In my limited understanding I had read or heard that Einstein developed GR from the Principle of Equivalence and conservation of energy perspective while the mathematician David Hilbert almost derived the same field equations from the Principle of Least Action. How do these fundamental physical principles arise within QM which Planck began around 1900 simply by making an off the cuff assumption that energy frequencies of radiation must be quantized in order to fit experimental data to explain the ultraviolet catastrophe? Could this be the underlying cause of incompatibility between GR and QM? GR is grounded in well established proven conservation laws while QM started off as an ad hoc guess simply to fit empirical data? Also while GR predicts a dynamically active curved spacetime which can be interpreted as gravity doesn't the current accepted QM framework act on a flat non dynamic spacetime?
2 Answers
Why are "well established proven conservation laws" in any way more fundamental than "empirical data"? All of physics is ultimately based on empirical data, we develop mathematical models and theories which explain experimental data. If it doesn't agree with experiments, it isn't a good theory (although it may still be a useful model)
Quantum field theory obeys the "principle of equivalence", i.e. it is a fully relativistic theory which respects the Poincaré symmetry of Minkowski spacetime. Its most famous incarnation in the Feynman path integral is also entirely based on a principle of least action, i.e. quantum theory allows for fluctuations away from the classical minimum of the action.
There are many fundamental difficulties in writing down a fully consistent theory of quantum gravity, i.e. a theory which explains both the large-scale universe captured by GR and the subatomic scale universe described by QFT and the Standard Model, too numerous to list here. I will give just a small taste of one corner of quantum gravity: General Relativity can be formulated via an action principle (the Einstein-Hilbert action), i.e. in a Lagrangian formulation. However, GR does not admit (to my knowledge) a Hamiltonian formulation, i.e. a time-evolution picture. This is deeply related to the fact that there is no unique timelike vector field that can be used to "foliate" spacetime (the existence of such a vector field would violate the equivalence principle and general covariance). The consequence of this is that the Hamiltonian of GR (c.f. ADM formalism) is a constraint, i.e. it is identically zero. Therefore GR cannot be quantized in the manner formulated originally by Dirac (see e.g. here). In the quantum theory, the Hamiltonian must annihilate physical states and therefore cannot serve as a generator of time-evolutions. On the other hand, if one starts from the action and attempts a naive path integral quantization, you find that the resulting quantum field theory is non-renormalizable.
- 3,930
- 1
- 15
- 29
The "off-the-cuff" assumption of Planck still stands as one of the most profound advances in our understanding of nature. It is an inductive step. Our understanding of physics advances through such inductive steps. In fact, the equivalence principle is also such an inductive step. What's more, the consequences of Planck assumption has been tested quite extensively and found to be correct. The the incompatibility between GR and QM is not because there something wrong in either of the two. It is simply because it is incomplete. We need another inductive step (or perhaps even more than one).
- 16,824