return to homepage String Theory
Part I
Since the late 1980's string theory has continued to gain in popularity, until now it has become a sort of fashion. Brian Greene puts it this way in his book The Elegant Universe: "[In 1984] there was a pervasive feeling among the older graduate students that there was little or no future for particle physics. The standard model was in place and its remarkable success at predicting outcomes indicated that its verification was merely a matter of time and details. . . . [Then] the success of Green and Schwarz finally trickled down even to first-year graduate students, and an electrifying sense of being on the inside of a profound moment in the history of physics displaced the previous ennui." Part II
As promised, I will now critique the math of string theory. String theory has, in the last six or seven years, graduated into M-theory, an 11-dimensional math that attempts to join together the six major 10-dimensional string theories. M-theory has 10 space dimensions and one time dimension, we are told. It is in this matter of dimensions that I have a bombshell to drop. All the extra-dimensional theories that have been proposed since the time of Kaluza in 1919 have contained a basic misunderstanding of the dimensions they described. No one has seen this before now, but the added dimensions, whether they are Kaluza's one extra dimension or M-theory's seven extra dimensions, are all time dimensions. Part III
I will end by analyzing a short quote by David Gross, which I also steal from Greene's book. "It used to be that as we were climbing the mountain of nature the experimentalists would lead the way. We lazy theorists would lag behind. . . . We all long for the return of those days. But we theorists might have to take the lead. This is a much more lonely enterprise."8 If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many noms de plume. If you are a Paypal user, there is no fee; so it might be worth your while to become one. Otherwise they will rob us 33 cents for each transaction.
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The Inelegant Universe
by Miles Mathis
Between foolish art and foolish science,
there may indeed be all manner of mischievous influence
John Ruskin
He was one of that countless and multifarious legion of nondescripts, putrescent abortions, and uninformed obstinate fools who instantly and infallibly attach themselves to the most fashionable current idea, with the immediate effect of vulgarizing it and of turning into
a ridiculous caricature any cause they serve, however sincerely
Feodor Dostoevsky
Readers coming here from a web-search or from other chapters of mine may assume I have nothing to say about the math of string theory. They will assume that since I am neither an insider nor a famous mathematician, the subtleties of 11-dimensional math are beyond me. And since the first part of this paper attacks the theory and not the math, many will assume that I am just making a philosophical critique. They are quite mistaken. In Part II of this paper I will make a foundational critique of the math, revealing some astonishing facts that even the princes of the theory will not want to miss. So if you tend to nod off at any stretch of sentences that fails to contain a number or a variable, there is something for you here, too. The big laughs are in the first part of this paper, but the lasting interest lies in the last part.
The Theory
Most will find nothing particularly revealing in this quote, I imagine. No doubt Greene believes he is just stating a fact, not baring his wicked soul. But I find in it the entire explanation for the movement in science in the 20th century. The keyword is "ennui". In the late 20th century it took a lot to interest the top graduate students like Brian Greene. They could see no quick road to fame by studying the boring past. What was wanted was an avant garde math or theory to latch onto. This is what had made Einstein famous, and after him Feynman and Hawking and all the rest. Mathematics had been the key, and it looked to continue to be the key in the near future. For Brian Greene and the other ambitious young physicists of our time, the job is not to try to discover why the old avant garde maths aren't working; no, the job is to create ever newer avant garde maths that are harder to test. This will automatically provide fewer empirical contradictions, and thereby a stronger theory.
In this paper I will use The Elegant Universe as my scratching post. I do this for a number of reasons, but the main reasons are: 1) It is a recent bestseller and has done as much as any book to popularize the theory, 2) It describes an almost unbelievably inelegant universe, 3) It is as transparent as thinnest glass, setting me up for easy scores on almost every page. As far as the last reason goes, I will show that it is probably a mistake for avant garde maths and theories to allow themselves to be presented to popular audiences, especially if the presentation is in a clear language. Brian Greene is a good science writer: good in the sense that a reader can penetrate what he is saying. But science used to understand that obscure theories should always remain in obscure language. That was the only hope for them, no matter the audience. An honest presentation of a dishonest theory is too dangerous. For one thing, it allows other scientists like me to find the flaws too easily. Fully cloaked in its armor of equations, it is not so easy to sort out, even for a mathematician. But stated baldly it becomes a sitting duck.
