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Finally the so promising "Theory of All" (that allegedly conciliates Einstein's Relativity with Quantum Physics, even if at the expense of our three-dimensional perception of reality), the String Theory has proven its worth. And it has done so in a crucial technological field: high-temperature superconductors.
A magnet levitating on a "warm" superconductor
As you probably know, superconductors (materials in which electrons travel without any resistence whatsoever) were initially found to work only at extremely low temperatures close to absolute zero but, recently, more and more cases of "warm" superconductors have been found. This was not possible to explain with quantum mechanics.
And here is where String Theory came to save the day: three physicists from Leiden University (Netherlands) decided to apply the controversial "Theory of All" to this problem and have been so successful that, initially, not even themselves could believe it too much.
But after proper revision, everything was right: the "quantum soup" state of warm superconductors is finally explained only by Maldacena's AdS/CFT correspondence within String Theory.
Hey... Maldacena!
According to co-researcher Jan Zaanen:
AdS/CFT correspondence now explains things that colleagues who have been beavering away for ages were unable to resolve, in spite of their enormous efforts. There are a lot of things that can be done with it. We don't fully understand it yet, but I see it as a gateway to much more.
Source: Science Daily.
Relevant paper: Mihailo Cubrovic, Jan Zaanen and Koenraad Schalm, String Theory, Quantum Phase Transitions, and the Emergent Fermi Liquid. Science 2009. DOI: 10.1126/science.1174962
.Relevant paper: Mihailo Cubrovic, Jan Zaanen and Koenraad Schalm, String Theory, Quantum Phase Transitions, and the Emergent Fermi Liquid. Science 2009. DOI: 10.1126/science.1174962
17 comments:
Good news for sting theory, I hope it will continue to make new predictions. The string equations derive 7 more dimensions of space (which I’m uncomfortable with). However,if the Hadron collider detects them that would be the ultimate conformation IMO.
Check out this sort of funny video: http://www.youtube.com/watch?v=DfPeprQ7oGc
It's not so much to make predictions but to see them confirmed, in fact.
Actually ST has already been hitting some nails as of late: in this article at New Sientists they review another one: ST was needed to explain the behaviour of gluon-plasm, that according to the Standard Model should behave like a gas but in fact does like liquids. Only ST could explain why.
(Note: I'm having isues with youtube videos: they ask me to install a more advanced java but when I try the installer says I have an even more advanced java installed. Mysteries of Ubuntu).
Hi Maju, regarding Youtube, try uninstalling one of the Javas or try a different browser. Youtube also need JavaScript enabled, so turn it on if it’s off.
The standard model (SM) is like a rough draft of particle physics IMO. There are probably more elegant mathematical formulations of Quantum field theory like string theory but SM takes the conservative approach. For example Einstein’s General relativity made better predictions about the bending of light by describing gravity as curved space/time. Those formulations required Riemann geometry which was not known at the time of Newton. 0D points lead to a lot of infinities that physicists have to remedy. The remedy makes use of a lot of approximations, making the solutions murky. String theory does away with this by not having OD matter in the first place.
I'm gonna do that. Uninstalling Java may work.
Not sure what OD stands for. I searched in Wikipedia and could not find anything that could match the context.
Anyhow, indeed it has a lot to do with maths but it's more something conceptual. Einstein, who was never too good at maths, learned Riemann geometry ex-professo in order to formulate and explain his pre existent theory.
Guess it has more to do with the ability to understand that there are no absolute reference points: that objectivity is always subjective. I understood basic relativity as a kid by going in old unprotected lifters, where you could see the world move relatively to you. There were no lifters (or trains) in Newton's time either and I don't think you can get that revelation by merely travelling at horse speed.
Similarly, we now can understand much better than people in the early 20th century that things need not to be strictly material, that they can also be virtual. We need videogames and computers in general to open our minds to that in full.
So we're ready now to discard matter if need be.
