I saw this in the latest Private Eye…

Follow @telescoper## Archive for Quantum Mechanics

## A Question for Prospective Physics Students

Posted in The Universe and Stuff with tags Many Worlds Interpretation, Private Eye, Quantum Mechanics on January 28, 2020 by telescoper## Making Better Sense of Quantum Mechanics

Posted in The Universe and Stuff with tags arXiv:1809.01639, David Mermin, Quantum Bayesianism, Quantum Mechanics on September 12, 2018 by telescoperThere is an interesting, pithy and polemical paper on the arXiv by David Mermin, with the abstract:

We still lack any consensus about what one is actually talking about as one uses quantum mechanics. There is a gap between the abstract terms in which the theory is couched and the phenomena the theory enables each of us to account for so well. Because it has no practical consequences for how we each use quantum mechanics to deal with physical problems, this cognitive dissonance has managed to coexist with the quantum theory from the very beginning. The absence of conceptual clarity for almost a century suggests that the problem might lie in some implicit misconceptions about the nature of scientific explanation that are deeply held by virtually all physicists, but are rarely explicitly acknowledged. I describe here such unvoiced but widely shared assumptions. Rejecting them clarifies and unifies a range of obscure remarks about quantum mechanics made almost from the beginning by some of the giants of physics, many of whom are held to be in deep disagreement. This new view of physics requires physicists to think about science in an unfamiliar way. My primary purpose is to explain the new perspective and urge that it be taken seriously. My secondary aims are to explain why this perspective differs significantly from what Bohr, Heisenberg, and Pauli had been saying from the very beginning, and why it is not solipsism, as some have maintained. To emphasize that this is a general view of science, and not just of quantum mechanics, I apply it to a long-standing puzzle in classical physics: the apparent inability of physics to give any meaning to “Now” — the present moment.

The `new perspective’ Mermin espouses is a form of `QBism (i.e. `Quantum Bayesianism’)‘. You can download the full article for free here.

## Cosmology: The Professor’s Old Clothes

Posted in Education, The Universe and Stuff with tags Big Bang, cosmology. cosmic microwave background, decoupling, neutrino oscillations, Physics, Quantum Mechanics, recombination, teaching on January 19, 2018 by telescoperAfter spending a big chunk of yesterday afternoon chatting the cosmic microwave background, yesterday evening I remembered a time when I was trying to explain some of the related concepts to an audience of undergraduate students. As a lecturer you find from time to time that various analogies come to mind that you think will help students understand the physical concepts underpinning what’s going on, and that you hope will complement the way they are developed in a more mathematical language. Sometimes these seem to work well during the lecture, but only afterwards do you find out they didn’t really serve their intended purpose. Sadly it also sometimes turns out that they can also confuse rather than enlighten…

For instance, the two key ideas behind the production of the cosmic microwave background are recombination and the consequent decoupling of matter and radiation. In the early stages of the Big Bang there was a hot plasma consisting mainly of protons and electrons in an intense radiation field. Since it was extremely hot back then the plasma was more-or-less fully ionized, which is to say that the equilibrium for the formation of neutral hydrogen atoms via

lay firmly to the left hand side. The free electrons scatter radiation very efficiently via Compton scattering

thus establishing thermal equilibrium between the matter and the radiation field. In effect, the plasma is *opaque* so that the radiation field acquires an accurate black-body spectrum (as observed). As long as the rate of collisions between electrons and photons remains large the radiation temperature adjusts to that of the matter and equilibrium is preserved because matter and radiation are in good thermal contact.

Eventually, however, the temperature falls to a point at which electrons begin to bind with protons to form hydrogen atoms. When this happens the efficiency of scattering falls dramatically and as a consequence the matter and radiation temperatures are no longer coupled together, i.e. decoupling occurs; collisions can longer keep everything in thermal equilibrium. The matter in the Universe then becomes transparent, and the radiation field propagates freely as a kind of relic of the time that it was last in thermal equilibrium. We see that radiation now, heavily redshifted, as the cosmic microwave background.

So far, so good, but I’ve always thought that everyday analogies are useful to explain physics like this so I thought of the following.

When people are young and energetic, they interact very extensively with everyone around them and that process allows them to keep in touch with all the latest trends in clothing, music, books, and so on. As you get older you don’t get about so much , and may even get married (which is just like recombination, not only that it involves the joining together of previously independent entities, but also in the sense that it dramatically reduces their cross-section for interaction with the outside world). As time goes on changing trends begin to pass you buy and eventually you become a relic, surrounded by records and books you acquired in the past when you were less introverted, and wearing clothes that went out of fashion years ago.

I’ve used this analogy in the past and students generally find it quite amusing even if it has modest explanatory value. I wasn’t best pleased, however, when a few years ago I set an examination question which asked the students to explain the processes of recombination and decoupling. One answer said

Decoupling explains the state of Prof. Coles’s clothes.

Anyhow, I’m sure there’s more than one reader out there who has had a similar experience with an analogy that wasn’t perhaps as instructive as hoped or which came back to bite you. Feel free to share through the comments box…

Follow @telescoper## The Quantum Mechanics of Voting

Posted in Politics, The Universe and Stuff with tags EU referendum, General Election, Physics, Politics, Quantum Mechanics, spin, voting on June 23, 2017 by telescoperNow that I’ve finished a marathon session of report-writing I thought I’d take a few minutes out this Friday afternoon, have a cup of tea and pass on a rather silly thought I had the other day about the relationship between Quantum Mechanics (and specifically the behaviour of spin therein) and voting behaviour in elections and referendums.

For a start here’s a brief summary of the usual quantum-mechanical context as it relates to, e.g., electrons (rather than elections). Being fermions, electrons possess half-integer spin. This attribute has the property that a measurement of its component in any direction has only two possible values, ±½ in units of Planck’s constant. In the Stern-Gerlach experiment illustrated above, which measures the spin in the vertical direction of silver atoms emerging from a source, the outcome is either “up” or “down”, not some spread of values in between. Silver has a single unpaired electron which is why its atoms behave in this respect in the same way as an individual electron.

The way this is often described in physics textbooks is to say that the operator corresponding to spin in the z-direction has only two eigenstates (call these ↑ and ↓) ; the act of measurement has to select one of them, not some intermediate state. If the source is thermal then the spins of individual atoms have no preferred direction so 50% turn out to be ↑ and 50% to be ↓ as shown in the cartoon.