I find it astonishing that string theory has made it this far. Greene says that the early years were a bit of a struggle, but I don't tend to believe it. The fact that a theory that is such a magnificent mess is on its feet at all is a very bad sign. It shows the uncritical nature of our milieu, not only in the public and publishing sector, but at the highest levels. The reason for this is clear: graduate students like Greene were well-trained in being uncritical, and they have been for more than half a century. The old uncritical graduate students are now deans and department chairs, and they are all very far gone down the road of non-discrimination. The list of things they have accepted at face value is long and shocking. Greene's first five chapters are a public airing of all the absurd things he has accepted without much analysis. It is clear that he has accepted them because he never really cared if they were true or made sense or not. He, like the others, has from the beginning judged each incoming piece of information based on its likelihood to add to his prestige, and anything that was already a settled question could not help in that area. What he and the other ambitious theoreticians were looking for all along was the end point. "Get me to the end-point as fast as possible." Because then they could begin making their personal contribution. "Put me as close to the front of the line as you can, where I can begin pushing."
For these brightest students, physics was no longer seen as a field they could add to, it was a field they could trump. Their greatest goal was to make all of the past immediately obsolete. Basic physics was digested like a breakfast at the drive-thru, Relativity was duly cut and pasted, and QED was memorized by rote. All this was done by the age of 24 or 25. Another year of all-nighters provided them with the latest hyper-maths and theories, so that they could immediately begin discussing ten-vector fields with full abandon at the coffeeshop and braintrust.
In this way science has become just like Modern Art. The contemporary artist and the contemporary physicist look at the world in much the same way. The past means nothing. They gravitate to novelty as the ultimate distinction, in and of itself. They do this because novelty is the surest guarantee of recognition. The contemporary artist always has his nose to the wind, sniffing the air for the next trend. As soon as he gets a whiff of it he is off running. He is always in a race with time, for it is no longer a matter of being best, it is a matter of being first. He therefore congregates with others of his type. They mass at the same hotspots, antennae erect.
The contemporary scientist is the same. He is a social creature, always trying to impress. Rigor impresses no one in the modern world, so he does not even have to fake it. What impresses are lots of difficult equations, with lots of new variables and terms. The ultimate distinction is coining new words for the new math and the new objects. Calabi-Yau shapes and 3-branes and orbi-folding: that is rich beyond anything.
The art departments have long since dismantled the old schedules: painting and sculpture are passe, studio art a dinosaur, drawing from the nude a sexist embarrassment. The physics and math departments will soon follow suit, no doubt. Mechanics and kinematics will be jettisoned as a theoretical nuisance, a blockage of creativity. Classical algebra and geometry will become an elective, taken only by historians and archivists. Instead, seventh graders will be offered "The Rudiments of Chaos Theory" and "Fun with Tensors" and "Computer Modelling with i."
Let me now show you a few examples of the absurdities that the standard model teaches. I do this to prove that by accepting these absurdities, it encourages a proliferation of more such absurdities. It teaches the graduate students, by example, that mathematical fuzziness pays and that conceptual rigor does not. Let us start with the "messenger particle,"1 a relatively new beast in the physical zoo. The messenger particle is a photon that tells another particle whether it should move away or move near. The messenger particle was invented to solve the problem of attraction. At some point it became clear to physicists that attraction couldn't logically be explained by a trading of particles. Their old blankets over this problem had begin to wear thin, so they needed a new concept. Enter the messenger particle. With the messenger particle, we no longer have to be concerned with explaining physical interactions mechanically. We don't even have to imagine that movement away in a field is caused by bombardment, which was such a simple concept. No, we can now explain both movement away in a field and movement toward in field as due to information in a messenger particle. This simultaneously explains both positive charge and negative charge. How easy: the photon just tells the particle what to do. Why did we not think of that before?
Once you accept that quantum particles are on speaking terms, physics is so much tidier. There is no end to what we can explain this way. We can have the particles trading recipes, emailing eachother, SMSing, watching TV. It is a theoretical goldmine.
Gluons, weak-gauge bosons, and gravitons are also messenger particles of their various forces. The problem of attraction is solved once and for all, for all possible fields. Gravity is not curved space or a physical force. It is a commandment.
The next absurdity is one of Feynman's famous absurdities.2 This one concerns letting an electron going through the two-slit experiment take all possible (infinite in number) paths simultaneously and then summing over these paths to find the wave function. Any idiot can see that this is just a mathematical consideration and has no physical implications, but Feynman was a special kind of idiot. He insisted for some reason that the math was the physics, and all the special idiots since then have taken his word for it. They love to quote or paraphrase him, as Greene does, "You must allow nature to dictate what is and what is not sensible." Which means, "You must allow me (Feynman) to dictate what is and what is not sensible. I am smarter than you are and if you don't allow me to dictate to you, I will browbeat you mercilessly." Even now that Feynman is long in the grave and incapable of personally browbeating anyone, the special idiots still quote and paraphrase and bow to his authority. Feynman himself was bowing to the authority of Heisenberg and Bohr, who first decided, by fiat, that the math of quantum mechanics was the physics. Or perhaps he was only learning from their example. Counterintuitive fiats had made them famous with all the toadies, why not make a few of his own counterintuitive fiats and toadies?
