A SHORT OVERVIEW OF SPIN REPRESENTATIONS. Contents

A SHORT OVERVIEW OF SPIN REPRESENTATIONS YUNFENG ZHANG Abstract. In this note, we make a short overview of spin representations. First, we introduce Clifford algebras and spin groups, and classify their finite dimensional representations. Next, we touch on infinite dimensional spin representations, introduce spinor bundles and make an explicit construction of SL(, C)- spinorial tensor fields on Minkowski spacetime. Finally, we briefly talk about the results of Bargmann-Wigner on explicit construction of irreducible unitary spin representations of the inhomogeneous Lorentz group via spinorial tensor field equations. Contents 1. Clifford algebra, Spin groups, and Representations 1 1.1. Clifford algebra 1.. The Pin and Spin groups 3 1.3. Representation of Spin groups 4. Spinor Bundles, Spinor fields on Minkowski spacetime 5.1. Spinor Bundle 5.. Spinors on Minkowski spacetime 6 3. Irreducible unitary representation of the inhomogeneous Lorentz group and Bargmann-Wigner equations 7 3.1. Lie algebra structure of ISO 1,3, classification of representations via Casimir elements 7 3.. Bargmann-Wigner equations 7 References 8 1. Clifford algebra, Spin groups, and Representations Spin groups Spin r,s are universal covering group of SO r,s, which is the special orthogonal group, namely, the matrix group that preserves the quadratic form on R r+s. We are interested in representations q r,s (x) = x 1 + + x r x r+1 x r+s ρ : Spin r,s GL(V ) of Spin r,s that does not factor through SO r,s, in other words, the kernel of ρ does not contain -1. Such representations are call spin representations. As it turns out, the spin group could be embedded within the group of units of an algebra called Clifford algebra, and from the representations of the Clifford algebra one gets all the spin representations. We remark that by semisimplicity of the spin group, every finite dimensional representation could be decomposed as direct sum of irreducible ones, thus it suffices to get all the irreducible representations. Date: 014. 1

1.1. Clifford algebra. In this section, we pick up basic concepts about Clifford algebra. Let V be a vector space over a field k(= R, C) and q is a quadratic form. The Clifford algebra Cl(V, q) associated to V and q is an associative algebra defined as follows. Let T (V ) = r V denote the tensor algebra of V and define I q to be the ideal in T (V ) generated by all elements of the form v v + q(v)1 for v V. Then the Clifford algebra is defined to be the quotient r=0 Cl(V, q) T (V )/I q (V ). One easily sees that V embeds in Cl(V, q) naturally and Cl(V, q) is generated by V subject to the relations v v = q(v)1. This leads to the following universal characterization of the Clifford algebra. Proposition 1.1. Let f : V A be a linear map into an associative k-algebra with unit, such that f(v) f(v) = q(v)1 for all v V. Then f extends uniquely to a k-algebra homomorphism f : Cl(V, q) A. Furthermore, Cl(V, q) is the unique associative k-algebra with this property. Here are some example of consequences of this characterization. (1)The orthogonal group O(V, q), the subgroup of GL(V ) that preserves that quadratic form, extends canonically to a group of automorphisms of Cl(V, q). ()We have the automorphism α : Cl(V, q) Cl(V, q) which extends the map α(v) = v. This in turn gives a decomposition Cl(V, q) = Cl 0 (V, q) Cl 1 (V, q), Cl i (V, q) being the eigenspaces of α that corresponds to eigenvalues ( 1) i. Elements in Cl 0 and Cl 1 are said to have pure even or odd degree. An algebra with such a decomposition is called a Z -graded algebra. We may define instead of the ordinary tensor product a Z -graded tensor product of two Z -graded algebras A and B, A ˆ B, via a Z -graded multiplication, (a b) (a b ) = ( 1) deg(b) deg(a ) (aa ) (bb ). The Z -graded tensor product has the following useful consequence. Proposition 1.. Let V = V 1 V be a q-orthogonal decomposition of the vector space V. Then there exists a natural isomorphism of Clifford algebras where q i denotes the restriction of q to V i. Cl(V, q) Cl(V 1, q 1 ) ˆ Cl(V, q ) Proof. The isomorphism is induced via universal