Once such measurement has been made, a given particle remains in the same eigenstate, which means that if it is passed through another similar measuring device it will always turn out to have spin pointing in the same direction. If you like, the particle has been `prepared’ in a given state by the act of measurement.

This applies as long as no attempt is made to make a measurement of the spin in a different direction, which is when the fun starts. If we start with a particle in the ↑ state and then pass it through an experiment that measures spin (say) with respect to the x-axis instead of the z-axis then the two allowed eigenstates are then not ↑ and ↓ but ← and →. A particle that was definitely spin-up would then be forced to decide between spin-left and spin-right (each would have a 50% probability).

Suppose now we took our long-suffering particle that began with spin ↑ after a measurement in the z-direction, then turned out to be spin → when we measured it in the x-direction. What would happen if we repeated the z-measurement? The answer is that “preparing” the particle in the → state destroys the information about the fact that it was previously prepared in the ↑ state – the outcome of this second z-measurement is that the particle that was previously definitely ↑ now has a 50% chance of being either ↑ or ↓.

So what does all this have to do with voting? It is clear than an election (or a referendum) is very far from a simple act of measurement. During the campaign the various sides of the debate make attempts to prepare a given voter in a given state. In the case of last year’s EU referendum the choice of eigenstates was `Leave’ or `Remain’; no other possibilities were allowed. The referendum then `prepared’ each voter in one or other of these possibilities.

If voters behaved quantum mechanically each would stay in their chosen state until some other measurement were attempted. But that’s exactly what did happen. Earlier this month there was a General Election. More than two parties were represented, but let’s simplify and assume there were only two options, `Labour’ and `Conservative’.

Now it is true that the `Leave’ camp was dominated by the right wing of the Conservative party, and the majority of Labour voters voted `Remain’, but there were a significant number of Labour Leave voters and a significant number of Tories voted Remain. While these pairs of states are therefore not exactly orthogonal, they are clearly not measuring the same thing so the situation is somewhat analogous to the spin measurement problem.

So along came the General Election result which `prepared’ voters in a state of `Labour’ or `Conservative’, with a slight preference for the latter whereas the earlier referendum had prepared a them in a state of `Leave’ versus `Remain’ with a slight preference for the former. From a quantum mechanical perspective, however, you can further argue that the General Election prepared the voters in such a way that should have erased memories of their vote in the referendum so the previous BrExit vote is now invalid.

There’s only one way to test this quantum-mechanical interpretation of voting patterns, and that is by repeating the EU Referendum…

Follow @telescoper## Wave Mechanics and Large-scale Structure

Posted in Books, Talks and Reviews, The Universe and Stuff with tags Cosmology, large-scale structure of the Universe, Madelung Transformation, Quantum Mechanics on May 24, 2017 by telescoperI thought I’d share the slides I used for the short talk I gave last Thursday at the Osservatorio Astronomico di Bologna, on the topic of *Wave Mechanics and Large-scale Structure*. I’ve posted about the general idea underpinning this workhere, and here are some links to references with more details of the cosmological setting, including a couple of papers by myself and Chris Short on some of whose old slides I based the talk.

http://adsabs.harvard.edu/abs/1993ApJ…416L..71W

http://adsabs.harvard.edu/abs/1997PhRvD..55.5997W

http://adsabs.harvard.edu/abs/2002MNRAS.330..421C

http://adsabs.harvard.edu/abs/2003MNRAS.342..176C

http://adsabs.harvard.edu/abs/2006JCAP…12..012S

http://adsabs.harvard.edu/abs/2006JCAP…12..016S

http://adsabs.harvard.edu/abs/2010MNRAS.402.2491J

I had a few problems with the movies during the actual talk, and they probably don’t work in this embedded version. There are a few formatting errors in the slideshare version too, but hopefully you can figure out what’s going on!

Follow @telescoper## Reflections on Quantum Backflow

Posted in Cute Problems, The Universe and Stuff with tags Gandalf Lechner, probability current, Quantum Backflow, Quantum Mechanics, Schrodinger Equation on November 10, 2016 by telescoperYesterday afternoon I attended a very interesting physics seminar by the splendidly-named Gandalf Lechner of the School of Mathematics here at Cardiff University. The topic was one I’d never thought about before, called *quantum backflow. *I went to the talk because I was intrigued by the abstract which had been circulated previously by email, the first part of which reads:

Suppose you are standing at a bus stop in the hope of catching a bus, but are unsure if the bus has passed the stop already. In that situation, common sense tells you that the longer you have to wait, the more likely it is that the bus has not passed the stop already. While this common sense intuition is perfectly accurate if you are waiting for a classical bus, waiting for a quantum bus is quite different: For a quantum bus, the probability of finding it to your left on measuring its position may increase with time, although the bus is moving from left to right with certainty. This peculiar quantum effect is known as backflow.

To be a little more precise about this, imagine you are standing at the origin (*x=0*). In the classical version of the situation you know that the bus is moving with some constant definite (but unknown) positive velocity *v. *In other words you know that it is moving from left to right, but you don’t know with what speed *v* or at what time *t _{0}* or from what position (

*x*) it set out. A little thought, (perhaps with the aid of some toy examples where you assign a probability distribution to

_{0}<0*v, t*and

_{0}*x*) will convince you that the resulting probability distribution for

_{0}*x*moves from left to right with time in such a way that the probability of the bus still being to the left of the observer,

*L(t),*represented by the proportion of the overall distribution that lies at

*x<0*generally

*decreases*with time. Note that this is not what it says in the second sentence of the abstract; no doubt a deliberate mistake was put in to test the reader!

If we then stretch our imagination and suppose that the bus is not described by classical mechanics but by quantum mechanics then things change a bit. If we insist that it is travelling from left to right then that means that the momentum-space representation of the wave function must be cut off for *p<0* (corresponding to negative velocities). Assume that the bus is a “free particle” described by the relevant Schrödinger equation.One can then calculate the evolution of the position-space wave function. Remember that these two representations of the wave function are just related by a Fourier transform. Solving the Schrödinger equation for the time evolution of the spatial wave function (with appropriately-chosen initial conditions) allows one to calculate how the probability of finding the particle at a given value of *x *evolves with time. In contrast to the classical case, it is possible for the corresponding *L(t) *does not always decrease with time.