Greene tells us outright: "Quantum mechanics requires that you hold such pedestrian complaints [about things making sense] in abeyance." What could be more convenient for a scientist? He is now in the position of a priest. The priests have always said the same thing to non-believers. "You must not expect it to make sense. You must have faith. Trust the Lord." Trust Feynman. He is smarter than you and understands what you should believe. He has filled the blackboards with Hamiltonians and has cracked safes. He has earned the right to say ridiculous things, like the Dalai Lama or the Buddha or the President.
This is the most important thing that string theorists have learned from quantum mechanics: you do not have to make sense anymore. Any contradiction can be relabeled a paradox, any infinity can be relabeled an axiom, any absurdity can be given to Nature herself, who is an absurd creature, in love with illogic and caprice.
I could go on indefinitely, listing other absurdities like the Twin Paradox and the singularity and so on, but I have analyzed these problems elsewhere in great detail; and besides, you already either accept them or don't accept them, so that my comments are nearly beside the point. You won't judge the concepts based on anything I could say of them, you will only judge me for what I say of them. Therefore, let me proceed to critique string theory, a theory that is not yet set in stone, even for the toadiest.
String theory begins by defining a string. In most instances a string is a one-dimensional loop, we are told. String theory is famous for its ever-increasing number of required dimensions, so that you would think that the theorists would have a pretty tight idea of what a dimension is. But if you think this you would be wrong. String theory is about math, not about concepts, and these brilliant mathematicians don't have a very clear idea what a dimension is or what a one-dimensional "thing" would be. In math, a one-dimensional thing is a line. It always has been, since the time of Euclid, and that has not changed recently. A zero-dimensional thing is a point, a two dimensional thing is a plane, and a three-dimensional thing is a cube or sphere or whatnot. But all of these things are mathematical abstractions. They don't exist and can't exist. Of all these mathematical things, only the three-dimensional things have a potential existence, and then only if you add time. There is a very simple reason for this that has nothing to do with gods or turning on the universe or anything else esoteric or metaphysical. Points, lines and planes cannot exist because they do not have any physical extension. A plane disappears in the z direction, a line disappears in the x and y direction, and a point disappears in all three directions. In mathematical terms, it means that the variable or field has hit a limit—a zero or infinity—at this point in the equations, making existence impossible.
Physicists used to understand simple concepts like this, but no more. Even mathematicians don't appear to understand them. These concepts just get in the way until some self-described genius somewhere finds a clever way around them, and we aren't bothered with them anymore. After that we are allowed to propose the existence of mathematical objects and no one blinks an eye. But it remains a (perhaps unpleasant) fact that a line cannot exist. Even in pure math, a "one-dimensional loop" cannot exist. A one-dimensional loop is false even as a mathematical abstraction. Why? Because a loop curves. Any curve is no longer one-dimensional. A curve is two-dimensional, by definition.
Greene and his heroes imagine that because you can, in a pinch, express a position on a curve with one variable, that it is a one-dimensional object. But it isn't. Greene proves this when he begins talking about his garden-hose world, where the position of a bug on the hose can be expressed with two variables. He then admits in an endnote that if the garden hose has an interior, we must have more dimensions. But when, in a physical situation, is it possible to imagine a garden hose, no matter how tiny, with no interior? It is not possible and his "two-dimensional" garden hose, if physical, must have three dimensions.
Greene makes the current confusion even more apparent when he begins increasing the Type IIA coupling constant.3 This allows strings to expand into two and three-dimensional objects. He says that the two-dimensional string is like a bicycle tire and the three-dimensional object is like a donut. So Greene thinks there is a dimensional difference between a bicycle tire and a donut! If a bicycle tire is not solid rubber through and through, then the third dimension has disappeared? We should at least have to suck the space out of it with some kind of space vacuum, right?
String theory is such a godawful mess right from the first concept that it is painful to go on. But I will. Once we have our impossible one-dimensional loops, we are to imagine that they are vibrating. To vibrate in the right way for the theory, they must be strung very, very, very tight. Now, a sensible person would already have several foundational questions. First of all, why are they vibrating? Second, why are some vibrating one way and some vibrating another way? Third, what causes the tension?
The first concept, basic vibration, we can give them. Vibration is far from being a basic motion, but there has to be some first cause, and so we will allow one unexplainable motion as first cause. But the difference in different vibrations cannot be uncaused. We cannot allow it to be a postulate. Different vibrations should have different mechanical causes. If one string is vibrating in a different way from another, there must be a reason. String theorists have already told us that strings are not made up of subparticles; they are absolutely indivisible. They should therefore be undifferentiated. Ultimate strings that are indivisible should act the same in the same circumstances. If they act differently, then the circumstances must differ. But we are not told what these different circumstances are. The vast variation in behavior is just another postulate.
