iStock/sdominick
The world of large things such as tables, planets, stars and galaxies, is extremely different from the world of small things such as electrons, protons, atoms, and photons. The most striking difference is that a table is never found in more than one place at the same time, whereas as electron or an atom can be in many places at the same time. Why should there be this difference? After all, a table is nothing but a collection of an extremely large number of atoms. Why is it that when a lot of atoms are put together to make a large object, the property of being in more than one place is lost?
In physics, simple sounding questions sometimes have far-reaching consequences, when their answers are found. That seems to be the case here too. When we understand why a table cannot be in more than one place at the same time, we also understand where space and time come from! We have recently shown that a remarkable new physical mechanism, known as spontaneous localization, is at play, and is responsible for the emergence of space-time, and also for the aforementioned property of large objects (Tejinder P. Singh, "Space and time as a consequence of GRW quantum jumps," arXiv:1806.01297 [gr-qc] (2018), to appear in Zeitschrift fur Naturforschung A).
Quantum theory, which gives the rules for the motion of microscopic objects such as electrons, does not provide an explanation for this difference between a table and an electron. This is all the more surprising considering that the theory is extremely successful in explaining a great variety of phenomena, and is not contradicted by any experiment. The motion of large objects is described by Newtonian mechanics, which is generally assumed to be a limiting case of quantum mechanics. There is however a catch here: Newtonian mechanics can be derived as a limiting case of quantum mechanics only if we additionally assume that large objects are localized, and cannot be in more than one place simultaneously. The unexplained difference between large and small remains unexplained; it has been with us ever since the birth of quantum mechanics. It is also at the heart of the so-called quantum measurement problem, and has been immortalized by the so-called Schrodinger's cat paradox.
Could it be that apart from quantum mechanics and Newtonian mechanics, there is a new mechanics, to which the former two are approximations? This new mechanics will have a built-in mechanism such that while any object can indeed be in a superposition of the states 'here' and 'there', the superposition does not last forever. The superposition is extremely short lived if the object is made of a very large number of atoms, but it lasts for enormously long times for small systems such as electrons and individual atoms. Such a new mechanics indeed exists and is known as the theory of Spontaneous Localisation. It was proposed by Ghirardi, Rimini, Weber and Pearle in the 1980s, and is popularly also known as the GRW theory.
In quantum theory, the state of a system is described by a complex mathematical object known as its wave function. The wave function evolves according to the Schrodinger equation, which is a deterministic and linear equation. The property of an electron being here and there at the same time is described by a state whose wave-function is a linear superposition of the two wave functions: electron here, and electron there. The theory of Spontaneous Localisation (SL) says that every superposition undergoes random and spontaneous collapse to one or the other alternatives (here, or there), with collapse happening very rarely for microscopic objects, and extremely frequently for macroscopic ones. Note that SL is different both from quantum theory as well as from Newtonian mechanics. In quantum theory, which obeys the Schrodinger equation, spontaneous collapse never takes place. In Newtonian mechanics, there are no superpositions in the first place, so there is no question of collapse. SL provides a middle ground, a bridge so to say, between quantum and classical mechanics. The collapse of the wave function during a quantum measurement is a special case of spontaneous collapse, resulting from a sudden interaction between a microscopic system and the macroscopic measuring apparatus. A table stays put in one place because its wave-function repeatedly and very rapidly keeps collapsing spontaneously, thus preventing superpositions. The experimental predictions of SL differ slightly from those of quantum theory, and laboratory tests of SL have presently entered an exciting stage. If confirmed, SL will be a radical generalization of quantum theory. Apart from solving the measurement problem, it will have far reaching implications for our understanding of space-time structure, and of quantum gravity.
Space, as we usually understand it, is a classical construct. It is that which is between objects, between tables and chairs, and between planets, stars and galaxies. But if all these classical objects are staying localized because of spontaneous collapse, is it not plausible that space by itself is a consequence of collapse of the wave-function? We might want to think of space as absolute, as a given, in which objects are embedded. But this viewpoint is challenged by quantum theory. Imagine a universe consisting only of quantum mechanical objects (and having nothing classical), each of which is 'everywhere'. What physical meaning there is then, to space? Furthermore, the time parameter that appears in the Schrodinger equation is a classical parameter, being a part of classical space-time geometry. There is a consequence of the so-called Einstein hole argument that in a universe in which there are no classical objects and everything is quantum, it is not physically meaningful to assume the point structure of a space-time manifold. From the point of view of GRW theory, suppose no spontaneous collapse has yet taken place in the universe: there would then be no classical objects, nor a classical space-time. For these reasons, it becomes evident that, GRW or otherwise, there ought to exist a formulation of quantum theory without classical space-time. From this formulation, space-time and classical objects are recovered via spontaneous collapse. Only, now the mechanism is more general than the GRW model, because the setting in which collapse now takes place does not have classical time to start with. We call this Relativistic Spontaneous Localisation (RSL).