To put all this another way, the probability current in the classical case is always directed from left to right, but in the quantum case that isn’t necessarily true. One can see how this happens by thinking about what the wave function actually looks like: an imposed cutoff in momentum can imply a spatial wave function that is rather wiggly which means the probability distribution is wiggly too, but the detailed shape changes with time. As these wiggles pass the origin the area under the probability distribution to the left of the observer can go up as well as down. The particle may be going from left to right, but the associated probability flux can behave in a more complicated fashion, sometimes going in the opposite direction.

Another other way of thinking about it is that the particle velocity corresponds to the *phase velocity* of the wave function but the probability flux is controlled by the *group velocity*…

For a more technical discussion of this phenomenon see this review article. The exact nature of the effect is dependent on the precise form of the initial conditions chosen and there are some quantum systems for which no backflow happens at all. The effect has never been detected experimentally, but a recent paper has suggested that it might be measured. Here is the abstract:

Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity may cause a current flow in the negative direction.

Fascinating!

Follow @telescoper

## Can single-world interpretations of quantum theory be self-consistent?

Posted in The Universe and Stuff with tags Quantum Mechanics, quantum theory on May 4, 2016 by telescoperI saw a provocative-looking paper on the arXiv the other day (by Daniela Frauchiger and Renato Renner) with the title *Single-world interpretations of quantum theory cannot be self-consistent.* No doubting what the authors think!

Here’s the abstract:

According to quantum theory, a measurement may have multiple possible outcomes. Single-world interpretations assert that, nevertheless, only one of them “really” occurs. Here we propose a gedankenexperiment where quantum theory is applied to model an experimenter who herself uses quantum theory. We find that, in such a scenario, no single-world interpretation can be logically consistent. This conclusion extends to deterministic hidden-variable theories, such as Bohmian mechanics, for they impose a single-world interpretation.

Since this is a subject we’ve had interesting debates about on this blog I thought I’d post a link to it here and see if anyone would like to respond through the comments. I haven’t had time to read it thoroughly yet, but I do have a bit of train travel to do tomorrow…

Follow @telescoper

## Do Primordial Fluctuations have a Quantum Origin?

Posted in The Universe and Stuff with tags Cosmology, Inflation, Perturbations, Primordial Density Fluctuations, Quantum Mechanics, quantum theory, the early Universe on October 21, 2015 by telescoperA quick lunchtime post containing a confession and a question, both inspired by an interesting paper I found recently on the arXiv with the abstract:

We investigate the quantumness of primordial cosmological fluctuations and its detectability. The quantum discord of inflationary perturbations is calculated for an arbitrary splitting of the system, and shown to be very large on super-Hubble scales. This entails the presence of large quantum correlations, due to the entangled production of particles with opposite momentums during inflation. To determine how this is reflected at the observational level, we study whether quantum correlators can be reproduced by a non-discordant state, i.e. a state with vanishing discord that contains classical correlations only. We demonstrate that this can be done for the power spectrum, the price to pay being twofold: first, large errors in other two-point correlation functions, that cannot however be detected since hidden in the decaying mode; second, the presence of intrinsic non-Gaussianity the detectability of which remains to be determined but which could possibly rule out a non-discordant description of the Cosmic Microwave Background. If one abandons the idea that perturbations should be modeled by Quantum Mechanics and wants to use a classical stochastic formalism instead, we show that any two-point correlators on super-Hubble scales can exactly be reproduced regardless of the squeezing of the system. The later becomes important only for higher order correlation functions, that can be accurately reproduced only in the strong squeezing regime.

I won’t comment on the use of the word “quantumness” nor the plural “momentums”….

My confession is that I’ve never really followed the logic that connects the appearance of classical fluctuations to the quantum description of fields in models of the early Universe. People have pointed me to papers that claim to spell this out, but they all seem to miss the important business of what it means to “become classical” in the cosmological setting. My question, therefore, is can anyone please point me to a book or a paper that addresses this issue rigorously?

Please let me know through the comments box, which you can also use to comment on the paper itself…

Follow @telescoper## Quantum Madness

Posted in The Universe and Stuff with tags Interpretations of Quantum Mechanics, Physics, Quantum Mechanics on September 18, 2015 by telescoperA very busy day lies in store so I only have time for a quick morning visit to the blog. If you enjoyed the recent guest post on the “hidden variables” interpretation of Quantum Mechanics, then you will probably enjoy reading a paper that recently appeared on the arXiv with the abstract:

Motivated by some recent news, a journalist asks a group of physicists: “What’s the meaning of the violation of Bell’s inequality?” One physicist answers: “It means that non-locality is an established fact”. Another says: “There is no non-locality; the message is that measurement outcomes are irreducibly random”. A third one says: “It cannot be answered simply on purely physical grounds, the answer requires an act of metaphysical judgement”. Puzzled by the answers, the journalist keeps asking questions about quantum theory: “What is teleported in quantum teleportation?” “How does a quantum computer really work?” Shockingly, for each of these questions, the journalist obtains a variety of answers which, in many cases, are mutually exclusive. At the end of the day, the journalist asks: “How do you plan to make progress if, after 90 years of quantum theory, you still don’t know what it means? How can you possibly identify the physical principles of quantum theory or expand quantum theory into gravity if you don’t agree on what quantum theory is about?” Here we argue that it is becoming urgent to solve this too long lasting problem. For that, we point out that the interpretations of quantum theory are, essentially, of two types and that these two types are so radically different that there must be experiments that, when analyzed outside the framework of quantum theory, lead to different empirically testable predictions. Arguably, even if these experiments do not end the discussion, they will add new elements to the list of strange properties that some interpretations must have, therefore they will indirectly support those interpretations that do not need to have all these strange properties.

You can download a PDF of the full paper here. It’s a short piece, but with a very good list of references for further reading.

Follow @telescoper## (Guest Post) – Hidden Variables: Just a Little Shy?

Posted in The Universe and Stuff with tags Anthony Garrett, Bell's Theorem, Quantum Mechanics, Quantum Weirdness, relativity, Socratic Dialogue on August 3, 2015 by telescoperTime for a lengthy and somewhat provocative guest post on the subject of the interpretation of quantum mechanics!

–o–

*Galileo advocated the heliocentric system in a socratic dialogue. Following the lifting of the Copenhagen view that quantum mechanics should not be interpreted, here is a dialogue about a way of looking at it that promotes progress and matches Einstein’s scepticism that God plays dice. It is embarrassing that we can predict properties of the electron to one part in a billion but we cannot predict its motion in an inhomogeneous magnetic field in apparatus nearly 100 years old. It is tragic that nobody is trying to predict it, because the successes of quantum theory in combination with its strangeness and 20th century metaphysics have led us to excuse its shortcomings. The speakers are Neo, a modern physicist who works in a different area, and Nino, a 19th century physicist who went to sleep in 1900 and recently awoke*. – *Anton Garrett*

*Nino*: The ultra-violet catastrophe – what about that? We were stuck with infinity when we integrated the amount of radiation emitted by an object over all wavelengths.

