The human memory works in funny ways. Yesterday, I thought of my first semester maths tutor, D. I really had a crush on him, awful. Some day, we happened to be alone together in the elevator. I had 25 floors to make a good impression.
We were just discussing complex numbers in maths. Parallel to this, the theoretical physicists had the harmonic oscillator on their schedule. And well, you know how it goes: plug in an exponential, find the complex solutions, take the real part. Sure, I could solve these equations, but I didn't understand where the imaginary part goes. If that question makes any sense. Plus I had learned Special Relativity with ict, which added to my confusion [1].
So, I asked D. why only the real numbers 'exist' and where the others are. (That means I must have found that a really good question.) 25 floors he had no way to get out of this.
This question came back to me yesterday when I read through your comments to The Mathematical Universe. See, as far as I know nobody has ever measured an observable to be a complex number. So whatever 'mathematical structure' constitutes the 'external physical reality' of our universe, complex numbers don't seem to be part of it [2].
However, we know that there are problems which can't be solved purely within the real numbers, say, take a square root of the Klein-Gorden equation. So the universe might try to evolve an initially real valued state, it wants to become complex, but the complex numbers aren't part of our 'external physical reality'. Then what? Do we get a cosmic error message? Does the unintelligent designer of our local patch in the multiverse get an F and fails the exam? Does the wave-function jump into another universe where the complex numbers exist, and then collapses back into ours?
Just asking.
The next time D. and I ran into danger of sharing the elevator, he had to use the bathroom really urgently. Gee, my whole life could have been different, if it wasn't for these complex numbers.
[1] Still today, Wick-rotations seem like magic to me. Is there any good reason why that works?
[2] Since we are dealing with complex numbers every day (well, some of us) this then means human thoughts are not real?
Does the Universe really care if it is real or complex? Does it even know what a real or complex number is?
ReplyDeleteWick rotation, quantum mechanics, and time are all interrelated -- this is the biggest unsolved conceptual mystery I know of.
ReplyDeleteI think complex numbers "exist" in the universe to the extent that many quantum possibilities "exist" with complex amplitudes. As far as I can tell, when we make a measurement the imaginary parts of these complex amplitudes cancel out because the action is time anti-symmetric. See, Wick rotation, time, and quantum mechanics all in one sentence. ;) But why? Thinking about it gives me quite the complex complex complex.
Hi Nanveeth:
ReplyDeleteAre you about to tell me the universe is self-aware? ;-)
Hi Garrett:
Yes, I tend to agree with you that the issue of not-real numbers, the measurement problem in qm (phase?), time, and Wick rotations are likely to be related. Since we're wildly speculating anyhow, I also think that the possibility to have such complex structures as us is linked to the question of time (a minus sign can make a big difference for a differential equation). The arrow of time is imo an illusion, but linked to the complex structures (i.e. the way we perceive time, or how we store memory).
But why? I don't know either. The metric signature around us being Lorentzian is about the only thing I'd be willing to 'explain' with an anthropic shoulder shrug.
Best,
B.
Hi Bee, there are many "holes" in many theories, these "holes" are really the interesting questions one ponders over, for instance the:http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
ReplyDeleteis physically re-arranging objects time symmetric or dimensionally symmetric?
For instance in:http://www.phys.ualberta.ca/~gingrich/phys512/latex2html/node63.html
one can ask of the "dimensionless" Electron:Why does the Electron not fall inwards to the Proton?
Feynman/Wheeler liked to think of Electrons occupying slots in Past or Future, but NEVER in the now?
What is the mathematical number for "time_now"?..this is the most complex initial number in existence, or just plain ZERO !
Questions, is 'zero' an imaginary number or real?..is the past as real as the present?..if the Electron really is never found in the present_time, what is is doing in the past or future?..is it creating or altering some aspects of the Universe, unknown to us humans, it may altering some mathematical sums, this may be thought of as a Hidden Sums Theory? ;)..best paul_v .
