A direct transformation between the Dirac equation and the Einstein field equations is not possible in the traditional sense because:
The Dirac equation describes fermionic quantum fields, while The Einstein field equations describe the classical curvature of spacetime.
However, quantum gravity aims to develop a formalism where both quantum matter and quantum spacetime can be treated consistently. In this sense, one might say that the coupling between Dirac fields and spacetime geometry (via tetrads, spin connection, etc.) allows us to study their interaction, but not a one-to-one transformation.
In quantum gravity frameworks like string theory or loop quantum gravity, both the spacetime geometry and the fermions are quantized, and their dynamics might be described by a unified formalism. However, this unification is still an area of active research, and no complete theory has yet emerged.
Conclusion:
While we cannot directly transform the Dirac equation into the Einstein field equations, quantum gravity approaches (such as loop quantum gravity, string theory, and others) attempt to describe a unified framework where quantum matter and quantum spacetime coexist and interact. These approaches seek to generalize both the Dirac and Einstein equations into a more fundamental quantum theory that captures the behavior of both matter and geometry at the smallest scales.
When mass movement in fabric of spacetime continum trigger ripple spacetime time, so that spacetime oscilate like quantum field, is it optional way to reconcile quantum field theory and general relativity theory?
Yes, the idea that mass moving through spacetime triggers ripples (gravitational waves) and that spacetime itself oscillates similarly to quantum fields is one potential pathway toward reconciling quantum field theory (QFT) with general relativity (GR). In essence, this refers to the broader concept of quantum gravity—a theory that attempts to describe gravity, not as a purely classical field as in general relativity, but as a quantized entity similar to other forces in nature.
Let's break down this idea and how it might help reconcile QFT and GR.
1. Spacetime Ripples and Gravitational Waves (General Relativity):
In classical general relativity, mass and energy distort the "fabric" of spacetime, and changes in the distribution of mass or energy can produce gravitational waves, ripples that propagate through spacetime itself. These were predicted by Einstein and confirmed by experiments such as LIGO's detection of gravitational waves from merging black holes.
Gravitational waves are analogous to ripples in spacetime, but in general relativity, spacetime is treated as a smooth, continuous entity, and these ripples are part of classical physics. Gravitational waves provide a direct link between the movement of mass in spacetime and oscillations in spacetime geometry, but this is still within the framework of a classical field.
2. Spacetime as a Quantum Field (Quantum Gravity):
To reconcile this with quantum field theory, one would need to quantize spacetime itself, treating it as a field that can exhibit quantum behavior, just like the electromagnetic field, for example. In quantum field theory, forces are mediated by particles (like photons for the electromagnetic force), and the idea would be to similarly quantize the gravitational field.
