Clifford Algebras and Spinors

Clifford Algebras

Today I'd like to write some brief notes about Clifford algebras and spinors. A classic reference is the paper "Clifford Modules" by Atiyah-Bott-Shapiro. Clifford algebras not only useful in algebra and geometry, but are essential for the construction of theories with fermions. Let \(V\) be a vector space with a non-degenerate symmetric bilinear form \(B\). We define the Clifford algebra \(Cl(V, B)\) to be the unital associative algebra generated by \(v \in V\) subject to the relation
\[ vw + wv = -2B(v,w) \]
Equivalently, the definiting relation is \(v^2 = -B(v,v)\).

The Clifford algebra inherits a \(\mathbb Z\)-filtration as well as a \(\mathbb Z_2\)-grading from the tensor algebra. In fact, we have an analogue of the Poincare-Birkhoff-Witt theorem for Lie algebras:

Theorem The associated graded algebra of \(Cl(V,B)\) is naturally isomorphic to the exterior algebra on \(V\).

In this way, we may view the Clifford algebra as a quantization of the exterior algebra, much in the same way that \(U(\mathfrak g)\) is a quantization of the Poisson algebra of functions on \(\mathfrak g^\ast\) for a Lie algebra \(\mathfrak g\).

Example. Take (V,B) to be the Euclidean space \(\mathbb E^1\). Then we have a single generator \(e\) satisfying the relation \(e^2 =  -1\). Hence
\[ Cl(\mathbb R) \cong \mathbb R \cdot 1 \oplus \mathbb R \cdot e \cong \mathbb C \]
Where the isomorpism is given by \(e \mapsto i = \sqrt{-1}\).

Example. Take \(\mathbb E^2\). We have generators \(e_1, e_2\) both squaring to -1, and additionally we have \(e_1 e_2 = e_2 e_1\). We can define an isomorphism from \(Cl(\mathbb E^2)\) to the quaternions \(\mathbb H\) by \(e_1 \mapsto i, e_2 \mapsto j\).

Spinors

Now consider the complexified Clifford algebra, denoted \(\mathbb{C}l(V)\). Since we can now take square roots of negative numbers, the complex Clifford algebra is insensitive to the signature (as long as our bilinear form is non-degenerate). Denote by \(C_n\) the complex Clifford algebra \(Cl(\mathbb C^n)\),
where \(\mathbb C^n\) is equipped with the standard bilinear form \((x,y) = \sum_{i=1}^n x_i y_i\).

Definition. A subspace \(W \subset \mathbb C^n\) is isotropic if the restriction of the standard bilinear form to \(W\) is identically 0. A maximal isotropic subspace is an isotropic subspace that is not properly contained in any other isotropic subspace.

Theorem. Let \(W\) be a maximal isotropic subspace, and let\( \{w_1, \dots, w_k\}\) be a basis of \(W\). Let \(\omega = w_1 \cdots w_k \in C_n\), and let \(S = C_n \cdot \omega\). If n is even, then \(S\) is an irreducible Clifford module. If n is odd, then \(S=S^+ \oplus S^-\) is a direct sum irreducible Clifford modules, and \(S^+ \cong S^-\).

Irreducible Clifford modules are called spinor modules. This description of spinor modules allows one to prove straightforwardly the following complete classification of complex Clifford algebras.

Corollary. We have \(C_{2m} \cong \mathrm{End}(\mathbb C^m)\) and \(C_{2m+1} \cong \mathrm{End}(\mathbb C^m) \oplus \mathrm{End}(\mathbb C^m)\).

Note that this classification depends on n mod 2, which is closely related to Bott periodicity. There is a similar classification of real Clifford algebras.

Dirac Operators

Now we come to the real importance of Clifford algebras. Consider Euclidean space \(\mathbb{E}^n\) and let \(S\) be a spinor module for its Clifford algebra. We define the Dirac operator acting on \(S\)-valued functions as
\[ D f = \sum_{i=1}^n e_i \cdot \partial_i f \]
Now the amazing property of \(D\) is the following:
\[ D^2 = \sum_{i,j} e_i e_j \partial_i \partial_j = \sum_i e_i^2 \partial_i^2 + \sum_{i,j} e_i e_j [\partial_i, \partial_j] = -\Delta \]
hence the Dirac operator provides an algebraic (as opposed to pseudodifferential) square root of the Laplacian.

To Be Added in an Update...

Supersymmetric point particle, Dirac operators on spin manifolds, Weitzenböck formula, spinor reps of Lorentz algebra, N=1 susy.