*Neo*: The radiation curve fell off at shorter wavelengths. We explained it with something called quantum theory.

*Nino*: That’s wonderful. Tell me about it.

*Neo*: I will, but there are some situations in which quantum theory doesn’t predict what will happen deterministically – it predicts only the probabilities of the various outcomes that are possible. For example, here is what we call a Stern-Gerlach apparatus, which generates a spatially inhomogeneous magnetic field.^{i} It is placed in a vacuum and atoms of a certain type are shot through it. The outermost electron in each atom will set off either this detector, labelled ‘A’, or that detector, labelled ‘B.’ All the electrons coming out of detector B (say) have identical quantum description, but if we put them through another Stern-Gerlach apparatus oriented differently then some will set off one of the two detectors associated with it, and some will set off the other.

*Nino*: Probabilistic prediction is an improvement on my 19th century physics, which couldn’t predict anything at all about the outcome. I presume that physicists in your era are now looking for a theory that predicts what happens each time you put a particle through successive Stern-Gerlach apparatuses.

*Neo*: Actually we are not. Physicists generally think that quantum theory is the end of the line.

*Nino*: In that case they’ve been hypnotised by it! If quantum mechanics can’t answer where the next electron will go then we should look beyond it and seek a better theory that can. It would give the probabilities generated by quantum theory as averages, conditioned on not controlling the variables of the new theory more finely than quantum mechanics specifies.

*Neo*: They are talked of as ‘hidden variables’ today, often hypothetically. But quantum theory is so strange that you can’t actually talk about which detector the atom goes through.

*Nino*: Nevertheless only one of the detectors goes off. If quantum theory cannot answer which then we should look for a better theory that can. Its variables are manifestly not hidden, for I see their effect very clearly when two systems with identical quantum description behave differently. ‘Hidden variables’ is a loaded name. What you’ve not learned to do is control them. I suggest you call them shy variables.

*Neo*: Those who say quantum theory is the end of the line argue that the universe is not deterministic – genuinely random.

*Nino*: It is our theories which are deterministic or not. ‘Random’ is a word that makes our uncertainty about what a system will do sound like the system itself is uncertain. But how could you ever know that?

*Neo*: Certainly it is problematic to define randomness mathematically. Probability theory is the way to make inference about outcomes when we aren’t certain, and ‘probability’ should mean the same thing in quantum theory as anywhere else. But if you take the hidden variable path then be warned of what we found in the second half of the 20th century. Any hidden variables must be nonlocal.

*Nino*: How is that?

*Neo*: Suppose that the result of a measurement of a variable for a particle is determined by the value of a variable that is internal to the particle – a hidden variable. I am being careful not to say that the particle ‘had’ the value of the variable that was measured, which is a stronger statement. The result of the measurement tells us something about the value of its internal variable. Suppose that this particle is correlated with another – if, for example, the pair had zero combined angular momentum when previously they were in contact, and neither has subsequently interacted with anything else. The correlation now tells you something about the internal variable of the second particle. For situations like this a man called Bell derived an inequality; one cannot be more precise because of the generality about how the internal variables govern the outcome of measurements.^{ii} But Bell’s inequality is violated by observations on many pairs of particles (as correctly predicted by quantum mechanics). The only physical assumption was that the result of a measurement on a particle is determined by the value of a variable internal to it – a locality assumption. So a measurement made on one particle alters what would have been seen if a measurement had been made on one of the other particles, which is the definition of nonlocality. Bell put it differently, but that’s the content of it.^{iii}

*Nino*: Nonlocality is nothing new. It was known as “action at a distance” in Newton’s theory of gravity, several centuries ago.

*Neo*: But gravitational force falls off as the inverse square of distance. Nonlocal influences in Bell-type experiments are heedless of distance, and this has been confirmed experimentally.^{iv}

*Nino*: In that case you’ll need a theory in which influence doesn’t decay with distance.

*Neo*: But if influence doesn’t decay with distance then everything influences everything else. So you can’t consider how a system works in isolation any more – an assumption which physicists depend on.

*Nino*: We should view the fact that it often is possible to make predictions by treating a system as isolated as a constraint on any nonlocal hidden variable theory. It is a very strong constraint, in fact.

*Neo*: An important further detail is that, in deriving Bell’s inequality, there has to be a choice of how to set up each apparatus, so that you can choose what question to ask each particle. For example, you can choose the orientation of each apparatus so as to measure any one component of the angular momentum of each particle.

*Nino*: Then Bell’s analysis can be adapted to verify that two people, who are being interrogated in adjacent rooms from a given list of questions, are in clandestine contact in coordinating their responses, beyond having merely pre-agreed their answers to questions on the list. In that case you have a different channel – if they have sharper ears than their interrogators and can hear through the wall – but the nonlocality follows simply from the data analysis, not the physics of the channel.

*Neo*: In that situation, although the people being interrogated can communicate between the rooms in a way that is hidden from their interrogators, the interrogators in the two rooms cannot exploit this channel to communicate between each other, because the only way they can infer that communication is going on is by getting together to compare their sets of answers. Correspondingly, you cannot use pre-prepared particle pairs to infer the orientation of one detector by varying the orientation of the second detector and looking at the results of particle measurements at that second detector alone. In fact there are general no-signalling theorems associated with the quantum level of description.^{v} There are also more striking verifications of nonlocality using correlated particle pairs,^{vi} and with trios of correlated particles.^{vii}

*Nino*: Again you can apply the analysis to test for clandestine contact between persons interrogated in separate rooms. Let me explain why I would always search for the physics of the communication channel between the particles, the hidden variables. In my century we saw that tiny particles suspended in water, just visible under our microscopes, jiggle around. We were spurred to find the reason – the particles were being jostled by smaller ones still, which proved to be the smallest unit you can reach by subdividing matter using chemistry: atoms. Upon the resulting atomic theory you have built quantum mechanics. Since then you haven’t found hidden variables underneath quantum mechanics in nearly 100 years. You suggest they aren’t there to be found but essentially nobody is looking, so that would be a self-fulfilling prophecy. If the non-determinists had been heeded about Brownian motion – and there were some in my time, influenced by philosophers – then the 21st century would still be stuck in the pre-atomic era. If one widget of a production line fails under test but the next widget passes, you wouldn’t say there was no reason; you’d revise your view that the production process was uniform and look for variability in it, so that if you learn how to deal with it you can make consistently good widgets.

