This small book consists of two distinct parts. The first entitled: Quantum Gravity: What and Why, expands on a lecture of the author to a general audience. It is on the level of a Scientific American article. For example there are no equations in it. There is no mathematical content, nor was there intended to be, in this article.
The second part has the same title as the book and is more technical in nature. It is concerned with developing a quantization procedure for the past and future null infinities \(I_{\pm}\) of an asymptotically flat, time oriented Lorentzian manifold (M,g) which represents either an isolated body which may emit gravitational radiation or a pure gravitational radiation field itself. The \(M\cup I_{\pm}\) are certain conformal completions of (M,g) in the ”infinite future and past”. The quantization will give the so-called out and in Hilbert spaces \({\mathcal H}_{\pm}\) associated with \(I_{\pm}\). One hopes to use these quantizations to construct the scattering matrix \(S: {\mathcal H}_-\to {\mathcal H}_+\) for quantum gravity in the situation described above.
The focus of attention, at least mathematically, for the author becomes a mathematical object which we now define. A null infinite is a pair (I,\(\{\) (q,N)\(\})\) where (a) I is a \(C^{\infty}\) manifold diffeomorphic to \(R\times S^ 2\); (b) \(\{\) (q,N)\(\}\) is a collection of pairs where q is a nonvanishing positive semi-definite tensor field on I and N is a non-vanishing vector field on I such that (i) \(i(V)q=0\) iff V is proportional to N; (ii) \({\mathcal L}_ Nq=0\); (iii) (q,N) and (q’,N’) are in the collection iff there exists a smooth function \(\omega\) on I such that \(q'=\omega^ 2q\), \(N'=\omega^{-1}\); (iv) the vector field N is complete and its manifold of orbits is diffeomorphic to \(S^ 2\). The infinite dimensional group B of diffeomorphisms of I leaving the family of pairs \(\{\) (q,N)\(\}\) invariant also plays an important role here. This is the so-called BMS group named after Bondi, Metzner and Sachs.
The author shows that there is a natural ”classical” symplectic manifold (\(\Gamma\),\(\omega)\) associated with \(\{\) (q,N)\(\}\). This is an infinite dimensional affine space consisting of certain equivalence classes of connections on I. In addition there are a family of ”classical” observables \({\mathbb{N}}(f)\) on \(\Gamma\) indexed by certain 2-tensor fields f on I which satisfy the canonical Poisson bracket relations \[ \{{\mathbb{N}}(f),{\mathbb{N}}(g)\}=\Omega (f,g). \] (The \(\{\) \({\mathbb{N}}(f)\}\) are quaintly called the smeared out news observables.) One then quantizes the canonical commutation relations by the usual Fock method. At this point one can show that the 1-particle sector represents particles with zero rest mass and spin (helicity) 2. These are called gravitons. The Fock representation is, however, too limited to handle the infra-red problem. Using the BMS group B one can define automorphisms of the \({}^*\)- algebra generated by the quantitized \({\mathbb{N}}(f)\) which are not unitarily implementable. These lead to a huge family of other non-isomorphic representations of the CCR which ”solve” the infra-red problem for gravitons (at least in this framework).