Jacques Pienaar’s guide to making physics (Pt.1)

PRINCIPLES AS TOOLS
(Not to be confused with using Principals as tools, which is what happens if your school Principal is a tool because he never taught you the difference between a Principal and a principle. Also not to be confused with a Princey-pal, who is a friend that happens to be a Prince).

`These principles are the boldly generalized results of experiment; but they appear to derive from their very generality a high degree of certainty. In fact, the greater the generality, the more frequent are the opportunities for verifying them, and such verifications, as they multiply, as they take the most varied and most unexpected forms, leave in the end no room for doubt.’ -Poincaré

 
One of the great things Einstein did, besides doing physics, was trying to explain to people how to do it as good as him. Ultimately he failed, because so far nobody has managed to do better than him, but he left us with some really interesting insights into how to come up with new physical theories.

One of these ideas is the concept of using `principles’. A principle is a statement about how the word works (or should work), stated in ordinary language. They are not always called principles, but might be called laws, postulates or hypotheses. I am not going to argue about semantics here. Just consider these examples to get a flavour:

The Second Law of Thermodynamics: You can’t build an engine which does useful work and ends up back in its starting position without producing any heat.

 
Landauer’s principle: you can’t erase information without producing heat.

 
The Principle of Relativity: It is impossible to tell by local experiments whether or not your laboratory is moving.

And some not strictly physics ones:

Shirky’s law: Institutions will try to preserve the problem to which they are the solution.

 
Murphy’s law: If something can go wrong, it will go wrong.

Stigler’s law: No scientific discovery is named after its original discoverer (this law was actually discovered by R.K. Merton, not Stigler).

 
Parkinson’s law: Work always expands to fill up the time allocated to doing it.
(See Wikipedia’s list of eponymous laws for more).

You’ll notice that principles are characterised by two main things: they ring true, and they are vague. Both of these properties are very important for their use in building theories.

Now I can practically hear the lice falling out as you scratch your head in confusion. “But Jacques! How can vagueness be a useful thing to have in a Principle? Shouldn’t it be made as precise as possible?”

No, doofus. A Principle is like an apple. You know what an apple is right?

hipstercat

Well, you think you do. But if I were to ask you, what colour is an apple, how sweet is an apple, how many worms are in an apple, you would have to admit that you don’t know, because the word “apple” is too vague to answer those questions. It is like asking how long is a piece of string. Nevertheless, when you want to go shopping, it suffices to say “buy me an apple” instead of “buy me a Malus domestica, reflective in the 620-750 nanometer range, ten percent sugar, one percent cydia pomonella“.

The only way to make a principle more precise is within the context of a precise theory. But then how would I build a new theory, if I am stuck using the language of the old theory? I can make the idea of an apple more precise using the various scientifically verified properties that apples are known to have, but all of that stuff had to come after we already had a basic vague understanding of what an “apple” was, e.g. a kind of round-ish thing on a tree that tastes nice when you eat it.

The vagueness of a principle means that it defines a whole family of possible theories, these being the ones that kind of fit with the principle if you take the right interpretation. On one hand, a principle that is too vague will not help you to make progress, because it will be too easy to make it fit with any future theory; on the other hand, a principle that is not vague enough will leave you stuck for choices and unable to progress.

The next aspect of a good principle is that it “rings true”. In other words, there is something about it that makes you want it to be true. We want our physical theories to be intuitive to our soft, human brains, and these brains of ours have evolved to think about the world in very specific terms. Why do you think physics seems to be all about the locations of objects in space, moving with time? There are infinitely many ways to describe physics, but we choose the ones we do because of the way our physical senses work, the way our bodies interact with the world, and the things we needed to do in order to survive up to this point. What is the principle of least action? It is a river flowing down a mountain. What is Newtonian mechanics? It is animals moving on the plains. We humans need to see the world in a special way in order to understand it, and good principles are what allow us to shoehorn abstract concepts like thermodynamics and gravitational physics into a picture that looks familiar to us, that we can work with.

That’s why a good principle has to ring true — it has to appeal to the limited imaginative abilities of us humans. Maybe if we were different animals, the laws of physics would be understood in very different terms. Like, the Newtonian mechanics of snakes would start with a simple model of objects moving along snake-paths in two dimensions (the ground), and then go from there to arbitrary motions and higher dimensions. So intelligent snakes might have discovered Fourier analysis way before humans would have, just because they would have been more used to thinking in wavy motions instead of linear motions.

Plissken

So you see, coming up with good principles is really an art form, that requires you to be deeply in touch with your own humanity. Indeed, principle-finding is part of the great art of generating hypotheses. It is a pity that many scientists don’t practice hypothesis generation enough to realise that it is an art (or maybe they don’t practice art enough?) It is also ironic that science tries so hard to eliminate the human element from the theories, when it is so apparent in the final forms of the theories themselves. It is just like an artist who trains so hard to hide her brush strokes, to make the signature of her hand invisible, even though the subject of the painting is her own face.

