Computer graphics

It is well-known in the field of computer graphics that quaternions are extremely useful. Not only are they extremely useful, but they save many billions of dollars in a huge industry that invests and makes money on a scale most of us can hardly comprehend. The reason is not hard to find. If you want to rotate an object on the screen with matrix methods, each timestep for each pixel requires two matrix multiplications, that is 144 flops, while with quaternion methods it requires two quaternion multiplications, that is 56 flops. There is no contest, which is why people who are interested in making money learnt this lesson many years ago.

It is is just a case of survival of the fittest: anyone who didn’t use quaternions simply went out of business. End of story.

Not so in theoretical physics. Here the law of survival of the fittest does not apply. Here it is the law of patronage that rules the day. Whoever has the richest friends wins. Whoever can blag most convincingly. Whoever can lie and cheat without getting caught. It is not what you know about physics that counts, but who you know, and whether they like you. That is why computer graphics is a success story and theoretical physics has failed to make significant progress in half a century.

Many people know that quaternions are essential to the understanding of quantum mechanics and particle physics. I have spoken to dozens of them, and read the work of dozens more. They are all side-lined, because the people with the money don’t like quaternions, because they don’t understand them. Quaternions are the only thing that can (a) quantise gravity, (b) explain the three generations of fermions, (c) account for the bizarre properties of entangled particles, be they electrons or polarised photons, (d) explain the chirality of the weak force, (e) explain where the 20-odd unexplained parameters of the standard model of particle physics come from, or (f) unify physics.

You would think that any one of these reasons would be enough for physicists to jump on the quaternion bandwagon. But no. It is survival of the fittest again: theoretical physicists rely for their continued employment on the continued failure of any else to solve these problems. This explains not only their extreme aggressiveness towards anyone who shows the slightest sign of making progress, but also their extreme timidity towards making such progress themselves, for fear of being sacrificed by the multitude in order to appease the gods.


14 Responses to “Computer graphics”

  1. Robert A. Wilson Says:

    Many human societies throughout history have practised human sacrifice, cannibalism, and other practices that enlightened people nowadays find abhorrent. Many people consider it a sign of progress when human sacrifice becomes a thing of the past, but while judiciaries around the world still enact the death penalty it cannot be said to have been completely eradicated. The continued practice of war, and many other rituals that result in death, whether intended or not, can be added to the score. One might argue, for example, that one or another government’s response to Covid might amount to the sacrifice of some lives in order to provide a better life for those who remain. It is just a case of where one draws the line. The community of theoretical physicists is a human society like any other, and it practices its own version of human sacrifice as ruthlessly as any that I have read about anywhere.

  2. Robert A. Wilson Says:

    The irony is the same that attends societies that resort to ever more extreme human sacrifice in order to persuade the gods to end the drought that has lasted for half a century – it doesn’t work, and ultimately it just hastens the end for a society in denial of the reality of the circumstances it is living under. Human sacrifice does not bring the rains back. Society has to move to where the rains are. That is the reality.

  3. Robert A. Wilson Says:

    History teaches us that, on the whole, the times of change, in particular from stagnation to innovation, occur when there is a sufficient turnover of people at the top of the pile. Stagnation occurs when the people at the top have been there for too long. That is the main reason why democracy works better than autocracy: the people at the top change more frequently. The reason why it doesn’t work all that well is because it is easier to get rid of the good people at the top that it is to get rid of the bad people at the top. Is that, or is it not, something that we have a large amount of topical evidence for?

    The people at the top of theoretical physics have been there for 40 years. It is time for a change.

  4. Robert A. Wilson Says:

    One might expect these problems to affect experimental physics in much the same way, given the fact that many experiments require collaborations of thousands of people, so that the leaders of these experiments might well be thought to be “top people” as well. But these seems not to be the case. The appropriate analogy seems to be that the theoretical physicists are politicians, while the experimentalists are the civil service, independent of the political biases of the politicians. The civil servants are in touch with reality, where the politicians are not. The civil servants test the theories of the politicians, and try as hard as they can to explain why the theories don’t work, but of course the politicians don’t listen. Clearly we do need the civil service to ensure that the country carries on running. But do we really need the politicians? The experience of Belgium seems to suggest that we do not.

  5. Robert A. Wilson Says:

    Ah, maybe now I understand what my position is. According to this analogy, as a mathematician I should be a civil servant. But I have become a politician. No wonder I am ill-equipped to deal with the vitriol that is directed my way as result. A civil servant who dares to *think* about what they are told to do, and dares to answer back to the politicians – oh dear!

  6. Robert A. Wilson Says:

    Talking of extreme aggressiveness, I have never experienced anything worse in an academic context than the aggressiveness of a certain SG who disrupted a serious lecture that a certain AGL was trying to give in Banff, purely on the grounds that SG thought that what AGL was saying was wrong. This is the only time in my entire mathematical career that I have ever experienced a member of the audience trying to prevent a lecturer from making a mathematical presentation on the grounds that it was “wrong”. That isn’t how we mathematicians decide what is right and what is wrong. We allow people to present their arguments first, and then we demolish them. We do not deny them the right to present their arguments in the first place.

    Even JPS allowed GM to present his lecture in Spetses without interruption, before taking the chalk and explaining how it should be done.