Ok, none of these conceptualizations strictly need of such advances and be familiar with them but helps a lot, sincerely. Also relativity itself challenges so much the idea of a material reality, replacing it by some sort of "ether": spacetime. And, while being correct, relativity is incompatible with the also "correct" quantum mechanics, itself already puzzling because of that annoying quantum uncertainty (not as purely material as they'd like it to be).
So well, all that opens our minds and allows us to go beyond the purely materialistic perception of reality.
And this is the real issue, IMO, as the main problem everybody sees with ST is that it also challenges the our three-dimensional perception of reality. In fact Relativity did it as well, by introducing time not as another dimension as some say, but as the imaginary part of the 3D space. But ST goes farther (by mere mathematical need) and this gets many materialists on their heels.
“Not sure what OD stands for. I searched in Wikipedia and could not find anything that could match the context. ”
Sorry, I should have explained, 0D means zero dimensional as a point, whereas a line and square are 1 and 2 dimensional respectively. In string theory particles are replaced by strings, vibrating 1D objects.
“Anyhow, indeed it has a lot to do with maths but it's more something conceptual. Einstein, who was never too good at maths, learned Riemann geometry ex-professo in order to formulate and explain his pre existent theory.”
I agree however I think that some theories are more conceptual that others. Special relativity is more conceptual than General relativity. Special relativity says:
1. Speed and position are relative to an observer (you mentioned this).
2. The speed of light (c) relative to any observer is the same.
The implication of these two principals is time dilation, length contracting, mass energy equivalence etc. Had Newton know these principals he could have easily derived special relativity via elementary algebra and calculus. But how does one explain gravity from the principals of special relativity? Mathematically space/time must be curved and Riemann geometry provides the tools to describe this effect. This mathematical modification (less conceptual) is called general relativity. Similarly how does one unify gravity with the other forces? Particles must be 1D strings and their properties (mass, charge etc.) explained as modes of vibrations. This is string theory.
“Ok, none of these conceptualizations strictly need of such advances and be familiar with them but helps a lot, sincerely. Also relativity itself challenges so much the idea of a material reality, replacing it by some sort of "ether": spacetime.”
Special relativity disposed of the idea of ether. See here:
http://www.gap-system.org/~history/HistTopics/Special_relativity.html
“And, while being correct, relativity is incompatible with the also "correct" quantum mechanics, itself already puzzling because of that annoying quantum uncertainty (not as purely material as they'd like it to be).”
Quantum mechanics (QM) is only compatible with special relativity (SR=flat space/time) not general relativity (GR=curved space/time). The mix of QM and SR is called quantum field theory (QFT). ST and SM are both QFT.
“And this is the real issue, IMO, as the main problem everybody sees with ST is that it also challenges the our three-dimensional perception of reality. In fact Relativity did it as well, by introducing time not as another dimension as some say, but as the imaginary part of the 3D space.”
According to string theory most of the dimensions are curled up in a ball and we can only experience the 4 dimensions. “Imaginary time” is a mathematical construct to handle the distance function of special relativity. “i” is an imaginary number see here:
http://en.wikipedia.org/wiki/Complex_number
Euclidian distance: S^2= X^2+Y^2+X^2
Relativity distance: S^2= X^2+Y^2+X^2-(cT)^2
This makes relativity distance look like the Euclidian distance by introducing the complex number “i”, since i^2 =-1
S^2= X^2+Y^2+X^2+(icT)^2
Similarly how does one unify gravity with the other forces? Particles must be 1D strings and their properties (mass, charge etc.) explained as modes of vibrations. This is string theory.
I'd dare say that particles have way too many inelegant (or even incomprehensible) properties when understood as particles, so I'd say that ST is a more "natural" way of understanding "particles" - or should we begin calling them "strings"?
After all the "particles" were soon understood not to be in a single place at any given moment (if I recall my school physics lessons properly) but in all their orbit or range at the same time. In that sense they were strings since the beginning, just that for some reason they insisted in considering them as 0D particles.
Special relativity disposed of the idea of ether.
