5. SPINORS 5.1. Prologue. 5.2. Clifford algebras and their representations. 5.3. Spin groups and spin representations. 5.4. Reality of spin modules. 5.5. Pairings and morphisms. 5.6. Image of the real spin group in the complex spin module. 5.7. Appendix: Some properties of the orthogonal groups. 5.1. Prologue. E. Cartan classified simple Lie algebras over C in his thesis in 1894, a classification that is nowadays done through the (Dynkin) diagrams. In 1913 he classified the irreducible finite dimensional representations of these algebras1. For any simple Lie algebra g Cartan's construction yields an irreducible representation canonically associated to each node of its diagram. These are the so-called fun- damental representations in terms of which all irreducible representations of g can be constructed using and subrepresentations. Indeed, if πj(1 j `) are the ⊗ ≤ ≤ fundamental representations and mj are integers 0, and if vj is the highest vector ≥ of πj, then the subrepresentation of m1 m` π = π⊗ ::: π⊗ 1 ⊗ ⊗ ` generated by m1 m` v = v⊗ ::: v⊗ 1 ⊗ ⊗ ` is irreducible with highest vector v, and every irreducible module is obtained in this manner uniquely. As is well-known, Harish-Chandra and Chevalley (independently) developed around 1950 a general method for obtaining the irreducible representa- tions without relying on case by case considerations as Cartan did. `+1 j If g = sl(` + 1) and V = C , then the fundamental module πj is Λ (V ), and all irreducible modules can be obtained by decomposing the tensor algebra over the 1 defining representation V . Similarly, for the symplectic Lie algebras, the decomposi- tion of the tensors over the defining representation gives all the irreducible modules. But Cartan noticed that this is not the case for the orthogonal Lie algebras. For these the fundamental representations corresponding to the right extreme node(s) (the nodes of higher norm are to the left) could not be obtained from the tensors over the defining representation. Thus for so(2`) with ` 2, there are two of these, ` 1 ≥ denoted by S±, of dimension 2 − , and for so(2` + 1) with ` 1, there is one such, ` ≥ denoted by S, of dimension 2 . These are the so-called spin representations; the S± are also referred to as semi-spin representations. The case so(3) is the simplest. In this case the defining representation is SO(3) and its universal cover is SL(2). The tensors over the defining representation yield only the odd dimensional irreducibles; the spin representation is the 2-dimensional representation D1=2 = 2 of SL(2). The weights of the tensor representations are integers while D1=2 has the weights 1=2, revealing clearly why it cannot be obtained from the tensors. However D1=2 gener-± ates all representations; the representation of highest weight j=2 (j an integer 0) 1=2 j 1=2 ≥ is the j-fold symmetric product of D , namely Symm⊗ D . In particular the 2 1=2 vector representation of SO(3) is Symm⊗ D . In the other low dimensional cases the spin representations are as follows. SO(4): Here the diagram consists of 2 unconnected nodes; the Lie algebra so(4) is not simple but semisimple and splits as the direct sum of two so(3)'s. The group SO(4) is not simply connected and SL(2) SL(2) is its universal cover. The spin representations are the representations D1×=2;0 = 2 1 and D0;1=2 = 1 2. The defining vector representation is D1=2;0 D0;1=2. × × × SO(5): Here the diagram is the same as the one for Sp(4). The group SO(5) is not simply connected but Sp(4), which is simply connected, is therefore the universal cover of SO(5). The defining representation 4 is the spin representation. The representation Λ24 is of dimension 6 and contains the trivial representation, namely the line defined by the element that corresponds to the invariant symplectic form in 4. The quotient representation is 5-dimensional and is the defining representation for SO(5). SO(6): We have come across this in our discussion of the Klein quadric. The diagrams for so(6) and sl(4) are the same and so the universal covering group for SO(6) is SL(4). The spin representations are the defining representation 4 of SL(4) and its dual 4∗, corresponding to the two extreme nodes of the diagram. The 2 2 defining representation for SO(6) is Λ 4 Λ 4∗. ' SO(8): This case is of special interest. The diagram has 3 extreme nodes and the group S3 of permutations in 3 symbols acts transitively on it. This means that S3 is the group of automorphisms