¯F) . Observe that

c ∼ Sp (U) = U(1)×˜ Sp(U) ≡ U(1) × Sp(U) /Z2 .

6.3.5 The Dirac mapγ ˜ can be viewed as an element

γ˜ ∈ H∗ ⊗ W ⊗ W F , whose expression in the Pauli and 4-spinor bases, (τλ) and (ζα) , is

α λ β γ˜ = γλ β t ⊗ ζα ⊗ z .

∗ Since H ⊂ U F ⊗ U F one also has

γ˜ ∈ U F ⊗ U F ⊗ (U ⊗ U F) ⊕ (U ⊗ U) ⊕ (U F ⊗ U F) ⊕ (U F ⊗ U) , 6.4 Dirac algebra in 2-spinor terms 67

and writes its expression in the 2-spinor basis ζA as

A A˙ B C BC˙ ¯ B˙ C C˙ B˙ ¯ γ˜ = z ⊗ ¯z ⊗ γAA˙ C ζB ⊗ z + γAA˙ ζB ⊗ ζC˙ + γAA˙B˙C ¯z ⊗ z + γAA˙B˙ ¯z ⊗ ζ C˙ .

Rewriting the deﬁnition ofγ ˜ in §1.5 as √ γ˜ , (p ⊗ q¯) ⊗ (λ, v¯) ⊗ (u, χ) = 2 hχ, q¯i hλ, pi + ε(p, u)ε ¯(¯q, v¯) one has the coordinate expression √ A A˙ ¯ A A˙ B˙ C γ˜ = 2 z ⊗ ¯z ⊗ ζA ⊗ ζA˙ + εAC ε¯A˙B˙z ⊗ ¯z ⊗ ¯z ⊗ z , that is √ √ BC˙ B C˙ B C˙ γAA˙ = 2 δA δA˙ , γAA˙B˙C = 2 εAC ε¯A˙B˙ , γAA˙ C = γAA˙B˙ = 0 . Thus one has γ˜ ∈ U F ⊗ U F ⊗ (U ⊗ U) ⊕ (U F ⊗ U F) and, up to index reordering, one could write simply √ √ γ˜ = 2 11W , ε ⊗ ε¯ ≡ 2 11W , g˜ .

← ← For A ∈ Aut U, using A¯ ∈ Aut U, AF ∈ Aut U F and A¯F ∈ Aut U F one getsγ ˜ transformed as √ 2 γ˜ 7→ 2 11W , | det A| ε ⊗ ε¯ .

In particular, the action induced by any K ∈ Slc(U) preservesγ ˜ , and so preserves the whole ← natural structure of W (note that nothing works if one took AF instead of AF as the induced action on U F ).

6.4 Dirac algebra in 2-spinor terms 6.4.1

Remember that the Dirac algebra is the real vector subspace D ⊂ W ⊗ W F multiplicatively generated by the vector subspace

γ˜(H) ⊂ (U ⊗ U) ⊕ (U F ⊗ U F) , and that C ⊗ D = W ⊗ W F . Also remember that D as a vector space is exactly the exterior algebra ∧[˜γ(H)] , which is on turn isomorphic to ∧H via the natural extensionγ ˆ pfγ ˜ (§B.1). Take an orthonormal (Pauli) basis (τλ) of H and select, of the two distinguished elements in ∧4H, the element

∗ # ∗ η = −η = τ0 ∧ τ1 ∧ τ2 ∧ τ3 ⇒ γη ≡ γˆ(η) = γ0 γ1 γ2 γ3 = γ0 ∧ γ1 ∧ γ2 ∧ γ3 .

Then for any ϑ ∈ ∧H one has γˆ(ϑ) γη = −γˆ(∗ϑ) 68 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS and, in particular

γ˘λ ≡ γλ γη = −γˆ(∗τλ) i.e.

γ˘0 = γ123 ≡ γ1 ∧ γ2 ∧ γ3 ,

γ˘1 = γ023 ≡ γ0 ∧ γ2 ∧ γ3 ,

γ˘2 = γ031 ≡ γ0 ∧ γ3 ∧ γ1 ,

γ˘3 = γ012 ≡ γ0 ∧ γ1 ∧ γ2 .

Moreover set 1 γλµ ≡ γλ ∧ γµ ≡ 2 (γλ γµ − γµ γλ) , which yields

γ01 γη = γ23 , γ02 γη = γ31 , γ03 γη = γ12 ,

γ23 γη = γ10 , γ31 γη = γ20 , γ12 γη = γ30 .

One then gets the basis 11 , γ0 , γ1 , γ2 , γ3 , γ01 , γ02 , γ03 , γ12 , γ13 , γ23 , γ˘0 , γ˘1 , γ˘2 , γ˘3 , γη +1 +1 −1 −1 −1 +1 +1 +1 −1 −1 −1 +1 −1 −1 −1 −1 of D , which coresponds viaγ ˆ to the basis of ∧ H induced by the cosen Pauli basis. The second line above shows the squares of the corresponding elements relatively to the pseudometric

1 ◦ (A, B) 7→ 4 Tr(A B) .

One then sees thatγ ˆ is not an isometry since relatively to the Lorentz metric one has43

2 2 2 2 (τ0 ∧ τ1) = (τ0 ∧ τ2) = (τ0 ∧ τ3) = (τ1 ∧ τ2 ∧ τ3) = −1 ,

2 2 2 2 2 2 (τ1 ∧ τ2) = (τ2 ∧ τ3) = (τ3 ∧ τ1) = (τ0 ∧ τ1 ∧ τ2) = (τ0 ∧ τ1 ∧ τ3) = (τ0 ∧ τ2 ∧ τ3) = 1 .

Of course, one can always deﬁne a diﬀerent metric on ∧H or D in such a way thatγ ˆ be an isometry.

6.4.2 The already considered decomposition

W ⊗ W F = (U ⊗ U F) ⊕ (U ⊗ U) ⊕ (U F ⊗ U F) ⊕ (U F ⊗ U) can be conveniently rewritten in a kind of “matricial” way as

U ⊗ U F U ⊗ U ! W ⊗ W F = . U F ⊗ U F U F ⊗ U

43These can also be checked by means of the formulas written in §B.2. 6.4 Dirac algebra in 2-spinor terms 69

It will also be convenient, for clarifying certain arguments, to use the provisional notation

U 0 ≡ U F ⇐⇒ U 0F ≡ U ⇐⇒ W ≡ U ⊕ U 0 ,

0 U ⊗ U F U ⊗ U F ! W ⊗ W F = , 0 0 0 U ⊗ U F U ⊗ U F

1 2 ζ10 ≡ ζ3 ≡ −¯z , ζ20 ≡ ζ4 ≡ −¯z ,

10 3 20 4 z ≡ z ≡ −ζ¯1 , z ≡ z ≡ −ζ¯2 . Correspondingly, any endomorphism Φ ∈ End(W ) can be written as a “matrix of tensors”

U U ! Φ U Φ 0 Φ = U , U0 U0 Φ U Φ U0 and its matrix can be written as the “matrix of matrices”

A A ! A AB˙! Φ Φ 0 Φ −Φ α B B B Φ β = = . A0 A0 −Φ Φ B˙ Φ B Φ B0 A˙B A˙

6.4.3

Consider now the decomposition of C ⊗ D ≡ End W into the direct sum (+) (−) (+) (−) C ⊗ D = C ⊗ D ⊕ C ⊗ D ≡ C ⊗ D ⊕ C ⊗ D , where U ⊗ U F {0} ! ⊗ D(+) ≡ , D(+) ≡ ⊗ D(+) ∩ D , C 0 0 C {0} U ⊗ U F

0 {0} U ⊗ U F! ⊗ D(−) ≡ , D(−) ≡ ⊗ D(−) ∩ D . C 0 C U ⊗ U F {0}

Then it is clear that γ(H) ⊂ D(−) and that D(+) and D(−) constitute respectively the subspaces of all even and odd elements of D ; also note that D(+) is a subalgebra, while D(−) is not. In practice, identifying γ(H) with H for simplicity, one actually writes (+) ∼ 2 4 D = R ⊕ ∧ H ⊕ ∧ H ,

D(−) =∼ H ⊕ ∧3H , where R corresponds to R 11W .

6.4.4 The above statements can be checked via the previously introduced γ-basis of D. Consider the matrices of the elements of the γ-basis of D in the Weyl basis (ζα) . One ﬁnds 1 0 0 0 ! 0 0 −1 0 ! 0 1 0 0 0 0 0 −1 11 = 0 0 1 0 , γ0 = −1 0 0 0 , 0 0 0 1 0 −1 0 0 70 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

0 0 0 −1 ! 0 0 0 i ! 0 0 −1 0 ! 0 0 −1 0 0 0 −i 0 0 0 0 1 γ1 = 0 1 0 0 , γ2 = 0 −i 0 0 , γ3 = 1 0 0 0 , 1 0 0 0 i 0 0 0 0 −1 0 0

0 −1 0 0 ! 0 i 0 0 ! −1 0 0 0 ! −1 0 0 0 −i 0 0 0 0 1 0 0 γ01 = 0 0 0 1 , γ02 = 0 0 0 −i , γ03 = 0 0 1 0 , 0 0 1 0 i 0 i 0 0 0 0 −1

−i 0 0 0 ! 0 1 0 0 ! 0 −i 0 0 ! 0 i 0 0 −1 0 0 0 −i 0 0 0 γ12 = 0 0 −i 0 , γ13 = 0 0 0 1 , γ23 = 0 0 0 −i , 0 0 0 i 0 0 −1 0 0 0 −i 0

0 0 i 0 ! 0 0 0 i γ˘0 = −i 0 0 0 , 0 −i 0 0

0 0 0 −i ! 0 0 0 −1 ! 0 0 −i 0 ! 0 0 −i 0 0 0 −i 1 0 0 0 i γ˘1 = 0 −i 0 0 , γ˘2 = 0 −1 0 0 , γ˘3 = −i 0 0 0 , −i 0 0 0 1 0 0 0 0 i 0 0

i 0 0 0 ! 0 i 0 0 γη = 0 0 −i 0 . 0 0 0 −i Then one writes the general coordinate expressions of elements of the various exterior degrees as

• exterior degree 0 :  X11 0 0 0  11 X11 11 = 0 X 0 0 ;  0 0 X11 0  0 0 0 X11

• exterior degree 1 :

 0 0 −X0−X3 −X1+i X2  1 2 0 3 Xλ γ = 0 0 −X −i X −X +X ; λ  −X0+X3 X1−i X2 0 0  X1+i X2 −X0−X3 0 0

• exterior degree 2 :

X λµ X γλµ = λ<µ

 −X03−i X12 (−X01+X13)+i (X02−X23) 0 0  01 13 02 23 03 12  −(X +X )−i (X +X ) X +i X 0 0   0 0 X03−i X12 (X01+X13)−i (X02+X23)  0 0 (X01−X13)+i (X02−X23) −X03+i X12

• exterior degree 3 :

 0 0 i (X˘ 0+X˘ 3) X˘ 2+i X˘ 1  ˘ 2 ˘ 1 ˘ 0 ˘ 3 λ 0 0 −X +i X i (X −X ) X˘ γ˘λ =   ;  i (−X˘ 0+X˘ 3) X˘ 2+i X˘ 1 0 0  −X˘ 2+i X˘ 1 −i (X˘ 0+X˘ 3) 0 0 6.4 Dirac algebra in 2-spinor terms 71

• exterior degree 4 : i Xη 0 0 0 ! η 0 i Xη 0 0 X γη = 0 0 −i Xη 0 . 0 0 0 −i Xη The endomorphisms X ∈ End W whose matrices have the above expressions in the Weyl basis are in D iﬀ all of the coeﬃcients (X11 ; Xλ ; Xλµ ; X˘ λ ; Xη) are real.

6.4.5 From the above coordinate expressions of elements in D one ﬁnds a simple characterization of the various exterior components of D in terms of endomorphisms of U. Consider the odd subspace ﬁrst.

• Exterior degree 1 : the matrix of X is of the type 0 M X = Xλ γ = , λ M ‡ 0

0 0 where 0 is a shortcut for ( 0 0 ) and − X0 − X3 −X1 + i X2 ! M = , −X1 − i X2 −X0 + X3

− X0 + X3 X1 − i X2 ! M ‡ = . X1 + i X2 −X0 − X3 Then one immediately sees that M and M ‡ are Hermitian matrices; if 0 6= det M = det M ‡ = (X0)2 − (X1)2 − (X2)2 − (X3)2 then M and M ‡ are related by44 ← M ‡ = (det M) M¯ F . Actually the “true” relation between M and M ‡, holding also if the determinant (which λ coincides with the Lorentz pseudonorm of the vector X τλ) vanishes, is clear if one looks at 0 0 these as matrices of components of elements in U ⊗ U F ≡ U ⊗ U and U ⊗ U F ≡ U F ⊗ U F, respectively: these two objects are actually elements of H and H∗, respectively, mutually related by the isomorphism determined byg ˜ and complex conjugation. Namely

A B0 AB˙ ¯ M B0 ζA ⊗ z = −M ζA ⊗ ζB˙ ,

‡A0 ‡ A˙ B 0 M B ζA ⊗ ζB = −MA˙B ¯z ⊗ z ,

‡ ¯ C˙D MA˙B =ε ¯C˙A˙εDB M ,

‡ ¯ A˙B ¯ C˙D ¯ A˙B ¯ MA˙B M =ε ¯C˙A˙εDB M M = 2 det M = 2 det M. Actually, all this is nearly obvious if one remembers the deﬁnition ofγ ˜ in 2-spinor terms.45

44Since the components Xλ are arbitrary, M is an arbitrary Hermitian matrix, while M ‡ is determined by M (or vice-versa). √ 45 Note the role of the√ 2 factor, in the deﬁnition ofγ ˜ , which remains hidden in this discussion because it cancels out with the 1/ 2 factor in the deﬁnition of τλ . 72 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

• Exterior degree 3 : the matrix of X is of the type

0 P X = X˘ λ γ˘ = , λ P ‡ 0

i (X˘ 0 + X˘ 3) X˘ 2 + i X˘ 1 ! P = , −X˘ 2 + i X˘ 1 i (X˘ 0 − X˘ 3)

i (−X˘ 0 + X˘ 3) X˘ 2 + i X˘ 1 X1 ! P ‡ = . − X˘ 2 + i X˘ 1 − i (X˘ 0 + X˘ 3)

Then one immediately sees that P and P ‡ are anti-Hermitian matrices, mutually related exactly like the matrices M and M ‡ just considered. This means that

γˆ(∧3H) =γ ˜(i H) = iγ ˜(H) ,

(−) D = C ⊗ γ˜(H) =γ ˜(H) ⊕ iγ ˜(H) .

6.4.6 The matrix of an arbitrary element

11 X λµ η (+) X = X 11W + X γλµ + X γη ∈ D λ<µ can be written in the form K 0 X = , 0 K‡

a+d b −a¯+d¯ −c¯ K = ,K‡ = , c −a+d −¯b a¯+d¯

a ≡ −X03 − i X12 , b ≡ (−X01 + X13) + i (X02 − X23) ,

c ≡ (−X01 − X13) − i (X02 + X23) , d ≡ X11 + i Xη .

In particular, the elements of exterior degree two constitute the subspace of D(+) characterized by d = 0 that is Tr K = Tr K‡ = 0 . If one only excludes the elements of degree zero then one gets the subspace of D(+) characterized by determinants do not vanish, are related by the relation ← K‡ = (det K¯ ) K¯ F 6.4 Dirac algebra in 2-spinor terms 73

(in this case det K and det K¯ need not be equal). In general one has

U A B Φ U = K B ζA ⊗ z ,

U0 ‡A0 0 ‡ B˙ A˙ B˙ 0 B Φ U0 = K B0 ζA ⊗ z ≡ K A˙ ¯z ⊗ ¯z ,

‡ B˙ ¯ C˙ D˙B˙ K A˙ =ε ¯C˙A˙K D˙ε¯ .

6.4.7

Summarizing, one has a real vector space isomorphism

KP (U ⊗ U F) ⊕ (U ⊗ U) → D :(K,P ) 7→ . P ‡ K‡

More precisely, the ﬁrst line of the above “matrix of matrices” depends C-linearly from (K,P ), while the second line depends anti-linearly from (K,P ) . Thus multiplying (K,P ) by i one gets i K i P i 11 0 KP (i K, i P ) 7→ = . −i P ‡ −i K‡ 0 −i 11 P ‡ K‡

Thus if (K,P ) corresponds to X ∈ D then (i K, i P ) corresponds to γη X ∈ D (it’s a diﬀerent “version” of a Hodge isomorphism, since the standard one is −X γη —namely it’s the same for odd elements and the opposite for even elements).

6.4.8

The above characterization of the Dirac algebra can be recast in a more elegant form by introducing a real endomorphism ‡ of W ⊗ W F as follows. Set46

F F ‡ [ # ¯ ¯ C˙ D˙B˙ A˙ ¯ ‡ : U ⊗ U → U ⊗ U : K 7→ K ≡ hε¯ ⊗ ε¯ , Ki =ε ¯C˙A˙K D˙ε¯ ¯z ⊗ ζB˙ ,

F F ‡ [ [ ¯ ¯C˙D A˙ B ‡ : U ⊗ U → U ⊗ U : P 7→ P ≡ hε¯ ⊗ ε , P i =ε ¯C˙A˙P εDB ¯z ⊗ z ,

F F ‡ # # ¯ CA ¯ D˙B˙ ¯ ‡ : U ⊗ U → U ⊗ U : Q 7→ Q ≡ hε ⊗ ε¯ , Qi = ε QCD˙ε¯ ζA ⊗ ζB˙ ,

F F ‡ # [ ¯ CA ¯ D B ‡ : U ⊗ U → U ⊗ U : J 7→ J ≡ hε ⊗ ε , Ji = ε J C εDB ζA ⊗ z .

Then, taking into account the natural inclusions of the above spaces into W ⊗ W F, one gets the real endomorphism

KP J ‡ Q‡ ‡ : W ⊗ W F → W ⊗ W F : 7→ . QJ P ‡ K‡

Since obviously one has (K‡)‡ = K ,(P ‡)‡ = P , one ﬁnds that ‡ is an involution, that is ‡ ◦ ‡ = 11. Thus W ⊗ W F can be decomposed into the direct sum of the (real) eigenspaces of ‡ corresponding to eigenvalues ±1 , that is exactly W ⊗ W F = D ⊕ i D.

46Note how this is independent of the choice of an arbitrary phase factor for ε . 74 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

I’ll also use the notation K‡ ≡ K¯˜ , P ‡ ≡ P¯˜ and the like, where the tilda denotes an operation called ε-transposition given by ← U ⊗ U F → U F ⊗ U : K 7→ K˜ ≡ hε[ ⊗ ε#,Ki = det K K F ,

U ⊗ U → U F ⊗ U F : P 7→ P˜ ≡ hε[ ⊗ ε¯[,P i =g ˜[P,

U F ⊗ U F → U ⊗ U : R 7→ R˜ ≡ hε# ⊗ ε¯#,Ri =g ˜#R,

← U F ⊗ U → U ⊗ U F : S 7→ S˜ ≡ hε# ⊗ ε[,Si = det S S F .

← ← Of course, the above characterisations K˜ = det K K F and det S S F only hold if K and S, respectively, are invertible. In any case, ε-transposition acts as a covariant functor, as one has

(K ◦ P )∼ = K˜ ◦ P,˜ and the like whenever composition is deﬁned.

6.4.9 As above, let

K ∈ U ⊗ U F ,P ∈ U ⊗ U ,J ∈ U F ⊗ U,Q ∈ U F ⊗ U F . The allowed compositions of two such elements are

KP ∈ U ⊗ U ,PJ ∈ U ⊗ U ,JQ ∈ U F ⊗ U F ,

QK ∈ U F ⊗ U F ,PQ ∈ U ⊗ U F ,QP ∈ U F ⊗ U . Then, of course, several more possibilities arise when one considers ε-transposition, standard transposition and complex conjugation. It may be useful to have ready a table of the index types of the various objects, in order to know at-a-glance which compositions are are allowed and which will be the type of the result. Indicating non-conjugate index types by a white square, and conjugate index types by a black square, one gets the following table

K ,K F , K˜ , K˜ F , K¯ , K¯ F , K¯˜ , K¯˜ F ,

P ,P F , P˜ , P˜F , P¯ , P¯F , P¯˜ , P¯˜F ,

Q ,QF , Q˜ , Q˜F , Q¯ , Q¯F , Q¯˜ , Q¯˜F ,

J ,J F , J˜ , J˜F J¯ , J¯F , J¯˜ , J¯˜F . In order to work with these objects one has to remember a few basic formulas:

˜ ∼ X˜ = X, (X˜)F = (X F) ,X FF = X,

Tr(XZ) = Tr(X˜ Z˜) = Tr(Z F X F) ,

(XY )∼ = X˜ Y,˜ (XY )F = Y F X F ,

X X˜ F = (det X) 11 , XX˜ F = (det X) 11 , whatever the type of X, and whatever the types of Y , Z such that XY is well-deﬁned and XZ is well-deﬁned and an endomorphism of a 2-spinor space. 6.4 Dirac algebra in 2-spinor terms 75

6.4.10

By direct calculations one checks that whenever A and B are tensors of the same of the above types, the determinant of their sum can be calculated as

det(A + B) = det(A) + det(B) + A ♣ B ≡

≡ det(A) + det(B) + Tr(AF ◦ B˜) , where the scalar product

♣ :(A, B) 7→ A ♣ B ≡ Tr(AF ◦ B˜) is symmetric. Note that ♣ is not Hermitian, and is deﬁned on each of the four subspaces of End(W ) . Its restrictions to U ⊗ U and U F ⊗ U F coincide with 2g ˜ and 2g ˜#, respectively. Then, by induction, one can express the determinant of a sum of an arbitrary number of terms. On ﬁnds

det(A + B + C) = det(A) + det(B) + det(C) + A ♣ B + B ♣ C + C ♣ A and the like.

6.4.11

By direct calculation47 one checks the following

KP Proposition 6.2 Let Φ = ∈ W ⊗ W F. Then48 QJ

det Φ = (det K) (det J) + (det P ) (det Q) − Tr(K F PJ˜ F Q˜) , ! ← (det J) K˜ F − Q˜F J P˜F (det P ) Q˜F − K˜ F P J˜F (det Φ) Φ = , (det Q) P˜F − J˜F Q K˜ F (det K) J˜F − P˜F K Q˜F where49

1 1 AB˙ CD˙ det P = 2 g˜(P,P ) = 2 εAC ε¯B˙D˙P P ,

1 [ F F 1 A˙C˙ BD det Q = 2 g˜ (Q ,Q ) = 2 ε¯ ε QA˙B QC˙D .

47A constructive proof is given in appendix A.6.1. 48By taking into account the properties of the trace relatively to transposition, ε-transposition and exchange of factors, one sees that Tr(KF PJ˜ F Q˜) can be equivalently written as the trace of any one of the following 16 endomorphisms:

KFPJ˜ FQ,˜ K˜ FP J˜FQ, KQ˜FJP˜F, KQ˜ FJP˜ F,P J˜FQK˜ F, PJ˜ FQK˜ F,P FKQ˜ FJ,˜ P˜FKQ˜FJ, QK˜ FP J˜F, QK˜ FPJ˜ F,QFJP˜ FK,˜ Q˜FJP˜FK,JP˜FKQ˜F, JP˜ FKQ˜ F,J FQK˜ FP,˜ J˜FQK˜ FP.

49See the remark in §1.4, page 6. 76 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

Moreover one ﬁnds50

det(det J) K˜ F − Q˜F J P˜F = (det J) (det Φ) ,

det(det P ) Q˜F − K˜ F P J˜F = (det P ) (det Φ) ,

det(det Q) P˜F − J˜F Q K˜ F = (det Q) (det Φ) ,

det(det K) J˜F − P˜F K Q˜F = (det K) (det Φ) .

Corollary 6.1 In particular, let ! KP KP D 3 Φ = ≡ ; P ‡ K‡ P¯˜ K¯˜ then

det Φ = | det K|2 + | det P |2 − Tr(K P¯F K¯˜ P˜F ) = 1 = det(det K¯ ) K˜ F − P¯F K¯˜ P˜F = det K¯ 1 = det(det P¯) P˜F − K¯ F P¯˜ K˜ F = det P 1 = det(det P¯) P˜F − K¯ F P¯˜ K˜ F = det P¯ 1 = det(det K) K¯ F − P˜F K P¯F , det K

 ˜  ← (det K¯ ) K˜ F − P¯F K¯ P˜F (det P ) P¯F − K˜ F P K¯ F (det Φ) Φ =   . (det P¯) P˜F − K¯ F P¯˜ K˜ F (det K) K¯ F − P˜F K P¯F

Remark. The three terms in the ﬁrst expression of det Φ above are all real; thus det Φ is real. The other expressions hold, of course, only if the respective denominators do not vanish.

6.5 Cliﬀord group in 2-spinor terms

6.5.1

Let D• ≡ D ∩ Aut W be the group of all invertible elements in D. The Clifford group51 Cl ≡ Cl(W ) is defined to be [Cr90, Gr78] the subgroup of D• under whose adjoint action H is stable. In other terms, Φ ∈ D• is an element of Cl iff

← Ad[Φ]v ≡ Φ γ(v) Φ ∈ γ(H) , ∀ v ∈ H .