Bee, I hate to push personal theories, but my reentrance into physics had to do with Wick rotations and all that. I invented a rewrite of special relativity where Wick rotations were the natural theory, and the usual theory was just a mathematical convenience.
ReplyDeleteAs it turns out with these sorts of ideas, a bunch of people had already thought that. The general term for it is "Euclidean Relativity".
The basic problem is that it requires that you add a hidden dimension to the four obvious ones, and you have to get rid of it one way or another.
I don't work on this sort of thing because it's not really very interesting. The only trace you will find of it in my recent work is that I took the assumption of one hidden dimension in addition to the four usual ones as a hint for choosing which Clifford algebra to generalize the Dirac matrices to.
As to the complex numbers, they are more deeply ingrained into the structure of quantum mechanics than most people realize. Feynman's layman book "QED" explaisn this well. Complex numbers are the last stage in a calculation before it becomes a probability. In E&M, the complex numbers can be removed by taking real parts everywhere. In QM, they cannot be removed so easily. The difference is that in QM, probabilities are nonlinear functions of the complex numbers.
From a Clifford algebra point of view, an imaginary unit is any element of the algebra that (a) squares to -1, and (b) commutes with everything else in the algebra. Such a thing is called a "geometric imaginary unit" in some geometric algebra papers
The Pauli algebra (the Clifford algebra generated by 3 spatial vectors) has a natural imaginary unit, namely the product of the three Pauli matrices. The Dirac algebra (i.e. the Clifford algebra generated by 3 spatial and one temporal vector) has none. I work in the Clifford algebra with 4 spatial vectors and one time vector, and this algebra again has a geometric imaginary unit.
The boss is screaming to me about the price of crude oil and corn, so please understand the likely inaccuracies while I type this.
Dear Bee,
ReplyDeleteComplex numbers are deeply embedded in every day electrical engineering. If your building has a 3 phase power supply or if you're considering why that travel transformer says, use for hair dryers only but not for motors, you're living the reality of complex numbers. It is very hard to think of imaginary numbers as imaginary. (Thinking of a undergrad analog filters course I took long ago :) )
I think in electrical engineering, complex numbers are more real than they are in quantum mechanics.
Best,
-Arun
Good old Control Theory: (joke remembered by self from undergrad days, copied from the web)
ReplyDeleteA bunch of Systems Engineers from Warsaw were flying out to attend a conference on “Automatic Control Systems” in Geneva. The weather conditions were perfect, no wind and blue sky with just a few light fluffy clouds. So once the plane reached cruising altitude the flight crew switched over to auto-pilot and settled in for a nice easy flight.
After some time one of the Engineers noticed a really strange cloud formation (resembling part of a women’s anatomy) off to the right hand side of the plane. He immediately alerted the others of this amazing curiosity and soon all the Engineers from both rows of seats had rushed over the take a look.
The sudden shift in weight caused the plane to pitch alarmingly to the right. The auto-pilot tried to correct but unfortunately it over-responded causing the plane to pitch ever more violently to the left. After several more failed over corrections the plane eventually spiralled out of control and crashed killing all on board.
Air-crash investigators examining the black box flight recorder eventually determined that the crash resulted from: “an instability in the auto-pilot system caused by” ...….punch line ….... “ too many Poles in the right half plane”.
----
Alain Connes
ReplyDeleteWhere a dictionary proceeds in a circular manner, defining a word by reference to another, the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their definition.
I thought this quote above kind of interesting. :)
When Stephen Hawking wrote about "imaginary time" in his famous book A Brief History of Time, he was referring to Wick rotation.
Yes I read complex number
As a layman it was always puzzling to me looking at Dirac's matrices and to see the "i" in there.
Then, when you think about anti-matter and Dirac's contribution. I then wondered if Feynman's toy models were based on "Dirac's insight" to envisioning such actions in the way he did.
This maybe seem easy thing for the educated people, but it was on my mind questionably.
