The Anticommutator
Field note on the Dirac gamma matrices.
You want to talk about spacetime? You want to write an equation that respects Einstein's geometry? Then you need the gamma matrices. They are the bridge between the algebra of matrices and the geometry of Minkowski space. They are the Clifford algebra of spacetime, written in the only language quantum mechanics understands.
Here is the defining relation, and it is all you need:
{γ^μ, γ^ν} = γ^μγ^ν + γ^νγ^μ = 2g^{μν}I
Four by four. Always four by four, in four-dimensional spacetime. The metric g^{μν} is the Minkowski metric — diag(1, −1, −1, −1) or diag(−1, 1, 1, 1) depending on your convention, and convention matters here like nothing else in physics. One choice and your hermiticity properties flip. Another and your γ⁵ definition changes sign.
The anticommutator is the key word. The gamma matrices do not commute. If they commuted, you'd get nothing but numbers. The whole structure — the spin, the Dirac equation, the existence of fermions — comes from the fact that γ^μγ^ν ≠ γ^νγ^μ when μ ≠ ν. The matrices anticommute. This is not a side effect. This is the entire point.
The Clifford algebra Cℓ(1,3) generated by these four matrices is richer than the algebra itself suggests. The four gamma matrices generate a 16-dimensional algebra. Not four. Sixteen. Because products of pairs, triples, and the quadruple product all produce independent elements. The complete basis of the Dirac algebra consists of:
- I — the identity (scalar, 1 element)
- γ^μ — four matrices (vector, 4 elements)
- σ^{μν} = i/2[γ^μ, γ^ν] — six antisymmetric combinations (tensor, 6 elements)
- γ^μγ⁵ — four matrices (pseudovector, 4 elements)
- γ⁵ = iγ⁰γ¹γ²γ³ — the chirality matrix (pseudoscalar, 1 element)
1 + 4 + 6 + 4 + 1 = 16. Exactly the dimension of 4×4 complex matrices. The Dirac algebra is M₄(ℂ).
γ⁵ deserves its own sentence. It anticommutes with all γ^μ. It commutes with σ^{μν}. Its square is the identity. It is the operator that measures chirality — whether a spinor is left-handed or right-handed. In the massless limit, γ⁵ becomes the helicity operator, and its eigenvalues ±1 separate the Dirac spinor into two chiral Weyl spinors. Mass couples these two chiralities. No mass, no coupling. Pure chirality. Pure Weyl.
Let me give you the most common representation, the Dirac (or standard) representation:
γ⁰ = [I 0] γⁱ = [ 0 σⁱ] [0 -I] [-σⁱ 0 ]
Where σⁱ are the Pauli matrices. γ⁰ is Hermitian. γⁱ are anti-Hermitian (in this convention, depending on metric signature). γ⁰^2 = I. γⁱ^2 = −I. This matches the Minkowski metric. The algebra is satisfied. The matrices do what they must.
But representations are not unique. A unitary transformation Uγ^μU⁻¹ gives you an equivalent set of matrices. The physics does not change. Only the Dirac algebra does. The specific matrices you choose are a matter of convenience — like choosing Cartesian or spherical coordinates. The algebraic relations are coordinate-independent. The Clifford algebra is the geometry. The matrices are just the coordinate system.
The trace identities are essential tools. Any trace of an odd number of gamma matrices vanishes. Trace of γ^μγ^ν = 4g^{μν}. Trace of γ^μγ^νγ^ργ^σ = 4(g^{μν}g^{ρσ} − g^{μρ}g^{νσ} + g^{μσ}g^{νρ}). These are not coincidences. They follow from the anticommutation relations and the dimension of the representation.
The Fierz identities — rearrangements of products of bilinears — complete the computational toolkit. They say that any product of Dirac matrices can be rewritten as a linear combination of the 16 basis elements. The algebra is closed. The basis is complete.
What does the anticommutator mean? It means that spacetime is encoded in matrix algebra. The geometric relation between two vectors — their inner product, dictated by the metric — is realized as the anticommutator of two operators acting on spinor space. Geometry becomes algebra becomes physics. The gamma matrices are the translation dictionary.
Every fermion in the Standard Model obeys the Dirac equation, which is built from these matrices. The weak interaction couples to γ^μ(1−γ⁵). The Higgs coupling is proportional to the identity in Dirac space. The photon couples through γ^μ. All of these interaction structures are expressed in the language of the Clifford algebra.
The anticommutator {γ^μ, γ^ν} = 2g^{μν} is not a formula you memorize and forget. It is a statement about how spacetime and quantum mechanics fit together. It says that the fundamental objects of fermionic matter — the spinors — live in a space whose geometry is governed by the Clifford algebra of Minkowski spacetime.
It is, in my view, the most profound equation nobody talks about at dinner parties.