Abstract The modern "pictures" of Euclidean geometry are incongruous with its modern analytical Cartesian understanding. Some details of this are explored, along with historical paths that led to this unfortunate situation which infects even topology and logic, and the mathematics upon which these depend. Spinors, in particular, are creatures of analytic geometry, and do not arise intrinsically from either quantum theory or relativity. There is a long way to go in understanding the fundamental aspects of "simple" Euclidean geometry. This is also a longwinded explanation of why and how, spin-1/2 particles must be understood, in terms of classical Euclidean geometry, to be pointlike, and even classically irreducible: why supposed classical models by angular momentum, though interesting, are doomed to failure as genuine models of the mathematical physics of spin-1/2. Spin-1/2 is already a matter of classical geometry and does not need attached fairy tales to explain it. Points of physical space are not well modeled by Euclidean points, even in a completely classical physics or Euclidean geometry. This is not something I had wanted, much less expected to show. Up to now, I had resisted the idea that an electron could be pointlike; now, I understand better the concept and picture of "point": it is not what I was taught in my mathematics or physics classes. A physical point is Q-smeared, and, even without any Q-smearing by intrinsic uncertainty, posseses algeraic structure. This should put an end to the vast number of physics papers over many decades devoted to such projects of explaining putatively quantum or relativistic particle spin in terms of constipated pictures of Euclidean geometry. Spin is already a matter of classical Euclidean geometry seen from a Cartesian point of view. It requires no applications of quantum or relativistic physics.
This is a companion to Origins of the Species of Time.
Classical physics and mathematics can be seen to have at least two different beginnings. Where perhaps most physicists' minds go is to the beginnings of Greek natural philosophy around 600 BCE, as in the traditions of classicists.
The beginnings might
equally be said to start with the Sumerians and Egyptians, around 3000 BCE,
continued in the Egypto+Sumer-Akkad-Babylonian tradition which
contains a good amount of the mathematics and astronomy that
influences our mathematics and time keeping even to this day
(360 degrees in a circle, 60 seconds in a minute, 60 minutes
in an hour and 24 hours in a day with ancient calendars having
360 days in a canonical year).
That older line
also contains a good amount of mathematics and science that
was either ignored or forgotten by the Greeks, much of which
involves number systems and their calculational techniques,
many of which were lost, forgotten or ignored from Greek
Archeologists (one might coin "archeoepistemologists")
have begun to recover some of them.
[Should you think, as a pure mathematician, that the representational system of numbers is somehow irrelevant to the abstract body of pure mathematics, I invite you to attempt algorithms for decimals and long division using only Roman numerals. Mathematics always comes down to actually computing numbers that correspond to measurements, and that is even more so in physics. Proofs by construction are far more fecund than proofs by Reductio ad Absurdum. It is completely cool to know that solutions to certain problems exist; it is better to have a specific solution in hand that can be constructed by algorithm; such things inevitably involve numerical calculation.]
A true beginning to point out with the Greeks is that mathematical logic distinguishes propositions that are "provably true." There is nothing that I am aware of predating the ideas of logic from Thales of Miletus (ca. 624 BCE - 547 BCE) and Pythagoras of Samos (ca. 572 BCE - 497 BCE). The famous Pythagorean Theorem was known and understood a millenium before Pathagoras. This is one of the first connections known between geometry and numbers that Descartes deals with many centuries later.
What physics (astronomy) and mathematics may have originally arisen in other earlier cultures around or even before the Egypto-Sumerian line seems, at the moment to be lost to us through lack of written records. On the basis of some Greek and Roman historical writings, there are possibilities in both Vedic and Keltic cultures. What after all were the circular megalithic structures of, e.g., Stonehenge all about? No written records are available that tell us anything about them, so many interesting and curious stories can be made of them, guessing by their current structures, and how their structures might have been at various times in the past.
In the later Greek classical era, the Greeks later invented a formal philosophical system of manipulation of linguistic symbols that we call "logic", the grand exponent of which is Aristotle (384 BCE - 322 BCE). , [Internet Encyclopedia of Philosophy], Aristotle [Wikipedia]. Aristotle. A great observer of the world, he attempted to organize and make sense of the world he saw, and in so doing laid the foundations of logic, physics, psychology and literary criticism, among others. Continuing the ancient line of thought contained in Greek (γεομετρια), he understood geometry of space as a part of the description of physical reality and not simply as an abstraction.
We now speak of deductive and inductive logic, the latter relying on a proper understanding of probability and statistics; but, it is not difficult to understand that the very creation of deductive logic was had by inductive means: it did not somehow descend from heaven magically as Athena was born fully grown from the head of Zeus by a simple stroke of the axe of Hephaestus. Analogously, the priest class of ancient Sumer developed the mythology that their written cuneiform writing was given by the gods, but its development over centuries belies that manipulative fantasy.
It is also not difficult to understand that deductive logic needed the sense of a spoken language that was evolved culturally: neither did language descend magically from heaven, nor did its written forms by which one generation can communicate to the next with more accuracy and precision than oral tradition provides.
This is not to say that oral traditions are then made irrelevant; the opposite of that conclusion is an important point being made here. Precision and accuracy can also often be better propagated by an oral and interactive process, teacher to student. Some understandings cannot, and are not written with precision and accuracy - as masters and good students of Zen Buddhism would understand immediately. They are induced thought patterns outside of language, and certainly outside of written symbols of language.
