Advanced quantum gauge field theory by van Nieuwenhuizen P.

By van Nieuwenhuizen P.

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QED Q and H take on a much simpler form, the so-called in- and out-states. We now proceed with a discussion of the relation between physical states, gauge conditions and BRST symmetry. This discussion is more technical than the rest of this chapter, but for a true assessment of the earlier work on QED from a modern perspective it is indispensable. In the Heisenberg picture states are time-independent while operators are in general time-dependent. Applying this to Q|ψ = 0, where also Q is time-independent, one can solve the equation Q|ψ = 0 most easily by using a basis in Hilbert space in terms of which fields take on a very simple form at asymptotic times t0 → ±∞.

This allowed one to use the manifestly relativistic Lagrangian approach instead of the older, more cumbersome Hamiltonian methods based on the Coulomb gauge. Thus around 1950 a shift occurred from the Hamiltonian to the Lagrangian approach, and the era of the Coulomb gauge came to an end. In the 1930’s calculations of radiative corrections to physical processes revealed what was to become a crisis: divergences. Dirac encountered them in 1934 when ¨ he calculated the polarization of the vacuum (extended by Uhling and Serber in Mesotron theories were attempts towards a field theory of the strong interactions by exchange of spin 0 and spin 1 bosons.

Heisenberg and Pauli took a completely opposite point of view. Both electrons and photons were described in terms of quantum field theory. In their approach, particle creation and annihilation emerged in a natural way, without the need of introducing a sea. Fermi developed a compromise: for pragmatic reasons he used quantum field theory for photons, but in his work in QED and weak interactions he treated electrons, photons and neutrons as particles. (He modeled his theory of the weak interctions on QED, and although a neutron could turn into a proton, the number of nucleonic particles did not change).

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