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Saturday, 14 July 2012

Dirac Fields (1)

Dirac Basis

The wave equation for integer spins are already known as Klein-Gordon equation: 2ϕ+m2ϕ=0We may develop half-ranked field to be (1)(iγμμm)ψ=0when (2+m2)ψ=(iγμμ+m)(iγμμm)ψTo satisfy the last equation, the Clifford algebra condition must be satisfied.{γμ,γν}=2ημνAlso the Lagrangian which derives the Dirac equation (1) can be written as L=ψ¯(im)ψwhere =γμμ. Then, we can see that the U(1) charge is ψ¯γ0ψd3xand usually we demand it to be ψψd3x to guarantee finiteness so ψ¯:=ψγ0

     Any basis of gamma matrices and their following theories are all equivalent, so first the Dirac basis is taken which is handy in nonrelativistic case.γ0=[11];γk=[σkσk]There is also Weyl basis which is written γ0=[11];γk=[σkσk]

Eigenspinors

As gamma matrices are constants, the plane wave solution kμ=iμψeikμxμ can also be a solution for the Dirac equation. By applying the solution to the equation, the equation may transform into momentum space. Then, by producting γ0 to the whole equation,
k0ψ=γ0(γk+m)ψwhere kμ=(k0,k). Here, the matrix on the right hand side is
γ0(γk+m)=[mσkσkm]The eigenvalue of k0 can be obtained as ±ω=±k2+m2.

     For k0=ω, ψ[m+ωσk]. For k0=ωψ[σkm+ω]. Attached plane wave part, ψ[m+ωσk]eiωt+ikx,[σkm+ω]eiωt+ikx=[m+ωσk]eiωt+ikx,[σkm+ω]eiωtikx=[m+ωσk]eikμxμ,[σkm+ω]eikμxμThese are the eigenfunction of the Dirac equation. As the first two and the second two have positive frequency and negative frequency respectively, the first eigenfunctions are usually interpreted as particle while the second are interpreted as antiparticle.

     Now, to normalise the functions in the same sense in the quantum mechanics, Noether charge may be normalised to 1. U(1) Noether charge of Dirac field is ψ¯iγ0δψd3x=ψ¯γ0ψd3x=ψψd3xFor both case, ψψ=(m+ω)2+(σk)2=m2+2mω+ω2+k2=2ω(m+ω)as m2+k2=ω2.

     Finally, the Dirac eigenfunctions are m+ω2ωV[αiσkm+ωαi]eikμxμ,m+ω2ωV[σkm+ωβjβj]eikμxμwhere αi's and βj's are the 2-component spinors for upper or lower space to handle 2×2 Pauli matrices and show the additional degrees of freedom which will correspond to classical spin.

     Or, the eigenspinors can be expressed with the scalar waves and spinors.u(i)(k)eikμxμ2ωV,v(j)(k)eikμxμ2ωVso defined isu(i)(k)=m+ω[αiσkm+ωαi];v(j)=m+ω[σkm+ωβjβj]andu(i+2)(k)=v(i)(k);v(j+2)(k)=u(j)(k)

Spin

Spinor indices come from space-time symmetry, so the space-time rotation may have corresponding transformation in spinor space. By considering the rotational transform xμxμ+ωμνxν on the gamma matrices, which is a converter between spinor indices and space-time indices, we may find the spinor representation of rotation.

     First, we may consider general fields f(xμ), and then for the infinitesimal angle ω, the scalar field may transforms to f(xμ)f(xμ+ωμν). Its generator may be Lμν=i(xμνxνμ)so that f(x)ei2Lμνωμνf(x), and additional space-time indices may give other infinitesimal transforms.

     As the gamma matrices has 1 vector index and is (1,1)-rank tensor (linear operator) in spinor space, the matrices may transform, under rotation, γμU(γμ+ωμνγν)U1Be ware of that U is not guaranteed to be unitary. We may express with generators: U=ei2Sρσωρσ (one half factor by analogy to Lμν). We expect that the gamma matrices is numeric tensor so invariant under such total transform. ei2Sρσωρσ(γμ+ωμνγν)ei2Sρσωρσ=γμ

     Applying Baker-Campbell formula eXYeX=Y+[X,Y]+O(X2)gives the equation of commutators. ωμνγν[i2Sρσωρσ,γμ+ωμνγν]+O(ω2)=ωμνγνi2[Sρσ,γμ]ωρσ+O(ω2)=0As Sρσ has 2 space-time indices and has commutation relation with gamma matrices, the ansatz can be taken Sρσ=Aγργσ. Then, as [γργσ,γμ]=γρ{γσ,γμ}{γμ,γρ}γσ=2ημσγρ2ημργσthe equation becomes ωμνγνiA(ωρμγρωμσγσ)=ωμνγν+2iAωμσγσ=0As A=i2 satifies the equation, the solution may be Sρσ=i2γργσ; however we demand the rotation generators to be antisymmetric generally, the solution may be antisymmetrised. Sρσ=i4[γρ,γσ]

     Therefore, Dirac fields transform by generator L+S. Now we may see the eigenvalues of the total generator in the nonrelativistic limit to match the quantum mechanics. Before that, we may explicitly obtain the generator to calculate; here, in Dirac basis. [γi,γj]=[σi,σj]=2iϵijkσk=4iϵijkSk(Euclidean)soSij=ϵijkSk(Euclidean)This is exactly the same with the familiar 3-d quantum mechanics. Thus, each upper and lower 2 component spinor in Dirac basis indicates the Pauli spinor for particle and antiparticle.

In Weyl basis, [γμ,γν]=[σ[μσ¯ν]σ¯[μσν]]
[γ0,γi]=[2σi2σi]
[γi,γj]=[σi,σj]

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