Please correct me if I am wrong here.
Well with that and looking further to understand Wick Rotations, I could'nt help to see Hawking's contributions to this matter.
Mathematicians are a clever lot. Just because a concept may not make sense at an intuitive level doesn�t mean that it can�t be used to help understand nature. Take imaginary numbers, for example. If you start with any �real� number and multiply it by itself, you get a positive number. For instance, 2 times 2 equals 4 but so does -2 times -2. That means the square root of 4 equals both 2 and -2. But what would the square root of -4 be? Mathematicians invented imaginary numbers to answer this question, defining the number i to equal the square root of -1 (making the square root of -4 equal to 2i).
Imaginary numbers can be used to help explain tunnelling, a quantum mechanical process in which, for instance, a particle can spontaneously pass through a barrier. In trying to unify general relativity with quantum mechanics, physicists used a related idea in which they would measure time with imaginary numbers instead of real numbers. By using this so-called imaginary time, physicists Stephen Hawking and Jim Hartle showed that the universe could have been born without a singularity.
Further thoughts were given on the "no boundary proposal."
But maybe I am moving to far away from the issue, while again wondering exactly what the trouble is?
Carl,
Paul Wesson's name looks familiar with regards to working on Alcuberrie warp drive?
Hi Bee,
ReplyDeleteHere's a discussion of the work of David Hestenes, by Gull, Lasenby and Doran. The title is "Imaginary Numbers are not Real - the Geometric Algebra of Spacetime." They note,
...it is possible to interpret the unit scalar imaginary number as arising from the geometry of real space.
And in their concluding remarks,
This leads us to say a few words about the widely-held opinion that, because complex numbers are fundamental to quantum mechanics, it is desirable to `complexify' every bit of physics, including spacetime itself. It will be apparent that we disagree with this view, and hope earnestly that it is quite wrong, and that complex numbers (as mystical uninterpreted scalars) will prove to be unnecessary even in quantum mechanics.
I'd love to hear your and Carl Brannen's views on the article.
Cheers,
Kris
Hi Bee,
ReplyDeleteInteresting question. I think that I have a very good answer to that question, but it would not fit in this comment :-)
Let me try anyway, and sorry if this is longer than my usual comments. Physics can be regarded as the science of finding working models for the universe. Today, we tend to express these models using mathematics, but this is not the only possible model. Wolfram for instance suggests using cellullar automatas (also known as programs) as an alternate way to explore the most complicated-looking forms of these regularities. Before science, god-based models were used throughout the world.
Historically, the first discovery is that there are mathematical regularities in the world around us. For instance: study how many objects there are (e.g. how many stones, how many horses, how many kids), and you quickly "discover" the same regularities we now call the group of integers. You start by learning how to count from these observations. Eons later, long after the bodies of those who built the mathematical theory of counting have turned into grass fertilizer, some clever guy starts wondering: "Hey, how come the universe knows how to count?". The answer is: of course it does: you built number theory to match the behavior of the universe...
But the theory is never perfect, even the simplest one. Mathematically, there is an infinite number of integers, but physically, there isn't an infinite number of horses. However, the "law of counting" remains a valid and useful approximation, even today. Similarly, when you write F=ma, this is an approximation valid for m "small enough" (e.g. it does not work for 10 times the mass of the universe).
Presumably, the second historical discovery is that the same laws repeat over and over again. This is the reason group theory is so prevalent in physics today: we identify the regularity once, and then we know how to recognize it everywhere. So this "law of counting" that works for concrete objects like horses also applies to abstract objects like footsteps between two locations (in that case what repeats is not a concrete physical object, but an abstract "translation"). In this case, it is a little less obvious why there would be an upper limit to the number of footsteps. But what matters is that we reused the same "group", the same mathematical regularity, for distinct physical entities.
