Gamma matrices

In mathematical physics, the gamma matrices, { γ 0 , γ 1 , γ 2 , γ 3 } {displaystyle {gamma ^{0},gamma ^{1},gamma ^{2},gamma ^{3}}} , also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles.This can be seen by exploiting the fact that all the four gamma matrices anticommute, so Take the standard anticommutation relation:Similarly to the proof of 1, again beginning with the standard commutation relation:To showIf μ = ν = ρ {displaystyle mu = u = ho } then ϵ σ μ ν ρ = 0 {displaystyle epsilon ^{sigma mu u ho }=0} and it is easy to verify the identity. That is the case also when μ = ν ≠ ρ {displaystyle mu = u eq ho } , μ = ρ ≠ ν {displaystyle mu = ho eq u } or ν = ρ ≠ μ {displaystyle u = ho eq mu } .From the definition of the gamma matrices, To showIf an odd number of gamma matrices appear in a trace followed by γ 5 {displaystyle gamma ^{5}} , our goal is to move γ 5 {displaystyle gamma ^{5}} from the right side to the left. This will leave the trace invariant by the cyclic property. In order to do this move, we must anticommute it with all of the other gamma matrices. This means that we anticommute it an odd number of times and pick up a minus sign. A trace equal to the negative of itself must be zero.To showFor the term on the right, we'll continue the pattern of swapping γ σ {displaystyle gamma ^{sigma }} with its neighbor to the left,To showFor a proof of identity 6, the same trick still works unless ( μ ν ρ σ ) {displaystyle left(mu u ho sigma ight)} is some permutation of (0123), so that all 4 gammas appear. The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so tr ⁡ ( γ μ γ ν γ ρ γ σ γ 5 ) {displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{ u }gamma ^{ ho }gamma ^{sigma }gamma ^{5} ight)} must be proportional to ϵ μ ν ρ σ {displaystyle epsilon ^{mu u ho sigma }} ( ϵ 0123 = η 0 μ η 1 ν η 2 ρ η 3 σ ϵ μ ν ρ σ = η 00 η 11 η 22 η 33 ϵ 0123 = − 1 ) {displaystyle left(epsilon ^{0123}=eta ^{0mu }eta ^{1 u }eta ^{2 ho }eta ^{3sigma }epsilon _{mu u ho sigma }=eta ^{00}eta ^{11}eta ^{22}eta ^{33}epsilon _{0123}=-1 ight)} . The proportionality constant is 4 i {displaystyle 4i} , as can be checked by plugging in ( μ ν ρ σ ) = ( 0123 ) {displaystyle (mu u ho sigma )=(0123)} , writing out γ 5 {displaystyle gamma ^{5}} , and remembering that the trace of the identity is 4.Denote the product of n {displaystyle n} gamma matrices by Γ = γ μ 1 γ μ 2 … γ μ n . {displaystyle Gamma =gamma ^{mu 1}gamma ^{mu 2}dots gamma ^{mu n}.} Consider the Hermitian conjugate of Γ {displaystyle Gamma } : In mathematical physics, the gamma matrices, { γ 0 , γ 1 , γ 2 , γ 3 } {displaystyle {gamma ^{0},gamma ^{1},gamma ^{2},gamma ^{3}}} , also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles. In Dirac representation, the four contravariant gamma matrices are γ 0 {displaystyle gamma ^{0}} is the time-like, hermitian matrix. The other three are space-like, antihermitian matrices. More compactly, γ 0 = σ 3 ⊗ I {displaystyle gamma ^{0}=sigma ^{3}otimes I} , and γ i = i σ 2 ⊗ σ i {displaystyle gamma ^{i}=isigma ^{2}otimes sigma ^{i}} , where ⊗ {displaystyle otimes } denotes the Kronecker product and the σ i {displaystyle sigma ^{i}} (for i = 1, 2, 3) denote the Pauli matrices. Analogous sets of gamma matrices can be defined in any dimension and for any signature of the metric. For example, the Pauli matrices are a set of 'gamma' matrices in dimension 3 with metric of Euclidean signature (3, 0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra. The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation where { , } {displaystyle {,}} is the anticommutator, η μ ν {displaystyle eta ^{mu u }} is the Minkowski metric with signature (+ − − −), and I 4 {displaystyle I_{4}} is the 4 × 4 identity matrix. This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by and Einstein notation is assumed. Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation: