Color and Reality

When I was a kid, I used to wonder if everyone saw the world in the same way. We can all look at the same grass, but maybe the color I called green showed up in my brain as the color my friend called blue. Maybe all of our colors were shifted around to the point where all the colors were accounted for, but how we perceived them was shuffled up. I thought it would be remarkably exciting, and hoped that I could see the world through someone else’s brain to see if, in fact, this was true.

meadows

My eight year old self would be bitterly disappointed technology today has not progressed far enough to make that wish a reality. At the time, we had to settle the debate by another manner – asking an adult, a source of concrete and immutable knowledge. The answer I was given was that everyone sees the same colors of course (although why this was so obvious was never really clear) and if they didn’t it wouldn’t matter much since we couldn’t tell. Color was “real” – bits of light had a color (later I found out we could call it the wavelength of a photon), it hit our eyes, and our brains converted it to a beautiful image.

The only problem is that this is wrong.

Color as Wavelength

Well, alright. Before you get upset, it isn’t completely wrong. We were all taught about Sir Isaac Newton who discovered that a glass prism can split white light apart into its constituent colors.

pink-floyd-dark-side-of-the-moon-crop

While we consider this rather trivial today, at the time you’d be laughed out of the room if you suggested this somehow illustrated a fundamental property of light and color. The popular theory of the day was that color was a mixture of light and dark, and that prisms simply colored light. Color went from bright red (white light with the smallest amount of “dark” added) to dark blue (white light with the most amount of “dark” added before it turned black).

Newton showed this to be incorrect. We now know that light is made up of tiny particles called photons, and these photons have something called “wavelength” that seems to correspond to color. Visible light is made up of a spectrum, a huge number of photons each with a different wavelength our eyes can see. When combined, we see it as white light.

visible_light_spectrum

So this appears to resolve my childhood debate. Light of a single wavelength (like that produced by a laser) corresponds to a single “real” color. The brain just translates wavelengths into colors somehow, and that is that. There’s just one problem.

We’re missing a color!

Color as Experience

To find out just what we’re missing, we have to consider how we can combine colors. For instance, you learned some basic color mixing rules as a kid. In this case, let’s use additive color mixing since we’re mixing light.

Additive_color_mixing

Let’s find two colors on the spectrum line, and then we can estimate the final color they’ll produce when you mix them by finding the midpoint.

Red and green make yellow.

red-green-yellow

Green and blue make turquoise.

blue-green-turquoise

Red and blue make…

red-blue-green

Green? What? That doesn’t seem to make any sense! Red and violet make pink! But where is pink in our spectrum? It’s not violet, it’s not red – it seems like it should be simultaneously above and below our spectrum. But it’s not on the spectrum at all!

So we’re forced to realize a very interesting conclusion. The wavelength of a photon certainly reflects a color – but we cannot produce every color the human eye sees by a single photon of a specific wavelength. There is no such thing as a pink laser – two lasers must be mixed to produce that color. There are “real” colors (we call them pure spectral or monochromatic colors) and “unreal” colors that only exist in the brain.

A Color Map

So what are the rules for creating these “unreal” colors from the very real photons that hit your eye? Well, in the 1920s W. David Wright and John Guild both conducted experiments designed to map how the brain mixed monochomatic light into the millions of colors we experience everyday. They set up a split screen – on one side, they projected a “test” color. On the other side, the subject could mix together three primary colors produced by projectors to match the test color. After a lot of test subjects and a lot of test colors, eventually the CIE 1931 color space was produced.

CIE-1931

I consider this to be a map of the abstractions of the human brain. On the curved border we can see numbers, which correspond to the wavelengths in the spectrum we saw earlier. We can imagine the spectrum bent around the outside of this map – representing “real” colors. The inside represents all the colors our brain produces by mixing – the “unreal” colors.