50For example, thse formulas can be checked by calculating the ﬁrst-hand sides by the above general formula for det(A + B) , and comparing that with the already found expression for det Φ . 51Here and in the next sections I’ll use the shorthand Cl ≡ Cl(W ) , as no confusion can arise. Similarly Pin ≡ Pin(W ), Spin ≡ Spin(W ) and the like. 6.5 Cliﬀord group in 2-spinor terms 77

One then has the standard result that Cl is multiplicatively generated by all γ[v] ∈ H such that g(v, v) 6= 0 ; namely, Cl is multiplicatively generated by all elements in D• which are of the type ! 0 V 0 V ≡ , V ‡ 0 V¯˜ 0 √ V ≡ 2 v ∈ U ∨¯ U ⊂ U ⊗ U , det V =g ˜(v, v) 6= 0 .

This obviously implies that any element in Cl , seen as an element in D, is either odd or even: no linear combination of an element in D(+) and of an element in D(−) belongs to the Cliﬀord group (unless one of the two terms vanishes, of course). Namely

Cl = Cl(+) ∪ Cl(−) ≡ (Cl ∩ D(+)) ∪ (Cl ∩ D(−)) .

Furthermore, these two components are mutually disconnected.52 Also, note that Cl(+) is a subgroup of Cl , while Cl(−) is not.

6.5.2 The above cited results about the structure of the Cliﬀord group are traditionally obtained by using further results about the orthogonal group of H and the way the adjoint action of H• ⊂ D• actually yields a particular family of orthogonal transformations (H• ⊂ H denotes the open submanifold of all non-isotropic elements). Here that result will be proved by independent arguments, in terms of two-spinor geometry. As a preliminary remark, observe that the adjoint action of D• on H can be written, in two-spinor terms, as   ! ! . . ˜ . . . . KP 0 V .K¯ . K˜ F − P¯F K¯ P˜F .P . P¯F − K˜ F P K¯ F .Φ. . Ad[Φ]v =   = P¯˜ K¯˜ V¯˜ 0 . . . .  .P¯. P˜F − K¯ F P¯˜ K˜ F .K. K¯ F − P˜F K P¯F

  P VX¯˜ + KV YP¯˜ VY¯˜ + KV X¯˜ =   , K¯˜ VX¯˜ + PV¯˜ Y¯˜ K¯˜ VY¯˜ + PV¯˜ X¯˜

. . √ where .X. is a shorthand for det X , V ≡ 2 v and

. . . . X ≡ .K¯ . K˜ F − P¯F K¯˜ P˜F ,Y ≡ .P . P¯F − K˜ F P K¯ F ,

. . . . X¯˜ = .K. K¯ F − P˜F K P¯F , Y¯˜ = .P¯. P˜F − K¯ F P¯˜ K˜ F .

Proposition 6.3 An element of D• which belongs to the Cliﬀord group is necessarily either odd or even, so that the Cliﬀord group is the disjoint union Cl = Cl(+) ∪ Cl(−) where Cl(+) ≡ Cl ∩ D(+) , Cl(−) ≡ Cl ∩ D(−) .

52In fact Cl(+) and Cl(−) are subsets, not containing the zero element, of two complementary subspaces in a ﬁnite dimensional space. 78 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS proof: In the expression for (det Φ) Ad[Φ]v written above consider the U ⊗ U F-component, namely . . . . P VX¯˜ + KV Y¯˜ = P V¯˜ (.K¯ . K˜ F − P¯F K¯˜ P˜F) + KV (.P¯. P˜F − K¯ F P¯˜ K˜ F) .

If Φ is in Cl , that is if Ad[Φ]v ∈ H for all v ∈ H, then this expression vanishes for all v ∈ H, that is KV Y¯˜ = −P VX,¯˜ ∀ V ∈ H .

Composing both sides with V˜ F K˜ F on the left, and remembering that K˜ F K = det K and the like, one gets (det K) (det V ) Y¯˜ = −V˜ F K˜ F P VX.¯˜

Now composing both sides with X˜ F on the right, and recalling det X = (det K¯ )(det Φ) , one gets (det K) (det V ) Y¯˜ X˜ F = −(det Φ)(det K¯ ) V˜ F K˜ F P V.¯˜ Now the above equality is certainly fulﬁlled in the particular case when det K = 0 . Suppose det K 6= 0 for the moment (the other case will be considered later). The left-hand side vanishes for all null (i.e. isotropic) vectors V ∈ H, thus also V˜ F K˜ F P V¯˜ vanishes for all null vectors V ; but this implies53 K˜ F P = 0 . Since 0 6= det K = det K˜ F, this on turn implies P = 0 . Summarizing, if Φ ∈ Cl and det K 6= 0 then P = 0 . Suppose now det P 6= 0 ; take again the relation KV Y¯˜ = −P VX¯˜ and this time compose on the left by V¯ F P˜F and on the right by Y¯ F, thus obtaining

(det P¯) (det Φ) V¯ F P˜F KV = −(det P ) (det V¯ ) X Y¯ F .

As above, this implies P˜F K = 0 and then K = 0 since det P 6= 0 . Summarizing, if Φ ∈ Cl and det P 6= 0 then K = 0 . The case which remains to be considered is that when det K = det P = 0 . Since det P = 1 2 g(P,P ), P is an isotropic element of U ⊗ U, and as such it is decomposable. Similarly, K is decomposable (see e.g. lemma 6.2). Namely one can write

K = k ⊗ λ , P = p ⊗ q¯ , V = s ⊗ s¯ , k, p, q, s ∈ U, λ ∈ U F

(remember that here k, p, q, λ are ﬁxed, while s is an arbitrary element). A little two-spinor algebra then yields

P¯ =p ¯⊗ q , P¯F = q ⊗ p¯ , P˜ = p[ ⊗ q¯[ , P¯˜ =p ¯[ ⊗ q[ , P˜F =q ¯[ ⊗ p[ ,

K¯ = k¯ ⊗ λ¯ , K¯˜ = k¯[ ⊗ λ¯# , K¯ F = λ¯ ⊗ k¯ , K˜ F = λ# ⊗ k[

V¯ =s ¯⊗ s , V¯˜ =s ¯[ ⊗ s[ ,

X = −P¯F K¯˜ P˜F = −(q ⊗ p¯)c(k¯[ ⊗ λ¯#)c(¯q[ ⊗ p[) = −ε¯(k,¯ p¯) hλ,¯ q¯i q ⊗ p[ ,

Y¯˜ = −K¯ F P¯˜ K˜ F = −(λ¯ ⊗ k¯)c(¯p[ ⊗ q[)c(λ# ⊗ k[) = −ε¯(¯p, k¯) hλ, qi λ¯ ⊗ k[ ,

53 In order to check this claim, consider the basis (w1 , w2 , w3 , w4 ) ≡ (τ0 + τ1 , τ0 + τ2 , τ0 + τ3 , τ0 − τ3 ) ⊂ H , which is constituted by isotropic vectors, and let Z ∈ U ⊗ U. A direct calculations shows that that the four F F F conditionsw ˜α Z w¯˜α , α = 1, 2, 3, 4 , imply Z = 0 . Also, note that actuallyv ¯ = v ,w ¯α = wα , since these are all Hermitian tensors (this substitution was not explicitely made in the above formulas). 6.5 Cliﬀord group in 2-spinor terms 79

P VX¯˜ + KV Y¯˜ = = −ε¯(¯p, k¯) hλ, qi (k ⊗ λ)c(s ⊗ s¯)c(λ¯ ⊗ k[) − ε¯(k,¯ p¯) hλ,¯ q¯i (p ⊗ q¯)c(¯s[ ⊗ s[)c(q ⊗ p[) = h i =ε ¯(k,¯ p¯) hλ, qi |hλ, si|2 k ⊗ k[ − hλ,¯ q¯i |ε(s, q)|2 p ⊗ p[ ,

det Φ = − Tr(K P¯F K¯˜ P˜F ) = − Tr[(k ⊗ λ)c(q ⊗ p¯)c(k¯[ ⊗ λ¯#)c(¯q[ ⊗ p[)] =

= − Tr[hλ, qi ε¯(k,¯ p¯) hλ,¯ q¯i k ⊗ p[] = |ε(k, p)|2 |hλ, qi|2 .

Now one sees that in order that det Φ 6= 0 one must have hλ, qi= 6 0 and ε(k, p) 6= 0 . Thus k ⊗ k[ and p ⊗ p[ are linearly independent elements of U ⊗ U F and, in order that P VX¯˜ + KV Y¯˜ vanishes for all V , one must have hλ, si = ε(q, s) for all s ∈ U , which implies λ = 0 and q = 0 that is K = 0 and P = 0 , a contradiction. Thus the case det K = det P = 0 cannot yield an element Φ ∈ Cl .

Proposition 6.4 (+) (+) a) Cl is the 7-dimensional real submanifold of D constituted of all elements in W ⊗ W F which are of the type K 0 ,K ∈ U ⊗ U F , det K ∈ \{0} 0 K‡ R

(namely all even elements in C ⊗ D characterized by K ∈ U ⊗ U F whose non-vanishing determinant is real, = det K = 0 ). (−) (−) b) Cl is the 7-dimensional real submanifold of D constituted of all elements in W ⊗ W F which are of the type 0 P ,P ∈ U ⊗ U , det P ∈ \{0} P ‡ 0 R

(namely all odd elements in C ⊗ D characterized by P ∈ U ⊗ U whose non-vanishing determinant is real, = det P = 0 ). proof: ! K 0 a) Let Φ = , K ∈ U ⊗ U F, det K 6= 0 . Then 0 K¯˜   ¯ ˜ F ! ← (det K) K 0 0 V √ (det Φ) Φ = , γ(v) =   ,V ≡ 2 v ∈ H , 0 (det K) K¯ F V¯˜ 0

  0 (det K) KV K¯ F (det Φ) Ad[Φ]v ≡ (det Φ) Ad[Φ]γ(v) =   . (det K¯ )K¯˜ V¯˜ K˜ F 0

For Ad[Φ]v to be in H, the two non-zero entries of the above matrix must be in H ≡ U ∨¯ U and in U F ∨¯ U F, respectively. Consider the U ⊗ U-entry. Since V¯ = V F because V is Hermitian, one ﬁnds F (det K) KV K¯ F = (det K¯ ) KV K¯ F , 80 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

and (det K) KV K¯ F is Hermitian for all V ∈ H iﬀ det K = det K¯ . Repeating this argument for the other non-zero entry gives the same result (actually the two entries are the same up to complex conjugation and ε-transposition). ! 0 P b) Let Φ = , P ∈ U ⊗ U, det P = 1 g(P,P ) 6= 0 . Then P¯˜ 0 2

¯F! ← 0 (det P ) P (det Φ) Φ = , (det P¯) P˜F 0

  0 (det P ) P V¯˜ P¯F (det Φ) Ad[Φ]v =   . (det P¯)PV¯˜ P˜F 0

¯ Using the same argument as before, one then sees that Φ ∈ Cl iﬀ det P = det P .

6.5.3

Lemma 6.1 [C. Franchetti] Every complex 2 × 2-matrix with real non-vanishing determinant can be written as a composition of three Hermitian matrices. proof: See appendix A.6.2, A.6.3.

Proposition 6.5 Cl is multiplicatively generated by H• ⊂ H, the subset of all elements in H with non-vanishing Lorentz pseudo-norm; in other terms, Cl is multiplicatively generated • K 0 (+) F by γ(H) ∩ D . More precisely, if 0 K‡ ∈ Cl then K ∈ U ⊗ U can be written in the form ˜ ˜ • K = P1 P¯2 P3 P¯4 ,Pi ∈ H ; 0 P (+) similarly, if P ‡ 0 ∈ Cl then P ∈ U ⊗ U can be written in the form

˜ • P = P1 P¯2 P3 ,Pi ∈ H .

So one recovers the well-known result (see also §B.3) that any element of Cl is of the form

Φ = v1 v2 . . . vn , vj ∈ H, g(vj, vj) 6= 0 ; its inverse is ← 1 Φ = v . . . v v , ν(Φ) n 2 1 where ν(Φ) ≡ g(v1, v1) g(v2, v2) . . . g(vn, vn) . √ Setting now Vi ≡ 2 vi one has ˜ det Vi = det V¯i = g(vi, vi)

˜ ˜ n ⇒ ν(Φ) = det V1 V¯2 V3 V¯4 ... = Π det(Vi) . i=1 6.5 Cliﬀord group in 2-spinor terms 81

K 0 (+) ‡ 0 P (−) Namely, if Φ = 0 K‡ ∈ Cl then ν(Φ) = det K = det K ; if Φ = P ‡ 0 ∈ Cl then ν(Φ) = det P = det P ‡ . The following shorthand may be useful:

 U K 0 (+) K ≡ Φ U , Φ = 0 K‡ ∈ Cl , Φ ≡ UUF 0 P (−) P ≡ Φ , Φ = P ‡ 0 ∈ Cl .

⇒ det Φ = ν(Φ)

2 Note that det Φ = (det Φ) , hence one cannot recover det Φ = ν(Φ) from det Φ alone. The adjoint action of any w ∈ H on H is easily checked to be the negative of the reﬂection through the hyperplane orthogonal to w ; in other terms, Ad[w]: H → H preserves the 1- dimensional subspace generated by w , while the directions orthogonal to w get inverted. In λ fact if v = v τλ then one has, for example,

0 1 2 3 Ad[γ0]v = v τ0 − v τ1 − v τ2 − v τ3 ,

0 1 2 3 Ad[γ1]v = −v τ0 + v τ1 − v τ2 − v τ3 ,

0 1 2 3 Ad[γ2]v = −v τ0 − v τ1 + v τ2 − v τ3 ,

0 1 2 3 Ad[γ3]v = −v τ0 − v τ1 − v τ2 + v τ3 .

It follows that Cl(+) is the subgroup of all elements in Cl whose adjoint action preserves the orientation of H. Moreover, the subgroup

↑ Cl ≡ {Φ ∈ Cl : ν(Φ) > 0 } = {Φ ∈ Cl : det Φ > 0 } is constituted of all elements of Cl whose adjoint action preserves the time-orientation of H. Its representation as Φ = v1 v2 . . . vn has an even number of spacelike factors and any number of timelike factors.54 Also, Cl(+)↑ ≡ Cl(+) ∩ Cl↑ is a subgroup of Cl(+) and of Cl↑ .

6.5.4 The ‘complexiﬁed’ Cliﬀord group Clc ≡ U(1)×˜ Cl is constituted of all endomorphisms of W which are either of the type

K 0 ei t , det K ∈ \{0} , t ∈ , 0 K‡ R R or of the type 0 P ei t , det P ∈ \{0} , t ∈ . P ‡ 0 R R

54 ↓ ↑ One could also consider the submanifold Cl ≡ Cl \ Cl = {Φ ∈ Cl : ν(Φ) < 0 } = {Φ ∈ Cl : det Φ < 0 } , which is not a subgroup of Cl and is constituted of all Φ ∈ Cl whose adjoint action on H changes time orientation. 82 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

Thus Clc is an 8-dimensional real manifold, but not a complex manifold (as it can also be seen from its Lie algebra, written below).[CTRL] Note that the non-zero entries in the matrix of an even element in Clc can be arbitrary automorphisms of U and U F, respectively, but their mutual relation is not given by ‡ ; an analogous observation is true about odd elements. In fact, Clc is not a subset of D. On the other hand, arbitrary elements in D(+) and D(−) can be written respectively in the form K0 0 ei t K 0 = , det K0 ∈ \{0} , det K ∈ \{0} , 0 K0‡ 0 e−i t K‡ C R

0 P 0 0 ei t P = , det P 0 ∈ \{0} , det P ∈ \{0} . P 0‡ 0 e−i t P ‡ 0 C R Also note that H is stable under the adjoint action of Clc, while it is not stable under the adjoint action of D•(+) ≡ D• ∩ D(+) or D•(−) ≡ D• ∩ D(−). Clearly one can also consider the complexiﬁed subgroups Cl(+)↑ c, Cl(+) c, Cl↑ c ⊂ Clc .

6.5.5 The unit element of Cl is 11 ∈ D(+) ⊂ D. Thus the Lie algebra of Cl is a 7-dimensional vector subspace (+) 2 4 2 4 LCl ⊂ D = R ⊕ ∧ H ⊕ ∧ H ≡ R 11 ⊕ γˆ(∧ H) ⊕ γˆ(∧ H) . Now observe that ∧4H is not contained in LCl , since i t i t 11U 0 e 11U 0 exp(t γη) = exp = −i t 0 −i t 11U F 0 e 11U F

i t F −i t is not in Cl because the two component endomorphsims e 11U ∈ U ⊗ U and e 11U F ∈ U F ⊗ U have non-real determinant. Hence, just by a dimension argument, one ﬁnds

2 LCl = R ⊕ ∧ H . Furthermore one has obviously

c 2 2 LCl = R ⊕ i R ⊕ ∧ H = C ⊕ ∧ H . In particular, let X λµ 2 X = X τλ ∧ τµ ∈ ∧ H . λ<µ K 0 Then one ﬁnds exp(X) = with 0 K‡

 03 12 01 13 02 23  1 α cosh α − (X + i X ) sinh α (−X +X +i X −i X ) sinh α K =   α (−X01−X13−i X02−i X23) sinh α α cosh α + (X03+i X12) sinh α

α ≡ [(X01 + X23)2 + (X02 − X13)2 + (X03 + X12)2 ]1/2 . Note that for this particular K one has det K = 1 . Moreover t exp(t, X) ≡ exp t 11W +γ ˆ(X) = e exp(X) . 6.6 4-spinor groups in 2-spinor terms 83

6.5.6 Since 11 ∈ Cl(+)↑ one has

(+)↑ (+) ↑ 2 LCl = LCl = LCl = LCl = R ⊕ ∧ H ,

(+)↑ c (+) c ↑ c c 2 LCl = LCl = LCl = LCl = C ⊕ ∧ H .

6.6 4-spinor groups in 2-spinor terms 6.6.1

If Φ ∈ Cl and a ∈ R \{0} then Ad[a Φ] = Ad[Φ] : H → H. It is then natural to consider the subgroup Pin ≡ {Φ ∈ Cl : det Φ = ±1} ≡ {Φ ∈ Cl : ν(Φ) = ±1} , which is multiplicatively generated by all elements in H whose Lorentz pseudo-norm is ±1 . It has the subgroups

(+) (+) (+) Spin ≡ Pin ≡ Pin ∩ Cl = {Φ ∈ Cl : det Φ = ±1} ,

↑ ↑ Pin ≡ Pin ∩ Cl = {Φ ∈ Cl : det Φ = 1} ,

↑ ↑ (+) Spin ≡ Spin ∩ Cl = {Φ ∈ Cl : det Φ = 1} . Then, recalling the previous discussion about LCl , one immediately sees that LPin = LSpin = LPin↑ = LSpin↑ = ∧2H . Furthermore one has the complexiﬁed groups Spinc , Pin↑ c , Spin↑ c , which share the same Lie algebra

c ↑ c ↑ c 2 LSpin = LPin = LSpin = i R ⊕ ∧ H .

6.6.2 Now remember (§6.1) Slc ≡ Slc(U) = { K ∈ Aut(U): | det K| = 1 } , LSlc = { A ∈ End(U): < Tr A = 0 } ,

Sl ≡ Sl(U) = { K ∈ Aut(U) : det K = 1 } , LSl = { A ∈ End(U) : Tr A = 0 } , The automorphisms of U which have positive real determinant constitute the group

+ R × Sl = (R \{0})×˜ Sl . c 55 Namely, if Φ ∈ Cl then Φ ∈ R × Sl , or

(+)↑ K 0 + Cl = 0 K‡ ∈ End W : K ∈ R × Sl ,

↑ K 0 Spin = 0 K‡ ∈ End W : K ∈ Sl .

55 (−) A similar statement can be made for Cl , viewing Φ ≡ P ∈ U ⊗ U as a real automorphism (complex anti-automorphism) of U through any isomorphism ε[ : U → U F. Note moreover that also the other groups besides Cl(+) and Spin↑ can be characterized in similar ways, but not as simply. 84 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

In other terms, if one deﬁnes

KP Π : End W → End U : QJ 7→ K then the restriction (+)↑ + Π: Cl → R × Sl is a group isomorphism, whose inverse is

← K 0 Π: K 7→ 0 K‡ . Similarly, the restriction Π: Spin↑ → Sl is a group isomorphism, whose inverse has the same expression as above. It was previously shown (§6.4) that

2 A 0 γˆ(∧ H) = 0 A‡ ∈ End W : Tr A = 0 ,

2 A 0 γˆ(R ⊕ ∧ H) = 0 A‡ ∈ End W : = Tr A = 0 ; moreover End U can be decomposed into the direct sum of the subspace of all traceless endomorphism, which is just LSl , and the subspace C 11generated by the identity. Then one has the Lie algebra isomorphisms

(+) 2 Π ◦ γˆ : LCl = LCl = R ⊕ ∧ H −→ (R 11) ⊕ LSl ,

Π ◦ γˆ : LPin = LSpin↑ = ∧2H −→ LSl .

The case of complexiﬁed groups is only slightly more involved. The restriction

(+) c • Π: Cl → C ×˜ Sl = Aut(U) is a group isomorphism, whose inverse is56

0 ! ← K 0 Π: K0 7→ , 0 e2iθ K0‡ where 0 2iθ 1 0 det K = e det K, that is θ = 2 arg det K . Similarly, Π: Spinc → Slc is a group isomorphism, whose inverse has the same expression as above. The corresponding isomorphisms of Lie algebras are simply

c (+) c 2 Π ◦ γˆ : LCl = LCl = C ⊕ ∧ H −→ (C 11) ⊕ LSl ,

↑ 2 Π ◦ γˆ : LPin = LSpin = i R ⊕ ∧ H −→ (i R 11) ⊕ LSl . 56In fact if eiθ K 0 = K0 0 then det K0 = det K00 = e2iθ det K ⇒ 2 θ = arg det K0 ⇒ K0‡ = e−iθ K‡ 0 K‡ 0 K00 ⇒ K00 = eiθ K‡ = e2iθ K0‡. 6.6 4-spinor groups in 2-spinor terms 85

6.6.3 Proposition 6.6 Let √ Φ = K 0 ∈ Cl(+) , v ∈ H , γ(v) = V 0 ≡ 2 v √ 0 . 0 K‡ 0 V ‡ 0 2 v‡ Then ¯ ! 1 0 [K⊗K](V ) Ad[Φ]γ(v) = . det K [K⊗K¯ ](V )‡ 0 proof: Remembering the previous results (§6.5) about the the 2-spinor expression of the adjoint action, and using det Φ = (det K)2 = (det K¯ )2 , one writes

¯ F! 1 0 (det K) KV K Ad[Φ]γ(v) = = det Φ (det K) K¯˜ V¯˜ K˜ F 0

¯ F! ¯ F! 1 0 KV K 1 0 KV K = = , ‡ det K K¯˜ V¯˜ K˜ F 0 det K KV K¯ F 0 where as usual V¯˜ ≡ V ‡ and the like. But

¯ F AA˙ A BB˙ ¯ F A˙ A BB˙ ¯ A˙ ¯ AA˙ BB˙ (KV K ) = K B V (K )B˙ = K B V K B˙ = (K ⊗ K) BB˙V .

Corollary 6.2 Let √ Φ = K 0 ∈ Spin , v ∈ H , γ(v) = V 0 ≡ 2 v √ 0 . 0 K‡ 0 V ‡ 0 2 v‡ Then 0 [K⊗K¯ ](V )! Ad[Φ]γ(v) = ± , [K⊗K¯ ](V )‡ 0 where the + sign holds iﬀ Φ ∈ Spin↑.

Remark. Since Ad[eiθ Φ] = Ad[Φ] , if Φ ∈ Cl(+) c one gets for Ad[eiθ Φ]γ(v) exactly the same expression as above, with the plus sign holding for the time-orientation preserving subgroup. In particular, this is true for elements in Spinc and Spin↑ c.

6.6.4 Now remember (§6.2) that the group

Aut(U) ∨¯ Aut(U) ≡ {K ⊗ K¯ : K ∈ Aut(U) } is constituted of automorphisms of U ⊗ U which preserve the splitting U ⊗ U = H ⊕i H and the causal structure of H. Its subgroup

Slc(U) ∨¯ Slc(U) = Sl(U) ∨¯ Sl(U) ≡ {K ⊗ K¯ : K ∈ Sl(U) } 86 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

↑ ↑ coincides with Lor+(H) . Thus the group isomorphism Sl → Spin determines the epimor- ↑ ↑ ↑ c ↑ phisms Spin Lor+ and Spin Lor+, namely one gets the exact group morphism sequences

↑ ↑ {11} −→ Z2 −→ Spin −→ Lor+ −→ {11} ,

↑ c ↑ {11} −→ U(1) −→ Spin −→ Lor+ −→ {11} .