The object is, of course, to grasp the language and the symbols by which it is represented in terms of a conceptual level of understanding.
The writing down of certain things was even forbidden in some cultures, perhaps because it degraded the art of memory. We are now so dependent on the writing of things because no human being being can remember the volume of what is called knowledge, even in a relatively small rarified area, e.g., analytic number theory. If one thinks of areas cut out with a more broad sword, like, mathematics, physics or philosophy, the memory requirements are almost beyond conception, much less human abilities.
But, the oral traditions have never disappeared, though they seem to be disappearing now when we need them desperately. Scientists and scholars tread narrow paths through forests of knowledge to get to an unknown frontier, and it is taking longer and longer to do that. Elder guides are needed, and they are disappearing for too many various reasons.
Euclid ca. (325 BCE - 265 BCE), summarized and organized the subject of Greek geometry in his systematic "Elements" on the subject, which also happened to include the foundations of number theory, and which became the foundation of the teaching of mathematics that persists to this day, but with about a 1000 year hiatus of the oral teacher-student tradition which can be traced historically as being essential in all the arts and sciences. I was rather amazed at one time to realize that my musical composition pedigree in these relationships goes back to Haydn.
Over the centuries of this new, derivative teaching, there arose a set of "pictures" of "Euclidean geometry" that are purported faithful descriptors of Euclid's combined axioms of points and lines (extended to surfaces) with an Aristotelean manipulation of the linguistics involved. This was not a mathematics that was as potently symbolic (and linguistically specialized) as is modern mathematics.
Mathematics has become so much of a language in itself that modern mathematicians can make sense to each other using standard written mathematical language without being able to speak each other's human language.
Mathematicians have long since extended the general ideas, axiomatics of geometrical pictures from 2 to n dimensions [Sommerville 1958] and even to the curved spaces which quite literally leave Euclidean geometry specifically flat, beginning with Bolyai (1802-1860) and Lobachevsky (Lobachewsky) (1792-1856) who simply began questioning Euclid's 5th axiom and its logical independence. The conceptual breakthrough to generally curved spaces reaches an apex in the work of Bernhard Riemann (1826-1866).
The important ideas of René Descartes (1596-1650), [Internet Encyclopedia of Philosophy], finally combined the notions of ancient Euclidean geometry, together with the developed pictures or descriptors with the older line of numerical calculations leading to the Islamic developments in algebra. It is unfortunate that the novel and inventive Descartes became intertwined with both continuing Western theism (spirit converted to law and logic) and the equally pernicious escape from it called "deism" (spirit converted to a machine). Neither supports reality, and both have the same level of epistemological enlightenment - commensurate with that of the tooth fairy, a product of Scholastic "thinking", the ultimate perversion of Aristotelean logic as instigated in Western theologized philosophy by Thomas Aquinas. There are those who would disagree with the perversion part; I do not care. See, perhaps Sketches in the History of Western Philosophy for an academic and detailed statement of the historical realities.
The Islamic line of mathematics and science, rather probably stemming from the Sumerian line was introduced to the Western world during the time of the so called "Reconquista" (718-1492) [Wikipedia]. and the fall of Toledo in 1085 CE. (The Reconquista is one of those great lies of history: there was no taking back of Iberian lands by Christians since land was occupied by Vandals, whose name has wrongly been taken pejoratively into English.) Islam Spain and the history of technology At the same time, many of the destroyed works of the ancient Greeks, perhaps especially those of Aristotle were also rediscovered mostly in their Arabic translations.
The continuing scholarship of Islamic civilization from both the Greek and Sumerian lines were essential to the "catching up" as it should be said, of previously destroyed knowledge in the West.
Artistically and intellectually speaking, the translation from the mostly unknown (to the West) Greek into Latin of the Corpus Hermeticum, commissioned by Cosimo de Medici in 1440 is the seminal, and mostly ignored, turning point of Western culture that together with the contents of the great library at Toledo leaves Western culture not still wallowing like pigs in mud holes. [There is a great deal of nonsense now surrounding the Corpus; beware.] See Marsilio Ficino (1433-1499). The cultural turning point was not simply a matter of the decline of medieval sociology and the rise of an economic middle class as many "history" textbooks would have one believe. Seek simplicity - and distrust it. The Renaissance was indeed, as the name says, a rebirth of and rediscovery of culture, art and science that had been deliberately suppressed and destroyed. Why would anyone think that all had been reclaimed?
Modern cultures, regardless of mother tongue, have adopted the rather perverse notion that language, and its methodical manipulations, is somehow the faithful encapsulator of all thought; what rubbish.
These materials learned through the Renaissance were "rediscoveries" of not only forgotten, but suppressed and destroyed knowledge of and from the ancient world. Much is still lost, unknown and most probably irreclaimable.
"Dark Ages" are where the history has been destroyed; there are a number of them, in various cultures. These are eras of glossed over history, all too gently placed bookmarks that ignore the fact that both history and knowledge is too frequently lost, and all too often by purposeful destruction.
During the ages from Euclid and Archimedes (287 BCE - 212 BCE) to the Renaissance and its intellectual origins in the library of Toledo, the languages, texts and translations of ancient texts had to some extent been regained, but the thought patterns were lost, and remained lost. They are still lost, in the same sense that the arts, the art of science or the arts of engineering can be lost.
