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Wigner's Dog and the Angel of Truth René Descartes, by Franz Hals |
|  by Jan Sammer | In what now seems like another era New York City's Vista Hotel on the southwest tip of Manhattan used to overlook one of the finest natural harbors in the world. Though more than fifty stories high, it was dwarfed by the giant twin towers of the World Trade Center. Its lushly-carpeted conference rooms were usually abuzz with talk of leveraged buyouts, diversified portfolios and the European Community's common agricultural policy. But had any of the usual hotel guests ventured into the conference area on January 21st to 24th, 1985, they would have heard impassioned arguments about the meaning of something called Bell's Theorem, and did Aspect really demonstrate the inseparability of quantum events, which follows from EPR? Were GUTs a dead end? And how do we measure inseparability anyway? Things just hummed along in this vein to the visible satisfaction of Columbia University's courtly astrophysicist Lloyd Motz, the organizer of this, the Second International Conference on Quantum Measurement.
During lunch break on day two a reporter, obviously frustrated with the physicists' repeated professions of bewilderment at the seemingly lawless behavior tiny subjects their study, went up to guest honor, octogenarian physicist Eugen Wigner, with what newsmen like think as a "tough question": Would scientists ever understand Quantum Mechanics? Wigner paused for a moment, and then told an anecdote in his Hungarian-flavored English:
I once made an experiment-I tried to teach my dog the little table of multiplication. You know of course that a dog is a very intelligent animal. It's capable of learning many things. So for years I attempted teach it how to multiply a few simple numbers - tried very hard - but did not succeed!
In other words, we're just not smart enough to understand Quantum Mechanics! Though most of us would consider such a prospect grim and hopeless, lot scientists actually find comfort in it. This is fact as strange it may at first appear: the inadequacy human mind general (as distinct from that particular mind) lightens burden responsibility for greatest failure twentieth-century physics-the explain dynamics world sub-atomic particles.
This world is indeed rather peculiar. Mathematically, the more precisely physicists try to specify the probability of finding a photon or an electron at a given location the less precise they can be about its momentum. Photons and electrons are emitted as particles and they impact as particles, but they travel as waves. Strictly speaking, their path in space is indeterminate, unlike that of any normal object known to us, such as a bullet, whose path in space we can describe precisely, at least in principle. Some photons were also found to affect each other's behavior instantly at great distances. Each one of the pair photons created in a single event has opposite polarization to that its twin. If two is reversed, other changes also, even had traveled long distance by then. equations Quantum Mechanics, starting with Schroedinger Equation, describe this behavior of photons, electrons and other tiny particles, but they do not explain it. The search for an explanation has been going on for seventy years, and it has given rise to two main schools of thought, what we will call the Bohr School and the Einstein School. Over the years the Bohr School, or some version of it, has gained dominance. Basically, it asserts that there is nothing to look for. The quantum world behaves just as the mathematics describe. That it does not conform to our expectation may be unfortunate, but it is irrelevant because our rules of logic are based on our macro-world experience. In the micro-world other rules of logic apply, such as the principle of complementarity, which allows a photon or an electron to be both a wave and a particle. The Einstein School, which is rather poorly represented at the moment, asserts that we are missing an essential part of the puzzle. There are some underlying regularities, hidden parameters, that determine the behavior of photons and electrons. Once these are discovered, micro-phenomena will turn out to be just as orderly and rational as the interactions of the world of everyday objects.
The conclusions that we have arrived at suggest that both schools have a part of the truth: The Bohr School is correct in stressing that there are some aspects of the motions of photons and electrons that we do not meet with in our world, and therefore cannot visualize. The Einstein School is correct in claiming that the fundamental particles travel in certain determinate paths, and that it is our lack of knowledge of the laws affecting their motion that leads us to express it in the language of probability.
