How to unify general relativity and quantum mechanics

Unifying quantum mechanics and general relativity has been a problem for decades. I believe that I have cracked that nut.

Special relativity:

Philosophical principle: Laws of physics should not depend on observer’s speed.

Math: Lorentz transform, new way to “add” speeds.

Issues it solved: Maxwell’s equations predict a value for the speed of light that does not depend on your own speed.

Physical observations: The speed of light is indeed independent on observers’ speed (Michelson and Morley’s experiment).

Counter-intuitive aspects: There is no absolute simultaneity and no absolute time. There’s an absolute speed limit for physical objects in the universe.

New requirements: Physicists must now pay attention to the “observer” or “referential”.

Thought experiment: Alice is in a train, while Bob is on the ground watching the train pass him by. What happens if Bob sees a flash hit the train “simultaneously” at both ends? Hint: what happens “at the same time” for Bob is not happening “at the same time” for Alice. That explains why we cannot consider simultaneity as absolute.

General relativity:

Philosophical principle: Laws of physics should not depend on observer’s state of motion, including acceleration.

Math: Non-euclidean geometry, tensor and metrics.

Issues it solved: Discrepancies in the trajectory of Mercury.

Physical observations: Gravitation has an impact on light rays and clocks.

Counter-intuitive aspects: Light has no mass, but is still subject to gravity. The presence of a mass “bends” space-time.

New requirement: Physicists must pay attention to the metric (including curvature) of a given region of space-time.

Typical thought experiment: Alice is in a box on Earth, Bob is in a similar box dragged by a rocket at 1 g. The similarity between their experience explains why we can treat gravitation as a curvature of space-time.

Quantum mechanics:

Philosophical principle: Several, “Shut up and calculate” being the top dog today (meaning: if math flies against your intuition, trust the math).

Math: Hilbert spaces, Hamiltonian.

Issues it solved: Black body radiation, structure of matter.

Physical observations: Quantization of light, wave-particle duality, Young’s slits experiment.

Counter-intuitive aspects: Observing something changes it. There are quantities we can’t know at the same time with arbitrary precision, e.g. speed and position of a particle.

New requirement: Physicists must pay attention to what they observe and in which order, as observation may change the outcome of the experiment.

Typical thought experiment: Schrödinger puts his cat in a box where a system built on radioactive decays can kill it at an unknown time in the future. From a quantum mechanical point of view, before you open the box, the cat is in a superposition of two states, alive and dead.

Theory of incomplete measurements:

Philosophical principle: Everything we know about the world, we know from measurements. Laws of physics should be independent from the measurements we chose.

Math: “Meta-math” notation to describe physical experiments independently from the mathematical or symbolical representation of the measurement results. The math of quantum mechanics and general relativity applies only to measurement results, the “meta-math” describes the experiments, including what you measure and what physical section of the universe you use to measure it.

Issues it solved: Unifying quantum mechanics and general relativity. Quantum measurement problem. Why is the wave function complex-valued. Why doesn’t quantum mechanics apply at macroscopic scale (the answer being that it does). Why are there infinities appearing during renormalization, and why is it correct to replace them with observed values?

Physical observations: Room-scale experiments with quantum-like properties. How to transition the definition of the “meter” from a solid rod of matter to a laser beam. Physically different clocks and space measurements diverge at infinity. How can we talk about the probability of a photon being “in the Andromeda galaxy” during a lab experiment? Every measurement of space and time is related to properties of photons. Space-time interpreted as “echolocation with photons”.

Counter-intuitive aspects: Quantum mechanics is the necessary form of physics when we deal with probabilistic knowledge of the world. In most cases, our knowledge of the world is probabilistic. All measurements are not equivalent, and a “better” measurement (i.e. higher resolution) is not universally better (i.e. it may not correctly extend a lower-resolution but wider scale measurement). Space-time (and all measurements) are quantized. There is no pre-existing “continuum”, the continuum is a mathematical simplification we introduce to unify physically different measurements of the same thing (e.g. distance measurements by our eye and by pocket rulers).