Remark. It has already been observed (§6.2) that others elements of Aut(U ⊗ U) preserving the splitting U ⊗ U = H ⊕ i H and the causal structure of H can be written in the form

V 7→ hF ⊗ F,V¯ Fi ,F ∈ U ⊗ U F ,

AB˙ C˙ that is (set F ≡ P[ = P ε¯B˙C˙ζA ⊗ ¯z )

V 7→ hP ⊗ P,¯ V˜ Fi ≡ h(P ⊗ P¯) ◦ g[,V Fi ,P ∈ U ⊗ U , det P 6= 0 .

This is related to the adjoint action of Cl(−)c in the same way as the action of Aut(U) ∨¯ Aut(U) is related to the adjoint action of Cl(+)c.

6.7 Bundle structure of W 6.7.1 The results of §3.5 show that the restrictions

p# : W \{0} −→ J ,

∗ p : W \{0} −→ J ,

∗ 57 are surjective (remember√ that J and J do not contain the zero element). Since the Lorentz “length” of p(u, χ) is 2 |hχ,¯ ui| one sees that the subset of all elements in W which project ∗ onto N (and N) is the 6-dimensional real submanifold

∗ W 0 ≡ [p#]−1(N) = p−1(N) = (u, χ) ∈ W \{0} : hχ,¯ ui = 0 ⊂ W .

∗ The subset of all elements in W which project onto K (and K) is the open submanifold

∗ W 8 ≡ [p#]−1(K) = p−1(K) = (u, χ) ∈ W : hχ,¯ ui= 6 0 ⊂ W , and one has W \{0} = W 0 ∪ W 8 . Finally, one considers the subsets W +, W − ⊂ W 8 deﬁned to be

± ± W ≡ (u, χ) ∈ W : hχ,¯ ui ∈ R . Recalling condition ii of proposition 3.1 and using the shorthands µ ≡ |hχ,¯ ui| , θ ≡ arg hχ,¯ ui , one has γ[p#ψ]ψ = µ (e−iθu, eiθ χ) ,

57Note that obviously p−1(0) = [p#]−1(0) = {0} ⊂ W , namely the zero element in W is the unique 4-spinor which is sent to the zero (co)vector by p or p# . 6.7 Bundle structure of W 87 which holds for every ψ ≡ (u, χ) ∈ W (if ψ ∈ W 0 then µ = 0). In particular W ± = ψ ≡ (u, χ) ∈ W \{0} : γ[p#ψ]ψ = ±µ ψ , µ ≡ |hχ,¯ ui| . Remark. W 8 is an open real submanifold of W , thus it is a real 8-dimensional manifold; W − ∪ W + is the 7-dimensional real submanifold of W 8 characterized by the condition =hχ,¯ ui = 0 .

6.7.2 Remember (§B.6, 6.5) that

↑ ↑ K 0 Spin ≡ Spin (U) = 0 K‡ : K ∈ Sl(U) is the subgroup of End W preserving (γ, k, g, η, ε) as well as time-orientation. It may be useful to review all those properties here, in terms of two-spinors.

↑ F K 0 • Obviously, Spin preserves the splitting W = U ⊕ U . If Φ = ‡ , K ∈ Sl(U) , then ← 0 K K˜ = K , so for ψ ≡ (u, χ), ψ0 ≡ (u0, χ0) ∈ W one gets ← ← ← ← k(Φψ, Φψ0) = k(K u, χ K¯ ), (K u0, χ0 K¯ ) = hχ¯ K, K u0i + hχ0 K,¯ K¯ u¯i =

= hχ,¯ u0i + hχ0, u¯i = k(ψ, ψ0) . • Since K ⊗ K¯ : U ⊗ U → U ⊗ U sends Hermitian tensors to Hermitian tensors and anti- Hermitian tensors to anti-Hermitian tensors, it preserves the splitting U ⊗ U = H ⊕ i H. Also, remember that K ⊗ K¯ = Ad[Φ] . ¯ ↑ • K ⊗ K = Ad[Φ] ∈ Lor+(H) , the subgroup of the Lorentz group which preserves orientation and time-orientation. • Φ preserves the Dirac map γ . In fact if y ∈ H then √ 0 2 y γ[y] = √ , y‡ ≡ y¯˜ =y ˜F , 2 y‡ 0 √ 0 2 [K ⊗ K¯ ] y Ad[Φ]γ[y] = √ = γ[K ⊗ K¯ ] y . 2 ([K ⊗ K¯ ] y)‡ 0

2 F K 0 ↑ • If K ∈ Sl then K preserves any simplectic form ε ∈ ∧ U . Hence Φ ≡ 0 K‡ ∈ Spin preserves the corresponding simplectic form (ε, ε¯#) ∈ ∧2W F and charge conjugation.

Remark. On the other hand, Spin↑c preserves (ε, ε¯#) only up to a phase factor. Also note ↑c iθ K 0 K 0 that elements in Spin are of the type e 0 K‡ with K ∈ Sl , thus not of the type 0 K‡ with K ∈ Slc .

6.7.3 Consider the subset W˜ 8 ≡ {(u, v): ε(u, v) 6= 0} ⊂ U × U , and note that when a normalized symplectic form ε ∈ ∧2U F is fixed, W˜ 8 can be identified ∗ ∗ with W 8 via the correspondencev ¯[ ↔ χ . W˜ 8 is a fibred set over K and K ; for each p ∈ K, the fibre of W˜ 8 over p (which is the same as the fibre of W˜ 8 over p#) is the subset

−1 ˜ 8 ˜ 8 √1 # Wp ≡ ˜p (p) = (u, v) ∈ W : 2 (u ⊗ u¯ + v ⊗ v¯) = p . 88 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

∗ ˜ # ˜ Proposition 6.7 The projections ˜p: W 8 K and ˜p : W 8 K are trivializable principal bundles with structure group U(2) .

0 0 0 0 proof: Let p = ˜p(u, v) = ˜p(u , v ) . From proposition 3.1 one then sees√ that (u, v) and (u , v ) are orthonormal bases of U relatively to the Hermitian metric h ≡ 2p/µ ¯ . Then there exists a unique transformation K ∈ U(U, h) such that

u0 = K(u) , v0 = K(v); ˜ 8 hence, Wp is a group-aﬃne space, with derived group U(2) . Let now (ζA) be an ε-normalized basis of U and (τλ) the associated Pauli frame. For each ∗ λ 58 # p = pλ t ∈ K let Lp be the boost such that Lpτ0 = p /µ (µ ≡ |p|); up to sign there is a unique Bp ∈ Sl(U) such that Lp = Bp ⊗ B¯p , and a consistent smooth way of choosing one such B for each p can be ﬁxed (see the remark at the end of §6.2.5). It turns out that B ζ p √ p A is a basis of U, orthonormal relatively to h ≡ 2p ¯[/µ , since 1 L τ = √1 (B ζ ⊗ B¯ ζ¯ ) + (B ζ ⊗ B¯ ζ¯ ) = p# = τ = ˜p#(B ζ ,B ζ ) . p 0 2 p 1 p 1 p 2 p 2 µ p 1 p 2 √ Taking instead the basis µ BpζA one gets

# √ √ # ˜p ( µ Bpζ1 , µ Bpζ2) = p .

∗ In this way one gets a trivialization W 8 → K × U(2) . # In the case of p the argument is almost identical.

Lemma 6.2 Let (u, v) , (u0, v0) be any two bases of U (not necessarily in the same ﬁbre of ˜p) and K ∈ Aut U the unique automorphism of U such that

K u = u0 , K v = v0 .

Then 1 K = ε(v, u0) u ⊗ v[ − ε(v, v0) u ⊗ u[ − ε(u, u0) v ⊗ v[ + ε(u, v0) v ⊗ u[ . [ε(u, v)]2 proof: The dual basis of (u, v) ⊂ U is (α, β) ⊂ U F where v[ u[ α = , β = . ε(v, u) ε(u, v)

a b Let c d be the matrix of K relatively to the basis (u, v) , that is u0 = K u = a u + c v , v0 = K v = b u + d v .

One has ε(v, u0) ε(v, v0) a = hα, u0i = , b = hα, v0i = , ε(v, u) ε(v, u)

ε(u, u0) ε(u, v0) c = hβ, u0i = , d = hβ, v0i = . ε(u, v) ε(u, v)

58 ↑ # This is the unique Lorentz transformation in Lor+(H) sending τ0 to p /|p| , whose restriction to the # 2-plane orthogonal to τ0 and p is the identity of that plane. 6.7 Bundle structure of W 89

Replacing the above expressions for α, β, a, b, c, d in

K = a u ⊗ α + b u ⊗ β + c v ⊗ α + d v ⊗ β one gets the stated expression for K. It is clear that that expression is invariant relatively to i θ the transformation ε → e ε .

Proposition 6.8 Let ψ ≡ (u, χ), ψ0 ≡ (u0, χ0) be elements of W 8, not necessarily in the same ﬁbre of p, and let K ∈ Aut U be the unique automorphism of U such that

K u = u , K χ¯# =χ ¯0# .

Then

1 K = hχ,¯ u0i u ⊗ χ¯ − ε#(¯χ, χ¯0) u ⊗ u[ + ε(u, u0)χ ¯# ⊗ χ¯ + hχ¯0, ui χ¯# ⊗ u[ . hχ,¯ ui2

Hence, it turns out that K is independent of the particular normalized symplectic form ε chosen. Moreover, one has χ0 = K¯˜ χ . proof: Since hχ,¯ ui= 6 0 (because ψ ≡ (u, χ) ∈ W 8), u and v ≡ −χ¯# are linearly independent and then constitute a basis of U. Moreover one has

v[ =χ ¯ ,

ε(u, v) = −ε(v, u) = −hv[, ui = −hχ,¯ ui ,

ε(v, u0) = hv[, ui = hχ,¯ u0i ,

ε(v, v0) = hv[, v0i = hχ,¯ −χ¯0#i = −ε#(¯χ0, χ¯) = ε#(¯χ, χ¯0) ,

ε(u, v0) = ε(u, −χ¯0#) = ε(¯χ0#, u) = −hχ¯0, ui .

Using the above as replacements in the expression of K from lemma 6.2, one gets the stated expression, which turns out to be invariant relatively to the transformation ε 7→ ei θε . Finally, one has

0 0[ [ # [ # B A B DC A χ¯ = v = (K v) = −(K χ¯ ) = −εBA (K χ¯ ) z = −εBA K C χ¯D ε z =

B CD ˜ D ˜ 0 ¯˜ = (εBA K C ε )χ ¯D = K A χ¯D = K χ¯ ⇒ χ = K χ .

Remark. Conversely, given ψ, ψ0 ∈ W 8, the conditions u0 = Ku and χ0 = K‡χ determine K uniquely. In fact, reversing the last calculation of the above proof, from χ0 = K‡χ one gets χ¯0 = −(K χ¯#)[ ⇒ χ¯0# = K χ¯# . 90 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

6.7.4

∗ ∗ # Consider now the fibred sets p : W 8 K and p : W 8 K. For each p ∈ K, the fibre of W 8 over p (which is the same as the fibre of W 8 over p#) is the subset

8 −1 8 √1 [ [ Wp ≡ p (p) = (u, χ) ∈ W : 2 (u ⊗ u¯ +χ ¯ ⊗ χ) = p .

When a normalized ε ∈ ∧2U F is given, one has a real vector bundle isomorphism W 8 ↔ W˜ 8. ∗ Through this correspondence, W 8 K and W 8 K turn out to be trivializable principal 0 8 bundles with structure group U(2) . If ψ, ψ ∈ Wp , let

a ¯b (K) = c ∈ U(2) , a, b, c ∈ : |a|2 + |b|2 = |c|2 = 1 , −b a¯ C be the matrix59 of K ∈ Aut U sending ψ to ψ0 (according to proposition 6.8) relatively to the basis (u, v) . Then

 0  0 # u = c (a u − b v) , u = c (a u + b χ¯ ) , ⇐⇒ v0 = c (¯b u +a ¯ v) , χ0 =c ¯(a χ + b u¯[) .

If you take a diﬀerent normalized symplectic form, sayε ˇ ≡ ei θε , then (lemma 6.2) K remains the same, while the corresponding element (K) ∈ U(2) changes, as it is the matrix of K in the “new” basis (u, vˇ) ,v ˇ ≡ −εˇ#(¯χ) . Actually, from

εˇ# = e−i θε# ⇒ vˇ = −εˇ#(¯χ) = e−i θ[−ε#(¯χ)] = e−i θv

⇒ v = ei θ vˇ , v0 = ei θ vˇ0 , one gets

u0 = c (a u − b ei θ vˇ) ,

ei θ vˇ0 = c (¯b u +a ¯ ei θ vˇ) ⇒ vˇ0 = c (¯b e−i θ u +a ¯ vˇ) .

Thus the matrix of K in the basis (u, vˇ) is ! a ˇ¯b (Kˇ ) = c ∈ U(2) , ˇb ≡ ei θ b . −ˇb a¯

Also note that withε ˇ and the new matrix the form of the relation between ψ and ψ0 remains unchanged, since one has60  0 # u = c (a u + ˇb χ¯ˇ ) , χ0 =c ¯(a χ + b u¯ˇ[) .

59Note that for given K the numbers a , b and c determining its matrix in a given basis of U are unique up to an overall sign change. In comparison with the general form of a U(2)-matrix given in §??, here I set c ≡ eiφ and exchanged b with ¯b for convenience. 60In fact ˇb εˇ# = b ei θ e−i θε# = b ε# , ˇb ε¯ˇ[ = b ei θ e−i θ ε¯[ = b ε¯[ . 6.7 Bundle structure of W 91

6.7.5

If (ζ1 , ζ2) is a normalized 2-spinor basis such that p = µ τ0 , then all p-generating couples can be expressed (see §??) as √ µ i φ i α −i β u = √ e cos γ e ζ1 − sin γ e ζ2 , 4 2 √ µ i φ i β −i α v = √ e sin γ e ζ1 + cos γ e ζ2 , 4 2 √ µ χ ≡ ε¯[(¯v) = √ e−i φ − cos γ ei α ¯z1 + sin γ e−i β ¯z2 , 4 2 with α, β, γ, φ ∈ R (of course the range of the parameters α, β, γ, φ must be restricted in order to have one-to-one correspondence between p-generating couples and parameter values). Evaluating ε(u, v) for a p generating couple (u, v) is also interesting. One ﬁnds µ ε(u, v) = u1 v2 − v2 u1 = √ e2 i φ 2 ε(v, u) ⇒ eiθ ≡ = −e2 i φ |ε(v, u)|

⇒ t = 2 φ + π mod 2π .

6.7.6 The above U(2)-action on W 8 does not preserve W ± ⊂ W 8. In fact: 0 + 0 − 0 0 0 Proposition 6.9 Let ψ, ψ ∈ Wp (resp. ψ, ψ ∈ Wp ), ψ ≡ (u, χ) , ψ ≡ (u , χ ) ; let K be ‡ 0 the unique√ automorphism of U such that Ku = u , K χ = χ . Then K ∈ SU(U, h) , where h ≡ 2p ¯[/µ . proof: From u0 = c (a u + b χ¯#) and χ0 =c ¯(a χ + b u¯[) one hasχ ¯0 = c (¯a χ¯ + ¯b u[) and hχ¯0, u0i = c2 |a|2 hχ,¯ ui + |b|2 hu[, χ¯#i =

= c2 (|a|2 + |b|2) hχ,¯ ui = c2 hχ,¯ ui where [ # A CB A C hu , χ¯ i = εAB u χ¯C ε = u χ¯C δA = hχ,¯ ui 0 0 was used. If hχ,¯ ui is real, then hχ¯ , u i is real and has the same sign iﬀ c = ±1 .

+ − Corollary 6.3 W K (resp. W K) is a trivializable principal bundle with structure group SU(2) . proof: As in the proof of proposition 6.7 fix an ε-normalized basis (ζA) of U and note that one can make a smooth choice K → Sl(U): p 7→ Bp in such a way that Lpτ0 = p/µ , where Lp ≡ Bp ⊗ B¯p . Because Bp is unimodular one has ε(Bpζ1 ,Bpζ2) = 1 , so that √ √ ˜ + ( µ Bpζ1 , µ Bpζ2) ∈ Wp . ˜ + In other terms one has a smooth choice of an “origin” element for each fibre of W K. Our statement then follows since these fibres, by virtue o proposition 6.9, are group affine with derived group SU(2) . 92 6 2-SPINOR GROUPS AND 4-SPINOR GROUPS

6.7.7 The ﬁbre of p : W 0 → N over k ∈ N is the subset

0 −1 √1 [ [ Wk ≡ p (k) = (u, χ) ∈ W : 2 (u ⊗ u¯ +χ ¯ ⊗ χ) = k .

0 [ There is an (up to sign, unique) element (u, χ) ∈ Wk such that χ =u ¯ , and any other element 0 0 0 2 2 (u , χ ) ∈ Wk can be written as (a u, b χ) , with a, b ∈ C such that |a| +|b| = 2 . Thus one sees that p : W 0 → N is a [trivializable?] bundle, whose ﬁbres are diﬀeomorphic to a 3-sphere. 93

A Spinors: further details and calculations

A.1 Complex spaces as real spaces

An n-dimensional vector space U over C can also be seen as a 2n-dimensional vector space over R, via the inclusion R ,→ C. Conversely, a complex vector space can be deﬁned as a real vector space U endowed with a linear automorphism ι : U → U such that ι2 = −11. Then it can be easily shown that the real dimension of U must be even, and that U can be seen naturally as a vector space over C of halved dimension via the relation ιu = i u , u ∈ U . A complex basis (bA) determines the real basis (wj), j = 1,..., 2n , given by

w2A−1 := bA , w2A = i bA ; its dual basis of U ∗, indicated by (wj), is given by:

2A−1 1 A ¯A 2A 1 A ¯A w = 2 (b + b ) , w = 2i (b − b ) .

The n-uple (bA): U → C constitutes a coordinate chart over the complex manifold U . Thus one may write61

A A z := b : U → C , xA :=

dzA = dxA + i dyA ; d¯zA = b¯A = dxA − i dyA .

Moreover one has ∂ ∂ = b , = i b . ∂xA A ∂yA A Let V be a second complex vector space, of complex dimension p. One has two tensor products, U ⊗R V and U ⊗C V ; when no confusion arises, the latter is indicated simply as U ⊗ V . The ﬁrst product is a real vector space of (real) dimension 4 n p , the second is a complex vector space of complex dimension n p . If (cB ) 1 ≤ B ≤ p , is a basis of V , then the induced bases of the above tensor products are, respectively, bA ⊗ cB , (i bA) ⊗ cB , bA ⊗ (i cB ) , (i bA) ⊗ (i cB ) ,

(bA ⊗ cB ) , while the induced basis of the (2 n p)-dimensional real space underlying U ⊗ V is

(bA ⊗ cB , i (bA ⊗ cB )) .

61This notation is redundant here, but it is useful for extending the discussion to the general case of complex manifolds. 94 A SPINORS: FURTHER DETAILS AND CALCULATIONS

There is a natural real projection

τ : U ⊗ V U ⊗ V , R C τ(i u) ⊗ v = τu ⊗ (i v) := i (u ⊗ v) .

Namely, τ essentially consists in ‘extracting’ the imaginary unit from the factors, or in the identiﬁcation (i u) ⊗ v ≡ u ⊗ (i v) ≡ i (u ⊗ v).

The real vector space LR(U, V ) of all R-linear maps U → V has a canonical splitting as the direct sum

LR(U, V ) = LC(U, V ) ⊕ AC(U, V ) , of the subspaces of all C-linear and C-antilinear maps (seen as real spaces). This splitting is given by

ϑ 7→ ϑL + ϑA ,

1 ◦ ◦ ϑL := 2 (ϑ − ιV ϑ ιU ) ,

1 ◦ ◦ ϑA := 2 (ϑ + ιV ϑ ιU ) .

∗ In particular one has, via the natural inclusion U ,→ LR(U, C) , the two maps

∗ F ∗ F φL : U → U , φA : U → U , which turn out to be real isomorphisms. ∼ ∗ Furthermore, considering the natural isomorphism LR(U, V ) = V ⊗R U , the projections LR(U, V ) → LC(U, V ) and LR(U, V ) → AC(U, V ) can be seen, respectively, as given by

∗ F τ ◦ (11V ⊗ φL): V ⊗ U → V ⊗ U , R C

∗ F τ ◦ (11V ⊗ φA): V ⊗ U → V ⊗ U . R C

F F F Finally observe that, for any f ∈ LC(U, V ) , the transpose map f ∈ LC(V , U ) and the F ¯F F F antitranspose map f ≡ f ∈ LC(V , U ) are exactly the restrictions of the real transpose morphisms f ∗, f¯∗ : V ∗ → U ∗ .

A.2 Real volume forms on spinor spaces A.2.1 Lemma A.1 Let A and B be (either real or complex) vector spaces. Then one has the natural isomorphism

p M ∧p(A ⊕ B) =∼ ∧p−qA ⊗ ∧qB= q=0

= ∧pA ⊕ ∧p−1A ⊗ B⊕∧p−2A ⊗ ∧2B⊕ · · · ⊕ A ⊗ ∧p−1B⊕∧pB .

In particular, if dim A = m and dim B = n , then

∧m+n(A ⊕ B) =∼ ∧mA ⊗ ∧nB . A.2 Real volume forms on spinor spaces 95

r s r+s proof: For r, s ∈ N ∪ {0} consider the linear map ∧ A ⊗ ∧ B → ∧ (A ⊕ B) which on decomposable elements is given by

(a1 ∧ a2 ∧ ... ∧ ar) ⊗ (b1 ∧ b2 ∧ ... ∧ bs) 7→ a1 ∧ a2 ∧ ... ∧ ar ∧ b1 ∧ b2 ∧ ... ∧ bs .

This map is clearly injective; in fact, if the right-hand side above is zero then either ai ∝ aj for some 1 ≤ i, j ≤ r or bi ∝ bj for some 1 ≤ i, j ≤ s ; namely, either a1 ∧ ... ∧ ar = 0 or 0 0 b1 ∧ ... ∧ bs = 0 , which means that the left-hand side is zero. Moreover, if (r , s ) 6= (r, s) then the above said map sends ∧rA ⊗ ∧sB and ∧r0 A ⊗ ∧s0 B to subspaces of ∧(A ⊕ B) having intersection {0} . Thus one gets a linear injection p M ∧p−qA ⊗ ∧qB,→ ∧p(A ⊕ B) , q=0 which turns out to be an isomorphism because the dimensions of the two spaces are the same: p m + n X m n = . p p − q q q=0

Moreover we observe that an analogous of the above result also holds for the symmetrized tensor product, namely after replacing ∧ by ∨ everywhere. The only diﬀerence regards the counting of the dimensions: we have m + p − 1 dim ∨pA = , p and one may check that actually p m + n + p − 1 X m + p − q − 1 n + q − 1 = . p p − q q q=0

A.2.2

Consider the R-linear map U × U × U × U → C :(a, b, c, d) 7→ ε(a, b)ε ¯(¯c, d¯) . 4 ∗ By antisymmetrization one obtains a natural real 4-form ϑ ∈ ∧ (UR) , namely 1 X ϑ(s , s , s , s ) := sign(p) ε(s , s )ε ¯(¯s , s¯ ) , 1 2 3 4 4! p1 p2 p3 p4 p∈P4 where P4 is the group of all permutations (§??) of the set {1, 2, 3, 4} . The fact that ϑ is real-valued can be straightforwardly checked: 24 ϑ(a, b, c, d) = ε(a, b)ε ¯(¯c, d¯) − ε(a, b)ε ¯(d,¯ c¯) − ε(a, c)ε ¯(¯b, d¯) + ε(a, c)ε ¯(d,¯ ¯b)+

+ ε(a, d)ε ¯(¯b, c¯) − ε(a, d)ε ¯(¯c, ¯b) − ε(b, a)ε ¯(¯c, d¯) + ε(b, a)ε ¯(d,¯ c¯)+

+ ε(b, c)ε ¯(¯a, d¯) − ε(b, c)ε ¯(d,¯ a¯) − ε(b, d)ε ¯(¯a, c¯) + ε(b, d)ε ¯(¯c, a¯)+

+ ε(c, a)ε ¯(¯b, d¯) − ε(c, a)ε ¯(d,¯ ¯b) − ε(c, b)ε ¯(¯a, d¯) + ε(c, b)ε ¯(d,¯ a¯)+

+ ε(c, d)ε ¯(¯a, ¯b) − ε(c, d)ε ¯(¯b, a¯) − ε(d, a)ε ¯(¯b, c¯) + ε(d, a)ε ¯(¯c, ¯b)+

+ ε(d, b)ε ¯(¯a, c¯) − ε(d, b)ε ¯(¯c, a¯) − ε(d, c)ε ¯(¯a, ¯b) + ε(d, c)ε ¯(¯b, a¯) = 96 A SPINORS: FURTHER DETAILS AND CALCULATIONS

= 4 ε(a, b)ε ¯(¯c, d¯) + ε(c, d)¯ε(¯a, ¯b) − 4 ε(a, c)ε ¯(¯b, d¯) + ε(b, d)ε ¯(¯a, c¯)+

+ 4 ε(a, d)ε ¯(¯b, c¯) + ε(b, c)ε ¯(¯a, d¯) .

