On the quantization of free fields of spin 1 and 2

The quantization of free fields is a standard topic in books on quantum field theory. The most simple procedure is to start with some 'elementary' particle, that is a projective unitary irreducible representation of the Poincare group (H,U) (here the first entry is the Hilbert space where the representation U acts), choose the appropriate statistics and apply the second quantization prescription: this is in mathematical language a functor attaching to H the corresponding Fock space Hphys ≡ F± (H), where the sign indicates the statistics; in this Hilbert space there exists a canonical procedure of constructing from U another representation of the Poincare group Γ (U) which is reducible and describes the symmetry properties of the multi-particle system of identical particles. In renormalisation theory one has to understand quite deeply the procedure of quantization with ghosts fields for particles of higher spin, like the electromagnetic, Yang-Mills or gravitational field. The idea is to extend the Fock space by adding fictitious particles (called ghosts). If the extended Hilbert space is Fgh then one tries to determine an operator Q, called supercharge which verifies Q2 = 0 and such that the physical Hilbert space is Hphys ≡ Ker(Q)/Im(Q). We will show below that there are still some aspects of these procedure not completely understood and proved into the literature. We will prove this in some detail for the case of the electromagnetic field (for which the elementary particle is the representation of the Poincare group characterized by zero mass and spin 1) and sketch the corresponding analysis for the massive spin 1 particle (heavy Bosons) and massless spin 2 particle (the graviton). In the case of the graviton we find some discrepancies with respect to the standard literature. As a consequence, we argue that the condition of gauge invariance which is generally postulated in these theories, is in fact not an independent axiom but the rather natural condition that the S-matrix factorizes to the physical Hilbert space Hphys ≡ Ker(Q)/Im(Q). (author)