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As usual, these notes are intended for use by class participants only, and are not for circulation.

Week 6: Lectures 11, 12

March 5, 2012

The and algebra

The Lagrange density for the Dirac equation is •

= ψ¯(i∂ γµ m) ψ(x) , (2) L µ − where we define

ψ¯ ψ† γ , (3) ≡ 0 so that

4 ψψ¯ ψ¯ ψ ≡ α α α!=1 2 2 a b ∗ = (ξa˙ )∗ η + η ξb˙ a!=1 b!=1 " # where we’ve inserted the sums for (hopefully) clarity. Notice that unlike indices, Dirac indices are not raised or lowered.

The fundamental relation for Dirac (or “gamma”) matrices, • γµγν + γν,γµ γµ,γν =2gµν (4) ≡{ } where the first line defines the anticommutator. These rules imply

γµ γν = γνγµ ,µ= ν µ 2 −µν # (γ ) = g I4 4 . × Thus, Dirac matrices anticommute if their indices are different, while 0 the square of γ is +I4 4 (the four-by-four identity ), while the × 6 square of any of the “spatial” gammas is I4 4. (Note, it is not con- × ventional to exhibit the , and− the right hand side of the second relation above is normally just written gµν.) Further, useful relations are

0 µ 0 µ γ γ γ = γ (2δµ,0 1) µ − = γ gµµ = γµ µ =(γ )† (5)

where the first line follows from the anticommutation relations, the sec- ond from the difference between covariant and contravariant indices, and the third follows by inspection of the explicit γµ in Weyl represen- tation.

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 1 times the monomial without the pair. ±

The is everything we get by multiplying and adding • gamma matrices together with scalar coefficients.

The anticommutation relations are invariant under any 4 4 unitary • µ µ 1 × transformation U where γ Uγ U − , ψ Uψ (so that also ψ¯ 1 → → → ψ¯U − . (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”,

1 0 σ 1 σ 0 γi = i γ0 = 0 √2 $ σi 0 % √2 $ σ0 % − − which is found from the Weyl representation by

1 σ σ U = 0 0 . √2 $ σ0 σ0 % − Roughly, we’ll find the Weyl representation more natural for relativistic and the Dirac representation for nonrelativistic spinors.

7 Other products of Dirac (“gamma”) matrices: • i σ [γ ,γ ],γ iγ0γ1γ2γ3 µν ≡ 2 µ ν 5 ≡ The full Dirac algebra is sums of only sixteen basic matrices, • 0 1 2 3 4 Iγµ σµν γµγ5 γ5

The transformation properties are seen most easily in the Weyl repre- • sentation. The general form is,

ψ$(Λx)=S(Λ) ψ(x) , where S(Λ) for the four-component spinor is uniquely defined by the transformations of the dotted and undotted two-component spinors a that make it up, ξa˙ and η , h (Λ 1)0 S(Λ) = † − . $ 0 h(Λ) % This defines a new representation of the . In the mathe- matical 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 transfor- mations, 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 δλµν. For the infinitesimal case, the definitions of the 2 2 transformations give, × 1 ) 1+ 2 ( δ)ω iδθ) )σ 0 S(1 + δΛ) = − − · 1 $ 0 1 + (δ)ω iδθ)) )σ % 2 − · 1 =1 iδλµν σ . − 4 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 σµν matrices of the Dirac algebra, defined just above.

8 We recall the general expansion of an arbitrary • near the identity for for a field with space-time index b, 1 S (1 + δλ) = 1 + iδµν (Σ ) . ab 2 µν ab In this notation, for a Dirac field, indices a and b are α,β =1... 4, and 1 (Σ ) = (σ ) . µν αβ −2 µν αβ

Finite Lorentz transformations for Dirac fields can now be written as, • i i S(Λ) = exp ωiσ0i exp θi /ijkσjk , &2 ' &−4 ' where we follow the convention of a rotation followed by a boost. This can represent any , which can also be repre- sented as a boost followed by a rotation.

Projecting Weyl spinors •

1 ξd˙ 1 0 (I γ5)ψ = , (I + γ5)ψ = c 2 − $ 0 % 2 $ η %

Form invariance, momenta and

Let’s recall the general form of a Noether current, • µ µ ∂Ja µ ∂δx ∂ ∂δ φi ∗ µ =0,Ja = L , (6) ∂x −L ∂βa − ∂(∂µφi) ∂δβa !i where δ φ(x) is the variation of the field “at a point”. ∗ For the Poincar´egroup, the explicit forms we’ll need are • µ µ µ ν δx = δa + δλ νx ,

µ ν µν ∂φi(x) 1 µν δ φi(x)= (δa gµ δλ xµ) + i (Σµν) δλ φj(x) , ∗ − − ∂xν 2 ij for the coordinates and fields.

9 For scalar fields, of course, Σ= 0. For Dirac and scalar fields, • 1 (Σ ) = (σ ) µν αβ −2 µν αβ (Σ ) = (m ) . µν λσ − µν λσ where we recall m0i = Ki and mij = /ijkJk. The Hamiltonian of the Dirac field found in this way is • P 0 = d3)x T 00 ( 3 = d )x ( + iψ†∂0ψ) ( −L = d3)x ψ¯(x)( i)γ ∂) + m)ψ(x) , ( − · where we note the lack of a time derivative in P 0. The angular momentum tensor for a general field, • 3 3 Jνλ = d )x (xνT0λ xλT0ν)+i d )xπi(Σνλ)ijφj , ( − ( where the first term, present for scalar fields as well, describes the “me- chanical” or “orbital” angular momentum, while the second describes the intrinsic angular momentum or spin. here πi is the (classical) con- jugate momentum of field πi. For the Dirac field, this turns out to be iψ†. The Paul-Lubanski vector isolates the intrinsic angular momentum, • 1 W = J νλP σ µ −2 1 3 νλ σ = i/µνλσ d )xπi(Σ )ijφjP . −2 ( Global symmetries and conserved currents • N = Ψ¯ [ i(γ ∂) mδ ]Ψ L α,i · αβ − αβ β,i !i=1 3 3 Q = d x Ψ¯ γ0Ψ Qa = d x Ψ¯ Taγ0Ψ ( (

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