  7. Robert A. Wilson Says:

    Twice in my career I was paid actual money for my input into commercial computer programs. I have no idea what the programs actually did, and cannot remember how they did it, and it was so long ago that the NDAs I signed no longer have any practical relevance: I couldn’t disclose anything if I tried. I am fairly sure that the quaternions were not relevant in either case. But I am absolutely sure that what I got paid for was not any technical expertise, which was and remains limited, and not even my ability to think outside the box, which may exist only in my imagination, but in fact my ability to ask lots of the right stupid questions at the right time in the right way. And even that ability is not necessarily a sign of intelligence – it is merely a sign of being sensitive to the weak points in an argument. It is well-known that people give away the weak points in their own arguments by their body language and other clues.

    I have found the weak points in theoretical physics by the same means. By observing the body language of theoretical physicists on the rare occasions when they are prepared to converse with me, I can distinguish between the cases where my argument is weak and the cases where their argument is weak, without the necessity for a huge investment of time into learning what they know that I don’t know. The important issue, after all, is not to find out what they know, it is to find out what they don’t know.

    By now, I have a deep understanding of what they don’t know, and why they don’t know it. The reason they don’t listen to me is that they don’t know what they don’t know, and they don’t know that they don’t know it.

    • Robert A. Wilson Says:

      Perhaps I ought also to extol the virtues of pattern-patching at this point. Some people like me who could not write a program in C++ to save his life, but understands the concept of grammar, can look at a C++ program and point out a bug that the programmer cannot see. I learnt this fact 30 years ago by observing a secretary whose job included typing examination papers, who, despite having limited mathematical knowledge, was able to spot errors in the mathematical formulae in the papers that the setter had not spotted. This ability seemed to come from a deep understanding of mathematical grammar, that was independent of a limited knowledge of mathematical vocabulary.

      There is a marked tendency for people, especially physicists, to think that meaning resides only in the vocabulary, and to forget that the grammar is also crucial. The grammar of physics is mathematics, and if you forget that then you lose the meaning of much of physics. Physicists tend to be very dismissive of mathematicians like me who try to correct their grammar, saying that everyone knows what they mean, even if it’s not grammatical. Unfortunately, they are wrong. Grammar exists for a purpose, to ensure that the meaning is clear. The meaning of much of theoretical physics is not clear, and this is a huge problem that has led to many thousands of pages of meaningless argument over many decades. It has led to arguments that ought to be mathematical being taken over by philosophers, and arguments that ought to be physical being taken over by journalists. This problem would not happen if we took proper care of the grammar.

  8. Math Światek Says:

    Out of curiosity, lately when i was wondering why Physicist say there is no wave function for the photon, i naively had the stupid though of assuming one could just use Psi =(E B)^T build of the two EM fields almost looks the same as one. One can even apply Borns rule to it and realize that it yields the energy density of the EM field which also satisfies the continuity equation. What’s more however, writing time evolution equation of Psi (Maxwell) yields something that has weirdly familiar structure of the Dirac equation when written for spinors (u_A u_B)^T.

    Now Maxwell for E B field has 6 real dimensions, but if one rewrites it in terms of quaternions it will change towards the same as 4 components as Dirac has. I just didn’t get around to check if the curls in Maxwell turn into this vector product of the Pauli matrix vector and the momentum operator – if so, those equation would have the same origin but differ only that one has a mass term where the other has a coupling to another field.

    However 2nd quantization is not done for the E B fields but the electromagnetic potential A. Not sure if the gauge properties of that cause issues that prohibit a formulation of a wave function for photons.

    Anyhow, do you maybe know where to look for a formulation of Maxwell via quaterions?

    • Robert A. Wilson Says:

      I believe it is the case that Maxwell himself wrote his equations in quaternionic form. It is only later generations who failed to understand the true significance of quaternions, and wrote them out of physics completely. Hamilton and Maxwell between them did more to revolutionise theoretical physics than anyone else between Newton and Einstein. Unfortunately, their work has been bowdlerised by prudish late Victorians who thought that quaternions were too sexy and should be abolished.

    • Robert A. Wilson Says:

      That is why I am so concerned to restore the quaternionic facts of life to theoretical physics, and to the education of physicists. No-one reads Hamilton or Maxwell in the original. But we should always read the classics in the original.

  9. Math Światek Says:

    While there are some big advantages of quaternions, i still think classic vector calculus has its merits but elsewhere. So usually it’s best to keep both versions and use the one depending on the purpose.

    But what i find most irritating that Dirac stumbled on quaternions merely by accident in trying to linearize an equation – and i am not sure Physicist fully understand that fact. After all they take Dirac and combine it with Maxwell quantized in the vector calculus form and call it QED. Like representing rotations via quaternions and vector calculus are quite different things and that they work together at all (even if not perfectly) in a single theory is actually rather very surprising to me.

    And yes, from what i read, Maxwell originally formulated it via quaternions though this has become so unpopular, that is is very hard to find a good reference to work with.

    Anyhow, initially i tried to get a pure vector calculus version of Dirac, but that seems to be actually quite hard to figure out (or more likely i am just too stupid for that). So i was looking to go the other way and get Maxwell into quaternions shape… but all that is also too unfamiliar for me to get it done properly, hence i was looking for some reference.

    But as i said: my initial impression is that if both equations are brought into the same framework they seem to be very much alike. I would actually like to see them in both in both frameworks to get a god idea what they are telling us and check if there is a difference if they are quantized properly within one of the frameworks only rather then a frankenstrein math.

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