Ok, now that we know what principles are, how do we find them? One of the best ways is by the age-old method of Induction. How does induction work? It really deserves its own post, but here it is in a nutshell. Let’s say that you are a turkey, and you observe that whenever the farmer makes a whistle, there is some corn in your bowl. So, being a smart turkey, you might decide to elevate this empirical pattern to a general principle, called the Turkey Principle: whenever the farmer whistles, there is corn in your bowl. BOOM, induction!

Now, what is the use of this principle? It helps you to narrow down which theories are good and which are bad. Suppose one day the farmer whistles but you discover there is not corn in the bowl, but rather rice. With your limited turkey imagination, you are able to come up with three hypotheses to explain this. 1. There was corn in the bowl when the farmer whistled, but then somebody came along and replaced it with rice; 2. the Turkey Principle should be amended to the Weak Turkey Principle, which states that when the farmer whistles, food, but not necessarily corn, will be in the bowl; 3. the contents of the bowl are actually independent of the farmer’s whistling, and the apparent link between these phenomena is just a coincidence. Now, with the aid of the Principle, we can see that there is a clear preference for hypothesis 1 over 2, and for 2 over 3, according to the extent that each hypothesis fits with the Turkey Principle.

This example makes it clear that deciding which patterns to upgrade to general principles, and which to regard as anomalies, is again a question of aesthetics and artistry. A more perceptive turkey might observe that the farmer is not a simple mechanistic process, but a complex and mysterious system, and therefore may not be subject to such strong constraints with regards to his whistling and corn-giving behaviour as are implied by the Turkey Principle. Indeed, were the turkey perceptive enough to guess at the farmer’s true motives, he might start checking the tool shed to see if the axe is missing before running to the food bowl every time the farmer whistles. But this turkey would no doubt be working on hypotheses of his own, motivated by principles of his own, such as the Farmer-is-Not-to-be-Trusted Principle (in connection with the observed correlation of turkey disappearances and family dinner parties).

An example more relevant to physics is Einstein’s Equivalence Principle: that no local experiment can determine whether the laboratory is in motion, or is stationary in a gravitational field. The principle is vague, as you can see by the number of variations, interpretations, and Weak and Strong versions that exist in the literature; but undoubtedly it rings true, since it appears to be widely obeyed all but the most esoteric phenomena, and it gels nicely with the Principle of Relativity. While the Equivalence Principle was instrumental in leading to General Relativity, it is a matter of debate how it should be formulated within the theory, and whether or not it is even true. Much like hammers and saws are needed to make a table, but are not needed after the table is complete, we use principles to make theories and then we set them aside when the theory is complete. The final theory makes predictions perfectly well without needing to refer to the principles that built it, and the principles are too vague to make good predictions on their own. (Sure, with enough fiddling around, you can sit on a hammer and eat food off a saw, but it isn’t really comfortable or easy).

For more intellectual reading on principle theories, see the SEP entry on Einstein’s Philosophy of Science, and Poincare’s excellent notes.

Wigner has no friends in space

The title phrase of this post is taken from an article by Seth Lloyd that appeared on today’s arXiv, entitled “Analysis of a work of quantum art“. Lloyd was talking about an artwork in collaboration with artist Diemut Strebe, called `Wigner’s friends‘ in which a pair of telescopes are separated, one remaining on Earth and the other going to the International Space Station. According to Lloyd, Strebe motivates the work by appealing to the concepts of quantum superposition and entanglement, referring to physicist Eugene Wigner’s famous thought experiment in which one experimenter, Wigner’s friend, finds herself in a superposition prior to Wigner’s measurement. In Strebe’s scenario, both telescopes are aimed at interstellar space, and it is the viewers of the exhibition that are held responsible for collapsing the superposition of the orbiting telescope by observing the image on the ground-based telescope. The idea is that, since there is nobody looking at the orbiting telescope, the image on its CCD array initially exists in a quantum superposition of all possible artworks; hence Wigner has no friends in space. Before I discuss this intriguing work, let me first start a new art movement.

I was doing my PhD at the University of Queensland when my friend Aggie (also a PhD at that time) came to me with an intriguing problem. She needed to integrate a function over a certain region of three-dimensional space. This region could be obtained by slicing corners off a cube in a certain way, but Aggie was finding it impossible to visualize what the resulting shape would look like. Even after doing a 3D plot in Mathematica, she felt that there was something missing from the flattened projections that one had to click-and-drag to rotate. She wanted to know if I’d ever seen this shape before, and if I could maybe draw it for her or make one out of paper and glue (Weirdly, I have always had an undeserved reputation for drawing and origami). I did my best with paper and sticky-tape, but it didn’t quite come out right, so I gave up. In the end, she went and bought some plasticine and made a cube, then cut off the corners until she got the shape she wanted. Now that she could hold it in her hands, she finally felt that she understood just what she was dealing with. She went back to her computer to perform the integration.

At the time, it did not occur to me to ask “Is it art?” While its form was elegant, it was there to serve a practical purpose, namely to help Aggie (who probably did not once suspect that she was doing Art) in her calculation by condensing certain abstract ideas into a concrete form.

