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These are the physical states. States with vanishing norm may or may not contain transverse gauge bosons but they are BRST exact: they can be written as |ψ = Q|ψ . For example |ψ = a†1 (k)Q|χ = Q(a†1 (k)|χ . Since Q2 = 0, BRST-exact states are orthogonal to themselves and to BRST-closed states: Qψ |Qψ = 0 and Qψ |ψ = 0 if Q|ψ = 0. (ii) the longitudinal and time-like polarizations together with the ghosts and anti-ghosts form multiplets called Kugo-Ojima quartets [62] (if one uses relativistic gauges with ∂ µ Aµ = 0).

However, the precise form of these BRST exact states is a complicated issue. For example, in the sector with two quartet modes the state (s† l† − b† c† )|0 > is BRST exact (s† denotes the scalar modes in A0 , l† the longitudinal modes, and b† and c† the antighost and ghost modes), but neither s† l† |0 > or b† c† |0 > are BRST closed. It should be stressed that the proof that Q becomes linear in asymptotic states at 1. QED 43 asymptotic times is not at all trivial for nonabelian gauge theories, see [62]24 , although the result is plausible if a conserved charge is not spontaneously broken, because in that case states form representations (“multiplets”) of the algebra satisfied by the charges.

36 Before the neutrino was conceived, nuclei were thought to be bound states of protons and electrons. For example, the nitrogen nucleus with A = 14 and Z = 7 required 14 protons and 7 electrons, and should thus have half-integer spin. However, band spectra showed that it had spin 1. Another problem with the proton-electron model of the nucleus was that nuclei would have far too large magnetic moments (a factor 1000 too large) if they contained electrons. See ref. b. 35 2. WEAK INTERACTIONS 53 concept in field theory was introduced by Fermi in the 1930’s: the notion of particle creation.