So let’s try this again – with a map of the brain instead of a map of photon wavelengths. Red and green make yellow.

cie-red-green-yellow

Green and blue make turquoise.

cie-blue-green-turq

Blue and red make…

cie-blue-red-magenta

Pink! Finally! Note that pink is not on the curved line representing monochromatic colors. It is purely a construction of your brain – not reflective of the wavelength of any one photon.

Is Color Real?

So is color real? Well, photons with specific wavelengths seem to correspond to specific colors. But the interior of the CIE 1931 color space is a representation of the a most ridiculously abstract concept, labels that aren’t even labels, something our brain experiences and calculates from averaged photon wavelengths. It is an example of what philosophers call qualia – a subjective quality of consciousness.

I later learned that my childhood argument was a version of the inverted spectrum argument first proposed by John Locke, and that the “adult” perspective of everyone seeing the same colors (and it not really mattering if they didn’t) was argued by the philosopher Daniel Dennett.

I have come no closer to resolving my question from long ago of “individual spectrums” – but for the future, I vow to pay more attention to the idle questions of children.

The Mystics and Realists of Quantum Physics

It is said that when the 20th century is long gone, it will be remembered for two revolutionary theories – those of relativity and quantum physics. While both have led to a deeper understanding of our world, quantum physics stands alone in its profound weirdness – the ability to accurately predict totally counter-intuitive aspects of the physical world. From the simple indisputable oddity of the double slit experiment to the philosophical implications of Schrodinger’s cat, it becomes clear that we still understand very little of the actual mechanics of our world.

When explanations are lacking, the mystical is often brought up to fill the void. This has degenerated today into complete pop-psychology crap such as The Secret or What the Bleep Do We Know, but the role that human consciousness plays as an “observer”, if any, was considered very early by the founders of these theories. These arguments brought forth by some of the finest thinkers of our time echo to this day.

Niels Bohr

Niels_Bohr

Winner of the Nobel Prize in Physics in 1922, employed by the Manhattan Project, and father of the Bohr model familiar to every high school student, Niels Bohr was first accused by Einstein of introducing “mystic” elements in his explanation of quantum physics – mystic elements which in Einstein’s view had no place in science.

This was part of the famous Bohr-Einstein debates, and was perhaps not a fair criticism. Bohr appeared to not worry excessively about the “reality” underpinning the equations of quantum theory, and was simply more concerned about the equations of quantum theory rather than their implications. He rejected the hypothesis that the wave function collapse requires a conscious observer, insisting that “It still makes no difference whether the observer is a man, an animal, or a piece of apparatus”.

His view is perhaps best summarized in the following quote recalled by Heisenberg:

This argument looks highly convincing at first sight. We can admittedly find nothing in physics or chemistry that has even a remote bearing on consciousness. Yet all of us know that there is such a thing as consciousness, simply because we have it ourselves. Hence consciousness must be part of nature, or, more generally, of reality, which means that, quite apart from the laws of physics and chemistry, as laid down in quantum theory, we must also consider laws of quite a different kind. But even here I do not really know whether we need greater freedom than we already enjoy thanks to the concept of complementarity.

In short, if the numbers work out, don’t worry too much.

Wolfgang Pauli

Wolfgang_Pauli

But some did worry. Pauli was a skeptic’s skeptic – a man so dedicated to rationality it led him down a strange path. In 1927 the Solvay Conference was busy reaching consensus that Bohr’s approach was the best way to regard quantum physics (the Cophenhagen Interpretation), but Pauli was equally confident in a different interpretation. He tried to trace out just what part of consciousness it is that seems to prevent an in-depth, rational understanding. Deeply influenced by Schopenhauer’s The World as Will and Representation, Pauli appropriated his concept of a will “which breaks through space and time”.

He viewed that the acquisition of knowledge in mathematics or quantum physics “gives rise, however, to a situation transcending natural science” that can even acquire a “religious function” in human experience. This is not a belief in the religions of old, but as Pauli states “I do not believe in the possible future of mysticism in the old form. However, I do believe that the natural sciences will out of themselves bring forth a counter pole in their adherents, which connects to the old mystic elements.”