property of Clifford algebra from the map f : V Cl(V 1, q 1 ) ˆ Cl(V, q ), f(v) = v 1 1 + 1 v where v = v 1 + v is the decomposition of v with respect to the splitting V = V 1 V. In particular, if we consider Cl r,s Cl(V, q) where V = R r+s and q(x) = x 1 + x r x r+1 x r+s, then we have an isomorphism Cl r,s = Cl 1,0 ˆ ˆ Cl 1,0 ˆ Cl 0,1 ˆ ˆ Cl 0,1 where Cl 1,0 appears r times and Cl 0,1 appears s times. It s easily seen that Cl 1,0 = C and Cl0,1 = R, and it follows that dim R (Cl r,s ) = r+s. Lastly, let s observe that we have a canonical vector space isomorphism V Cl(V, q) given by v 1 v r 1 sign(σ)v σ(1) v σ(r). r! This is not an isomorphism of algebras unless q = 0. σ

1.. The Pin and Spin groups. We now consider the multiplicative group of units in the Clifford algebra, which is defined to be the subset Cl (V, q) {φ Cl(V, q) : φ 1 with φ 1 φ = φφ 1 = 1}. This group clearly contains all the elements x V with q(v) 0. This is a Lie group with Lie algebra Cl(V, q), thus we have the adjoint representation give by and give by Ad : Cl (V, q) Aut(Cl(V, q)) Ad φ (x) = φxφ 1 ad : Cl(V, q) End(Cl(V, q)) ad y (x) = [y, x] = yx xy. One computes for v V Cl(V, q) with q(v) 0, that Ad v (w) = w q(v, w) v, ( ) q(v) which has a geometrical interpretation as reflection across the hyperplane v = {w V : q(w, v) = 0}. It follows that Ad v (V ) = V and Ad v preserves the quadratic form. Therefore, if we define P (V, q) to be the subgroup of Cl (V, q) generated by the elements v V with q(v) 0 then we have the representation Ad : P (V, q) O(V, q) where O(V, q) is the orthogonal group of the form q. The group P (V, q) has certain important subgroups. Definition 1.3. The Pin group of (V, q) is the subgroup Pin(V, q) of P (V, q) generated by the elements v V with q(v) = ±1. The associated Spin group is defined by Spin(V, q) = Pin(V, q) Cl 0 (V, q). That is, Pin(V, q) = {v 1 v r P (V, q) : q(v i ) = ±1 for all j}, Spin(V, q) = {v 1 v r Pin(V, q) : r is even}. Now we modify the adjoint representation using the Z -graded structure of Clifford algebra to eliminate the negative sign in front of the equation ( ). Consider the twisted adjoint representation by setting Ãd : Cl (V, q) GL(Cl(V, q)) Ãd φ (y) = α(φ)yφ 1. Ãd φ (y) = ±Ad φ (y) the sign depending on whether φ is an even or odd element. In particular, q(v, w) Ãd v (w) = w q(v) v. Assume from now V is finite dimensional and q is nondegenerate. For any φ Pin(V, q), φ = v 1 v r, q(v i ) 0, we have Ãd φ = Ãd v 1 Ãd v r. Since each Ãd v i is reflection, thus by the classical result that the orthogonal group is generated by reflections, the representation Ãd : P(V, q) O(V, q) is surjective. One could also prove that the kernel is the group k of nonzero multiples of 1. Restricting to SP(V, q) Cl 0 (V, q), we get the surjective representation Ãd : SP(V, q) SO(V, q) where SO(V, q) is the special orthogonal group. 3

Finally, restricting to the groups Pin(V, q) and Spin(V, q) which simply normalizes P(V, q) and SP(V, q), we have the following short exact sequences 0 F Spin(V, q) Ãd SO(V, q) 1 0 F Pin(V, q) Ãd O(V, q) 1 where F = {±1} if k = R or F = {±1, ± 1} if k = C. 1.3. Representation of Spin groups. Every nondegenerate quadratic form over R or C could be reduced to the standard q(x) = x 1 + + x r x r+1 x r+s and q(z) = z 1 + + z n respectively. Let Cl r,s and Cl n denote the Clifford algebras that corresponds to the these forms. It turns out Cl r,s and Cl n could be explicitly described as matrix algebras K( m ) or K( m ) K( m ) over K = R, C or H, where K( m ) is simply the algebra of n n K-matrices. But the representation theory of such algebras are relatively simple. Theorem 1.4. (1)The natural representation ρ of K(n) on K n is, up to equivalence, the only irreducible real representation of K(n). ()K(n) K(n) has exactly two equivalent classes of irreducible real representations given by ρ 1 (φ 1, φ ) ρ(φ 1 ) and ρ (φ, φ ) ρ(φ ). We have similar results for irreducible complex representations. Now let s define the spinor representations as the restriction of the irreducible real or complex representations of Clifford algebra to the spin group. From these spinor representations one gets