Clifford Algebras

This document collects the abstract definition and structure theory of Clifford algebras — the algebraic objects underlying gamma matrices, Dirac spinors, Spin groups, and the relativistic-fermion fields used throughout QFT. It is intended as a reference companion: physics docs link here for the algebra they use (e.g. the Dirac algebra $\mathrm{Cl}(1,3)$ in QED/historical.md § 1.3), and this page stays general.

For the Lie group / Lie algebra prerequisites (representations, universal covers, $SL(2,\mathbb{C})$, $\mathrm{Spin}(n)$) see group-theory.md.

1. Abstract Definition

1.1 Tensor-algebra construction

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$ (here $\mathbb{R}$ or $\mathbb{C}$) equipped with a non-degenerate symmetric bilinear form $g:V\times V\to \mathbb{F}$, equivalently a quadratic form $Q:V\to \mathbb{F}$ with $Q(v)=g(v,v)$ and polarization $g(u,v)=\frac{1}{2}[Q(u+v)-Q(u)-Q(v)]$.

The Clifford algebra $\mathrm{Cl}(V,Q)$ is the associative unital $\mathbb{F}$-algebra

$$\mathrm{Cl}(V,Q)\equiv T(V)/⟨v\otimes v-Q(v)\cdot 1:v\in V⟩,$$

where $T(V)={⨁}_{k=0}^{\infty }{V}^{\otimes k}$ is the tensor algebra over $V$ and the brackets denote the two-sided ideal generated by the displayed elements. Concretely: take all formal words in vectors of $V$, multiply by concatenation, and impose

$$\boxed{v\cdot v=Q(v)1\text{for all }v\in V.}$$

Polarizing this relation (apply it to $v=u+w$, expand, and subtract):

$$\boxed{u\cdot v+v\cdot u=2g(u,v)1\text{for all }u,v\in V.}$$

This is the defining anticommutation relation: vectors do not commute, but their failure to commute is exactly twice the inner product. The two boxed forms are equivalent.

1.2 Universal property

$\mathrm{Cl}(V,Q)$ is characterized — up to unique isomorphism — by the following universal property:

For any associative unital $\mathbb{F}$-algebra $A$ and any linear map $\varphi :V\to A$ satisfying $\varphi (v{)}^2=Q(v)1_A$ for all $v\in V$, there exists a unique algebra homomorphism $\tilde{\varphi }:\mathrm{Cl}(V,Q)\to A$ extending $\varphi$.

In other words, $\mathrm{Cl}(V,Q)$ is the "freest" associative algebra in which the boxed relation holds. Every concrete realization of vectors-as-things-squaring-to-$Q$ — gamma matrices, complex numbers, quaternions, etc. — is a representation (i.e. an algebra homomorphism from $\mathrm{Cl}$ to some matrix algebra) of one of these.

1.3 Dimension and basis

Pick an orthogonal basis $\{e_1,\dots ,e_n\}$ of $V$ with $Q(e_i)={ϵ}_i\in \{+1,-1\}$ (always possible by Gram–Schmidt for non-degenerate $Q$ over $\mathbb{R}$; over $\mathbb{C}$ all ${ϵ}_i$ can be taken $+1$). The defining relation reduces to

$$e_i^2={ϵ}_{i}1,e_i{e}_j=-e_j{e}_i(i\ne j).$$

Any product of the $e_i$ can therefore be reordered (with sign flips) so each $e_i$ appears at most once and in a fixed canonical order. The independent monomials are indexed by subsets $S\subseteq \{1,\dots ,n\}$:

$$e_S\equiv e_{i_1}{e}_{i_2}\cdots e_{i_k},S=\{i_1<i_2<\cdots <i_k\}\subseteq \{1,\dots ,n\},$$

with $e_{\varnothing }=1$. These form a basis of $\mathrm{Cl}(V,Q)$ as a vector space, so

$${\dim }_{\mathbb{F}}\mathrm{Cl}(V,Q)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)=2^n,n=\dim V.$$

The grading by $|S|=k$ recovers the $k$-fold antisymmetric products — see § 2 for the relation to the exterior algebra.