Soft Cube
© Malcolm Wright

Disclaimer: Before continuing, please note that I reject the idea that there can be a universal definition of Art. I further reject the (often claimed) corollary that therefore anything and everything can be Art. Instead, I posit that there are many different Arts, and just like living species, they are continually springing into existence, evolving into new forms, and going extinct. Just as a discussion about “what is a species” can lead to interminable and never-ending arguments, I posit that it is much better and more constructive to discuss “what is a lion”? Here, I am going to talk about, and attempt to define, something that might be called Science-Art, Technologism, Scientism, or something like that. Let’s go with `Zappism’, because it reminds me of things that supposedly go `zap’, but really don’t, like lasers.

So what is Zappism? Let me give some examples of what it is and what it is not. Every now and then, there are Art in Science exhibitions where academic researchers submit images of pretty things that they encountered in the course of their research. I include in this category colourful images of fractals, decorated graphs of pretty mathematical functions, astrophysical images of planets and stars and things, and basically anything where a scientist was just mucking around and noticed something beautiful and then made it into a graphic. For this stuff I would suggest the name “Scientific Found Art”, but it is not Zappism.

© Jonathan McCabe. An example of scientific found art.

Aggie’s shape might seem at first to fit the bill of found art, but there is a crucial difference: were the shape not pretty, it still would have served its purpose, which was to explore, in material form, scientific ideas that would otherwise have been elusive and abstract. A computer simulation of a fractal does not serve this purpose unless one also comes to understand the fractal better as a consequence of the simulation, and I’m not convinced this is true any more than one can understand a sentence better by writing it out in binary and then colouring it in.

Zappism is the art of using some kind of medium — be it painting, film, music, literature or something else — and using it to transform some ethereal and ungraspable Platonisms of science into things the human mind can more readily play with. Sometimes something is lost in translation, like adding unscientific `zap’ sounds to lasers, but this is acceptable as long as the core idea is translated — in the case of lasers, the idea that light can be focused into beams that can burn through things.

Many episodes of Star Trek exhibit Zappism. In the episode `Tuvix‘, the transporter merges two crew members into a single person, an incident that is explicitly explained by appealing to the way the transporter recombines matter. Similarly, Cronenberg’s film The Fly is classic Zappism, as is Spielberg’s Jurassic Park. Indeed, almost any science fiction that uses science in an active way almost can’t help but be Zappist. Science fiction can still fail to be Zappist if it uses the science as a kind of gloss or sugar-coating, instead of engaging with the science as a main ingredient. Star Wars is not really Zappist because it is not concerned with the mechanisms of the technology invoked. Luke and Darth might as well be using swords and riding on flying horses for all the story cares, making it is more like Science Fantasy (Why do lightsabers simply stop at a convenient sword-length?)

A science fiction movie can always ignore inconvenient facts, like conservation of momentum, or how there is no sound in space. These annoying truths are often seen as getting in the way of good action and drama. The truth is the opposite: it takes a creative leap of genius to see how to use these facts to the advantage of dramatic effects. The recent film Coherence does a brilliant job of using the idea of Schrodinger’s Cat to create a tense and frightening scenario. When film, art and storytelling are able to incorporate physical law in a natural and graspable way, we are one step closer to connecting the public to cutting-edge science.

Screen Shot 2015-01-08 at 9.57.10 PM
Actress Emily Baldoni grapples with Schrödinger’s equation in Coherence.

On the non-cinematic side, Koen Vanmechelen’s breeding program for cosmopolitan chickens, Maguire and collaborator’s epic project `Dr. Brainlove‘, and Theo Jansen’s Strandbeest could all be called examples of Zappism. But perhaps the most revealing examples are those that do not explicitly use physical technology for the scientific motive, but instead use abstract ideas. For these I cite Dali’s Persistence of Memory (and its Disintegration) with their roots in Relativity theory and Quantum Mechanics; the book Flatland by Edwin Abbott; Alice in Wonderland by Carroll; Gödel, Escher, Bach: An Eternal Golden Braid by Hofstadter, and similar books that bring abstract scientific or mathematical ideas into an imaginable form. A truly great work of Zappism was the invention of the Rubik’s Cube, by the Hungarian sculptor and mathematician Erno Rubik. Rubik conceived the cube as a solution to a more abstract structural design problem of how to rotate the parts of a cube in all three dimensions while keeping the parts connected.

Returning now to Strebe’s artwork `Wigner’s friends’, it should be remarked that the artwork is not a scientific experiment and there is no actual demonstration of quantum coherence between the telescopes. However, Seth Lloyd for some reason seems intent on defending the idea that maybe, just maybe, there is some tiny smidgen of possibility that there is something quantum going on in the experiment. I understand his enthusiasm: I also think it is a very cool artwork, and somehow the whole point of the artwork is its reference to quantum mechanics. But in order to plausibly say that something quantum was really going on in Strebe’s artwork, Lloyd is forced to invoke the Many Worlds interpretation, which to me is tantamount to begging the question — under that assumption isn’t my cheese sandwich also in a quantum superposition?