New requirement: Physicists must specify which measurement they use and how two measurements of the “same thing” (e.g. mass) are calibrated to match one another.

Typical thought experiment: Measure the earth surface with the reference palladium rod, and then with a laser. Both methods were at some point used to define the “meter” (i.e. distance). Why don’t they bend the same way under gravitational influence? In that case, the Einstein tensors and metrics would be different based on which measurement “technology” you used.

More details: IntroductionShort paper.

So how does the unification happen?

To illustrate how the unification happens without too much math, imagine a biologist trying to describe the movement of ants on the floor.

The “quantum mechanical” way to do it to compute the probability of having an ant at each location. The further away from the ants’ nest, the lower the probability. Also, the probability to find an ant somewhere is related to the probability of finding it someplace near a short time before. When you try to setup the “boundary conditions” for these probabilities, you will say something like: the ant has to be somewhere, so the probability summed over all of space is one; and the probability becomes vanishingly small “at infinity”.

The general-relativistic way to do it will consider the trajectories of the ants on the 2D surface. But to be very precise, it will need to take into account the fact that ants are on a large-scale sphere, and deduce that the 2D surface they walk on is not flat (euclidean) but curved. For example, if an ant travelled along the edges of a 1000km square (from its point of view), it would not return exactly where it left off, therefore proving that the 2D surface is not flat.

At a relatively small scale, the two approaches can be made to coincide almost exactly. But they diverge in their interpretation of “at infinity”. Actually, assuming observed ants stay within a radius R of the nest, there are an infinite number of coordinate systems that are equal on that radius R, but diverge beyond R. Of course, the probabilities you compute depend on the coordinate system.

In particular, if you take a “curved” coordinate systems that loops around the earth to match the “general relativistic” view of the world, the physically observed probability does not match the original idea we have that probability becomes vanishingly small at infinity and that the sum is one. In that physical coordinates system, the probability to see ants is periodically non-zero (every earth circumference, you see the same ant “again”). So your integral and probability computation is no longer valid. It shows false infinities that are not observed in the physical world. You need to “renormalize” it.

In the theory of incomplete measurements, you focus on probabilities like in quantum mechanics, but only on the possible measurement results of your specific physical measurement system. If your measurement system follows the curvature of earth (e.g. you use solid rods of matter), then the probabilities will be formally different from a measurement system that does not follow it (e.g. you use laser beams). Key topological or metric properties therefore depend on the measurement apparatus being chosen. There is no “x” in the equations that assumes some underlying space-time with specific topology or metric. Instead, there is a “x as measured by this apparatus”, with the topology and metric that derives from the given apparatus.

Furthermore, all the probabilities will be computed using finite sums, because all known measurement instruments give only finite measurement results. There may be a “measurement not valid” probability bin. But if you are measuring the position of a photon in a lab, there cannot be a “photon was found in the Andromeda galaxy” probability bin (unlike in quantum mechanics), because your measurement apparatus simply cannot detect your photon in the Andromeda galaxy. Such a probability is non-sensical from a physical point of view, so we build the math to exclude it.

So in the theory of incomplete measurements, you only have finite sums that cannot diverge, and renormalisation is the mathematical equivalent of calibrating physically different measurement instruments to match one another.

The analogy is not perfect, but in my opinion, it explains relatively well what happens with as little math as possible.


No little thing is to small for grandiose words chiseled by some marketing war machine.

Seen on a Lampe Berger anti-mosquito product this morning:

Parfum “Absolu de vanille”

Vanilla Gourmet Scent

Not only is this ridiculously hyperlative, but they also have a different “tint” for the Engish and French version. English reader will notice that the French version sounds more like “Absolute Vanilla”, because that’s basically what it means. Who on Earth paid people to tell their customers that their anti-mosquito drug had a “Vanilla Gourmet scent?”