The continuing lines of understanding, thought and "tricks of the trade" that pass only from teacher to student had not been merely severed; that line had been forcibly annihilated. We can only guess at what came before.
The teaching of geometry is the subject, and it was a resuscitated subject, raised from the dead many centuries after Euclid. (One could say the same of reason, logic and Aristotelean philosophy, philosophy generally, optics, biology, and a scientific concept generally, not to mention simple curiosity.) There is no available record of how our cultured ancestors may have thought about their formalisms, or no record of their results. We have documents on ancient Greek musical theory, but have little idea of what the practices were, and there is also little idea of what the nouns actually symbolized. What was, really, what we call a "mode"? Only the destroyed line of student to teacher connections might have told. Words can have subtle, and not so subtle distinctions in their use, depending on where and when they are used. The word "raga" has rather different meanings in Northern and Southern traditions of classical Indian musical theory: it does not signify by itself, a unified and well defined concept, as it might appear to a westerner.
Possibly, if that line of connections had not been broken, it would not have taken 1000 years for Descartes to have caught up with the connection of geometric thought with the beginnings of algebraic thought that is also, at least conceptually but not exclusively, connected with the schools of mathematics and astronomy that existed in India (Kerala) at least as far back as 310 BCE with Indian mathematics - and I suspect the existence goes considerably further back. As a matter of lost connections, one might simply notice that Elamite is a Dravidian language [Wikipedia], as is Malayalam, the language of Kerala, intimately related to Tamil.
A point of this small history is that when the ancient mathematics and science had to be reconstructed from manuscripts and translations of translations, the thought patterns that lay in back also had to be reconstructed - which is almost to say fabricated from thin air (or fat air?). While many of the early writings can be read literally and translated, the corpus of interpretive thought in back of them is lost to us.
There are two pertinent peculiar parts of this story. One is that the pictorial descriptors are fabricated, much later than the Greek mathematical texts and that we mistake them as truly representing Greek thought. The second, which is truly rather amazing, is that these fabricated descriptors have for the most part dominated all mathematical and physical thought, even beyond the work of Descartes, despite the fact that this combining of algebra and geometry opens up entirely new worlds of mathematics and actually does it in unequivocal terms lead to quite different descriptors of geometry.
It might be worth mentioning that while Descartes did indeed work out the utility of using algebra to solve geometric problems, in Le Geometrie, an appendix to his Discourse on Method, he did not conceive of analytic geometry, or even the celebrated "Cartesian plane" as we know them today. Descartes' Life and Works Arguably, Fermat (1601-1665) had as much to do with such developments. They were not exactly on friendly terms, Descartes being the arrogant prick that he was. Even well before this, Al-Mahani, ca. (820-880), was engaged in the solution of geometric problems by algebraic means. Relationships between the algebra of Al-Khwarizmi ca. (780-850), number theory and geometry were in fact signal aspects generally of this much earlier Arabic mathematics that itself was well aware of prior mathematical and astronomical works from India.
The idea that geometry was a theory of the physical world and not merely some cute mathematical structure was part of ancient Greek physics, and that idea remains in modern physics, without bothering to explore the new worlds of physical geometry opened up by Descartes, et al. The word itself, (γεομετρια), meaning "earth measurement" should be clue enough.
This is not to say that mathematicians have not done the exploring, because they have; I simply mention algebraic geometry and geometric algebra along with the writings of W. K. Clifford (1845-1929) [Wikipedia], The Work of W.K. Clifford B. Riemann (1826-1866) and A. Einstein (1879-1955) [Wikipedia] that lead to the Clifford-Riemann-Einstein program of geometrizing physics which is quite impossible without understanding connections between algebra and geometry. We do not get a conceptual leg up on this program if we noodle about with, wrong and restricted pictures that ignore and defy those very connections.
Physicists' notions of space and time geometry with the small exception of the relativistic spacetime, have not changed much (in operative consensus) to include the Cartesian relations in centuries.
Because of this, physicists perhaps, have strange and antiquated working pictures and notions of space and time that produce confusion and paradox where there should be none.
The clearest example of such confusion is with the idea of spinor, (which we devoutly wish should have the universal spelling "spinnor", so that it is truly pronounced as it is spelled) over which much fuss and many incantations have been said repeatedly, both relativistically and quantum mechanically. This mystical nonsense and confusion continues in physics textbooks to this day.
Spinors, and indeed Clifford algebras [Wikipedia] (an article with incisive mathematical particularity) have absolutely nothing to do with relativity theory or quantum theory per se. They are purely aspects of classical geometry as algebraically suggested in principle by Descartes, and which can be developed with no additional physical assumptions, through a little more thought and even less formal algebra.