Perhaps the most interesting viewpoint on Pauli was that of Heisenberg, who viewed Pauli’s paradigm as even more rational than Bohr’s equation-focused approach because only he acknowledged that the scientific evidence pointed to something pre-rational or ‘mystical’. Pauli claimed that consciousness was presented philosophically by mystics and studied scientifically by psychologists. With the advent of quantum mechanics, physicists should then be able to unify both approaches. Unfortunately, we continue to wait.

Albert Einstein

Einstein 1921

Einstein was a scientific superstar, with fame not equalled to this day. One day, a quote was making the round in British newspapers that Einstein subscribed to the theory that “the outer world is a derivative of consciousness”. His response was swift and critical.

No physicist believes that. Otherwise he wouldn’t be a physicist. Neither do [Eddington and Jeans]. . . . These men are genuine scientists and their literary formulations must not be taken as expressive of their scientific convictions. Why should anybody go to the trouble of gazing at the stars if he did not believe that the stars were really there?

Einstein’s opposition to Bohr’s view or more “mystical” approaches is often cast as the great divide between the philisophies of idealism and those philosophies based on realism. Pauli often referred to Einstein’s “philosophical prejudice” assuming that reality is independent of any mind. In fact, his approach and objections were far more subtle. His major concern was the incredibly subjective aspect of consciousness introducing an unmeasurable “geist”, and this clash with the precise and well defined philosophical principles of physics such as locality or determinism.

This led to Einstein’s famous attempt at “breaking” quantum physics, the EPR paradox. At first a thought experiment which appeared to demonstrate quantum physics violating the seemingly well established principle of locality, later experiments showed that quantum physics instead proved locality false.

Violations of locality and determinism seemed to bother Einstein greatly, and this can be seen in his famous quote objecting to the randomness involved in wave function collapse under Bohr’s interpretation, that “God does not play dice”.

Bohr, summing up the debate perfectly, replied “Einstein, stop telling God what to do with his dice.”

John von Neumann

JohnvonNeumann-LosAlamos

The “last of the great mathematicians”, von Neumann solved one of the great problems of quantum theory. While the theory itself was established and experimentally verified, it lacked a “deep” mathematical understanding based on an axiomatic approach. He treated the world as a Hilbert space, an infinite dimensional structure.

While classical mechanics approached the world as a collection of points with six different characteristics (position and momentum along the x, y, and z axis), von Neumann’s approach considers a quantum system as a point in infinite dimensional space, corresponding to the infinite amount of possible outcomes. This led to very interesting implications in terms of “measurement”. While the “measurement” of a classical system simply involved finding one or more of those six values, the “measurement” of a quantum system involved mathematical operators acting on an infinite amount of values to produce a finite measurement.

The interesting conclusion arises when we consider the “real” interpretation of these mathematical operators. While we may say that an scientific instrument has caused wave function collapse, we run into the problem that no physical system (and a scientific instrument is a physical system completely described by quantum mechanics) can cause wave function collapse. We can describe the entire ensemble perfectly as a Hilbert space. But we do not experience this Hilbert space – we measure and experience only finite values.

The conclusion von Neumann reached is that consciousness, whatever it is, appears to be the only thing in physics that can ultimately cause this collapse or observation. This does not mean that consciousness is “required” for the universe to work, but that wave function collapse appears to be caused by consciousness and we observe only a tiny slice. We are therefore an “abstract ego” acting as a measurement device on the infinite values of true reality.

Today

Today, the argument has largely died down, a combination of practicality and lack of any suitably shocking experimental results. The majority of physicists today take the approach of “it works”, namely that quantum physics produces accurate predictions of the real world and that the mathematical formalism is just that – a mathematical formalism that produces accurate results.

