Feynman Path Integral Formulation

1.7 Higher Derivative Terms 25
case, in fact one can prove that higher derivative gravity (to be defined below) is
perturbatively renormalizable to all orders in four dimensions.
At the same time new issues arise, which will be detailed below. The first set
of problems has to do with the fact that, quite generally higher derivative theories
with terms of the type φ∂ 4 φ suffer from potential unitarity problems, which can
lead to physically unacceptable negative probabilities. But since these are genuinely
dynamical issues, it will be difficult to answer them satisfactorily in perturbation
theory. In non-Abelian gauge theories one can use higher derivative terms, instead
of the more traditional dimensional continuation, to regulate ultraviolet divergences
(Slavnov, 1972), and higher derivative terms have been used successfully for some
time in lattice regulated field theories (Symanzik, 1983a,b). In these approaches the
coefficient of the higher derivative terms is taken to zero at the end. The second set of
issues is connected with the fact that the theory is asymptotically free in the higher
derivative couplings, implying an infrared growth which renders the perturbative
estimates unreliable at low energies, in the regime of perhaps greatest physical interest.
Note that higher derivative terms arise in string theory as well (Forger, Ovrut,
Theisen and Waldram, 1996).
Let us first discuss the general formulation. In four dimensions possible terms
quadratic in the curvature are
∫
d 4 x √ gR 2
∫
d 4 x √ gR μν R μν
∫
d 4 x √ gR μνλσ R μνλσ
∫
d 4 x √ gC μνλσ C μνλσ
∫
d 4 x √ g ε μνκλ ε ρσωτ R μνρσ R κλωτ = 128π 2 χ
∫
d 4 x √ g ε ρσκλ R μνρσ R μν κλ = 96π2 τ , (1.127)
where χ is the Euler characteristic and τ the Hirzebruch signature. It will be shown
below that these quantities are not all independent. The Weyl conformal tensor is
defined in d dimensions as
C μνλσ = R μνλσ − 2
d−2 (g μ[λ R σ]ν − g ν[λ R σ]μ )
+
2
(d−1)(d−2) Rg μ[λ g σ]ν , (1.128)
where square brackets denote antisymmetrization. It is called conformal because
it can be shown to be invariant under conformal transformations of the metric,
g μν (x) → Ω 2 (x)g μν (x). In four dimensions one has
C μνλσ = R μνλσ − R λ[μ g ν]σ − R σ[μ g ν]λ + 1 3 Rg λ[μg ν]σ . (1.129)