Topic

Why can't gravity be unified with Quantum Mechanics?

Gravity QFT Foundations Open Problems

The Short Answer

Gravity can be combined with quantum mechanics at low energies—this is called Quantum Field Theory on curved spacetime. What we cannot do is make gravity itself a quantum field theory in the same way as the other forces. This is the problem of quantum gravity.

What Works: QFT on Curved Spacetime

We can do quantum field theory on a curved spacetime background. This means:

Spacetime is a curved Lorentzian manifold $(M, g)$
The metric $g$ is fixed (classical, not quantized)
Quantum fields (electrons, photons, etc.) propagate on this background
The fields are quantized, the geometry is not

This works perfectly well and gives predictions like Hawking radiation—black holes emit thermal radiation due to quantum effects near the horizon.

The Limitation

The metric is treated as a fixed background. We don’t ask: “What happens if we try to put the metric itself into a superposition?” This semiclassical approach breaks down when:

Spacetime curvature becomes extreme (Planck scale)
We need to describe the gravitational field quantum mechanically

What Doesn’t Work: Quantizing Gravity

When we try to quantize gravity using the same techniques as QED or QCD, we encounter fundamental problems.

The Perturbative Approach

In standard QFT, we expand around a background and compute corrections order by order. For gravity:

$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$

where $h_{\mu\nu}$ is a small perturbation (the graviton field).

The linearized theory works—gravitons are spin-2 massless particles. But when we compute quantum corrections:

Non-Renormalizability

The gravitational coupling constant has dimensions:

$[G_N] = [\text{length}]^2 = [\text{mass}]^{-2}$

In 4 dimensions, this makes gravity non-renormalizable. At each order in perturbation theory, new divergences appear that require new counterterms. We would need infinitely many parameters to absorb all divergences.

Compare with QED: $[\alpha] = [\text{dimensionless}]$

Dimensionless coupling → renormalizable → finite number of parameters.

What Non-Renormalizability Means

It doesn’t mean the theory is wrong—it means perturbation theory breaks down at high energies. Specifically, at the Planck scale:

$E_{Planck} = \sqrt{\frac{\hbar c^5}{G_N}} \approx 10^{19} \text{ GeV}$

Above this energy, our perturbative description fails. We need new physics.

Why Gravity is Fundamentally Different

Background Dependence

In gauge theories:

Spacetime is fixed
Fields live on spacetime
Gauge transformations are symmetries of the fields

In gravity:

Spacetime itself is dynamical
There’s no fixed background
Diffeomorphisms (coordinate changes) are gauge symmetries

This is called background independence—the theory should not depend on any fixed background structure.

The Metric vs The Connection

Gauge Theory	Gravity
Connection $A_\mu$ is dynamical	Metric $g_{\mu\nu}$ is dynamical
Metric is fixed background	No fixed background
Curvature = field strength	Curvature = geometry of spacetime
Internal symmetry	Spacetime symmetry

The Problem of Time

In quantum mechanics, time is a parameter—it tells us how states evolve. In general relativity, time is part of the dynamical geometry. When we try to quantize:

What is the “time” in the Schrödinger equation?
How do we define “equal time” when spacetime is dynamical?

This is the problem of time in quantum gravity.

Approaches to Quantum Gravity

String Theory

Replace point particles with 1-dimensional strings. Gravity emerges automatically:

Closed strings have a massless spin-2 excitation = graviton
Consistent only in 10 or 11 dimensions
Extra dimensions compactified on small manifolds

Loop Quantum Gravity

Quantize spacetime geometry directly:

Area and volume are quantized
Spacetime has discrete structure at Planck scale
Background independent by construction

Asymptotic Safety

Perhaps gravity is non-perturbatively renormalizable:

There exists a UV fixed point
Only finitely many parameters needed
Still an active research area

Emergent Gravity

Maybe spacetime geometry emerges from something more fundamental:

Entanglement entropy of quantum systems
Holography (AdS/CFT)
Spacetime from quantum information

Summary

Regime	Description	Status
QFT on flat spacetime	Standard Model	Well-understood
QFT on curved spacetime	Quantum fields + classical gravity	Works, gives Hawking radiation
Perturbative quantum gravity	Gravitons as spin-2 fields	Non-renormalizable
Full quantum gravity	Spacetime itself quantum	Open problem

The statement “QFT uses Riemannian geometry when gravity is considered” refers to the semiclassical regime—quantum fields on a classical curved background. This is well-defined and useful. The full problem of making gravity quantum is unsolved.

References

S. W. Hawking, “Particle Creation by Black Holes,” Communications in Mathematical Physics 43, 199–220 (1975) — Original Hawking radiation paper
R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994)
C. Kiefer, Quantum Gravity, 3rd ed. (Oxford University Press, 2012)
C. Rovelli, Quantum Gravity (Cambridge University Press, 2004)
J. Polchinski, String Theory, Vols. 1–2 (Cambridge University Press, 1998)
S. Carlip, “Quantum Gravity: a Progress Report,” Reports on Progress in Physics 64, 885 (2001) — arXiv:gr-qc/0108040
Living Reviews in Relativity — Open access reviews on gravity research