New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

22 J.M. Gracia-BondíaIn particular, first-order CGI meanssT 1 (x) = i(∂ · T 1/1 )(x).For T 1 , let us try a general Ansatz containing cubic terms in the fields and leadingto a renormalizable theory. At our disposal there are three field sets: h, u, ũ. Themost general coupling with vanishing ghost number without derivatives is of theformaϕ 3 + b ϕh νμ h νμ + ch μν h ν γ hγμ + (u ·ũ)ϕ + eh νμ u ν ũ μ .Correspondingly, with ghost number one since the action of the BRS operatorincreases ghost number by one, we can have (with an obvious simplified notation)T μ 1/1 = a′ u μ ϕ 2 + b ′ u μ h · h + c ′ (u · h) μ ϕ + d ′ u α h αβ h βμ + e ′ u(u ·ũ).Forlorn hope. It must bes(∂ · T 1/1 ) = 0.This condition has only the trivial solution T 1/1 = 0.Since one cannot form scalars with one derivative, we are forced to considercubic couplings with two derivatives. This is the root of “non-normalizability” (inEpstein – Glaser jargon) of gravitation. There are 12 possible combinations in T 1involving only h with two derivatives, and 21 combinations in T 1 involving h, u, ũ,with two derivatives and zero total ghost number. At the end of the day, one obtainsT 1 = T1 h + T 1 u,with T 1 h uniquely proportional to L(1) (modulo physically irrelevantdivergences), andT u 1 = a( − u α (∂ β ũ ρ )∂ α h βρ + (∂ β u α ∂ α ũ ρ − ∂ α u α ∂ β ũ ρ + ∂ ρ u α ∂ β ũ α )h βρ) .The calculations are excruciatingly long, and of little interest. They, as well as theexplicit expression of T 1/1 , can be found in [37], to which we remit. By the way,had we tried to useg μν = η μν + λh μνinstead of (18), then T1 h turns out much more complicated – even after eliminationof a host of divergence couplings.More intrinsically interesting are the calculations of CGI at second order, alsodone in [37], which indeed reproduce L (2) . For the higher order analysis, one needssome (rather minimal) familiarity with the Epstein – Glaser method to inductivelyrenormalize (i.e. to define) the time-ordered products T n based on splitting of distributions.This requires use of antichronological products corresponding to the expansionof the inverse S-matrix. If we write the inverse power series