By van Nieuwenhuizen P.

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**Example text**

They showed that this condition led to normalizable physical states. 13) k is shorthand notation for a†1 (k1 )n1 a†1 (k2 )n2 · · · a†1 (km )nm and cl are constants. 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.

The BRST operator (or, in fact, any other conserved charge [62] unless spontaneously broken) becomes bilinear in asymptotic fields, and the BRST transformations reduce to linear transformations on the in- and out fields. The S matrix connects in- and out-states: S|ψ, out = |ψ, in , and since the set of in-fields is the same as the set of out-fields, one may identify the in- and out states (and fields) and speak of asymptotic states and fields. The condition Q|ψ = 0 (BRST closed states) selects then asymptotic states with the following properties: (i) states |ψ have either strictly positive norm or vanishing norm.

QED 49 present before,32 and now he also found a numerical result for the energy shift that agreed with experiment. These calculations of the anomalous magnetic moment and the Lamb shift established QED as a consistent field theory [75]. For more details, see the chapter on the anomalous magnetic moment of the muon and electron. Schwinger pioneered many other developments in quantum field theory [81]. He introduced the proper time technique (based on the heat kernel method which was developed by Fock [82]) in a study of the gauge invariance of vacuum polarization (perhaps his best paper).