Higher-Derivative Quantum Gravity

Abstract

Let M = ϕⁱ be the configuration space of a boson gauge field and S( ϕ ) be a classical action functional that is invariant with respect to local gauge transformations

$$ \delta \varphi ^i = R^i _\alpha \left( \varphi \right)\xi ^\alpha , $$

(4.1)

forming the gauge group G. Here ξ^α are the group parameters and Rⁱ _α(ϕ) are the local generators of the gauge transformations that form a closed Lie algebra

$$ \left[ {\hat R_\alpha ,\hat R_\beta } \right] = C^\gamma _{\alpha \beta } \hat R_\gamma , $$

(4.2)

where

$$ \hat R_\alpha \equiv R^i _\alpha \left( \varphi \right)\frac{\delta } {{\delta \varphi ^i }}, $$

(4.3)

and $C_{\alpha \beta }^{^\gamma }$ are the structure constants of the gauge group satisfying the Jacobi identity

$$ C^\alpha _{\mu \left[ \beta \right.} C^\mu _{\left. {\gamma \delta } \right]} = 0. $$

(4.4)

The classical equations of motion determined by the action functional S(ϕ) have the form

$$ \varepsilon _i \left( \varphi \right) = 0, $$

(4.5)

where ε_i = S,i is the “extremal” of the action. The equation (4.5) defines the “mass shell” in the quantum perturbation theory.