1 1 2 1 2 Recalling that ε(a, b) = 2 (a b − b a ) one has 24 ϑ(a, b, c, d) = a1 b2 c¯1 d¯2 − a1 b2 d¯1 c¯2 − a1 c2 ¯b1 d¯2 + a1 c2 d¯1 ¯b2+

+ a1 d2 ¯b1 c¯2 − a1 d2 c¯1 ¯b2 − b1 a2 c¯1 d¯2 + b1 a2 d¯1 c¯2+

+ b1 c2 a¯1 d¯2 − b1 c2 d¯1 a¯2 − b1 d2 a¯1 c¯2 + b1 d2 c¯1 a¯2+

+ c1 a2 ¯b1 d¯2 − c1 a2 d¯1 ¯b2 − c1 b2 a¯1 d¯2 + c1 b2 d¯1 a¯2+

+ c1 d2 a¯1 ¯b2 − c1 d2 ¯b1 a¯2 − d1 a2 ¯b1 c¯2 + d1 a2 c¯1 ¯b2+

+ d1 b2 a¯1 c¯2 − d1 b2 c¯1 a¯2 − d1 c2 a¯1 ¯b2 + d1 c2 ¯b1 a¯2 . Then, by abuse of language one writes ϑ = z1 ∧ z2 ∧ ¯z1 ∧ ¯z2 = −4 b1 ∧ b2 ∧ b3 ∧ b4 , where the four covectors

1 1 1 1 2 1 2 2 3 1 1 1 4 1 2 2 b ≡ 2 (z + ¯z ) , b ≡ 2 (z + ¯z ) , b ≡ 2i (z − ¯z ) , b ≡ 2i (z − ¯z ) , ∗ constitute (§A.1) a basis of (UR) , dual to b1 , b2 , b3 , b4 ≡ ζ1 , ζ2 , i ζ1 , i ζ2 , and 1 2 3 4 1 1 1 2 2 1 1 2 2 b ∧ b ∧ b ∧ b = − 16 (z + ¯z ) ∧ (z + ¯z ) ∧ (z − ¯z ) ∧ (z − ¯z ) =

1 1 2 1 2 1 2 1 2 = − 16 z ∧ z ∧ ¯z ∧ ¯z + z ∧ ¯z ∧ (−¯z ) ∧ ¯z + + ¯z1 ∧ z2 ∧ z1 ∧ (−¯z2) + ¯z1 ∧ ¯z2 ∧ z1 ∧ z2 =

1 1 2 1 2 = − 4 z ∧ z ∧ ¯z ∧ ¯z .

Remark. In order to interpret the above passages correctly, note that if α , β ∈ LR(U, C) then α ⊗ β is a real tensor product, that is a real linear map

α ⊗ β : U ⊗ U → C ⊗ C . R On the other hand, the universal property of the tensor product

C × C @ @R C ?

C ⊗R C 62 yields a linear real map C ⊗R C → C ; by composition with this one sees α ⊗ β (as well as α ∧ β) as valued in C (actually this is a diﬀerent operation which is denoted by the same symbol by abuse of language).

62 This is obviously, in the particular case U ≡ V ≡ C , the real linear projection τ : U ⊗R V → U ⊗C V which was already seen in §A.1 to hold for any two complex vector spaces U and V . A.2 Real volume forms on spinor spaces 97

∼ F F Remark. Also remember from §A.1 that LR(U, C) = U ⊕U , 8-dimensional as a real vector space, and one has

∧4(U F ⊕ U F) = ∧4U F ⊕ ∧3U F ⊗ U F ⊕ ∧2U F ⊗ ∧2U F ⊕ U F ⊗ ∧3U F ⊕ ∧4U F . R R R R R R R

On the other hand, U F ⊕ U F is also a 4-dimensional complex space, and one has

∧4(U F ⊕ U F) =∼ ∧2U F ⊗ ∧2U F .

Now ε ⊗ ε¯ ∈ ∧2U F ⊗ ∧2U F; when antisymmetrized to be seen as an element of ∧4(U F ⊕ U F), it determines a real 4-form on UR .

A.2.3 One has ∧2W ≡ ∧2(U ⊕ U F) =∼ ∧2U ⊕ (U ⊗ U F) ⊕ ∧2U F . Actually, ∧2W can be seen as the subspace of

∧2U ⊕ (U ⊗ U F) ⊕ (U F ⊗ U) ⊕ ∧2U F ⊂

⊂ (U ⊗ U) ⊕ (U ⊗ U F) ⊕ (U F ⊗ U) ⊕ (U F ⊗ U F) = W ⊗ W constituted by all (w, t, t0, ω) such that t0 = −tF (namely only one of the “components” t ∈ U ⊗ U F and t0 ∈ U F ⊗ U can be arbitrarily chosen). Of course, this is consistent with

2 2 F 2 F 6 = dimC ∧ W = 1 + 4 + 1 = dimC ∧ U + dimC(U ⊗ U ) + dimC ∧ U .

A.2.4 One has ∧3W =∼ (∧2U ⊗ U F) ⊕ (U ⊗ ∧2U F) , which has complex dimension 2 + 2 = 4 .

A.2.5 One has ∧4W =∼ ∧2U ⊗ ∧2U F . Thus there is no natural complex 4-form on W , but rather a family of complex 4-forms related by a phase factor; actually, if ω = c ε with c ∈ C , then c¯ ω# ⊗ ω¯ = ε# ⊗ ε¯ = e−2iθ ε# ⊗ ε¯ , θ ≡ arg c . c

A.2.6 One has ∧8W =∼ ∧4U ⊗ ∧4U F . R R R Thus (the antisymmetrization of) ϑ ⊗ ϑ# yields a real volume form on W . 98 A SPINORS: FURTHER DETAILS AND CALCULATIONS

A.2.7 Since U ⊗ U = H ⊕ i H, one has

∧4(U ⊗ U) = ∧4H ⊕ (∧3H ⊗ iH) ⊕ (∧2H ⊗ ∧2iH) ⊕ (H ⊗ ∧3iH) ⊕ ∧4iH , R with the real dimension

8 70 = ( 4 ) = 1 + 4 · 4 + 6 · 6 + 4 · 4 + 1 .

Also note that the real isomorphism i : H → iH determines isomorphisms ∧pH → ∧piH for all integer p . Moreover, the real projection τ from real to complex tensor products (§A.1) determines real isomorphisms

∧4iH → ∧4H , ∧3iH → i∧3H , ∧2iH → ∧2H , only the ﬁrst of which coincides with the isomorphism determined by i .

A.3 Two-spinor exchange formulas The fact that U is two-dimensional yields some identities which are used in a number of calculations. First, observe that

ε(u, v) s + ε(v, s) u + ε(s, u) v = 0 , u, v, s ∈ U . (2) proof: If u ∝ v ∝ s this statement is trivially true; if not, then two of the three vectors are linearly independent, so the third is a linear combination of them, say s = a u + b v, a, b ∈ C. Thus

ε(u, v) s + ε(v, s) u + ε(s, u) v =

= a ε(u, v) u + b ε(u, v) v + a ε(v, u) u + b ε(v, v) u + a ε(u, u) v + b ε(v, u) v = 0 .

Alternatively, from s = a u + b v one has ε(s, u) = b ε(v, u) , and ε(s, v) = a ε(u, v). Let now63

(u, χ) , (s, σ) ∈ W , v := −χ¯# ⇐⇒ v¯[ := χ ,

⇒ ε(u, v) = −ε(v, u) = ε(¯χ#, u) = −hχ, ui ,

hχ, σ#i ≡ hσ#, χi ≡ ε¯∗(σ, χ) = −ε¯∗(χ, σ) .

Moreover, let ε(u, v) 6= 0 (so that (u, v) is a basis of U ; one has

1 1 s = hχ,¯ si u + ε(u, s)χ ¯# = ε(v, s) u + ε(s, u) v , (3) hχ,¯ ui ε(v, u) 1 1 s[ = hχ,¯ si u[ + ε(s, u)χ ¯ = ε(v, s) u[ + ε(s, u) v[ , (4) hχ,¯ ui ε(v, u)

63Remember u[ ≡ ε(u, ), χ# ≡ ε¯∗(χ, ) and so on; hence (u[)# = −u ,(χ#)[ = −χ and the like. A.4 Pauli matrices 99

1 σ¯# = hχ,¯ σ¯#i u + ε(u, σ¯#)χ ¯# = hχ,¯ ui (5) 1 1 = ε∗(¯σ, χ¯) u + hσ,¯ ui χ¯# = ε∗(¯σ, χ¯) u − hσ,¯ ui v , hχ,¯ ui ε(v, u) 1 σ¯ = −(¯σ#)[ = − ε∗(¯σ, χ¯) u[ − hσ,¯ ui χ¯ = hχ,¯ ui (6) 1 1 = ε∗(¯χ, σ¯) u[ + hσ,¯ ui χ¯ = −hσ,¯ vi u[ + hσ,¯ ui v[ , hχ,¯ ui ε(v, u) 1 1 σ = ε¯∗(χ, σ)u ¯[ + hσ, u¯i χ = −hσ, v¯i u¯[ + hσ, u¯i v¯[ . (7) hχ, u¯i ε¯(¯v, u¯) proof: It follows from identity 2, replacing v → −χ¯# and the like. Alternatively, setting s = a u + b χ¯# , one has ε(u, s) = b ε(u, χ¯#) = b hχ,¯ ui , hχ,¯ si = a hχ,¯ ui , hence the formulas for the decomposition of s (and s[) in terms of u and χ (or v). Moreover, replacing s → σ¯# one ﬁnds the decomposition formulas for σ .

A.4 Pauli matrices A.4.1 The Pauli matrices are deﬁned as

1 0 0 1 0 −i 1 0 (σλ) := 0 1 1 0 i 0 0 −1 .

These matrices are used for introducing convenient bases in various spaces. In the case of the ¯ F AA˙ space H := U ∨ U (§1.4) one considers them as matrices (σλ ) of components of elements in H. Thus, given a 2-spinor basis of U one obtains the orthonormal basis

√1 AA˙ ¯ τλ := 2 σλ ζA ⊗ ζA˙ of H. The index rising and lowering formalism yields matrices:

AA˙ 1 0 0 1 0 −i 1 0 (σλ ) = 0 1 1 0 i 0 0 −1 ,

λ AA˙ 1 0 0 −1 0 i −1 0 (σ ) = 0 1 −1 0 −i 0 0 1 ,

λ 1 0 0 1 0 i 1 0 (σ AA˙) = 0 1 1 0 −i 0 0 −1 ,

1 0 0 −1 0 −i −1 0 (σλ AA˙) = 0 1 −1 0 i 0 0 1 . Note how the third line implies that the dual basis of τλ is

λ √1 λ A A˙ τ = 2 σ AA˙z ⊗ ¯z , namely one obtains the matrices of the elements of the dual basis by index lowering and rising via ε and the Lorentz metric. 100 A SPINORS: FURTHER DETAILS AND CALCULATIONS

A.4.2 The Pauli matrices fulﬁl

k k σi σj = δij 11+ i εij σk ≡ δij σ0 + i εij σk ,

k [σi , σj] = 2 i εij σk ,

Tr(σi σj) = 2 δij ,

k hk 1 where εij := εijh δ (it may be seen as index raising through the metric 2 K , in which the basis (σi) of iSU is orthonormal; see §??).

A.4.3

One can use the Pauli matrices in order to introduce a convenient basis in the space S ⊗ SF = U ⊗ U F of all endomorphisms of the 2-spinor space. Denote this basis as

A B σˆλ = σλ B ζA ⊗ z . If U is endowed with a Hermitian metric h and ζA is h-orthonormal, then σˆλ is a basis of A AA˙ the real vector space of all h-Hermitian endomorphisms. In that case one has σλ B = σλ hA˙B . F 1 In all cases, σˆλ is an orthonormal basis of S ⊗ S relatively to the “Killing” metric 2 K, namely K(ˆσλ , σˆµ) ≡ Tr(ˆσλ ◦ σˆµ) = 2 δλµ . Next we set 1 ◦ 1 ◦ ◦ Σλµν := 2 K σˆλ σˆµ , σˆν = 2 Tr σˆλ σˆµ σˆν , Then we find Σλµν = ηλµν + i ε0λµν , where 0 0 0 0 0 0 ηλµν := δλµδν + δνλδµ + δµνδλ − 2δλδµδν is a totally symmetric symbol. Note that under exchange of two nearby indices Σλµρ transforms into its complex conjugate. By convention, indices in the symbols ηλµν , ελµνρ and Σλµν 1 λµ are raised and lowered via via 2 K , namely via the coefficients δ and δλµ . Now from the definition of Σλµν we immediately get

ρ ◦ σˆλ σˆµ = Σλµ σˆρ , Some further useful formulas:

λµρ ρ ρ 0 ηλµν η = 2(δν − δ0 δν) , 0λµρ ρ ρ 0 ε0λµν ε = 2(δν + δ0 δν) , ¯ λµρ ρ Σλµν Σ = 4δν .

Through the coeﬃcients Σλµρ, with the above said conventions, one can also express contractions between σ-objects of diﬀerent kinds. Actually one obtains

AA˙ µ µν A σλ σ BA˙ = Σλ σν B , AA˙ µ ¯ µν ¯ A˙ σλ σ AB˙ = Σλ σˆν B˙ . A.5 Spinors as endomorphisms 101

A.4.4 Using the above formulas, one can check again that the mapγ ˜ introduced in §1.5 is actually λ AA˙ A A˙ a Dirac map. In fact fromγ ˜(y)(u, χ) = y (σλ χA˙ζA + σλAA˙u ¯z ) one ﬁnds

◦ µ λ BA˙ A AA˙ B˙ γ˜(y) γ˜(y)(u, χ) = y y (σµ σλAA˙u ζB + σµAB˙σλ χA˙¯z ) = λ µ BA˙ ρ A AA˙ ρ B˙ = y y (gλρ σµ σ AA˙u ζB + gµρ σλ σ AB˙χA˙¯z ) = λ µ ρν A A ¯ ρν ¯ A˙ B˙ = y y (gλρ Σµ σν B u ζB + gµρ Σλ σˆν B˙χA˙¯z ) . Replacing the Σ coeﬃcients with their expression in the above formula yields, after some calculations, ¯ γ(y) ◦ γ(y) = g(y, y) (ˆσ0 ⊕ σˆ0) = g(y, y) 11W as expected. Remark.

A.5 Spinors as endomorphisms The assignment of a Hermitian metric on S is equivalent to the assignment of an ε-normalized Hermitian metric h on U , ε ∈ ∧2U F being any element of the normalized U(1)-subset (§1.3). This h is a timelike unit element in H∗, and can be seen as an ‘observer’ (§3.1). Thus there exists an ε-normalized basis (ζA) of U such that √ 0 √1 0 A A˙ # AA˙ ¯ h = t = 2 σ AA˙z ⊗ ¯z , h = 2 τ0 = 2 σ0 ζA ⊗ ζA˙ .

One observes that (ζA) is orthonormal relatively to √ ˜ 0 A A˙ ˜# √1 # 0 AA˙ ¯ h := 2 h = σ0 = σ AA˙z ⊗ ¯z , h = 2 h = s = σ0 ζA ⊗ ζA˙ . When h is given one also has an isomorphism √ [ ˆ ˆ 2 11U ⊗ h : H → H = R·11 ⊕ H0 , where Hˆ ⊂ End(U) is the real vector subspace of all h-Hermitian endomorphisms, and Hˆ 0 ⊂ Hˆ is the subspace of all traceless Hermitian endomorphisms. Then one also has Hˆ 0 = iSU(U) , where SU(U) is the Lie algebra (relatively to the commutator of endomorphisms) of the group SU(U) of special unitary automorphisms of U (namely SU(U) is constituted by all traceless anti-Hermitian endomorphisms). ˆ ˆ The isomorphism H → H sends theg ˜-orthonormal basis (τλ) to a basis (ˆτλ) of H , given by √1 A A τˆλ = 2 σλ B ζA ⊗ z ,

A where the matrices (σλ B ) are exactly the same Pauli matrices introduced in §A.4. The dual basis (ˆtλ) ⊂ U F ⊗ U is given by

ˆλ √1 λ B A t = 2 s A z ⊗ ζB , with

λ B λ ˜A˙B λ A˙B (s A ) = (σ AA˙h ) ≡ (σ AA˙σ0 ) =

1 0 0 1 0 i 1 0 = 0 1 1 0 −i 0 0 −1 , 102 A SPINORS: FURTHER DETAILS AND CALCULATIONS as it can be checked from λ ˆλ ˆλ B A δ µ = ht , τˆµi = t A τˆµ B . The metricg ˜ itself is transformed to a Lorentz metricg ˆ on Hˆ , namely

gˆ(u ⊗ α, v ⊗ β) = ε(u, v) ε#(α, β) , which is well-deﬁned since ε0 = ei tε ⇒ ε0# = e−i tε# . In particular,

gˆ(ϕ, ϕ) = 2 det ϕ , ϕ ∈ Hˆ , and one easily checks that the basis (ˆτλ) isg ˆ-orthonormal. One may compareg ˆ with the Euclidean metric64 (§??, ??, ??)

K:(ϕ, ϑ) 7→ Tr(ϕ ◦ ϑ) .

ˆ ⊥ ⊥ Since (ˆτλ) is K-orthonormal, the restriction of K to H0 coincides with −gˆ , whereg ˆ is the ˆ ⊥ ˆ √1 restriction ofg ˆ to the subspace H = H0 orthogonal to the observer τ0 = 2 11, while the 65 restrictions of K andg ˆ to R·11coincide. A Let Q ∈ SU(U) , and let (Q B ) ∈ SU(2) be its matrix in the basis (ζA) . Let moreover

¯ ↑ ˆ ˆ −1 ∗ ↑ ˆ R ≡ R(Q) := Q ⊗ Q ∈ Lor+(H) , R ≡ R(Q) := Q ⊗ (Q ) ∈ Lor+(H) . Then (§6.2) µ µ AA˙ B ¯B˙ R λ = τ BB˙τ λ Q A Q A˙ and, similarly, one ﬁnds D E ˆµ ˆµ ˆµ B A C D R λ = t ,R(ˆτλ) = t A z ⊗ ζB , τˆλ D Q(ζC ) ⊗ Q(z ) = D E ˆµ B A C E −1 D F = t A z ⊗ ζB , τˆλ D Q C (Q ) F ζE ⊗ z =

ˆµ B C E −1 D A F = t A τˆλ D Q C (Q ) F δ E δB =

ˆµ B C A −1 D = t A τˆλ D Q C (Q ) B .

A.6 Spin groups A.6.1 Inverse in End W in 2-spinor terms Here I’ll show how one can construct the inverse of an element KP Φ ≡ ∈ Aut W ,K ∈ U ⊗ U F ,P ∈ U ⊗ U ,Q ∈ U F ⊗ U F ,J ∈ U F ⊗ U QJ and ﬁnd the expression stated in §6.4. Write

. . ← XY .Φ. . Φ = ,X ∈ U ⊗ U F ,Y ∈ U ⊗ U ,Z ∈ U F ⊗ U F ,T ∈ U F ⊗ U , ZT

64Observe (see also §??), that one has an isomorphism Hˆ ↔ Hˆ ∗ given by ordinary transposition, and this isomorphism can be seen as induced by K ; for any (Φ,A) ∈ Hˆ ∗ × Hˆ one has hΦ,Ai = K(ΦF,A). 65It is well-known that an observer in a Minkowski space determines the Euclidean metric gk − g⊥ . A.6 Spin groups 103

. . where .Φ. . ≡ det Φ (and the like). Then ! ! ! ! . . 11U 0 KP XY KX + PZKY + PT .Φ. . = = 0 11U F QJ ZT QX + JZQY + JT

 . . KX + PZ = .Φ. 11 (a)  KY + PT = 0 (b) ⇒ QX + JZ = 0 (c)   . . QY + JT = .Φ. . 11 (d) . . Composing the equation (b) on the left by K˜ F, since K˜ F K = det K ≡ .K. one gets

. . . .−1 .K. Y + K˜ F PT = 0 ⇒ Y = −.K. K˜ F PT. (e) Replacing this value in equation (d) one gets

. .−1 . . . .−1 . . −.K. Q K˜ F PT + JT = .Φ. . 11 ⇒ −.K. Q K˜ F P + J T = .Φ. . 11 . (f) Composing equation (f) on the left by

. .−1 ∼ . .−1 −.K. Q K˜ F P + J F = −.K. P˜FK Q˜F + J˜F one gets . .−1 . . . .−1 det−.K. Q K˜ F P + J T = .Φ. . −.K. P˜FK Q˜F + J˜F , and, since . .−1 . .−2 . . det−.K. Q K˜ F P + J = .K. det−Q K˜ F P + .K. J , one also gets

. .−2 . . . .−1 . . . . .K. det−Q K˜ F P + .K. J T = −.K. .Φ. . −P˜FK Q˜F + .K. J˜F . . . . .K. .Φ. . . . ⇒ T = .K. J˜F − P˜FK Q˜F . (g) . . det(.K. J − Q K˜ F P ) Replacing this value in equation (e) one gets

. .−1 Y = −.K. K˜ F PT =   ...... −1 .K. .Φ. . . = −.K. K˜ F P  .K. J˜F − P˜FK Q˜F  = . . det(.K. J − Q K˜ F P )

. . .Φ. . ...... = − .K. K˜ F P J˜F − .K. .P . Q˜F ⇒ . . det(.K. J − Q K˜ F P )

. . . . .K. .Φ. . . . ⇒ Y = .P . Q˜F − K˜ F P J˜F . (h) . . det(.K. J − Q K˜ F P ) 104 A SPINORS: FURTHER DETAILS AND CALCULATIONS

Similarly, by composing equation (c) on the left by Q˜F one gets

. . . .−1 .Q. X + Q˜F JZ = 0 ⇒ X = −.Q. Q˜F JZ. (i)

Replacing this in equation (a) one gets

. .−1 . . . .−1 . . −.Q. K Q˜F JZ + PZ = .Φ. . 11 ⇒ −.Q. K Q˜F J + P Z = .Φ. . 11 , (j)

Composing equation (j) on the left by

. .−1 ∼ . .−1 −.Q. K Q˜F J + P F = −.Q. J˜FQ K˜ F + P˜F one gets

. .−2 . . . .−1 . . . . .Q. det−K Q˜F J + .Q. P Z = −.Q. .Φ. . −J˜FQ K˜ F + .Q. P˜F . . . . .Q. .Φ. . . . ⇒ Z = .Q. P˜F − J˜FQ K˜ F . (k) . . det(.Q. P − K Q˜F J)

Replacing this value in equation (i) one gets

. .−1 X = −.Q. Q˜F JZ =   ...... −1 .Q. .Φ. . . = −.Q. Q˜F J  .Q. P˜F − J˜FQ K˜ F  = . . det(.Q. P − K Q˜F J)

. . .Φ. . ...... = − .Q. Q˜F J P˜F − .Q. .J. K˜ F ⇒ . . det(.Q. P − K Q˜F J)

. . . . .Q. .Φ. . . . ⇒ X = .J. K˜ F − Q˜F J P˜F . (l) . . det(.Q. P − K Q˜F J)

By direct calculation one checks that the determinant fulﬁll the relations

...... det.J. K˜ F − Q˜F J P˜F = .J. .Φ. . , ...... det.P . Q˜F − K˜ F P J˜F = .P . .Φ. . , ...... det.Q. P˜F − J˜F Q K˜ F = .Q. .Φ. . , ...... det.K. J˜F − P˜F K Q˜F = .K. .Φ. .). , A.6 Spin groups 105 and then, ﬁnally,

. . X = .J. K˜ F − Q˜F J P˜F , . . Y = .P . Q˜F − K˜ F P J˜F , . . Z = .Q. P˜F − J˜FQ K˜ F , . . T = .K. J˜F − P˜FK Q˜F .

A.6.2 Products of Hermitian matrices

Here is the proof, due to Carlo Franchetti, that an arbitrary 2 × 2 complex matrix with nonvanishing real determinant can be written as a product of three Hermitian matrices. Write three arbitrary Hermitian matrices as

α u¯ a z¯ b w¯ H = , K = , L = , u β z c w d where α, β, α, b, c, d ∈ R , u, z, w ∈ C . The product

α a b + α w z¯ + b z u¯ + c w u¯ α a w¯ + α d z¯ + c d u¯ +w ¯ z u¯ HKL =   u a b + u w z¯ + β b z + β c w a w¯ u + d z¯ u + β c d + β w¯ z is the general form of the product of three Hermitian matrices. Our task is to ﬁnd values for the components of H , K and L in such a way that HKL = M , where

AB M = , A, B, C, D ∈ ,AD − BC = γ ∈ \{0} CD C R is any given matrix with real non-vanishing determinant. Remark. In general, an arbitrary 2 × 2 complex matrix with nonvanishing real determinant cannot be written as a product of just two Hermitian matrices. In fact the general form of such a product is a b + w z¯ a w¯ + d z¯ KL =   , b z + c w c d +w ¯ z

0 B which can’t reproduce the case (M) = CD with D not real. Actually, since a b, c d ∈ R , one has a b + w z¯ = 0 ⇒ w z¯ ∈ R ⇒ c d +w ¯ z ∈ R

It will be necessary to split the discussion into a number of cases, according to possible diﬀerent conditions fulﬁlled by the entries of M . 106 A SPINORS: FURTHER DETAILS AND CALCULATIONS

Case A = 0 We ﬁrst consider the case A = 0 , namely 0 B M = ,B,C,D ∈ , − BC = γ ∈ \{0} . CD C R Note that the condition BC ∈ R \{0} can be written as C = r B¯ , r ∈ R \{0} . We consider three subcases.