Besides, even if we admitted the impossible—that a one-dimensional loop could exist—once you give it a vibration it automatically gains a dimension. All you have to do is look at the direction it is postulated to vibrate in. Does it vibrate lengthwise? Of course not. How could it? It is undifferentiated lengthwise, meaning that it is not made up of subparticles. There is no way a pulse could travel lengthwise in a string that was not divisible. So the theorists propose sideways vibrations, of different sizes and wavelengths. In technical terms, we are talking about transverse waves, not longitudinal waves. A transverse wave will automatically push the string into a second dimension. So all talk of one-dimensional strings is a wash from the beginning, for two fundamental reasons, not one.
This brings us to another question: is it even possible for a one-dimensional string to vibrate sideways? I have reminded the reader that a longitudinal wave is impossible to imagine without some subdivision of the string. There has to be some sort of longitudinal variation to propose compression; but this variation is not possible without subdivision. In the end this is because without subdivision you cannot insert any space into the string. You need space in between the particles making up the string in order to propose variation in compression. But a closer analysis shows the same problem with transverse waves on a one-dimensional string. How is a one-dimensional string bendable without some "give" between particles making up the string? If the string is absolutely indivisible and undifferentiated, then it is not clear that we can bend it. A bend would occur at the bond between particles, in a macroscopic string. In a string-theory string, there is no bond between particles, since there are no particles making up the string. Bending or vibrating a string-theory string is like proposing to bend or vibrate a cube or a cone or a sphere. If our fundamental particle were any of these instead of the string, you would laugh if someone proposed that it bent. Imagine a cube bending. How would a fundamental, undifferentiated cube bend? Or a fundamental, undifferentiated sphere? But bending a fundamental, undifferentiated string is just as silly. It is just another postulate that is impossible to explain or justify.
Likewise, tension is a pretty complex concept. It is not a fundamental motion or event. In fact, tension is a force. But string theory is supposed to be explaining the four fundamental forces, not creating more. What causes the tension? How is it possible to have a tension across an undifferentiated ultimate string? How is it possible to have tension in a closed loop, unless that loop is being expanded by some outside force? None of this is explained. Tension is just an assumption, another axiom.
After a first reading, I had already discovered that string theory has more basic postulates than any theory I had ever seen or imagined. To any logical person from past centuries, string theory would look like a comedy of errors. Newton has been all but laughed at by string theorists for not giving a mechanical explanation of force at a distance. But these theorists are in no position to throw stones. Newton would look at string theory and say something like, "Well, of course, if you are allowed to make enough unprovable assumptions at the beginning, you can formulate a theory to contain anything. Especially if you are allowed to beg the question so egregiously. String theory is the attempt to unify the four basic force fields. To do this it creates, as a postulate, a huge force of uncaused tension. Then it adds to that a basic 'particle' that can morph into anything, just by changing its 'tune.' All these morphs are uncaused and act as further postulates—as postulates they do not require proof or any justification. Then, whenever the math stops spitting out numbers they want, they postulate new branes, donuts, tubes, three-holed buttons, frisbees, and anything else that tickles their fancy. None of these new objects has to be justified beyond the fact that they needed them to fill a hole in the math. 'It fit the hole, therefore it must be real!' Then, when the going gets really tough, they add a new dimension. M-theory gives them the 11th dimension, and why stop there? I predict that, like Feynman, they will finally understand that the sky is the limit. Why not predict an infinite number of curled up dimensions, sum over them in some fudgy way, and achieve any answer you like, to fit any occasion. Only then will the madness come to its illogical end."
This is the basic technique of string theory: if you run into some dead-end at any point in the math, transport that dead-end back to the string. For example, perhaps you find the need for a new particle but your math at that level of size or theory does not allow it. Well, simply make it another axiom of string theory. Postulate that your basic string takes that shape under the circumstances you have discovered, and your work is done. In this way, every conceivable problem can be collected at the foundational level and made into an axiom. Since you don't have to prove axioms, you will never be pestered to supply a proof or explain anything. All problems can be collected, reinserted at the axiomatic level, and treated ever after as assumptions. In this way string theory really is the perfect theory. Using this technique, nothing is beyond mathematical expression.
String theory is actually even more inelegant than QED, and QED is not exactly a poster child for elegance or simplicity. Greene tells us that string theory was invented to simplify the huge number of "elementary" particles in QED, as well as to combine QED and Relativity. But he seems oblivious to the fact that string theory has a record-setting number of axioms and an ever-increasing number of vibrations, dimensions, blobs, branes, and jellies. The only object not yet incorporated into string theory is the moss-covered three-handled family gradunza. It also has a truly impressive number of manufactured manipulations, such as the set of instructions for orbi-folding a Calabi-Yau shape or the tearing of space in a flop transition. These manipulations come provided with no theory, and are basically added to the list of postulates: postulate #89,041—we can flop-tear an orbi-folded 3-brane goofus as long as we can say afterwards that the math made us do it (and provide a sexy little computer-generated diagram).