2. Relation to the Exterior Algebra

The exterior algebra ${\Lambda }^{\bullet }(V)={⨁}_{k=0}^n{\Lambda }^k(V)$ is the special case $Q=0$:

$${\Lambda }^{\bullet }(V)\cong \mathrm{Cl}(V,0).$$

When $Q=0$ the relation $uv=-vu$ is exact, so all products are antisymmetric; this is the Grassmann algebra used in differential geometry (differential forms) and in fermionic path integrals.

For general $Q$, define the antisymmetric piece of a product:

$$u\wedge v\equiv \frac{1}{2}(uv-vu),u\cdot v\equiv \frac{1}{2}(uv+vu)=g(u,v)1.$$

So Clifford multiplication splits as

$$uv=g(u,v)1+u\wedge v,$$

a symmetric (scalar) piece plus an antisymmetric (2-form) piece. Higher products extend this: every Clifford product is a sum of $k$-form parts. As a vector space (forgetting the multiplication)

$$\mathrm{Cl}(V,Q)\cong {\Lambda }^{\bullet }(V)=\underset{k=0}{\overset{n}{⨁}}{\Lambda }^k(V),$$

with the same total dimension $2^n$. Turning $Q$ on "deforms" the multiplication so that products absorb scalar pieces, but the underlying vector space is unchanged. This is sometimes called the Chevalley identification.

3. Standard Notations and Signatures

Over $\mathbb{R}$, by Sylvester's law of inertia, every non-degenerate quadratic form has a signature $(p,q)$ with $p$ positive and $q$ negative squares ($n=p+q$). Write

$$\mathrm{Cl}(p,q)\equiv \mathrm{Cl}({\mathbb{R}}^{p+q},Q_{p,q}),Q_{p,q}(x)=x_1^2+\cdots +x_p^2-x_{p+1}^2-\cdots -x_{p+q}^2.$$

Equivalently, the orthogonal generators $e_1,\dots ,e_n$ satisfy $e_i^2=+1$ for $i\le p$ and $e_i^2=-1$ for $i>p$. The Lorentzian conventions of physics:

• $\mathrm{Cl}(1,3)$: signature $(+,-,-,-)$, the Dirac algebra with ${\gamma }^{\mu }$ satisfying $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }1$, $\eta =\mathrm{diag}(+,-,-,-)$. Used in this repo. See QED/historical.md § 0.1.
• $\mathrm{Cl}(3,1)$: signature $(-,+,+,+)$, the "mostly-plus" convention common in GR-flavored texts. Not isomorphic to $\mathrm{Cl}(1,3)$ over $\mathbb{R}$ (they are different real algebras), but isomorphic after complexification.

Over $\mathbb{C}$, signature stops mattering: any quadratic form can be diagonalized to $Q=\sum x_i^2$, so there is a unique complex Clifford algebra in each dimension,

$${\mathrm{Cl}}_n(\mathbb{C})\equiv \mathrm{Cl}({\mathbb{C}}^n,Q_n),\mathrm{Cl}(p,q{)}_{\mathbb{C}}\cong {\mathrm{Cl}}_{p+q}(\mathbb{C}).$$

This is the version directly relevant to QFT representations (since spinors are complex).

4. Small-Dimensional Examples

Two structural takeaways from the small cases:

• Complex numbers, quaternions, and split-complex numbers are all Clifford algebras — historically, this is how Clifford (1878) found his algebras: by generalizing Hamilton's quaternions and Grassmann's exterior algebra into a single framework.
• The 16-dim Dirac algebra $\mathrm{Cl}(1,3)$ is isomorphic to $M_2(\mathbb{H})$ over $\mathbb{R}$ — i.e. $2\times 2$ matrices of quaternions — and to $M_4(\mathbb{C})$ after complexification. The latter is what guarantees the Dirac, Weyl, and Majorana representations of § 0.1 of QED-historical exist and are all equivalent.

