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 2 A. Moreover one sets <f ≡ 2 (f + f) (real part) and =f ≡ 2i (f − f) (imaginary part). Throughout this section V and W will stand for finite-dimensional complex vector spaces, V ⊗ W for their complex 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 2 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 2 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 ¯ 2 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 (x??). 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 fbA_g is a basis of V and fb g its dual basis of V . For v 2 V and λ 2 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! wy, which on decomposable tensors reads (u ⊗ v¯)y = 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 2 V ⊗ V such that wy = w, while an arbitrary w is uniquely decomposed into the sum of an Hermitian and an anti-Hermitian tensor as 1 y 1 y w = 2 (w + w ) + 2 (w − w ) : 1 Note that w is Hermitian (resp. anti-Hermitian) iff its real part <w = 2 (w +w ¯) is symmetric 1 (resp. antisymmetric) and its immaginary part =w = 2i (w − w¯) is antisymmetric (resp. AB_ ¯ symmetric). In coordinates one has a corresponding situation: let w = w bA ⊗ bB_; then 1.2 Hermitian tensors 3 w is Hermitian (anti-Hermitian) iff the matrix (wAB_) of its components is of the same type, namelyw ¯B_A = ±wAB_. Obviously V F ⊗ V F decomposes in the same way, and one has the natural isomorphisms (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 2 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).