A Brief Guide to the Great Equations

by Robert Crease

  Late in 1927, however, Bohr was scheduled to take a trip to the U.S., and he and Heisenberg were anxious to finalize an interpretation before his departure, so the two agreed to a truce, mainly on Bohr’s terms. The truce was made public that September, at a celebration in Lake Como of the hundredth anniversary of Alessandro Volta’s death. Bohr gave a speech proposing an awkward accommodation of wave and matrix mechanics, and Heisenberg stood up at the end to signal his approval. Waves and particles, Bohr said in effect, are ways we speak about events in the atomic realm. Neither way is entirely accurate, but the two ways have overlapping but restricted spheres of application. They are, Bohr declared, complementary ways of speaking about something of which we can have no direct knowledge. As he once put it, ‘There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’47

  Thus the origin of what has become known as the Copenhagen interpretation of quantum mechanics. It was not universally appreciated. Einstein called it ‘shaky’, adding that ‘The Heisenberg-Bohr tranquilizing philosophy – or religion? – is so delicately contrived that, for the time being it provides a gentle pillow for the true believer from which he cannot very easily be aroused.’48 In complete theory, he wrote years later in 1935, with Boris Podolsky and Nathan Rosen, ‘every element of the physical reality must have a counterpart in the physical theory.’ Einstein tried to argue, unsuccessfully, that the incompleteness of quantum mechanics was a flaw revealing that there had to be more to be discovered, so-called hidden variables, the discovery of which will make its formulations refer directly to the real world. He pressed the argument for years, with Bohr countering that position and momentum were inherently classical concepts, inapplicable to events in the microworld except in loose and, strictly speaking, inaccurate ways.

  The Copenhagen interpretation – that somewhere beyond or beneath the macroscopic world lurks something that we cannot visualize, and that is made visualizable by an ensemble or arrangement of things whose behaviour is macroscopic – amounts to a clear, logical interpretation, and appears to be the simplest one consistent with all experimental and theoretical constraints. It is an interpretation that makes us all uncomfortable, but that is a psychological phenomenon, not an argument for or against the interpretation.

  Interlude

  THE YOGI AND THE QUANTUM

  The idea of intermediate kinds of reality was just the price one had to pay.

  – Werner Heisenberg

  In 1929, 2 years after the appearance of the uncertainty principle, a physicist at Harvard University named Percy Bridgman – a future Nobel laureate – published an article in Harper’s Magazine about the meaning of the uncertainty principle. The implications are far-reaching, he said, even for the public. ‘The immediate effect will be to let loose a veritable intellectual spree of licentious and debauched thinking.’ For, Bridgman continued, the nonscientist is apt to conclude from the uncertainty principle, not that it stated ‘the end of meaning’, but rather that ‘there is something beyond the ken of the scientist.’ In a remarkably prophetic passage, Bridgman wrote:

  This imagined beyond, which the scientist has proved he cannot penetrate, will become the playground of the imagination of every mystic and dreamer. The existence of such a domain will be made the basis of an orgy of rationalizing. It will be made the substance of the soul; the spirits of the dead will populate it; God will lurk in its shadows; the principle of vital processes will have its seat here; and it will be the medium of telepathic communication. One group will find in the failure of the physical law of cause and effect the solution of the age-long problem of the freedom of the will; and on the other hand the atheist will find the justification of his contention that chance rules the universe.1

  Eighty years later, we see that Bridgman was correct: each of these views has indeed been advanced. Bridgman went on to point to a positive side, saying that eventually, we can develop the ‘new methods of education’ to inculcate into people the ‘habits of thought’ required to reshape the thinking we use in ‘the limited situations of everyday life.’ The end result, Bridgman concluded, will be salutary:

  [S]ince thought will conform to reality, understanding and conquest of the world about us will proceed at an accelerated pace. I venture to think that there will also eventually be a favorable effect on man’s character; the mean man will react with pessimism, but a certain courageous nobility is needed to look a situation like this in the face. And in the end, when man has fully partaken of the fruit of the tree of knowledge, there will be this difference between the first Eden and the last, that man will not become as a god, but will remain forever humble.

  Eighty years later, we are still working on acquiring courageous nobility, and remaining humble. But we are also still working on how to talk about the physical interpretation of quantum mechanics, on how it connects with other, more familiar and visualizable features of the world.

  Of all the founders of quantum mechanics, Niels Bohr was the most insistent that we should try to fully express the quantum world in the framework of ordinary language and classical concepts. ‘[I]n the end’, as Michael Frayn has Bohr’s character say in the play Copenhagen, ‘we have to be able to explain it all to Margrethe’, his wife and amanuensis who serves as the onstage stand-in for the ordinary (that is, ‘classically thinking’) person.

