The incompatibility between the theory of relativity (specifically, general relativity) and quantum mechanics arises from their differing mathematical frameworks and conceptual foundations. Here are a few key reasons why these theories have been challenging to reconcile:
Mathematical Formulations: General relativity is formulated within the framework of classical physics and is described by smooth, continuous spacetime geometry. On the other hand, quantum mechanics uses a probabilistic wave function description and operates in a discrete, quantized manner. The mathematical structures of the two theories are fundamentally different, making it difficult to merge them into a unified framework.
Scale Discrepancy: General relativity describes gravity and the behavior of massive objects on cosmological and astrophysical scales, while quantum mechanics governs the behavior of particles at the microscopic level. Attempts to combine these theories typically encounter difficulties when trying to describe phenomena that involve both very large masses and very small distances, such as black holes or the early moments of the universe.
Renormalization and Infinities: Quantum field theories, which are used to describe elementary particles and their interactions, often encounter infinities in their calculations. Renormalization techniques are employed to remove these infinities and extract meaningful predictions. However, when applying these techniques to gravity (as described by general relativity), infinities arise in a way that has not been successfully renormalized. This poses a challenge for incorporating gravity into a quantum framework.
Measurement and Observables: Quantum mechanics emphasizes the role of measurement and the collapse of the wave function when an observation is made. In contrast, general relativity is more concerned with the overall geometry of spacetime, and the concept of measurement is not as central. Reconciling the fundamentally different ways these theories handle observations and measurements remains an open question.
Lack of Experimental Verification: The extreme conditions where the effects of both quantum mechanics and general relativity are significant, such as the interior of black holes or the very early universe, are currently beyond our experimental reach. The lack of experimental data in these regimes makes it challenging to test and refine potential theories that aim to unify quantum mechanics and general relativity.
While there have been various attempts to develop a theory that unifies quantum mechanics and general relativity, such as string theory or loop quantum gravity, a fully satisfactory and experimentally supported theory of quantum gravity is still an ongoing area of research.