Then, we discover more complicated regularities, like the Pythagorean theorem. We know today that on the surface of Earth, it is only an approximation, but it was "good enough" when it was discovered. So we have this law x^2+y^2=d^2 in a number of cases, any kind of circular movement, etc. We may think of it, again, as exact, but it is not. Consider any distance measurement apparatus, for instance one with a resolution of 1mm. Now, build a square of side 1000mm, and measure its diagonal. Mathematically, the number we are trying to measure is irrational, so it is clear that our finite resolution apparatus cannot verify Pythagoras exactly. Like the infinite number of horses, the infinite precision is an illusion (albeit a more persistent one).
Finally, comes the time where we identify a physics law that instead obeys x^2-t^2=d^2. And to model that particular representation and reuse the old techniques, you invent a number i such that i^2=-1. Inventing that number is not that different from inventing the number 1 initially. The only difference is the physical scenarios in which it applies. Unsurprisingly, every time an equation in this form appears (waves, electricity, periodic movements), you can model its solutions using complex numbers, since they were precisely built to model the solutions to this class of equations, just like integers were built for counting.
Einstein discovered a number of relations that were considered totally counter-intuitive, but that held solidly from a mathematical standpoint. So we have taken the bad habit of considering that mathematics are a truer representation of the world than our perceptions. This is not true. Random mathematics do not make good physics. It's always mathematics corresponding to the actual regularities of the universe.
Now, maybe you think I did not answer the question and only pushed it aside: "but why does the universe obey any mathematical regularities"? I believe this is where it's hard to not invoke either an anthropic or religious principle. The anthropic answer is: because a sentient being could probably not exist in a totally chaotic universe, or at least, not without internal "regularities" that match those of the universe it leaves in. The judeo-christian answer, for instance, is: because God created the universe that way, and it was "good". I am not convinced that science will be able to go much beyond that.
I hope this helps.
Bee said:
ReplyDelete"The arrow of time is imo an illusion"
Do you do requests? Please write a blog entry about this!
You're right of course. I have long believed that all those line appearing around my eyes are "in some sense" illusory. Similarly the idea that "young" women are "too young" for me is likewise an utter delusion and completely unscientific.
David Hestenes:physicists quickly become impatient with any discussion of elementary concepts'
ReplyDeleteAlain Connes may have capture the quoted problem here above?:)
Infinite regress had to move through all the thoughts first, which by the way is chaotically complex in it's own right?
Could, if one had thought about it, quickly adopted to a position as an anthropic argument?
Christophe de Dinechin said...
I believe this is where it's hard to not invoke either an anthropic or religious principle. The anthropic answer is: because a sentient being could probably not exist in a totally chaotic universe, or at least, not without internal "regularities" that match those of the universe it leaves in. The judeo-christian answer, for instance, is: because God created the universe that way, and it was "good". I am not convinced that science will be able to go much beyond that.
That's a cop out if I ever heard one.:)It's the very act of infinite regress that we would know what is "self evident."
These define the parameters with which any movement forward can be made scientifically?
It requires "new thinking" and observant eye on the irregularities observed in that "infinite regress." This is always a critical examination of what is current, and problematic.
Paul Dirac is played by the hyper-expressive Schmid. He wrestles with the equations of quantum mechanics and relativity, and his succession of triumphs and deadlocks are played out as a narrated - but fictional - letter to Werner Heisenberg.
ReplyDeleteOf course one has to set the stage. Sometimes it is by coincidence that the jest of the problem was always just right there.
Excerpts of the letter to Heisenberg keep us up to date with Dirac's musings, and the contrasting worlds are interpreted by a dance sequence in which two dancers - depicted as darkness and light - mirror each other on opposite sides of the stage.
I thought you might like that Bee.
Paul Dirac:
ReplyDeleteWhen one is doing mathematical work, there are essentially two different ways of thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.
Dear Arun:
ReplyDeleteComplex numbers are without doubt an useful tool to deal with phase differences. However, in electrodyn you can do without, even if inconvenient. Not so in QM. Well, you can talk about im and re part instead of course, and understand C as vector space with a funny multiplication law, but you don't get rid of the complexity.