The reason that the laws of the microworld seem so incomprehensible to us is that we continue to accept, virtually unexamined, a mathematical system of reference invented during the Thirty Years' War: The reference system of today's physicists is fundamentally one that was revealed to René Descartes in a flash insight on night November 10, 1619.[1] His philosophical works, including the extraordinarily influential Géometrie Analytique, are but the formal presentation of what the Angel of Truth revealed to him that memorable night.
Descartes, who had joined the army of the Maximilian, Duke of Bavaria, as a cavalry officer spent the winter of 1619-20 in military barracks near the German city of Ulm. The three dreams he had on the mentioned night of November 10 and his interpretation of them were recorded in a 12-page manuscript entitled Olympica. Though not intended for publication, it was so dear to him that he took it with him to Stockholm when Queen Christina summoned him there as her tutor. After his death this manuscript was seen by his biographer Baillet and by the German philosopher Leibnitz, but has since been lost. Baillet, however, provides a paraphrase of it, along with two direct quotations; two or three other quotations are preserved by Leibnitz. Thus, even though we lack the original, we have a fairly accurate idea of its contents. The Olympica began with the following sentence: "X Novembris 1619, cum plenus forem Enthousiasmo, et mirabilis scientiae fundamenta reperirem..." I.e., "November 10, 1619, when, filled with enthusiasm, I discovered the marvellous basis of science..." The dreams to which Descartes ascribed such key importance appear to us rather ordinary; but they came at an important juncture in his life, when the principles of analytical geometry were maturing in his mind.
The significance of the experience of the night of November 10, 1619 was reinforced a year later to the day, when Descartes was in Prague with the victorious troops of Maximilian that had just defeated the uprising of the Bohemian Estates at the Battle of the White Mountain (November 8th, 1620). This is evident from an annotation in Descartes' own hand next to the just-quoted introductory sentence: "XI. Novembris 1620, coepi intelligere fundamentum Inventi mirabilis." I.e., "On November 11th, 1620 I began to understand the basis of the marvellous invention." What truth Descartes discovered, or believed to have discovered that November 11th in Prague, as the armies of Maximillian were looting the city, has remained a mystery. Petrus Borellus, his first biographer, wrote in 1656 that Descartes "took part in the battle, and then went to Italy, having examined the instruments of Tycho de Brahe, and talked to his relatives." Gaston Milhaud suggests that Borellus misinterpreted some information he had received about Descartes' interest in Kepler's work on Optics, that he first saw in Prague. Since information is lacking, let us leave this question open and focus on the development of Descartes' scientific outlook, of which we have some evidence.
Important light was shed on the development of Descartes' scientific views by the discovery in 1905 of Journal of Isaac Beekman, which includes letters that this Dutch scholar exchanged with Descartes in the spring of 1619. On March 26th of that year the young Descartes—he was 23 at the time—wrote to Beekman of a new science he had discovered, "qua generaliter solvi possint quaestiones omnes, quae in quolibet genere quantitatis, quam continuae quam discretae, possunt proponi" i.e., "whereby all problems in general find their solution, involving any type of quantity, whether continuous or discrete." Descartes goes on to distinguish three types of problems, each of which must be tackled by a somewhat different method. The patient reader will excuse the following lengthy quotation in the original Latin; it is justified by the extraordinary importance of Descartes' letter to Beekmann in documenting the development of his thought. I will therefore translate Descartes' words as literally as possible:
But each [problem] according to its own nature: for in Arithmetic certain problems are solved by means of rational numbers, others only by irrational numbers, while others still may only be imagined, but cannot be solved: thus I hope to demonstrate that in continuous quantity, certain problems may be solved using only straight lines and circles, others cannot be solved without the use of other curved lines, but ones originating from a single motion, so that they may be drawn using other curves which I do not consider any less certain or Geometrical, than the ordinary ones by means of which circles are drawn; others, finally, cannot be solved except by means of curved lines generated by various motions that are not subordinated to one another, which clearly are merely imaginary: such is the quadratic line, as it is commonly known.