Ok. I used a bad comparison. I just meant that spacetime is some sort of "ethereal" kind of "object" - immaterial.
ST and SM are both QFT.
I'd say that, while using the same general mechanics as QTF, ST goes beyond QTF a bit. If nothing else because it manages to integrate General Relativity and gravity in the system, right?
According to string theory most of the dimensions are curled up in a ball and we can only experience the 4 dimensions. “Imaginary time” is a mathematical construct to handle the distance function of special relativity. “i” is an imaginary number...
Sure but that originates in the assumption that maths do not really mean what they express but that they are mere tools without meaning. It's a position I have often found in people with a materialist (or hyper-rationalist) kind of intelligence but that I do not really share.
For me the relationship of our 3D+i reality with the larger spacetime of ST may well be similar to that a cartoon has with our 3D+i world: in a cartoon the third dimension is also hyper-compacted (and irrelevant) and the cartoon's time (timeline, sequence) has no strict relation with our time (we can go forth and back at whim, even if the characters can't.
From that viewpoint time is effectively imaginary and higher dimensions are in fact compacted. I think our 3D+i (or is it rather 3D.i?) reality is also that way in relation with the other extra dimensions and their time.
This actually has important metaphysical implications and that really scares many people, both atheists and religious. You really need to be very open-minded to consider that without feeling uncomfortable about your "truths" of life.
“I'd dare say that particles have way too many inelegant (or even incomprehensible) properties when understood as particles, so I'd say that ST is a more "natural" way of understanding "particles" - or should we begin calling them "strings"?”
The thing about O-D points is that they lead to zero division which causes infinities. The infinities muddle up the equations that unnaturally have to be cured. An elementary particle such as an electron may be thought of as a closed string whereas something like a proton is more akin to a ball of yarn. I do find it more natural to imagine energy as loops.
“I'd say that, while using the same general mechanics as QTF, ST goes beyond QTF a bit. If nothing else because it manages to integrate General Relativity and gravity in the system, right?”
Definitely; when ST was in its beginning stages of development physicist were afraid the theory would fall apart because there was a mystery particle that they couldn’t get rid of. It took several years to figure out that the particle was actually the graviton which could never be worked into the SM. That paved the way for string theory to unify all 4 forces whereas the standard could only handle 3.
“Sure but that originates in the assumption that maths do not really mean what they express but that they are mere tools without meaning. It's a position I have often found in people with a materialist (or hyper-rationalist) kind of intelligence but that I do not really share. ”
I go with Minkowski space which does away with the imaginary scale factor on the time dimension. Either way “time component” of a space/time 4-vector is “cT” and measured as a unit of length. You do know that imaginary numbers have no relationship with issues with non existence and social imaginary, right? Also do you know about complex/imaginary numbers? If not I can explain.
“See Minkowski space”
http://en.wikipedia.org/wiki/Minkowski_space
“For me the relationship of our 3D+i reality with the larger spacetime of ST may well be similar to that a cartoon has with our 3D+i world: in a cartoon the third dimension is also hyper-compacted (and irrelevant) and the cartoon's time (timeline, sequence) has no strict relation with our time (we can go forth and back at whim, even if the characters can't.”
So an ant that has only 2 degrees (up/down, left/right) of movement on the earth has a different sense of time that an earthworm that has 3(up/down, left/right, top/bottom)? If the "times" are independent then string theory would predict >2 dimensions of time, but from my understanding those dimensions are actually spacelike dimensions, significantly less than 1mm across.
you do know that imaginary numbers have no relationship with issues with non existence and social imaginary, right? -
I know well that i is the square root of -1, which is different from any other "real" number, that it's been called imaginary because they can't see it anywhere in reality but that was before it became the "unit" of time in General Relativity.
Right now imaginary numbers are time or at least that is one of their potential translations in real "objects". Time is an "object" of our reality, even if very different from the others. And time is now expressed (at least in GR) in imaginary units.
That for me means that time is always multiplied by the root of -1 and that's something we can't understand though we can go around calling that i, as we actually do.