5. Classification: Bott Periodicity

The complete classification of real Clifford algebras (Atiyah–Bott–Shapiro 1964, building on earlier work) shows that $\mathrm{Cl}(p,q)$ is always a matrix algebra (or a direct sum of two) over $\mathbb{R},\mathbb{C}$, or $\mathbb{H}$, and the pattern is periodic in $q-p\mathrm{mod}8$ for real algebras and periodic in $n\mathrm{mod}2$ for complex algebras. The full table:

with $n=p+q$. For complex Clifford algebras the period is 2:

$${\mathrm{Cl}}_n(\mathbb{C})\cong \{\begin{array}{ll}{M}_{2^{n/2}}(\mathbb{C})& n\text{ even}\\ M_{2^{(n-1)/2}}(\mathbb{C})\oplus M_{2^{(n-1)/2}}(\mathbb{C})& n\text{ odd}.\end{array}$$

The minimum faithful matrix dimension (the spinor dimension) is therefore $2^{\lfloor n/2\rfloor }$ over $\mathbb{C}$, and may be larger or smaller over $\mathbb{R}$ depending on signature:

This pattern — when each kind of spinor (Dirac/Weyl/Majorana/Majorana–Weyl) exists — is governed entirely by the algebraic classification above, and is a non-trivial input to supergravity and string theory.

6. Pin and Spin Groups

Sitting inside $\mathrm{Cl}(V,Q{)}^{\times }$ (the multiplicative group of invertible elements) are two distinguished subgroups built from products of unit vectors.

6.1 Definitions

A unit vector is $u\in V$ with $Q(u)=\pm 1$ (i.e. $u^2=\pm 1$).

• $\mathrm{Pin}(V,Q)$ = subgroup of $\mathrm{Cl}(V,Q{)}^{\times }$ generated by all unit vectors of $V$.
• $\mathrm{Spin}(V,Q)$ = subgroup of $\mathrm{Pin}(V,Q)$ generated by products of an even number of unit vectors.

Equivalently, $\mathrm{Spin}(V,Q)=\mathrm{Pin}(V,Q)\cap {\mathrm{Cl}}^0(V,Q)$, where ${\mathrm{Cl}}^0$ is the even subalgebra (linear combinations of $e_S$ with $|S|$ even).

6.2 Connection to the orthogonal group

Every element $\rho \in \mathrm{Pin}(V,Q)$ acts on $V\subset \mathrm{Cl}(V,Q)$ by the twisted adjoint action

$$v\mapsto \alpha (\rho )v{\rho }^{-1},$$

where $\alpha$ is the grading automorphism $\alpha (e_S)=(-1{)}^{|S|}{e}_S$. This map sends $\mathrm{Pin}(V,Q)$ to the orthogonal group $\mathrm{O}(V,Q)$ and $\mathrm{Spin}(V,Q)$ to the special orthogonal group $\mathrm{SO}(V,Q)$:

$$1\to \{\pm 1\}\to \mathrm{Pin}(V,Q)\to \mathrm{O}(V,Q)\to 1,1\to \{\pm 1\}\to \mathrm{Spin}(V,Q)\to \mathrm{SO}(V,Q)\to 1.$$

These are double covers (kernel $\{\pm 1\}$). The Spin one is the universal cover of $\mathrm{SO}(V,Q{)}^0$ in dimensions $\ge 3$ — see group-theory.md § Connectedness and discrete components.

6.3 Worked example: $\mathrm{Spin}(1,3)\cong SL(2,\mathbb{C})$

For the physically central case $\mathrm{Cl}(1,3)$:

• The even subalgebra ${\mathrm{Cl}}^0(1,3)$ has dimension $2^{4-1}=8$.
• ${\mathrm{Cl}}^0(1,3)\cong M_2(\mathbb{C})$ (4×4 real-dim = 8 complex-dim).
• $\mathrm{Spin}(1,3)=\{x\in {\mathrm{Cl}}^0(1,3):x\bar{x}=1\}\cong \{M\in M_2(\mathbb{C}):\det M=1\}=SL(2,\mathbb{C})$.

So $\mathrm{Spin}(1,3)\cong SL(2,\mathbb{C})$ — exactly the universal cover used in QFT/preliminaries.md § Universal cover $SL(2,\mathbb{C})$ and foundations-modern.md to handle spinor representations. The Clifford-algebra construction is where this universal cover comes from algebraically; the Lorentz-group story only sees it as "the simply connected group with the same Lie algebra as $S{O}^{+}(1,3)$".