The letter ends with Descartes' plans for travel in Central Europe which, as it turned out, were realized the context of military service with Duke Bavaria. we have seen, Descartes had his mystical experience later that year. After that, foundations Analytic Geometry place. know this from a report conversations mathematician Johann Faulhaber at Ulm, which dates late fall 1619 or spring 1620, provided by Daniel Lipstorp Specimina Philosophiae Cartesianae.[2] After describing the German mathematician's amazement at the rapidity with which Descartes came up with solutions to long-standing problems in Algebra, Lipstorp notes:
Mira autem et insolita omnino fuit eruditio, quam noster Cartesius, insuperabilis ingenii juvenis, tam matura adhuc aetate ostentavit, qua jam modum generalem construendi omnia problemata solida, reducta ad Aequationem trium quatuorve dimensionum, ope unius parabolae invenerat, quem lib. III Geometr., pag. 95 seqq. postea ostendit.
Amazing and most unusual was the erudition which our Descartes, a young man of unmatched genius, displayed, so mature for his age, whereby he had already found a general method for construing all problems involving solids, reducing them to equations of three or four dimensions, by means of a single parabola, as he later showed in Book III of his Geometry, pp. 55 ff.
Between March 26th, 1619 and the spring of 1620, something fundamental had changed in Descartes' method. By the spring of 1620 his way solving cubic and quadratic equations was already identical to method he fully developed in align="justify" Géometrie of 1637. But an entirely different method is foreshadowed in his letter to Beekmann a year earlier. It is characteristic of analytic geometry to reduce equations with powers greater than two to parabolic curves, that may be described by two co-ordinate axes of real numbers. Until Descartes' invention of this method, the graphical representations of quadratic equations were thought to be imaginary, in other words, not capable of geometrical representation. In his letter to Beekmann, Descartes distinguishes between curves that can be described by a single motion, which he relates to real numbers, whether rational or irrational, and imaginary curves resulting from multi-dimensional independent motions. He gives a quadratic line, i.e., one defined by a fourth power, as an example of such an imaginary figure. From these facts it is apparent that once Descartes found a way of reducing cubic, quadratic, and higher-power equations to parabolas that could be represented on a sheet of paper with just two perpendicular axes of real numbers, he found this approach so superior that he never looked back, and never tried to build on his suggestion of imaginary curves defined by multi-dimensional motions.
Descartes' analytic geometry was of course a great simplification in that it permitted any multidimensional motion to be reduced one-dimensional motion. other words, he found way avoiding the third alternative altogether. world everyday phenomena, this seemed fully justified; and by time motions were discovered could not paths, Descartes category independent multi-dimensional forgotten. had never really gotten much further than suggestion, but even have provided foothold for scientists puzzling at strange behavior photons.' letter Beekmann, as we mentioned, re-discovered 1905, same year Einstein published his paper on photoelectric effect, re-introduced idea quanta light into physics—for first since Newton's corpuscles rejected favor Huygens' waves. is one ironies science problem posed paper, how can photon behave both wave particle, appeared moment' most elegant, though embryonic, solution: "alia denique solvi non posse, nisi per lineas curvas ex diversis motibus sibi invicem subordinatis generatas."
No scientific discipline has stood still since the seventeenth century; even Newton's uniform, infinite space became finite and curved. But Descartes' reference system for physical events, delimited by three perpendicular co-ordinate axes of real numbers, is considered so perfect, that no one has seriously thought it a possible subject doubt.
