Schrodinger's Cat, and Einstein's Space-time, in the 21st Century: November 2019

Thursday, November 28, 2019

Limitations of the theory of spontaneous localisation

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.

Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory
____________________________________________

November 28, 2019

Limitations of the theory of spontaneous localisation

Tejinder Singh

The GRWP theory of spontaneous localisation is a falsifiable phenomenological theory. It is designed to provide a dynamical solution to the quantum measurement problem, and to provide a cover for Newtonian mechanics that agrees with quantum mechanics for microscopic systems. The theory is precise enough for experimentalists to be able to test it, and confirm it or rule it out. That is why experimentalists are testing it. For knowing about some of the latest experimental developments on this front, the reader can visit tequantum.eu The most direct way to test GRWP is to verify if the principle of linear superposition holds for large objects. As we saw before, quantum mechanics predicts that a superposition of two position states of an object lasts forever. On the other hand GRWP predict that the superposition lasts for a time T/N, with N being the number of particles in the superposed object. So experimentalists prepare a superposed state, say by using a diffraction grating, and watch if it decays during the time of observation. If it does not, quantum mechanics wins, and one puts a lower bound on T. These are the so-called interferometric tests of (spontaneous) collapse models.

However, in recent years, the so-called non-interferometric tests of collapse models have moved centre-stage. Every time an object in a superposed state undergoes spontaneous collapse to some random location in space, its wave function expands again, and then again it collapses, with the mean life-time between collapses being T/N. These repeated random collapses amount to a random walk, with which is associated a tiny amount of kinetic energy. Spontaneous collapses cause the quantum object to gain a very tiny amount of energy. After cooling the object to extremely low temperature - few milli-Kelvins - and low pressure, one can attempt to look for this random walk, which obviously is in violation of quantum mechanics. Such experiments are currently in an exciting stage, and we might hear of some exciting results in the next few years.

The GRWP theory mathematically amounts to a modification of the Schrodinger equation. One adds a non-Hermitian (random) term to the Hamiltonian of the quantum system. Random because we want the resulting spontaneous collapse to result randomly. Non-Hermitean because we want one of the superposed states to grow exponentially, and the other one to decay exponentially (i.e. destroy superposition). Now, adding such a term implies that evolution no longer preserves norm of the quantum state. However, if the Born probability rule has to be obtained, norm must be preserved. Thus, a new quantum state is defined, by scaling with the norm of the old state. The new state now obeys a non-linear and non-Hermitian stochastic differential equation, which describes spontaneous collapse theory. The equation has the standard linear and Hermitean part which describes Schrodinger evolution, and in addition it has a non-linear, non-Hermitian part which describes non-unitary evolution, which causes spontaneous localisation and breaks position superposition. It is this equation which the experimentalists are testing. For microscopic objects, the predictions of this equation are extremely close to that of the Schrodinger equation (the non-linear part is negligible), but for macroscopic objects the deviations from quantum mechanics become significant. Here, the predictions of the new equation differ from those of quantum mechanics, and are falsifiable.

A theorist can raise a whole lot of questions and criticisms against the theory of spontaneous collapse, and these need to be addressed and resolved, so that the theory becomes more credible. As it stands, the theory is ad hoc in various ways. What is the origin of the random noise which has been added to the Schrodinger equation? What is the spectrum of this noise? Why should the collapse rate parameter T have this particular value of 1017 sec, and no other value? What causes spontaneous collapse in the first place? Why should the norm of the state vector be preserved, in spite of the evolution being non-unitary?

Perhaps the most serious criticism against collapse models is that they are non-relativistic. And attempts to make a relativistic Lorentz-invariant theory of spontaneous collapse have not been successful. Now, our most successful physical theories are relativistic quantum field theories, which describe the standard model of particle physics, and show excellent agreement with experiments. The Schrodinger equation is readily shown to be the non-relativistic approximation to the Dirac equation, which is relativistic. How then are we adapt the stochastic corrections provided by GRWP to the context of a quantum field theory?