It may not reflect the true reality of the world (whatever it is), but it is suitably accurate to any level of precision that we are physically able to obtain. One may stay awake at night wondering “why”, but one will not get much work done with this approach. Perhaps more clarity lies in the future, but in the meantime – we will have to tolerate crap that tells us we can “will” our way to riches with quantum mechanics (and coincidentally make the authors rather rich, will indeed) instead of a rational approach dedicated to the pursuit of truth.

Koide’s Formula

Finding a beautiful and simple equation for something in the natural world is fascinating to me – it’s like picking at a corner of loose wallpaper in your room and suddenly seeing the scrolling green text of the Matrix on the wall behind it. Often these relations lead to a deeper understanding, but sometimes an indisputably true and simple relation will remain maddeningly confounding.

In 1981 Yoshio Koide was researching leptons, a family of fundamental particles that includes the familiar electron. There are three leptons which are “charge carriers” (they have mass) – the electron, the muon, and the tauon.

Koide was wondering if there was a way to relate the masses of these three particles with one another. He developed the following equation (related to the eigenvectors of the democratic matrix, here’s a review paper if you want more detail):

koide_1

Nothing too wild mathematically here. If we assume our three lepton masses are positive (pretty reasonable) then the value of Q can range from 1/3 (all the masses are the same) to 1 (the masses vary wildly from each other). So what is the value of Q? Well, when Koide first proposed this equation, the masses of the leptons were thought to be as follows:

  • Electron: 0.511 MeV/c2
  • Muon: 105.658 MeV/c2
  • Tauon: 1,784.2 MeV/c2

If we plug these values into Koide’s equation, we get a value of 0.667074 – incredibly close to 2/3, which would be precisely halfway between our upper (1) and lower (1/3) bounds we figured out before! This seems like a ridiculous coincidence.

Things like this make you wonder… well, is it exactly 2/3? Or is it just “kind of” close? The mass of the electron and the muon had been measured to a rather high level of accuracy, but the accuracy of the tauon measurements had been lagging behind due to the higher energies required. Perhaps the measurement of the tauon was wrong! It’s a hell of a hunch – but let’s go with it. Assume that the tauon mass has been measured incorrectly, we can set Q = 2/3, input the masses of the electron and muon, and see what the tauon mass “should” be. It turns out that Koide’s equation says the mass of the tauon “should” be 1777 MeV/c2.

Well that’s wonderful, but nature doesn’t seem to care how you think it “should” behave. The only test was to wait for more accurate measurements of the tauon mass and see if this was a neat coincidence based on measurement error or whether there may be something more interesting going on. The mass of the tauon was later revised with better measurements, and… drumroll…

Old Measurement Koide’s Prediction New Measurement
1,784.2 MeV/c2 1,777 MeV/c2 1,776.9 MeV/c2

Whoa. Our simple little equation, using nothing more than grade school arithmetic, has accurately predicted the mass of a fundamental physical particle years in advance of having this measurement confirmed by the best research labs on earth.

And now the question becomes why – why does this work at all? We have three seemingly random lepton masses, measurements of the most complicated physical system we know – our universe. We then input them into a ridiculously simple equation, and the most ridiculously simple answer pops out.

We can gain a tiny bit of insight by figuring out what exactly this equation is telling us.

koide_cone

Basically, we can calculate Q for a given set of three lepton masses. This Q will tell us where a three-dimensional vector specified by the square roots of our three lepton masses will end up.

Q = 1/3 The set of all vectors that form an angle of zero with the unit vector (multiples of the unit vector).
Q = 2/3 The cone seen above which fits perfectly into the “corner” created by our three axes. The set of all vectors that form an angle of pi/4 with the unit vector.
Q = 1 The set of vectors that form an angle of zero with our basis vectors. These vectors lie along one of our three axes.

So it appears that our lepton masses have been chosen in some magical manner as to fall perfectly in the middle of these two extremes. The concept appeals to our perception of the universe as a finely tuned apparatus, but gets us nowhere closer to an interpretation based in physical reality.

It’s a maddening equation. Beautiful. Simple. True. And no one knows what the hell to do with it.