But for me that is as much as being in fact imaginary, because you can't ever see or comprehend the root of -1, the same you can't see or comprehend a fairy or a god. We can give them names, study their properties and use them to explain the universe but we can't interact with them anywhere in reality. They are all fantastic: imaginary.
WTF would physicists and mathematicians do if they'd be forced to consider all problems that give absurd numbers like i as solution invalid? They do that with infinite and I don't think root of -1 is much better than that, though at least you can treat it as a real number in equations, something that infinite does not allow for.
But that means you can use i in equations but you cannot still accept i in the solutions unless you are ready to accept the imaginary quality of reality.
If you lack imagination, so to say, 317i is not a valid solution as 317 alone would be. I is still an incognite and really makes the solution absurd and unreal.
Just like time.
See Minkowski space.
Yah, I'm gonna have to re-read all that (not just the Minkowski space article but Einstein himself on General Relativity) if we're gonna have this discussion at high mathematical levels.
But the case is that we think and understand in Euclidean terms in which time is not present or is represented by several snapshots of the Euclidean spacial reality. Talking of time as a normal dimension is misleading because we can't mostly travel through time (we can maybe accelerate it but we can never move through it as we do through space, in all directions equally). Time is in this sense more a force like gravity, that makes for us to travel locally in the third dimension rather costly.
But time is of course a dimension or, in Euclidean reality, something imaginary present in all three dimensions equally.
For me time is so elusive that calling it "imaginary" just makes total sense. And again you have the time of 2 dimensional realities (movies, comics, etc.) which is clearly imaginary for us, "superior" 3D beings that can push the pause, rewind and fast forward buttons.
So an ant that has only 2 degrees (up/down, left/right) of movement on the earth has a different sense of time that an earthworm that has 3(up/down, left/right, top/bottom)? -
Sorry but ants definitively have and use the third dimension. That our head is as high to them as the clouds are to us does not mean what you say. They climb trees and dig deep nests, for example.
There are no real, living, 2D beings. We have to go to the cartoon metaphor to understand that. We do not know anything of a possible 4D or 7D reality either. It may not exist at all but... it may exist too and, then, we 3D beings may be for its dwellers somewhat like cartoon characters are for us.
If the "times" are independent then string theory would predict >2 dimensions of time, but from my understanding those dimensions are actually spacelike dimensions, significantly less than 1mm across.
ST does not really deal with the problem of time, AFAIK.
Anyhow, notice that the "time" of a cartoon is just space in our 3 reality, with a "vector" on it indicating the correct way of reading (numeration of pages, sequence of the film...).
IDK how would "time" be at the largest possible dimensioned space for sure but I can imagine that in such "high dimensions" time would just not exist and at that level the Universe (or Metaverse or whatever you want to call it) is eternal and immutable. Much like some of the concepts about God, when people go philosophical instead of just mythological (for example the super-God of some versions of Zoroastrianism, that cannot be worshipped nor even said much about, or the God of Spinoza who is just infinite and has ALL attributes simultaneously).
“I know well that i is the square root of -1, which is different from any other "real" number, that it's been called imaginary because they can't see it anywhere in reality but that was before it became the "unit" of time in General Relativity.”
Actual the imaginary numbers are quite easy too see, the “pure” imaginary axis of orthogonal to the real axis.
See here:
http://www.usna.edu/MathDept/CDP/ComplexNum/Module_3/ComplexPlane_files/image012.gif
Every point on that plane is a “complex number”. The “pure imaginary” and “pure real” are mere subsets of the complex numbers. The complex number is the more fundamental number unit since it solves every polynomial equation and factors linearly which reals can not. For example:
X^2+4=0
How do you factor it over the reals? You can’t. What solutions does it have in the reals? None. However over the complex numbers X^2+4 factors as (X+2i)(X-2i) and the solutions are 2i and -2i. In fact every polynomial factors into linear factors as above. This property is called algebraic closure as there is no bigger set that complex numbers belong to. The complex number is more fundamental type of number.