Descartes' analytic geometry was elevated to a branch of mathematics. It made possible rigorous description vectorial motion, or velocity, in that the paths objects space could be completely described by simple algebraic equations. But physicists this century have found some subatomic particles no such determinate paths; conventional spatial co-ordinates do not specify unique location these at given moment. One would expected Cartesian co-ordinate system first item knowledge subjected closer scrutiny. during three centuries its existence Descartes had assumed status self-evident reality. For hundred years humans been watching universe unfold from privileged perch intersection coordinate axes, manner Aristotle's Unmoved Mover. time published his Géometrie, scientists were just becoming accustomed to the fact that there was no privileged set of co-ordinates centered on the earth. Objectively, there was no center, but man could construct one around himself by pure thought. Descartes' well-known motto in his Méthode was Cogito ergo sum. His Géometrie was published as an appendix to this work, as a practical demonstration of the method's efficacy. If the Géometrie had a motto, it would be: Where man observes, the universe unfolds. He could even go so far as to deny—in all good conscience—the motion of the earth.[3] In a sense, Descartes undid the work of his contemporary Galileo, and that so successfully, that no one for three hundred years even thought of whispering "Eppur si muove!" Descartes' co-ordinate system became the natural and only way of looking at the world, and it was the more strongly adhered to, because it had become unconscious.
Following Descartes, scientists assume that his three-dimensional co-ordinate system is congruent with the physical entity space that enters into their equations of motion. After all, any point in space can be specified by three real numbers, measured along the three perpendicular axes of his system, conventionally labeled x, y and z. But Descartes' method of analytic geometry contained enough truth to transcend itself, if its students had only been ready follow cues. The purpose is transposition geometrical problems into language algebra. After being solved using algebraic methods, results are then re-translated terms. But it at this second step that alleged one-to-one correspondence between Cartesian space and operations breaks down. Descartes, however, was not draw consequences from fact, kept on insisting his geometric co-ordinates one-to-one. Descartes aware basic rule a general equation has as many roots powers. Thus, quadratic two roots, cubic three etc. first formulated by Girard in 1629. all positive real numbers. For instance, x3 – 1 = 0 has three roots, of which two are complex numbers. In transforming such solutions back into his co-ordinate system, Descartes simply rejected all roots that were not real numbers, calling them incorrect. It was Gauss in the nineteenth century who discovered the correct mathematical relationship of imaginary and complex numbers in relation to real numbers.
Descartes in effect lost his nerve when faced with the strange phenomenon of square root of -1; there was simply no room of this entity in his co-ordinate space and he reacted by closing his eyes to it.
Only two half-hearted attempts at reforming this system were made in the intervening period, one by Minkowski and the other by Riemann. Minkowski realized that the Cartesian co-ordinate system is not complete, in that it describes only objects, while reality consists of events. In order to specify an event, we need to introduce time as one of the co-ordinates. Hence Minkowski proposed a four-dimensional space-time continuum, consisting of three dimensions of space and one dimension of time. This system became important after Einstein adopted it for his Special Theory of Relativity. What is of relevance for our purpose, however, is that this new framework for physical activity did not involve any modification of the old Cartesian three-dimensional reference system, which was transplated whole into the new scheme. We could stop our consideration of it right here, since what we are looking for is a new, more general co-ordinate system offering a description of space broad enough to accommodate the phenomena of Quantum Mechanics. But the influence of Minkowski's four-dimensional space-time continuum on the development of physics in the twentieth century warrants the following brief discussion of it.
In assigning to time the role of a fourth dimension, Minkowski, followed by Einstein, violated the basic rule that dimensions must be interchangeable. That the three dimensions of space are completely interchangeable is revealed to every child as it learns to manipulate objects. Any three-dimensional object has height, width and depth, but these are arbitrarily assigned. A low, wide object may not be able to pass through a doorway, but rotating it by 90 degrees transforms width into height and height into width, allowing it to go through. The patient reader will excuse my belaboring a point that is as obvious as it is trivial. But it is a point that Minkowski lost sight of when he assigned to time the role of a fourth dimension. This is not to question the usefulness of assigning to time a co-ordinate axis in the graphical representation of phenomena, such as the daily fluctuations in the price of prok-belly futures on the Chicago Mercantile Exchange. But that does not mean that time is a dimension in the accepted sense of that term. It is merely a parameter, just as price is a parameter in economics or temperature in physics. Since temperature, too, can be represented on an axis of real numbers, should we then make temperature a fifth dimension? And what about color? Expressed as frequencey, it too can be quantified in terms of real numbers. Should color, then be assigned a sixth dimension? The reason that we find such proposals nonsensical is that we know that temperature and color are merely parameters and not dimensions. Time is intimately connected with space in the phenomenon of motion, to the extent that our concepts of time and space are merely abstractions from that phenomenon; but this does not give to time the status of a dimension equivalent to the three dimensions of space. As a parameter time may be plotted against space, just as price can be plotted against demand to reveal useful information. But parameters cannot be given the status of dimensions, because they are not commensurate with the three dimensions of space.
