The difficulty in incorporating electromagnetism and gravity into a single quantum field theory arises primarily from the differences in their fundamental nature and the mathematical frameworks used to describe them.
Electromagnetism is described by quantum electrodynamics (QED), which successfully combines quantum mechanics and special relativity. QED treats electromagnetic interactions as the exchange of virtual photons between charged particles. It is a well-developed and highly successful theory, accurately predicting phenomena and matching experimental results with remarkable precision.
On the other hand, gravity is described by general relativity, which provides a classical theory of gravity based on the curvature of spacetime. General relativity successfully explains the behavior of gravity on large scales, such as the motion of celestial bodies and the bending of light around massive objects.
The challenge lies in reconciling the principles of quantum mechanics, which govern the microscopic world of particles, with general relativity, which describes gravity in a classical, continuous framework. These two theories have fundamentally different mathematical structures and assumptions, making their direct combination problematic.
One significant difficulty is that general relativity treats spacetime as a smooth, continuous manifold, while quantum mechanics deals with discrete, quantized quantities. Combining the two requires a framework that can reconcile the discreteness of quantum mechanics with the smoothness of general relativity.
Additionally, the mathematical formulation of general relativity involves a tensorial description of gravity, while quantum field theories like QED rely on the principles of local gauge invariance and quantization of fields. Naively extending these principles to gravity encounters mathematical inconsistencies and divergences that are difficult to resolve.
The search for a consistent theory of quantum gravity, often referred to as a theory of everything or a theory of quantum gravity, is an active area of research in theoretical physics. Various approaches, such as string theory, loop quantum gravity, and other quantum gravity models, have been proposed, but a definitive and experimentally validated theory has not yet been established.
Developing a complete and consistent theory of quantum gravity that incorporates both electromagnetism and gravity remains a major challenge in theoretical physics. It requires reconciling the discrete nature of quantum mechanics with the smooth spacetime of general relativity, resolving mathematical inconsistencies, and ultimately producing predictions that can be tested through experiments or observations.