“We can give them names, study their properties and use them to explain the universe but we can't interact with them anywhere in reality. They are all fantastic: imaginary.”
Modern scientists and mathematician no longer think of “Imaginary numbers” as fantastic an imaginary. In fact the term “complex number” was adopted to get away from those perceptions of the 16th century that complex numbers are of no use therefore and "imaginary". In fact much use of complex numbers goes into physics/engineering and computer science.
“WTF would physicists and mathematicians do if they'd be forced to consider all problems that give absurd numbers like i as solution invalid? They do that with infinite and I don't think root of -1 is much better than that, though at least you can treat it as a real number in equations, something that infinite does not allow for.”
Well sometime it’s the purely complex number solutions they are looking other times it is the purely real one. Every situation is different and it up to a rational person to make sense of reality; you might consider both solutions of none. If a square has an area of 9. I’s it length 3 or -3? Obviously we have to throw away the negative. Solutions to an equation DONT always guarantee all solutions are valid for explaining all situations.
“Sorry but ants definitively have and use the third dimension. That our head is as high to them as the clouds are to us does not mean what you say. They climb trees and dig deep nests, for example.”
LOL , let me put it another way. If I share 2 dimensions of space with a cartoon why would I not share the time dimension with them as well?
Actual the imaginary numbers are quite easy too see, the “pure” imaginary axis of orthogonal to the real axis.
Yah, I knew that. But I have always been confused if that applies to each of the spatial dimensions or rather to the 3D space as whole (from memory, Einstein seemed to suggest it was to each of the dimensions - but I may have got that wrong).
In any case, that is not something wholly real that you can see, specially for the negative figures, much less manage like any other dimension. You do not move through time: you are somehow pushed through it (or if you wish time moves through you... and everything else).
This is in fact more similar to the folded "other dimensions" that we can't easily (if at all) manage, than to the usual spatial dimensions. And certainly it is too similar to the way "time" moves through a movie, even if it's not our "real" (oops, "imaginary") time.
The complex number is more fundamental type of number.
Then it must encode something real.
Modern scientists and mathematician no longer think of “Imaginary numbers” as fantastic an imaginary. In fact the term “complex number” was adopted to get away from those perceptions of the 16th century that complex numbers are of no use therefore and "imaginary". In fact much use of complex numbers goes into physics/engineering and computer science.
Sure but what do they mean in plain terms? They must mean something that is beyond our quotidian experience. Call them imaginary or complex or whatever: what I mean is that you need to reach out beyond the usual in our limited 3D.i experience in order to grasp their meaning. That's something that mathematicians who think of them as mere abstract meaningless entities dp not want to deal with.
If a square has an area of 9. I’s it length 3 or -3? Obviously we have to throw away the negative.
No. We just use absolute and not relative (+/-) measure. Squares with negative values do exist in an euclidean space. This would be denoted as |l|, where l is the length of the side of the square and the bars around mean "the absolute value of..."
There's no such big problem with negative values in fact because they are just relative to where the euclidean origin is, and this can be set arbitrarily (and often we do need to do that, as there is no "natural" origin we can use). For example we do use negative values in the Celsius scale of temperature but could never use them in the Kelvin one.
Solutions to an equation DONT always guarantee all solutions are valid for explaining all situations.
If the solutions have a practical value then they mean something real. If they include necessarily complex numbers such as i, then these complex numbers express something real.
LOL , let me put it another way. If I share 2 dimensions of space with a cartoon why would I not share the time dimension with them as well? -
We can more or less synchronize them in our mind, our imaginary... but actually we can also desynchronize them totally and even ignore that timeline altogether. The 2D time is independent from ours, even if we usually want them to be related, not necessarily identical but in coherent sequence anyhow, so we can follow the 2D
events.
We are not 2D. 2D space (plane) is as much an abstraction to us like a 4D space. It's easier to manage because it's a subset of our space, our dimensional reality, but it's not it. 4D or 5D are supersets and it really gets us thinking hard to even get an approximation to how they could be. Maths help but otherwise we are lost with that.