Another of the inelegances of string theory is the required energy of a string. The unbelievable amount of tension [1039 tons] on a single string gives it a mass of some 1019 protons. This is about the mass of a grain of dust. The theorists need all this force on the string since they have gathered all the other forces here at the axiomatic level. This has the added benefit, they think, of making the mass too great to be discovered in an accelerator. Unfortunately, the mass is so huge that it should make the string discoverable by macroscopic means. I might suggest a sieve. Seriously though, the theorists admit that "all but a few of the vibrational patterns will correspond to extremely heavy particles," meaning particles many times heavier than a grain of dust.4 It is hard to believe that masses of this sort floating around are undetectable. Greene says that they are unlikely because "such super-heavy particles are usually unstable."5 It is interesting to note that string theory never says why all such super-heavy particles should be unstable. In fact, there is no theoretical reason they all should be. It is another postulate: postulate #76,904—super-heavy particles are all unstable because if they weren't we might be able to find one. The instability is another axiomatic convenience of the theory.
But let us reverse for just a moment. The tension is even harder to believe than the mass. Try imagining putting a thousand trillion trillion trillion tons of tension on a grain of dust. Talk about tensile strength. Talk about potential energy. And you thought the atom had a lot of stored up energy. What a bomb you could make out of one string! Get one little string to relax for a moment, and you could blow up the entire galaxy. I suspect someone may have miscalculated by a teeny bit. You will say, “C'mon, a whole galaxy? Isn't that hyperbole?” No, 1039 tons is actually more than the weight of 2 trillion suns, which would be four Milky Way galaxies. All that tension on one string.
Here's another inelegance. In a subchapter ironically entitled, "The More Precise Answer,"6 Greene develops this idea: the "violent quantum jitters" can be quieted by imagining a collision of point particles as a collision of strings instead. One string represents an electron, say, and the other a positron. The two strings join for a moment as a string that represents a photon and then re-separate as two new strings. The reason this is an improvement, we are told, has to do with Relativity. Greene uses his two observers George and Gracie (the Burns and Allen ghosts are due massive apologies for being brought into this mess, I think) to "slice" his strings into different events. George sees the strings meeting at one time and Gracie see the meeting at another time. Among all possible observers the meeting point will be smeared out over some time. This smearing calms the quantum jitters.
This is among the most dishonest uses of Relativity and diagramming I have ever seen. In order for his argument to work, Greene has to diagram the strings as three-dimensional objects. For it is not the length of the strings that causes the Relativistic difference in his argument, it is the thickness. But he began the subchapter by admitting that the strings were one-dimensional. He brags that one-dimensional strings can do what zero-dimensional points cannot. Remember that strings have only a length dimension. They have no thickness. As a matter of width or thickness or radius, they act just like points. They have zero radial dimension. This means that Greene's Relativistic slicing is flat wrong. His diagrams are a big fat lie, since they cause you to visualize something that cannot be happening. His words are saying one thing and his diagrams are saying the opposite. If the strings are one-dimensional lines, then the fork where they meet will also be one-dimensional. If you slice it at a dt, then the fork will be in the same exact place for all viewers. String theory adds absolutely nothing to QED or the point problem. It simply adds another layer of lies to cover it up.
The Math
To understand why this must be the case, we must go back to the basic calculus. All the higher maths that are used by string theory are based on the calculus. Calculus itself is a math that is based on comparing rates of change. My long paper on the calculus makes this crystal clear, but it has always been understood in some form or another. Velocity is a rate of change of distance and acceleration is a rate of change of velocity. That is why velocity is the first derivative of distance and acceleration is the second. I showed that you can also find third and fourth derivatives of distance, and so on. The third derivative is a change in acceleration and the fourth derivative is a change in that change. These multiple accelerations can really happen: they are physical. I also showed that you could write a velocity in one of two ways, either of which was mathematically acceptable: Δx/Δt or ΔΔx. Likewise, acceleration can be written as Δx/Δt2 or ΔΔΔx. A second-degree acceleration can be written Δx/Δt3 or ΔΔΔΔx, and so on.
You can see that even here there is some sort of mathematical equivalence between x and t. Einstein showed us that this equivalence goes far beyond anything Newton could have imagined, but even in Newton's calculus equations, there was a hidden equivalence. The variables Δx and Δt are the inverse of eachother, in some sense. In the equations above, the x goes in the numerator when the t goes in the denominator. This is because, as variables, they always change in inverse proportion, even when no transformational changes are involved. Remember that Einstein showed that as time dilated, length contracted. One gets bigger as the other gets smaller. This is clear in the transform equations of Special Relativity, and it is clear in the equations above as well. A Δx in the numerator is, in some very important sense, the same as a Δt in the denominator.
