Abstract
We discuss the calculation of one-loop effective actions in Lorentzian spacetimes, based on a very simple application of the method of steepest descent to the integral over the field. We show that for static spacetimes this procedure agrees with the analytic continuation of Euclidean calculations. We also discuss how to calculate the effective action by integrating a renormalization group equation. We show that the result is independent of arbitrary choices in the definition of the coarse-graining and we see again that the Lorentzian and Euclidean calculations agree. When applied to quantum gravity on static backgrounds, our procedure is equivalent to analytically continuing time and the integral over the conformal factor.
Highlights
The path integral of a Lorentzian Quantum Field Theory (QFT) is ZL ( g) ≡ eiΓ L ( g) =Z dφ eiSL (φ,g), (1)where g may denote external parameters or background fields
The formula (26) can be seen as a consequence of this relation, since the effective action are proportional to the derivative evaluated for z = 0 of the zeta function associated with the Hessian of the action
If we use dimensional regularization to isolate the poles in this expression, we find that for a given dimension d there is only one divergent term in the sum: in particular for even d there is a dimensional pole when j = d2, coming from the zeta functions, while for odd d there is a pole for j = d−
Summary
This in itself is not problematic, because the spin-zero field does not propagate, but it means that while the Euclidean integral over the spin-two degrees of freedom is exponentially damped, the integral over the spin-zero degrees of freedom is exponentially divergent This can be seen at non-perturbative level: the spin-zero field is related to the conformal part of the metric, and the Hilbert action can be made arbitrarily negative by performing a highly oscillating conformal transformation [2]. The main point we would like to make is that the functional integral is not ill-defined because of the oscillatory character of the integrand, but because of the presence of infinitely many degrees of freedom This issue is exactly the same in the Euclideanized theory and in the Lorentzian theory treated by the steepest-descent method.
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