In the opinion of this author, there is a convincing reason why one cannot have a relativistic theory of collapse, without making additional conceptual changes. Recall that spontaneous collapse takes place in position space: the position operator of a particle jumps to a specific eigenvalue, causing spontaneous localisation. Now, in special relativity, we expect position and time to be treated in a symmetric fashion. Hence, in order to make a relativistic theory of spontaneous collapse, we must allow also for spontaneous localisation in time! For that to happen, time will have to be treated as an operator, just like position is an operator in quantum mechanics. In that case, time loses its role as a parameter for defining evolution, and we are then compelled to introduce into relativistic quantum mechanics a new absolute and universal time parameter, which can be used to define evolution. To summarise, in order to have a relativistic theory of spontaneous localisation, space-time coordinates must be turned into space-time operators in quantum theory, which can undergo spontaneous collapse, and time evolution has to be described by a new absolute time parameter.

There is a lot that has been said and encoded in the previous paragraph, so we now dwell carefully on the various issues that arise. Firstly, why is it that relativistic spontaneous collapse forces us to treat ordinary time as an operator, whereas no such compulsion arises in standard relativistic quantum field theory? The answer is subtle. So long as spontaneous collapse in position can be ignored [as of course is the case for QFT], spontaneous collapse in time can be ignored as well, and we have our Lorentz invariant quantum field theory. In non-relativistic quantum mechanics, switching on collapse in position space does not compel us to switch on collapse in time space. Because the theory is Galilean invariant; it is not Lorentz invariant, and time is absolute. On the other hand, in the relativistic case, Lorentz invariance compels us to introduce spontaneous collapse in time, soon as we introduce spontaneous collapse in position. In turn, that forces us to introduce an absolute universal time parameter.

It should be mentioned though, that such a (covariant) formulation of relativistic quantum field theory, which treats position as well as time as operators, does exist. It is known as the Horwitz-Stueckelberg theory and the reader can read more about in Lawrence Horwitz’s book `Relativistic Quantum Mechanics’ [Springer, 2015]. The book also discusses the phenomenon of `quantum interference of time’ which will inevitably arise once time has been made an operator. It means that a quantum particle can be at more than one time, at a given universal time. Sone researchers claim that experimental evidence for quantum interference of time already exists. In any case, it is of great importance to perform experiments to look for quantum interference of time. Spatial quantum interference is comparatively much easier to detect, but as and when quantum interference of time is detected, QFT and relativistic quantum mechanics will have to be written in the language of the Horwitz-Stueckelberg theory.

Spontaneous collapse in time may appear to be a bizarre phenomenon, but we have been led to it in a logical inescapable manner. In order to have a cover theory of Newton mechanics which agrees with quantum mechanics for microscopic systems, we are compelled to introduce spontaneous collapse in position. In order to make this collapse theory relativistic, we are compelled to introduce spontaneous collapse in time. Experimentalists ought to look for collapse in time, just as they are testing the GRWP theory.

We can also ask: just as a chair is never found in two places at the same time, why is the chair never found in two times at the same place?! This maybe attributed to rapid spontaneous collapse in time, caused by the chair being made of many many particles. Spontaneous collapse in space as well as time together define classical events. We expect a quantum particle such as an electron to be at more than one time at the same place (as already hinted at by the path integral formulation of relativistic quantum mechanics) in a very real and physical sense. A quantum particle senses the past as well as the future `simultaneously’. What implications does this have for our understanding of physical reality?

Lastly we mention that the universal absolute time parameter which relativistic collapse theories compel us to introduce, turns out to be rooted in non-commutative geometry, and in the theory of spontaneous quantum gravity, which we will take up in subsequent posts. But we can already see that the need to introduce space-time coordinate operators already takes us towards quantum gravity, and away from classical space-time geometries. And later we will see how and why spontaneous collapse is an inevitable consequence of spontaneous quantum gravity. The ad hoc nature of the GRWP theory is removed then, because it emerges from an underlying physical theory.