Why is this important? It is important because what all the big maths of Maxwell's equations, General Relativity, Quantum Mechanics, Kaluza's five-dimensional theory, string theory, and M-theory all do is express fields. All these fields are force fields and force fields are based on some acceleration. By the old equation F = ma, if you have a force you have an acceleration. The reason that Kaluza's fifth dimension helped so much at first is that it allowed the expression of both the gravitational field and the electromagnetic field, the only two of the major four that were known at the time. Using the vector fields as they have been defined since the end of the 19th century, the four-vector field could contain only one acceleration. If you tried to express two acceleration fields simultaneously, you got too many (often implicit) time variables showing up in denominators and the equations started imploding. The calculus, as it has been used historically, couldn't flatten out all the accelerations fast enough for the math to make sense of them. What Kaluza did is push the time variable out of the denominator and switch it into another x variable in the numerator, just as I did above. Minowski's new math allowed him to do this without anyone seeing what was really going on.
String theory and M-theory continue to pursue this method. They have two new fields to express, so they have (at least) two new time variables to transport into the numerators of their math. Every time you insert a new variable, you insert a new field. Since they insert the field in the numerator as another x-variable, they assume that it is another space field. But it isn't. It is a transported time variable.
Readers will no doubt be reeling at this information. It was difficult enough to imagine extra space dimensions, most of them curled up like little pillbugs. But how do we make sense out of eight simultaneous time dimensions? It is actually a lot easier than you think, since, once understood, it is easy to visualize. It doesn't take any leaps of faith or warnings that "you can't possibly diagram this, but you must accept it anyway." To show this, I will use the visualization I used in my calculus paper. Let us say you are at the airport, walking along normally. In addition, let us say that you are walking in a perfectly straight line and that your stride is perfect. Each step is the same as every other step. You therefore have a constant velocity, and your stride is, in a sense, measuring off the ground. If you are very retentive, you might even be counting as you walk: 1, 2, 3, 4, and so on. Well then, let us give you a watch, too. Some Swiss-quartz stunner than never misses a beat. So time, the retentive so-and-so that he is, is also counting off his numbers: 1, 2, 3, 4 and so on. Next you come to a moving sidewalk. You step on and for one interval you are accelerated. It is not an instantaneous interval, since some amount of time passes between your initial speed and your final speed. But there is an acceleration over only one interval. It stops at the end of that interval and you have a new constant velocity, a velocity found by adding your own velocity and the velocity of the sidewalk.
Since there was an acceleration over that interval, then by the standard way of expressing acceleration, we how have a t2 in the denominator: a = Δx/Δt2. As we know, that can also be written a = Δx/Δt/Δt. Either way we have a second time variable. Therefore we might say that we have a second time field and a second time dimension. Now, we must study that interval. What happened over that interval? Did you step into another dimension? Did another dimension open up under you? In a very limited sense, yes. That interval is a sort of sub-interval to the ones you were measuring off with your feet and your watch. But it is not mysterious in any way. It is not curled up anywhere. In fact, you can measure both time dimensions with your watch. That is why we usually just square the time dimension. It is a second measurement over the same interval. If the two time dimensions weren't directly related, we obviously couldn't square them. We would have to call them Δt1 and Δt2 or something, and keep them separate.
What I mean when I say that we are measuring two things simultaneously is that we are measuring how far the sidewalk goes in Δt and how far the walker goes in Δt. An acceleration is these two velocities measured over the same interval. So you can see that what we really have is two Δx's and one Δt. But since, in the real world of airports and things like that, we measure strides and lengths on sidewalks with the same measuring rods, it is easier to write the equation with one Δx and two Δt's. Therefore, we get the equation a = Δx/Δt2.
All this is very elementary, of course. But everyone seems to have lost sight of it at this late date in history. Because what it means, especially when you have a math that is expressed by a lot of superscripted dx's, is that those dx's are not mysterious extra space dimensions, they are equivalent to velocities being measured simultaneously to achieve complex accelerations. These complex accelerations express the meeting of multiple fields over the same (perhaps infinitesimal) interval.
To see this even more clearly, let us say that our man on the moving sidewalk hits yet another field. When he first stepped onto the sidewalk, we might say that he passed through a one-interval field. Well, if we are mischievous, we can plaster that interval with so many fields it will make the man's head spin. We can move him sideways, up, down, or we can spin him any number of revolutions we like. And we can do them all at once. Again, not all at an instant, but all over the same finite interval. Every time we add a motion, whether is a linear motion or a spin, we have hit him with another force. We have also hit him with another variable. We have also hit him with another field. We have also hit him with another dimension. We have done a finite number of things to him during a finite time. Therefore we have created a one-interval field of multiple forces and dimensions.