Clifford algebras can arise in several different ways, all of which have origins in classical Cartesian-Euclidean geometry. One way is by simply considering the factorization of a quadratic form, which is after all exactly what Dirac (1902-1984) (Paul A.M. Dirac - Biography and Nobel Lecture) did to derive his equation for the electron with spin-½ from the Klein-Gordon equation. [Wikipedia] However it may seem, there is nothing magical here. The idea is routinely propagated that the electron spin is somehow a relativistic phenomenon. [Dirac equation - Wikipedia] when very clearly, it is not: the very same kind of factorization of the Laplace operator (a quadratic differential operator) [Laplace's equation - Wikipedia] in the nonrelativistic Schrödinger equation [Wikipedia] can be done, and it yields the Pauli algebra, [Pauli matrices - Wikipedia] an irreducible representation of the Lie algebra Lie algebra [Wikipedia] su(2), [Special unitary group - Wikipedia] whose irreducible representations describe quantum spin [Spin (physics) - Wikipedia] generally, as it is understood in an Euclidean space of three dimensions. One can, in fact, pull this very same "factorization trick" in any inner product space over R or C in any dimension, and with any signature, and so discuss the spin representations of SO(p, q) and SU(p, q), and associated Clifford algebras generally.
Physically, Dirac's equation which takes the 2x2 ad hoc Pauli algebra to the 4x4 Dirac algebra merely adds the difficulty of ± signs for E by virtue of taking a square root; both the relativistic Klein-Gordon equation and the nonrelativistic Schrödinger equation factorize to expose spin-½.
Moreover, both these equations are equations expressing a conservation of energy, as are all the core equations of physics.
Regarding Dirac's factorization, we might just as well have written
and factorized the quadratic form of the LHS of the equation. The appropriate Clifford algebra would still have materialized by purely algebraic means in a nonrelativistic context. Now figure out what that means - *geometrically*. This is not difficult if you transcend the erroneous idea that mathematical points are structureless. (Physical "points" as irreducible geometric atomics are even more complicated.)
There are many ways of conceptualizing spinors (a hint, perhaps to their necessary ubiquity), one is as "square root of a vector", which seems only to be reasonable in a 3-Dim space. Spinors, can also be understood sturcturally as two sided ideals in Clifford algebras, becoming the elements of the carrier space of the representations of the Clifford algebras, and then of orthogonal Lie groups. There is no magic; just simple algebra that has nothing to do with quantum mechanics, and has also nothing to do with relativity theory.
Returning to the idea of square root of a vector, and generalizing via axial vectors to bivectors, understand spinors as eigendirections of antisymmetric forms that represent bivectors. I thank R. M. Kiehn for this understanding and connection.
Any student of quantum physics who has looked at the meanings of spinors has encountered the "belt trick", nicely explained by John Baez in week61 of his always enlightening "Finds in Mathematical Physics". It is an explicitly macro topological phenomenon, not a matter of quantum fiddling.
There is nothing, either relativistic or quantum mechanical about this; it is a macroscopic, yet non global topological property of the physical E3 in which we seem to exist. We seemed to have developed, on the basis of our false descriptors, the idea that topology is only considered as local or global, and that there is no significant understanding inbetween. The belt trick actually puts the lie to that assumption, since it exists, is demontrable and *is* in between.
With regard to the rotational symmetry of E3 the appropriate connected Lie group, SO(3) is *doubly* connected, and this is true for any SO(n), n > 2. This may have something to do with the mathematical existence of the spinor representations of rotations; on the other hand, the Lie groups SU(n) also generally have spin representations which are extentions by complexification from SO(n). An important aspect of spinor representations is a confluent isomorphism between associated Lie algebras, e.g., the isomorphism between the Lie algebras so(3) and su(2). But, while the associated spin group for SO(3) happens to have a nice confluence with a classical SU(2), this is not the state of affairs with SO(n) for n > 2. Spin groups are generally not classical Lie groups; the case of SO(3) is an isolated accident, but given the stress that this relationship is given, it can lead to gross misunderstandings of the mathematical reality. [This description does not answer all questions, nor does it pretend that all relevant questions can be answered by existing mathematics or by me; many geometric questions of the Cartesian viewpoint remain.]
There is nothing magical about any this; it is all "classical" (but unfinished and possibly lost) mathematical understanding, and also most interestingly, a new understanding of classical physics. Physical reality does seem to take advantage of possibilities of the mathematical model that combines the logic of pure Euclidean geometry with the natural algebraic extensions by Descartes - which is interesting, bemusing and almost annoying because of the simplicity. If you can get the mathematics understood, the physics is not far behind. This is a bit more Platonic than I was prepared for.
What is the problem, and why does this stuff seem mysterious? It *seems* that way, because we have paid entirely too much attention to and placed entirely too much stock in those long ago fabricated descriptors, and authoritarian text books.
We have become bovinely wedded to various descriptorial notions, in a 1000 year disconnection of thought patterns, of Euclidean geometry that refuse the Cartesian understandings, e.g., the idea that "physical" points have no structure - because that is how we have been taught to picture them, and *not* because the mathematics tells us otherwise, which, in fact, it does. It is also the model that our vision supports.
The question is, what does the physics tell us about the mathematical models of geometry and the pictures that we "glue on" to them? The physics tells us that the developed Cartesian mathematics is quite right, and that the simplistic Medieval geometrical pictures that do not faithfully represent the Cartesian-Euclidean formulation in algebraic mathematics are wrong; they are overly simplistic to the point of serious mathematical error in physical theory.
Why should there be a problem? The only problem is that we have persisted in "picturing" Euclidean geometry wrongly while the mathematics has told us rightly that our pictures are wrong, and have been, for well over a century. The correct ideas have been available all along; they have simply been ignored. They need to be refigured and reinstalled in the minds of young mathematicians and mathematical/theoretical physicists.