Let’s not get used to this kind of marketing hyperbole…

Hyperbole in science

In despair, I turned to a slightly more serious text, the first page of this month’s issue of Science et Vie. And here is what I read there about faster than light neutrinos:

Incroyable? Alors là oui, totalement! Et même pis. Que la vitesse de la lumière puisse être dépassée, ne serait-ce que de très peu, n’est pas seulement incroyable, mais totalement impensable. Absolument inconcevable. […] c’en serait fini d’un siècle de physique. Mais, et ce serait infiniment plus grave, c’en serait aussi fini avec l’idée selon laquelle la matière qui compose notre univers possède des propriétés, obéit à des lois. Autant dire que la quête de connaissance de notre monde deviendrait totalement vaine.

Incredible? Absolutely! And even worse. That the speed of light can be exceeded, even a little, is not only unbelievable, but totally unthinkable. Absolutely inconceivable. […] This would end a century of physics. Even more serious, we would be done with the the idea that matter making up our universe has properties, obeys laws. This would mean that the quest for knowledge in our world would become totally hopeless.

Whaaaaat? I really don’t like this kind of pseudo-science wrapped in dogma so pungent to be the envy of the most religious zealots. How can anybody who understood anything about Einstein’s work write something like that? Let’s backpedal a little bit and remember where the speed of light limit comes from.

Where does the speed of light limit come from?

At the beginning was Maxwell’s work on the propagation of electromagnetic waves, light being such a wave. These equations predicted a propagation of light at a constant speed, c, that could be computed from other values that were believed at the time to be physical constants (the “epsilon-0″ and “mu-0″ values in the equations). The problems is that we had a physical speed constant, in other words a speed that did not obey the usual law of speed composition. If you walk at 5 km/h in a train that runs at 200 km/h, your speed relative to the ground is 205 km/h or 195 km/h depending on whether you walk in the same direction as the train or in the opposite direction. We talk about an additive composition rule for speed. That doesn’t work with a constant speed: if I measure the speed of light from my train, I won’t see c-200 km/h, since c is constant. The Michelson-Morley experiment proved that this was indeed the case. Uh oh, trouble.

For one particular speed to be constant, we need to change the law of composition. Instead of adding speeds, we need a composition law that preserves the value of c. It’s the Lorentz transformation. What Einstein acknowledged with his special relativity theory is that this also implied a change in how we consider space and time. Basically, Lorentz transformation can be understood as a rotation between space and time. And in this kind of rotation, the speed of light becomes a limit in a way similar to 90 degrees being the “most perpendicular direction you can take”. Nothing more, nothing less. Of note, that “c” value can also be interpreted as the speed at which we travel along time when we don’t move along any spatial dimension.

There are limits to limits

Once you understand that, you realize how hyperbolic what Science et Vie wrote is.

First, the value of c was computed as a speed of light, for equations designed for electromagnetism. It was never intended to say anything about neutrinos. We don’t know how to measure space and time without electromagnetic interactions somewhere. So the speed of light limit is a bit like the speed of sound limit for bats who would measure their world using only echo-location. It doesn’t necessarily mean nothing can travel faster than light, it only means that no measurement or interaction based on electro-magnetic interactions can ever measure it. I have tried to elaborate a bit on this in the past.

Second, Einstein revised his initial view to include gravity, and this made the world much more complex. Now space-time could be seen as modified locally by gravity. Now imagine how solid your “90 degrees is the most perpendicular direction” argument is if you look at a crumpled sheet of paper. The reasoning doesn’t mean much beyond very small surfaces. Remember that in the neutrinos experiments, we are in a very complex gravitational environment (mountains, …) and you’ll see that this “crumpled sheet of paper” analogy may not be so far off.

In short, it we find conditions where something appears to travel faster than light, it is exciting, it is interesting, it is worth investigating, but it’s certainly not the End of Science as Science et Vie claimed. Let’s not get used to this kind of crap.

