The incompatibility between General Relativity and Quantum Field Theory (QFT) is one of the biggest open questions in theoretical physics. Both theories are extremely successful in their respective domains, but they are based on different mathematical frameworks and physical principles. When physicists try to combine them into a single theory, they encounter various conceptual and technical problems.

Here are some of the main issues:

1. Different Mathematical Structures:
   • General Relativity is a classical theory based on the geometry of spacetime. It describes gravity as the curvature of spacetime caused by mass and energy. The equations of General Relativity are continuous and deterministic.
   • Quantum Field Theory is a quantum theory that describes particles as excitations of underlying fields. It incorporates the principles of quantum mechanics, which are probabilistic and involve discrete quantities (quanta).
2. Scale of Applicability:
   • General Relativity is most applicable on large scales, such as the motion of planets, stars, and galaxies. It becomes crucial when dealing with massive objects like black holes.
   • Quantum Field Theory is most applicable on small scales, like the interactions of subatomic particles. It becomes crucial when dealing with phenomena at the atomic level or smaller.
3. Conceptual Differences:
   • General Relativity treats spacetime as a smooth, continuous manifold. In contrast, QFT, being a quantum theory, inherently deals with discrete quantities and probabilistic outcomes.
   • In General Relativity, gravity is not a “force” in the traditional sense but a curvature of spacetime. In QFT, forces are mediated by force-carrying particles (gauge bosons), but a satisfactory quantum description of a force-carrying particle for gravity (the graviton) has not been achieved.
4. Technical Problems:
   • When trying to combine the two theories, physicists encounter severe mathematical difficulties. For example, calculations in a quantum theory of gravity often lead to infinities that cannot be easily dealt with (as opposed to QFT, where similar infinities can be handled with a process called renormalization).
   • The probabilistic nature of quantum mechanics clashes with the deterministic nature of General Relativity.

Physicists have been working on various approaches to reconcile General Relativity and QFT. One of the most promising approaches is string theory, which posits that fundamental particles are one-dimensional “strings” rather than point particles. Another approach is loop quantum gravity, which attempts to quantize spacetime itself. However, as of now, a fully satisfactory theory of quantum gravity that combines General Relativity and QFT remains elusive.