These ideas are not at all new, and were essentially understood by W. R. Hamilton [Biography of Hamilton] and J. C. Maxwell (1831-1879), though they did not enunciate them so forcefully: it was an embryonic time for the resurgence of these concepts, and they were unsure.
In the first edition of Maxwell's great treatise on electromagnetics (1873), he did indeed flirt with idea of formulating electromagnetic (EM) theory in terms of Hamilton's William Rowan Hamilton (1805-1865) [Wikipedia] Hamilton quaternions, [Quaternion, Wikipedia] and did a number of translations of the equations into a quaternionic language. In the following edition of 1884, by Oliver Heaviside (1850-1925), the quaternionic sections were removed, and this probably was a great infortuity. They were almost side notes in the general progression done in Gibbsian [Josiah Willard Gibbs (1839-1903) Willard Gibbs, Wikipedia] vector language, and may have seemed to get in the way.
O. Heaviside edited like some modern movie editors who have the strange idea that "moving the action along" is somehow always more important than actually making sense in telling a cohesive story. You can get Maxwell's first edition from good libraries (interlibrary loan) and see that I have described the historical and mathematical realities reasonably accurately.
It should be noted, however, that Maxwell did write a manuscript in 1870 on the application of quaternions to electromagnetism that is reprinted in Vol. II of Maxwell's collected works. Maxwell was not thinking in field theoretic terms, and did assume a physically real aether with classically physical proprties, and had the mathematical machinery available for waves in an elastic medium, both vector and scalar parts, as given in [Morse 1953], pp 142-144, where it is shown that superluminal scalar waves are possible only if the medium is compressible. If spacetime is actually compressible, it should only be so at energies that we have not yet achieved, somewhere in the neighborhood of greater than 250 GeV.
True historical reality is much like the reality of physics in that we never get to perceive it directly, but only indirectly through constructed models and theory applied to them. Since historical models are rarely deniable and often fabricated, they have an innate dubiousness, and do not suggest a convergence on any truth, and so should be taken only with the proverbial grain of salt.
There are those who have claimed that Maxwell originally cast EM theory in quaternions, that the equations were somehow more general, and that some sort of conspiratorial suppression of the "real truth" was engaged in, particularly by Heaviside. They speak wrongly; the quaternionic equations were not more general, and Heaviside simply did not understand the purpose of quaternions; nor was Maxwell particularly clear in their significance, supremely careful mathematician and scientist though he was.
Written history does have a clearly conspiratorial aspect, but this one is pure confabulated nonsense. The first edition of Maxwell's treatise on EM used Gibbsian vectors throughout, with quaternionic afterthoughts, and, as Maxwell himself shows in this treatise, the quaternionic formulation is isomorphic to his own vector formulation - which to a large extent missed some of the very interesting points of modern EM formulations by David Hestenes (1933-) [Wikipedia] in terms of Clifford algebras, or as Hestenes would put it in "geometric calculus". Further developments in understanding classical EMT can be found in the works of [E. J. Post 1963] and R. M. Kiehn: Maxwell Theory and Differential Forms
Prof. Kiehn also gives another way of understanding spinors as eigendirections of an antisymmetric matrix that has implications for the understanding of the Maxwell equations and its spinorial solutions.
I seem to remember also an earlier work of Penrose on spinorial solutions to the Maxwell equations related to his twistor theory [Wikipedia], in the Journal of Mathematical Physics, but I have not yet rediscovered the citation.
Maxwell's use of quaternions is considerably more messy and less elegant than modern formulations, showing, I would think, that Maxwell was greatly intrigued by quaternions, but that he was also not exactly facile with them; on the other hand, neither was anybody else. Hamilton himself continued to search for their logic and meaning through to his death. Quaternions are not a part of the general education of mathematicians, even today; a pitiable condition.
Maxwell was definitely not talking about functions of a quaternionic variable, as say, noncommutative extensions of the theory of analytic functions of a complex variable. This mathematical theory of noncommutative analysis was not yet a developed subject at the time of Maxwell, and is only now a subject of research. See also Notes on Noncommutative Geometry
All of quaternions (for DIM=4) and Clifford algebras have now elucidated a generalized meaning of EM theory, as has also the approach to the Maxwell equations though differential forms. EM is clearly a topological (not metric) theory that expresses itself in antisymmetric forms (not symmetric forms). You can express the same sorts of equations on discrete, simplicial [Notes on Simplicial Homology], networks using coboundary operators [Sorkin 1975], or Kuratowski closure operators. RMK Articles: Specials and Freeform Index Page [R. M. Kiehn]
One of the extended meanings of the Cartesian-Euclidean understanding is that the geometry of physical space and time should most generally be understood in at least locally complex coordinates, in addition to the addition of the noncommutative Cliffordian and nonlinear spinor structures associated with physical "points". Physical points do indeed have physical structure, and that structure seems best described by both a complex and Cliffordian nature. This is an understanding that I believe Clifford himself had, but that has been only sporadically picked up on.