  Many physicists, finding this task irrelevant or impossible, were satisfied with partial explanations – and Heisenberg argued that the mathematics works: that’s enough! Bohr rejected such dodges, and rubbed physicists’ noses in what they did not understand or tried to hide. He did not have an answer himself – and knew it – but he had no reason to think one could not be found. His closest answer was the doctrine of complementarity, an ordinary-language way of saying that quantum phenomena behave, apparently inconsistently, as waves or particles depending on how the instruments are set up, and that you need both concepts to fully grasp the phenomena. While this provoked debate among physicists on the ‘meaning’ of quantum mechanics, the doctrine – and discussion – soon all but vanished.

  Why? A large part of the answer is that, by 1930, physicists found a perfectly adequate way to represent classical concepts within the quantum framework involving a special abstract mathematical language called Hilbert (infinite-dimensional) space. In this space, the concepts of position and momentum are associated with different sets of coordinate axes that do not line up with each other, resulting in the situation captured in ordinary language terms by complementarity.2 While Bohr used the notion of complementarity to say that quantum phenomena were both particles and waves – somewhat confusingly, and in ordinary language terms – the notion of Hilbert space provided an alternate and much more precise framework in which to say that they are neither. But it was not a language that Margrethe understood; for her, quantum mechanics would have to remain esoteric and she would have to cope with understanding it as best she could. This is what has left the door open for the kinds of fantastic interpretations of meaning for human life mentioned at the beginning of this chapter, and by Bridgman.

  What makes the interpretation of quantum mechanics difficult to talk about? It is that we expect a complete theory to fall short of fully describing nature, but in a particular and well-defined way, for it provides a model that is an ideal limit of measurement, with any gaps or discrepancies between it and what we encounter in the laboratory arising from errors and imperfections in the measuring equipment. Many other theories and equations that physicists teach and use have other kinds of gaps and discrepancies, if they omit aspects of nature in the interests of a good approximation. An example is F = ma, which leaves out mass-energy conversion in the cause of buying ease of application. These are what we might call ‘harmlessly fudging or incomplete descriptive theories.’ Any gaps between the theory and the world are epistemological; that is, they have to do with our knowledge of th
e world, or the gap between our representations of the world and what it represents.

  The uncertainty principle is incomplete in a different sense. It is a mathematical relation, and a feature of the statistical interpretation of the wave function in quantum mechanics. It makes no reference to any underlying physical picture; there are no references to waves or particles, nor to physical experiments. It is not obvious what it refers to, except possibly the clicks of a detector. Yet it is about gaps in the world itself. These gaps are not epistemological but ontological; having to do not with our knowledge but with the world.

  This is strange, but why? It is important to see what the strangeness is not due to. The strangeness of the uncertainty principle is not due to the measurement process disturbing the object measured, which would be a feature of any Newtonian theory involving exchange of particles. Nor is it due to the presence of statistics. Rather, the strangeness of quantum mechanics is that quantum formulations are not ‘about’ a real or ideal object in the conventional sense.

  In classical physics, deviations of measured quantities from ideal norms are treated independently in a statistically based theory of errors. But the variations – statistical distributions – of quantum measurements are systematically linked in a single formalism. It tells you that all you can know precisely is the width of a distribution, and that you cannot make individual predictions. The superpositions are possibilities in the world, possibilities of observations. The wave formulas of quantum mechanics are thus neither about an ideal object, nor about a real object, but about a special kind of semiabstract object that admits numerous potential experimental realizations in becoming a real object. This special kind of semiabstract object is incomplete if we try to think of it the way we do other more familiar elements of the real world. One needs to add something to the abstract object to bring it into the world, and the choices that one makes regarding what to measure and how to measure it affect what one is measuring. This abstract object can appear wavelike or particle-like, for instance, depending on the kind of situation we put it in, waves and particles being models in visualizable space-time.

  This, then, is the ‘intermediate kind of reality’ that Heisenberg said was the price one had to pay to have quantum phenomena. It has the funny kind of incomplete, semiabstract reality that scripts or scores have – they are programs, as it were, for real things in the world (the produced play, the performed music) that require adding a context, and decisions about that context affect the whole of the abstract object. It brings back the role of human purposes and decisions that Newton left out.

  The challenge in explaining the meaning of the uncertainty principle to nonscientists lies in trying to explain this new kind of semiabstract object. And it is important to try, for otherwise there will continue to be information loss and distortion in the public understanding of the uncertainty principle.

  Heisenberg proved that. Just not mathematically.

  Conclusion

  BRINGING THE STRANGE HOME

  We can bring the strange home, and bring it home with precision.

  – Stephen Dunn, Walking Light:

  Memoirs and Essays on Poetry

  I have referred to the paths to these equations as journeys, but that metaphor can be misleading. It can mislead because it suggests smooth and steady progress toward a stable and predetermined destination, whereas the path to understanding that culminated in most of these equations was uneven and the travelers often wound up in a different place from where they thought they were heading. The metaphor also falsely suggests that the travelers were spectators taking in a vision of nature, rather than active participants in interactions with it who learned from their changing interactions and often changed their ideas in response.