*Neo*: But production lines aren’t based on quantum processes!

*Nino*: But I’m not wedded to quantum mechanics! I am making a point of logic, not physics. Quantum mechanics has answered some questions that my generation couldn’t and therefore superseded the theories of my time, so why shouldn’t a later generation than yours supersede quantum mechanics and answer questions that you couldn’t? It is scientific suicide for physicists to refuse to ask a question about the physical world, such as what happens next time I put a particle through successive Stern-Gerlach apparatuses. You say you are a physicist but the vocation of physicists is to seek to improve testable prediction. If you censor or anaesthetise yourself, you’ll be stuck up a dead end.

*Neo*: Not so fast! Nolocality isn’t the only radical thing. The order of measurements in a Bell setup is not Lorentz-invariant, so hidden variables would also have to be acausal – effect preceding cause.

*Nino*: What does ‘Lorentz-invariant’ mean, please?

*Neo*: This term came out of the theory that resolved your problems about aether. Electromagnetic radiation has wave properties but does not need a medium (‘aether’) to ‘do the waving’ – it propagates though a vacuum. And its speed relative to an observer is always the same. That matches intuition, because there is no preferred frame that is defined by a medium. But it has a counter-intuitive consequence, that velocities do not add linearly. If a light wave overtakes me at *c* (lightspeed) then a wave-chasing observer passing me at *v* still experiences the wave overtaking him at *c*, although our familiar linear rule for adding velocities predicts *(c – v)*. That rule is actually an approximation, accurate at speeds much less than c, which turns out to be a universal speed limit. For the speed of light to be constant for co-moving observers then, because speed is distance divided by time, space and time must look different to these observers. In fact even the order of events can look different to two co-moving observers! The transformation rule for space and time is named after a man called Lorentz. That not just the speed of light but all physical laws should look the same for observers related by the Lorentz transformation is called the relativity principle. Its consequences were worked out by a man called Einstein. One of them is that mass is an extremely concentrated form of energy. That’s what fuels the sun.

*Nino*: He was obviously a brilliant physicist!

*Neo*: Yes, although he would have been shocked by Bell’s theorem.^{viii} He asserted that God did not play dice^{ix} – determinism – but he also spoke negatively of nonlocality, as “spooky actions at a distance.” ^{x} Acausality would have shocked him even more. The order of measurements on the two particles in a Bell setup can be different for two co-moving observers. So an observer dashing through the laboratory might see the measurements done in reverse order than the experimenter logs. So at the hidden-variable level we cannot say which particle signals to which as a result of the measurements being made, and the hidden variables must be acausal. Acausality is also implied in ‘delayed choice’ experiments, as follows.^{xi} Light – and, remarkably, all matter – has both particle properties (it can pass through a vacuum) and wave properties (diffraction), but only displays one property at a time. Suppose we decide, after a unit of light has passed a pair of Young’s slits, whether to measure the interference pattern – due to its diffractive properties as a wave propagating through both slits – or its position, which would tell us which single slit it traversed. According to quantum mechanics our choice seems to determine whether it traverses one slit or both, even though we made that choice after it had passed through! Acausality means that you would have to know the future in order to predict it, so this is a limit on prediction – confirming the intuition of quantum theorists that you can’t do better.

*Nino*: That will be so in acausal experimental situations, I accept. I believe the theory of the hidden variables will explain why time, unlike space, passes, and also entail a no-time-paradox condition.

*Neo*: Today we say that a theory must not admit closed time-like trajectories in space-time.

*Nino*: But a working hidden-variable theory would still give a reason why the system behaves as it did, even if we can’t access the information needed for prediction in situations inferred to be acausal. You can learn a lot from evolution equations even if you don’t know the initial conditions. And often the predictions of quantum theory are compatible with locality and causality, and in those situations the hidden variables might predict the outcome of a measurement exactly, outdoing quantum theory.

*Neo*: It also turned out that some elements of the quantum formalism do not correspond to things that can be measured experimentally. That was new in physics and forced physicists to think about interpretation. If differing interpretations give the same testable predictions, how do we choose between them?

*Nino*: Metaphysics then enters and it may differ among physicists, leading to differing schools of interpretation. But non-physical quantities have entered the equations of a theory before. A potential appears in the equations of Newtonian gravity and electromagnetism, but only differences in potential correspond to something physical.

*Neo*: That invariance, greatly generalised, lies behind the ‘gauge’ theories of my era. These are descriptions of further fundamental forces, conforming to the relativity principle that physics must look the same to co-moving observers related by the Lorentz transformation. That includes quantum descriptions, of course.^{xii} It turned out that atoms have their positive charge in a nucleus contributing most of the mass of an atom, which is orbited by much lighter negatively charged particles called electrons – different numbers of electrons for different chemical elements. Further forces must exist to hold the positively charged particles in the nucleus together against their mutual electrical repulsion. These further forces are not familiar in everyday life, so they must fall off with distance much faster than the inverse square law of electromagnetism and gravity. Mass ‘feels’ gravity and charge feels electromagnetic forces, and there are analogues of these properties for the intranuclear forces, which are also felt by other more exotic particles not involved in chemistry. We have a unified quantum description of the intranuclear forces combined with electromagnetism that transforms according to the relativity principle, which we call the standard model, but we have not managed to incorporate gravity yet.

*Nino*: But this is still a quantum theory, still non-deterministic?

*Neo*: Ultimately, yes. But it gives a fit to experiment that is better than one part in a thousand million for certain properties of the electron – which it does predict deterministically.^{xiii} That is the limit of experimental accuracy in my era, and nobody has found an error anywhere else.

*Nino*: That’s magnificent, and it says a huge amount for the progress of experimental physics too. But I still see no reason to move from can-do to can’t-do in aiming to outdo quantum theory.