all the irreducible real or complex representations of the spin groups which do not factor though the orthogonal group. Let s simply record here one case in these results, namely the case when the Clifford algebra is Cl n,0 over the reals and n = 4m. Proposition 1.5. Consider 4m : Spin 4m GL(S) given by restricting an irreducible real representation ρ : Cl 4m,0 Hom R (S, S) to Spin 4m Cl 4m,0. Then we have a decomposition 4m = + 4m 4m where ± 4m are the inequivalent irreducible representations of Spin 4m. Proof. The key role in the analysis is played by the volume element in Cl n,0 ω = e 1 e n where {e i } is an orthonormal basis of the base space R n. It has the following basic properties Thus ρ(ω) = 1, we may decompose ω = 1, φω = ωα(φ) for all φ Cl n,0. S = S + S where S ± are eigenspaces of ρ(ω) corresponding to eigenvalues ±1 respectively. Since ω commutes with Cl 0 n,0, S ± are invariant under Cl 0 n,0. Consider the map f : R n 1 Cl 0 n,0 by setting f(e i ) = e n e i, i < n where R n 1 = span{e i } i n 1. One easily checks that f(x) = q(x)1, thus by universal property of Clifford algebra, f extends to an algebra homomorphism f : Cl n 1,0 Cl 0 n,0 which is in fact an isomorphism. f maps the volume element ω = e 1 e n 1 in Cl n 1,0 to the volume element ω Cln,0. 0 Via the isomorphism f : Cl n 1,0 = Cl 0 n,0, we may view the representation restricted on Cln,0 0 as representations of Cl n 1,0 on S ± and it follows that ω acts as Id on S + and Id on S, thus these two representations are inequivalent which correspond to the two irreducible representation classes of Cl n 1,0. Finally, we observe that any irreducible representation of Cln,0 0 restricts to an irreducible representation of Spin n,0 because Spin n,0 contains an additive basis for Cln,0. 0 We usually call the vectors in a spin representation space as spinors. 4

. Spinor Bundles, Spinor fields on Minkowski spacetime Now we turn to infinite dimensional (irreducible) (unitary) representations of the spin groups. A natural construction goes as follows. Let ρ : G Gl(V ) be a finite dimensional representation of G and consider an infinite dimensional linear space of functions on V, denoted F (V ). Then we may define a representation P : G Aut(F (V )) via (P g f)(x) = f(ρ g 1x), x V, f F (V ). In particular, we may choose V to be G itself, and G acts on G via left multiplication, and the function space chosen to be L (G). When G is a compact Lie group, for example the spin groups Spin n,0, this is the setting for Peter-Weyl theorem. It states that, the above particular unitary representation, as well as every unitary representation on a Hilbert space splits into irreducible finite dimensional unitary representations. But for non-compact groups, one could show that no nontrivial irreducible finite dimensional unitary representation exists. And it is an unsolved problem how to effectively classify irreducible unitary representations. Known approaches include the Langlands classification of admissible Harish-Chandra modules and the cases SL (R) and SL (C) are solved. We will not touch these general approaches but will talk about the SL (C) case, which is Spin 3,1 the universal cover group of the Lorentz group SO 3,1, the construction of which irreducible representations via relativistic wave equations will be seen. The basic idea of this construction is to consider a possibly infinite dimensional representations of the spin group SL (C) on a bundle of finite dimensional spinors, in this case the spinors being C on which SL (C) naturally acts as matrice multiplication, over the base manifold the Minkowski spacetime M. Now SL (C) acts on M via Lorentz transformation. Thus one may define (gφ) A (x) = g A Bφ B [g 1 (x)] where g = gb A SL (C), φ A (x) is a spinor field, i.e. a section of the bundle of spinors, x M, and elements in SL (C) acts on the Minkowski spacetime by Lorentz transformation. The Lie algebra simultaneously acts as differential operators on the spinor fields φ. This Lie algebra action plays a key rule of specifying irreducible representations in the following way. We define the Casimir elements as the elements in the universal enveloping