Physicists do indeed work with quantum mechanics in curved spacetime in certain contexts. This field of study is known as quantum field theory on curved spacetimes. It is an important framework for understanding the behavior of quantum fields (such as the electromagnetic field or the Higgs field) in the presence of curved spacetime, as described by general relativity.
However, the challenge that physicists face is that general relativity and quantum mechanics are formulated using different mathematical frameworks and have different conceptual foundations. General relativity describes gravity as the curvature of spacetime caused by matter and energy, while quantum mechanics provides a framework for understanding the behavior of particles and fields on microscopic scales.
The problem arises when one attempts to combine these two theories into a consistent framework known as quantum gravity. Such a theory would be able to describe the behavior of matter and gravity at extremely small scales, such as those found near black holes or during the early stages of the universe. However, despite decades of research, a fully satisfactory theory of quantum gravity remains elusive.
The challenge lies in the fact that the mathematical formulations of general relativity and quantum mechanics are incompatible at their core. General relativity relies on smooth and continuous spacetime, while quantum mechanics introduces discrete and probabilistic behavior at the microscopic level. Reconciling these two frameworks requires a deeper understanding of the fundamental nature of spacetime and the development of new mathematical tools.
Efforts to unify general relativity with quantum mechanics, such as string theory and loop quantum gravity, aim to provide a consistent and comprehensive theory of quantum gravity. These approaches go beyond quantum field theory on curved spacetime and attempt to address the fundamental incompatibilities between the two theories. By seeking a unified framework, physicists hope to resolve longstanding questions about the nature of spacetime, the behavior of matter and gravity on small scales, and the origins of the universe.
While quantum field theory on curved spacetime is a valuable tool in certain contexts, a complete understanding of the quantum nature of gravity requires a more profound synthesis of general relativity and quantum mechanics, which is still an active area of research.