*Neo*: Let me explain some quantum mechanics.^{xiv} The variables we observe in regular mechanics, such as momentum, correspond to operators in quantum theory. The operators evolve deterministically according to the Hamiltonian of the system; waves are just free-space solutions. When you measure a variable you get one of its eigenvalues, which are real-valued because the operators are Hermitian. Quantum mechanics gives a probability distribution over the eigenspectrum. After the measurement, the system’s quantum description is given by the corresponding eigenfunction. Unless the system was already described by that eigenfunction before the measurement, its quantum description changes. That is a key difference from classical mechanics, in which you can in principle observe a system without disturbing it. Such a change (‘collapse’) makes it impossible to determine simultaneously the values of variables whose operators do not have coincident eigenfunctions – in other words, non-commuting operators. It has even been shown, using commuting subsets of the operators of a system in which members of differing sets do not commute, that simultaneous values of differing variables of a system cannot exist.^{xv}

*Nino*: Does that result rest on any aspect of quantum theory?

*Neo*: Yes. Unlike Bell setups, which compare experiment with a locality criterion, neither of which have anything to do with quantum mechanics (it simply predicts what is seen), this further result is founded in quantum mechanics.

*Nino*: But I’m not committed to quantum mechanics! This result means that the hidden variables aren’t just the values of all the system variables, but comprise something deeper that somehow yields the system variables and is not merely equivalent to the set of them.

*Neo*: Some people suggest that reality is operator-valued and our perplexities arise because of our obstinate insistence on thinking in – and therefore trying to measure – scalars.

*Nino*: An operator is fully specified by its eigenvalues and eigenfunctions; it can be assembled as a sum over them, so if an operator is a real thing then they would be real things too. If a building is real, the bricks it is constructed from are real. But I still insist that, like any other physical theory, quantum theory should be regarded as provisional.

*Neo*: Quantum theory answered questions that earlier physics couldn’t, such as why electrons do not fall into the nucleus of an atom although opposite charges attract. They populate the eigenspectrum of the Hamiltonian for the Coulomb potential, starting at the lowest energy eigenfunction, with not more than two electrons per eigenfunction. When the electrons are disturbed they jump between eigenvalues, so that they cannot fall continuously. This jumping is responsible for atomic spectrum lines, whose vacuum wavelength is inversely proportional to the difference in energy of the eigenvalues. That is why quantum mechanics was accepted. But the difficulty of understanding it led scientists to take a view, championed by a senior physicist at Copenhagen, that quantum mechanics was merely a way of predicting measurements, rather than telling us how things really are.

*Nino*: That distinction is untestable even in classical mechanics. This is really about motivation. If you don’t believe that things ‘out there’ are real then you’ll have no motivation to think about them. The metaphysics beneath physics supposes that there is order in the world and that humans can comprehend it. Those assumptions were general in Europe when modern physics began. They came from the belief that the physical universe had an intelligent creator who put order in it, and that humans could comprehend this order because they had things in common with the creator (‘in his image’). You don’t need a religious faith to enter physics once it has got going and the patterns are made visible for all to see; but if ever the underlying metaphysics again becomes relevant, as it does when elements of the formalism do not correspond to things ‘out there,’ then such views will count. If you believe there is comprehensible and interesting order in the material universe then you will be more motivated to study it than others who suppose that differentiation is illusion and that all is one, i.e. the monist view held by some other cultures. So, in puzzling why people aren’t looking for those not-so-hidden variables, let me ask: Did the view that nature was underpinned by a divine creator get weaker where quantum theory emerged, in Europe, in the era before the Copenhagen view?

*Neo*: Religion was weakening during your time, as you surely noticed. That trend did continue.

*Nino*: I suggest the shift from optimism to defeatism about improving testable prediction is a reflection of that change in metaphysics reaching a tipping point. Culture also affects attitudes; did anything happen that induced pessimism between my era and the birth of quantum mechanics?

*Neo*: The most terrible war in history to that date took place in Europe. But we have moved on from the Copenhagen ‘interpretation’ which was a refusal of all questions about the formalism. That stance is acceptable provided it is seen as provisional, perhaps while the formalism is developed; but not as the last word. Physicists eventually worked out the standard model for the intranuclear forces in combination with electromagnetism. Bell’s theorem also catalysed further exploration of the weirdness of quantum mechanics, from the 1960s; let me explain. Before a measurement of one of its variables, a system is generally not in an eigenstate of the corresponding operator. This means that its quantum description must be written as a superposition of eigenstates. Although measurement discloses a single eigenvalue, remarkable things can be done by exploiting the superposition. We can gain information about whether the triggering mechanisms of light-activated bombs are good or dud triggers in an experiment in which light might be shone at each, according to a quantum mechanism.^{xvi} (This does involve ‘sacrificing’ some of the working bombs, but without the quantum trick you would be completely stuck, because the bomb is booby-trapped to go off if you try to dismantle the trigger.) Even though we have electronic computers today that do millions of operations per second, many calculations are still too large to be done in feasible lengths of time, such as integer factorisation. We can now conceive of quantum computers that exploit the superposition to do many calculations in one, and drastically speed things up.^{xvii} Communications can be made secure in the sense that eavesdropping cannot be concealed, as it would unavoidably change the quantum state of the communication system. The apparent reality of the superposition in quantum mechanics, together with the non-existence of definite values of variables in some circumstances, mean that it is unclear what in the quantum formalism is physical, and what is our knowledge about the physical – in philosophical language, what is ontological and what is epistemological. Some people even suggest that, ultimately, numbers – or at least information quantified as numbers – are physics.

*Nino*: That’s a woeful confusion – information about what? As for deeper explanation, when things get weird you either give up on going further – which no scientist should ever do – or you take the weirdness as a clue. Any no-hidden-variables claim must involve assumptions or axioms, because you can’t prove something is impossible without starting from assumptions. So you should expose and question those assumptions (such as locality and causality). Don’t accept any axioms that are intrinsic to quantum theory, because your aim is to go beyond quantum theory.

*Neo*: Some people, particularly in quantum computing, suggest that when a variable is measured in a situation in which quantum mechanics predicts the result probabilistically, the universe actually splits into many copies, with each of the possible values realised in one copy.^{xviii} We exist in the copy in which the results were as we observed them, but in other universes copies of us saw other results.

*Nino*: We couldn’t observe the other universes, so this is metaphysics, and more fantastic than Jules Verne! What if the spectrum of possible outcomes includes a continuum of eigenvalues? Furthermore a measurement involves an interaction between the measuring apparatus and the system, so the apparatus and system could be considered as a joint system quantum-mechanically. There would be splitting into many worlds if you treat the system as quantum and the apparatus as classical, but no splitting if you treat them jointly as quantum. Nothing privileges a measuring apparatus, so physicists are free to analyse the situation in these two differing ways – but then they disagree about whether splitting has taken place. That’s inconsistent.

