Non-commutative Gravity, a Quantum-Classical Duality, and the Cosmological Constant Puzzle
Tejinder Pal Singh
[Every year since 1949, the Gravity Research Foundation honors best submitted essays in the field of Gravity. This year's list of awardees has something unique about it. While the first prize for the award winning essay goes to T. Padmanabhan, the second prize goes to his former Ph.D student, Tejinder Pal Singh. The five award-winning essays will be published in the Journal of General Relativity and Gravitation (GRG) and subsequently, in a special issue of the International Journal of Modern Physics D (IJMPD). Today we present here an invited article from Prof. Singh on his current work.
-- 2Physics.com ]
Author: Tejinder Pal Singh
Affiliation: Tata Institute of Fundamental Research, India
The evolution of a system in quantum mechanics is described by the Schrodinger equation. What happens to this quantum system when a measurement is made on it by a classical measuring apparatus? What we have learnt from standard text-books in quantum mechanics is that the wave-function for the quantum system ‘collapses’ into one of the eigenstates of the observable being measured. For instance, if a double slit interference experiment is performed on a beam of photons, one observes an interference pattern on the photographic screen. The interference pattern arises because the wave-function of a photon is a linear superposition of two wave-functions: one corresponding to its passing through the upper slit, and the other corresponding to its passing through the lower slit. It is as if the photon is simultaneously passing through both the slits . However, if now a detector is placed behind one of the slits (this is a measurement) the interference pattern disappears, and the photon is interpreted as having passed through one or the other of the two slits, depending on whether the detector has clicked or not. The wave-function of the photon is said to have collapsed, from being originally in a linear superposition, to being a wave-function corresponding to the photon passing through only one of the two slits, not both.
What is often not emphasized in text-books is that this so-called collapse of the wave-function cannot be explained by the Schrodinger equation. This is because the Schrodinger equation is linear in the wave-function, and preserves superposition during evolution. The collapse process, on the other hand, breaks superposition, because the system goes from being in a superposition of many states (before measurement), to being in only one of those states (after measurement). What is the physical process which causes this collapse to take place? The honest answer is that as of today we do not know the correct answer, although an enormous effort has been invested, for nearly a century, in finding the answer. It is not some vague issue of ‘interpreting’ quantum mechanics; rather we are looking for a physical answer, based on sound mathematics, to the question: if we treat the original quantum system, along with the classical measuring apparatus, as one larger quantum system, why does this larger (macroscopic) system not obey the linear superposition principle of quantum mechanics? It is a physical question in precisely the same sense in which understanding planetary motion was a physical question : long ago, people did not know what caused planets to wander in the sky; until through observation and theory it became established that planets revolve around the Sun, and their motion is explained by Newton’s law of gravitation. Today we do not understand what causes the wave-function to collapse, but one day, through experiment and theory, we hope to have a clear understanding of the physics involved.
A remarkable aspect of the collapse process is the Born probability rule. During a measurement, when the wave-function collapses to one of the eigenstates, which eigenstate does it collapse to? This is where probabilities enter quantum mechanics, and this is the only place where they do (The Schrodinger evolution, prior to the measurement, is completely deterministic). The probability that the wave-function goes into one particular eigenstate is proportional to the square of the absolute magnitude of the wave-function for that eigenstate. Repeated experimental measurements on the same quantum system will produce different outcomes, always in accordance with this Born probability rule. There is no explanation in standard quantum mechanics for this rule, and the correct explanation of the collapse process must also include a derivation of this probability rule.
The possible explanations of the collapse process broadly fall into two classes. The first is the Everett many-worlds interpretation  of quantum mechanics, according to which the collapse never really takes place in fact, and is in essence an illusion. According to this explanation, at the time of a quantum measurement, the Universe (this includes the measuring apparatus and the observer) splits into many branches, and one outcome is realized in one branch, and a different outcome in another branch. For our double slit experiment, this means that when the detector is placed behind the slit (say the upper slit), then in one branch of the Universe (say ours) it will click, and the photon will have gone through the upper slit. In another branch of the Universe, a ‘different copy’ of the observer will find that the detector did not click, and the photon went through the lower slit. Linear superposition is preserved, and Schrodinger evolution continues to be preserved during and after the measurement. The different branches of the Universe do not interfere with each other because of the (experimentally observed) phenomenon of decoherence . This is the process wherein, because of the interaction of a macroscopic system with its environment, interference between different outcomes is strongly destroyed, even though superposition among the outcomes continues to be preserved. This would explain why in the double slit experiment the detector either clicks or it does not, but is never seen in a superposition of the two states `detector clicks’ and ‘detector does not click’ even though the superposition is in reality present.