In building classical spinors on an E3 e.g., by starting with Eli Cartan's map of E3 to Hermitean 2x2 matrices, [Contexts for Spinor Algebra], we discover the Pauli matricies, and also that the building of spinors as null vectors, that we must allow that the real vector coordinates of E3 must embrace a complexification that must be meaningfully geometric. One might also simply recall See [Goldstein 1950]. "Cayley-Klein" - Google Search in E3 and their purely classical origins, and uses. This is mathematics that begins in the the 19th century, before the physical quantum and relativistic theories - and has everything to do with spinors, and the Pauli algebra, su(2) of the covering group of so(3).
A similar pattern of necessary extensions from the old Euclidean descriptors of E3 to their complexification and Cliffordization can happen in any E(p,q), p+q=n. It might be worth noting that when a real Clifford algebra Cl(p,q) associated to an E(p,q) is complexified that the inner product signature (p,q) is essentially lost since the generating basis elements with suitable factors of 'i', can all be legitimately made to square to ±1, uniformly, and that while real Clifford algebras have "periodicity structure" mod 8, complex Clifford algebras have periodicity structure mod 2, and so have a simpler structure theory.
Independently of the physically meaningful and necessary complexification, also consider the local analysis of curvature forms in differential geometry, e.g., Petrov's classification of Einstein spaces, [Petrov 1969] (in which there is a mathematical error in Type III spaces that I will get around to defining sometime, the meantime, see [Hammel 1974]), or the similar analysis of the electromagnetic field tensor, which shows the complexification to be physically meaningful in perfectly classical senses that are not directly connected to any relativistic assumptions. See again, the works of R. M. Kiehn, RMK Articles: Specials and Freeform Index Page.
Once again, in both these cases, the physics tells us that physical geometry cannot simply be described by the old Euclidean pictures, and that these pictures are entirely too simple and simpleminded in thinking about the physical geometry of space, time and spacetime. There is much more there, physically; it is the case that we have not been looking at physical geometry with this necessary mathematical understanding, and have become confused, making the seemingly mysterious mystical, when the mystery is rather a concocted illusion in the first place.
Instead, Wheeler's notions "pregeometry", and "geometry without geometry" have taken hold in physics, mostly because J. A. Wheeler (1911-2008) [Wikipedia] was a brilliant guy, and has shown the way in many areas. Anybody can miss the obvious, and a conceptual elision does not detract from Wheeler's genius. The return and connection with Wheeler's ideas might be through the concepts of topology and new understandings in differential forms that go beyond the usual language of tensor analysis of classical fields worked out by R. M. Kiehn. See the wealth of this at Cartan's Corner, and more particularly at RMK Articles: Specials and Freeform Index Page.
The error of the old pictures can be, and likely is, besides theoretical and philosophical inertia, a continued error of language: After all this time, i=sqrt(-1), is still called "imaginary". The complex field is the smallest algebraically closed field. There is nothing imaginary or mystical about this, though it seemed there was at the time of its invention or calling into being; yet, the designation "imaginary" remains, and befuddles students even now. Why still, "imaginary"? History! Tradition! Toscanini described tradition as "the last bad performance". Enough said? We also prejudice our thinking simply by saying that a point "has dimension 0", and conclude from that its alleged structurelessness, and triviality. This picture is based on erroneously reconstructed Euclidean geometry.
The technomages of mathematics should already have had enough fun messing with the heads of the acolytes on that score, and equally so in the matter of "classical geometry". Both a classical Euclidean point as understood through Descartes, and a classical *physical* point do have structure, and it is both complex and spinorial/Cliffordian.
This means that classical physical points, having spinorial structure are, in a language of quantum theory, Fermionic objects. This is not a trivial conclusion, or understanding. See, e.g. Introduction to quantum set theory and its (incomplete) sequel Set theory, quantum set theory & Clifford algebras. In the last reference, the exploration suggests that even a classical point in a space 3 dimensions is well described by the 8 dimensional Lie algebra gl(2, C), and that a classical point in 4 dimensional classical spacetime (of which we are not thoroughly convinced) should be described (perhaps, within a topological neighborhood) by the 16 complex dimensional complex space of the Lie algebra gl(4, C).
Perhaps, to beat a dead horse in the matter of complex numbers being required, in any formulation of QM, which always needs an expression of interfering alternative outcomes, a complex structure cannot be avoided. [Mackey 1968] One cannot, e.g., in Q statisical mechanics, willy-nilly, separate q-space from p-space; they must be considered together. Then the complex structure on phase space is unavoidable, as it is also essentially unavoidable in QM. The essential noncommutativity of Q phase space is also unavoidable; so it presents itself as a prototypical noncommutative geometry.
It is often suggested in QM texts that the complex projective Hilbert space is somehow the essential element of QM; this is patently wrong on two counts.
First, all separable (all one needs for QM) infinite dimensional Hilbert spaces are isometrically isomorphic; it is so then also for the associated projective Hilbert spaces. This leaves no room for distinuishing physically different systems.
Second, and complementarily, the physics is actually captured in the structure of the algebra of "observables" corresponding to a *-algebra of linear operators (some, necessarily unbounded, if you believe in CCR) acting on the Hilbert space, or more likely on a common domain within the Hilbert space of both the q and p operators. In any case, the kinematics is defined by the operator algebra, and the dynamics is defined by a semigroup of operators acting on the *-algebra. This was already made conceptually clear in [Neumann 1931].