*Neo*: The two descriptions must be reconciled. As I said, a system left to itself evolves according to the Hamiltonian of the system. When one of its variables is measured, it undergoes an interaction with an apparatus that makes the measurement. The system finishes in an eigenstate of the operator corresponding to the variable measured, while the apparatus flags the corresponding eigenvalue. This scenario has to be reconciled with a joint quantum description of the system and apparatus, evolving according to their joint Hamiltonian including an interaction term. Reconciliation is needed in order to make contact with scalar values and prevent a regress problem, since the apparatus interacts quantum-mechanically with its immediate surroundings, and so on. Some people propose that the regress is truncated at the consciousness of the observer.

*Nino*: I thought vitalism was discredited once the soul was found to be massless, upon weighing dying creatures! The proposal you mention actually makes the regress problem worse, because if the result of a measurement is communicated to the experimenter via intermediaries who are conscious – who are aware that they pass on the result – then does it count only when it reaches the consciousness of the experimenter (an instant of time that is anyway problematic to define)? If so, why?

*Neo*: That’s a regress problem on the classical side of the chain, whereas I was talking about a regress problem on the quantum side. This suggests that the regress is terminated where the system is declared to have a classical description.^{xix} I fully share your scepticism about the role of consciousness and free will. Human subjects have tried to mentally influence the outcomes of quantum measurements and it is not accepted that they can alter the distribution from the quantum prediction. Some people even propose that consciousness exists because matter itself is conscious and the brain is so complex that this property is manifest. But they never clarify what it means to say that atoms may have consciousness, even of a primitive sort.

*Nino*: Please explain how our regress terminates where we declare something classical.

*Neo*: For any measured eigenvalue of the system there are generally many degrees of freedom in the Hamiltonian of the apparatus, so that the density of states of the apparatus is high. (This is true even if the quantum states are physically large, as in low temperature quantum phenomena such as superconductivity.) Consider the apparatus variable that flags the result of the measurement. In the sum over states giving the expectation value of this variable, cross terms between quantum states of the apparatus corresponding to different eigenvalues of the system are very numerous. These cross terms are not generally correlated in amplitude or phase, so that they average out in the expectation value in accordance with the law of large numbers.^{xx} Even if that is not the case they are usually washed out by interactions with the environment, because you cannot in practice isolate a system perfectly. This is called decoherence,^{xxi} and nonlocality and those striking quantum-computer effects can only be seen when you prevent it.

*Nino*: Remarkable! But you still have only statistical prediction of which eigenvalue is observed.

*Neo*: Your deterministic viewpoint has been disparaged by some as an outmoded, clockwork view of the universe.

*Nino*: Just because I insist on asking where the next particle will go in a Stern-Gerlach apparatus? Determinism is a metaphysical assumption; no more or less. It inspires progress in physics, which any physicist should support. Let me return to nonlocality and acausality (which is a kind of directional nonlocality with respect to time, rather than space). They imply that the physical universe is an indivisible whole at the fundamental level of the hidden variables. That is monist, but is distinct from religious monism because genuine structure exists in the hidden – or rather shy – variables.

*Neo*: Certainly space and time seem to be irrelevant to the hidden interactions between particles that follow from Bell tests. As I said, we have a successful quantum description of electromagnetic interactions and have combined it with the forces that hold the atomic nucleus together. In this description we regard the electromagnetic field itself as an operator-valued variable, according to the prescription of quantum theory. The next step would be to incorporate gravity. That would not be Newtonian gravity, which cannot be right because, unlike Maxwell’s equations, it only looks the same to co-moving observers who are related by the Galilean transform of space and time – itself only a low-speed approximation to the correct Lorentz transform. Einstein found a theory of gravity that transforms correctly, known as general relativity, and to which Newton’s theory is an approximation. Einstein’s view was that space and time were related to gravity differently than to the other forces, but a theory that is almost equivalent to his (predicting identically in all tests to date) has since emerged that is similar to electromagnetism – a gauge theory in which the field is coupled naturally to matter which is described quantum-mechanically.^{xxii} Unlike electromagnetism, however, the gravitational field itself has not yet been successfully quantised, hindering the marrying of it to other forces so as to unify them all. Of course we demand a theory that takes account of both quantum effects and relativistic gravity, for any theory that neglects quantum effects becomes increasingly inaccurate where these are significant – typically on the small scale inside atoms – while any theory that neglects relativistic gravitational effects becomes increasingly inaccurate where they are significant – typically on large scales where matter is gathered into dense massive astronomical bodies. Not even light can escape from some of these bodies – and, because the speed of light is a universal speed limit, nor can anything else. Quantum and gravitational effects are both large if you look at the universe far enough back in time, because we have learned that the universe was once very small and very dense. So a complete theory is indispensable for cosmologists who seek to study origins. The preferred program for quantum gravity today is known as string theory. But it has a deeply complicated structure and is infeasible to test experimentally, rendering progress slow.

*Nino*: But it’s still not a complete theory if it’s a quantum theory. Please say more about that very small dense stage of the universe which presumably expanded to give what we see today.

*Neo*: We believe the early part of the expansion underwent a great boost, known as inflation, which explains how the universe is unexpectedly smooth on the largest scale today and is also not dominated by gravity. Everything in the observed universe was, in effect, enormously diluted. Issues of causality also arise. But the mechanism for inflation is conjectural, and inflation raises other questions.

*Nino*: Unexpected large-scale smoothness sounds to me like a manifestation of nonlocality. Furthermore the hidden variables are acausal. Perhaps you cannot do without them at such extreme densities and temperatures. Then you wouldn’t need to invoke inflation.

*Neo*: We believe that inflation took place after the ‘Planck’ era in which a full theory of quantum theory of gravity is indispensible for accuracy. In that case our present understanding is adequate to describe the inflationary epoch.

*Nino*: You are considering the entire universe, yet you cannot predict which detector goes off next when consecutive particles having identical quantum description are fired through a Stern-Gerlach apparatus. Perhaps you should walk before you run. Then your problems in unifying the fundamental forces and applying the resulting theory to the entire universe might vanish.

*Neo*: That’s ironic – the older generation exhorting the younger to revolution! To finish, what would you say to my generation of physicists?