of SO(8) modulo the group of inner au- 2 tomorphisms, and so S3 acts on the set of irreducible modules also. The vector representation 8 as well as the spin representations 8± are all of dimension 8 and S3 permutes them. Thus it is immaterial which of them is identified with the vector or the spin representations. This is the famous principle of triality. There is an octonionic model for this case which makes explicit the principle of triality8;13. Dirac's equation of the electron and Clifford algebras. The definition given above of the spin representations does not motivate them at all. Indeed, at the time of their discovery by Cartan, the spin representations were not called by that name; that came about only after Dirac's sensational discovery around 1930 of the spin representation and the Clifford algebra in dimension 4, on which he based the relativistic equation of the electron bearing his name. This circumstance led to the general representations discovered by Cartan being named spin representations. The elements of the spaces on which the spin representations act were then called spinors. The fact that the spin representation cannot be obtained from tensors meant that the Dirac operator in quantum field theory must act on spinor fields rather than tensor fields. Since Dirac was concerned only with special relativity and so with flat Minkowski spacetime, there was no conceptual difficulty in defining the spinor fields there. But when one goes to curved spacetime, the spin modules of the orthogonal groups at each spacetime point form a structure which will exist in a global sense only when certain topological obstructions (cohomology classes) vanish. The structure is the so-called spin structure and the manifolds for which a spin structure exists are called spin manifolds. It is only on spin manifolds that one can formulate the global Dirac and Weyl equations. Coming back to Dirac's discovery, his starting point was the Klein-Gordon equation 2 2 2 2 2 @ (@ @ @ @ )' = m ' @µ = 0 − 1 − 2 − 3 − @xµ where ' is the wave function of the particle (electron) and m is its mass. This equation is of course relativistically invariant. However Dirac was dissatisfied with it primarily because it was of the second order. He felt that the equation should be of the first order in time and hence, as all coordinates are on equal footing in special relativity, it should be of the first order in all coordinate variables. Translation invariance meant that the differential operator should be of the form D = γ @ X µ µ µ where the γµ are constants. To maintain relativistic invariance Dirac postulated that D2 = @2 @2 @2 @2 (1) 0 − 1 − 2 − 3 3 and so his equation took the form D' = im': ± Here the factor i can also be understood from the principle that only the i@µ are self adjoint in quantum mechanics. Now a simple calculation shows that no scalar 2 2 2 2 γµ can be found satisfying (1); the polynomial X X X X is irreducible. 0 − 1 − 2 − 3 Indeed, the γµ must satisfy the equations 2 γ = "µ; γµγν + γν γµ = 0(µ = ν)("0 = 1;"i = 1; i = 1; 2; 3) (2) µ 6 − and so the γµ cannot be scalars. But Dirac was not stopped by this difficulty and asked if he could find matrices γµ satisfying (2). He found the answer to be yes. In fact he made the discovery that there is a solution to (2) where the γµ are 4 4 matrices, and that this solution is unique up to similarity in the sense that any× 1 other solution (γµ0 ) of degree 4 is of the form (T γµT − ) where T is an invertible 4 4 matrix; even more, solutions occur only in degrees 4k for some integer k 1 and× are similar (in the above sense) to a direct sum of k copies of a solution≥ in degree 4. Because the γµ are 4 4 matrices, the wave function ' cannot be a scalar any- more; it has to have 4 components× and Dirac realized that these extra components describe some internal structure of the electron. In this case he showed that they indeed encode the spin of the electron. It is not immediately obvious that there is a natural action of the Lorentz group on the space of 4-component functions on spacetime, with respect to which the Dirac operator is invariant. To see this clearly, let g = (`µν ) be an element of the Lorentz group. Then it is immediate that D g 1 = g 1 D ;D = γ @ ; γ = ` γ : − − 0 0 µ0 µ µ0 X µν ν ◦ ◦ ν Since 2 1 2 2 D0 = (g D g− ) = D ◦ ◦ it follows that 1 γµ0 = S(g)γµS(g)− for all µ, S(g) being an invertible 4 4 matrix determined uniquely up to a scalar multiple.