I don’t see why all this is necessary: when Dali painted the Disintegration of the Persistence of Memory, nobody was scrambling to argue that his oil paint was in a quantum superposition on the canvas. It would be just as absurd as insisting that Da Vinci’s portrait of the Mona Lisa actually contained a real person. There is a sense in which the artistic representation of a person is bound to physics — it is constrained to some extent by the way physical masses compose in three dimensional space — but the art of correct representation is not to be confused with the real thing. Even Mondrian, whose works were famously highly abstract, insisted that he was bound to the true representation of Nature as he saw it [1]. To me, Strebe’s artwork is a representation of quantum mechanics, put into a physical and graspable form, and that is what makes it Zappism. But is it good Zappism? That depends on whether the audience feels any closer to understanding quantum mechanics after the experience.

[1] “The masses generally find my work rather vague. I construct lines and color combinations on a flat surface, in order to express general beauty with the utmost awareness. Nature (or that which I see) inspires me . . . but I want to come as close as possible to the truth…” Source: http://www.comesaunter.com/2012/02/piet-mondrian-on-his-art.html

Ten Rules for Research

I see a lot of articles out there giving advice in the form of a list of rules. People have a fascination with rule lists. You’ve got the rules of Fight Club, the writer who uses a personal formula, policemen who follow “The Book” to the letter, gangsters with a personal code of ethics, and so on. So here’s my list of rules for being a scientist.

1. Keep reading everything.

2. The value of public speaking skills cannot be underestimated.

3. Remember the big questions that got you here in the first place.

4. Take philosophy seriously, but only the parts you can understand.

5. Sometimes, you just have to shut up and calculate.

6. Don’t distract yourself from the things you don’t know by working on things you do know.

7. The best defense against politics is integrity and a smile.

8. The more certain you are of a result, the more you should double check it.

9. If you aren’t curious to know the result of a calculation, it isn’t worth doing it.

10.  Ask dumb questions. If you are truly an idiot, you’ll be found out eventually, so you might as well satisfy your curiosity in the meantime.

In the end, I think Rule 1 is most important.  So, you should go and read Michael Nielsen’s classic advice to researchers, which is far more eloquent than the garbage you read on my blog.

Calvin and Hobbes
© 2013 Bill Watterson

Time-travel, decoherence, and satellites.

I recently returned to my roots, contributing to a new paper with Tim Ralph (who was my PhD advisor) on the very same topic that formed a major part of my PhD. Out of laziness, let me dig up the relevant information from an earlier post:

“The idea for my PhD thesis comes from a paper that I stumbled across as an undergraduate at the University of Melbourne. That paper, by Tim Ralph, Gerard Milburn and Tony Downes of the University of Queensland, proposed that Earth’s own gravitational field might be strong enough to cause quantum gravity effects in experiments done on satellites. In particular, the difference between the strength of gravity at ground-level and at the height of the orbiting satellite might be just enough to make the quantum particles on the satellite behave in a very funny non-linear way, never before seen at ground level. Why might this happen? This is where the story gets bizarre: the authors got their idea after looking at a theory of time-travel, proposed in 1991 by David Deutsch. According to Deutsch’s theory, if space and time were bent enough by gravity to create a closed loop in time (aka a time machine), then any quantum particle that travelled backwards in time ought to have a very peculiar non-linear behaviour. Tim Ralph and co-authors said: what if there was only a little bit of space-time curvature? Wouldn’t you still expect just a little bit of non-linear behaviour? And we can look for that in the curvature produced by the Earth, without even needing to build a time-machine!”

Artistic view of matter in quantum superposition on curved space-time. Image courtesy of Jonas Schmöle, Vienna Quantum Group.

In our recent paper in New Journal of Physics, for the special Focus on Gravitational Quantum Mechanics, Tim and I re-examined the `event formalism’ (the fancy name for the nonlinear model in question) and we derived some more practical numerical predictions and ironed out a couple of theoretical wrinkles, making it more presentable as an experimental proposal. Now that there is growing interest in quantum gravity phenomenology — that is, testable toy models of quantum gravity effects — Tim’s little theory has an excitingly real chance of being tested and proven either right or wrong. Either way, I’d be curious to know how it turns out! On one hand, if quantum entanglement survives the test, the experiment would stand as one of the first real confirmations of quantum field theory in curved space-time. On the other hand, if the entanglement is destroyed by Earth’s gravitational field, it would signify a serious problem with the standard theory and might even confirm our alternative model. That would be great too, but also somewhat disturbing, since non-linear effects are known to have strange and confusing properties, such as violating the fabled uncertainty principle of quantum mechanics.

You can see my video debut here, in which I give an overview of the paper, complete with hand-drawn sketches!

PicC

(Actually there is a funny story attached to the video abstract. The day I filmed the video for this, I had received a letter informing me that my application for renewal of my residence permit in Austria was not yet complete — but the permit itself had expired the previous day! As a result, during the filming I was half panicking at the thought of being deported from the country. In the end it turned out not to be a problem, but if I seem a little tense in the video, well, now you know why.)

Why does matter curve space and time?

This is one of those questions that has always bugged me.
black-hole
Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?

Skin Deep, by Xetobyte
Image Credit: Xetobyte

 

Stop whining and accept these axioms.

One of the stated goals of quantum foundations is to find a set of intuitive physical principles, that can be stated in plain language, from which the essential structure of quantum mechanics can be derived.