Other physically relevant cases:

7. Spinor Representations

A spinor representation of $\mathrm{Spin}(V,Q)$ is an irreducible representation that does not descend to $\mathrm{SO}(V,Q)$ — i.e. it is genuinely double-valued on the orthogonal group, single-valued only on the cover. The available spinor types in signature $(1,d-1)$ are governed by the Bott classification of § 5:

• Dirac spinor. A faithful representation of the full complex Clifford algebra ${\mathrm{Cl}}_d(\mathbb{C})$, restricted to $\mathrm{Spin}(1,d-1)$. Always exists; dimension $2^{\lfloor d/2\rfloor }$.
• Weyl spinor. When $d$ is even, ${\mathrm{Cl}}_d(\mathbb{C})$ has a central element ${\gamma }^{\ast }\equiv i^{(d-2)/2}{e}_1\cdots e_d$ with $({\gamma }^{\ast }{)}^2=1$. Its $\pm 1$ eigenspaces give two inequivalent half-dimensional irreps of $\mathrm{Spin}(1,d-1)$ — the left- and right-handed Weyl spinors. For $d=4$: ${\gamma }^{\ast }={\gamma }^5$, projectors $P_{L,R}=(1\mp {\gamma }^5)/2$.
• Majorana spinor. When a real structure (charge conjugation $C$) on the spinor space commutes with the $\mathrm{Spin}$ action and squares to $+1$, we can impose $\psi ={\psi }^c$, halving the real dimension. Exists in certain signatures by the Bott table — including $(1,3)$ (real Majorana, 4 real components).
• Majorana–Weyl spinor. Both conditions simultaneously. Requires Weyl projector and charge conjugation to be compatible. By the Bott table this exists only in signatures with $p-q\equiv 0\mathrm{mod}8$, including $d=10$ Minkowski — the dimension that makes superstring theory consistent.

The story in $d=4$ specifically is exactly the table in QED/historical.md § 0.1: the Dirac (standard), Weyl (chiral), and Majorana representations of the $\gamma$-matrices are three concrete realizations of the same abstract Clifford module on ${\mathbb{C}}^4$, related by similarity transformations.

8. Bilinear Covariants (Dirac field, $d=4$)

The 16-dim basis of $\mathrm{Cl}(1,3{)}_{\mathbb{C}}$ classifies the independent Lorentz-covariant bilinears $\bar{\psi }\Gamma \psi$ that can be built from a single Dirac field:

Total $1+4+6+4+1=16=\dim \mathrm{Cl}(1,3{)}_{\mathbb{C}}$. These are the building blocks of every Lorentz-invariant fermion interaction:

• $\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ in QED (vector × vector).
• $\bar{\psi }{\gamma }^{\mu }(1-{\gamma }^5)\psi W_{\mu }$ in the weak interaction (V−A coupling, parity-violating).
• $\bar{\psi }{\sigma }^{\mu \nu }{F}_{\mu \nu }\psi$ as the magnetic-moment term (Pauli term).
• $(\bar{\psi }\Gamma \psi )(\bar{\psi }{\Gamma }^{\prime }\psi )$ four-fermion contact interactions (Fermi theory, NJL, EFT).

The 16-dim count is therefore not abstract trivia — it is exactly the dimension of the operator basis you build interactions from.

9. Pointers and References

This document is a reference; for derivations and worked examples in the physics context:

• QED/historical.md § 0.1 — Gamma matrices, Dirac spinors, notation: the three concrete $\gamma$-matrix representations and bilinear notation conventions used throughout the QFT docs.
• QED/historical.md § 1.3 — Derivation of the Clifford algebra and gamma matrices: how $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }1$ falls out of squaring the Dirac operator (Route A above).
• QFT/preliminaries.md § Universal cover $SL(2,\mathbb{C})$: the Lorentz/Spin story from the Wigner-classification side.
• group-theory.md § Connectedness and discrete components: the general universal-cover machinery that $\mathrm{Spin}$ instantiates.

External references:

• Lawson & Michelsohn, Spin Geometry — comprehensive mathematical reference (Cl, Spin, Dirac operators, K-theory).
• Hamilton, Mathematical Gauge Theory — Chapters 8–9 give a physics-friendly Clifford / Spin treatment.
• Atiyah, Bott, Shapiro, "Clifford modules" (Topology 3, 1964) — the original classification paper.
• Doran & Lasenby, Geometric Algebra for Physicists — Clifford algebra as a unified language for classical mechanics, EM, and relativity (a different pedagogical tradition).
• Polchinski, String Theory Vol. 2, App. B — Clifford-algebra / spinor tables in arbitrary dimensions, indispensable for SUSY / string spectrum.