[Paper Review] One-loop quantum gravity from a worldline viewpoint

Fiorenzo Bastianelli, Roberto Bonezzi|arXiv (Cornell University)|Apr 26, 2013

Black Holes and Theoretical Physics17 references4 citations

TL;DR

This paper develops a worldline path integral approach to one-loop quantum gravity in D=4 by deriving a particle action from quadratic fluctuations of the gravitational field, enabling background-covariant gauge-fixed computation of the effective action. It successfully reproduces known one-loop divergences in Einstein gravity with cosmological constant, offering a simpler alternative to standard QFT methods for computing one-loop amplitudes in quantum gravity.

ABSTRACT

We develop a worldline approach to quantum gravity in D=4. Using the background field method we consider the covariantly gauge fixed Einstein-Hilbert action with cosmological constant, and find a worldline representation of the differential operators identified by its quadratic approximation. We test it by computing the correct one-loop divergencies. Alternative worldline methods, such as the use of the O(4) spinning particle that is known to describe correctly the propagation of a massless spin 2 particle in D=4, find obstructions in the coupling to an arbitrary background metric, apparently preventing a more extensive use in perturbative descriptions of quantum gravity. We expect that our model might simplify calculations of one-loop amplitudes with respect to standard quantum field theoretical methods.

Motivation & Objective

• To develop a worldline formalism for one-loop quantum gravity in four dimensions, avoiding the limitations of existing spinning particle models.
• To overcome the obstruction in coupling the O(4)-extended spinning particle to arbitrary curved backgrounds, which limits its use in generic perturbative quantum gravity.
• To derive a worldline action directly from the quadratic approximation of the Einstein-Hilbert action with cosmological constant.
• To test the method by reproducing known one-loop divergencies in Einstein gravity, validating its correctness for future applications.
• To provide a flexible path integral representation of the one-loop effective action that could simplify calculations compared to standard quantum field theory techniques.

Proposed method

• Apply the background field method to the Einstein-Hilbert action with cosmological constant, splitting the metric into classical background $g_{\mu\nu}$ and quantum fluctuation $h_{\mu\nu}$.
• Expand the action to quadratic order in $h_{\mu\nu}$, identifying the differential operators that govern the quadratic fluctuations.
• Construct a worldline particle action whose path integral reproduces the heat kernel of these differential operators via dimensional regularization.
• Use dimensional regularization by extending the worldline time to $\mathbb{R}^n \times [0,1]$ to regulate ill-defined products of distributions and delta functions.
• Implement ghost fields $(a,b,c)$ to ensure gauge invariance and remove divergent terms arising from $\dot{q}\dot{q}$ propagators.
• Utilize fermionic propagators twisted by a $U(1)$ modulus $\phi$, matching known results from previous worldline studies in $D=4$.

Experimental results

Research questions

• RQ1 Can a worldline path integral formalism be constructed for one-loop quantum gravity in D=4 that avoids the limitations of the O(4)-spinning particle model?
• RQ2 Does the derived worldline action correctly reproduce the standard one-loop divergencies of Einstein gravity with cosmological constant?
• RQ3 Can the method be generalized to compute one-loop amplitudes in quantum gravity more efficiently than standard quantum field theory methods?
• RQ4 Why does the O(4)-spinning particle fail to couple to arbitrary background metrics, and can a simpler alternative be derived directly from the quadratic action?
• RQ5 How can regularization techniques like dimensional regularization be applied to worldline path integrals involving non-trivial propagators and delta-function products?

Key findings

• The worldline path integral formulation successfully reproduces the correct one-loop divergencies in Einstein gravity with cosmological constant, validating the method against known results.
• The method correctly captures both logarithmic and quadratic divergences, which were not previously tested in similar worldline models, confirming consistency with Christensen:1979iy and deBerredoPeixoto:2001qx.
• The use of dimensional regularization in extended spacetime ($\mathbb{R}^n \times [0,1]$) allows safe manipulation of divergent distributions like $\delta(\tau,\tau)$ and $\delta(\tau,\sigma)^2$.
• The ghost system $(a,b,c)$ effectively removes divergent contributions from $\dot{q}\dot{q}$ propagators, ensuring gauge invariance and consistency of the path integral.
• The fermionic propagator with $U(1)$ modulus $\phi$ matches the form used in prior worldline studies, allowing reuse of established regularization and computation techniques.
• The derived worldline action provides a viable alternative to standard QFT methods for computing one-loop amplitudes in quantum gravity, with potential for simplification and broader applicability.