Because spontaneous collapse is essential for localisation of macroscopic objects and resolution of the quantum measurement problem, and because localisation of macroscopic objects is essential for the existence of space-time [Einstein hole argument] and because space-time emerges from quantum gravity, we conclude that the solution of the quantum measurement problem comes from a quantum theory of gravity. Thus one cannot construct a quantum theory of gravity by quantising classical gravity, because doing so does not give a quantum theory of gravity which will dynamically explain absence of superposition of classical space-time geometries. In subsequent posts, we will clearly see how and why Planck length appears in the stochastic part of the non-linear Schrodinger equation which explains spontaneous collapse.

Saturday, November 23, 2019

The quantum measurement problem, and its solution via spontaneous collapse theory

Friday, November 22, 2019

From quantum foundations to quantum gravity

From quantum foundations to quantum gravity

Tejinder Singh

Quantum theory was invented to explain experimental data which could not be explained by Newton’s mechanics. There is no such clear-cut compelling observational evidence to suggest that gravity must be quantised. It could be said that the classical general theory of relativity agrees with every experiment / observation carried out till date. It may or may not turn out to be the case that understanding dark energy, cosmological constant, dark matter, require us to unify quantum and gravity. It may or may not turn out to be the case that the gravitational singularities that arise in general relativity require us to quantise the theory.

However, there are reasons within quantum (field) theory which compel us to consider a non-classical description of space-time and of space-time geometry. Quantum theory needs a time parameter, so as to describe the evolution of quantum systems. This time parameter is a part of a classical space-time, whose geometry is determined by classical bodies according to the laws of general relativity. But classical bodies are a limiting case of quantum systems. It should be possible to describe the dynamics of a quantum system without having any dependence [direct or indirect] on classical bodies. And yet, in the absence of classical bodies [i.e. if all matter were quantum], one cannot have a classical space-time geometry, nor a classical space-time manifold. This is a consequence of the so-called Einstein hole argument, which you can learn more about, from say this video: The problem of time in quantum theory [https://www.youtube.com/watch?v=fGdOTokept8]

Thus, we must have a formulation of quantum theory which does not depend on classical space-time. This will be our sought for quantum theory of gravity. We do not quantise gravity or space-time. Rather, we remove space-time from quantum (field) theory.

Can this goal be achieved by applying the rules of quantum theory to a classical theory of gravity? The answer is no. Firstly, the quantum rules are written down assuming classical time to exist. How then can we apply these rules to quantise the very time parameter whose classical existence was in the first place assumed, for writing these rules? There is no guarantee that this [admittedly illogical] step will lead us to the correct theory.

But secondly, there is an even more serious reason for the answer to be no. A classical theory of gravity does not permit superposition of space-time geometries: such superpositions are never observed, just as a chair is never observed in more than one place at the same time. On the other hand, a quantum gravity theory resulting from quantising classical gravity will naturally admit superpositions of geometries. And the theory will predict a superposition of geometries even when the bodies producing these geometries become large and classical. Same way as quantum theory predicts that a chair can be here and there at the same time. This is the Schrodinger cat paradox in the context of spacetime geometries. In the language of the previous post, such a quantum gravity theory is not the cover of classical general relativity.

To recover classical general relativity from quantum gravity, the sought for quantum gravity theory must admit a spontaneous collapse of superposed geometries, precisely in the spirit of the GRWP theory discussed in the previous post. Let us name such quantum gravity, which admits spontaneous collapse of geometry, as spontaneous quantum gravity. From here it is easy to reason that the absence of macroscopic position superpositions in the classical world is a consequence of spontaneous quantum gravity. Classical space emerges from quantum gravity, and moreover for classical space to exist, macroscopic bodies must be classical (not quantum). Thus the GRWP theory is a consequence of quantum gravity. This is readily seen in another way. Imagine a situation in which no quantum object has yet undergone spontaneous collapse: then there is only quantum matter and quantum space-time - the domain of quantum gravity. It follows that GRWP must arise from quantum gravity. Hence the name spontaneous quantum gravity.