So what exactly is wrong with the axioms proposed by Chiribella et. al. in arXiv:1011.6451 ? Loosely speaking, the principles state that information should be localised in space and time, that systems should be able to encode information about each other, and that every process should in principle be reversible, so that information is conserved. The axioms can all be explained using ordinary language, as demonstrated in the sister paper arXiv:1209.5533. They all pertain directly to the elements of human experience, namely, what real experimenters ought to be able to do with the systems in their laboratories. And they all seem quite reasonable, so that it is easy to accept their truth. This is essential, because it means that the apparently counter intuitive behaviour of QM is directly derivable from intuitive principles, much as the counter intuitive aspects of special relativity follow as logical consequences of its two intuitive axioms, the constancy of the speed of light and the relativity principle. Given these features, maybe we can finally say that quantum mechanics makes sense: it is the only way that the laws of physics can lead to a sensible model of information storage and communication!

Let me run through the axioms briefly (note to the wise: I take the `causality’ axiom as implicit, and I’ve changed some of the names to make them sound nicer). I’ll assume the reader is familiar with the distinction between pure states and mixed states, but here is a brief summary. Roughly, a pure state describes a system about which you have maximum information, whereas a mixed state can be interpreted as uncertainty about which pure state the system is really in. Importantly, a pure state does not need to determine the outcomes to every measurement that could be performed on it: even though it contains maximal information about the state, it might only specify the probabilities of what will happen in any given experiment. This is what we mean when we say a theory is `probabilistic’.

First axiom (Distinguishability): if there is a mixed state, for which there is at least one pure state that it cannot possibly be with any probability, then the mixed state must be perfectly distinguishable from some other state (presumably, the aforementioned one). It is hard to imagine how this rule could fail: if I have a bag that contains either a spider or a fly with some probability, I should have no problem distinguishing it from a bag that contains a snake. On the other hand, I can’t so easily tell it apart from another bag that simply contains a fly (at least not in a single trial of the experiment).

Second axiom (Compression): If a system contains any redundant information or `extra space’, it should be possible to encode it in a smaller system such that the information can be perfectly retrieved. For example, suppose I have a badly edited book containing multiple copies of some pages, and a few blank pages at the end. I should be able to store all of the information written in the book in a much smaller book, without losing any information, just by removing the redundant copies and blank pages. Moreover, I should be able to recover the original book by copying pages and adding blank pages as needed. This seems like a pretty intuitive and essential feature of the way information is encoded in physical systems.

Third axiom (Locality of information): If I have a joint system (say, of two particles) that can be in one of two different states, then I should be able to distinguish the two different states over many trials, by performing only local measurements on each individual particle and using classical communication. For example, we allow the local measurements performed on one particle to depend on the outcomes of the local measurements on the other particle. On the other hand, we do not need to make use of any other shared resources (like a second set of correlated particles) in order to distinguish the states. I must admit, out of all the axioms, this one seems the hardest to justify intuitively. What indeed is so special about local operations and classical communication that it should be sufficient to tell different states apart? Why can’t we imagine a world in which the only way to distinguish two states of a joint system is to make use of some other joint system? But let us put this issue aside for the moment.

Fourth axiom (Locality of ignorance): If I have two particles in a joint state that is pure (i.e. I have maximal information about it) and if I measure one of them and find it in a pure state, the axiom states that the other particle must also be in a pure state. This makes sense: if I do a measurement on one subsystem of a pure state that results in still having maximal information about that subsystem, I should not lose any information about the other subsystems during the process. Learning new information about one part of a system should not make me more ignorant of the other parts.

So far, all of the axioms described above are satisfied by classical and quantum information theory. Therefore, at the very least, if any of these axioms do not seem intuitive, it is only because we have not sufficiently well developed our intuitions about classical physics, so it cannot really be taken as a fault of the axioms themselves (which is why I am not so concerned about the detailed justification for axiom 3). The interesting axiom is the last one, `purification’, which holds in quantum physics but not in probabilistic classical physics.

Fifth axiom (Conservation of information) [aka the purification postulate]: Every mixed state of a system can be obtained by starting with several systems in a joint pure state, and then discarding or ignoring all except for the system in question. Thus, the mixedness of any state can be interpreted as ignorance of some other correlated states. Furthermore, we require that the purification be essentially unique: all possible pure states of the total set of systems that do the job must be convertible into one another by reversible transformations.

As stated above, it is not so clear why this property should hold in the world. However, it makes more sense if we consider one of its consequences: every irreversible, probabilistic process can be obtained from a reversible process involving additional systems, which are then ignored. In the same way that statistical mechanics allows us to imagine that we could un-scramble an egg, if only we had complete information about its individual atoms and the power to re-arrange them, the purification postulate says that everything that occurs in nature can be un-done in principle, if we have sufficient resources and information. Another way of stating this is that the loss of information that occurs in a probabilistic process is only apparent: in principle the information is conserved somewhere in the universe and is never lost, even though we might not have direct access to it. The `missing information’ in a mixed state is never lost forever, but can always be accessed by some observer, at least in principle.