Stated in this way, there is nothing mysterious about it at all. When we add a dimension, all we have to do is add another Δx to the numerator or another Δt to the denominator (but not both). But none of these extra dimensions is strange or difficult to imagine. We have just imagined it, physically. We could also draw it and diagram it. All we need is tracing paper and overlays. Nothing is curled up. Nothing is unexpressed, nothing only comes out at night.
An 11-dimensional math can be expressed using all the ridiculous equations of M-theory, where super-computers are wasted just storing the postulates. Or you can express it like this: A = Δx, Δy, Δz/ Δt8.
A perceptive reader may say, "Wait, didn't you just say that we could express a new field or force by either a new space variable in the numerator or a new time variable in the denominator? If this is true, then why can't M-theory call all the new dimensions space dimensions if it wants to?"
Well, it can, but it has to be very careful what this implies. I have shown how multiple time dimensions are really a rather simple idea, with no mystery involved. Likewise, seven new space dimensions, correctly interpreted, are also not mysterious or esoteric or difficult to understand at all. A new dimension is a new force applied over a finite interval. If this force is continuous, then it causes a continuous field. By continuous field I mean a field that spreads across an extended set of intervals, not just one interval. Over one interval, a force causes a velocity. Over an extended set of intervals, a continuous force causes a continuous acceleration. We have known this for a long time. But what all this means is that the new dimension is not a new direction in space. Every time we add a new dimension to our math, it does not mean that a new, autonomous x-variable has been invented, going off in some strange new path, like the path of i or the path of a curled up pillbug. It just means that we have a new velocity happening simultaneously to all our other velocities over the interval in question. This velocity does not have to have a direction in space all to itself. It does not have to be at a right angle to all previous dimensions. There is nothing to say that forces cannot overlap, or that directions cannot overlap or that times and subtimes cannot superimpose. In fact, they must superimpose. When we create these large-dimensional maths, we are doing so precisely to ask how the various accelerations and forces superimpose. That is what we are seeking. We are seeking the total field at dt, and that total field is a superposition of all the velocities caused by all the force fields present at that interval.
There are two basic and separate ways to "unify" the four known forces and all the spins. One is to find a math that contains them all. The other is to show how one force field is equivalent to another field, thereby simplifying our equations. If we can show that the same basic motion causes two separate fields, we will have unified those two fields. String theory attempts to unify in both ways, but does neither. It tries to unify all fields by expressing them as various vibrations of an ultimate particle, but this part of the theory is just gibberish, as I have shown. It also tries to contain all the fields using a multi-dimensional math, and in this it has made one tiny step in the right direction. If the dimensions are interpreted in the way I have interpreted them above, then they start to make some sense. The first step toward the mathematical expression of a unified total field is to add up all the accelerations and to express them as separate dimensions. But I think it is clear from my airport sidewalk example that a completely successful math must, in the end, recognize the ultimate equivalence of all the time variables. The man in the airport could measure all the various velocities with the same watch, and so can the scientist computing the unified field. When physics understands how all these fields superimpose, it will be able to simplify its equations back down to x. y, z, t. It can do this because all the t's are equivalent. Once again, the total physical field, in the presence of eight degrees of linear and angular acceleration, would be A = Δx, Δy, Δz/Δt8. That is (a maximum of) 8 fields, but only 4 dimensions7.
Some will complain that all the time variables can't be equivalent due to Relativity. But I refer them to my airport example once more. In that example we are studying the field over one interval. In all the maths that are based on the calculus, including tensor calculus and the math of M-theory, we would make that interval an infinitesimal interval or dt. Relativity can't find any variance at dt or dx, for the very simple reason that the observer cannot be any distance away from the phenomena at dx, dt. Notice that in the airport example we have the walker measuring himself using his own watch. He is therefore at no distance from the event and the speed of light has nothing to do with it. The time variables must be equivalent, both physically and definitionally.
A reader will have one final question: why Δt8? Might that be telling us something fundamental about the number of real accelerations that exist in the universe? Meaning, if five-dimensional math was used for gravity and electromagnetism, then shouldn't the (limited) success of 11-dimensional math be telling us we have 8 fundamental accelerations going on simultaneously, and therefore 8 fundamental force fields?
Maybe. It is possible that we can get to 11 dimensions by adding spin as a dimension wherever we find it. The spin of the electron may be caused by one separate force and the spin of the quark may be caused by another, and so on. Or, some of the accelerations we already know about may be second or third-degree accelerations. No one has ever considered the possibility that the strong force or E/M may be Δx/Δt3 instead of Δx/Δt2.