The Hilbert space can actually be conceptually eliminated by hiking it up into the C*-algebra, using projection operators, as the boundary (set of extremal points) of the forward cone of the algebra, whose interior describes the density operators of states in quantum statistical mechanics. The algebra of operators containing both the observables and the states of the system is the thing, not the Hilbert space, a point made long ago, and I think originally by I. Segal. This does not seem to have caught on.
Yet another clue from physics that the fundamental picture tools of Euclidean geometry need to be reconsidered is in the concept of Supersymmetry - Wikipedia].
The distinctions among Bosonic (symmetry of interchange) and Fermionic (antisymmetry of interchange) particles with their different statistical behaviors are separate symmetries. Though there is little in the way of physical evidence to suggest it, it would be completely cool to have a theoretical framework which combines these two "particle types" into simply "particles". This did lead to the idea and investigation of superalgebras; it turns out that Clifford algebras are, in fact, superalgebras. Viewing Euclidean geometry in proper Cartesian fashion then provides evidence on the mathematical level for the essential validity of superalgebras which express supersymmetry. While physical evidence suggests the mathematical language in which physics is expressed, it is also true that the language suggests how to view the meaning of the physical evidence and mathematical language. Implications in both directions are operative here.
Perhaps, it is no great shock to see that these extended Cartesian descriptors of physical geometry fit nicely into Klein's Erlanger program for geometry where a geometry is specified by the action of a group on a space together with a set of geometric invariants of the group action.
One way of understanding "structure of a point" in a classical sense of pictures goes like this: In E3, for example, any point can be considered a Euclidean ball with antipodal points identified.
Physicaly and classically, we would not observe this substructure directly, and space would appear to be conforming to the old picture. Passage of anything "through the ball" is not apparent from the outside. Such a ball is topologically equivalent to an SO(3) manifold, or a "contracted toroid with a half twist": take a finite cylinder or radius r, score it with lines parallel to its centroid. Holding one end fixed, twist the other by π radians. Now, glue the two disc ends together. There are "geodesics" on the surface of this solid toroid which must go around the central hole of the "doughnut" twice before returning to the starting point. Take the radius of the circle that is now the centroid of the Urcylinder to be R. If we contract the toroid's major radiu R→0, allowing the substance of the toroid to pass through itself, the result is a ball with its antipodal points identified. The manifold of the Lie group SO(3) can be understood as the same thing by allowing two of its angular parameters to have range [-π/2, +π/2] and [0, 2π] covering a spherical surface, and a third with range [0, π] that as a radial coordinate with the two surface coordinates finally determines a ball in E3 of radius π with antipodal points identified. Think Cayley-Klein parameters. See [Goldstein 1950].
If one thinks of a physical space being made up of such structured physical points, an idea resembling Bose-Einstein condensation of the space itself is not too far behind - given enough points. Simply put two such projective spheres in contact, and consider the identifications; continue the process. How many such classical angelic points may dance on the surface of another? As many as want to, within some finite limit?
Limitations on that arise when points acquire quantum bulk, and some principle (Pauli, e.g.) that holds them apart, and from this notion of dimension can arise in the way that dimension is related to the "Kissing Problem", of how many unit spheres in n dimensions can be tangent to a central unit sphere. [Conway 1993] If the general concepts of geometry need some rethinking, then most probably do the general concepts of its abstraction, topology.
Keep in mind that a unitary quantum theory is spoken of in terms of a projective Hilbert space, and its sphere where antipodal points are identified.
There are fundamental geometric problems with combining the Poincaré symmetry of a relativistic spacetime and the internal symmetries of the Standard Model [Wikipedia] in elementary particle physics. The suggestion is, of course, that these problems stem from ignoring the necessities of the very classical geometry upon which the physical model is predicated.
This extended way of seeing E3 involves to start, only an additional structure of points that is physically consistent with the primitive Scholastic descriptors. One only gets to "see" such structure physically when the quantum (fine grained) nature of physical geometry is probed or taken into account, and the classical points are fuzzed according to the basic necessities of quantum theory.
Within the Brouwer-Urysohn concept of dimension, [Hurewicz 1948] one passes hierarchically in three dimensions from point to line (path), to plane (surface) to volume. We have already discussed the necessary complication of the concept of Euclidean "point" from a classical viewpoint.
From a bottom up viewpoint, the obvious suggestion is that a classical 0-dimensional geometrical point, regardless of the space of which it is a member, should be replaced mathematically and conceptually with the smallest complex Clifford algebra. The question, of course, is what is the "smallest"? In one sense, the smallest is simply the complex numbers; but these are commutative, while the fundamental conceptions of quantum theory are noncommutative. So, the next and only possible choice of smallest "quantum point" is the complex Clifford algebra of complex dimension 4, and real dimension 8, which happens to be represented by the algebra of complex 2x2 matrices. This also happens to be the definining and lowest dimensional faithful IRREP of the Lie algebra gl(2, C).
This geometrical viewpoint implies that all fermionic spin 1/2 particles are essentially geometrically irreducible, and "quantum pointlike". Thus, explanations of their existence in terms of wrong Euclidean pictures will fail.
If one seeks further explanation, it must be found within a quantum theoretical language within the constraint that this quantum point with structure is still essentially irreducible, from the viewpoints of classical geometry and from quantum theory. The further subtleties are in what quantum theory may contribute beyond the classical geometry.
