Why `quantising' a classical system is like going halfway and then stopping!

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; Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity; black holes, gyromagnetic ratio 
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Why `quantising' a classical system is like going halfway and then stopping!

When we quantise a classical system, we raise the canonical variables, q and p, to the status of operators, and assign quantum commutation relations to them: [q, p] = ih

How to recover the classical limit back? It is impossible! One cannot un-quantise a quantised system so as to make it classical again. Because the quantum system obeys linear superposition, and it does so irrespective of the size of the quantum system. Even large quantum systems are demanded by theory to obey superposition, even though they are in fact observed to be classical (no superposition).

The way out of this mess is not that we invent some `interpretation' of quantum mechanics!! The way out is to re-examine how we could have done things differently in the first place.

One very promising way is: ok let us make q and p into operators, but not impose *quantum* commutation relations. The commutation relations are arbitrary, and evolve with time, and are determined by dynamical laws. The dynamical laws are similar to those of classical dynamics, but now adapted to operator variables. This is the theory of Trace Dynamics.

It turns out that for microscopic systems, trace dynamics reduces to quantum theory, quantum commutation relations emerge, and the quantum superposition principle holds. But for large systems, trace dynamics reduces to classical mechanics, and superpositions do not hold [because of the mechanism known as spontaneous localisation, which we discussed earlier].

So, instead of quantising a classical system, one should `operatorise' it: make the commutators arbitrary. Quantisation is like going only half the distance. Trace dynamics is the full story. Thus

Trace Dynamics = Quantum Theory + Spontaneous Localisation.

For small systems, the last bit is negligible. For large systems it is very important.

It is possible to include gravity also in trace dynamics. This leads to a quantum theory of gravity:

Trace Dynamics + Trace Gravity = Quantum gravity + Spontaneous Localisation

For large systems, the last bit is important, and responsible for the emergence of classical space-time and laws of general relativity, from quantum gravity.

Spontaneous Localisation is what un-quantises a quantum system and makes it classical again. Experimentalists are carrying out experiments to find out if spontaneous localisation occurs in nature.

Why does a charged rotating black hole have the same gyromagnetic ratio as an electron?!

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; 
Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity 
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January 11, 2020


Why does a charged rotating black hole have the same gyromagnetic ratio as an electron?!


We know that a charge moving in a magnetic field experiences a force. A moving charge is of course a current. So, a current carrying wire experiences a force in a magnetic field. The same would be true if the wire forms a closed loop, say for simplicity a circular loop with some radius, in which a charge is going around with some angular velocity. The response of such a charge to an external magnetic field depends on the value of the charge, the angular speed, and the area of the loop. These quantities combine to define the magnetic moment of the current loop, which is simply current times the area. The response of a current in a loop depends on the magnetic moment, and is essentially magnetic moment times the external magnetic field.

It is easy to show that the magnetic moment is proportional to the orbital angular momentum of the charge in the loop. The ratio of the magnetic moment to the angular momentum is a constant q/2m, independent of the parameters of the orbit, where q is the magnitude of the charge, and m the mass of the particle. This ratio, q/2m, known as the gyromagnetic ratio, is the object of our interest in today’s post.

So if we have an electron with a charge e and mass m going around in a loop, it will have a g-ratio of e/2m. Now, an electron also has a spin, I.e. an intrinsic angular momentum, distinct from orbital angular momentum. It is a purely quantum mechanical effect, arising out of the Dirac equation, and the spin of an electron is one-half the Planck constant. The electron also has an intrinsic magnetic moment because it is a charge with angular momentum. Again, this magnetic moment is proportional to intrinsic angular momentum, but the g-ratio is now e/m, not e/2m. This can be proved from the Dirac equation. The g-ratio of a Dirac fermion is twice the classical value - this can be proved from the Dirac equation. Nothing terribly surprising if the quantum value is twice the classical value. The surprise comes now, when we talk of black holes.

