To reconcile QFT and GR, we need to extend the formalism of spacetime geometry to account for quantum properties of spacetime at the Planck scale. This might involve modifying Ricci geometry to include quantized lengths, times, and possibly even a discrete structure of spacetime.
Potential Approaches:
1. Effective Field Theory and Quantum Corrections to GR: One approach is to treat general relativity as an effective field theory that works well at large scales but requires quantum corrections at smaller (Planck) scales. In this framework:
At scales much larger than the Planck length, spacetime appears smooth, and the classical Einstein field equations hold.
As we approach the Planck scale, quantum fluctuations in spacetime geometry would introduce corrections to the Einstein equations, modifying the behavior of spacetime curvature (Ricci tensor) and the metric.
These corrections could be described using higher-order terms in the curvature (such as  or  terms) and possibly other quantum effects. This is an approach used in quantum gravity as an effective field theory, where the leading-order behavior is classical, and quantum corrections become important at very small scales.
2. Quantization of Spacetime: Discretizing Geometry: At the Planck scale, spacetime may not be smooth and continuous as in Ricci geometry but instead take on a discrete, quantum structure. This is the focus of approaches such as loop quantum gravity (LQG):
Loop Quantum Gravity proposes that spacetime is composed of discrete "chunks" or quanta of space. These are described mathematically as spin networks, and the areas and volumes of spacetime are quantized.
The challenge is to link this discrete quantum geometry with the smooth, continuous geometry of GR at larger scales. The quantum states of geometry in LQG would recover classical Ricci geometry in the limit of large distances (much larger than the Planck scale), but at the smallest scales, spacetime would behave very differently.
In this framework, length and time at the Planck scale are no longer continuous variables but take on discrete values, which would fundamentally alter how we think about spacetime geometry.
3. String Theory and the Holographic Principle: String theory offers another approach to reconciling QFT and GR by introducing a new formalism for spacetime:
In string theory, all particles (including the graviton, which mediates gravity) are viewed as different vibrational modes of fundamental strings.