algebra that commutes with all the Lie algebra elements. Thus the representatives of the Casimir elements, called Casimir operators, commutes with all representatives of the Lie algebra, hence by Schur s lemma on irreducible representation, must be a multiple of the identity operator. These numbers provides convenient labels of so as to classify the irreducible representations, and more importantly, they may be interpreted as the conserved quantities when the spinor field is interpreted as a physical system. One may set up differential equations of the form of a eigenvalue equation Cφ = cφ where C stands for the Casimir operator and c is a constant, and expect the solutions to be desired constructions of irreducible representations. This turns out to be the case. In the following sections, we will first introduce a rigorous definition of spinor bundles and then give an explicit construction of spinor fields and spinorial tensor fields on Minkowski spacetime..1. Spinor Bundle. To define a spinor bundle, one needs to define the notion of a principal fiber bundle. A principal fiber bundle (B, G, M, φ) consists of a manifold B (called the bundle manifold), a Lie group G (called the fiber group), a manifold M (called the base manifold), and a free left action φ : G B B satisfying the following local trivialization: for each x M there exists an open neighborhood, U, of x such that there is a diffeomorphism, ψ, taking π 1 (U) B into G U such that the action of G on π 1 (U) corresponds to left multiplication on G U, i.e. if ψ(p) = (g, x), then ψ(φ g (p)) = (g g, x). An important example of a principal fiber bundle is the bundle of bases of tangent space, and the group action is the natural action of GL(n) on the bases. If we require the bases to be orthonormal, assuming the base manifold M with a metric, we may construct the bundle of orthonormal bases via action of the orthogonal group. Definition.1. Let P SO be the principle fiber bundle of orthonormal bases on the base pseudo Riemannian manifold (M, g). Let ξ 0 : Spin SO be the universal covering homomorphism. Then a spin structure on M is a principal Spin-bundle P Spin together with a -sheeted covering ξ : P Spin (E) P SO (E) 5

such that ξ(gp) = ξ 0 (g)ξ(p) for all p P Spin and g Spin. In other words, the spin structures on E are in one-to-one correspondence with -sheeted coverings of P SO which are nontrivial on the fibers. We remark that the spin structure might not always exist. Consider the special case when P SO is simply-connected. (Even though the fiber SO is not simply connected, one could still have a simply-connected fiber bundle P SO.) In this case, there is no nontrivial -sheeted covering of P SO. It turns out that a topological invariant of the tangent bundle, namely the second Stiefel-Whitney class is the obstruction to existence of spin structures. To define spinor bundles, let s briefly describe the associated bundle construction. Let (B, G, M, φ) be a principal fiber bundle, and F another manifold, and a left action χ : G F F of G on F, there is a general procedure for constructing from B a new fiber bundle over M with fiber F as follows. Consider the left action ψ of G on B F by ψ g (b, f) = (φ g (b), χ g (f)) for b B, f F. We define B χ F M to be the set of orbits of G on B F. It is called the bundle associated to B by χ. Definition.. Let (M, g) be an oriented pseudo Riemannian manifold with a spin structure ξ : P Spin P SO. A spinor bundle of M is a bundle of the form where µ : Spin GL(S) is a spin representation. S(M) = P Spin (E) µ S.. Spinors on Minkowski spacetime. Let s consider the special case when M is the Minkowski spacetime, with the natural spin representation of SL (C) on C. We would love to explicitly describe the spinor fields and more generally the spinorial tensor fields. Let s start with W = C. Define a tensor T of type (k, l, k, l ) over W as a multilinear map from direct products of k copies of W, l copies of W, k copies of W, l copies of W to C. Here W is the complex conjugate space of W. We shall adopt index notation for T, e.g. TC ABD denotes a tensor of type (, 1; 1, 0). The complex conjugation map of vectors extends to