Celebrating the 100th anniversary of the general theory of relativity in the quantum (or at least noncommutative) way
The most glaring incompatibility of concepts in contemporary physics is that between Einstein's principle of general coordinate invariance and all the modern schemes for a quantum-mechanical description of nature. Einstein based his theory of general relativity  on the principle that God did not attach any preferred labels to the points of space-time. This principle requires that the laws of physics should be invariant under the Einstein group E, which consists of all one-to-one and twice-differentiable transformations of the coordinates. By making full use of the invariance under E, Einstein was able to deduce the precise form of his law of gravitation from general requirements of mathematical simplicity without any arbitrariness. He was also able to reformulate the whole of classical physics (electromagnetism and hydrodynamics) in E-invariant fashion, and so determine unambiguously the mutual interactions of matter, radiation and gravitation within the classical domain. There is no part of physics more coherent mathematically and more satisfying aesthetically than this classical theory of Einstein based upon E-invariance.
On the other hand, all the currently viable formalisms for describing nature quantum-mechanically use a much smaller invariance group. The analysis of Bacry and Lévy-Leblond  indicates the extreme range of quantum-mechanical kinematical groups that have been contemplated. In practice all serious quantum-mechanical theories are based either on the Poincaré group P or the Galilei group G. This means that a class of preferred inertial coordinate-systems is postulated a priori, in flat contradiction to Einstein's principle. The contradiction is particularly uncomfortable, because Einstein's principle of general coordinate invariance has such an attractive quality of absoluteness. A physicist's intuition tells him that, if Einstein's principle is valid at all, it ought to be valid for the whole of physics, quantum-mechanical as well as classical. If the principle were not universally valid, it is difficult to understand why Einstein achieved such deeply coherent insights into nature by assuming it to be so.
To make the mathematical incompatibility more definite, I will focus attention on one of the competing schemes for describing a quantum-mechanical universe. I choose the scheme which is most carefully based on rigorous mathematical definitions and which is also general enough to encompass a wide variety of physical systems. This scheme is the "Algebra of Local Observables" of Haag and Kastler ...
These axioms, taken together with the axioms defining a C*-algebra , are a distillation into abstract mathematical language of all the general truths that we have learned about the physics of microscopic systems during the last 50 years. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. In some sense, the axioms represent the most serious attempt that has yet been made to define precisely what physicists mean by the words "observability, causality, locality, relativistic invariance," which they are constantly using or abusing in their everyday speech.
If we look at the axioms in detail, we see that (1), (2), (3) and (6) are consistent with Einstein's general coordinate invariance, but (4) and (5) are inconsistent with it. Axioms (4) and (5), the axioms of Poincaré invariance and local commutativity, require the Poincaré group to be built into the structure of space-time. If we try to replace the Poincaré group P by the Einstein group E, we have no way to define a space-like relationship between two regions, and axiom (5) becomes meaningless. I therefore propose as an outstanding opportunity still open to the pure mathematicians, to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.
I had better warn any mathematician who intends to respond to my challenge that his task will not be easy. No merely formal rearrangement of the Haag-Kastler axioms can possibly be sufficient. For we know that Einstein could construct his E-invariant classical theory of 1916 only by bringing in the full resources of Riemannian differential geometry. He needed a metric tensor to give his space-time a structure independent of coordinate-systems. Therefore an E-invariant axiom of local commutativity to replace axiom (5) will require at least some quantum-mechanical analog of Riemannian geometry. Some analog of a metric tensor must be introduced in order to give a meaning to space-like separation. The answer to my challenge will necessarily involve a delicate weaving together of concepts from differential geometry, functional analysis, and abstract algebra With these words of warning I leave the problem to you.
Freeman J. Dyson ,(September 1972)
Extending the Riemann manifold paradigm...
Riemann was well aware of the limits of his own point of view as is clearly expressed in the last page of his inaugural lecture; ()
”Questions about the immeasurably large are idle questions for the explanation of Nature. But the situation is quite different with questions about the immeasurably small. Upon the exactness with which we pursue phenomenon into the infinitely small, does our knowledge of their causal connections essentially depend. The progress of recent centuries in understanding the mechanisms of Nature depends almost entirely on the exactness of construction which has become possible through the invention of the analysis of the infinite and through the simple principles discovered by Archimedes, Galileo and Newton, which modern physics makes use of. By contrast, in the natural sciences where the simple principles for such constructions are still lacking, to discover causal connections one pursues phenomenon into the spatially small, just so far as the microscope permits. Questions about the metric relations of Space in the immeasurably small are thus not idle ones.
If one assumes that bodies exist independently of position, then the curvature is everywhere constant, and it then follows from astronomical measurements that it cannot be different from zero; or at any rate its reciprocal must be an area in comparison with which the range of our telescopes can be neglected. But if such an independence of bodies from position does not exist, then one cannot draw conclusions about metric relations in the infinitely small from those in the large; at every point the curvature can have arbitrary values in three directions, provided only that the total curvature of every measurable portion of Space is not perceptibly different from zero. Still more complicated relations can occur if the line element cannot be represented, as was presupposed, by the square root of a differential expression of the second degree. Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and that of a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena.
The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the basis for the metric relations of space. In connection with this question, which may indeed still be ranked as part of the study of Space, the above remark is applicable, that in a discrete manifold the principle of metric relations is already contained in the concept of the manifold, but in a continuous one it must come from something else. Therefore, either the reality underlying Space must form a discrete manifold, or the basis for the metric relations must be sought outside it, in binding forces acting upon it.
An answer to these questions can be found only by starting from that conception of phenomena which has hitherto been approved by experience, for which Newton laid the foundation, and gradually modifying it under the compulsion of facts which cannot be explained by it. Investigations like the one just made, which begin from general concepts, can serve only to insure that this work is not hindered by too restricted concepts, and that progress in comprehending the connection of things is not obstructed by traditional prejudices.
This leads us away into the domain of another science, the realm of physics...”.
(Submitted on 23 Nov 2000)
Updated on November 29 2015
Distillating a quantum mechanical analog from the harvest of particle physics
Noncommutative Geometry, Quantum Fields and Motives (p177 to 182)
Alain Connes and Matilde Marcolli
//Last update on December 1
Last words to Dirac as a safeguard
We ... heard an anecdote from Roger Penrose, in response to the first question from the audience which was along the lines of 'which came first quantum mechanics or general relativity?'. Penrose replied by telling of a time he had listened to a wonderfully animated lecture by John Wheeler and at the end there came a similar question from the audience, which came first general relativity or the quantum principle? Penrose said that a small voice in the front of the audience piped up and asked 'what is the quantum principle?' The small voice belonged to Dirac.
Quantization is still a mystery indeed...