Spontaneous quantum gravity [SQG] is the cover for classical general relativity, same way as GRWP is cover for Newton’s mechanics. SQG is falsifiable, because it predicts spontaneous collapse, and the latter is falsifiable. Recall that in GRWP, spontaneous collapse is proposed in an ad hoc manner. But in SQG, spontaneous collapse is not ad hoc. It is a consequence of the structure and dynamics of the theory.

One could well ask, quantum (field) theories of other interactions, such as QED, are not covers of their classical counterparts, such as Maxwell’s electrodynamics. Yet, why is QED such a successful theory, even though it does not explain the absence of superpositions in the classical electro-magnetic world? The answer is that QED is not a quantum theory of spacetime. It is the quantum theory of a field which lives on spacetime, and of the electric charges which produce these fields. Quantum gravity has to explain how classical space emerges, and since classical space is tied to absence of position superpositions in macroscopic bodies, quantum gravity has to explain why macroscopic bodies are classical. Once the position of macroscopic bodies is localised, their mass is localised, and their electric charge is localised too, and hence the associated electromagnetic fields are classical. Electromagnetic fields live on classical space-time, and require space-time to pre-exist. Space-time does not live on a classical electromagnetic field! Hence the buck stops with gravity.

We saw in the previous post that GRWP theory is the cover for Newton’s mechanics, and for small systems GRWP reduces to quantum theory, because the rate of spontaneous localisation is negligible for small systems. In the present post we see that spontaneous quantum gravity is the cover for classical general relativity, and for small objects it reduces to …? Reduces to what? We expect it to reduce to quantum gravity, because now the rate of spontaneous collapse of geometries is negligible, where by quantum gravity we mean quantisation of classical general relativity. [Incidentally, when we talk of rate of collapse of superposed space-time geometries, how is rate defined? What is this time parameter which keeps the rate? We will take up this deep question subsequently]. Thus, in all likelihood, we expect that the limit of SQG for small objects is related to loop quantum gravity. So we can say:

GRWP theory = Quantum theory + Spontaneous collapse

SQG = Quantum gravity + Spontaneous collapse

The GRWP theory already exists and is well defined and is being tested in the laboratory. How do we mathematically formulate spontaneous quantum gravity? We will take this up in a forthcoming post.

Thursday, November 21, 2019

Schrodinger's cat in the 21st century

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.

Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory

____________________________________________________

November 21, 2019

Schrodinger's cat in the 21st century

Tejinder Singh

If a physical theory agrees with some of the data, but not with all the data, it should be replaced by a falsifiable new theory which agrees with all of the data. The new theory should reduce to the old theory in those domains where the old theory agrees with the data.

Such a strategy worked successfully in the transition from Newtonian mechanics to special relativity, and it worked successfully in the transition from Newtonian gravitation to Einstein's general relativity. We can say that special relativity is a cover for Newton's mechanics, and general relativity is a cover for Newtonian gravitation. But the transition from Newtonian mechanics to quantum mechanics is a different story altogether!! Quantum mechanics is not a cover for classical mechanics.

Quantum mechanics was invented to explain data such as the black-body radiation spectrum, atomic spectra, and the photoelectric effect, which Newton's mechanics fails to explain. And quantum theory explains these, and much much more, beautifully. But quantum theory fails to explain the data that Newton's mechanics explains! So we need a new theory which will agree with both quantum mechanics, as well as with classical mechanics.

At the heart of the disagreement between classical and quantum mechanics is the very elegant quantum linear superposition principle, which says that if a quantum system can be in State A and if it can be in State B, then it can also be in the superposed state A+B. In particular, the superposition principle is known to hold for positions of a particle. If an electron can be here, and if the same electron can be there, it can also simultaneously be here as well as there. This is how we understand the appearance of interference fringes on the screen in a double-slit experiment with electrons.

The quantum superposition principle has been experimentally verified to hold for photons, neutrons, atoms, small molecules, and is known to hold for objects as heavy as 25,000 a.m.u. That is, an object made of 25,000 nucleons. Experimentalists would love to test the principle for even heavier objects, but its technologically extremely challenging. They are at it.