Hi Dr. Who,
I do requests... theoretically... practically the list just gets longer. Either way, I don't feel really comfortable writing about things I didn't have time to think about. So, it might take some more while, but I'll keep it in mind. Best,
B.
"So, it might take some more while, but I'll keep it in mind. Best,
ReplyDelete"
Thanks!
ps I note the care with which you avoided the word "time" in that sentence....
Not just electrical circuits - polarized light is crying out to be described with complex numbers. Two components at right angles and one has a magnitude and phase with respect to the other.
ReplyDeleteAs for quantum mechanics - even though individual phases are unobservable, they crop up very often when one has a system with more than one particle or parameter. Aharonov-Bohm, Berry, ...
The mystery is rather why phases disappear in observable quantities. Wave function collapse, decoherence and all that?
Dear Bee,
ReplyDeleteSimilarly we don't need multiplication, repeated addition will do :)
Phase is very real classically, listen to your FM radio and think about it :)
Dear Arun:
ReplyDeleteA vector space comes by definition with a multiplication law with a scalar. What you need for a complex structure is an multiplication among elements of the space. You can get multiplication with a real number out of addition, but what you actually need is multiplication with a complex number.
Sure, phase differences exist, and as I said above complex numbers are without doubt an enormously useful tool to deal with them. But the question is (as Thomas said above), where does the (overall) phase go, i.e. why is the wave-function complex valued but we can only observe real outcomes. This is not the same as in classical ed where you can in principle constrain yourself to real fields.
Best,
B.
Hi Who: got me ;-)
ReplyDeleteArun,
ReplyDeleteYou're saying classical phase implies complex numbers? Get real!
Regarding the Cambridge intro to geometric algebra (GA). I would guess that everyone else in the industry has also read all these things at one time or another. Looking at it again, a couple things come to mind:
ReplyDelete(1) Adding scalars to vectors and the like. This is more natural to elementary particle physicists than other branches, especially if they think about stuff like the V-A and the weak force. In addition, the elements of the Pauli algebra (that is, the Pauli matrices) form a vector, and no one thinks much about it when you add the unit matrix to a Pauli matrix. My version of what is going on here is that GA deals with transformations, not geometry per se. And using it, one does not split up a transformation in the manner that one does when ones tool is tensor notation.
(2) There is a fascinating paper by Hestenes on the crystal groups which gives another way of looking at geometric algebra; the vectors are reflections. I think that this is the deepest meaning of the stuff. Perhaps this has something to do with my youthful fascination with geometry.
(3) One of the bad things we tend to do with GA, and especially one sees this in string theory, is we treat it as "geometric algebra", when it really needs to be "geometric calculus". That is, we tend to separate the derivatives from the geometry. The geometric algebra arises from connecting the Clifford algebra up with the partial derivatives in the same manner as is done with the Dirac operator. I don't think we should separate these things, though it makes stuff harder to do.
Also, the reason "physicists quickly become impatient" with discussions of elementary concepts is that everyone has an opinion, but almost no one has (a) written any papers on the subject, or (b) taken any classes on the subject from someone who wrote papers on it. The nice thing about Clifford algebra or quantum field theory is that it raises the bar high enough that not everyone can jump over it.
Hi Thomas,
ReplyDeletepolarized light is crying out to be described with complex numbers.
that reminds me... there is a way to describe polarised light by complex two-dimensional vectors, related to the the Stokes parameters, and there is this connection with the Poincaré sphere...
I am always a bit confused when trying to understand these formalisms, I have the impression that again and again, the wheel is reinvented, and it all boils down to the algebra of Pauli matrices, or, more formally, to some Clifford algebra, and to this funny double covering of the rotations by SU(2)... I would be glad to know some systematic exposition - maybe I should have a look at the Hestenes stuff some time.