tensors and maps a tensor T of type (k, l; k, l ) into a tensor, denoted T, of type (k, l ; k, l). Note that T = T. Note that the vector space of antisymmetric tensors of type (0, ; 0, 0) is one-dimensional. Let s choose a particular tensor ɛ AB = ɛ BA. Since ɛ AB is nondegenerate we obtain via ξ A ɛ AB ξ A an isomorphism of W and W. Following standard conventions, we define ɛ AB to be minus the inverse of ɛ AB, i.e. ɛ AB is the antisymmetric tensor of type (, 0; 0, 0) which satisfies ɛ AB ɛ BC = δc A, where δa C denotes the identity map on W. Now consider the natural action of the matrix group SL(, C) on tensors on W. Note in particular that L A CL B Dɛ AB = ɛ CD for L A B SL(, C), which is equivalent to the condition that LA B has unit determinant. This equation means that ɛ AB is preserved under the action of L A B. The relation between SL(, C) and the Lorentz group now may be established. The tensors of type (1, 0; 1, 0) comprise a four complex dimensional vector space, Y. Let o A, ι A be a basis of W satisfying o A ι A = 1. Then we have the following basis of Y : t AA = 1 (o A ō A + ι A ῑ A ), x AA = 1 (o A ῑ A + ι A ō A ), y AA = i (o A ῑ A ι A ō A ), z AA = 1 (o A ō A ι A ῑ A ). Now, complex conjugation maps Y into itself, and we call an element that is taken into itself under complex conjugation a real element. We can easily check that the above basis for Y are real and they span all the real elements of Y. Let V denote this four dimensional real vector space. 6

The tensor g AA BB = ɛ AB ɛ A B defines a multilinear map V V R and has a nondegenerate signature + by checking it on the basis. Now associate with each map L A B SL(, C) the map λ : V V defined by λ AA BB = LA A B L B. Since L A B preserves ɛ AB, it follows that λ AA BB preserves g AA BB. This means λaa BB is a Lorentz transformation on V. A more careful computation, e.g. in exponent form, yields that λ AA BB in fact belongs to the special orthogonal group on the Minkowski spacetime V. This gives an explicit universal covering map SL(, C) SO 1,3. Finally, we define the spinor field on the Minkowski spacetime to be simply a map of spacetime into spinor space W. Similary, a spinorial tensor field of a specified type is defined as a map of spacetime into the tensor space over W of that type. We define the representation of g ISL(, C), i.e., SL(, C) added with translations, on spinor fields via ψ A (x) L A B ψb [P 1 (x)] where L A B SL(, C) is the homogeneous part of g and P is the inhomogeneous Lorentz group (denoted ISO 1,3 ) element associated with g. This representation does not descends to a true representation of the inhomogeneous Lorentz group, thus a spin representation. One could see this by making the identification of the tangent space of the Minkowski spacetime with the space V defined earlier. Via the identification of the four spinorial tensor fields, t AA, x AA, y AA, z AA, of type (1, 0; 1, 0) constituting the basis of V with the basis of tangent space, we may consistently identify real spinorial tensor fields of type (1, 0; 1, 0) with vector fields, and more generally, spinorial tensor fields of type (k, l; k, l) with ordinary tensor fields of type (k, l). One realizes that the ISL(, C) representations of spinorial tensor fields of the type (k, l; k, l) descends to true representations of ISO 1,3 on ordinary tensor fields of type (k, l), but not for spinorial tensor fields of other types. 3. Irreducible unitary representation of the inhomogeneous Lorentz group and Bargmann-Wigner equations 3.1. Lie algebra structure of ISO 1,3, classification of representations via Casimir elements. Let g kl be the standard Minkowski metric tensor of signature +. The Lie algebra of ISO 1,3 is generated by p k and M kl, where p k is translation in the x k direction and M kl is a rotation in the (x k, x l ) plane with M kl = M lk. We record here the Lie bracket structures. Define [M kl, M mn ] = i(g lm M kn g km M ln + g kn M lm g ln M km ), [p k, p l ] = 0, [M kl, p m ] = i(g lm p k g km p l ). P = p k p k, W = 1 M klm kl p m p m M km M lm p k p l. One could compute to see that these two elements commutes with all M kl and p k, thus they are