The new business is then the concept of line/path which has to do with either or both the fitting together of quantum points, or from an opposed direction, the boundaries of quantum surfaces. I will come back to the "line problem" soon, I hope.
Whether or not I live to complete this concept related to terms of Brouwer dimensions greater than zero, this is not some sort of new "nutsy" stuff: it is rather the digging out of aspects of the mathematical understanding of Euclidean spaces that "should" have been understood and well known centuries ago.
Seeing this now, after all this studious time has passed, I am quite a bit mortified, and feel like an idiot for not having seen the obvious years ago.
There is nothing at all peculiar here, even in modern days, and maybe
even especially in modern days, about rediscoveries in mathematics,
physics and other sciences.
The nicely attributed "Ising Model" of spontaneous magnetization
as a phase transition in statistical mechanics comes easily to mind.
Ising model - Wikipedia.
Ising originally used the concept of nearest neighbor interactions of spin 1/2 entities in a lattice to explain spontaneous magnetization. He worked out the model in one dimension and found that there was no such behavior. The model was generally abandoned for a while, but revived by Heisenberg. The first problem turned out to be that in one dimension, there are not enough (2) nearest neighbors, but that in two dimensions, 4 nearest neighbors are enough, and so certainly in three dimensions with 8 nearest neighbors there are certainly enough. Onsager, a chemist, showed in a most clever but tedious way that in two dimensions, spontaneous magnetisation does indeed exist; just solving the problem and showing this was very difficult, and no one had done this before Onsager. [Huang 1963], and references therein.
The models of the physics of the universe in which we live is has been highly restricted by the barriers of concepts based on the illusions of our common perceptual neurology, so much so that our intellectual concepts even neglect primitive errors that obviously contradict reality. We rely entirely too much on direct visual perception.
Because we are so physically large, we have no perception of the ultimately small. Our visual perception is only of molecular order. Discovery of the quantum regime has broken one barrier of what is possible and what is also necessary in the small for our own existence. Were it not for the existence of quantum indeterminancies, this mysterious universe in which we live could not exist: it would be a dead thing that would not allow the birth of anything new beyond its primordial existence. Time could not have any meaning, or exist in any sense.
We construct a 3+1 dimensional existence because of our size and our inherited neurological substructure, and base our mathematics and mathematical models of reality on those things; we have "discovered" molecules, atoms and certain elementary particles. What else may be discovered?
Even logically and mathematically it is clear that neither infinitesimals nor infinities can exist in reality; yet, all current physics is based on these 19th century interpolational delusions. Why? Because that monumental hierarchy of of mathematics and mathematical physics exists, and it is how we were instructed to think, despite the fact that it is clearly wrong.
All of physics has manifestly gone awry, begiginning with the incompatibilities of quantum theory and relativity theory. Those began about a century ago! Ptolomeic epicycles come to mind.
This a minimal reconstruction of geometrical pictures, but there is no reason to suspect that complex Clifford algebras are the end of story. Spinors in higher dimensions than 3 are more complicated, though still connected with Clifford algebras. There are at least further algebraic and geometric connections with division algebras, Lie, Jordan and Malcev algebras. This is still an open door to epiphanies in the physics of space.
In considering the various possible connections from complex Clifford algebras for space itself, one should be mindful of the "time" concept and its multiplicities of meanings, but also of the tantalizing SU(3)xSU(2)xU(1) symmetry of the "standard theory" or elementary particle physics, and that while EMT is an essential topological expresion of space and time, concerning antisymmetric tensorial entities, general relativity theory is a metric theory concerning symmetric tensorial entities. Symmetry and antisymmetry are algebraic qualia, independent of "pictures".
To interpolate a mathematical structure between the complex Clifford algebra at a physical point and the mathematical point of continuous manifold containing the metrical substance of GR, we will need something akin to a pseudohermitean complex manifold, where the complex structures in the local tangent spaces are not necessarily integrable.
This pseudohermitean complex manifold need not even be a manifold as such: it could be discrete in the quantum sense. Both EMT and GR can be done on simplicial complexes, but the 0-dimensional structurless mathematical point must be replaced with a physical point modeled on its most elementary level by a complex Clifford algebra.
"The truth points to itself" -- Kosh Naranek
Many thanks Prof. R. M. Kiehn for very helpful discussion, for asking pregnant questions, and for leading me to his work that made so many connections for me. Thanks also to Mitch Smith for prior discussions on continua, models and ancient mathematics, to Richard B. Carter for encouraging me to read Descartes, and for old and seminal discussion, to Elihu Lubkin, to Leonard Parker and Dale Snider for discussion and comment on the geometrical meanings of spinors, and last, and certainly least, to my old physics teacher Harvey Kramer for repeatedly angrily telling me that every unorthodox thing I thought was perfectly insane, including my naïve reinvention and working out of fractional differential calculus that was a new (useless and irritating) idea to him. It is so wonderful to have knowledgeable teachers; in total, I think there were five or six: two were in musical composition. Others were mentors of one kind or another, Don Gelman, for one who guided me as an undergrad. Love on their heads.
Nothing here has been supported by governmental or criminal corporate pseudoscience; so, no results, conclusions or opinions have been paid for.
"We live for the one; we die for the one." -- Zathras
"Zathras is used to being beast of burden to other people's needs. Very sad life, ... probably have very sad death, but at least there is symmetry."
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