Hi Dr. Who,
"The arrow of time is imo an illusion"
Do you do requests? Please write a blog entry about this!
I guess we should wait at least until after the Workshop on the Arrow of Time ;-)
Best, Stefan
Dear Bee,
ReplyDeleteWell, you can talk about im and re part instead of course, and understand C as vector space with a funny multiplication law
but this funny multiplication law for pairs of real numbers seems somehow to "exist" in nature - in the sense that it provides an economic description of the relation among quantities measured by real numbers? Then, the question is perhaps not "Why complex numbers?", but "Why this funny multiplication law?"
Are there not some theorems that you cannot have much more consistent multiplication laws in a number field - complex numbers equivalent to some real quaternions, the octonions, and that's it? Sorry, my "knowledge" about this is quite vague. But maybe just every multiplication law that can be realised consistently is actually realised in nature?
Best, Stefan
The mystery is rather why phases disappear in observable quantities.
ReplyDeleteThe phases do show up, as cos() or sin(). The whole mysterious two-slit experiment is about how phases show up, even when only one particle is transiting the interference apparatus.
Hi Bee,
ReplyDeleteThe Arun comment of 7:16 PM, October 14, 2007 is very close to the likely insight of the misnomer “imaginary numbers” by Leibniz [or whomever] as I understand this concept.
“Invisible numbers” is probably a more appropriate term since these numbers are detectable but not visible.
Paul J Nahin [PhD EE, former chair, now emeritus UNH-US] has an easy to read series of books on “imaginary numbers” and other topics [need not be read in this order]:
a - with #2 dealing with EE Fourier transforms / pairs and ideal unobtainable curves:
1 - An Imaginary Tale: The Story of "i" [the square root of minus one]
2 - Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills
b - and on extrema and game theory:
3 - When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible
4 - Chases and Escapes: The Mathematics of Pursuit and Evasion
Caspar Wessel in 1797 demonstrated the existence of “imaginary” numbers.
Charles Proteus Steinmetz [“wizard who created electricity from i“, German immigrant to US and chief EE for GE under Thomas Edison] associated “imaginary” numbers with electromagnetism through phasor equations based on Grassmann Algebra.
Historically EE separated from physics at MIT about 1882.
David Hestenes has knowledge of Wessel’s work.
http://modelingnts.la.asu.edu/
John Baez contrasts Hestenes’ Geometric Algebra / Calculus with GRT in October 12, 2007 post on Geometric Representation Theory [GRT] (Lecture 4), comment October 15, 2007 3:03 AM.
If you think the complex numbers are weird (and I remember my sense of awe, looking at my sister's math textbook when I was 8 or 9), then you should get a really big kick out of the "surreal numbers" and such as that. Check it out. I still don't know if I can really get on board with that.
ReplyDeleteIn his work "An Imaginary Tale", Paul Nahin says a number of amusing things about the square root of -1. Here's one that's especially amusing:
ReplyDelete"The intimate connection of the square root of -1 to physical reality is still not always appreciated, however, even by educated people who claim to know quite a bit about math. Consider, for example, these words from celebrity intellectual Marilyn vos Savant:...'The square root of +1 is a real number because +1 x +1 = +1; however, the square root of -1 is imaginary because -1 x -1 would also equal +1, instead of -1. This appear to be a contradiction. Yet it is accepted, and imaginary numbers are used routinely. But how can we justify using them to prove a contradiction?'... Vos Savant's words reveal a curious lack of sophistication in her understanding of the complex plane. The real numbers, of which she apparently has no fear, are no more (or less) trustworthy than the complex ones."
Bee,
ReplyDeleteAbout those pesky imaginaries
I think that the real problem here is not that the imaginary numbers are more “imaginary” than the reals, but that the reals are less “real” than we imagine. We think of reals as being more concrete because we get used to putting them on a line, and because they look more like the natural “counting” numbers. Associating any numbers with distances, velocities, or field strengths is making a choice about how we represent our geometry and our theory. Real numbers fill out a line nicely. Complex numbers do the same for a plane.