It is curious that probabilistic classical physics does not obey this property. Surely it seems reasonable to expect that one could construct a probabilistic classical theory in which information is ultimately conserved! In fact, if one attempts this, one arrives at a theory of deterministic classical physics. In such a theory, having maximal knowledge of a state (i.e. the state is pure) further implies that one can perfectly predict the outcome of any measurement on the state, but this means the theory is no longer probabilistic. Indeed, for a classical theory to be probabilistic in the sense that we have defined the term, it necessarily allows processes in which information is irretrievably lost, violating the spirit of the purification postulate.

In conclusion, I’d say this is pretty close to the mystical “Zing” that we were looking for: quantum mechanics is the only reasonable theory in which processes can be inherently probabilistic while at the same time conserving information.

The cracked mirror

“I know who I WAS when I got up this morning, but I think I must have been changed several times since then.”
— Alice Through the Looking-Glass

Magritte - La Reproduction Interdite
Magritte – La Reproduction Interdite

Professor Lee Tsung gripped the edge of the sink and stared into the eyes of her reflection. In the Physics department’s toilets, all was silent, except for the dripping of a single tap, and her terse breathing. Finally, the door swung open and someone else entered, breaking the spell. The intruder was momentarily baffled by the end of a phrase that Lee whispered to herself on her way out. Something about her eyes being switched.

“I’m a mess,” she confessed later to Yang Chen, her friend and colleague in the particle physics department. Both women had been offered jobs at the American Institute of Particle Physics at the same time, and had become close friends and collaborators in their work on extensions to the Standard Model. Lee Tsung’s ambition and penetrating vision had overflowed into her personal life. It infiltrated her mannerisms, giving her nervous ticks, fiery and unstable relationships and a series of dramatic break-downs and comebacks that littered her increasingly illustrious career. Chen was the antidote to Lee’s anxious and hyperactive working style; calm, meditative and thorough, her students’ biggest complaint was that she was boring. But united by a common goal, the two scientists had worked perfectly together to uncover some of the deepest secrets of nature, pioneering the field of research on neutrino oscillations and being heavily involved with experimentalists and theoreticians in the recently successful hunt for the Higgs Boson.

“I have to quit physics. I can’t look at another textbook. I don’t dare.”

Chen was visibly shocked. In all of the ups and downs she had experienced with Tsung, the idea of quitting physics altogether had never arisen.

“What’s going on? What happened?”

Lee fixed her friend with a two-colour gaze. Her eyes, normally hidden in brooding black shadows beneath her eyebrows, now reflected the dim light of the university cafe. One eye was green and the other brown.

“You’ll think I’m crazy.”

“I already do. You’ve got nothing to lose.”

Lee studied her friend a moment longer, then sighed in resignation.

“Well if I don’t tell you, I suppose I’ll go mad anyway. But before I tell you, you have to promise me something. Promise me that you won’t tell me, not even drop a hint, which way the bias went in the Wu experiment.”

“What, you mean the parity violation experiment?”

“Don’t even talk about it! It is too dangerous. You might let something slip. I don’t want to know anything about their result.”

“But you already know the result — you lectured on particle theory back when you were an associate professor.”

“Shush already! Just listen. Last night I was working late at the cyclotron. Nothing to do with the machine itself, I was just hanging out in the lab trying to finish these damn calculations — that blasted theoretical paper on symmetry breaking. Anyway, I could barely keep my eyes open. You know how the craziest ideas always come just when you are dozing off to sleep? Well, I had one of those, and it was, –” Lee broke off to give a short laugh, “– well, it was as crazy as they come. I was thinking, isn’t it weird that the string theorists tell us every particle has a heavier super-partner, so that their calculations balance out. Well, why don’t we do the same thing for parity?”

“You lost me already.”

“You know, what if there is another universe that exists, identical to ours in every way — except that it is flipped. Into its mirror image.”

“Like in Alice through the looking-glass?”

“Exactly. In theory, there is nothing against having matter that is ordinary in every respect, except with left and right handedness reversed. The only question would be, if such a mirror-image universe existed, where is it? So I did some back-of-the-envelope calculations and found out that the coupling between mirror-matter and ordinary matter would be extremely weak. The two worlds could co-exist side by side, and we wouldn’t even know!”

Chen stirred her coffee, looking almost bored, but she didn’t drink it for a full minute. In any case, it didn’t need stirring — there was no sugar in it. Lee could almost see her friend’s mind working. At last she said:

“That’s pretty unlikely. We have some really sensitive instruments downstairs, and some really high energies. Are you telling me even they wouldn’t be able to pick something up?”

Lee’s eyes glittered with excitement.

“Exactly! That’s what I thought. It turns out that, with just a small modification of the collider experiment — the one we’re running right this minute — we could instantiate a reaction between our universe and the mirror universe. We could even exchange a substantial amount of matter between the two worlds!”

Chen’s looked up sharply.

“Lee,” she said, “Please tell me you didn’t already do it.”

Lee shrugged guiltily.

“It only cost them a few hours of data. Nobody will care – they’ll assume the diversion of the beam was due to a mistake by one of the grad students. But it worked Chen! I opened up the mirror world.”

“That’s great! You’ll be famous!”

But Lee stared down at her fingers, the nails chewed raw, and said nothing.

“That’s a Nobel prize right there! Why are you freaking out?”

“Just wait. We’re about to get to the really weird part,” said Lee.