Casimir elements of the Lorentz group. Thus for every irreducible representation of ISO 1,3, it corresponds to an irreducible representation of iso 1,3, and the representatives of the Casimir elements are constant multiple of identity. Thus we may classify the representations according to the values of P = m as well as W = m S. Representations corresponding to all these cases are obtained by Bargmann and Wiger, and in particular for the cases m > 0 we have the realization of the representations on spinorial tensor fields on spacetime, which are generalizations of the Dirac equation. In the following exposition, we will sketch the construction of these spinorial tensor field equations, which corresponds to cases when m > 0, S = s(s + 1) and s = N, N a nonnegative integer. This corresponds to relativistic quantum particles of positive mass and N -spin. 3.. Bargmann-Wigner equations. As mentioned in the first paragraph of this section, to find a good candidate for the spinorial tensor field equation, one may try to set up an eigenvalue equation of the form P φ = m φ where φ is a spinorial tensor field, P = p k p k is one of the Casimir operators, m is a positive constant. To guess a possible form of P = p k p k, we could borrow some physical intuition, one being that the p k are momentum operators. One knows from basic quantum mechanics that the momentum operators are often 7

realized as differentiation, in fact, we define the momentum operator as multiplication by the frequency (which is the momentum) on the frequency space, thus on the physical space, they are differentiations. Thus p k p k may be realized as the Laplacian. We have ( x 1 + x + x 3 t )φ = m φ. Inheriting the notations from the last section, we could also rewrite it as AA AA φ = m φ. Now let φ be a spinor field of type (n = s, 0, 0, 0). The cases s = 0, 1, 1 correspond to the Klein-Gordon, Dirac, and Proca equations, respectively. From previous discussions, we see that s being a half integer corresponds to spinor representations while the integer s cases descend to true representations of the Lorentz group on ordinary tensor fields. For s > 0, the Bargmann-Wigner equation could also be expressed in the following form. Let the spinorial tensor field be φ A1 An A,,An. We define the auxiliary variable σ by A 1 A 1 φ A1 An then the Bargmann-Wigner equation becomes A 1 A1 σ A An A 1 A 1 = m σ A An A, 1 = m φ A1 An. From these coupled equations for the case n = 1, denoting the four components of φ A, σ A as ψ 0, ψ 1, ψ, ψ 3 respectively, one gets the more commonly known form of Dirac equation iγ µ µ ψ = mψ where γ µ are the matrix generators of the Clifford algebra Cl 1,3, which should be of little surprise. In fact, very real spin bundle is also a bundle of modules over the bundle of Clifford algebras, thus the Clifford acts on spinors naturally. Historically, Dirac discovered this equation by trying to take the square root of the Laplacian x 1 + x + x 3 t. One could generalize the notion of Dirac operator to be any first-order differential operator acting on a vector bundle (e.g. spinor bundle) with a Riemannian connection such that D =. For a famous example, for a spin manifold M, i.e. a manifold M with a spin structure, we could define the Atiyah-Singer- Dirac operator locally as follows: for x M and e 1 (x),, e n (x) a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is n e j (x) Γ ej(x), j=1 where Γ is a lifting of the Levi-Civita connection on M to the spinor bundle over M, and denotes the Clifford algebra multiplication. We omit the process of equipping spinorial tensor fields a ISL(, C)-invariant inner product thus making the representations unitary. References [1] H. Blaine Lawson and JR. Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. [] Robert M. Wald, General Relativity, The University of Chicago Press, 1984. [3] E. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Mathematics, Second Series, Vol.40, No 1, 1939. [4] V. Bargmann and E. P. Wigner, Group Theoretical Discussion of Relativistic Wave Equations, Princeton University, 1947 [5] Richard Borcheds, https://math.berkeley.edu/ reb/courses/61/37.pdf, Lecture notes, UCB, 01. 8