[Paper Review] Three functions in dilaton gravity: The good, the bad and the muggy

Daniel Grumiller|ArXiv.org|May 9, 2003

Quantum and Classical Electrodynamics3 references3 citations

TL;DR

This paper reinterprets two-dimensional dilaton gravity through three distinct dilaton-dependent functions: 'the good' (w), which governs classical spacetime structure and scattering; 'the muggy' (I), which controls semi-classical and quantum effects like Hawking radiation; and 'the bad' (Z), which may underlie nonperturbative quantum phenomena but complicates first-order formulations. The key contribution is a novel classification that clarifies the roles of these functions in classical, semi-classical, and quantum regimes, offering a unified framework for understanding dilaton gravity models including CGHS and Jackiw-Teitelboim theories.

ABSTRACT

Dilaton gravity in two dimensions is briefly reviewed from the perspective of three dilaton potentials: One determines classical physics ("the good", denoted by w), the second is relevant for semi-classical (and quantum) effects ("the muggy", denoted by I) and the third could be responsible for nonperturbative quantum effects ("the bad", denoted by Z). This paper is based upon lectures given in Cernowitz in October/November 2002 at The XIV International Hutsulian Workshop Mathematical Theories and their Physical and Technical Applications.

Motivation & Objective

• To clarify the distinct roles of three dilaton-dependent functions in two-dimensional dilaton gravity: w (classical), I (semi-classical/quantum), and Z (nonperturbative quantum).
• To reframe the standard action using integrated combinations of potentials rather than the original functions, enhancing conceptual clarity and computational utility.
• To demonstrate how 'the good' function w determines causal structure and classical vertices, while 'the muggy' I governs Hawking radiation and test particle dynamics.
• To argue that 'the bad' Z, though often neglected due to technical obstructions in first-order formulations, may be essential for nonperturbative quantum effects.

Proposed method

• The paper rewrites the dilaton gravity action in terms of three integrated functions: w(X), I(X), and Z(X), derived from the original potentials Z(X), U(X), and V(X).
• It employs the first-order formulation of gravity using Cartan variables (zweibein e^a, spin connection ω, and auxiliary field X^a), leading to a Poisson-σ model structure.
• Classical physics is governed by w(X), which defines the causal structure and generates vertices for scattering processes.
• Semi-classical and quantum behavior are controlled by I(X), which appears in the Hawking flux and affects test particle motion.
• The function Z(X) is analyzed for its role in nonperturbative effects; its non-invertibility obstructs standard first-order quantization, hence its 'bad' moniker.
• Path integral quantization is revisited, with new one-loop results derived in Appendix B, and the S-matrix is shown to be nonlocal in momentum space despite being energy-independent.

Experimental results

Research questions

• RQ1 How do the three functions w(X), I(X), and Z(X) collectively determine the classical, semi-classical, and quantum behavior of two-dimensional dilaton gravity?
• RQ2 Why is the function I(X) referred to as 'muggy'—what physical and technical roles does it play?
• RQ3 What is the significance of Z(X) being non-invertible, and how might it still contribute to nonperturbative quantum gravity?
• RQ4 In what sense is the S-matrix derived from this framework nonlocal, and how does this relate to the effective action?
• RQ5 How do boundary conditions and topology changes arise in this formulation, and what is their physical interpretation?

Key findings

• The function w(X) fully determines the classical causal structure and scattering vertices in dilaton gravity, making it central to classical physics.
• The function I(X) governs the semi-classical Hawking radiation flux and influences the dynamics of test particles, playing a key role in quantum effects.
• The function Z(X), though called 'bad' due to its obstruction to standard first-order formulation, may be essential for nonperturbative quantum phenomena.
• The S-matrix derived from the path integral is nonlocal in momentum space, as evidenced by non-polynomial dependence on momenta, despite being independent of total energy.
• Topology changes, such as those in the Vaidya-Bañados-Teitelboim (VBH) geometry, can be interpreted as fluctuations in the Euler characteristic, though physical observers do not detect them due to boundary conditions.
• A renormalized boundary term in the effective action, resembling Kuchař's result, can be recovered indirectly from the path integral, though the derivation is less direct than in canonical approaches.