As usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12

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1 As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation is L = ψi µ γ µ m ψx, where we define ψ ψ γ 0, 3 so that ψψ = 4 ψ α ψ α α= ξȧ η a + a= b= η b ξḃ where we ve inserted the sums for hopefully clarity Notice that unlike spinor indices, Dirac indices are not raised or lowered The fundamental relation for Dirac or gamma matrices, γ µ γ ν + γ ν,γ µ {γ µ,γ ν } =g µν 4 where the first line defines the anticommutator These rules imply γ µ γ ν = γ ν γ µ, µ ν γ µ = g µν I 4 4 Thus, Dirac matrices anticommute if their indices are different, while the square of γ 0 is +I 4 4 the four-by-four identity matrix, while the 6

2 square of any of the spatial gammas is I 4 4 Note, it is not conventional to exhibit the identity matrix, and the right hand side of the second relation above is normally just written g µν Further, useful relations are γ 0 γ µ γ 0 = γ µ δ µ,0 = γ µ g µµ = γ µ = γ µ 5 where the first line follows from the anticommutation relations, the second from the difference between covariant and contravariant indices, and the third follows by inspection of the explicit γ µ in Weyl representation By using the anticommutation relations 5, we can reduce the power of any monomial involving more than four γ s, since in this case, at least two must be the same, and any pair can be anticommuted past other gammas to give a square, which results in ± times the monomial without the pair The Dirac algebra is everything we get by multiplying and adding gamma matrices together with scalar coefficients The anticommutation relations are invariant under any 4 4 unitary transformation U where γ µ Uγ µ U, ψ Uψ so that also ψ ψu The last relation is why we want U to be unitary This is said to be a change in the representation of the Dirac matrices The main alternative to the Weyl representation given above is the Dirac representation, γ i = 0 σi σ i 0 γ 0 = σ0 0 σ 0 which is found from the Weyl representation by U = σ0 σ 0 σ 0 σ 0 Roughly, we ll find the Weyl representation more natural for relativistic spinors and the Dirac representation for nonrelativistic spinors 7

3 Other products of Dirac gamma matrices: σ µν i [γ µ,γ ν ], γ 5 iγ 0 γ γ γ 3 The full Dirac algebra is sums of only sixteen basic matrices, I γ µ σ µν γ µ γ 5 γ 5 The transformation properties are seen most easily in the Weyl representation The general form is, ψ Λx =SΛ ψx, where SΛ for the four-component spinor is uniquely defined by the transformations of the dotted and undotted two-component spinors that make it up, ξȧ and η a, h SΛ = Λ 0 0 hλ This defines a new representation of the Lorentz group In the mathematical sense, it is reducible because the two two component spinors don t mix under Lorentz transformations As usual, however, we can look at infinitesimal transformations to identify the generators for this transformation There will be two ways of representing these transformations, in terms of the three boost and three rotational parameters, ω and θ, respectively, or in terms of the six independent components of an antisymmetric matrix δλ µν For the infinitesimal case, the definitions of the transformations give, + S + δλ = δ ω iδ θ σ δ ω iδ θ σ = 4 iδλµν σ µν Notice that the only difference in the two block diagonal terms is the sign on the boost vector, confirming that there is only one independent representation of rotations, and that it is only the possibility of boosts that require two representation In the second equality, we specify the relationship to the σ µν matrices of the Dirac algebra, defined just above 8

4 We recall the general expansion of an arbitrary transformation matrix near the identity for for a field with space-time index b, S ab + δλ = + iδµν Σ µν ab In this notation, for a Dirac field, indices a and b are α, β = 4, and Σ µν αβ = σ µν αβ Finite Lorentz transformations for Dirac fields can now be written as, i SΛ = exp ω iσ 0i exp i 4 θ i ɛ ijk σ jk, where we follow the convention of a rotation followed by a boost This can represent any Lorentz transformation, which can also be represented as a boost followed by a rotation Projecting Weyl spinors I γ 5ψ = ξ d 0, 0 I + γ 5ψ = η c Form invariance, momenta and spin Let s recall the general form of a Noether current, J µ a x µ =0, Jµ a = L δxµ β a i L µ φ i δ φ i δβ a, 6 where δ φx is the variation of the field at a point For the Poincaré group, the explicit forms we ll need are δx µ = δa µ + δλ µ νx ν, δ φ i x = δa µ g µ ν δλ µν x µ φ ix x ν + i Σ µν ij δλ µν φ j x, for the coordinates and fields 9

5 For scalar fields, of course, Σ = 0 For Dirac and scalar fields, Σ µν αβ = σ µν αβ Σ µν λσ = m µν λσ where we recall m 0i = K i and m ij = ɛ ijk J k The Hamiltonian of the Dirac field found in this way is P 0 = d 3 x T 00 = d 3 x L + iψ 0 ψ = d 3 x ψx i γ + mψx, where we note the lack of a time derivative in P 0 The angular momentum tensor for a general field, J νλ = d 3 x x ν T 0λ x λ T 0ν +i d 3 x π i Σ νλ ij φ j, where the first term, present for scalar fields as well, describes the mechanical or orbital angular momentum, while the second describes the intrinsic angular momentum or spin here π i is the classical conjugate momentum of field π i For the Dirac field, this turns out to be iψ The Paul-Lubanski vector isolates the intrinsic angular momentum, W µ = J νλ P σ = iɛ µνλσ d 3 x π i Σ νλ ij φ j P σ Global symmetries and conserved currents N L = Ψ α,i [ iγ αβ mδ αβ ]Ψ β,i Q = i= d 3 x Ψγ 0 Ψ Q a = d 3 x ΨT a γ 0 Ψ 0