But the superposition principle fails for large objects that we see in our day to day life, obviously. We never see a chair to be here and there at the same time. Nor do we see a planet to be in the north and in the south at the same time. Yet the motions of chairs and planets is successfully described by Newton's mechanics. This is what we mean when we say that quantum mechanics fails for large objects and disagrees with classical mechanics. In fact, Schrodinger's equation predicts that a chair can be here and there simultaneously. The mass of the object described by the Schrodinger equation is completely arbitrary. This mass can be as large as we please. Hence Schrodinger's equation should hold for a chair, and superposition should have been observed, but its not observed. Another way to appreciate the problem is that the chair is made of elementary particles which themselves obey the superposition principle. Why is it that when we put many such particles together, superposition breaks down?

What is the way out of this contradiction between quantum mechanics and Newtonian mechanics? We need a cover for Newton's mechanics, which will agree with quantum mechanics for small objects. Such a new theory was proposed by physicists Ghirardi, Rimini, Weber and Pearle (GRWP) in the 1980s. Their idea is beautiful and simple. They said, look - quantum mechanics says that a superposition, once created, lasts forever. This is because the Schrodinger equation is linear and deterministic. On the other hand, Newton's mechanics does not allow for position superpositions at all. GRWP said, let's make a very very small modification to quantum theory. Let us propose that superposition of two position states of a particle, say a proton, does not last forever, but lasts for an extremely long time T. That is, the mean superposition life-time of two position states of a proton is not infinite, but a large number T. For definiteness, they proposed T to have the value 10^17 seconds, (which also happens to be the age of the universe) for a nucleon. After a time T, the superposition is assumed to spontaneously collapse to one or the other states which were superposed (here or there). This is the GRWP theory of spontaneous collapse. Remember, T is the mean lifetime, and collapse is a random (Poisson) process in time. There is always a tiny probability for spontaneous collapse to take place in a time much smaller than T.

This small modification to quantum theory suffices to provide us with a new theory which reduces to Newton's mechanics for large objects. Again, this works in a very elegant manner. Consider, for a start, a deutron - the nucleus of the deutirium atom, which is a bound state [also an entangled state] of a neutron and a proton. Now, for a deutron to undergo a spontaneous collapse, starting from a superposed state, it is enough for either the proton to undergo a spontaneous collapse, or for the neutron to undergo spontaneous collapse. One particle will take the other with it, because they are bound (entangled). You can then reason that the superposition lifetime for the deutron is halved, it is T/2, because there are two independent ways in which the collapse can happen.

There, now you have it. A large object such as a chair is made is of an enormous number of nucleons and electrons. If there are N particles in the chair, the chair can be in a superposed state (here and there) only for a time T/N. [Because any one particle collapsing will take the whole chair with it]. But N for a chair is huge, say 10^23. Since T is assumed to be 10^17 seconds, T/N is a mere millionth of a second. The superposed state for a chair lasts for such a short time, that we do not even notice it. Schrodinger's cat is dead as well as alive for a millionth of a second; after that it is dead, or alive. That is why large objects appear to obey Newton's mechanics.

In this way, the theory of spontaneous collapse is the cover for Newton's mechanics. The cover theory reduces to quantum mechanics for small objects. This is because for small objects, the superposition life-time is so enormous as to be practically infinite, as demanded by quantum mechanics.

The GRWP theory would occupy the same place of pride as special relativity and general relativity, if it were to be confirmed by experiment. Experimentalists are working hard to test it. The current experimental bound on T is that T > 10^8 seconds. Recall that GRWP say that T=10^17 seconds. and quantum mechanics says that T is infinite. Still nine more orders of magnitude to go before GRWP is ruled out. Note that a confirmed detection of spontaneous collapse below GRWP value will also prove the theory of spontaneous collapse. The theory will be ruled out if experiments will push the bound on T beyond the GRWP value. If T is higher than the GRWP value, then for large objects T/N will approach time scales larger than a millionth of a second, which means we would see a chair here and there at the same time. Thus, values of T larger than the GRWP value do not provide a cover theory for Newton's mechanics.

Pages