Primordial basis: How many angels can dance on the head of a pin?

Reading Gerard 't Hooft ['t Hooft 1999; 't Hooft 2000] is a pleasant and rewarding intellectual exercise ['t Hooft 1996; 't Hooft 1997]. At first sight, it seems that his efforts to postulate a deterministic hidden-variable theory employing dissipation of information at Planck scale, with built-in information loss and a special basis for Hilbert space called 'primordial basis', has resulted into a very ambitious and sophisticated conjecture. Is it grounded on rocks, however?

First about the hypothetical information loss due to the hypothetical black holes absorbing "negative amounts of energy, allowing positive energy to escape to infinity". Well, that's quite a challenge.

Since Gerard 't Hooft has clearlty declared his intention to develop some fundamental theory, I think the first and foremost task should be to resolve the basic problems with the so-called black holes -- they might not exist at all [Loinger 2000a; Loinger 2000b; Mashkevich 2000; Singh 1998; Singh 2000].

A second and parallel task should be to cure the string theory from some built-in inconsistencies [Woit 2001].

I have no idea whether his theory would survive after completing these first off tasks, but it will be certainly very useful if Gerard 't Hooft could suggest a solution to the cosmological constant problem, which requires new physics. I believe new physics will be needed to explain why we don't observe naked singularities [Joshi et al., 2001].

There must be at least one testable prediction from his new theory, otherwise it will be as academic as arguments about how many angels can dance on the head of a pin [Hawking 1995].

Let's wish him best of luck in his endeavors.

Dimiter G. Chakalov

April 6, 2001
Last update: October 4, 2001


References and notes

1. Gerard 't Hooft. Quantum Gravity as a Dissipative Deterministic System. Thu, 1 Apr 1999 15:52:56 GMT,

"We have learned to live with the curious phenomenon that our wave functions can be eigenstates of operators which at different space-time points usually do not commute. A "physical state" can be an eigenstate of an arbitrary set of mutually commuting operators, but then other operators are not diagonalized, and so, these observables tend to be smeared, becoming "uncertain". The idea that such uncertainties may be due to nothing other than our limited understanding of what really is going on, has become unpopular, for very good reasons. Attempts at lifting these uncertainties by constructing theories with 'hidden variables', have failed.

"It is the author's suspicion, however, that these hidden variable theories failed because they were based far too much upon notions from everyday life and 'ordinary' physics, and in particular because general relativistic effects have not been taken into account properly. The interpretation adhered to by most investigators at present is still not quite correct, and a correct interpretation is crucial for making further progress at very technical levels in quantum gravity.
"The most important distinction between gravitational and non-gravitational models is that, in gravitational models, information loss naturally occurs, since black holes may be formed. Indeed, it will be hard to avoid the development of coordinate singularities, but quite generally, one expects such singularities to be hidden behind horizons. So we have black holes.
"We now see that, since the black hole must loose weight, the primordial model must also have local fluctuations with negative "curvature energy". Black holes absorb negative amounts of energy, allowing positive energy to escape to infinity.
"The idea that there might exist a deterministic law of physics underlying all of this essentially amounts to nothing more than the suggestion that there exists a 'primordial basis', a preferred  basis of states in Hilbert space with the property that any operator that happens to be diagonal in this basis, will continue to be diagonal during the evolution of the system. None of the operators describing present-day atomic and subatomic physics will be completely diagonal in this basis. This enables us to accept both quantum mechanics with its usual interpretation and to assume that there is a deterministic physical theory lying underneath it.
"Due to information loss, Planck scale degrees of freedom must be combined into equivalence classes, and it is these classes that will form a special basis for Hilbert space, which we refer to as the 'primordial basis'.

"It is of interest to observe that, in constructing models with a deterministic interpretation for quantum states, the restriction to  1+1  dimensions is usually quite helpful. This is a reason to suspect that a deterministic interpretation of string theory is possible. In Appendix A, a construction is shown. Here, we succeeded in producing a model in  3+1  dimensions, but its ultraviolet cut-off is fairly artificial. In  1+1   dimensions, the cut-off is straightforward."

2. Gerard 't Hooft. Determinism and Dissipation in Quantum Gravity, Erice lecture. Tue, 16 May 2000 13:40:51 GMT,

"(iii) Even at a local scale (i.e. not cosmological), there are problems that we could attribute to a clash with Quantum Mechanics. Apart from the question of the cosmological principle, these are: 

- the non-renormalizability of gravity; 

- the fact that the gravitational action (the Einstein-Hilbert action) is not properly bounded in Euclidean space, while the Maxwell and Yang-Mills actions are. This is  related to the fundamental instability of the gravitational force.

- topologically non-trivial quantum fluctuations. They could destroy the causal coherence of any theory. Perhaps most such fluctuations may have to be outlawed, as they would also require the boundary conditions to fluctuate into  topologically non-trivial ones.

- black holes cause the most compelling conflicts with local quantum mechanics.

- there still is the mystery of the cosmological constant. It appears to require a reconsideration not only of physical principles at the Planck scale, but also at cosmological scales, since we are dealing here with an infrared divergence that appears to be cancelled out in a way that requires new physics.
"The new ingredient needed might be information loss [8]. At first sight this is surprising. One would have thought that, with information loss, the evolution operator will no longer be unitary, and hence no quantum mechanical interpretation is allowed. [Fig. 1, Transition rule with information loss.]
"Thus, the constraint appears to correspond to limiting oneself to the stable orbits only. Note that, with  Eq. (5.14), the hamiltonian can obtain any kind of eigenvalue spectrum, as opposed to the equidistant lines of the harmonic oscillator.