Similarly for a pure ideal 2D cartoon character managing 1D should hold no big problem, even if they are "larger" than that. But managing our 3D is beyond their scope because they can never leave the 2D plane they "live" in.
“Yah, I knew that. But I have always been confused if that applies to each of the spatial dimensions or rather to the 3D space as whole (from memory, Einstein seemed to suggest it was to each of the dimensions - but I may have got that wrong).”
Hi Maju, the complex numbers are on a plane and therefore belong to a space that is 2-D. The dimension of a space is always described in real numbers. However if you try to graph a complex version of a parabola (y=x^2), you will get a surface in 4-space which can’t be visualized.
“In any case, that is not something wholly real that you can see, specially for the negative figures, much less manage like any other dimension. You do not move through time: you are somehow pushed through it (or if you wish time moves through you... and everything else).”
Yes you can see the negative figures too. For example, -2-2i is a complex number in the third quadrant (2 units back, 2 units down) .
“Then it must encode something real”
Yes, it’s the modulus the real distance the complex number is from the origin. For example the modulus of 3+4i is 5. We say |3+4i| = 5
“Sure but what do they mean in plain terms? They must mean something that is beyond our quotidian experience. Call them imaginary or complex or whatever: what I mean is that you need to reach out beyond the usual in our limited 3D.i experience in order to grasp their meaning. That's something that mathematicians who think of them as mere abstract meaningless entities dp not want to deal with.”
Complex numbers reach out beyond the typical experience of 1-D numbers and extend the notion of number to 2D, but we are already familiar with 2-Space which is a subset of 3-space. Mathematicians are perfectly happy to work with “abstract entities” but they also have to have meaning otherwise it wouldn’t be mathematics. The problem with theories of physics is that abundance of mathematical formulations. How do you decide if one is better than the other? For example 3D+i space vs. Minkowski space vs. quaternion (look it up) space are all ways to explain the physics of relativity.
Hi Maju, the complex numbers are on a plane and therefore belong to a space that is 2-D. The dimension of a space is always described in real numbers. However if you try to graph a complex version of a parabola (y=x^2), you will get a surface in 4-space which can’t be visualized
Our real numbers are in fact 3D, we may say that 2³=8 but if the first unit is meters (m), the result is cubic meters (m³), and they are clearly different.
If you prefer to use an abstract unity (u) the result will be the same cubic units (u³).
So when we add time to the equation we got imaginary numbers and u^4 units. It can't be visualized in a single photogram because it's nothing but a cube "moving" through time: you actually need a film, a film of exactly the time length of the imaginary units involved, to visualize it.
You can project that also to three dimensions, at least some have done it, but the real thing is through time. Unless you'd be talking of some other spatial fourth dimension, that would not need i units.
How do you decide if one is better than the other? For example 3D+i space vs. Minkowski space vs. quaternion (look it up) space are all ways to explain the physics of relativity.
Not sure about the quaternion but Minkowski space and (3+i)D spacetime are the same thing, right?
Re. quaternion, I've been reading and while I do not understand it well enough, I see that i (a number whose square is -1) shows up again when they go "realistic".
i seems to be there for real, what is kind of funny for a number called "imaginary". :)
Let me illustrate with a concrete example for Y=X^2, graphing one point.
Let X=1+2i the X^2=-3+4i
First component (real): 1
Second component (imaginary): 2i
Third component (real): -3i
Fourth component (imaginary): 4i
The result is a point in a 4 space.
Not related to above:
“Our real numbers are in fact 3D, we may say that 2³=8 but if the first unit is meters (m), the result is cubic meters (m³), and they are clearly different.
If you prefer to use an abstract unity (u) the result will be the same cubic units (u³).”
