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Dirac Matrices and Lorentz Spinors

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Dirac Matrices and Lorentz Spinors Dirac Matrices and Lorentz Spinors 1 Background: In 3D, the spinor j = 2 representation of the Spin(3) rotation group is constructed from the Pauli matrices σx, σy, and σz, which obey both commutation and anticommutation relations i j ijk k i j ij [σ ; σ ] = 2i σ and fσ ; σ g = 2δ × 12×2 : (1) Consequently, the spin matrices i 1 S = − 4 σ × σ = 2 σ (2) commute with each other like angular momenta, [Si;Sj] = iijkSk, so they represent the generators of the rotation group. In this spinor representation, the finite rotations R(φ, n) are represented by M(R) = exp−iφn · S; (3) while the spin matrices themselves transform into each other as components of a 3{vector, M −1(R)SiM(R) = RijSj: (4) In this note, I shall generalize this construction to the Dirac spinor representation of the Lorentz symmetry Spin(3; 1). The Dirac Matrices γµ generalize the anti-commutation properties of the Pauli matrices σi to the 3 + 1 Minkowski dimensions: µ ν ν µ µν γ γ + γ γ = 2g × 14×4 : (5) The γµ are 4 × 4 matrices, but there are several different conventions for their specific form. In my class I shall follow the same convention as the Peskin & Schroeder textbook, namely 1 the Weyl convention where in 2 × 2 block notations ! ! 02×2 12×2 02×2 +~σ γ0 = ; ~γ = : (6) 12×2 02×2 −~σ 02×2 Note that γ0 is an hermitian matrix while γ1, γ2, and γ3 are anti-hermitian matrices. Apart from that, the specific forms of the matrices are not important, the Physics follows from the anti-commutation relations (5). i 1 The Lorentz spin matrices generalize S = − 2 σ × σ rather than S = 2 σ. In 4D, the vector product becomes the antisymmetric tensor product, so we define µν def νµ def i µ ν S = −S = 4 [γ ; γ ]: (7) Consequently, µ ν 1 µ ν 1 µ ν µν µν γ γ = 2 fγ ; γ g + 2 [γ ; γ ] = g × 14×4 − 2iS : (8) Thanks to the anti-commutation relations (5) for the γµ matrices, the Sµν obey the commu- tation relations of the Lorentz generators J^µν = −J^νµ. Moreover, the commutation relations of the spin matrices Sµν with the Dirac matrices γµ are similar to the commutation relations of the J^µν with a Lorentz vector such as P^µ. Lemma: [γλ;Sµν] = igλµγν − igλνγµ: (9) Proof: In light of eq. (8) and the fact that the unit matrix commutes with everything, we have µν i µ ν [X; S ] = 2 [X; γ γ ] for any matrix X: (10) Moreover, the commutator on the RHS may often be obtained from the Leibniz rules for the commutators or anticommutators: [A; BC] = [A; B]C + B[A; C] = fA; BgC − BfA; Cg; (11) fA; BCg = [A; B]C + BfA; Cg = fA; BgC − B[A; C]: 2 In particular, [γλ; γµγν] = fγλ; γµgγν − γµfγλ; γνg = 2gλµγν − 2gλνγµ (12) and hence λ µν i λ µ ν λµ ν λν µ [γ ;S ] = 2 [γ ; γ γ ] = ig γ − ig γ : (13) Quod erat demonstrandum. Theorem: The Sµν matrices commute with each other like Lorentz generators, Sκλ;Sµν = igλµSκν − igλνSκµ − igκµSλν + igκνSλµ: (14) Proof: Again, we use the Leibniz rule and eq. (8): γκγλ;Sµν = γκ γλ;Sµν + γκ;Sµν γλ = γκigλµγν − igλνγµ + igκµγν − igκνγµγλ = igλµγκγν = gκν − 2iSκν − igλνγκγµ = gκµ − 2iSκµ + igκµγνγλ = gλν + 2iSλν − igκνγµγλ = gλµ + 2iSλµ = igλµgκν − igλνgκµ + igκµgλν − igκνgλµ hh note cancellation ii + 2gλµSκν − 2gλνSκµ − 2gκµSλν + 2gκνSλµ = 2gλµSκν − 2gλνSκµ − 2gκµSλν + 2gκνSλµ (15) and therefore κλ µν i κ λ µν λµ κν λν κµ κµ λν κν λµ S ;S = 2 γ γ ;S = ig S − ig S − ig S + ig S : (16) Quod erat demonstrandum. In light of this theorem, the Sµν matrices represent the Lorentz generators J^µν in the 4-component spinor multiplet. 3 Finite Lorentz transforms: Any continuous Lorentz transform | a rotation, or a boost, or a product of a boost and a rotation | obtains from exponentiating an infinitesimal symmetry 0µ µ µν X = X + Xν (17) where the infinitesimal µν matrix is antisymmetric when both indices are raised (or both µν νµ µ lowered), = − . Thus, the L ν matrix of any continuous Lorentz transform is a matrix exponential µ µ µ µ 1 µ λ 1 µ λ κ L ν = exp(Θ) ν ≡ δν + Θ ν + 2 Θ λΘ ν + 6 Θ λΘ κΘ ν + ··· (18) of some matrix Θ that becomes antisymmetric when both of its indices are raised or lowered, Θµν = −Θνµ. Note however that in the matrix exponential (18), the first index of Θ is raised while the second index is lowered, so the antisymmetry condition becomes (gΘ)> = −(gΘ) instead of Θ> = −Θ. The Dirac spinor representation of the finite Lorentz transform (18) is the 4 × 4 matrix i αβ MD(L) = exp − 2 ΘαβS : (19) The group law for such matrices + 8L1;L2 2 SO (3; 1);MD(L2L1) = MD(L2)MD(L1) (20) follows automatically from the Sµν satisfying the commutation relations (14) of the Lorentz generators, so I am not going to prove it. Instead, let me show that when the Dirac matrices µ γ are sandwiched between the MD(L) and its inverse, they transform into each other as components of a Lorentz 4{vector, −1 µ µ ν MD (L)γ MD(L) = L νγ : (21) This formula makes the Dirac equation transform covariantly under the Lorentz transforms. 