How To Teach Special Relativity

How to Teach Special Relativity is a famous article by John Bell where he advocates that the way we teach relativity does not give good results. He describes an experiment now known as Bell’s spaceship paradox (even if Bell did not invent it):

In Bell’s version of the thought experiment, two spaceships, which are initially at rest in some common inertial reference frame are connected by a taut string. At time zero in the common inertial frame, both spaceships start to accelerate, with a constant proper acceleration g as measured by an on-board accelerometer. Question: does the string break – i.e. does the distance between the two spaceships increase?

Considered a difficult problem

The correct answer is that the string does break, even if the spaceships appear to always be at the same distance from one another as seen from an observer who did not accelerate with the spaceships. Yet, according to Wikipedia:

Bell reported that he encountered much skepticism from “a distinguished experimentalist” when he presented the paradox. To attempt to resolve the dispute, an informal and non-systematic canvas was made of the CERN theory division. According to Bell, a “clear consensus” of the CERN theory division arrived at the answer that the string would not break.

In other words, this problem was considered hard by a majority of serious physicists at the time Bell raised the question in 1976. I would venture to say that this remains the case today, except that this particular paradox is probably well known now. But the teaching of special relativity has not changed. This is how we explain the paradox today. I have a lot of admiration for John Baez in general, his “blog” is even in the sidebar of this one. But with all due respect, the explanation of the paradox posted on his web site is utterly complicated (I know he gives credit to someone else for it, but by hosting it on his web site, I would say that he condones it).

It should be easy

This particular formulation of the paradox was not known to me until someone recently asked me if the string would break. Using my little technique, it took me less than one minute to have the correct answer, without looking it up, obviously, but also without any computation or complicated diagram. Here is the mental diagram I used (click to see it in high resolution):

On this diagram, time is represented horizontally, and the two space ships are represented by the green and red curves, which are identical but separated by a distance along the vertical “spatial” axis. The distance at rest is represented by the blue arrow. The distance as measured between the two ships after they started moving is measured by the green and red arrows. The distance as measured “from the ground” is along the vertical axis, and remains constant.

Remember the only trick is that a “cosine” contraction on this diagram corresponds to a dilatation in relativity and conversely. On the diagram, the red and green arrows are obviously shorter than the blue arrow. The contraction factor is the cosine of the angle between these arrows and the vertical (space) axis, which is the same as the angle between the red or green curve and the horizontal axis. Therefore, relativity predicts that the distance between the two ships, as seen from the ship, will increase. Specifically, it increases by a factor usually denoted “gamma” (but which I prefer writing as the cosine of an angle myself), which can also be seen as a hyperbolic cosine, and which plays in Minkowski geometry the exact same role as the cosine in the Euclidean diagram above. You can find the precise mathematical relationship here.

Consequently, the string will break.

Accelerated solids in relativity

Another interesting observation one can make from the diagram is that you cannot draw a straight line that is perpendicular to both curves. What is “space” for one ship is not just “space” for the other. You need to draw a curved line between the two rockets if you want to always be perpendicular to the local “time” direction. In other words, the “time” direction for the string is not constant along the way, so all parts are not moving at the same speed. Someone sitting anywhere on the string will see other parts of the string move relative to him. That’s another way to explain why the string will break.

You can easily verify that this problem exists for any kind of accelerated solid. All parts of an accelerated solid in special relativity do not move at the same speed.

My own puzzle

Here is the interesting other thing that I realized within this short moment of reflection: there is a way for the two ships to accelerate “identically” (for a suitable definition of identically which remains to be given) so that the string will not break. Can you find it?

C’est tout de même dur à avaler.
Comment voulez vous qu’on enseigne cela à des élèves de Terminale?

Jean-Claude Carrière, in Entretiens sur la multitude du monde with Thibault Damour