But before we run off pell-mell in search of some giant equations to express this, I think we should back up a bit and reassess the entire road to how we got here. In this paper I have shown that string theory is criminally confused about almost everything. Years have been wasted chasing curled up dimensions that don't even exist. There are no Calabi-Yau shapes clinging to the corners of x, y, z. The orbi-folding and all the rest was just mental masturbation. The string theorists have invented shapes and folds and histories and ancestries for these pillbugs nesting in the crannies, and now they find they are completely uninfested. Like some nefarious chemical company, they have soaked the foundations of the communal house in order to roust out the bugs, and now we find that we are all poisoned.
I think it is time to declare that string theory is worthy of a Superfund site and move into a new house. In this house our first order of business is in truly understanding the physical heritage that has come down to us. I have shown in my various papers that there is plenty of work to do in this regard. All the misguided theorists of the past century have quite simply been wrong when they stated, with maximum hubris, that classical and quantum physics was over. Neither classical nor quantum physics is anywhere near finished. We have only touched the surface, even regarding linear maths and "poolball" mechanics. Post quantum theories, whatever they are, will never be possible until QED is corrected and filled out. And QED will never be corrected and filled out until some of the elementary concepts I have spent such a vast amount of time exploring are better understood. Until we understand how our maths are working we can never hope to understand how the universe is working. And this is only the beginning. Velocity, acceleration, circular motion, rate of change, and many other basic physical concepts are not understood to this day. All our physical "knowledge" is dominated by heuristics. Whether we are studying quantum interaction, orbits, or cosmological origins, our equations are overwhelmed by nescience. The best thing to do in this situation is admit the fact and get to work.
A primary piece of this work will be in re-establishing QED without the point particle. One of the only places I agree with string theory is in its critique of the point particle. Quantum math was never able to express its field using an extended particle. String theory realized the problem here and the need to correct it, but it only corrected it by burying it below the Planck limit where no one can see it. In this way the point is given extension, but the extension is only another postulate. Postulate #5 or so: the loop has extension but it is so small that 1) it can't be detected physically, 2) it can't be detected mathematically. Therefore we can fudge over it by misusing the calculus for the millionth time. To my mind this is not a great advance over QED. The only solution is to return to the beginning of QED and start over. We have a lot of very useful heuristics that we can use to guide us, and lots of experimental data. But the math needs a thorough cleaning. The place to begin is in a better understanding of the calculus. Establishing the calculus on the constant differential instead of the diminishing differential will change every physical and mathematical concept of the last 300 years, and will impact all our theories of motion, force, and action. Only once we have rebuilt the old theories from the ground up can we begin counting the force fields and dimensions we will need for a unified math. We may find that we need 11 dimensions. But we may not. We may find that the house looks very different after we have cleared away all the garbage.
Conclusion
You can almost hear the violins. Those poor put-upon theorists, saving us from the past, leading us bravely into the future. I am not an experimentalist, but when I read this quote my eyes rolled so far back in my head I nearly broke into St. Vitus' dance. The dishonesty literally pours off the page. The string theorist pretending to be an unwilling leader, a humble servant. When in fact he is little more than a shallow revolutionary, a completely monomaniacal, delusional person who has convinced himself that by hoodwinking us he has done us some great favor. Salesmanship posing as magnanimity.
I think you can tell by the tone of this paper that I am angry at string theory, and I don't deny it. The last century would try any honest person's patience, in any number of fields. In my opinion we are past the point of a mild rebuke. The physics department needs a good kick in the pants, and the math department too. Both have degenerated nearly past the point of recognition, and they might as well join up with the art department and begin putting on Dali-esque plays and masked balls. I had hoped that QED would someday develop some humility and that we, as physicists, would get back to work. That we would recognize the huge gaps in our theories, going all the way back to Euclid, and make some effort to fill them. Instead young physicists have continued to learn all the wrong lessons from the recent past and to fail to learn the most-needed lessons. What they have taken from QED is only its Berkeleyan idealism and its intellectual dishonesty. They have remained buried so far under their esoteric maths that they cannot see daylight. And they have continued to dig. They are now at a depth that apparently precludes all cries of logic, all ropes of humility, all ladders of embarrassment. It seems likely that they will continue to dig until the air runs out. Or until they hit the baby black hole at the center of the earth, and the self-created chasm at the center of their own theory sucks them into a well-earned hell.
1The Elegant Universe, p. 123.
2Ibid, p. 111.
3Ibid, p. 311.
4Ibid, p. 151.
5Ibid, p. 152.
6Ibid, p. 158.
7In fact, it is only three dimensions, since time is operationally just a second measurement of one of the others. See my paper on time for a clarification of this assertion.
8The Elegant Universe, p. 214.