4 Proof: In light of the exponential form (19) of the matrix MD(L) representing a finite Lorentz transform in the Dirac spinor multiplet, let's use the multiple commutator formula (AKA the Hadamard Lemma ): for any 2 matrices F and H, 1 1 exp(−F )H exp(+F ) = H + H; F + 2 H; F ;F + 6 H; F ;F ;F + ··· : (22) µ i αβ −1 In particular, let H = γ while F = − 2 ΘαβS so that MD(L) = exp(+F ) and MD (L) = exp(−F ). Consequently, −1 µ µ µ 1 µ 1 µ MD (L)γ MD(L) = γ + γ ;F + 2 γ ;F ;F + 6 γ ;F ;F ;F + ··· (23) where all the multiple commutators turn out to be linear combinations of the Dirac matrices. Indeed, the single commutator here is µ i µ αβ 1 µα β µβ α µα β µ λ γ ;F = − 2 Θαβ γ ;S = 2 Θαβ g γ − g γ = Θαβ g γ = Θ λγ ; (24) while the multiple commutators follow by iterating this formula: µ µ λ µ λ ν µ µ λ ρ ν γ ;F ;F = Θ λ γ ;F = Θ λΘ νγ ; γ ;F ;F ;F = Θ λΘ ρΘ νγ ;:::: (25) Combining all these commutators as in eq. (23), we obtain −1 µ µ µ 1 µ 1 µ MD γ MD = γ + γ ;F + 2 γ ;F ;F + 6 γ ;F ;F ;F + ··· µ µ ν 1 µ λ ν 1 µ λ ρ ν = γ + Θ νγ + 2 Θ λΘ νγ + 6 Θ λΘ ρΘ νγ + ··· (26) µ µ 1 µ λ 1 µ λ ρ ν = δν + Θ ν + 2 Θ λΘ ν + 6 Θ λΘ ρΘ ν + ··· γ µ ν ≡ L νγ : Quod erat demonstrandum. 5 Dirac Equation and Dirac Spinor Fields History: Originally, the Klein{Gordon equation was thought to be the relativistic version of the Schr¨odingerequation | that is, an equation for the wave function (x; t) for one relativistic particle. But pretty soon this interpretation run into trouble with bad probabilities (negative or > 1) when a particle travels through high potential barriers or deep potential wells. There were also troubles with relativistic causality, and a few other things. Paul Adrien Maurice Dirac had thought that the source of all those troubles was the ugly form of relativistic Hamiltonian H^ = pp^2 + m2 in the coordinate basis, and that he could solve all the problems with the Klein-Gordon equation by rewriting the Hamiltonian as a first-order differential operator @ H^ = p^ · ~α + mβ =) Dirac equation i = −i~α · r + mβ (27) @t where α1; α2; α3; β are matrices acting on a multi-component wave function. Specifically, all four of these matrices are Hermitian, square to 1, and anticommute with each other, 2 fαi; αjg = 2δij ; fαi; βg = 0; β = 1: (28) Consequently 2 1 2 ~α · p^ = αiαj × p^ip^j = 2 fαi; αjg × p^ip^j = δij × p^ip^j = p^ ; (29) and therefore 2 ^ 2 2 2 2 ^2 2 HDirac = ~α · p^ + βm = ~α · p^ + fαi; βg × p^im + β × m = p + 0 + m : (30) This, the Dirac Hamiltonian squares to p^2 + m2, as it should for a relativistic particle. The Dirac equation (27) turned out to be a much better description of a relativistic 1 electron (which has spin = 2 ) than the Klein{Gordon equation. However, it did not resolve the troubles with relativistic causality or bad probabilities for electrons going through big 2 potential differences e∆Φ > 2mec . Those problems are not solvable in the context of a relativistic single-particle quantum mechanics but only in the quantum field theory. 6 Modern point of view: Today, we interpret the Dirac equation as the equation of motion for a Dirac spinor field Ψ(x), | which comprises 4 complex component fields Ψα(x) arranged in a column vector 0 1 Ψ1(x) B C B Ψ2(x) C Ψ(x) = B C ; (31) B Ψ (x) C @ 3 A Ψ4(x) 0µ µ ν and transforms under the continuous Lorentz symmetries x = L νx according to 0 0 Ψ (x ) = MD(L)Ψ(x): (32) The classical Euler{Lagrange equation of motion for the spinor field is the Dirac equation @ i Ψ + i~α · rΨ − mβΨ = 0: (33) @t To recast this equation in a Lorentz-covariant form, let β = γ0; αi = γ0γi ; (34) it is easy to see that if the γµ matrices obey the anticommutation relations (5) then the ~α and β matrices obey the relations (28) and vice verse.
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