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The Lamb shift was subsequently correctly computed by Fukuda, Miyamoto and Tomonoga, and Kroll and Lamb, and French and Weisskopf. 31 The results he initially obtained for the Lamb shift did not agree with the experimental results. In the meantime, others [80] using covariant or noncovariant methods, found derivations of the Lamb shift which agreed with the experimental data. There occurred some unfortunate misunderstandings [75] but in the end all scientific questions were resolved. In early 1948, Schwinger worked out a manifestly relativistic and gauge invariant formulation, developing a Schr¨odinger equation on a general spacelike surface which was manifestly Lorentz invariant.

23 Fortunately, there exists a basis in the space of fields on which Another argument for adding a gauge fixing term to the classical gauge action came from a canonical analysis of QED in the 1950’s, which showed that the Maxwell field equations are inconsistent as operator equations, but adding a gauge-fixing term by hand, the corresponding Heisenberg field equations become consistent [60]. 22 But at finite temperature one even needs ghosts in QED. For example in the calculation of the free energy the field Aµ yields four times the result for one real scalar, and the ghosts subtract twice the result of one real scalar.

Because the operators a0 (k) + a3 (k) in (∂ µ Aµ )+ commute with the operators a†0 (k) + a†3 (k) in |phys , it is clear that these states are solutions of the Gupta-Bleuler condition. It is not difficult to prove that they are the only solutions. For l = 0 the states N |0 are null states: they are orthogonal to themselves, to other null states N |0 of the same form, and to purely transverse states. Physical states For QED with interactions, the positive frequency part of the field Aµ itself is difficult to write down in closed form, but the field ∂ µ Aµ is a free field satisfying ∂ · A = 0.

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Advanced quantum gauge field theory by van Nieuwenhuizen P.

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