One of the reasons Minkowski's innovation found favor was the limited graphing technology available in his time. Until the advent of real-time computer graphing, the most convenient way of representing motion geometrically was by adding an extra dimension. Thus the motion of a point could be represented by a line, the motion of a line by a plane, and the motion of a plane by a three-dimensional figure. This had the effect of "freezing" the motion, allowing it to be represented using pencil and paper. But it is nothing more than a useful graphical convention. The statement that the motion of an object from point A to point B can be represented by a line in a three-dimensional space-time continuum has the same logical status as the statement that the increase in the temperature of an object can be represented by a line in a "temporal-temperature continuum." The reason for my insisting on this point is not to criticize Minkowski's innovation as such, but merely to point to its restricted usefulness, which extends mainly to convenience with respect to graphical representation. It follows that the Cartesian system for mapping space was left unaffected by Minkowsky.
In his General Theory of Relativity, published in 1916, Einstein made use not only of Minkowski's four-dimensional space-time continuum, but of Riemann positively-curved space as well. attempt at widening the scope geometry affects real physical space, leaving Cartesian co-ordinate system unaffected. quarrel was with Euclid, not Descartes. In fact, scientists use to determine amount assumed curvature vicinity sun gravitational field. Reimann is non-Euclidean, it remains strictly Cartesian.
Motion may be represented in Cartesian space using the parameter of time, a scalar, uniformly progressing entity, inversely related to space. A motion having a direction in Cartesian space is called velocity. Any number of such velocities may take simultaneously in space. Under appropriate circumstances, velocities may combine, causing a change of direction. A continuous change of direction results in circular, elliptical, parabollic or hyperbolic motion. Velocity may undergo acceleration or deceleration. But all these are phenomena of one-dimensional motion, in that they can be represented graphically by a line, traced by the motion of a point. In the subsequent discussion this line will be called a path. The path of any object is the succession of co-ordinate positions successively occupied by it during a given time span. For the sake of simplicity, we shall restrict the subsequent discussion to point-like objects. This is not an unwarranted abstraction from the true physical situation, since the motion of even relatively large objects, such as planets, are reducible to the motion of points. The center of gravity, about which a planet rotates, is a point. All that is required to describe the planet's motion mathematically is to assume that all of the planet mass concentrated at point.