If real numbers are 3D, how do you represent them only in 3-space? How do you represent -5 in 3 space? It’s more correct to say the three -tuplets as a whole form a 3D space, eg all numbers in the appearance X=(x,y,z). But the question arises what is x y or z? To the average person it just mean real numbers but to mathematically inclined it could be any “scalar field” (see wiki). What I was saying is that the scalar field of real numbers is a 1-dimentionsal while the scalar field of complex numbers is 2D which is different form the dimension of the space as a whole. Also, concepts like distance, area, and volume are measures (see measures theory) of sets in space. Sure different measures also have the notion of dimension but they are not directly related the dimension of the scalar field.
“Not sure about the quaternion but Minkowski space and (3+i)D spacetime are the same thing, right?”
Minkowski space is not the same in that all the components are real numbers. The other way used is 3 space and one it component, which you seem to like a lot :-) but yet another way is with Quaternions. Quaternion have 3 dimensions hyper -complex eg (i,j,k) and the time as real. Note that the “i” hyper complex element is not the complex “I”. Also note that quaternion’s have a different algebraic structure than the real’s and complexes. A*B=B*A for complex/reals but A*B=-B*A for quaternion’s.
“Re. quaternion, I've been reading and while I do not understand it well enough, I see that i (a number whose square is -1) shows up again when they go "realistic".”
They have that magic property of being squared to produce a real. They are also interesting from a mathematical standpoint as being one of the four “division” algebras, which are algebras where division is possible.
http://en.wikipedia.org/wiki/Normed_division_algebra
The result is a point in a 4 space.
IMO, it's a temporal line, because I is involved (and was before you squared it).The representation in a 2D space is just a convenient simplification.
It's interesting though because it means that when you square time you get regular space: i²=-1, even if negative. This must have some logical implication that right now I cannot really grasp (probably because my mind tends to run away when maths are involved).
If real numbers are 3D, how do you represent them only in 3-space? How do you represent -5 in 3 space?
It's arbitrary, relativistic so to say: -5 is in 3D (actually in 1D) because it just means 5 to the left, instead to 5 to the right, which by convention is the positive side. Negative numbers do not exist in reality, only in accounting (and then you are trouble).
Ok. arguably in electromagnetism too... but that's surely a simplified convention anyhow. As we have two exactly opposite variables we us that convention but in nuclear force we need to use the color (flavour) convention instead. It's like North and South, that we put the North on top doesn't mean the South is negative: it's just a convention for mathematical convenience.
The origin or zero is relative, except in cases like temperature when measured with absolute degrees (Kelvin scale) but then there are no negative temperatures anymore.
What I was saying is that the scalar field of real numbers is a 1-dimentionsal while the scalar field of complex numbers is 2D which is different form the dimension of the space as a whole.
Ok. But that's an abstraction and anyhow assimilates to space for any representation on paper or in the mind.
Sure different measures also have the notion of dimension but they are not directly related the dimension of the scalar field.
Are you telling me that distance or volume are not measures of space? We are still talking of euclidean space anyhow, right?
Academics may want to explain that it's not real space because they are not directly dealing with it, but real numbers represent spatial measures and nothing else.
A point can be described as a vector and that is nothing but a distance with a directional arrow, meaning maybe from here (the origin) to there (the point). It's all space...
But i is not. It's an oddity. Though maybe in higher dimensional spaces it is reduced to just a spatial formulation of some sort, like our movies used to be sorted in long linear sequences of photos.
Minkowski space is not the same in that all the components are real numbers.
But that's nothing but a convenient custome: as soon as you operate a square root, i comes up again. It's just that the units of 3D+i have been replaced by their squares. It has no real implications.
Quaternion now:
They have that magic property of being squared to produce a real.
See: it's nothing but a mathematical trick to make the figures more convenient and hopefully easier to manage.
But time is not squared in reality but stubbornly manifest in the form of x.i seconds/minutes/hours/years/milennia...
What do you get when you square time? Correct me if I'm wrong but there is just nothing real that is measured in seconds squared (SI). Second squared makes part of other units like that of acceleration but itself measures nothing of our reality.
Until we find something like that we will not be able to manage time, not like we do with space certainly. It'd be a cool discovery, really. :)
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