Black holes are solutions of Einstein equations, and their properties are determined by the laws of classical general relativity. These laws are *classical*, not quantum, and black holes are classical objects. A black hole is described by at most three parameters - its mass, charge, and angular momentum, let’s label them M, Q and J respectively. The composition of this mass is not relevant for describing a black hole. Already we see something striking - an electron is also described by its mass, charge, and (spin) angular momentum! A black hole with no charge or angular momentum is described only by its mass M, and is called the Schwarzschild black hole. A Reissner-Nordstrom black hole has mass and charge, but no angular momentum. The most general black hole is the Kerr-Newman black hole, where M, Q and J are all non-zero. This is the charged rotating black hole. It has a magnetic moment, and a g-ratio, which can be calculated from Einstein equations. The g-ratio turns out to be Q/M, not Q/2M, agreeing with the Dirac quantum value for the electron, and disagreeing with the classical value! Even though black holes are classical objects.

This should set alarm bells ringing. In my view, this is a great hint that, despite appearances. Einstein equations have something to do with the Dirac equation, and black holes have something to do with Dirac fermions. Thus, since classical general relativity (with matter) is an approximation to a quantum gravity theory (along with matter), the latter theory must explain why a black hole has a g-ratio twice the classical value. This should be a requirement for a quantum gravity theory to be viable.

There is already considerable evidence in known physics that Einstein equations are deeply connected with the Dirac equation. Firstly, if we ask how to describe the dynamics of a relativistic particle of some mass m, the answer is ambiguous. We could describe it by the Dirac equation, which claims to hold for arbitrary values of mass; or we could describe it by Einstein equations, which also claim to hold for arbitrary values of the mass. Obviously, both descriptions cannot be correct. From experience we know that Dirac equation works for small particles, which are quantum in nature, and Einstein equations work for large objects, which are classical in nature. But how small is small, and how large is large? Neither the Dirac equation nor Einstein equations have  a mass scale. Obviously then, there ought to exist an underlying theory with a mass scale (or equivalently a length scale), such that for masses much smaller than this scale, the theory reduces to Dirac equation for quantum systems. And for masses much larger than this scale, the theory reduces to Einstein equations for classical systems.

The next big hint that Einstein equations are connected with Dirac equations, comes from geometry - something we discussed in an earlier post. Given  a Riemannian manifold, and the standard Dirac operator on it, the sum of the eigenvalues of the square of the Dirac operator is proportional to the Einstein-Hilbert action on this manifold. This remarkable result again suggests an Einstein-Dirac connection, provided we figure out a way to include Dirac fermions in the theory.

This is precisely what has been done in our recently proposed theory of spontaneous quantum gravity. An atom of space-time-matter has an associated length scale L, which is interpreted as its Compton wave length. If L is much larger than Planck length the theory reduces to quantum theory and the Dirac equation for a fermion. If L is much smaller than Planck length, the equations describing the STM atoms reduce to Einstein equations of general relativity, and the collection of STM atoms can be shown to be a classical black hole! One way to understand this is to note that given L, one can define another length from it, namely the square of Planck length divided by L. This is a quantity equal to the Schwarzschild radius, and it exceeds Planck length when L goes below Planck length. Thus when L crosses from a value larger than Planck length to a value smaller than Planck length, there is a cross-over from the Dirac fermion phase to the Einstein black hole phase. The net mass in the system crosses over from less than Planck mass to greater than Planck mass. A collection of entangled STM atoms behaves quantum mechanically, obeying the Dirac equation, if the total mass is less than Planck mass. The collection behaves like a black hole if he total mass exceeds Planck mass.

There in fact is a duality which maps a Dirac fermion to a black hole. It can be shown that if a solution describes an STM atom with Compton wavelength L, the adjoint of this solution describes a black hole with Schwarzschild radius L and Compton length L’=L_P^2/L. Now, if we associate an electric charge e with the Dirac fermion, we can associate a dipole moment eL with the STM atom. There is strong evidence that this dipole moment remains unchanged under the said duality map, which maps the Dirac fermion to a black hole with charge Q and mass M=1/L’ in such a way that eL = QL’. But this product is nothing but the gyromagnetic ratio. Hence this duality between Dirac fermions and black holes explains why a charged rotating black hole has the same gyromagnetic ratio as the electron. Black holes and electrons are simply two different states of atoms of space-time-matter. We consider this to be compelling evidence that spontaneous quantum gravity is a viable theory of quantum gravity.