The many-worlds interpretation is completely consistent with standard quantum mechanics, but it is not clear how it can be experimentally tested, because by construction one is not supposed to be able to observe the other branches of the Universe. Also, it is not yet clear how the Born probability rule will be arrived at within the framework of this explanation of a quantum measurement.
The second class of explanations of the collapse process assumes that there is only one branch of the Universe, not many branches, and that collapse is a real physical process, not an illusion. It is then immediately obvious that the Schrodinger equation, and hence quantum mechanics, must be modified  in order to explain the collapse process, because only then will it become possible to break linear superposition during the measurement process. For instance, it could be that the Schrodinger equation that we know of is only a linear approximation to a more general, non-linear, Schrodinger equation. The non-linearity might become significant only during a quantum measurement, and be responsible for breakdown of superposition, driving the quantum system to one of the eigenstates, in accordance with the Born rule.
As it turns out, as of today there is absolutely no experimental evidence that the Schrodinger equation needs to be modified. We thus find ourselves in this unpalatable position that if the Schrodinger equation is not modified, we must accept the many-worlds interpretation, but there seems to be no way to experimentally test this interpretation! So, does the collapse take place or not? Do we have to wait for more and more precise experimental tests of quantum mechanics to know the answer? Or is there some theoretical reason, over and above quantum mechanics as we know it, which favours collapse over no collapse, or vice versa? Fortunately, the answer to this question seems to be yes, and there is a theoretical argument suggesting that collapse does take place . Furthermore, it may be possible to test this argument experimentally.
The theoretical argument is based on another incompleteness in quantum mechanics, more serious but much less appreciated in comparison with the quantum measurement problem. Quantum systems evolve with time; but this time is a classical concept. Time is a part of space-time, whose geometry is determined by classical bodies such as stars and galaxies, through the Einstein equations of the general theory of relativity. If there were no classical bodies in the Universe, there would be no classical time – this is a consequence of something known as the Einstein hole argument . But even in such a situation, one should be able to describe quantum systems – there must exist a reformulation of quantum mechanics which does not refer to an external classical time. In looking for such a reformulation, one is led to the conclusion that standard linear quantum mechanics is a limiting case of a more general non-linear quantum theory. The non-linearity becomes significant when the mass-energy of the quantum system becomes comparable to or larger than Planck mass, but is completely negligible for smaller systems such as atoms. Planck mass is a fundamental unit of mass made out of Planck’s constant, speed of light, and Newton’s gravitational constant, and its numerical value is about a hundred-thousandth of a gram. Since this non-linearity in the Schrodinger equation becomes significant in about the same mass range where quantum measurement takes place, it suggests the possibility that linear superposition might break down during a measurement. Hence the many-worlds interpretation is disfavoured, as a consequence of the theoretical arguments described in this paragraph.
A programme, still tentative, is being developed to arrive at such a reformulation of quantum mechanics, and at the consequent non-linear Schrodinger equation . One starts by noting that in the absence of a classical space-time, the point structure of space-time is lost, and space-time points are themselves subject to quantum fluctuations. An inevitable mathematical way to express such fluctuations is to impose commutation relations amongst these coordinates, and also amongst the components of momenta of a particle in the presence of such spacetime fluctuations. The branch of mathematics which can naturally accommodate these features is known as noncommutative geometry . In such a geometry, which is a natural extension of the Riemannean geometry of general relativity, space-time coordinates do not commute with each other.
