Quantum mechanics and general relativity are considered incompatible because they have different mathematical frameworks and conceptual foundations, making it challenging to directly merge them into a single consistent theory. Here are a few reasons why they are difficult to reconcile:
Different mathematical formalisms: Quantum mechanics is formulated using principles such as wave functions, superposition, and probabilistic outcomes. It relies on complex numbers and the mathematical framework of Hilbert spaces. On the other hand, general relativity describes gravity as the curvature of spacetime caused by matter and energy. It is based on differential geometry and the mathematics of tensors. The mathematical structures and equations used in these theories are fundamentally different, making their direct combination difficult.
Scale disparity: Quantum mechanics operates at the microscopic scale, describing the behavior of particles such as atoms, electrons, and photons. On the other hand, general relativity applies to the macroscopic scale, describing the behavior of massive objects and the curvature of spacetime. The theories are formulated to address phenomena at vastly different scales, and reconciling their predictions in regimes where both are relevant (e.g., the behavior of particles in strong gravitational fields) poses significant challenges.
Treatment of spacetime: In quantum mechanics, spacetime is considered fixed and background-independent. Quantum fields propagate on this fixed spacetime background. In general relativity, however, spacetime is dynamic and affected by the presence of matter and energy. Gravity is not described as a force acting on particles but as the curvature of spacetime itself. This different treatment of spacetime makes it difficult to directly incorporate it into the framework of quantum mechanics.
Renormalizability and infinities: In quantum field theories, such as quantum electrodynamics, it is possible to use renormalization techniques to remove infinities that arise in calculations. However, when applying these techniques to gravity, infinities persist, indicating that the theory is nonrenormalizable. This suggests that additional fundamental principles are needed to make sense of gravity on the quantum level.
These challenges highlight the need for a theory of quantum gravity, a unified framework that can describe both quantum mechanics and general relativity consistently. Several approaches, such as string theory, loop quantum gravity, and causal set theory, among others, are being pursued to reconcile these theories and provide a more comprehensive understanding of the fundamental nature of the universe. However, finding a complete and experimentally validated theory of quantum gravity remains an ongoing endeavor in theoretical physics.