From Special Relativity to Feynman Diagrams.pdf

# Can be interpreted at the lagrangian level as the

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12.8 A Pedagogical Introduction to Renormalization 525 e 2 | k | 2 e 2 | k | 2 ( 1 + ( c ) ( −| k | 2 )) , where we have defined e 2 = Z 3 e 2 0 and we have set k 2 = −| k | 2 since we are in the non-relativistic limit in which k 0 = 0 . The factor 1 /( 1 + ( c ) ( −| k | 2 )) behaves much like a dielectric constant ( k ) since, as we show below, it reduces the effective charge (in absolute value) ‘seen’ at a given | k | as | k | decreases (i.e. as the distance from the charge increases) it lowers the Coulomb in the same way as it happens for charges in a dielectric material. Pictorially we may say that the vacuum polarization creates electron–positron virtual pairs circulating in the loop with a resulting partial screening of the electric charge, as it happens for a charge in a polar dielectric material. The actual value of ( c ) ( −| k | 2 )) can be computed explicitly by appropriate regularization of μν ( k 2 ). One finds that for | k | 2 m 2 (the threshold for the pair production e + e ) e 2 | k | 2 1 + ( c ) ( −| k | 2 ) e 2 | k | 2 1 + α 15 π | k | 2 m 2 . (12.293) Consider now a Hydrogen-like atom and let us Fourier transform the corresponding potential energy to configuration space. We have: eV ( x ) = d 3 k ( 2 π) 3 e i k · x e 2 | k | 2 ( 1 + ( c ) ( −| k | 2 )) e 2 4 π r α 15 π e 2 m 2 δ ( 3 ) ( x ), where r = | x | . This change indicates that the electromagnetic force becomes stronger at small distances. 37 This effect can be measured in hydrogen-like atoms, where the wave function is non-zero at the origin for s -waves. In fact this produces a shift of the 2 s 1 2 level given by E = d 3 x | ψ( x ) | 2 α 15 π e 2 m 2 δ ( 3 ) ( x ) = − 4 α 2 15 m 2 | ψ( 0 ) | 2 , and using | ψ( 0 ) | 2 = α 3 m 3 8 π for the 2s state, we get E = − 1 . 123 × 10 7 eV .
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