“At first I thought the experiment wasn’t working. I had expected to see a definite interaction region, generating all kinds of particles. But I guess I hadn’t really worked out the details of what to expect. Nothing seemed to be happening to the beam. I was puzzled for a few seconds. But then there was a weird change in the light of the whole building – it seemed to get slightly brighter. And everything doubled — even the humming of the machine seemed to get twice as loud. And the numbers coming up were really bizarre. I started seeing not just the normal data stream, but additional characters superimposed on top. I don’t know that much about the displays we have, but I’m pretty sure they can’t make figures like that.”

“What do you mean?”

“They were backwards. You know, like when you try to read a book in the mirror. When I looked at the clock on the wall, it had six hands, and the two second-hands were ticking in opposite directions. The three wrong hands seemed to flicker in and out of existence, along with the extra light and sound that was coming through. ”

“Wait a second,” said Chen breathlessly, “you’re saying the interaction region encompassed the whole building?”

“Yes. Well, at least the lab and most of the surrounding infrastructure. I don’t know if it extended outside. Nobody would have noticed, I think. It was 3am.”

Chen exhaled and leaned back in her chair.

“Wow. Okay, that does sound pretty nuts. In fact, it sounds just like a crazy dream. Have you seen a therapist?”

“It felt pretty real to me. And it gets crazier. I got this overwhelming sense of dread. Fear, like you can’t imagine. I couldn’t figure out where it was coming from, and then I realised that my PC screen had gone dark, and I could see my own reflection in it.”

She swallowed.

“Not just everything in the room, but me too — I was also doubled up. There were two versions of me coexisting at once. One of them was facing me, but the other, sitting in the same place — I could only see the back of her head. I don’t know why I felt so afraid. But I did.”

“But if the two worlds were really superimposed on each other like you said, and the interaction region was so large, than the coupling must have still been very weak. Probably mostly electromagnetic. So it would have ended the party pretty fast, right? And with a very low rate of matter exchange.”

“Yes, that’s what I expected too. But it didn’t happen. The double-images persisted for at least a minute. It seemed like forever, just me sitting there staring at the back of my own head bizarrely superimposed on my face. I kept wondering, if I were in the other world, would I also just be sitting here? But it was the strangest thing … it was quiet enough that I could hear us both breathing. Her and me — or me and me, I suppose. But I could also hear buttons being pressed at the console, very slowly and softly. I recognised the `click’ they made. It was unmistakable. I swear she had an arm up somewhere where I couldn’t see in the reflection, controlling the beam.”

“Wait. What does that mean? What are you saying?”

Lee leaned forward, almost hesitating to speak. When she did, it was in a low voice, uncharacteristic of her.

“Don’t play dumb Chen. I’m saying, maybe that interaction was exactly as strong as I expected. But maybe all the matter exchange was happening at a single location. It’s against protocol, but I know that in principle we can localise the beam almost anywhere within the facility … including the console where I was sitting.”

“You think your mirror-self was focusing the interaction on herself? On you? Swapping your matter with hers between the two universes?”

“Well, I don’t know really – I sort of blacked out. I woke up after 5am, slumped at the desk by the beam controls, drooling like a baby. First thing I did was look at the clock. The second hand was moving clockwise, like always. I checked every document I could find, it was all written left to right. I still have the birthmark on my left hand, not the right. You can verify that.”

Lee held up her hand. Chen said:

“So, everything’s fine then! You’re not in the mirror world. You’re in the real world. I’m the real me, you’re the real you. And you’ll be famous for discovering mirror matter!”

Lee shook her head.

“There’s more. I was looking at myself in the mirror just before I called you today. Staring at my eyes. Of course, as you might guess, the left one was still brown and the right one was still green, as it always was. But a thought occurred to me. These eyes being examined — they themselves are the examiners. The world looks the right way around, alright, but what if I’m seeing it through mirrored eyes? Sure, when you look in the mirror everything looks flipped left and right from your perspective. But what if your perspective also gets flipped?”

“You mean, if a mirror image reads a book, does it look backwards to her?”

“Exactly. You look at your reflection and you see its left and right hands are switched. But if you point to the hand that you think is your right hand, the image points to its left hand. The image perceives it’s left hand as its right hand. It looks at you, and from its perspective, you are the one who has got it wrong. The mirror image can’t tell that it is a mirror image.”

Chen was silent.

“Imagine that, next time you wake up, you have a fifty-fifty chance of waking up as your own mirror image, in the mirror universe. Could you leave any kind of trace or signal for yourself that you could use to tell if it happened or not? You can try to use books, clocks, birthmarks or even tattoo yourself with big signs saying `left hand’ and `right hand’ — but it won’t do any good, because the switch is always compensated by your own switch in perspective. You lose your reference point.”

“Come on. There has to be some way to tell. Gyroscopes? Spinning stars? Light polarisation?”

Lee shook her head.

“Not quite. We know that classical physics is parity invariant. Newton’s equations, even electromagnetism, all invariant under flipping left and right. Think more fundamental. More modern.”

Chen sucked on the end of her spoon, frowning at the ceiling of the cafe. She still hadn’t touched her coffee. Then she gasped:

“Beta decay! Weak interactions violate parity! You could just dig up the paper by Wu and check whether the direction of emission was — mmf!”