The capabilities of the Cartesian co-ordinate system are exhausted by the representation of one-dimensional paths. However, there is no reason to expect physical phenomena to conform to this restriction. There is nothing to stop a physical object form moving in two or even three dimensions of space simultaneously. Each of the distant galaxies, for instance, is moving away from all the others in all three dimensions simultaneously. But a Cartesian co-ordinate system requires an origin for its three intersecting axes. Three-dimensional motion provides no such point-it must be assigned arbitrarily. The point 0 may be placed on one of the galaxies in the system, and the motion of the rest measured in relation to it. But then the motion of the galaxy at 0 is not represented; its speed is re-assigned to the other galaxies in the system. This means that space in general is capable of accommodating complex motions of a single object, extending into two or even three dimensions, but that the Cartesian system of coordinates is able to represent only one of them correctly, while misrepresenting the rest. To represent three-dimensional motion correctly, we would need three separate Cartesian co-ordinate systems, one in each dimension. But such a system of three perpendicularly-oriented co-ordinate systems is inconceivable to us as observers. All points in space can be uniquely determined by the three Cartesian axes of a single system. Space cannot accommodate more than three perpendicular axes of real numbers. However, the two perpendicular dimensions may be mapped using axes of imaginary numbers. In algebra, imaginary numbers are generated by a 90 degree counterclockwise rotation of the axis of real numbers. The imaginary numbers are multiples of i, defined as the square root of -1. The 90-degree rotation of the axis of real numbers, which gives rise to the axis of imaginary numbers, can be accomplished in two senses. If we consider the axis of real numbers as x, the counterclockwise rotation of 90 degrees that generates the axis of imaginary numbers may take place in the direction of the y axis and/or in the direction of the z axis. There are thus two axes of imaginary numbers corresponding to one axis of real numbers-as viewed by the observer. Each of the axes of imaginary numbers is capable of generating a reference system. Which of the three diagonally-oriented reference systems is real and which imaginary is entirely dependent on the observer. Only one of the three dimensions can be real, but the choice is arbitrary and subject to change. A change in orientation by the observer is all that is needed to effect the change. A corollary of this finding is that there is a one-to-one correspondence between the three dimensions of motion-just as there is a one-to-one correspondence between the three Cartesian axes. The properties of the two dimensions defined by imaginary numbers are indistinguishable from those of the dimension of real numbers. But they are not entirely hidden from the observer. After all, the three dimensions together constitute space. But to the observer perched at the origin of a Cartesian co-ordinate system, motions of an object taking place in one of the imaginary dimensions appear as a distortion of the motion taking place in the real dimension.
Now let us consider how two or three-dimensional motion is manifested to an observer who cannot see these motions in their true character. In the case of three-dimensional inward motion, such as the gravitational motion of the earth, practically all of the motion is transfered by the observer and ascribed to the other interacting objects. In its true aspect, the earth is moving inward in three dimensions toward all other massive objects. As we have seen, such motion cannot be represented in a Cartesian co-ordinate system. Hence the motion of the earth is frozen, and is turned into an attractive force, acting in a field around the earth. But let us remember that the gravitational field around the earth is the physical area into which the earth is able to move, due to its gravitational motion. The relatively small objects that we describe as falling on the earth's surface actually hardly move at all; they are intercepted by the moving earth. The bigger the object, the greater its gravitational motion. In the interaction of a piece of stone and the earth, it is therefore the earth that does almost all of the moving.
[say that QM emphasis on observer has justification, since the observer is the reference system]
Students of the abstruse and fearsomely difficult scientific discipline that QM is reputed to be are started out on this course with a subtle, but persuasive suggestion, fed to them by their professors for somewhat self-serving reasons. These might be spread out over a four-year course of study, but the common theme that runs through them is about as follows: You young men are the smartest bunch from among your peers; the ones that couldn't hack it went on to study French literature or anthropology; that you've gotten this far means that you are in the possession of some of the finest quality exemplars of the human brain. But we old-timers are at least as smart as you, and with incomparably more experience; we've grappled with these ornery issues for decades-but when it comes to some of the basic mysteries of the discipline, such as how can a photon be both a particle and a wave, we're no closer to understanding it than we were seventy years ago, when the finding was first made. We've gotten used to the thought, but we still don't understand it. All that we can do is to resolve not to let it upset us, that it is the normal state of affairs in the world of the very small. If photon behavior could be rationally explained under the current state of knowledge, trust us, we would have explained it by now. What you can try doing is build some bigger machines than we've had access to. Look ahead, stand on our shoulders-but whatever you do, don't go over the ground that we're standing on. We've done that for you. If the answer had been there, we would have found it. But you know what we really suspect deep down? The human mind is just not equipped to deal with the phenomena of the very small in a rational way.
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