Brief introduction to Quantum Foundations

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


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.

Brief Introduction to quantum foundations

Shivnag 
IISc, Bangalore

In this post, I shall attempt to briefly examine some of the foundational issues of quantum mechanics (which I will abbreviate as QM for convenience). I think it is a good idea to remind the well-experienced practitioners of QM every now and then the fact that they don't really understand what they are working with. That is, this post serves to remind us of the fact that QM is, at its heart, quite messed up. On a more pragmatic note, with all the talk about entanglement-this and entanglement-that (every guy on the street who has heard the word “quantum” has probably heard of it attached to the word “entanglement”), it is perhaps a good idea to understand what exactly entanglement is and what's the big deal about it. If you do wish to pursue these issues in more details, you inevitably arrive at the doorstep of quantum computation and quantum information, but that's for a later day (and a later post). So, let's get started!

Setting the Stage

First off, what are the features of quantum mechanics you found most annoying when learning about it in your sophomore (or was it junior) year? Most of us would agree that it was rather unsettling to see that QM answers to questions like “What precise outcome do we observe when we make a measurement?” by giving a set of possible outcomes and telling us the likelihood of each outcome. That is, if you were to perform the same experiment a million times under identical conditions, you could use QM to conclude how many of those times you'd get outcome A or outcome B, or you might be able to say that outcome C is ruled out because it is not in the set of allowed outcomes predicted by QM. While this is progress in itself, it begs the question, “Why can't I predict the precise outcome of any one experiment?” The first thing that we are told in our QM courses is that the problem is qualitatively different from say, the problem with predicting whether you get heads or tails on a coin toss. In principle, you could predict the outcome of a toss if you knew the exact mass distribution of the coin, the exact angular velocity with which it is flipped, the exact pressure distribution of air molecules which create drag on the coin etc. In addition to knowing every last detail about the coin, its surroundings and how you toss it, you'd need to have a computer powerful enough to process all that information (and preferably do it before the coin landed back in your hand). Needless to say, we'll not be discarding our method of using coin tosses to decide who bats first anytime soon. However, what's important is that it can be done.

However, we are told that the situation is different in QM. That is, it isn't a technological limitation that is preventing us from predicting the outcome of a single run of an experiment. The uncertainty is built into the physical laws. That is, no matter how powerful a computer you build, you'll not be able to predict the outcome of a quantum measurement precisely. Naturally, the first question that comes to mind is, “How are we sure that QM is the whole story?” And I am not posing this question in any deep, philosophical sense (that's for later). The most pedestrian objection one can have to this line of thought is, “Isn't it possible that QM is not capable of understanding some interaction (or missing something else) and that's the reason why it fails to predict precise outcomes?” Or one may wonder, “Maybe nature has some additional variables that need to be measured and fed into our mathematical machinery to get precise predictions?” Both these questions are important, and both bothered physicists and philosophers alike over the past century. Also, if the answer to either question is a yes, then that would mean that QM is incomplete or worse, wrong. The latter is unlikely because we have verified the predictions of QM countless times in numerous situations. You are doing it right now as you read this - for instance, QM provides the theoretical framework for understanding and working with the semiconductor electronics which power your computer. However, it surely can be incomplete, and one school of thought is to try and probe (no pun intended!) the question of how measurement works in quantum mechanics. This might not be as futile a venture as people these days make it to be. After all, Einstein arrived at special relativity, by examining the question of how you measure distances and times (that's what led him to conclude that simultaneity is not an absolute concept). The same could be true for quantum mechanics too, because the truth is, we don't have a good idea as to what exactly constitutes a measurement in quantum mechanics.

I will try to make some progress in this direction by trying answer the two questions I posed above. I will start with the second one - that is, the question of whether there are variables in Nature which haven't been observed and which, if observed, could eliminate the uncertainties in quantum mechanics. In fact, Einstein & co. believed that this was indeed the case and the great lengths they went to in order to prove their point is the stuff of legends. The argument that Nature has hidden variables that we have not (or cannot) observe go by the rather unimaginative name of “hidden variable theories”. I will examine their argument in the next post and see how (if at all) this argument can be verified.