Case i1 : A = 0 , C 6= B¯ , D 6= 0 . Taking the special values α = 1 , β = b = d = 0 , z = −c u , namely 1u ¯ a −c u¯ 0w ¯ H = , K = , L = , u 0 −c u c w 0 one gets    2 ¯ 0w ¯ (a − c |u|2) w a − w c |u| = B,  HKL =   ⇒ −w c |u|2 = C, 2 −c |u| w w¯ a u w¯ a u = D. One then ﬁnds the solution D B¯ − C u = , w = , B − C¯ a where a, c 6= 0 are subjected to the condition c |D|2 = a (C¯ − B) .

Case i2 : A = 0 , C 6= B¯ , D = 0 . Taking the special values α = c = 1 , b = d = u = z = 0 , namely 1 0 a 0 0w ¯ H = , K = , L = , 0 β 0 1 w 0 one gets    0 a w¯ a w¯ = B,  HKL =   ⇒ β w 0 β w = r B,¯ where (remember) r ≡ C/B¯ ∈ R \{0} . One then ﬁnds the solution r B¯ B C B 2

Case i3 : A = 0 , C = B¯ . Here we can assume D 6= 0 (otherwise M is already a Hemitian matrix). Taking the special values α = a = b = c = d = 0 , β = w = 1 , namely 0u ¯ 0z ¯ 0 1 H = , K = , L = , u 1 z 0 1 0 one gets     0 z u¯ z u¯ = B, u = B/¯ D,¯   HKL =   ⇒ ⇒ u z¯ z z = D, z = D.

Case A 6= 0 Finally we consider the case A 6= 0 . We write D as (γ+BC)/A , where γ ≡ det M ∈ R\{0} .

Case ii1 : A 6= 0 , C = B¯ . In this particular case C = B¯ , hence D = (γ + |B|2)/A . Taking the special values

α = w = 1 , a = b = c = d = 0 , namely 1u ¯ 0z ¯ 0 1 H = , K = , L = , u β z 0 1 0 one gets     z = A,¯ z¯ z u¯ z¯ = A,    ¯ HKL =   ⇒ z u¯ = B, ⇒ u = B/A , u z¯ β z  2  β z = (γ + |B| )/A β = (γ + |B|2)/|A|2 .

Case ii2 : A 6= 0 , C 6= B¯ . As before we write D ≡ (γ +BC)/A , where γ ≡ det M ∈ R\{0} . Taking the special values

α = d = 1 , a = b = c = 0 , namely 1u ¯ 0z ¯ 0w ¯ H = , K = , L = , u β z 0 w 1 one gets  w z¯ w¯ z u¯ +z ¯  HKL =   . w u z¯ w¯ β z +z ¯ u By substitutions one ﬁnds the solution

C A γ + |C|2 u = , w = , z = B¯ − C , β = . A B − C¯ |A|2 108 A SPINORS: FURTHER DETAILS AND CALCULATIONS

A.6.3 Products of Hermitian matrices (2)

In this section I’ll give an alternative proof of the result that any automorphism K ∈ U ⊗ U F ¯˜ ¯˜ such that det K ∈ R \{0} can be seen as a product K = P1 P2 P3 P4 with Pi ∈ H . This proof used the fact that any automorphism in a complex vecotr space is diagonalizable; namely, there is a 2-spinor basis in which the matrix of K is diagonal,66 namely

1 1 1 K 1 0 A 1 + i B 1 0 (K) = 2 ≡ 2 2 , 0 K 2 0 A 2 + i B 2

1 1 2 2 where the four real components A 1 , B 1 , A 2 and B 2 obey the condition = det K = 0 that is

1 2 2 1 A 1 B 2 + A 2 B 1 = 0 . Suppose for the moment that each of the above said four real components is diﬀerent from zero. Then the product of matrices

1 1 S ≡ 2 (r+s) σ0 + 2 (r−s) σ3 a σ1 + α σ2 σ1 =

r 0 0 a + i a 0 1 r (a + i α) 0 = = 0 s a − i a 0 1 0 0 s (a − i α) coincides with (K) for suitable choices of the real parameters r , s , a and α . For example,67

 2 1 2 r = −A 2 B 1/B 2 ,   2 1 2 1   2 −A 2 B 1/B 2 + i B 1 0 s = A 2 , ⇒ S =   . a = 1 , 2 2  0 A 2 + i B 2  2 2 α = −B 2/A 2

Thus, passing from the σ-matrices to the corresponding elements σλ ∈ H (via the chosen 2-spinor basis of eigenvectors of K), and observing that σ¯˜0 = σ0 , σ¯˜i = − σi , i = 1, 2, 3 ,

66 1 p 2 K has the eigenvalues 2 Tr K ± (Tr K) − 4 det K , which do not vanish since det K 6= 0 . 67Similarly, resolving the condition = det K = 0 with respect to the other real components, one ﬁnds

r = −A1 B2 /A2 ,  1 2 2  1 1 2 2   2 A 1 − i A 1 B 2/A 2 0 s = −B 2 , 2 2 ⇒ S =   , a = A 2/B 2 , 2 2  0 A 2 + i B 2 α = 1

r = A1 ,  1  1 1   1 2 1 A 1 + i B 1 0 s = −A 1 B 2/B 1 , ⇒ S =   , a = 1 , 1 2 1 2  0 −A 1 B 2/B 1 + i B 2  1 1 α = B 1/A 1

r = B1 ,  1  1 1   2 1 1 A 1 + i B 1 0 s = A 2 B 1/A 1 , 1 1 ⇒ S =   . a = A 1/B 1 , 2 2 1 1  0 A 2 − i A 2 B 1/A 1 α = 1 A.7 4-momentum and spinors 109 one has ˜ ˜ K = P1 P¯2 P3 P¯4 ,

1 with P1 = 2 [(r+s) σ0 + (r−s) σ3] ,P2 = −(a σ1 + α σ2) ,P3 = σ3 ,P4 = σ0 .

1 2 2 1 Now observe that the real-determinant condition A 1 B 2 +A 2 B 1 = 0 and the non-vanishing- 1 2 1 2 1 2 determinant condition A 1 A 2 − B 1 B 2 6= 0 together imply A 1 = 0 ⇐⇒ A 2 = 0 and 1 2 1 1 2 2 B 1 = 0 ⇐⇒ B 2 = 0 ; namely, if any of the components A 1 , B 1 , A 2 and B 2 vanish, then

1 1 A 1 0 i B 1 0 either K = 2 or K = 2 . 0 A 2 0 i B 2 In the ﬁrst case one has K = P1 σ¯˜0 = P1 σ¯˜0 σ0 σ¯˜0 1 2 where P1 is the same as above with r = A 1 , s = A 2 . In the second case

1 2 K = P1 σ¯˜1 σ2 σ¯˜0 , r = −B 1 , s = B 2

1 2 1 2 1 2 1 (in particular, if A 1 = A 2 ≡ r and B 1 = B 2 = 0 then K = r 11; if A 1 = A 2 = 0 and B 1 = 2 B 2 ≡ r then one can also write K = r σ0 σ¯˜1 σ2 σ¯˜3 = −r σ1 σ¯˜2 σ3 σ¯˜0 , since σ¯˜1 σ2 σ¯˜3 = i σ¯˜0 ). Remark. In a 2-spinor basis in which the matrix of K is diagonal one is always able to write ˜ ˜ K = P1 P¯2 P3 P¯4 with P4 = σ0 ; representing the fourth factor in this way (its matrix being the identity matrix) is a feature of that basis. Remark. When one changes the basis, so that K is no more diagonal, the matrices of the Hermitian factors remain Hermitian. From the above argument, without using the result of §A.6.2, one may deduce that any 0 P (−) odd-degree element P ‡ 0 ∈ Cl can be written as the product of ﬁve Hermitian factors. 1 • In fact, let P ∈ U ⊗ U with det P = 2 g(P,P ) ∈ R \{0} , and let V ∈ H . Because P V¯˜ ∈ U ⊗ U F has again non-vanishing real determinant, by using the previous results one has ¯˜ ¯˜ ¯˜ 1 ¯˜ ¯˜ ¯ F • P V = P1 P2 P3 P4 ⇒ P = det V P1 P2 P3 P4 V ,Pi ∈ H , having composed both sides with V¯ F on the right, and remembered V¯˜ V¯ F = det V¯ = det V . 1 ¯ F • Setting P5 := det V V ∈ H one gets the stated result.

A.7 4-momentum and spinors This section gives some further details about the matter of §3.5; notations are the same.

A.7.1 Let √ 1 1 h¯# := 2 τ = (u ⊗ u¯ + v ⊗ v¯) = (u ⊗ u¯ +χ ¯# ⊗ χ#) , |ε(u, v)| |hχ,¯ ui| √ 2 1 1 h := p¯[ = (¯u[ ⊗ u[ + χ ⊗ χ¯) = (λ¯ ⊗ λ + χ ⊗ χ¯) . µ |hχ,¯ ui| |ε#(¯χ, λ)| 110 A SPINORS: FURTHER DETAILS AND CALCULATIONS with v =∼ −χ¯# , λ =∼ u[ . Then one sees that h and h# are mutually inverse Hermitian metrics. In fact one has

1 hχ, u¯i Tr1(h ⊗ h#) = hu¯[, χ#i u[ ⊗ χ¯# + hχ, u¯i χ¯ ⊗ u = u[ ⊗ χ¯# +χ ¯ ⊗ u = 1 |hχ,¯ qi|2 |hχ,¯ ui| 1 1 = u[ ⊗ χ¯# +χ ¯ ⊗ u = −u[ ⊗ v + v[ ⊗ u = hχ,¯ ui ε(v, u)

= 11 ∈ U F ⊗ U , since, for any s ∈ U and ν ∈ U F one ﬁnds

1 sc Tr1(h ⊗ h#) = ε(s, u) v + ε(v, s) u = s (see §A.3), 1 ε(v, u) 1 Tr1(h ⊗ h#)cν = ε∗(¯χ, ν) u[ + hν, ui χ¯ = ν . 1 hχ,¯ ui

A.7.2

The parity operator (§3.3) associated with τ is P ≡ γ˜[τ] and acts as68

1 1 γ˜[τ](u, χ) = hχ, u¯i u , hχ,¯ ui χ = ε¯(¯v, u¯) u , ε(v, u)v ¯[ . |hχ,¯ ui| |ε(u, v)|

Next, remember (§3.2) that for each normalized 2-form ω = eit ε one deﬁnes

−it # [ Cω : W → W :(s, σ) 7→ e ε (¯σ), −ε¯ (¯s) .

In particular, when u and v have been assigned, one considers ω deﬁned by69

2 2 ω = λ ∧ χ¯ ⇒ ω∗ = u ∧ v , µ := |hχ,¯ ui| = |ε(u, v)| , µ µ ε(u, v) ⇒ e−it = hε, ω∗i = , |ε(u, v)|

68The last line can be obtained from the demi-last line by the replacementv ¯[ → χ , v → −χ¯# ; note that the possible the phase factor eiϕ in the relation between v and χ disappears in this expression. So one has hσ, v¯i = hσ, −χ#i = −hχ#, σi = −ε∗(χ, σ) , and ε(v, s) = ε(−χ¯#, s) = hχ,¯ si . 69 1 From §1.3, remember hε, u ∧ vi = 2 ε(u, v). A.7 4-momentum and spinors 111

Recalling the endomorphismγ ˜η :(s, σ) 7→ i (s, −σ) one now writes down the time reversal morphism associated with τ and ω as

T (s, σ) =γ ˜η γ˜[τ] Cω(s, σ) =

−it # [ = e γ˜η γ˜[τ] (¯σ , −s¯ ) =

e−it = γ˜ −hs¯[, u¯i u − hs¯[, v¯i v , ε(u, σ¯#)u ¯[ + ε(v, σ¯#)v ¯[ = |ε(u, v)| η

e−it = γ˜ ε¯(¯u, s¯) u +ε ¯(¯v, s¯) v , hσ,¯ ui u¯[ + hσ,¯ vi v¯[ = |ε(u, v)| η

i e−it = −ε¯(¯u, s¯) u − ε¯(¯v, s¯) v , hσ,¯ ui u¯[ + hσ,¯ vi v¯[ = |ε(u, v)|

i e−it = −ε¯(¯u, s¯) u + hχ, s¯i χ¯# , hσ,¯ ui u¯[ + ε∗(¯σ, χ¯) χ . |hχ,¯ ui| In particular i e−it T (u, v¯[) = ε¯(¯v, u¯) v , ε(u, v)u ¯[ , |ε(u, v)|

i e−it T (u, χ) = hχ, u¯i χ¯# , hχ,¯ ui u¯[ . |hχ,¯ ui| From the above results one sees that p applied to each of the 4-spinors C(u, χ), P(u, χ) ≡ γ˜[τ](u, χ) ,γ ˜η(u, χ) and T (u, χ) always gives p(u, χ).

A.7.3 One can directly check that, if K ∈ U(2) and u0 = K(u), v0 = K(v) , then p(u, v) = p(u0, v0). Recalling §?? one writes u0 = ei φ cos γ ei α u − sin γ e−i β v ,

v0 = ei φ sin γ ei β u + cos γ e−i α v , so that u0 ⊗ u¯0 + v0 ⊗ v¯0 = cos γ ei α u − sin γ e−i β v ⊗ cos γ e−i α u¯ − sin γ ei β v¯ +

+ sin γ ei β u + cos γ e−i α v ⊗ sin γ e−i β u¯ + cos γ ei α v¯ =

= cos2 γ u ⊗ u¯ − sin γ cos γ e−i (α+β) v ⊗ u¯−

− sin γ cos γ ei (α+β) u ⊗ v¯ + sin2 γ v ⊗ v¯+

+ sin2 γ u ⊗ u¯ + sin γ cos γ e−i (α+β) v ⊗ u¯+

+ sin γ cos γ ei (α+β) u ⊗ v¯ + cos2 γ v ⊗ v¯ =

= u ⊗ u¯ + v ⊗ v¯ . 112 A SPINORS: FURTHER DETAILS AND CALCULATIONS

The expression of p = µ τ0 as represented by a couple (u, v) can be written in terms of the λ λ λ λ A A˙ Pauli basis expressions u ⊗ u¯ = (u ⊗ u¯) τl and v ⊗ v¯ = (v ⊗ v¯) τl , with (u ⊗ u¯) = tAA˙u u¯ and the like (§A.4); in this way one sees, in usual spacetime terms, how τ0 can be expressed as the sum of two null vectors according to the choice of (u, v) as in the above proposition. One gets

1 µ (u ⊗ u¯)0 = √ s0 uA u¯A˙ = , 2 AA˙ 2 1 (u ⊗ u¯)1 = √ s1 uA u¯A˙ = −µ sin γ cos γ cos(α + β) , 2 AA˙ 1 (u ⊗ u¯)2 = √ s2 uA u¯A˙ = −µ sin γ cos γ sin(α + β) , 2 AA˙ 1 µ (u ⊗ u¯)3 = √ s3 uA u¯A˙ = (cos2 γ − sin2 γ) , 2 AA˙ 2 1 µ (v ⊗ v¯)0 = √ s0 vA v¯A˙ = , 2 AA˙ 2 1 (v ⊗ v¯)1 = √ s1 vA v¯A˙ = µ sin γ cos γ cos(α + β) , 2 AA˙ 1 (v ⊗ v¯)2 = √ s2 vA v¯A˙ = µ sin γ cos γ sin(α + β) , 2 AA˙ 1 µ (v ⊗ v¯)3 = √ s3 uA u¯A˙ = (sin2 γ − cos2 γ) . 2 AA˙ 2

Then one sees that, in all cases, µ τ0 is expressed as

µ µ µ τ = u ⊗ u¯ + v ⊗ v¯ = τ + w + τ − w , 0 2 0 2 0

⊥ ⊥ where w := (u ⊗ u¯) = −(v ⊗ v¯) is a spacelike vector orthogonal to τ0 , with Lorentzian pseudo-norm given by

2 2 2 2 2 2 2 1 2 2 2 g(w, w) = −µ sin γ cos γ cos (α+β) + sin γ cos γ sin (α+β) + 4 (cos γ − sin γ) = µ2 = − . 4

Finally, one is interested in seeing how the couple (u, χ) , with χ =ε ¯[(¯v) as in the above construction, splits in terms of the decomposition of the 4-spinor space into electron and [ 2 [ 1 positron subspaces. Fromε ¯ (ζ1) =z ¯ ,ε ¯ (ζ2) = −z¯ , one has √ µ χ ≡ ε¯[(¯v) = √ e−i φ − cos γ ei α z¯1 + sin γ e−i β z¯2 = 4 2 √ µ −i φ i α −i β = √ e cos γ e ζ3 − sin γ e ζ4 , 4 2 A.7 4-momentum and spinors 113

√ µ i (φ+α) i (φ−β) i (−φ+α) −i (φ+β) (u, χ) = √ cos γ e ζ1 − sin γ e ζ2 + cos γ e ζ3 − sin γ e ζ4 = 4 2 √ µ = √ cos γ ei (φ+α) (ζ0 + ζ0 ) − sin γ ei (φ−β) (ζ0 + ζ0 )+ 4 2 1 3 2 4

i (−φ+α) 0 0 −i (φ+β) 0 0 + cos γ e (−ζ1 + ζ3) − sin γ e (−ζ2 + ζ4) = √ µ = √ cos γ ei α (ei φ − e−i φ) ζ0 − sin γ e−i β (ei φ − e−i φ) ζ0 + 4 2 1 2

i α i φ −i φ 0 −i β i φ −i φ 0 + cos γ e (e + e ) ζ3 − sin γ e (e + e ) ζ4 = √ 2 µ = √ i sin φ cos γ ei α ζ0 − sin γ e−i β ζ0 + cos φ cos γ ei α ζ0 − sin γ e−i β ζ0 . 4 2 1 2 3 4 + 0 0 The (positive energy) electron subspace W0 ⊂ W is represented by (ζ1 , ζ2) , hence an electron momentum is characterized by a couple (u, χ) as above such that cos φ = 0 . Similarly, the − 0 0 (negative energy) positron subspace W0 ⊂ W is represented by (ζ3 , ζ4) , hence a positron momentum is characterized by a couple (u, χ) as above such that sin φ = 0 .

A.7.4

Let (u, χ), (s, σ) ∈ W be such that p ∈ Pm , m ∈ M , is represented by both couples as m m p = √ (u ⊗ u¯)[ +χ ¯ ⊗ χ = √ (s ⊗ s¯)[ +σ ¯ ⊗ σ . 2 |hχ,¯ ui| 2 |hσ,¯ si| Then several identities hold, which can be reduced to |hχ,¯ ui hσ, s¯i| = |ε(s, u)|2 + |hσ,¯ ui|2 = (8)

= |hχ,¯ si|2 + |ε¯∗(χ, σ)|2 , (9)

hχ,¯ ui hσ, s¯i hχ, s¯i = hσ,¯ ui , (10) |hχ,¯ ui| |hσ, s¯i|

hχ,¯ ui hσ, s¯i ε¯∗(χ, σ) = ε(u, s) . (11) |hχ,¯ ui| |hσ, s¯i| proof: The identities are proved by comparing the two expressions for hp, a ⊗ ¯bi , with a, b ∈ {u, χ¯#, s, σ¯#} . Of course, replacing a ⊗ ¯b by b ⊗ a¯ one just gets the conjugate identity; there are other symmetries of the given setting, so that the indipendent identities√ are only four. 2 Consider for example the case a = b = u . Then writing down m hp, u ⊗ u¯i for the two expressions of p one gets 1 D E 1 D E (u ⊗ u¯)[ +χ ¯ ⊗ χ, u ⊗ u¯ = (s ⊗ s¯)[ +σ ¯ ⊗ σ, u ⊗ u¯ |hχ,¯ ui| |hσ,¯ si| 1 1 ⇒ hχ,¯ ui hχ, u¯i = |ε(s, u)|2 + |hσ,¯ ui|2 |hχ,¯ ui| |hσ,¯ si|

⇒ |hχ,¯ ui hσ, s¯i| = |ε(s, u)|2 + |hσ,¯ ui|2 . 114 A SPINORS: FURTHER DETAILS AND CALCULATIONS

Similarly, by setting a = b =χ ¯# , one gets 1 1 hu[, χ¯#i hu¯[, χ#i = hs[, χ¯#i hs¯[, χ#i + hσ,¯ χ¯#i hσ, χ#i |hχ,¯ ui| |hσ,¯ si|

A.7.5 Let again p = √ m (u ⊗ u¯)[ +χ ¯ ⊗ χ = √ m (s ⊗ s¯)[ +σ ¯ ⊗ σ ∈ P . 2 |hχ,u¯ i| 2 |hσ,s¯ i| m ± ± ± Then (§3.5, proposition 3.1) (u, χ) ∈ Wp (resp. (s, σ) ∈ Wp ) iﬀ hχ,¯ ui ∈ R (resp. hσ,¯ si ∈ ± ± R ). When this is the case, since Wp are vector spaces, any non-vanishing linear combination of (u, χ) and (σ, s) , say (u0, χ0) , must yield again p = √ m (u0 ⊗ u¯0)[ +χ ¯0 ⊗ χ0 . This 2 |hχ¯0,u0i| fact can be directly checked, by thorough (though boring) calculations, as follows. 0 0 × proof: If (u , χ ) = c (u, χ) with c ∈ C then the above statement is obviously true. Then, + − one assumes that hχ,¯ ui and hσ,¯ si are either both in R or both in R , and shows that the above statement is true with (u0, χ0) = (u+s, χ+σ). First, one shows that hχ0, u¯0i = hχ, u¯i + hσ, s¯i + hχ, s¯i + hσ, u¯i + − + − is again in R (resp. R ). The two ﬁrst terms in the right-hand side are in R (resp. R ) by hypothesis; as for the third and fourth terms, they are mutually complex conjugate, since identity 10 gives, in the present case, hχ, s¯i = hσ,¯ ui . Then write hχ0, u¯0i = hχ, u¯i + hσ, s¯i + 2 2 |

= hχ, u¯i2 + hσ, s¯i2 + 2 hχ, u¯i hσ, s¯i − 4 (

= hχ, u¯i2 + hσ, s¯i2 + 2 |ε(s, u)|2 + |hσ,¯ ui|2 − 4 (

≥ hχ, u¯i2 + hσ, s¯i2 + 2 |ε(s, u)|2 + 2 |hσ,¯ ui|2 − 4 |hσ, u¯i|2 =

= hχ, u¯i2 + hσ, s¯i2 + 2 |ε(s, u)|2 − 2 |hσ, u¯i|2 >

> hχ, u¯i2 + hσ, s¯i2 − 2 |ε(s, u)|2 − 2 |hσ, u¯i|2 =

= hχ, u¯i2 + hσ, s¯i2 − 2 |ε(s, u)|2 + 2 |hσ, u¯i|2 =

= hχ, u¯i2 + hσ, s¯i2 − 2 hχ, u¯i hσ, s¯i = (hχ, u¯i − hσ, s¯i)2 ≥ 0 .

Then, consider

P := (u0 ⊗ u¯0)[ +χ ¯0 ⊗ χ0 = (u + s)[ ⊗ (¯u +s ¯)[ + (¯χ +σ ¯) ⊗ (χ + σ) .

The task consists now of showing that P ∝ p , which follows by using the two-spinor ‘exchange formulas’ stated in §A.3. One ﬁnds hχ,¯ si ε(s, u) hχ, s¯i ε¯(¯s, u¯) P = u[ + u[ + χ¯ ⊗ u¯[ + u¯[ + χ + hχ,¯ ui hχ,¯ ui hχ, u¯i hχ, u¯i ε∗(¯χ, σ¯) hσ,¯ ui ε¯∗(χ, σ) hσ, u¯i + χ¯ + u[ + χ¯ ⊗ χ + u¯[ + χ = hχ,¯ ui hχ,¯ ui hχ, u¯i hχ, u¯i

A.8 World-spinors In this section I look, somewhat more closely, to the relation between my treatment of 2- spinors and Minkowski spacetime, and the traditional one by Penrose et al [PR84]. There will be some repetitions of previous results.

A normalized basis (ζA) of U determines the basis

¯ ¯ ¯ ¯ ¯ (ζAA˙) := (ζA ⊗ ζA˙) = ζ1 ⊗ ζ1 , ζ1 ⊗ ζ2 , ζ2 ⊗ ζ1 , ζ2 ⊗ ζ2 116 A SPINORS: FURTHER DETAILS AND CALCULATIONS of U ⊗ U . The induced basis of the underlying real vector space is ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (ζAA˙, i ζAA˙) = ζ1 ⊗ ζ1 , ζ1 ⊗ ζ2 , ζ2 ⊗ ζ1 , ζ2 ⊗ ζ2 , i ζ1 ⊗ ζ1 , i ζ1 ⊗ ζ2 , i ζ2 ⊗ ζ1 , i ζ2 ⊗ ζ2 .

