From Special Relativity to Feynman Diagrams.pdf

# Which has the form of the potential energy of a

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which has the form of the potential energy of a magnetic dipole in an external magnetic field with: μ s = e mc s = g e 2 mc s , (10.225) representing the electron intrinsic magnetic moment . The factor g = 2 is called the g-factor and the gyromagnetic ratio associated with the spin, defined as | μ s | / | s | , is g | e | /( 2 mc ). Recall that the magnetic moment associated with the orbital motion of a charge e reads μ orbit = e 2 mc M , (10.226) M being the orbital angular momentum. The gyromagnetic ratio | μ s | / | s | = | e | /( mc ) is twice the one associated with the orbital angular momentum. This result was found by Dirac in 1928. 10 Finally we note that in the present non-relativistic approximation, taking into account that the small components χ can be neglected, the probability density ψ ψ = ϕ ϕ + χ χ reduces to ϕ ϕ as it must be the case for the Schroedinger equation. Let us write the Lagrangian density for a fermion with charge e , coupled to the electromagnetic field: L = ¯ ψ( x ) i cD mc 2 ψ( x ). (10.227) The reader can easily verify that the above Lagrangian yields ( 10.212 ), or, equiva- lently ( 10.213 ). Just as we did for the scalar field, we can write L as the sum of a part describing the free fermion, plus an interaction term L I , describing the coupling to the electromagnetic field: L = L 0 + L I , L 0 = ¯ ψ( x ) i c mc 2 ψ( x ), L I = A μ ( x ) J μ ( x ) = eA μ ( x ) ¯ ψ( x μ ψ( x ), (10.228) where we have defined the electric current four vector J μ as: J μ ( x ) ej μ ( x ) = e ¯ ψ( x μ ψ( x ). (10.229) In Sect.10.4.2 we have shown that, by virtue of the Dirac equation, J μ is a conserved current, namely that it is divergenceless: μ J μ = 0 . 10 We recall that the Zeeman effect can only be explained if g = 2 . We see that this value is correctly predicted by the Dirac relativistic equation in the non-relativistic limit.

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352 10 Relativistic Wave Equations 10.8 Parity Transformation and Bilinear Forms It is important to observe that the standard representation of the γ -matrices given in (10.4.1) is by no means unique. Any other representation preserving the basic anticommutation rules works exactly the same way. It is only a matter of convenience touseoneor theanother. Inparticular theexpression( 10.97 ) of theLorentzgenerators μν in terms of γ μ -matrices is representation-independent. In this section we introduce a different representation, called the Weyl represen- tation , defined as follows: γ 0 = 0 1 2 1 2 0 ; γ i = 0 σ i σ i 0 ; i = 1 , 2 , 3 . (10.230) It is immediate to verify that the basic anticommutation rules ( 10.61 ) are satisfied. Defining σ μ = ( 1 2 , σ i ) ; ¯ σ μ = ( 1 2 , σ i ), (10.231) equation ( 10.230 ) can be given the compact form γ μ = 0 σ μ ¯ σ μ 0 . (10.232) The standard (Pauli) and the Weyl representations are related by a unitary change of basis: γ μ Pauli = U γ μ Weyl U . Decomposing as usual the spinor ψ into two-dimensional spinors ξ e ζ : ψ = ξ ζ , (10.233) one can show that, in the Weyl representation, the proper Lorentz transformations act separately on the two spinors, without mixing them. As we are going to show below, this means that the four-dimensional spinor representation, irreducible with
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