If space-time results from collapse of the wave functions of macroscopic objects, what is there, in place of space-time, prior to collapse? Ignoring gravity for now, we recall that ordinary space-time is described by the Minkowski line-element of special relativity, for the coordinates (x,y,z,t). Here, there is the elegant symmetry principle, namely that physical laws are the same for all inertial observers, and their respective coordinates are related to each other via Lorentz transformations. We propose a minimal generalization of this symmetry principle: physical laws are the same for all inertial observers, and their respective coordinates are related to each other via Lorentz transformations, but the coordinates no longer commute with each other. They become operators (equivalently matrices) (x,y,z,t), which have arbitrary commutation relations amongst them. Since time has become an operator, and is no longer a scalar parameter, it cannot be used to described evolution. Instead, evolution is described by the so-called Trace time, which is arrived at by taking the matrix trace of the operator Minkowski line-element. Matter degrees of freedom are also non-commuting; they 'live' on this operator spacetime, and the dynamics is exactly analogous to that of ordinary special relativity. We refer to this as noncommutative special relativity.
It turns out, remarkably, that in a certain thermodynamic approximation, whose details we do not discuss here, noncommutative special relativity is identical to relativistic quantum mechanics on an operator Minkowski space-time. Space-time coordinates continue to be operators, but these now commute with each other, and with the matter degrees of freedom as well. Each 'particle' is represented by a four- operator qia and in 'position' representation the wave function, evolving according to a Lorentz invariant Schrodinger equation, depends on the eigenvalues xia of the position four-operator, there being four such eigenvalues for every particle. This is the sought for quantum theory without classical time, and here evolution takes place in trace time. We could as well have taken this as the starting point for invoking relativistic spontaneous localization; however starting from a non-commutative special relativity underscores the underlying symmetries of the theory.
The beauty of this formulation is that the state vector lives in an extended Hilbert space, i.e. one that is endowed with the operator space-time metric. And this is the whole physical universe! There is no longer any external 3-space, nor external time, which somehow are otherwise uneasily latched on to the conventional Hilbert space of quantum mechanics.
When spontaneous localization takes place in this extended Hilbert space, every macroscopic matter degree of freedom is localized to some or the other eigenvalue of the space-time operator, as a result of spontaneous collapse of their wave-function. The full set of 'signposts' provided by these collapsed objects gives the extended Hilbert space a semblance of a classical space-time, in which macroscopic systems are embedded. In this sense collapse of the wave-function is responsible for the emergence of space-time, and we see that there is a deep connection between the problem of time in quantum theory, and the measurement problem. The microscopic degrees of freedom continue to live in the extended Hilbert space--their true home--though from the vantage point of the classical space-time produced from collapse of macroscopic objects, their dynamics appears the same as ordinary quantum mechanics, supplemented by the possibility of spontaneous collapse in space via the GRW mechanism.
How might we be sure that the idea of an extended Hilbert space, and space-time emerging because of collapse of the wave-function, is correct? There are at present two pieces of evidence for operator time and extended Hilbert space. Firstly, it helps understand the EPR paradox and peculiar nature of quantum non-locality. The physics of quantum measurement on correlated entangled particle pairs can be correctly understood only in the extended Hilbert space, because it involves collapse, and collapse takes place in operator space-time. When Alice makes a measurement on one of the particles in the pair, it instantaneously and simultaneously affects the other particle--but simultaneously in trace time. There being no notion of distance and separation in operator space-time, no influence travels from the first to the second particle. Hence, when Bob makes a measurement outside Alice's light cone, and we refer the physics to ordinary space-time, there is indeed a causality puzzle, but it is only because we have chosen the wrong frame to describe collapse. In reality, when quantum theory, operator space-time and collapse are combined, there is nothing unphysical or puzzling about quantum non-locality.
Secondly, in the extended Hilbert space, the wave-function of a particle has a non-zero amplitude to be at more than one time, for a given trace time, because time is now an operator. As a result, we predict that there will be a quantum interference in time. A particle can go through a slit now, and come back later to go through it again, and these two states (same place, different times) will interfere with each other! Interestingly, atomic physics experiments done in the past have reported results which can be interpreted sensibly only as quantum interference in time. This is a promising avenue which needs to be investigated carefully in the future, as possible evidence for relativistic spontaneous localization.
Outstanding remaining challenges in this program include generalization to quantum field theory, and incorporating gravity. Nonetheless, it appears satisfying that the tension between quantum theory and relativity can be released by generalizing to a non-commutative special relativity, and then invoking collapse of the wave-function to recover ordinary space-time. Our physical universe is a collection of wave-functions residing in the extended Hilbert space, and space-time is its emergent macroscopic limit.
(Edited on 12 September 2018 to add that in a follow-up paper I predict a new effect, quantum interference and spontaneous localisation in time. This should be testable in the lab: arXiv:1809.03441 [gr-qc] (2018).)
[Edited on 10th April 2019: these above ideas have now lead to a proposal for a new quantum theory of gravity: arXiv:1903.05402 [gr-qc] (2019).]
Update 22nd November, 2019
New Blogspot: Schrodinger's cat, and Einstein's space-time, in the 21st century
--
Tejinder Singh is an FQXi member and a physicist at the Tata Institute of Fundamental Research, Mumbai, India.
Reference: Space and time as a consequence of Ghirardi-Rimini-Weber quantum jumps; arXiv:1806.01297v4 [gr-qc] (2018), to appear in Zeitschrift fur Naturfortschung A. This work was supported by an FQXi mini-grant.