Quantum mechanics was not derived directly by minimizing an action in the same way as general relativity.
While classical mechanics and general relativity derive their equations by minimizing action, quantum mechanics uses this principle to understand the probability amplitudes of different paths, with the classical path of least action dominating in the classical limit.
In contrast, general relativity is derived by minimizing the Einstein-Hilbert action, which directly leads to the Einstein field equations.
However, the principle of minimizing action plays a role through Feynman's path integral formulation. This approach considers all possible paths a particle can take, each weighted by a complex phase related to the action along that path
No. Because quantum mechanics describes a certain class of physical systems, in equilibrium with quantum fluctuations and general relativity doesn't. The two don't have anything to do with each other. They are completely different subjects.
How to describe the quantum fluctuations of non-gravitational systems is known; how to describe the quantum fluctuations of gravitational systems is not.
Mr Nicolis,
Thanks for the reply/. What about Loop Quantum Gravity (LQG), which is a quantum gravity theory which includes fluctuations (a theory which proposes a discrete structure of spacetime at the Planck scale), in which quantum fluctuations in geometric observables, such as universe volume, are studied ?
Can you can you suggest an approach to extension of general relativity akin to the "quantum fluctuations" type ?
That's not known, for the moment. Loop quantum gravity does not propose a discrete structure of spacetime, but of phase space; however it has issues with accounting for anomalies, that arise from quantum fluctuations.
P. P. Afxenti
Building on the questions you ask here it seems that you want to jump directly from the elementary theories which are taught in introduction lectures to the cutting-edge models. This has never worked out for anyone and the only possible outcome of it is frustration with the state of physics.
As in any scientific discipline, there is a large middle ground between introduction and cutting edge and getting through it is tedious and occasionally boring even for the biggest enthusiast. Being in this middle ground may take years of building up contextual knowledge, but in order to make contributions that will be considered substantial, this will have to be invested.
One reason ``big name problems" haven't been solved yet is that the correct way to formulate them takes time and effort to understand and develop. The Standard Theory of particle physics took almost a century. Fermat's last theorem was proved only when it was realized how it could be related to an, apparently, unrelated issue about elliptic curves. And so on. How to understand quantum fluctuations of spacetime is in a similar situation.
That's why asking how to describe quantum fluctuations of spacetime, for the moment, can only receive the answer that it is not known; people are working to understand what it can mean.
Jürgen Weippert,
You may be right and thanks for the remark.
What I believe is that elementary theories have foundational truths that are sometimes overlooked in the advanced theories and that re-linking these may work. At least conceptually elementary theories or ideas from " introduction lectures" have affinity and "genetic" relevance
Yeah, I got what you want. That approach has lead too many people who could have become viable scientists down an unpleasant slope, I thought I'd remark that before you're fully frustrated.
You may be interested to know that Quantum Mechanics was established on an erroneous phase wave velocity due to a wrong frequency assumed for the energy induced in the electron in the ground state of the hydrogen atom. Analyzed and numerically established in this paper published in 2024
Article Critical Analysis of the Origins of Heisenberg's Uncertainty Principle
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