Projecting this set into H := U ∨¯ U and iH and taking all independent elements one obtains bases ¯ ¯ ¯ ¯ ¯ ¯ (aλ ) := ζ1 ⊗ ζ1 , (ζ1 ⊗ ζ2 + ζ2 ⊗ ζ1) , i (ζ1 ⊗ ζ2 − ζ2 ⊗ ζ1), ζ2 ⊗ ζ2 ,

¯ ¯ ¯ ¯ ¯ ¯ (i aλ ) := i ζ1 ⊗ ζ1 , i (ζ1 ⊗ ζ2 + ζ2 ⊗ ζ1) , (ζ2 ⊗ ζ1 − ζ1 ⊗ ζ2), i ζ2 ⊗ ζ2 of the Hermitian and anti-Hermitian subspaces 0 ≤ λ ≤ 3 . Their dual bases are the bases ∗ ∗ (aλ ) of H = U F ∨¯ U F and (i aλ ) of iH , where

λ 1 1 1 1 2 2 1 1 1 2 2 1 2 2 (a ) = z ⊗ ¯z , 2 (z ⊗ ¯z + z ⊗ ¯z ) , 2i (z ⊗ ¯z − z ⊗ ¯z ) , z ⊗ ¯z .

Of course, (aλ ) can also be viewed as a complex basis of U ⊗ U = H ⊕ (iH) . The matrix of the bilinear formg ˜ in this basis is 0 0 0 1 0 −2 0 0  (˜g) =   . 0 0 −2 0  1 0 0 0

This is also the matrix in the basis (aλ ) of the real restriction ofg ˜ to H (while the matrix of the restriction ofg ˜ to iH , in the basis (i aλ ) , is just the opposite of this). By a diagonalizing procedure one ﬁnds the orthonormal, mutually dual bases

√1 AA˙ ¯ (τλ) := ( 2 σλ ζA ⊗ ζA˙) = √1 ¯ ¯ √1 ¯ ¯ = 2 (ζ1 ⊗ ζ1 + ζ2 ⊗ ζ2) , 2 (ζ1 ⊗ ζ2 + ζ2 ⊗ ζ1) , √i ¯ ¯ √1 ¯ ¯ 2 (−ζ1 ⊗ ζ2 + ζ2 ⊗ ζ1) , 2 (ζ1 ⊗ ζ1 − ζ2 ⊗ ζ2) ,

λ √1 λ A A˙ (t ) := ( 2 σ AA˙z ⊗ ¯z ) = √1 1 1 2 2 √1 1 2 2 1 = 2 (z ⊗ ¯z + z ⊗ ¯z ) , 2 (z ⊗ ¯z + z ⊗ ¯z ) , √i 1 2 2 1 √1 1 1 2 2 2 (z ⊗ ¯z − z ⊗ ¯z ) , 2 (z ⊗ ¯z − z ⊗ ¯z ) , already introduced in §1.4. With this orthonormal basis, one checks that the signature ofg ˜ restricted to the Hermitian subspace is (+ − − −) , while the signature of the restriction to the anti-Hermitian subspace is (− + + +) . In the physics literature (see for example [HT85]) one ﬁnds the following notation. A symplectic 2-spinor basis is called a dyad and indicated by70

(o, ι) ≡ (ζ0 , ζ1) . This gives rise to the ‘tetrad’ (basis of U ⊗ U)

(l , n , m , m¯ ) ≡ (o ⊗ o¯ , ι ⊗ ¯ι , o ⊗ ¯ι , ι ⊗ o¯) ,

70The Greek letters o and ι are reminders of the indices 0 and 1 . A.9 Tensor algebra of world-spinors 117 where l and n are future-pointing timelike vectors and m is an isotropic ‘complex’ vector, i.e. not belonging to H ≡ U ∨¯ U . Then m¯ is the complex conjugate of m when H is considered as the real subsapce of U ⊗ U . Moreover one has

g˜(l, n) = −g˜(m, m¯ ) = 1 .

AA˙ ¯ λ An element w ∈ U ⊗ U can be written as w = w ζA ⊗ bB˙ = w τλ , with

λ √1 λ AA˙ (w ) = 2 (σAA˙w ) = √1 11˙ 22˙ 12˙ 21˙ 12˙ 21˙ 11˙ 22˙ = 2 (w + w ) , (w + w ) , i (w − w ) , (w − w ) ,

w0 + w3 w1 − i w2 (wAA˙) = √1 (σAA˙wλ) = √1 . 2 λ 2 w1 + i w2 w0 − w3

Then, one also ﬁnds

g˜(w, w) = (w0)2 − (w1)2 − (w2)2 − (w3)2 = 2 det(wAA˙) .

The matrix ofg ˜ in the two bases bAA˙ and (τλ) is given, respectively, by

gAA˙,BB˙ = εAB ε¯A˙B˙ ,

gλµ = ηλµ (of Minkovski).

Note how all this is consistent with index raising and lowering. For example, the components ∗ of the covectorg ˜(w, ) ∈ H = U F ∨¯ U F with respect to the two dual bases are given by

AA˙ AA˙ wBB˙ := w gAA˙,BB˙ = w εAB ε¯A˙B˙ , λ wµ := w gλµ .

Note also that all the above constructions and results have nothing to do with a Hermi- tian metric. If a hermitian metric h on U is given, then one has a richer structure (§3.1); in particular, h determines an observer in H , and one has the orthogonal subspace of spacelike world-spinors, which can be characterized as having vanishing trace relatively to h . Further- more, one can now equivalently treat world spinors as endomorphism of U .

A.9 Tensor algebra of world-spinors A.9.1 In the standard 2-spinor literature one uses the convention of grouping all factors of a given type (relatively to conjugation). Namely, two tensor spaces having the same number of indices of each type are considered equivalent, even if the indices are arranged in a diﬀerent order. According to this convention, the tensor algebra of U is

(r,s) r s p F q F ⊗(p,q)U := (⊗ U) ⊗ (⊗ U) ⊗ (⊗ U ) ⊗ (⊗ U ) :=

:= U ⊗ · · · ⊗ U ⊗ U ⊗ · · · ⊗ U ⊗ U F ⊗ · · · ⊗ U F ⊗ U F ⊗ · · · ⊗ U F . | {z } | {z } | {z } | {z } r factors s factors p factors q factors 118 A SPINORS: FURTHER DETAILS AND CALCULATIONS

(r,s) In components, the expression of τ ∈ ⊗(p,q)S is:

A1...ArB˙1...B˙s τ = τ C1...CpD˙1...D˙q ×

C1 Cp D˙1 D˙q × ζA1 ⊗ ... ⊗ ζAr ⊗ ζB˙1 ⊗ ... ⊗ ζB˙s ⊗ z ⊗ ... ⊗ z ⊗ z ⊗ ... ⊗ z . Non-dotted indices raising and lowering is performed by contraction with the ﬁrst index71 AB of ε and εAB ; similarly, dotted indices raising and lowering is performed by contraction A˙B˙ with the ﬁrst index ofε ¯ andε ¯A˙B˙.

A.9.2 A non-vanishing tensor cannot have antisymmetric groups of more than two indices. If τ is antisymmetric in two indices, then its expression in components can be simpliﬁed, since it equals ε times a tensor with two indices less. Also for symmetric tensors there is a simpliﬁcation: they are decomposable.

Proposition A.1 a) One has

···[AB]··· 1 AB ··· C··· τ = 2 ε τ C ,

1 C τ···[AB]··· = 2 εAB τ···C ··· ,

···[A˙B˙]··· 1 A˙B˙ ··· C˙··· τ = 2 ε¯ τ C˙ ,

1 C˙ τ···[A˙B˙]··· = 2 ε¯A˙B˙τ···C˙ ··· . b) A symmetric tensor is decomposable as

(A1···Ar) τ := τ ζA1 ⊗ ... ⊗ ζAr = u1 ∨ · · · ∨ ur , u1, . . . , ur ∈ U , and similar formulas hold for symmetric tensors of other types. proof: CD C D D C a) From εAB ε = δA δB − δA δB one gets

CD 1 C εAB ε τ···[CD]··· = 2 τ···[AB]··· ⇒ τ···[AB]··· = 2 εAB τ···C ··· , and similar calculations for the other index types. 1 2 b) For any given z ∈ C consider λz := z + z z ∈ U F . Then

A1···Ar 11...1 11...12 2 22...2 r τ(λz, . . . , λz) = τ (λz)A1 ... (λz)Ar = τ + r τ z + ··· + τ z =

X r = τ (k,r−k) zr−k , k 0≤k≤r where τ (k,r−k) stands for the component of τ having k indices equal to 1 and (r − k) indices equal to 2 . Thus τ(λz, . . . , λz) is a complex polynomial of degree r in the complex variable z , and as such it is factorizable as

τ(λz, . . . , λz) = a (z − z1)(z − z2) ... (z − zr) , a, zi ∈ C , 1 ≤ i ≤ r . 71Forgetting this rule may lead to sign inconsistencies. A.9 Tensor algebra of world-spinors 119

For each value of the above introduced index i there is a unique ui ∈ U such that hλz , uii = z−zi for all z ∈ C ; in fact, this condition can be written as 1 2 (ui) + z (ui) = z − zi ∀ z ∈ C , 1 2 and has the unique solution (ui) = −zi ,(ui) = 1 , that is

ui = −zi ζ1 + ζ2 . Thus one also has

τ(λz, . . . , λz) = (a u1 ⊗ · · · ⊗ ur)(λz, . . . , λz) , ∀ z ∈ C . 1 2 Observing that every element in U F is proportional to λz := z + z z for some z ∈ C , one concludes that τ = a u1 ∨ · · · ∨ ur .

A.9.3 (1,1) Observe now that U ⊗ U ≡ ⊗(0,0)U . More generally, the above said index grouping convention yields the identiﬁcations (r,r) r ⊗(s,s)S ≡ ⊗s(U ⊗ U) . When dealing with the tensor algebra of U ⊗ U or H , this grouping convention may be not the most convenient, since indices come naturally into pairs (each one constituted by two indices of diﬀerent type). It can be useful, however, as it yields various types of interesting tensor decompositions [HT85] by taking into account (anti)symmetry properties. In any case, one should be careful and state explicitely which is the used convention. (r,0) If τ = u1 ∨ · · · ∨ ur ∈ ⊗(0,0)U is a completely symmetric tensor, then the r isotropic elements uj ⊗ u¯j ∈ U ∨¯ U , 1 ≤ j ≤ r are called the principal null directions of τ . The anti-isomorphisms † : U ⊗ U → U ⊗ U : u ⊗ v¯ 7→ v ⊗ u¯ , † : U F ⊗ U F → U F ⊗ U F : λ ⊗ µ¯ 7→ µ ⊗ λ¯ , restrict to real automorphisms of the Hermitian and anti-Hermitian subspaces. On turn, † r can be extended to the whole tensor algebra ⊗s(U ⊗ U) as These give rise naturally to the anti-endomorphism:

r r r r † †s := ⊗s† : ⊗s(U ⊗ U) → ⊗s(U ⊗ U): τ 7→ τ . r One then deﬁnes the hermitian subspace of ⊗s(U ⊗ U) as r r r † ∨¯ s(U ⊗ U) ≡ ⊗s(U ∨¯ U) := τ ∈ ⊗s(U ⊗ U): τ = τ . In terms of components, the matrices of hermitian tensors are hermitian, that is

AB˙ BA˙ τ··· ···CD˙··· =τ ¯··· ···DC˙··· .

r Various types of tensors of ⊗s(U ∨¯ U) can be decomposed in interesting ways [HT85] taking into account the (anti)symmetry properties stated in §A.8. 120 A SPINORS: FURTHER DETAILS AND CALCULATIONS

A.9.4 As already stated in §??, one has

εAA˙BB˙CC˙DD˙ = i (εAC εBD ε¯A˙D˙ε¯B˙C˙ − εAD εBC ε¯A˙C˙ε¯B˙D˙) .

A.9.5 The following decomposition of an antisymmetric 2-tensor, depending on the choice of ε ∈ ∧2U , is used in the study of the electromagnetic ﬁeld (§??) and in the extension of the Fermi transport to spinors (§4.6). Let F ∈ ∧2H∗ . Then F = Φ ⊗ ε¯+ ε ⊗ Φ¯ ¯ that is FAA˙BB˙ = ΦAB ε¯A˙B˙ + εAB ΦA˙B˙ ,

1 # 2 F where Φ := 2 ε¯ cF ∈ ∨ U has the coordinate expression

1 A˙B˙ 1 B˙A˙ ΦAB = 2 ε¯ FAA˙BB˙ = 2 ε¯ FBB˙AA˙ = ΦBA . proof: The following proof is taken from Penrose and Rindler [PR84], 3.4 pp.149–150. First, one notes that any antisymmetric tensor ψ ∈ ∧2U F is proportional to ε hence it can be written in the form 1 C 1 CD ψAB = 2 ψ C εAB ≡ 2 ε ψCD εAB , as it can be checked by contracting both sides with εAB ; if τ ∈ U F ⊗ U F then

C CD τAB − τBA = τ C εAB ≡ ε τCD εAB .

Next, on uses FAA˙BB˙ = −FBB˙AA˙ in order to write 1 1 FAA˙BB˙ = 2 (FAA˙BB˙ − FAB˙BA˙) + 2 (FAB˙BA˙ − FBB˙AA˙) = 0 = ΦAB ε¯A˙B˙ + εAB ΦA˙B˙ , where

1 A˙B˙ ΦAB = 2 ε¯ FAA˙BB˙ , 0 1 AB 1 AB ¯ ΦA˙B˙ = 2 ε FAB˙BA˙ = 2 ε FAA˙BB˙ = ΦA˙B˙ .

A.10 Spinor connection A.10.1 In some cases it may be useful to have an explicit expression of the Dirac operator. If ψ = (u, χ) one ﬁnds ∇/ ψ = (∇0 − ∇3)χ1 − (∇1 − i ∇2)χ2 ζ1+ + −(∇1 + i ∇2)χ1 + (∇0 + ∇3)χ2 ζ2+

1 2 1 + (∇0 + ∇3)u + (∇1 − i ∇2)u ¯z +

1 2 2 + (∇1 + i ∇2)u + (∇0 − ∇3)u ¯z = A.10 Spinor connection 121

3 4 = − (∇0 − ∇3)ψ − (∇1 − i ∇2)ψ ζ1+

3 4 − −(∇1 + i ∇2)ψ + (∇0 + ∇3)ψ ζ2+

1 2 3 − (∇0 + ∇3)ψ + (∇1 − i ∇2)ψ ζ +

1 2 4 − (∇1 + i ∇2)ψ + (∇0 − ∇3)ψ ζ .

A.10.2 I give an alternative description of the relations between the spinor connection and the induced connections on bundles derived from S. First, write the coefficients of Γ as Γ A := Γλσ A , a B a λ B with Γλ : M → , where (σ A ) denotes the Pauli matrices (§A.4). By a straightforward a C λ B calculation one finds a a ∇l = −2Ga dx ⊗ l , ∇ε = 2i Ya dx ⊗ ε , where Y = 1 (Γ A − Γ¯ A˙ ) = 1 (Γ0 − Γ¯0) , a 4i a A a A˙ 2i a a G = 1 (Γ A + Γ¯ A˙ ) = 1 (Γ0 + Γ¯0) . a 4 a A a A˙ 2 a a Since Ga and Ya are real, one sees that Γ actually yields a linear connection on L, and that the induced linear connection on ∧2U is Hermitian (preserves its natural Hermitian structure). Let Γ0 and Γ˜ be the connections induced on S ⊗ S and U ⊗ U, respectively. Then their coefficients in the frame (τλ) turn out to be given by 0 µ ν ¯ µ ¯ν µ Γ a λ = Γa Σνλ + Γa Σνλ , ˜ µ 0 µ µ 0 µ 0 ¯0 µ Γa λ = Γ a λ − 2 Ga δ λ = Γ a λ − ( Γa + Γa) δ λ , µ ˜ µ where (§A.4)σ ˆνσˆλ = Σνλ σˆµ . Note that the coefficients Γa λ are real. We can show these coefficients in a matrix:  0 ¯0 1 ¯1 2 ¯2 3 ¯3  Γa + Γa Γa + Γa Γa + Γa Γa + Γa   Γ1 + Γ¯1 Γ0 + Γ¯0 −i (Γ3 − Γ¯3) i (Γ2 − Γ¯2)  0 µ  a a a a a a a a  Γ =   . a λ  2 ¯2 3 ¯3 0 ¯0 1 ¯1   Γa + Γa i ( Γa − Γa) Γa + Γa −i ( Γa − Γa)   3 ¯3 2 ¯2 1 ¯1 0 ¯0 Γa + Γa −i ( Γa − Γa) i ( Γa − Γa) Γa + Γa A further calculation gives ∇[Γ]˜˜ g = 0, hence the connection induced on U ⊗ U by any 2-spinor connection is metric, and is reducible to a metric connection on the Hermitian bundle H := U ∨¯ U. ˜ µ Because of the metricity, the coefficients Γa λ are antisymmetric. Their expression above can be rewritten, by separating the ‘timelike’ index 0 from ‘spacelike’ indices (indicated by latin letters p, q, r, s, ..), as ˜ r p ¯p rs Γa q = −i( Γa − Γa)εsqpδ , ˜ r r ¯r Γa 0 = Γa + Γa , ˜ 0 p ¯p Γa q = ( Γa + Γa)δpq . 122 A SPINORS: FURTHER DETAILS AND CALCULATIONS

λ In either form, one sees that the relation between the coeﬃcients Γa and the coeﬃcients Ga , ˜ µ 72 Ya and Γa λ can be inverted, giving

A A ˜ A A 1 ˜ λ µp A Γa B ≡ (Ga + i Ya)δ B + Γa B = (Ga + i Ya) δ B + 4 Γa µ Σλ σp B = A i qsp ˜ r 1 ˜ p A = (Ga + i Ya)δ B + ( 4 δqr ε Γa s + 2 Γa 0) σp B .

Explicitely: ˜ 1 ˜ 1 ˜ 2 ! 1 i Γa 2 −Γa 3 + i Γa 3 Γ˜ A = . a B 2 ˜ 1 ˜ 2 ˜ 1 Γa 3 + i Γa 3 −i Γa 2

Moreover one has the similar relation

A A 1 ˜ λ µp A Rab B = −2 (dG + i dY )ab δ B + 4 Rab µ Σλ σp B =

A i qsp ˜ r 1 ˜ p A = −2 (dG + i dY )ab δ B + ( 4 δqrε Rab s + 2 Rab 0) σp B between the curvature tensors of Γ and Γ.˜ ˜ λ Remark. If a 2-spinor connection Γ fulfills ∇τ0 = 0 , that is Γa 0 = 0 , then one has r ¯r λ the relations Ga = 0 and Γa + Γa = 0 ; hence the coefficients Γa are all immaginary. More generally, this is a characterization of a Hermitian linear connection on a complex vector bundle with 2-dimensional fibres, whose coefficients are written as a linear combination of Pauli matrices.

A.10.3 The 4-spinor coordinate expression of Γ

α α 1 ˜ λµ α Γa β = fa δ β + 4 Γa (γλ ∧ γµ) β can be directly recovered by using just the property ∇γ = 0 . Write

λ λµ ˘λ ˘ Γa = fa 11+ fa γλ + fa γλ ∧ γµ + fa γλ γη + fa γη , λ λµ ˘λ ˘ fa , fa , fa , fa , fa : M → C ,

λµ and fa is antisymmetric in the upper indices. For any θ : M → End(W ) one has ∇aθ = ∂aθ + θ Γa − Γaθ . λ ˘λ λ ˘λ From ∇η = 0 it follows ∇γη = 0, namely 2(fa γηγλ +fa γλ) = 0 which implies fa = fa = 0 . Since ∂aγ = 0 (working in the appropriate ‘gauge’), the condition ∇γ = 0 can be expressed ˜ ρ ˜ ρ ˘ λµ as ∇a[ Γ]γν = −Γa νγρ . Then one gets Γa νγρ = Γaγν −γν Γa = 2faγηγν +2fa (gνµγλ −gνλγµ). ˘ ˜ ρ λµ ρλ 1 ˜ ρλ Hence fa = 0 and Γa νγρ = 2fa (gνµγλ − gνλγµ). The latter relation yields fa = 4 Γa . No condition is to be imposed on fa .

72 λµ By convention (§A.4) indices in the Σ-symbols are raised and lowered via δ and δλµ . A.11 Spinor Laplacian 123

A.10.4 One has the ‘commutation’ formulas ν ν ˜ ν µ γ Γa = Γa γ + Γa µ γ , ρ ν ρ ν ˜ ν ρ µ ˜ ρ µ ν γ γ Γa = Γa γ γ + Γa µ γ γ + Γa µ γ γ . proof: The fundamental Cliﬀord product property (§B.1) can be rewritten ν ν ν γ γλ = 2 δ λ 11 − γλ γ . Hence ν ν ν ν ν γ γλ γµ = (2 δ λ 11 − γλ γ ) γµ = 2 δ λ γµ − γλ γ γµ = ν ν ν = 2 δ λ γµ − γλ (2 δ µ 11 − γµ γ ) = ν ν ν = 2 δ λ γµ − 2 δ µ γλ + γλ γµ γ , and ν ν ν ν γ (γλ γµ − γµ γλ ) = 4 δ λ γµ − 4 δ µ γλ + (γλ γµ − γµ γλ ) γ . Then one ﬁnds ν ν 1 ˜ λµ γ Γa = γ fa 11+ 8 Γa (γλ γµ − γµ γλ ) = ν 1 ˜ λµ ν ν ν = fa γ + 8 Γa 4 δ λ γµ − 4 δ µ γλ + (γλ γµ − γµ γλ) γ = ν 1 ˜ νµ 1 ˜ λν ν ˜ νµ ν = fa γ + 2 Γa γµ − 2 Γa γλ + Γa γ = Γa γµ + Γa γ , and ρ ν ρ ν ˜ ν µ ˜ ρ µ ρ ν ˜ ρ µ ν γ γ Γa = γ ( Γa γ + Γa µ γ ) = (Γa µ γ + Γa γ ) γ + Γa µ γ γ = ˜ ν ρ µ ˜ ρ µ ν ρ ν = Γa µ γ γ + Γa µ γ γ + Γa γ γ .

A.11 Spinor Laplacian this section is to be thoroughly reviewed for possible mistakes

2 λ µ Consider the spinor Laplacian ∇/ ψ = γ ∇λ(γ ∇µψ) ; taking G = 0 one has the expression / 2 1 # # 1 # # ∇ ψ = ∆ψ − 2 T cc(γ ∧ γ )c∇ψ + 2 (γ ∧ γ )ccR(ψ) =

ab 1 h λ µ 1 λ µ = g ∇a∇bψ − 2 T λµ γ ∧ γ ∇hψ + 2 γ ∧ γ Rλµψ =

ab 1 h a b = g ∇a∇bψ − 2 T ab γ ∧ γ ∇hψ+ a b 1 + −i dYab γ ∧ γ + 4 hRi+ b b b b h b h b h a d + (∇aT bd − ∇dT ba − ∇bT ad + T bh T ad − T ah T bd + T dh T ba) γ ∧ γ +

1 abcd h h k + 8 ε gch (∇aT bd − T ak T bd) γη ψ , where hRi is the scalar curvature. proof: As a ﬁrst step, taking into account the expression of the torsion in terms of Θ calculate the Lie bracket ← a ← h ← a ← h [Θλ , Θµ] = Θλ ∂aΘµ − Θµ ∂aΘλ ∂xh = 124 A SPINORS: FURTHER DETAILS AND CALCULATIONS

← a h ← b ← a h ← b = Θλ δ b ∂aΘµ − Θµ δ b ∂aΘλ ∂xh =

← a ← h ν ← b ← a ← h ν ← b = Θλ Θν Θb ∂aΘµ − Θµ Θν Θb ∂aΘλ ∂xh =

← a ← h ν ← b ← a ← h ν ← b = −Θλ Θν ∂aΘb Θµ + Θµ Θν ∂aΘb Θλ ∂xh =

← h ← a ← b ν ν = Θν Θλ Θµ −∂aΘb + ∂bΘa ∂xh =

← h ← a ← b ν α ˜ ν α ˜ ν = −Θν Θλ Θµ T ab − Θa Γb α + Θb Γa α ∂xh =

← a ← b h ← h α ˜ ν ← h α ˜ ν = −Θλ Θµ T ab − Θν Θa Γb α + Θν Θb Γa α ∂xh . Hence λ µ λ µ h ← h ← a ˜ ν γ ∧ γ ∇[Θλ ,Θµ]ψ = −γ ∧ γ T λµ + 2 Θν Θλ Γa µ ∇hψ =

λ µ h ˜ ν = −γ ∧ γ T λµ∇hψ + 2 Γλ µ ∇νψ .