  But the metaphor does capture the way each step of the journey readjusted the perception of the travelers as new sights appeared and others disappeared, and as the overall landscape reorganized around new landmarks. What the travelers thought was important subtly changed as a new world slowly came into view. Such changes were not due to any specific development – to any single distinction, discovery, technique, or person – but to the journey itself. This is what philosophers mean by the historicity of human action. Each group of travelers inherits a landscape, a way of thinking, an accompanying set of dissatisfactions, and a direction to head in to resolve these dissatisfactions, and in the resulting journey the landscape is taken up and transformed. At each step along the way the world seems to have a wild heterogeneity, possessing one order that does not seem inherent in the way the world appears to us – for the order we see in nature is due to our previous explorations and journeys – but to possess hints of another, inherent order that we might be able to see more clearly through inquiry. What Heaviside said of Maxwell’s work – ‘[I]t was only by changing its form of presentation that I was able to see it clearly’ – could be said by any of the individuals mentioned in this book. These individuals were discontent with what they saw, had an anticipatory vision of what might be, and the ability to organize an inquiry to seek it (philosophers call this process the hermeneutic circle). There will be no ‘final’ stopping point to the journey, for each new discovery – not to mention changing practical, instrumental, and theoretical contexts – works changes in the landscape. We will never stop being discontent, never stop anticipating, never stop organizing inquiry. Science could not happen in any other way, or it would be trivial or impossible.

  Most of the time, however, we care more about the equations, and about the things they help us do, rather than about the journeys that led to them. We tend to pay attention to the part of the world directly beneath our gaze. This is understandable and there are good reasons for doing so. But we can also learn much from studying the journeys – paths from ignorance to knowledge – that the scientific community, and individual scientists, took to these equations.

  First of all, we learn that the journeys are very different. Some journeys described in this book are short enough to take an individual just a few minutes. The Pythagorean theorem is an example; it allows someone with no mathematical training not only to grasp it but also to experience the thrill of discovery. Other journeys are extended: the journey to F = ma and Fg = Gm1m2/r2 justly can be said to have taken hundreds, even thousands, of years. Some journeys were taken in effect by a community of scientists constantly talking to one another, such as those to E = mc2, to the second law of thermodynamics, and to the uncertainty principle. Other journeys were traveled more or less solo, such as Einstein’s to his general equation for gravitation and Schrödinger’s to his wave equation, though such individuals in effect carried on conversations with colleagues even when working alone. There is, in short, no single road to discovery.

  We also learn that equations are not simply scientific tools but have ‘social lives’, so to speak. We tend to view equations as inert and mute instruments, able to affect the world only when wielded by scientists and engineers. But equations are active and can exert an educational and even cultural force, instructing us about the world and occasionally reshaping the human perception of it. The Pythagorean theorem teaches each new generation of schoolchildren about the meaning of proof, while Newton’s law of gravitation taught certain political thinkers about the meaning of laws. The second law of thermodynamics helps keep in check humanity’s utopian visions of free energy, while Einstein’s E = mc2 and his general equation for gravitation reshaped the human understanding of space and time on a fundamental level. Schrödinger’s equation and Heisenberg’s uncertainty principle force us to rethink what being a ‘thing’ means.

  Yet another thing we learn from these journeys is about the nature of scientific concepts. It is tempting to think that there is some pre-existing structure embedded in nature that we are only discovering and translating into mathematical language – that equations are descriptions rather than interpretations or creations. But how we translate depends on the journey we have already taken, on our dissatisfactions with i
t, and on how we responded to those dissatisfactions. We ‘fall up’, to adapt a phrase of George Steiner. It is thus misleading to picture science as proceeding solely by scientists producing new concepts, then testing and revising them. Two things are wrong with this picture. One is that the meaning of one concept depends on the meaning of all the others; a concept is one element of that fishbowl-like world that Newton discovered at the heart of the world we live in, and needs everything else in that fishbowl for its meaning. Testing one concept thinking you are testing it and not everything else is like asking, Is New York to the right or left of Boston? without knowing where you are; without having the rest of the map. And we not only need the rest of the fishbowl, but the rest of our experience of the world as well. A scientific concept that we trust is really a concept plus that experience, and when our experience changes – new practices, new technologies – so does how the concept applies to the world. That’s why concepts never stay put, and always change or are being elaborated; a concept that tests right at one time can be inadequate at another. There is no right way to say something that does not include our experience with it. Concepts are thus not determinative but indicative; they ‘point’ to something based on our experience, in full awareness that what they point to is going to change with further inquiry. Philosophers call this ‘formal indication.’ Concepts are formal because we can evaluate them rigorously and test them as being adequate or inadequate; they belong to a closed system. Concepts are indicative because they point to and depend on other things for their adequacy – all our experiments and definitions and technology and open-ended connection to the world – and when these change so can the formal elements as well. In fact, we expect that it will. Historian of science Peter Galison has a wonderful description of this. It is the theorist’s experience, he writes, that: