The problem of reconciling quantum mechanics with general relativity and formulating a theory of quantum gravity is a topic of ongoing research and debate in theoretical physics. One approach to this problem is through the process of quantization, which involves treating gravity as a quantum field theory.
It is true that spacetime, as described by general relativity, is a continuous manifold. However, the quantization of gravity does not necessarily require spacetime itself to be discrete. Instead, the discreteness arises in the quantization of the gravitational field.
In the framework of quantum field theory, the gravitational field is described by particles called gravitons. These gravitons are the quanta of the gravitational field, similar to how photons are the quanta of the electromagnetic field in quantum electrodynamics.
The quantization of gravity involves assigning operators to the dynamical variables of the gravitational field, such as the metric tensor. These operators act on a quantum state and give rise to observable quantities, such as particle interactions and gravitational waves.
The challenge in formulating a consistent theory of quantum gravity lies in the fact that the quantization of gravity introduces infinities and non-renormalizability. This means that straightforward application of standard quantization techniques leads to mathematical inconsistencies.
Several approaches have been proposed to tackle this issue, such as string theory, loop quantum gravity, and causal dynamical triangulations, among others. These approaches provide different mathematical frameworks and conceptual ideas for addressing the quantization of gravity while respecting the principles of quantum mechanics and general relativity.
It's important to note that the precise details of how gravity is quantized and how spacetime emerges from such a theory are still active areas of research. The search for a complete theory of quantum gravity is a challenging and ongoing endeavor in theoretical physics.