Now, using γλγµ =g ˜λµ + γλ ∧ γµ , and observing that

2 ∇λ∇µψ = ∇λ(Θµc∇ψ) = ∇λ(Θµ)c∇ψ + ΘλcΘµc(∇ ψ) = ˜ ν 2 = −Γλ µ ∇νψ + ΘλcΘµc(∇ ψ) , one gets

λ µ λ µ λ µ γ ∇λ(γ ∇µψ) = γ (∇λγ )∇µψ + γ γ ∇λ∇µψ =

˜ µ λ ν λµ λ µ = Γλ ν γ γ ∇µψ + (˜g + γ ∧ γ )∇λ∇µψ =

˜ µλ ˜ µ λ ν ˜ νλ = Γλ ∇µψ + Γλ ν γ ∧ γ ∇µψ − Γλ ∇νψ + ∆ψ+

1 λ µ + 2 γ ∧ γ (∇λ∇µψ − ∇µ∇λψ) =

˜ ν λ µ 1 λ µ = ∆ψ + Γλ µ γ ∧ γ ∇νψ + 2 γ ∧ γ Rλµψ + ∇[Θλ ,Θµ]ψ =

1 λ µ h = ∆ψ + 2 γ ∧ γ Rλµψ − T λµ ∇hψ . The stated expression now follows by elaborating the curvature term. Remembering the identities found in §B.2 one ﬁnds

1 λ µ 1 λ µ 1 ν ρ 2 γ ∧ γ Rλµ = 2 γ ∧ γ −2i dYλµ 11+ 4 Rλµνρ γ ∧ γ =

λ µ 1 λρ µν λν µρ = −i dYλµ γ ∧ γ + 8 Rλµνρ g g − g g − − gµρ γλ ∧ γν + gµν γλ ∧ γρ + gλρ γµ ∧ γν − gλν γµ ∧ γρ + γλ ∧ γµ ∧ γn ∧ γρ = 125

λ µ 1 λ µ λ µ ν ρ = −i dYλµ γ ∧ γ + 8 2 hRi + 4 Ricλµ γ ∧ γ + Rλµνρ γ ∧ γ ∧ γ ∧ γ .

The antisymmetric part of the Ricci tensor has been already calculated. As for the last tern, one gets

λ µ n ρ a b c d Rλµνρ γ ∧ γ ∧ γ ∧ γ = Rabcd γ ∧ γ ∧ γ ∧ γ =

1 abcd i j h k = 4! Rabcd ε εijhk γ ∧ γ ∧ γ ∧ γ = abcd h abcd = Rabcd ε γη = gch Rab d ε γη =

1 h abcd = 3! gch R[ab d] ε γη =

1 abcd h h k = 3! ε gch A[abd] ∇aT bd − T ak T bd γη = abcd h h k = ε gch ∇aT bd − T ak T bd γη , where A[abd] stands for the antisymmetrization operator relatively to the indices a, b, d without dividing by 3! .

B Cliﬀord algebra

B.1 Clifford algebra Let V be a real vector space endowed with a non-degenerate symmetric, possibly non-positive, 2-form g ∈ V ∗ ⊗ V ∗ . The Clifford algebra C(V , g), henceforth denoted simply by C, is the associative algebra generated by V where the product of any u, v ∈ V is subjected to the condition u v + v u = 2 g(u, v) 11 , u, v ∈ V , or, equivalently, v v = g(v, v) 11 , v ∈ V . The Clifford algebra fulfills the following universal property: if A is an associative algebra with unity and γ : V → A is a Clifford map, namely a linear map such that

γ(v) γ(v) = g(v, v) 11 , v ∈ V , then γ extends to a unique homomorphismγ ˆ : C → A. Namely the image ofγ ˆ, together with the restriction of the algebra product of A, turns out to be an algebra isomorphic to C. It can be proved that C is isomorphic, as a vector space, to the vector space underlying the exterior algebra ∧V . The isomorphism is characterized by the identiﬁcation

A(v1 . . . vp) ≡ v1 ∧ ... ∧ vp , where A stands for the antisymmetrisazion operator deﬁned by

1 X A(v . . . v ) = ε(π) v . . . v ; 1 p p! π(1) π(p) π the sum is extended to all permutations π of the set {1, . . . , p} and ε(π) denotes the permu- tation’s sign. In other terms, one has two distinct algebras on the same underlying vector 126 B CLIFFORD ALGEBRA space. Any element of C can be uniquely expressed as a sum of terms, each of well-defined exterior degree. In particular u v = u ∧ v + g(u, v) , u, v ∈ V . From this one sees that the Clifford algebra product does not preserve the exterior algebra degree, but only its parity. Namely, C is Z2-graded. Also, let φ ∈ ∧rV , θ ∈ ∧sV . Then the Clifford product φ θ turns out to be a sum of terms of exterior degree r+s, r+s−2 ,..., |r−s| (see §B.2). In particular, for any chosen orientation of V consider the g-unimodular positively oriented volume form η (unique up to sign); then ± ∗v = v η# = ±η# v , v ∈ V , η# η# = ±1 , the signs depending on the dimension of V and on the signature of g . If γ : V → A is a Clifford map, thenγ ˆ, being an injective vector space morphism, transfers onto its image also the exterior algebra structure. In particular, for decomposable elements of exterior degree 2 one obtains

1 1 γu ∧ γv :=γ ˆ(u ∧ v) = 2 [γu , γv] := 2 (γu γv − γv γu) . Also, note thatγ ˆ yields a homomorphism C[ , ] → A[ , ] of the commutator-induced Lie algebras.

B.2 Practical Clifford product calculations All Clifford products can be calculated in terms of the underlying exterior algebra by repeatedly applying the fundamental rule u v := u ∧ v + g(u, v) : hence the statement that the Clifford algebra is generated by V by this rule or, equivalently, that C is the exterior algebra ∧V as a vector space, with the further structure of associative algebra determined by the rule. For a decomposable element v1 v2 ··· vp , vj ∈ V , the calculation goes as follows. First, consider the antisymmetrization v1 ∧ v2 ∧ · · · ∧ vp , and write it explicitely as a sum of p! terms, corresponding to all permutations of Np ; next, by repeated application of the fundamental rule (in the form u v = 2 g(u, v) − v u), in each term exchange nearby elements which do not have the initial ordering, generating at the same time terms of lower exterior degree p−2 , p−4 and so on; eventually one expresses v1 ∧ v2 ∧ · · · ∧ vp as v1 v2 ··· vp plus a sum of terms of lower exterior degree; these further terms can be treated in the same way (it is a recursive procedure which can be easily implemented by a computer program), and one ends up with the above said decomposition (after moving v1 v2 ··· vp to the left-hand side of the equality and all the exterior products to the right-hand side). For example, consider u v w , u, v, w ∈ V . Write

1 u ∧ v ∧ w = 3! (u v w + v w u + w u v − v u w − u w v − w v u) =

1 h = 3! u v w + v 2 g(u, w) − u w + 2 g(u, w) − u w v− i − 2 g(u, v) − u v w − u 2 g(v, w) − v w − 2 g(v, w) − v w u =

1 h = 3! u v w − v u w − u w v + u v w + u v w + v w u+ i + 2 g(u, w) v + 2 g(u, w) v − 2 g(u, v) w − 2 g(v, w) u − 2 g(v, w) u = B.2 Practical Cliﬀord product calculations 127

1 h = 3! 3 u v w − 2 g(u, v) − u v w − u 2 g(v, w) − v w + v 2 g(u, w) − u w + i + 4 g(u, w) v − 2 g(u, v) w − 4 g(v, w) u =

1 h i = 3! 5 u v w − v u w + 6 g(u, w) v − 4 g(u, v) w − 6 g(v, w) u =

= u v w + g(u, w) v − g(u, v) w − g(v, w) u . Hence u v w = u ∧ v ∧ w + g(v, w) u − g(u, w) v + g(u, v) w . Repeating this procedure for u v w z , u, v, w, z ∈ V one has u v w z = u ∧ v ∧ w ∧ z+ + g(w, z) u ∧ v − g(v, z) u ∧ w + g(v, w) u ∧ z+ + g(u, z) v ∧ w − g(u, w) v ∧ z + g(u, v) w ∧ z+ + g(u, z) g(v, w) − g(u, w) g(v, z) + g(u, v) g(w, z) . For non decomposable elements one applies the procedure to each term, getting (u ∧ v) w = u ∧ v ∧ w + g(v, w) u − g(u, w) v ;

u (v ∧ w) = u ∧ v ∧ w − g(u, w) v + g(u, v) w ;

(u ∧ v) w z = u ∧ v ∧ w ∧ z+ + g(w, z) u ∧ v − g(v, z) u ∧ w + g(v, w) u ∧ z+ + g(u, z) v ∧ w − g(u, w) v ∧ z+ + g(u, z) g(v, w) − g(u, w) g(v, z);

u v (w ∧ z) = u ∧ v ∧ w ∧ z− − g(v, z) u ∧ w + g(v, w) u ∧ z+ + g(u, z) v ∧ w − g(u, w) v ∧ z + g(u, v) w ∧ z+ + g(u, z) g(v, w) − g(u, w) g(v, z);

(u ∧ v)(w ∧ z) = u ∧ v ∧ w ∧ z− − g(v, z) u ∧ w + g(v, w) u ∧ z+ + g(u, z) v ∧ w − g(u, w) v ∧ z+ + g(u, z) g(v, w) − g(u, w) g(v, z);

u (v ∧ w ∧ z) = u ∧ v ∧ w ∧ z+ + g(u, z) v ∧ w − g(u, w) v ∧ z + g(u, v) w ∧ z ;

(u ∧ v ∧ w) z = u ∧ v ∧ w ∧ z+ + g(u, v) w ∧ z − g(v, z) u ∧ w + g(u, z) v ∧ w .

In particular let (eλ) be a (not necessary orthonormal) basis of V , λ = 1,..., dim(V ). Then one has the following formulas (which will be useful for calculations):

eλ eµ = eλ ∧ eµ + gλµ ; 128 B CLIFFORD ALGEBRA

eλ eµ eν = eλ ∧ eµ ∧ eν + gµνeλ − gλνeµ + gλµeν ;

eλ eµ eν eρ = eλ ∧ eµ ∧ eν ∧ eρ+

+ gνρ eλ ∧ eµ − gµρ eλ ∧ eν + gµν eλ ∧ eρ+

+ gλρ eµ ∧ eν − gλν eµ ∧ eρ + gλµ eν ∧ eρ+

+ gλρ gµν − gλν gµρ + gλµ gνρ .

(eλ ∧ eµ) eν = eλ ∧ eµ ∧ eν + gµν eλ − gλν eµ ;

eλ (eµ ∧ eν) = eλ ∧ eµ ∧ eν − gλν eµ + gλµ eν ;

(eλ ∧ eµ) eν eρ = eλ ∧ eµ ∧ eν ∧ eρ+

+ gνρ eλ ∧ eµ − gµρ eλ ∧ eν + gµν eλ ∧ eρ+

+ gλρ eµ ∧ eν − gλν eµ ∧ eρ+

+ gλρ gµν − gλν gµρ ;

eλ eµ (eν ∧ eρ) = eλ ∧ eµ ∧ eν ∧ eρ+

− gµρ eλ ∧ eν + gµν eλ ∧ eρ+

+ gλρ eµ ∧ eν − gλν eµ ∧ eρ + gλµ eν ∧ eρ+

+ gλρ gµν − gλν gµρ ;

(eλ ∧ eµ)(eν ∧ eρ) = eλ ∧ eµ ∧ eν ∧ eρ−

− gµρ eλ ∧ eν + gµν eλ ∧ eρ+

+ gλρ eµ ∧ eν − gλν eµ ∧ eρ+

+ gλρ gµν − gλν gµρ ;

eλ (eµ ∧ eν ∧ eρ) = eλ ∧ eµ ∧ eν ∧ eρ+

+ gλρ eµ ∧ eν − gλν eµ ∧ eρ + gλµ eν ∧ eρ ;

(eλ ∧ eµ ∧ eν) eρ = eλ ∧ eµ ∧ eν ∧ eρ+

+ gλµ eν ∧ eρ − gµρ eλ ∧ eν + gλρ eµ ∧ eν .

One may also prove general formulas for the Cliﬀord product of exterior forms (see [HS84]). For example one has

X j u (v1 ∧ ... ∧ vr) = u ∧ v1 ∧ ... ∧ vr − (−1) g(u, vj)(v1 ∧ ... ∧ vˆj ∧ ... ∧ vr) , 1≤j≤r where a hat ‘ ˆ ’ means ‘dropped’. Then for any u ∈ V and θ ∈ C one has

u θ = u ∧ θ + u|θ , where contraction is made via the metric. Moreover if θ ∈ ∧rV then

θ u = (−1)r(u ∧ θ − u|θ) .

By repeatedly applying the above found rules one obtains, for example,

(u1 ∧ u2) θ = u1 ∧ u2 ∧ θ + u1 ∧ (u2|θ) − u2 ∧ (u1|θ) − (u1 ∧ u2)|θ . B.3 Cliﬀord group 129

In general, if φ ∈ Cr, θ ∈ Cs, r ≤ s, then the Clifford algebra product φ θ will be a sum of terms of exterior degree r+s, r+s−2 ,..., −r+s, from φ ∧ θ to (−1)r(r−1)/2 φ|θ. A Clifford algebra, as well as the underlying exterior algebra, has two canonical elements: the unit 11V , identified withthe number 1 , and the volume form η determined, up to sign, by the metric g. From the above expressions one obtains, for all u ∈ V ,

u η = ∗ u , η u = −(−1)n u η = −(−1)n ∗ u , where ∗ is the Hodge isomorphism, and

η2 := η η = (−1)n(n−1)/2 ∗η = (−1)q+n(n−1)/2 , where q is the number of negative elements in the signature (p, q) of g. In coordinates one has73 p|g| η = ηλ1...λn e ∧ ... ∧ e := ελ1...λn e ∧ ... ∧ e ; λ1 λn n! λ1 λn

p|g| η := e η = g ελ1...λn−1 e ∧ ... ∧ e = µ µ (n − 1)! µλ1 λ1 λn−1

p µλ1...λn−1 = n |g| gµλ1 η eλ1 ∧ ... ∧ eλn−1 ;

λ u η = u ηλ .

B.3 Clifford group Let C× be the group of all invertible elements of C. The adjoint action of C× on C is defined by Ad(Λ)(Φ) := ΛΦΛ−1 , Λ ∈ C× , Φ ∈ C . The Clifford group Cl ≡ Cl(V , g) is defined to be the group of all invertible elements of C for which V ⊂ C is stable under the adjoint action. In other terms, θ ∈ C is an element of Cl iff θ v θ−1 ∈ V , ∀v ∈ V . It turns out [Cr90, Gr78] that Cl is (multiplicatively) generated by all non isotropic elements in V , namely any element of Cl is of the form

θ = v1 v2 . . . vn , vj ∈ V , g(vj, vj) 6= 0 , and its inverse is given by 1 θ−1 = v . . . v v , ν(θ) n 2 1

ν(θ) := g(v1, v1) g(v2, v2) . . . g(vn, vn) .

The subgroup Cl↑ ⊂ Cl is deﬁned as

Cl↑ := {θ ∈ Cl : ν(θ) > 0} ,

73This holds if µ runs from 1 to n. If µ runs from 0 to n − 1, as it is usual in relativity, then one has to replace −(−1)µ with +(−1)µ. 130 B CLIFFORD ALGEBRA namely θ ∈ Cl is an element of Cl↑ iﬀ it has an even number of Cliﬀord factors with negative square (and any number of factors with positive square). The subgroup

Cl+ ⊂ Cl is constituted by all even-degree elements. Moreover we set

Cl+↑ := Cl+ ∩ Cl↑ .

We have the subgroups

Pin := {θ = v1 v2 . . . vn , vj ∈ V , g(vj, vj) = ±1} ⊂ Cl , Spin ≡ Pin+ := Pin ∩ Cl+ , namely Spin is the subgroup of Pin constituted by all even-degree elements. Then

+ Cl = R × Pin , + + Cl = R × Spin . Note that the adjoint action of rθ ≡ (r, θ) ∈ Cl is the same as the adjoint action of θ ∈ Pin . One also sets

Pin↑ := Pin ∩ Cl↑ , Spin↑ := Spin ∩ Cl↑ = Pin ∩ Cl+↑ .

−1 ↑ If θ = v1 v2 . . . vn ∈ Pin then θ = ±vn . . . v2 v1 , and the plus sign holds if θ ∈ Pin . Clearly, all the above introduced groups are Lie groups. Denote by L(G) the Lie algebra of the Lie group G. Then (see [Cr90], Ch. 6)

L(C×) = C[ , ] ,

L(Cl) = L(Cl+) = L(Cl↑) = L(Cl+↑) = 2 [ , ] = R ⊕ ∧ V ⊂ C , L(Pin) = L(Spin) = L(Pin↑) = L(Spin↑) = = ∧2V ⊂ C[ , ] .

B.4 Dirac algebra In the case of Minkowski spacetime, namely when the vector space V , now denoted by H, is 4-dimensional, and g is a Lorentz metric with signature (1, 3), the Cliﬀord algebra is also called the Dirac algebra, and denoted by D . Since D is isomorphic (as a real vector space) to the exterior algebra ∧H , it has (real) dimension 24 = 16 . One has (see §6.4 for details)

v η# = −η# v = ∗v , v ∈ V ,

η# η# = η∗ η∗ = −1 , η∗ = −η# .

Denote by Lor ≡ Lor(H) := O(H, g) the (full) Lorentz group and by Lor+ := SO(H, g) its special (i.e. orientation preserving) subgroup. If v ∈ H is such that g(v, v) 6= 0 , then one easily sees that Ad(v) is the negative of the reflection through the hyperplane perpendicular to B.5 4-spinors 131 v . From a well-known theorem [Cr90, Gr78] it then follows that the adjoint action restricted to Cl is a group epimorphism onto the Lorentz group. The restriction to Pin turns out to be a two-to-one epimorphism Pin Lor , while the restriction to Spin turns out to be a + two-to-one epimorphism Spin Lor . Furthermore, from the above recalled interpretation of Ad(v) as a reflection one sees that Ad(v) preserves time-orientation iff g(v, v) > 0 . It follows that Cl↑ is the subgroup of Cl which preserves time-orientation, and the same holds for the other ‘up arrow’ subgroups. In particular, Spin↑ turns out to be the double covering of the special orthochronous Lorentz group Lor+↑ . The double covering Pin Lor determines a Lie algebra isomorphism L(Pin) ↔ L(Lor): l ↔ ˜l ,

If (eµ) is a basis of V then l and ˜l are related by 1 ˜µ νρ l = 4 l ρ g eµ ∧ eν .

B.5 4-spinors Let again H be a Minkowski space. Let W be a 4-dimensional complex vector space. Let

γ : H → EndW : v 7→ γv := γ(v) be a Cliﬀord map. Then Dγ :=γ ˆ(D) ⊂ EndW is a real vector algebra, the Dirac algebra generated by γ , obviously isomorphic to D . One has

C ⊗ Dγ := Dγ ⊕ i Dγ = EndW . The Dirac algebra has the canonical element ∗ # γη :=γ ˆ(η) = −γˆ(η ) . 2 Since γη = −1 one has a splitting W = U ⊕ U 0 into the direct sum of the (complex) eigenspaces of iγη with eigenvalues ±1, the projections 1 0 being given 2 (11 ∓ iγη). This is called the chiral splitting, and U and U are called the chiral subspaces of W . One finds that the Clifford map γ exchanges the chiral subspaces, namely 0 0 γv(U) ⊂ U , γv(U ) ⊂ U , v ∈ H . 0 If g(v, v) 6= 0, then the restrictions of γv to U and U are isomorphisms. It follows immediately that the odd part of Dγ exchanges the chiral subspaces, while the even part leaves them invariant. If (eλ) is a basis of H then one sets γλ := γ(eλ). One usually takes an orthonormal and positively oriented basis, and sets γ5 := −i γ0γ1γ2γ3 . Then γ5 = −i γη . The essential algebraic structure necessary for doing physics is a 4-spinor structure,74 defined to be a couple (γ, k) where γ is a Dirac map as above, and k ∈ W F ∨¯ W F is a Hermitian 2-form on W fulfilling

k(γvφ, ψ) = k(φ, γvψ) , v ∈ V , φ, ψ ∈ W . One easily proves 74Detailed proofs of the statements of this and the next sections, as well as a more complete description of the family of spinor structures in a given space W , can be found in [CJ97a]. 132 B CLIFFORD ALGEBRA

Proposition B.1 Each chiral subspace is a maximal totally k-isotropic subspace.

Proposition B.2 The maps

0F [ 0 F 0 [ 0 U → U : u 7→ k (u)|U 0 , U → U : u 7→ k (u )|U , are anti-isomorphisms.

0 Then one can identify U with U F , and gets the 2-spinor setting W ≡ U ⊕ U F which in §1 will be introduced as the fundamental one.

Proposition B.3 Let L ⊂ U be a 1-dimensional subspace. Then there exists a unique 1- dimensional subspace L0 ⊂ U 0, called k-conjugated of L , such that k(L, L0) = 0.

Set Q2 := ∧2U , a 1-dimensional complex vector space. There is a distinguished U(1)- −2 subset of normalized elements ε ∈ Q ≡ ∧2U F , characterized by the property75

−1 1 ◦ ε¯ = g(v,v) ε (γv, γv) , v ∈ H .

Hence, one has a distinguished Hermitian metric on Q2. A charge conjugation is deﬁned to be an antilinear map C : W → W fulﬁlling

C2 = 11 , k(Cψ, ψ) = 0 , ψ ∈ W ,

C ◦ γv = γv ◦ C , v ∈ H .

Then C(U) = U 0, C(U 0) = U and C sends any 1-dimensional subspace L ⊂ S to its k- conjugated subspace L0 ⊂ S0, and vice-versa. Note also that, given the ﬁrst assumption above, the second is equivalent to

k ◦ (C × C) = −k¯ .

It turns out that the family of charge conjugations, for a given spinor structure, is a U(1)- family, namely any two charge conjugations are diﬀerent by a phase factor. Actually, there is a one-to-one correspondence between charge conjugations and normalized symplectic forms ε ∈ Q−2 (see also §3.2). In 4-spinor terms this correspondence reads

◦ # ◦ [ C = −i γη k εW ,

εW = i k ◦ (Cγη × 11) , where 2 −1 −2 2 F εW := ε ⊕ ε¯ ∈ Q ⊕ Q ⊂ ∧ W is the symplectic form induced by ε on W .

2 −2 75Hereε ¯−1 ∈ Q denotes the inverse ofε ¯ ∈ Q relatively to the contraction which is deﬁned to be 1/4 standard exterior algebra contraction (§1.3). B.6 Complexiﬁed spinor groups 133

B.6 Complexiﬁed spinor groups Let γ : H → W be a Dirac map. Then its natural extensionγ ˆ sends the subgroups of C× to subgroups of the general linear group Gl(W ) of all complex automorphsims of W . When no confusion arises, one can just identify Pin ≡ γˆ(Pin), Spin ≡ γˆ(Spin) and the like. Let G ⊂ C× be any one of these subgroups. One has the natural ‘complexiﬁed’ extension Gc ⊂ Gl(W ) constituted by all elements of G multiplied by a phase factor, namely

c G := U(1) ×˜ G := (U(1) × G)/∼ = (U(1) × G)/Z2 , where ∼ is the equivalence relation (λ, θ) ∼ (λ0, θ0) ⇔ λθ = λ0θ0. The last equality follows from considering the subgroup of U(1)×G generated by (−1, −11),which is a normal subgroup 76 isomorphic to Z2 . Explicitely, c + + c Cl := U(1) ×˜ Cl := R × U(1) ×˜ Pin := R × Pin , + c + + + c Cl := U(1) ×˜ Cl := R × U(1) ×˜ Spin := R × Spin , ↑ c ↑ + ↑ + ↑ c Cl := U(1) ×˜ Cl := R × U(1) ×˜ Pin := R × Pin , +↑ c +↑ + ↑ + ↑ c Cl := U(1) ×˜ Cl := R × U(1) ×˜ Spin := R × Spin . One has the Lie algebras L(Clc) = L(Cl+ c) = L(Cl↑ c) = L(Cl+↑ c) = 2 [ , ] = C ⊕ γˆ( ∧ H) ⊂ End (W ) , L(Pinc) = L(Spinc) = L(Pin↑ c) = L(Spin↑ c) = 2 [ , ] = i R ⊕ γˆ( ∧ H) ⊂ End (W ) . L(Pin) = L(Spin) = L(Pin↑) = L(Spin↑) = =γ ˆ( ∧ 2H) ⊂ End[ , ](W ) . The isomorphism between the Lie algebras of Pin ≡ γˆ(Pin) and of Lor(H) can be read 1 ˜µ νρ l = 4 l ρ g γµ ∧ γν . In physics texts this is usually written as i ˜µν i l = − 4 l σµν , σµν := 2 [γµ , γν] = i γµ ∧ γν . From the deﬁnitions it follows that Clc is the group of all elements of Gl(W ) for which γ(H) is stable under the adjoint action. Then the map ϕ : Clc → End H deﬁned by

γ(ϕ(Λ)v) = Ad(Λ)(γ(v)) := Λγ(v)Λ−1 , v ∈ H, Λ ∈ Clc . turns out to be a group epimorphism Clc → Lor, as well as its restriction to Pinc . The restrictions of ϕ to Cl+ c and Spinc turn out to be group epimorphisms onto Lor+, and the c c subgroup Lor ×γ Cl ⊂ Lor × Cl constituted by all elements of the form (ϕ(Λ), Λ) can be c + + c ∼ + c identiﬁed with Cl itself. Similarly Lor ×γ Cl = Cl . Note that γ can be seen as an element of H∗ ⊗ EndW . Then the natural action of (Λ˜, Λ) ∈ Lor × Clc on γ is given by

(Λ˜, Λ)∗(γ) = Λ−1 γ ◦ Λ˜ Λ .

