Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called a Clifford algebra. 1 In particular, (γ i ) = I and γ i γ j = γ j γ i, if i j 1. If we have a representation of the Clifford algebra, we could form the matrices Σ ij = 1 4i (γ iγ j γ j γ i ) 1 4i [γ i, γ j ] () Show that these matrices have the same algebra as the generators in the defining representation of so(n) where we defined those matrices as [ ω ij ] kl = i ( δ i kδ j l δi lδ j k) (3) 1 For those of you who have studied relativistic quantum mechanics or quantum field theory, the γ i are the Dirac matrices that you would need to write down the Dirac equation on N-dimensional Euclidean space. If you wanted Minkowski space, you could simply substitute γ N iγ N γ 0, the Kronecker delta on the right-hand-side of equation (1) would be replaced by the Minkowski metric and the construction in what follows would give you a spinor representation of the Lorentz group in N spacetime dimensions. We will not pursue this program here. 1
Note that the quantities written in equation (3) are the matrix elements of the N(N 1)/ linearly independent generators of the so(n) Lie algebra in the N-dimensional defining representation. The matrices written in equation () are the same generators in a different representation, one that is sometimes called a spinor representation of the same so(n) Lie algebra. We have not yet determined the dimension or other properties of this spinor representation. The exercise above is simply to show that it is indeed a representation.. Is the spinor representation a Hermitian representation of the so(n) Lie algebra? Remember that a Hermitian representation of the Lie algebra generators would give us a unitary representation of the Lie group. We might ask the question as to whether we can find a representation of the Clifford algebra, and what that representation might look like. Even dimensions: N = n Consider the following pairings of the gamma-matrices a 1 = 1 (γ 1 + iγ ), a 1 = 1 (γ 1 iγ ) a = 1 (γ 3 + iγ 4 ), a = 1 (γ 3 iγ 4 )... a n = 1 (γ n 1 + iγ n ), a n = 1 (γ n 1 iγ n ) 3. Show that each of these pairs of matrices, a i and a i have the same anticommutators as we would expect to find for a single fermionic harmonic oscillator, that is a i = 0, a i = 0, ai a i + a i a i = 1 4. Show that, the creation and annihilation operators for different oscillators anti-commute, a i a j + a j a i = 0, i j
5. For this collection of oscillators, we can define a vacuum state, 0 >, which is annihilated by all of the annihilation operators, a i 0 >= 0, < 0 0 >= 1 Then we can create the rest of a basis of the vector space on which these matrices act by simply operating with creation operators: a i a i a i 3 0 >,... (4) where, to prevent double counting, on a given basis state, i i > i > i 3 >.... How many such basis states are there? Are they orthogonal? Are they normalized? What is the dimension of the representation? In analogy with fermions, we will call the span of the set of states in (??) the Hilbert space. 6. Now, we can recover the gamma-matrices as operators in our fermionic Hilbert space γ i 1 = a i + a i, i = 1,..., n γ i = 1 ( ) a i a i, j = n + 1,..., n i This allows us to write the generators Σ ij in terms of the fermion creation and annihilation operators. Recall that the Cartan subalgebra which we chose for the Lie algebra so(n) was the maximal set of mutually commuting generators Σ 1, Σ 34 n 1,..., Σ n} There are n elements and the Lie algebra has rank n. Show that the Cartan generators are given in terms of fermion creation and annihilation operators by a 1a 1 1, a a 1,..., a na n 1 } Since the eigenvalues of the number operator a i a i are plus one and zero, the n eigenvalues of the Cartan generators are given by all choices of plus and minus signs in the n-component vector ± 1 }, ±1,..., ±1 3
These are the weights of the spinor representation of so(n). How many weights are there, that is, what is the dimension of this representation? Does this match your answer to a previous question where you counted the number of fermion states? 7. Recall that we deduced the roots of so(n) as some of the vectors which connected different weights in the defining representation. They were ±ê i ± e j, where ê i is the unit vector in the i th direction and i, j run from 1 to n. Use this fact to see that the spinor representation that we have constructed is reducible. We could characterize the two irreducible representations as acting solely on subspaces of the vector space, one where the total fermion number of all states is even, the other where the total fermion number of all states is odd. One way to see these subspaces are not mixed is to note that all of the spin operators Σ ij = 1 [γ 4i i, γ j ], i, j = 1,,..., n+1 change the fermion number by an even amount, they all have only terms which change the number of fermions by, 0,. Recalling our convention for ordering weights, the highest weights of the two representations are ( 1, 1,..., 1 ) n 1 = êi and i=1 ( 1, 1 ),..., 1 n 1 1 = 1 i=1 êi ên Are these fundamental weights, that is, are these the highest weights of fundamental representations? 3 Odd dimensions, N = n + 1 In n + 1 dimensions, we can use the same fermionic oscillator construction for the first n gamma matrices as we did in n dimensions above. That leaves us needing to find a representation for γ n+1. Its essential properties are (γ n+1 ) = I and it must anti-commute with all of the other gamma-matrices. We can construct such an operator 4
from the fermionic oscillators as γ n+1 = e iπ n 1 a i a i 8. Show that γ n+1 given above has the property (γ n+1 ) = I. 9. Show that γ n+1 given above has the property γ n+1 = γ n+1. 10. Show that γ n+1 given above has the property γ n+1 γ k = γ k γ n+1 for k = 1,,..., n. 11. Does Σ ij = 1 4i [γ i, γ j ], i, j = 1,,..., n + 1 now yield a representation of so(n + 1)? 1. If so, what is the dimension of this representation? 13. If so, is this representation reducible or irreducible? 14. If it is reducible, are the highest weights, and if it is irreducible, is the highest weight a fundamental weight? That is, is it the highest weight of a fundamental representation? 5