Gamma Matrices

Gamma matrices arise because a relativistic wave equation first order in time and space must still reproduce the quadratic mass shell. Writing

$$H=c\boldsymbol\alpha\cdot\mathbf p+\beta mc^2$$

and demanding $H^2=\mathbf p^2c^2+m^2c^4$ for every momentum forces a matrix algebra. In covariant form it is

$$\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}I_4.$$

This Clifford algebra is representation independent. Particular Dirac or chiral matrices are coordinate choices on the spinor space and live on the Gamma-Matrix Conventions lookup page. Here the objective is to derive the algebra and the conclusions that do not depend on a chosen basis.

Required background. Metric and Units, Four-Vectors, and the Energy–Momentum Relation supply the convention ledger, Lorentz invariants, and dispersion relation. This page also assumes complex linear algebra and elementary Pauli-matrix identities.

Linearizing the relativistic Hamiltonian

The positive square-root Hamiltonian

$$H=\sqrt{\mathbf p^2c^2+m^2c^4}$$

is nonlocal in position space and separates one energy branch before a local relativistic wave equation has been constructed. Dirac’s algebraic strategy is to seek a matrix-valued expression linear in momentum,

$$H=c\sum_{i=1}^3\alpha_i p_i+\beta mc^2,$$

acting on a multicomponent amplitude. The coefficients are independent of $\mathbf p$. Squaring gives

$$\begin{aligned} H^2 &= c^2\sum_{i,j}\alpha_i\alpha_jp_ip_j +mc^3\sum_i(\alpha_i\beta+\beta\alpha_i)p_i +\beta^2m^2c^4. \end{aligned}$$

Because $p_ip_j$ is symmetric in $i,j$, the antisymmetric part of $\alpha_i\alpha_j$ drops out. Equality with the scalar dispersion relation for every $\mathbf p$ and $m$ requires

$$\{\alpha_i,\alpha_j\}=2\delta_{ij}I, \qquad \{\alpha_i,\beta\}=0, \qquad \beta^2=I.$$

These conditions cancel every mixed term and leave

$$H^2=(\mathbf p^2c^2+m^2c^4)I.$$

The square-root relation has not disappeared: it has been factored into a first-order matrix operator whose eigenvalues occupy both energy branches.

From alpha and beta to gamma matrices

Define

$$\gamma^0=\beta, \qquad \gamma^i=\beta\alpha_i.$$

Then $\alpha_i=\gamma^0\gamma^i$ and the relations above imply

$$(\gamma^0)^2=I, \qquad (\gamma^i)^2=-I, \qquad \gamma^\mu\gamma^\nu=-\gamma^\nu\gamma^\mu \quad(\mu\ne\nu).$$

Combining the cases gives

$$\boxed{ \{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}I }.$$

The box records the defining relation, not a choice of matrices. Lowered gamma matrices are obtained with the spacetime metric:

$$\gamma_\mu=\eta_{\mu\nu}\gamma^\nu, \qquad \gamma_0=\gamma^0, \qquad \gamma_i=-\gamma^i.$$

Lowering a Lorentz label is unrelated to Hermitian conjugation.

For any covector $a_\mu$, define slash notation by

$$a\!\!\!/\equiv\gamma^\mu a_\mu.$$

The Clifford algebra immediately yields

$$(a\!\!\!/)^{2} = \frac12\{\gamma^\mu,\gamma^\nu\}a_\mu a_\nu = a_\mu a^\mu I.$$

The commutator part vanishes because $a_\mu a_\nu$ is symmetric. More generally,

$$a\!\!\!/\,b\!\!\!/+b\!\!\!/\,a\!\!\!/ = 2a\cdot b\,I.$$

This is the representation-independent factorization behind the free Dirac operator.

Why four components are minimal in 3+1 dimensions

The matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ must be invertible and pairwise anticommute while having the prescribed squares. One-dimensional matrices cannot anticommute nontrivially. Two-dimensional complex matrices can realize at most three independent pairwise-anticommuting directions using the Pauli matrices; there is no fourth nonzero $2\times2$ matrix that anticommutes with all three.

The complexified Clifford algebra in four spacetime dimensions is

$$\mathrm{Cl}_{1,3}(\mathbb C)\cong M_4(\mathbb C).$$

Its irreducible complex representation therefore has dimension four. A minimal Dirac amplitude has four complex components. This statement concerns the representation dimension of the Clifford algebra; it does not say that a spinor is a four-vector or that its entries transform with a Lorentz matrix.

Reducible representations of dimension $8,12,\ldots$ are possible by taking direct sums, but they repeat the minimal algebra rather than describe the ordinary irreducible Dirac spinor.

Hermiticity and the Dirac adjoint

For $H=c\boldsymbol\alpha\cdot\mathbf p+\beta mc^2$ to be Hermitian in the standard spinor inner product, one may choose

$$\alpha_i^\dagger=\alpha_i, \qquad \beta^\dagger=\beta.$$

It follows that

$$(\gamma^0)^\dagger=\gamma^0, \qquad (\gamma^i)^\dagger=-\gamma^i.$$

Both cases are summarized covariantly by

$$(\gamma^\mu)^\dagger = \gamma^0\gamma^\mu\gamma^0.$$

This identity motivates the Dirac adjoint of a spinor,

$$\bar\psi=\psi^\dagger\gamma^0.$$

The ordinary row $\psi^\dagger$ does not transform covariantly under boosts; $\bar\psi$ is the combination that forms Lorentz-covariant bilinears. The current and its positive time component are derived on the covariant Dirac page rather than assumed here.