76These complexiﬁed subgroups are also called ‘torogonal groups’ [Cr90]. 134 B CLIFFORD ALGEBRA

In particular, the action of (ϕ(Λ), Λ) =∼ Λ ∈ Clc on γ is given by

Λ∗(γ) = Λ−1 γ ◦ ϕ(Λ) Λ = Λ−1Λ γ Λ−1Λ = γ .

Namely

Proposition B.4 a) The group of all automorphisms of (g, γ) is Clc. b) The group of all automorphisms of (g, η, γ) is Cl+ c.

Moreover one has the time-orientation preserving subgroups Cl↑ c and Cl+↑ c. One also ﬁnds:

Proposition B.5 c) The group of all automorphisms of (g, γ, k) is Pin↑ c . d) The group of all automorphisms of (g, η, γ, k) is Spin↑ c . If C is a charge-conjugation, then e) The group of all automorphisms of (g, η, γ, k, C) is Spin↑ . REFERENCES 135

References

[AM78] Abraham, R. and Marsden, J.E.: Foundations of Mechanics, Addison-Wesley, Redwood City (1978).

[Ar89] Arnold, V.I.: Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag, New York (1989).

[BT84] Balachandran, A.P. and Trahern, C.G.: Lectures on group theory for physicists, Bibliopolis, Napoli (1984).

[Ba80] Barut, A.O.: Electrodynamics and Classical Theory of Fields and Particles, Dover, New York (1980).

[BT87] Benn, I.M. and Tucker, R.W.: An Introduction to Spinors and Geometry with Applications in Physics, Adam Hilger, Bristol and Philadelphia (1987).

[BD82] Birrel, N.D. and Davies, P.C.W.: Quantum ﬁelds in curved space, Cambridge University Press, Cambridge (1982).

[BLM89] Blaine Lawson, H. and Michelson, M.-L.: Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989.

[Bl81] Bleecker, D.: Gauge Theory and Variational Principles, Addison-Wesley, Reading, Massachusets (1981).

[BLT75] Bogolubov, N.N., Logunov, A.A. and Todorov, I.T.: Introduction to Axiomatic Quantum Field Theory, Benjamin, Reading (1975).

[Bo67] Boman, J.: ‘Diﬀerentiability of a function and of its composition with functions of one variable’, Math. Scand. 20 (1967), 249–268.

[BTr87] Budinich, P. and Trautman, A.: ‘An introduction to the spinorial chessboard’, J. Geom. Phys. 4 (1987), 361–390.

[CC91I] Cabras, A. and Canarutto, D.: ‘Systems of Principal Tangent-valued Forms’, Rendiconti di Matematica VII 11 (1991), 471–493.

[CC91II] Cabras, A. and Canarutto, D.: ‘The System of Principal connections’, Rendiconti di Matematica VII 11 (1991), 849–871.

[CK95] Cabras, A. and Kol´aˇr,I.: ‘Connections on some functional bundles’, Czech. Math. J. 45, 120 (1995), 529–548.

[CK97] Cabras, A. and Kol´aˇr,I.: ‘On the iterated absolute diﬀerentiation on some functional bundles’, Arch. Math. Brno 33 (1997), 23–35.

[CK98] Cabras, A. and Kol´aˇr,I.: ‘The universal connection of an arbitrary system’, Ann. Mat. Pura e Appl. (IV) Vol. CLXXIV (1998), 1–11.

[CCKM90] Cabras, A., Canarutto, D., Kol´aˇr, I. and Modugno, M.: Structured bundles, Pitagora, Bologna (1990), 1–100. 136 REFERENCES

[C98] Canarutto, D.: ‘Possibly degenerate tetrad gravity and Maxwell-Dirac ﬁelds’, J. Math. Phys. 39, N.9 (1998), 4814–4823.

[C00a] Canarutto, D.: ‘Smooth bundles of generalized half-densities’, Archivum Mathe- maticum, Brno, 36 (2000), 111–124.

[C00b] Canarutto, D.: ‘Two-spinors, ﬁeld theories and geometric optics in curved spacetime’, Acta Appl. Math. 62 N.2 (2000), 187–224.

[C01] Canarutto, D.: ‘Generalized densities and distributional adjoints of natural operators’, Rendiconti del Seminario Matematico dell’Universit`ae del Politecnico di Torino, 59 N.4 (2001), 27–36.

[C02] Canarutto, D.: ‘Quantum connections on distributional bundles’, Proc. XV Italian Meeting on General Relativity and Gravitation, Frascati 9-12/9/02 (to appear).

[C04a] Canarutto, D.: ‘Connections on distributional bundles’, Rend. Semin. Mat. Univ. Padova 111 (2004), 71–97.

[C04b] Canarutto, D.: ‘Quantum connections and quantum ﬁelds’, Rend. Ist. Mat. Univ. Trieste, 36 (2004), 1–21.

[C05] Canarutto, D.: ‘Quantum bundles and quantum interactions’, Int. J. Geom. Met. Mod. Phys., 2 N.5, (2005), 895–917. arXiv:math-ph/0506058v2.

[C07] Canarutto, D.: ‘“Minimal geometric data” approach to Dirac algebra, spinor groups and ﬁeld theories’, Int. J. Geom. Met. Mod. Phys., 4 N.6, (2007), 1005– 1040. arXiv:math-ph/0703003.

[C09a] Canarutto, D.: ‘Fermi transport of spinors and free QED states in curved spacetime’, Int. J. Geom. Met. Mod. Phys., 6 N.5 (2009), 805–824; arXiv:0812.0651v1 [math-ph].

[C10a] Canarutto, D.: ‘Tetrad gravity, electroweak geometry and conformal symmetry’, Int. J. Geom. Met. Mod. Phys., 8 N.4 (2011), 797–819; arXiv:1009.2255v1 [math- ph].

[C12a] D.Canarutto: Positive spaces, generalized semi-densities and quantum interactions. J. Math. Phys. 53 (3), 032302 (2012); http://dx.doi.org/10.1063/1.3695348 (24 pages).

[CJ97a] Canarutto, D. and Jadczyk A.: ‘Fundamental geometric structures for the Dirac equation in General Relativity’, Acta Appl. Math. 50 N.1 (1998), 59–92.

[CJ97b] Canarutto, D. and Jadczyk, A.: ‘Two-spinors and Einstein-Cartan-Maxwell-Dirac ﬁelds’, Il Nuovo Cimento 113 B (1997), 49–67.

[CJM95] Canarutto, D., Jadczyk A. and Modugno, M.: ‘Quantum mechanics of a spin particle in a curved spacetime with absolute time’, Rep. Math. Phys. 36 (1995), 95–140. REFERENCES 137

[CM85] Canarutto, D. and Modugno, M.: ‘Ehresmann’s connections and the geometry of energy-tensors in Lagrangian ﬁeld theories’, Tensor 42 (1985), 112–120.

[Ch54] Chevalley, C.: The algebraic theory of spinors, Columbia University Press (1954).

[CdW82] Choquet-Bruhat, Y. and DeWitt-Morette, C.: Analysis, Manifolds and Physics, North-Holland, Amsterdam (1982).

[Cr90] Crumeyrolle, A.: Orthogonal and Symplectic Cliﬀord Algebras, Kluwer, Dordrecht (1990).

[DeGe2013] Derezi´nski,J. and G´erard,C.: Mathematics of Quantization and Quantum Fields, Cambridge University Press (2013).

[Di64] Dirac, P.: Lectures on quantum mechanics, Dover, New York (1964).

[Do80] Dodson, C. T. J.: Categories, Bundles and Spacetime Topology, Shiva Publishing Ltd., Orpington, Kent (1980).

[DV87] Dubois-Violette, M.: ‘Syst`emesdynamiques constraints: l’approche homologique’, Ann. Inst. Fourier, 37, n.4, p.45–57.

[Fe83] Ferraris, M.: ‘Affine Unified Theories of Gravitation and Electromagnetism’, in Proceedings of the JourneésRelativistes, 5–8 May 1983, Torino, Pitagora Editrice, Bologna (1983), 125–153.

[Fe84] Ferraris, M.: ‘Fibered connections and global Poincar´e-Cartanforms in higher- order calculus of variations’, Quaderni di Matematica dell’Universit di Torino, n.80, ottobre 1984, 1–49.

[FK82] Ferraris, M. and Kijowski J.: ‘Uniﬁed Geometric Theory of Electromagnetic and Gravitational Interactions’, Gen. Rel. Grav. 14, 1 (1982), 37–47.

[FK83] Ferraris, M. and Kijowski J.: ‘On the equivalence between the purely metric and the purely aﬃne theories of gravitation’, Rend. Sem. Mat. Univers. Politecn. Torino, 41, 3 (1983), 169–201.

[Fr82] Fr¨olicher, A.: Smooth structures, LNM 962, Springer-Verlag (1982) 69–81.

[FK88] Fr¨olicher, A. and Kriegl, A.: Linear spaces and diﬀerentiation theory, John Wiley & sons (1988).

[FN56] Fr¨olicher, A. and Nijenhuis, A.: ‘Theory of vector valued diﬀerential forms, I’, Indag.Math.˜ 18 (1956), 338–360.

[FN60] Fr¨olicher, A. and Nijenhuis, A.: ‘Invariance of vector form operations under map- pings’, Communicationes Mathematicae Helveticae 34 (1960), 227–248.

[Ga72] Garc´ıa, P.L.: ‘Connections and 1-jet ﬁbre bundle’, Rend. Semin. Mat. Univ. Padova 47 (1972), 227–242.

[Ga74] Garc´ıa,P.L.: ‘The Poincar´e-CartanInvariant in the Calculus of Variations’, Sym- posia Mathematica 14 (1974), 219–246. 138 REFERENCES

[GS64] Gel’fand, I.M. and Shilov, G.E.: Generalized functions, Academic Press, New York (1964).

[Ge68] Geroch, R.P.: ‘Spinor structures of space-times in general relativity’ I, J. Math. Phys. 9 (1968), 1739–1744.

[Ge70] Geroch, R.P.: ‘Spinor structures of space-times in general relativity’ II, J. Math. Phys. 11 (1970), 343–348.

[Gh07] Ghiotti, M.: ‘Gauge ﬁxing and BRST formalism in non-Abelian gauge theories’, Ph.D. thesis, University of Adelaide (2007), arXiv:0712.0786v1.

[Go69] Godbillon, C.: GéométrieDifférentielle et Mécanique Analitique, Hermann, Paris (1969).

[GS73] Goldsmith, H. and Sternberg, S.: ‘The Hamilton-Cartan formalism in the calculus of variations’, Ann. Inst. Fourier, Grenoble 23 (1973), 203–267.

[Gr89] Greiner, W.: Quantum Mechanics, An Introduction, Springer-Verlag, Berlin Hei- delberg (1989).

[GM89] Greiner, W. and M¨uller,B.: Quantum Mechanics, Symmetries, Springer-Verlag, Berlin Heidelberg (1989).

[Gr90] Greiner, W.: Relativistic Quantum Mechanics, Springer-Verlag, Berlin Heidelberg (1990).

[Gr78] Greub, W.: Multilinear Algebra, Springer, New York (1978).

[GP75] Greub, W. and Petry, H.R.: ‘Minimal coupling and complex line bundles’, J. Math. Phys. 16 (1975), 1347–1351.

[GP82] Greub, W. and Petry, H.R.: ‘The curvature tensor of Lorentz manifolds with spin structure’. Part I, C.R. Math. Rep. Acad. Sci. Canada IV (1982), 31–36. Part II, C.R. Math. Rep. Acad. Sci. Canada V (1982), 217–222.

[GH96] Gronwald, F. and Hehl, F.W.: ‘On the Gauge Aspects of Gravity’, Proc. of the 14th Course of the School of Cosmology and Gravitation on Quantum Gravity, held at Erice, Italy, May 1995, P.G. Bergamann, V. de Sabbata and H.-J. Treder eds. World Scientiﬁc, Singapore (1996).

[HCMN95] Hehl, F.W., McCrea, J.D., Mielke, E.W. and Ne’eman, Y.: ‘Metric-aﬃne gauge theory of gravity: ﬁeld equations, Noether identities, world spinors, and breaking of dilaton invariance’, Phys. Rep. 258 (1995), 1–171.

[HE73] Hawking, S.W. and Ellis, G.F.R.: The large scale structure of space-time, Cam- bridge Univ. Press, Cambridge (1973).

[HS84] Hestenes, D. and Sobczyk, G.: Cliﬀord Algebra to Geometric Calculus, D. Reidel Publishing Company, Dordrecht (1984).

[Hu65] Hu, S.-T.: Elements of Modern Algebra, Holden-Day, S. Francisco (1965). REFERENCES 139

[HT85] Huggett, S.A. and Tod, K.P.: An Introduction to Twistor Theory, Cambridge Univ. Press, Cambridge (1985).

[Hu72] Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin (1972).

[IW33] Infeld, L. and van der Waerden, B.L.: ‘Die Wellengleichung des Elektrons in der Allgemeinem Relativit¨atstheorie’,Sitz. Ber. Preuss. Akad. Wiss., Phys.-Math. Kl. 9 (1933), 380–401.

[Is95] Isham, C.J.: ‘Structural issues in quantum gravity’, gr-qc/9510063 v2, 1–42.

[IZ80] Itzykson, C. and Zuber, J.-B.: Quantum Field Theory, McGraw-Hill, New York (1980).

[J83] Jadczyk, A.: ‘Conservation laws and string-like matter distributions’, Ann. Inst. H. Poincar´e 38, 2 (1983), 99–111.

[J84] Jadczyk, A.: ‘Symmetry of Einstein-Yang-Mills system and dimensional reduction’, J. Geom. Phys. 1, N.2 (1984), 97–126.

[JM92] Jadczyk, A. and Modugno, M.: ‘An outline of a new geometrical approach to Galilei general relativistic quantum mechanics’, in Proc. XXI Int. Conf. on Dif- ferential Geometric Methods in Theoretical Physics, Tianjin 5-9 June 1992, ed. Yang, C.N. et al., World Scientiﬁc, Singapore (1992), 543–556.

[JM93] Jadczyk, A. and Modugno, M.: ‘Galilei general relativistic quantum mechanics’, Preprint Dipartimento di Matematica Applicata ”G. Sansone”, Florence (1993).

[JM94] Jadczyk, A. and Modugno, M.: ‘A scheme for Galilei general relativistic quantum mechanics’, in General Relativity and Gravitational Physics, M. Cerdonio, R. D’Auria, M. Francaviglia, G. Magnano eds., World Scientiﬁc, Singapore (1994), 319–337.

[JJM] Jadczyk, A., Janyˇska, J. and Modugno, M.: ‘Galilei general relativistic quantum mechanics revisited’, Preprint Dipartimento di Matematica Applicata ”G. Sansone”, Firenze, N.8 (1997), 1–58.

[JK85] Jakubiec, A. and Kijowski, J.: ‘On Interaction of the Uniﬁed Maxwell-Einstein Field with Spinorial Matter’, Lett. Math. Phys. 9 (1985), 1–11.

[JM96] Janyˇska, J. and Modugno, M.: ‘Phase space in general relativity’, in Differential Geometry and its applications, Proceedings of the 6th International Conference held in Brno, Czech Republic, 28 August–1 September 1995, Janyˇska, J., Koláˇr,I. and Slovák,J. editors, Masaryk University, Brno (1996), 573–602.

[JM02] Janyˇska, J. and Modugno, M.: ‘Covariant Schr¨odingeroperator’, J. Phys. A 35 (2002), 8407–8434.

[JM06] J. Janyˇska and M. Modugno: ‘Hermitian vector ﬁelds and special phase functions’, Int. J. Geom. Met. Mod. Phys., 3 N.4 (2006), 1–36; arXiv:math-ph/0507070v1.

[JMV08] Janyˇska, J., Modugno, M. and Vitolo, R.: ‘Semi–vector spaces and units of measurement’, Preprint (2008), arXiv:0710.1313v1. 140 REFERENCES

[JMV10] J. Janyˇska, M. Modugno and R. Vitolo: ‘An algebraic approach to physical scales’, Acta Appl. Math. 110 N.3 (2010), 1249–1276; arXiv:0710.1313v1 [math.AC].

[Ka93] Kaku, M.: Quantum Field Theory, A Modern Introduction, Oxford University Press (1993).

[Ka61] Kastler, D.: Introduction a l’´electrodynamique quantique, Dunod, Paris (1961).

[Ko84a] Kolaˇr,I.: ‘Higher order absolute diﬀerentiation with respect to generalised connections’, Diﬀ. Geom. Banach Center Publications 12 (1984), 153–161.

[Ko84b] Kolaˇr,I.: ‘A geometrical version of the higher order Hamilton formalism in ﬁbred manifolds’, J. Geom. Phys. 1 n.2 (1984), 127–137.

[KMS93] Koláˇr,I., Michor, P. and Slovák,J: Natural Operations in Differential Geometry, Springer-Verlag (1993).

[KM97] Kriegl, A. and Michor, P.: The convenient setting of global analysis, American Mathematical Society (1997).

[La62] Lang, S.: Introduction to Diﬀerentiable Manifolds, John Wiley & Sons, New York (1962).

[Li60] Lichnerowicz, A.: ‘Ondes et radiations électromagnétiqueset gravitationelles en relativitégénérale’,Annali di Matematica 50 (1960), 1–95.

[Li64] Lichnerowicz, A.: ‘Propagateurs, Commutateurs et Anticommutateurs en Rel- ativit´eGenerale’, in Relativity, Groups and Topology, ed. DeWitt, C. and De- Witt, B., pp 821–861, Gordon and Breach, New York (1964).

[MS84] Mandl, F. and Shaw, G.: Quantum Field Theory, John Wiley & Sons, Chichester (1984).

[MM83a] Mangiarotti, L. and Modugno, M.: ‘Fibered spaces, jet spaces and connections for ﬁeld theory’, in Proc. Int. Meeting on Geom. and Phys., 135–165, Pitagora Ed., Bologna 1983.

[MM83b] Mangiarotti, L. and Modugno, M.: ‘Some results on the calculus of variations on jet spaces’, Ann. Inst. H. Poinc. 39 (1983), 29–43.

[MM84] Mangiarotti, L. and Modugno, M.: Graded Lie algebras and connections on ﬁbred spaces, J. Math. Pures Appl. 83 (1984), 111–120.

[MM91] Marathe, K.B. and Modugno, M.: ‘Polynomial connections on aﬃne bundles’, Tensor 50 (1991), 35–46.

[MR99] Marsden, J.E. and Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd Edition, Springer, New York (1999).

[Mi01] Michor, P.W.: ‘FrlicherNijenhuis bracket’, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer (2001).

[MTW] Misner, C.W., Thorne, K.S. and Wheeler, J.A.: Gravitation, Freeman and Co., San Francisco (1973). REFERENCES 141

[Mo91] Modugno, M.: ‘Torsion and Ricci tensor for non-linear connections’, Diﬀ. Geom. and Appl. 2 (1991), 177–192.

[MK98] Modugno, M. and Kol´aˇr,I.: The Fr¨olicher-Nijenhuisbracket on some functional spaces, Ann. Pol. Math. LXVIII.2 (1998), 97–106.

[MST05] M. Modugno, D. Saller, J. Tolksdorf: ‘Classiﬁcation of inﬁnitesimal symmetries in covariant classical mechanics’, J. Math. Phys. 47 062903 (2006)

[MV96] Modugno, M. and Vitolo, R.: ‘Quantum connection and Poincar´e-Cartanform’, Proceedings of the meeting in honour of A. Lichnerowicz, held in Frascati (Rome), ed. by G. Ferrarese, Pitagora Editrice, Bologna (1995).

[Nu96] Nurowski, P.: ‘Optical geometries and related structures’, J. Geom. Phys. 18 (1996), 335–348.

[OR04] Ortega, J.-P. and Ratiu, T.: Momentum Maps and Hamiltonian Reduction, Birkh¨auser,Boston (2004).

[PC96] de la Pe˜na,L. and Cetto, A.M.: The Quantum Dice, Kluwer, Dordrecht (1996).

[Pe71] Penrose, R.: ‘Angular momentum: an approach to combinatorial space-time’, in Quantum Theory and Beyond—essays and discussions arising from a colloquium, Bastin T. editor, Cambridge Univ. Press, Cambridge (1971), 151–180.

[PR84] Penrose, R. and Rindler, W.: Spinors and space-time. I: Two-spinor calculus and relativistic ﬁelds, Cambridge Univ. Press, Cambridge (1984).

[PR88] Penrose, R. and Rindler, W.: Spinors and space-time. II: Spinor and twistor methods in space-time geometry, Cambridge Univ. Press, Cambridge (1988).

[Pe86] Percacci, R.: Geometry of nonlinear ﬁeld theories, World Scientiﬁc, Singapore (1986).

[Ro94] Rovelli, C.: ‘Half way through the woods’, Lecture presented at the ‘34th Annual Lecture Series’ of the Center for the Philosophy of Science of the University of Pittsburgh (1994).

[Ro99] Rovelli, C.: ‘The century of the incomplete revolution: searching for general relativistic quantum ﬁeld theory’, hep-th/9910131 (1999).

[Pr95] Prugove˘cki,E.: Principles of Quantum General Relativity, World Scientiﬁc, Sin- gapore (1995).

[Re83] Regge, T.: ‘The group manifold approach to uniﬁed gravity’, in Relativity, groups and topology II, Les Houches 1983, B.S. DeWitt and R. Stora editors, North- Holland, Amsterdam (1984), 933–1005.

[Ru91] Rudin, W.: Functional Analysis, McGraw-Hill, New York (1991).

[Ry85] Ryder, L.H.: Quantum ﬁeld theory, Cambridge Univ. Press, Cambridge (1985).

[SW77] Sachs, R.K. and Wu, H.: General Relativity for Mathematicians, Springer-Verlag, New York (1977). 142 REFERENCES

[Sa85] Sakurai, J.J.: Modern Quantum Mechanics, Benjamin, New York (1985).

[SN00] Samuel, J. and Nityananda, R.: ‘Transport along null curves’, J. Phys. A, 33 (2000), 2895-2905.

[Sa89] Saunders, D. J.: The Geometry of Jet Bundles, Cambridge University Press (1989).

[Sc66] Schwartz, L.: Th´eoriedes distributions, Hermann, Paris (1966).

[Sl86] Slov´ak,J.: Smooth structures on ﬁbre jet spaces, Czech. Math. J. 36, 111 (1986), 358–375.

[So74] Souriau, J.-M.: ‘Modèlede particule àspin dans le champ électromagnétiqueet gravitationnel’, Ann. Inst. H. Poincaré 20, 4 (1974), 315–364.

[St79] Stakgold, I.: Green’s Functions and Boundary Value Problems, John Wiley & Sons, New York (1979).

[St94] Sternberg, S.: Group theory and physics, Cambridge University Press, Cambridge (1994).

[SV00] D. Saller, R. Vitolo: ‘Symmetries in covariant classical mechanics’, J. Math. Phys. 41, 10 (2000), 6824–6842, arXiv:math-ph/0003027v2.

[Tr67] Trautman, A.: ‘Noether Equations and Conservation Laws’, Commun. Math. Phys. 6 (1967), 248–261.

[Tr97] Trautman, A.: ‘On Complex Structures in Physics’, Vienna, Preprint ESI 470 (1997), 1–16.

[TT94] Trautman, A. and Trautman, K.: ‘Generalized pure spinors’, J. Geom. Phys. 15 (1994), 1–22.

[Va84] Varadarajan, V.S.: Lie groups, Lie algebras and their representations, Springer- Verlag, N.Y. (1984).

[Ve94] Veltman, M.: Diagrammatica, Cambridge University press (1994).

[Vi99] R. Vitolo: ‘Quantum structures in Galilei general relativity’, Annales de l’Institut H. Poincar´e, 70 (1999), 239–258.

[Vi00] R. Vitolo: ‘Quantum structures in Einstein general relativity’, Lett. Math. Phys. 51 (2000), 119–133.

[Wa84] Wald, R.M.: General Relativity, The University of Chicago Press, Chicago (1984).

[Wa83] Warner, F.W.: Foundations of diﬀerentiable manifolds and Lie groups, Scott, Foresman and Company, Glenview IL (1983).

[We96] Weinberg, S.: The Quantum Theory of Fields, Vol. I-II, Cambridge University press (1996).

[We80] Wells, R.O.: Diﬀerential Analysis on Complex Manifolds, Springer-Verlag, New York (1980). REFERENCES 143

[YLV] Yutsis, A.P., Levinson, I.B. and Vanagas, V.V.: Mathematical apparatus of the theory of angular momentum, Israel Program for Scientiﬁc Translations, Jerusalem (1962).