The hitchhiker's guide to the whole quantum galaxy
The road to the resolution of the grand problem of theoretical physics – the search for a unified theory of all fundamental forces – does not come with many road signs. The work by Connes and coworkers on the standard model of particle physics, where the standard model coupled to general relativity is reformulated as a single gravitational model written in the language of noncommutative geometry, appears to be such a road sign. From the road sign, which we believe this formulation of the standard model is, we read of three travel advices for the road ahead:
1. It is a formulation of fundamental physics in terms of pure geometry. Thus, it suggests that one should look for a unified theory which is gravitational in its origin.
2. The unifying principle in Connes formulation of the standard model hinges completely on the noncommutativity of the algebra of observables. Thus, it suggests that one should search for a suitable noncommutative algebra.
We pick up the third travel advice from the fact that Connes work on the standard model coupled to general relativity is essentially classical. With its gravitational origin this is hardly a surprise: if the opposite was the case it would presumably involve quantum gravity and the problem of finding a unified theory would be solved. This, however, suggests:
3. That we look for a theory which is quantum in its origin.
If we combine these three points we find that they suggest to look for a model of quantum gravity that involves an algebra of observables which is sufficiently noncommutative and subsequently arrive at a principle of unification by applying the machinery of noncommutative geometry. The aim of this review paper is to report on efforts made in this direction. In particular we shall report on efforts made to combine noncommutative geometry with canonical quantum gravity.
(Submitted on 28 Mar 2012)
... saying goodbye to our classical space-time relationship ?
Despite the realization that important conceptual problems are still affecting quantum mechanics as a fundamental theory of nature (the problem of measurement, the problem of time for covariant quantum theories, the conflicts with classical determinism) and despite the recurrent claims of a need for a theory that supersedes, modifies or extends quantum theory (hidden variables, several alternative interpretations, collapse of wave function, deterministic derivations of quantum theory), very few people have pointed out (perhaps because of a misinterpretation of N. Bohr’s correspondence principle) that quantum theory still now is essentially an incomplete theory, incapable of “standing on its own feet”.
In all the current formulations of quantum theory, the basic degrees of freedom of a theory are specifically introduced “by hand” and make always reference to a classical underlying geometry.
The choice of the degrees of freedom is usually done in elementary quantum mechanics through “quantization procedures” via the imposition of Weyl (or Heisenberg) commutation relations for conjugated observables starting from a classical pair of position and momentum as in Dirac canonical quantization and, more generally, as in Weyl quantization, associating quantum observables starting from functions living on a classical phase-space.
In quantum field theory of free fields again, the local Weyl algebras are obtained by second quantization from symplectic spaces that originate from propagators of hyperbolic operators living on a classical Lorentzian manifold.
Even in algebraic quantum field theory, the most abstract axiomatization available, we have seen that there is always an underlying classical space-time as an indexing base of the net of quantum observables or a category of classical space-time geometries as a domain of the C∗-algebra-valued functor that defines the theory.
What is even more incredible is that, although it is widely recognized that many of the problems encountered in quantum field theory can be traced back to “unhealthy usage” of classical notions of space-time manifolds (divergences, convergence of Feynman n integrals), the previous problem of intrinsic characterization of degrees of freedom “inside quantum theory” has rarely, if ever, been seriously considered (with some notable exceptions) and that most of the attempts to cure the problem are simply trying to substitute “by hand” commutative classical geometries with non-commutative counterparts without addressing the issue from within quantum physics.
Among those really notable exceptions in this direction (see the companion review [30, Section 5.4] for more details), we have to mention the few attempts to spectrally reconstruct spacetime from operationally defined data of observables and states in algebraic quantum field theory starting from U. Bannier  and culminating in S.J. Summers, R. White  and, at least form the ideological point view, the efforts in information theoretical approaches for an operational definition of quantum theory as, for example, in A. Grinbaum [127, 128, 129].
A very interesting passage, asserting that in a fundamental theory the notion of space-time must be derived a posteriori and understood through relations between “quantum events”, can be found in R. Haag [132, Section VII, Concluding remarks]. Our main ideological motivation for the work on “modular algebraic quantum gravity” presented here comes from the view [25, 27, 28, 30] that:
space-time should be spectrally reconstructed a posteriori from a basic operational theory of observables and states; A. Connes’ non-commutative geometry provides the natural environment where to attempt an implementation of the spectral reconstruction of space-time; Tomita–Takesaki modular theory should be the main tool to achieve the previous goals, associating to operational data, spectral non-commutative geometries.
A very interesting recent work by C. Rovelli, F. Vidotto [218, see Section III B] is making some clear progress in the direction of the first two claims above obtaining relations between entropy of graphs describing the quantum geometry of loop gravity and the “spectral geometry” given by the non-relativistic Hamiltonian of a single particle interacting with the gravitational field. Still the deepest and at the same time most visionary proposal for a program aiming at the reconstruction of space-time via interacting events from purely quantum theoretical constructs is the one that has been given by C. Rovelli [214, Section 5.6.4] inside the framework of his relational theory of quantum mechanics [215, 237].
The modest ideas suggested here might be seen as just a very partial attempt to set up a mathematical apparatus capable of implementing C. Rovelli’s intuition, merging it with noncommutative geometry and utilizing instruments from category theory, higher C∗-categories and Fell bundles, in order to formulate a theory of relational quantum mechanics. [...]
Finally we must stress that, contrary to most of the proposals for fundamental theories inphysics that are usually of an ontological character, postulating basic microscopic degrees of freedom and their dynamics with the goal to explain known macroscopic behaviour, our approach (if ever successful) will only provide an absolutely general operational formalism to model information acquisition and communication/interaction between quantum observers (described via certain categories of algebras of operators) and to extract from that some geometrical data in the form of a non-commutative geometry of the system. Possible connections of these ideas to “quantum information theory” and “quantum computation” are also under consideration .
(Submitted on 23 Jul 2010 (v1), last revised 19 Aug 2010 (this version, v2))