Knowlpedia

Definition

Gamma matrices

Matrices representing Clifford multiplication in a chosen basis of a spinor module.

Definition

Let (V,g)(V,g) be a finite-dimensional real or complex with basis eae_a, and let c:Cl⁡(V,g)→End⁡(Δ)c:\operatorname{Cl}(V,g)\to\operatorname{End}(\Delta) be a written in a basis of Δ\Delta. The gamma matrices are

γa=c(ea).\gamma_a=c(e_a).

Using the geometric Clifford convention v2=−g(v,v)1v^2=-g(v,v)1, they satisfy

γaγb+γbγa=−2gabI.\gamma_a\gamma_b+\gamma_b\gamma_a=-2g_{ab}I.

Thus gamma matrices are the basis-dependent matrices of , not additional invariant geometric objects.

Raised indices

Writing gabg^{ab} for the inverse matrix, set γa=gabγb\gamma^a=g^{ab}\gamma_b. Contracting gamma matrices with the components of a vector or covector is the .

Minkowski-signature convention

For the Lorentzian knowls in this collection, take

η=diag⁡(−1,+1,+1,+1).\eta=\operatorname{diag}(-1,+1,+1,+1).

Under the geometric convention used here,

γ02=+I,γj2=−I(j=1,2,3).\gamma_0^2=+I, \qquad \gamma_j^2=-I\quad(j=1,2,3).

Many physics texts instead use the opposite metric signature ηphys=diag⁡(+1,−1,−1,−1)\eta_{\mathrm{phys}}=\operatorname{diag}(+1,-1,-1,-1) together with

{γμ,γν}=+2ημνI.\{\gamma^\mu,\gamma^\nu\}=+2\eta^{\mu\nu}I.

With ηphys=−η\eta_{\mathrm{phys}}=-\eta, this has the same numerical right-hand side as the convention in the core. If one changes only the Clifford sign while holding the metric fixed, multiplying all complex gamma matrices by ii passes between the two relations. Metric signature and Clifford sign must therefore be recorded separately.

Change of basis and spin covariance

Changing the basis of the spinor module conjugates every gamma matrix simultaneously. A spin transformation ss satisfies

ρ(s)c(v)ρ(s)−1=c(svs−1),\rho(s)c(v)\rho(s)^{-1}=c(svs^{-1}),

so the matrices transform compatibly with the orthogonal action on VV. Different-looking gamma-matrix realizations can therefore describe equivalent Clifford representations.

Related constructions

In even dimension, the normalized product of all gamma matrices represents the . The normalization is convention-dependent, whereas the resulting grading of the complex spin representation is invariant.

References
  1. H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I.
  2. Daniel S. Freed, Five Lectures on Supersymmetry, American Mathematical Society, 1999. Relevant: Lectures 1–2.
  3. Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison–Wesley, 1995. Relevant: Sections 3.1–3.4 for the common physics sign convention.