The aforesaid reformulation is motivated by the following new proposal : basic laws of physics are invariant under general coordinate transformations of non-commuting coordinates. This seems like a natural step forward from the general theory of relativity, which is based on the principle of invariance under general coordinate transformations of (commuting) coordinates. Standard linear quantum mechanics is reformulated as a non-commutative special relativity. As and when an external classical time becomes available, the reformulation reduces to the standard linear quantum theory. The generalization from non-commutative special relativity to non-commutative general relativity leads to a non-linear quantum mechanics. The latter reduces to the former when the mass-energy of the quantum system is much less than Planck mass. The relation between the non-linear quantum theory and its linear limit is the same as the relation between general relativity and special relativity. The second is recovered from the first in the limit in which Newton’s gravitational constant goes to zero. When the mass-energy of the system is much larger than Planck mass, the non-linear quantum theory goes over to standard classical mechanics.
The non-linear Schrodinger equation which arises here can in principle explain the collapse of the wave-function, under a further assumption whose validity remains to be established. The essential idea is that at the onset of quantum measurement the non-linearity drives the quantum system to one or the other outcomes, depending on certain initial conditions in the quantum system (for instance the phase of the wave-function) at the time when the measurement begins. Superposition is thus broken. One can also give a quantitative estimate of the life-time of a quantum superposition – predictably this life-time goes from astronomically large values to extremely small values as the number of degrees of freedom in the system is increased.
An interesting fall-out of this study is that one might obtain some understanding of the origin of the observed acceleration of the Universe, and of dark energy, for which the most likely explanation is a non-zero value for the cosmological constant. Why is this constant non-zero, and yet so small when expressed in fundamental units? In the present study, it appears that the dynamics of a quantum particle whose mass m1 is much less than Planck mass can be recovered from the knowledge of the dynamics of a classical particle whose mass m2 is much greater than Planck mass. We call this a quantum-classical duality . The product of the masses m1 and m2 is equal to the square of Planck mass. If one assumes that the classical ‘particle’ is the whole observed Universe, then the cosmological constant can be shown to be equal to the (finite) zero-point energy of the dual quantum field, and this matches with the value currently seen in cosmological observations.
The programme described here should strictly be described as ‘work in progress’, and there is still quite some way to go before these ideas can be put on a firm footing, and before one knows that this is the right track. Nonetheless, the ideas appear aesthetically appealing and natural, and a distinct advantage of the programme is that it is experimentally falsifiable. The non-linear theory agrees with standard quantum mechanics for small masses such as atomic masses, and it agrees with classical mechanics for large macroscopic masses. However its predictions differ from those of linear quantum mechanics in the mesoscopic mass range, which very crudely could be taken to be the mass range 10-20 grams to 10-8 grams. It is a significant fact that quantum mechanics has not been experimentally verified in this vast mass range, simply because such experiments are very difficult to perform with the currently available technology. The non-linear Schrodinger equation that we have predicts that the lifetime of a quantum superposition will decrease with increasing mass of the system. If the disturbing effects of the environment could be shielded (avoidance of decoherence) such a dependence of the superposition life-time on mass could be experimentally tested. Avoiding decoherence is however a great experimental challenge. An easier class of experiments is one for which the predictions of the non-linear theory for some measurable constant differ from that of the linear theory. For instance, the non-linear theory predicts a different value of the ratio h/m in the mesoscopic range, as compared to the linear theory, and this should be testable. Another possible prediction of the non-linear theory is that the outcome of a quantum measurement is not probabilistic, but deterministic, and possibly depends on the phase of the wave-function at the onset of measurement. Suitable correlation experiments might be able to test this by making fast successive measurements on a quantum system.
 "Feynman Lectures in Physics", Vol. III, Chapter I",
R. P. Feynman, R. B. Leighton and M. Sands, (Addison-Wesley, Reading, 1965).
 " 'Relative State' Formulation of Quantum Mechanics",
Hugh Everett, III, Reviews of Modern Physics 29, 454 (1957). Abstract Link.
 "Decoherence and the appearance of a classical world in quantum theory",
E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch and I.-O. Stamatescu, (Springer, New York) 2nd Edn.
 "Collapse Models", P. Pearle, http://in.arxiv.org/abs/quant-ph/9901077 .
 "Quantum measurement and quantum gravity : many-worlds or collapse of the wave-function?"
T. P. Singh, http://arxiv.org/abs/0711.3773.
 "An introduction to non-commutative differential geometry and its physical applications",
J. Madore (Cambridge University Press, 1999).
 "Noncommutative gravity, a `no strings attached' quantum-classical duality, and the cosmological constant puzzle", T.P. Singh, http://arxiv.org/abs/0805.2124.