*Nino*: It is magnificent that you can predict properties of the electron to nine decimal places, but that makes it more embarrassing that you cannot tell something as basic as which way a silver atom will pass through an inhomogeneous magnetic field, according to its outermost electron. That incapability should be an itch inside your oyster shell. Seek a theory which predicts the outcome when systems having identical quantum specification behave differently. Regard all strange outworkings of quantum mechanics as information about the hidden variables. Purported no-hidden-variables theorems that are consistent with quantum mechanics must contain extra assumptions or axioms, so put such theorems to work for you by ensuring that your research violates those assumptions. Ponder how to reconcile the success of much prediction upon treating systems as isolated with the nonlocality and acausality visible in Bell tests. Don’t let anything put you off because, barring a lucky experimental anomaly, only seekers find. By doing that you become part of a great project.

*Anthony Garrett has a PhD in physics (Cambridge University, 1984) and has held postdoctoral research contracts in the physics departments of Cambridge, Sydney and Glasgow Universities. He is Managing Editor of Scitext Cambridge (www.scitext.com), an editing service for scientific documents.*

^{i} Gerlach, W. & Stern, O., “Das magnetische Moment des Silberatoms”, Zeitschrift für Physik 9, 353-355 (1922).

^{ii} Bell, J.S., “On the Einstein Podolsky Rosen paradox”, Physics 1, 195-200 (1964).

^{iii}Garrett, A.J.M., “Bell’s theorem and Bayes’ theorem”, Foundations of Physics 20, 1475-1512 (1990).

^{iv} The most rigorous test of Bell’s theorem to date is: Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S.-W., Ursin R. & Zeilinger, A., “Bell violation using entangled photons without the fair-sampling assumption”, Nature 497, 227-230 (2013). For a test of the 3-particle case, see: Bouwmeester, D., Pan, J.-W., Daniell, M., Weinfurter, H. & Zeilinger, A., “Observation of three-photon Greenberger-Horne-Zeilinger entanglement”, Physical Review Letters 82, 1345-1349 (1999).

^{v} Bussey, P.J., “Communication and non-communication in Einstein-Rosen experiments”, Physics Letters A123, 1-3 (1987).

^{vi}Mermin, N.D., “Quantum mysteries refined”, American Journal of Physics 62, 880-887 (1994). This is a very clear tutorial recasting of: Hardy, L., “Nonlocality for two particles without inequalities for almost all entangled states”, Physical Review Letters 71, 1665-1668 (1993).

^{vii} Mermin, N.D., “Quantum mysteries revisited”, American Journal of Physics 58, 731-734 (1990). This is a tutorial recasting of the ‘GHZ’ analysis: Greenberger, D.M., Horne, M.A. & Zeilinger, A., 1989, “Going beyond Bell’s theorem”, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. M. Kafatos (Kluwer Academic, Dordrecht, Netherlands), p.69-72.

^{viii} Einstein, A., Podolsky, B. & Rosen, N., “Can quantum-mechanical description of physical reality be considered complete?”, Physical Review 47, 777-780 (1935).

^{ix}Einstein, A., Letter to Max Born, 4th December 1926. English translation in: The Born-Einstein Letters 1916-1955 (MacMillan Press, Basingstoke, UK), 2005, p.88.

^{x} Einstein, A., Letter to Max Born, 3rd March 1947. Ibid., p.155.

^{xi}Wheeler, J.A., 1978, “The ‘past’ and the ‘delayed-choice’ double-slit experiment”, in Mathematical Foundations of Quantum Theory, ed. A.R. Marlow (Academic Press, New York, USA), p.9-48. Experimental verification: Jacques, V., Wu, E., Grosshans, F., Treusshart, F., Grangier, P., Aspect, A. & Roch, J.-F., “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment”, Science 315, 966-968 (2007).

^{xii} Weinberg, S., 2005, The Quantum Theory of Fields, vols. 1-3 (Cambridge University Press, UK).

Hanneke, D., Fogwell, S. & Gabrielse, G., “New measurement of the electron magnetic moment and the fine-structure constant”, Physical Review Letters 100, 120801 (2008); 4pp.

^{xiii} Dirac, P.A.M., 1958, The Principles of Quantum Mechanics (4th ed., Oxford University Press, UK).

^{xiv} Mermin, N.D., “Simple unified form for the major no-hidden-variables theorems”, Physical Review Letters 65, 3373-3376 (1990); Mermin, N.D., “Hidden variables and the two theorems of John Bell”, Reviews of Modern Physics 65, 803-815 (1993). This is a simpler version of the ‘Kochen-Specker’ analysis: Kochen, S. &

^{xv}Specker, E.P., “The problem of hidden variables in quantum mechanics”, Journal of Mathematics and Mechanics, 17, 59-87 (1967).

^{xvi} Elitzur, A.C. & Vaidman, L., “Quantum mechanical interaction-free measurements”, Foundations of Physics 23, 987-997 (1993).

^{xvii} Mermin, N.D., 2007, Quantum Computer Science (Cambridge University Press, UK).

^{xviii} DeWitt, B.S. & Graham, N. (eds.), 1973, The Many-Worlds Interpretation of Quantum Mechanics (Princeton University Press, New Jersey, USA). The idea is due to Hugh Everett III, whose work is reproduced in this book.

^{xix}This immediately resolves the well known Schrödinger’s cat paradox.

^{xx}Van Kampen, N.G., “Ten theorems about quantum mechanical measurements”, Physica A153, 97-113 (1988).

^{xxi} Zurek, W.H., “Decoherence and the transition from quantum to classical”, Physics Today 44, 36-44 (1991).

^{xxii} Lasenby, A., Doran, C. & Gull, S., “Gravity, gauge theories and geometric algebra”, Philosophical Transactions of the Royal Society A356, 487-582 (1998). This paper derives and studies gravity as a gauge theory using the mathematical language of Clifford algebra, which is the extension of complex analysis to higher dimensions than 2. Just as complex analysis is more efficient than vector analysis in 2 dimensions, Clifford algebra is superior to conventional vector/tensor analysis in higher dimensions. (Quaternions are the 3-dimensional version, a generalisation that Nino would doubtless appreciate.) Nobody uses the Roman numeral system any more for long division! This theory of gravity involves two fields that obey first-order differential equations with respect to space and time, in contrast to general relativity in which the metric tensor obeys second-order equations. These gauge fields derive from translational and rotational invariance and can be expressed with reference to a flat background spacetime (i.e., whenever coordinates are needed they can be Cartesian or some convenient transformation thereof). Presumably it is these two gauge fields, rather than the metric tensor, that should satisfy quantum (non-)commutation relations in a quantum theory of gravity.