The Euclidean formulation of quantum gravity, obtained via Wick rotation (t →−iτ), is often regarded as a mere computational device to regularize oscillating path integrals. However, recent developments suggest that the Euclidean regime may be more fundamental than the Lorentzian spacetime we perceive. This paper reviews the arguments for this shift in perspective. We examine the AdS/CFT correspondence, where Euclidean geometry is required to define the holographic dictionary. Furthermore, we explore the intersection of Euclidean gravity with Unimodular Gravity and the Asymptotic Safety program, demonstrating how restricting the conformal mode in a Euclidean setting resolves mathematical pathologies and supports the existence of a non-perturbative ultraviolet fixed point.

1. Euclidean Quantum Gravity and Physical Reality:
From Holography to Asymptotic Safety
November 26, 2025
Abstract
The Euclidean formulation of quantum gravity, obtained via Wick rotation (t → −iτ),
is often regarded as a mere computational device to regularize oscillating path integrals.
However, recent developments suggest that the Euclidean regime may be more fundamental
than the Lorentzian spacetime we perceive. This paper reviews the arguments for this shift
in perspective. We examine the AdS/CFT correspondence, where Euclidean geometry is
required to define the holographic dictionary. Furthermore, we explore the intersection of
Euclidean gravity with Unimodular Gravity and the Asymptotic Safety program, demon-
strating how restricting the conformal mode in a Euclidean setting resolves mathematical
pathologies and supports the existence of a non-perturbative ultraviolet fixed point.
1 Introduction: Math Trick or Physical Foundation?
The central object in quantum gravity is the Feynman path integral. In the Lorentzian signature
(− + ++), the partition function sums over metrics weighted by an oscillating phase:
ZL =
Z
Dgµν eiS[g]/ℏ
(1)
This integral is mathematically ill-defined due to lack of convergence. The standard resolution
is Wick Rotation, substituting time t with imaginary time τ = it. This transforms the metric
to Euclidean signature (+ + ++) and converts the path integral into a statistical mechanics
partition function:
ZE =
Z
Dgµν e−SE[g]/ℏ
(2)
While this renders the integral convergent (conceptually similar to a Boltzmann distribution),
it raises a profound ontological question: Is Euclidean space merely an auxiliary tool, or does
it represent the fundamental state of the universe from which time and causality emerge?
2 The Holographic Evidence: AdS/CFT
The strongest argument for the fundamental nature of Euclidean geometry arises from the
AdS/CFT Correspondence (Holographic Principle). This framework posits an equivalence
between Quantum Gravity in a bulk Anti-de Sitter (AdS) space and a Conformal Field Theory
(CFT) on its boundary.
2.1 The Geometric Transformation
The correspondence is most rigorously defined in the Euclidean frame.
• Lorentzian AdS: The geometry is an infinite cylinder in time. The boundary conditions
at temporal infinity are complex to define.
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2. • Euclidean AdS: Upon Wick rotation, the geometry becomes a hyperbolic ball (repre-
sented by the Poincaré disk). The boundary becomes a compact sphere (Sn).
This transformation allows for a precise mapping of partition functions:
ZGravity[ϕ0] ≡
D
e
R
∂AdS ϕ0O
E
CFT
(3)
Here, the Euclidean path integral of gravity in the bulk (ZGravity) is identically equal to the
generating function of the quantum field theory on the boundary.
2.2 Gravity without Gravity
In this view, the ”Euclidean” frame is the bridge. Phenomena such as Black Hole entropy are
equivalent to the thermal entropy of the boundary fluid. The Euclidean “cigar” geometry, which
fixes the Hawking temperature via periodicity (T = ℏ/β), corresponds exactly to the temper-
ature of the CFT. This suggests that the “math trick” is actually the dictionary translating
between the hologram (CFT) and the bulk reality (Gravity).
3 Unimodular Asymptotic Safety
While the Euclidean path integral is powerful, it suffers from the Conformal Factor Problem:
the Euclidean action SE is unbounded from below. If the metric is scaled by a factor Ω(x), the
action can diverge to −∞, rendering the integral unstable.
A promising resolution lies in the synthesis of Unimodular Gravity and the Asymptotic
Safety program.
3.1 The Conformal Factor Problem
Standard General Relativity allows the metric determinant g to fluctuate. In the Euclidean
action, the kinetic term for the conformal mode has the wrong sign (negative kinetic energy),
causing the instability.
3.2 The Unimodular Remedy
Unimodular Gravity imposes a constraint on the path integral: the determinant of the metric
must be fixed (classically
√
g = const).
Dgµν → Dgµν δ(
√
g − ω) (4)
Geometrically, this restricts the symmetry group to volume-preserving diffeomorphisms. Phys-
ically, it freezes the conformal mode. By removing the degree of freedom responsible for
the unbounded action, the path integral becomes stable without ad-hoc contour rotations.
3.3 Asymptotic Safety and the UV Fixed Point
Asymptotic Safety posits that gravity is renormalizable due to the existence of an interacting
Ultraviolet (UV) Fixed Point. To test this, physicists use the Functional Renormalization Group
(FRG), which relies heavily on Euclidean Heat Kernel methods.
When Unimodular Gravity is combined with FRG methods:
1. The elimination of the conformal factor stabilizes the flow equations.
2. Explicit calculations show that the UV Fixed Point persists in the Unimodular setting.
3. This implies that the fixed point is a robust physical feature of gravity, not an artifact of
the conformal instability.
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3. 4 Conclusion
The role of Euclidean Quantum Gravity has evolved from a computational convenience to a
candidate for fundamental reality.
• Thermodynamics: It correctly computes Black Hole entropy and temperature where
Lorentzian methods struggle.
• Holography: It provides the necessary geometric structure (compact manifolds) to define
the AdS/CFT dictionary.
• Renormalization: When constrained by Unimodular symmetry, it cures the conformal
factor instability, providing strong evidence for Asymptotic Safety.
We may summarize the relationship between these modern approaches as follows: Eu-
clidean Quantum Gravity is the arena; Asymptotic Safety is the hypothesis; and
Unimodular Gravity is the constraint that stabilizes the arena. Together, they suggest
a universe where ”time” is an emergent approximation of a deeper, timeless Euclidean geometry.
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