Fig. 3, Stable orbits, a) for the harmonic oscillator, b) an anhormonic oscillator. After switching on a dissipative term, the regions in between these trajectories will have only non-periodic solutions, tending towards the stable attractors.
"6. Conclusions
"In this lecture we investigated classical, deterministic, dissipative models, and we found that, in general, they develop distinct stable orbits. The mathematics for analyzing these models requires that we first introduce non-dissipative equations, which allow a formalism using quantum mechanical notation, but, without dissipation, it cannot be understood why the hamiltonian would be bounded from below. Then we find that dissipation imposes constraints on the solutions, which appear to provide bounded hamiltonians. It is remarkable that dissipation also leads to an apparent quantization of the orbits, and this quantization indeed resembles the quantum structure seen in the real world. 

"The next step, yet to be taken, is to couple infinite numbers of dissipating oscillators to form models of quantum field theories. This may appear to be a very difficult task, but we do notice that in classical general relativity black hole formation is inevitable, and black holes indeed absorb information. This would imply that the distance scale at which dissipation plays a role must be the Planck scale.
"It is far too early to ask for tangible results and firm testable predictions of the approach that we have in mind. A very indirect prediction may perhaps be made. We conjecture that the apparently quantum mechanical nature of our world is due to the statistics of fluctuations that occur at the Planck scale, in terms of a regime of completely deterministic dynamics."

3. Gerard 't Hooft. In Search of the Ultimate Building Blocks. Cambridge: Cambridge UP, 1996.

Frank Wilczek (review for Nature): "One finds on every page of this book sharp statements and novel formulations that show the workings of a first-rate, confident and original mind. It deserves attention."

4. Gerard 't Hooft. Distinguishing causal time from Minkowski time and a model for the black hole quantum eigenstates. Tue, 18 Nov 1997 09:54:57 GMT,

"Of even more importance, however, appears to be the fact that assuming the preservation of quantum information leads to interesting new insights in the forces of nature. Conservation of quantum informationis likely to demand a new kind of conspiracy, and the resolution of the paradox may well lead to important new physics.
"Fig. 3. The multi-valued time parameter

"In Fig. 3, this is further illustrated. The collapse is observed at the point  S . The ingoing observer follows the path  B  towards the Schwarzschild singularity. The outside observer continues along the path  A . After the collapse, Hilbert space is not described by the products of the states along path  A  and the ones along path  B , but it is spanned either by the states at  A  alone, or by the states at  B  alone. The Hamiltonian  H  is  defined along  A  and along  B , and it dictates the evolution everywhere along the curve. Thus, we see that time has become a manifold more complicated than just a single line."

5. Angelo Loinger. On continued gravitational collapse. Wed, 26 Jan 2000 12:50:43 GMT,

Abstract: "According to a widespread idee fixe, the spherically-symmetric collapse of a sufficiently massive celestial body of spherical shape should generate a black hole. I prove that this process generates simply an ordinary point mass. My argument is model-independent."

Angelo Loinger: "The conclusion is obvious. Vain is the chase of the black holes."

6. Angelo Loinger. On the concept of mass point in general relativity. Sat, 10 Jun 2000 04:55:42 GMT,

(To see the list of all papers by A. Loinger posted at Los Alamos E-print archive, click here.)

7. Vladimir S. Mashkevich. Unboundable Spacetimes with Metric Singularities and Matching Metrics and Geodesics: A Black-White Hole and a Big Crunch-Bang. Sat, 25 Nov 2000 04:02:00 GMT,

"Spacetime singularities are inherent in general relativity, or more specifically in gravitational collapse and cosmology. The analysis of spacetimes with singularities is one of the most principal and difficult problems in general relativity. Singularity theorems of general relativity utilize the notion of causal geodesic incompleteness as a criterion for the presence of a singularity. (A comprehensive presentation and discussion is given in [1].) The incompleteness of a causal, i.e., timelike or null curve implies physically the end and/or beginning of the existence of a particle, which are undeniably events. In the commonly accepted approach, singularities are not incorporated into a spacetime manifold. Thus spacetime turns out to be event-incomplete, i.e., does not include all events.

"Furthermore, the beginning and end of the existence of a free particle means that there is creation from nothing and extinction into nothing. Those phenomena are in conflict with conservation laws and appear physically pathological. Maybe it is possible to put up with extinction into nothing, arguing that nature is so structured. At least extinction follows a clear-cut law: Arriving at a singularity results in extinction. With creation from nothing, the situation is much worse. In this role, (naked) singularities are sources of lawlessness. All sorts of nasty things -- green slime, Japanese horror movie monsters, etc. -- may emerge  helter-skelter from a singularity [1]. To get rid of that nightmare, Penrose proposed the cosmic censorship hypothesis. But cosmic censorship may be legislated only by a fiat, it does not follow from known physical laws."

8. T.P. Singh. Gravitational Collapse, Black Holes and Naked Singularities. Mon, 18 May 1998 13:04:53 GMT,

9. T.P. Singh. Comparing quantum black holes and naked singularities. Thu, 21 Dec 2000 11:55:48 GMT,

10. Peter Woit. String Theory: An Evaluation. Fri, 16 Feb 2001 18:03:42 GMT,

11. Pankaj S. Joshi, Naresh Dadhich, Roy Maartens. Why do naked singularities form in gravitational collapse? Fri, 14 Sep 2001 16:04:06 GMT,

12. Stephen W. Hawking. Virtual Black Holes. Fri, 6 Oct 1995 16:09:21 +0100 (BST),

Stephen W. Hawking: "Unless quantum gravity can make contact with observation, it will become as academic as arguments about how many angels can dance on the head of a pin."