Similarity transformations and representation independence

Let $S$ be any invertible constant matrix and define

$$\gamma'^\mu=S\gamma^\mu S^{-1}.$$

Then

$$\begin{aligned} \{\gamma'^\mu,\gamma'^\nu\} &= S\{\gamma^\mu,\gamma^\nu\}S^{-1} \\ &= 2\eta^{\mu\nu}I. \end{aligned}$$

If spinor components are changed simultaneously by $\psi'=S\psi$, the abstract equation and all consistently transformed bilinears describe the same physics. A trace, determinant, characteristic polynomial, or completely contracted gamma identity is similarity invariant.

Not every arbitrary similarity matrix preserves the simple Hermiticity conditions with the same Euclidean component inner product. Unitarily related standard representations do. For a general similarity transformation, the spinor metric used in the adjoint must be transformed consistently.

Lorentz generators on spinors

The antisymmetric products

$$\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]$$

generate the spinor representation of the Lorentz algebra, conventionally through $\sigma^{\mu\nu}/2$. Their commutator with a gamma matrix is

$$[\sigma^{\mu\nu},\gamma^\rho] = 2i\left( \eta^{\nu\rho}\gamma^\mu -\eta^{\mu\rho}\gamma^\nu \right).$$

This identity ensures that conjugating the gamma matrices by the spinor transformation reproduces the vector transformation of their Lorentz label. Rotations are represented unitarily, while finite-dimensional boost matrices are not unitary with respect to the ordinary positive-definite component inner product. Lorentz covariance instead uses the Dirac adjoint.

The precise exponential sign depends on whether one writes active or passive Lorentz transformations and how the antisymmetric parameters are defined. The representation-independent authority is the intertwining relation

$$S(\Lambda)^{-1}\gamma^\mu S(\Lambda) = \Lambda^\mu{}_{\nu}\gamma^\nu,$$

which will be used to prove covariance of the Dirac equation.

What belongs on the convention page

The Clifford relation determines an algebra but not a unique basis. The companion Gamma-Matrix Conventions page fixes:

• the Dirac basis used for the nonrelativistic limit;
• a chiral-basis comparison;
• $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$;
• $\sigma^{\mu\nu}=i[\gamma^\mu,\gamma^\nu]/2$;
• adjoint, slash, trace, epsilon, and source-translation identities.

Keeping the derivation here and tables there makes representation independence visible rather than forcing every calculation through one matrix basis.

Common pitfalls

Treating the Lorentz index as a matrix index. In $(\gamma^\mu)^a{}_b$, $\mu$ labels a spacetime vector component while $a,b$ are spinor-space matrix indices. They transform in different representations.

Equating $\gamma_\mu$ with $(\gamma^\mu)^\dagger$. The first lowers a Lorentz index with $\eta_{\mu\nu}$. The second takes a matrix adjoint. Their component formulas happen to share signs in this convention but express different operations.

Checking a basis and declaring the result basis dependent. Matrix entries change under similarity transformations; the Clifford relation and fully contracted identities do not.

Importing Euclidean formulas unchanged. Euclidean gamma matrices square to $+I$ in every direction. Here spatial gamma matrices square to $-I$ because the metric has signature $(+---)$.

Exercises

1. Squaring the linear Hamiltonian

Starting from $H=c\alpha_ip_i+\beta mc^2$, derive the three algebraic conditions on $\alpha_i$ and $\beta$.

Solution

Expanding $H^2$, symmetrize the $p_ip_j$ coefficient:

$$c^2\alpha_i\alpha_jp_ip_j = \frac{c^2}{2}\{\alpha_i,\alpha_j\}p_ip_j.$$

Matching $c^2\delta_{ij}p_ip_jI$ requires $\{\alpha_i,\alpha_j\}=2\delta_{ij}I$. The coefficient linear in momentum vanishes only if $\{\alpha_i,\beta\}=0$, and the mass term matches only if $\beta^2=I$.

2. Slash factorization

Prove

$$(p\!\!\!/-mc)(p\!\!\!/+mc) =(p^2-m^2c^2)I.$$

Solution

The scalar $mc$ commutes with every gamma matrix, so the mixed terms cancel. The remaining first term is

$$(p\!\!\!/)^{2} = \frac12\{\gamma^\mu,\gamma^\nu\}p_\mu p_\nu =p^2I.$$

Subtracting $m^2c^2I$ gives the result.

3. Similarity invariance

Show that $\gamma'^\mu=S\gamma^\mu S^{-1}$ preserves both the Clifford relation and $\operatorname{tr}(\gamma^\mu\gamma^\nu)$.

Solution

Conjugation preserves products and sums, giving

$$\{\gamma'^\mu,\gamma'^\nu\} = S\{\gamma^\mu,\gamma^\nu\}S^{-1} =2\eta^{\mu\nu}I.$$

Cyclicity of the trace gives

$$\operatorname{tr}(S\gamma^\mu\gamma^\nu S^{-1}) = \operatorname{tr}(\gamma^\mu\gamma^\nu).$$

4. Hermiticity identity

Use $(\gamma^0)^\dagger=\gamma^0$, $(\gamma^i)^\dagger=-\gamma^i$, and the Clifford algebra to prove $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$.

Solution

For $\mu=0$, the right side is $(\gamma^0)^3=\gamma^0$. For $\mu=i$, anticommutation and $(\gamma^0)^2=I$ give

$$\gamma^0\gamma^i\gamma^0=-\gamma^i,$$

which equals $(\gamma^i)^\dagger$.

References

• P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117, 610–624, 1928, doi:10.1098/rspa.1928.0023.
• W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, 2000, doi:10.1007/978-3-662-04275-5.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.