Lee lunged over the table, upsetting a glass of water, to block Chen’s mouth forcibly with her hand.

“Sorry!” Chen whispered breathlessly. Lee glared at her.

“I told you! I don’t want to know which way those experiments came out.”

There was a pause as a waiter came to mop up the spill, glancing nervously at the two women before moving on.

“But why not?” said Chen when he was gone.

“It will resolve your question once and for all. In fact, –” she rummaged around in her bag, “I have my lecture notes here that I prepared for the particle physics course. It’s on page 68 … or maybe 69. One of those.”

She carefully placed the paper on the table, her lips pursed to prove to Lee that she wasn’t going to accidentally mention the answer. Lee took the paper and placed her coffee cup solidly on top.

“You have to understand,” said Lee, “what it would mean if I was right. If I find out that the direction of parity violation in this universe is different to what it is in my memory, that means that I live in the wrong world. Could I live with that knowledge?”

Chen shrugged.

“Why not? You said everything is the same between the worlds, right? Who cares if your left hand turned into your right?”

“Yes. That’s the point. Doesn’t it all seem rather convenient? That the mirror world should exactly match our universe, in all details, apart from things like the direction of beta decay. Don’t forget, we are talking about physics here. Simply reversing the handedness of matter doesn’t say anything about what kind of world is constructed out of it. In all probability, I should have coupled through to just empty space — after all, most of the universe is empty space. Why should the orbits of the two Earth’s match up so precisely, and their rotations, right down to the location of my laboratory? Think about it Chen. What kind of powerful force could arrange everything just so that, if one person should find a way to open up a gateway between the worlds, it would be possible to switch her with her mirror copy, and she wouldn’t notice a thing?”

“I don’t know, Lee!” said Chen, now visibly exasperated.

Lee slumped in her chair.

“I don’t know either,” she said at last.

“That’s the thing that bothers me. Maybe there really is a perfectly good reason why there should be another copy of Earth in the mirror universe, and another copy of me. But why would the people on the other side try to hide it? It is like they were waiting for it. Like they didn’t want me to know the difference. So that I wouldn’t try to go back after the switch was made. But if that’s true, then who — or what — took my place?”

Chen held her friend’s hand tight.

“Okay. I believe you. But please listen to me. It might seem like a fifty-fifty chance whether you are in the real world or the mirror world. But that’s not how probability works. Taking into account human psychology, you have to admit, it’s far more likely that you are suffering from delusions. I beg you, read the results of the Wu experiment. Set your mind at rest. The odds aren’t even — they’re a billion to one against. Do it right now, and end this.”

Lee looked at the document in front of her. She sighed. What was worse, knowing or not knowing? She leafed over to page 68.

“Hmm.”

Chen waited eagerly.

“Just preliminaries.”

“Must be the next page!”

Chen nearly tore the page while turning it, and pointed to the relevant paragraph.

“Look. That’s where I describe the set-up.”

Lee was looking curiously at her friend.

“What? Don’t you want to know?”

“I don’t know. Why are you so eager to show me? Is this part of the plan?”

Chen was caught off guard.

“What plan? Come on, Lee. I just want you to stop acting crazy so we can go back to how things were before.”

Lee held Chen’s eyes a moment longer, then finally looked down at the page.

“Okay. From what I remember, the electrons were biased parallel to the direction of current through the coil.”

She skimmed through the paragraph with her fingertip and then froze. For a moment, she seemed unable to lift her head, and when she did, her eyes held a peculiar expression.

“Chen…” she said, and her voice barely came out, “this says the bias was against the current.”

A strange blank stare had come over Chen’s face.

“Who … what are you?” said Lee. Her voice rose and her eyes flickered over the people around her in the cafe:

“Why did you do this? What world is this?”

She stood up and her chair scraped backwards and almost toppled. At the sound, Chen’s peculiar paralysis seemed to break. She held up both her hands.

“Woah woah woah. Calm down.”

“What! Explain to me what the hell is going on!”

“Listen to me,” said Chen urgently, “what charge convention were you using?”

Lee stared, speechless. Chen blustered on:

“I know you like to be unconventional. I remember you often saying that sometimes you define current as being in the direction of positive charge, just to keep your students on their toes. If so, then your memory of the result would be that the electrons were emitted parallel to positive current — which means against the actual flow of electrons. Just like it says in my notes.”

Lee stood still for a full ten seconds, and the entire cafe had gone quiet, waiting for a scene to break out. But after what seemed like an eternity, she fell back into her seat and buried her face in her arms.

“I can’t remember.”

And all the while, as Chen tried to console her, as Lee slowly came to terms with the fact that with the one escaped memory of that lecture she had given, ten years ago, where she may or may not have used an unconventional definition of electric current, another memory kept returning to her. It was a memory, not form ten years ago, but in fact from that very same day, after she had woken up — from what might have been a dream — and staggered down to the bathroom to stare at herself so long in the mirror. The memory of what might have been an involuntary twitch of the eye of her reflection. Nothing to notice at the time, but on reflection, now that she thought about it, it seemed almost, unmistakably: a suppressed wink.