Bee says that whenever we make a measurement we get a “real” number, but in many cases it could more naturally be considered a complex number, a vector, or a quaternion. It’s often convenient to project any of the latter into components, i.e. real numbers. The numbers on the dial of my speedometer – are they real? They look more like angles to me, or phases of a complex number, but that’s another choice of representation.
The problem of complex numbers in quantum mechanics is more fundamental. The evolution of the wave equation is most conveniently represented with more degrees of freedom than we can measure. That’s a puzzle, all right, but only incidentally related to the question of the “reality” of complex numbers.
Wick rotation is yet another issue, but here the point is that algebra and calculus have more content in the complex plane than in the reals. Incidentally, a former physicist many of us know wrote a very nice article on the subject back in the days when he wrote more about physics. The comments are also very informative.
Hi Stefan,
ReplyDeleteSorry that I was so sloppy with mentioning the 'funny multiplication law'. The complex numbers are, as you say, a field, i.e. a vector space with a multiplication (among elements of the space, not a scalar multiplication, the vector space already has this). The multiplication has, roughly spoken, to work with the addition of the vector space (associative). The real numbers are a field as well, but unlike R, C is algebraically closed. You get the multiplication law if you require i^2 = -1, in addition to being a field, there's no ambiguity in that.
Now to come to the quaternions, the multiplication in C is commutative. if you drop this requirement you can get more general structures.
Dear Arun:
The mystery of the double slit experiment is not the interference itself - you get these (phase differences) for classical waves. The mystery is that you get the interferences for single particles. That brings us back to the wave-particle problem, i.e. quantum mechanics.
Best,
B
Plato wrote: That's a cop out if I ever heard one.:)It's the very act of infinite regress that we would know what is "self evident."
ReplyDeleteNo, it's just pointing out the difference between "describing" and "explaining". If I open my hand and you see an apple fall from it, you can describe the movement of the apple using Newton's gravitation law. You cannot explain why the apple fell. There is an initial condition here, "I held the apple, and at some point, I decided to drop it", which is not described by Newton's gravitational model.
All I'm saying is that I do not hold for probable that we will one day find some meta-rule a la Tegmark that will describe all the particular regularities we observe without some pretty big initial condition. This is also the primary objection to the "landscape" and associated anthropic "solution" in string theory.
Bee wrote: The real numbers are a field as well, but unlike R, C is algebraically closed. You get the multiplication law if you require i^2 = -1, in addition to being a field, there's no ambiguity in that.
ReplyDeleteThis is how you get complex numbers in mathematics, but I don't think this is how you get them in physics.
Why do we use real numbers to measure distances? Because our original measurements of distance were coarse compared to the resolution of our eye. So it looked to us as if there was a continuum between two tick marks on a ruler. And so integers seemed appropriate, it "looked smooth", and we started postulating that space was a "continuum".
But just like the infinite number of horses, the existence of an infinite resolution is a reasoning error. And so is the "existence" of real numbers in physics.
If you want to convince me otherwise, you will need to point a way for example a distance measurement that gives a meaning to 10^-123456789 meter, or to sqrt(2) meter (not 1.414 meter, but exactly sqrt(2)).
It does not mean that real or complex numbers are not good approximations. They are. A complex unit circle is a good ideal model for a physical circle of radius 1 unit. I can easily map a physical rotation to a change in polar angle and the constant distance 1 unit to the constant mathematical distance one. The structures match, and so a lot of the reasoning I can do on the mathematical "thing" will apply to the physical "thing" as well.
To me, the puzzling thing is not why complex numbers work in physics, but why so many physicists are utterly convinced that real numbers are real :-)
Bee:The mystery of the double slit experiment is not the interference itself - you get these (phase differences) for classical waves. The mystery is that you get the interferences for single particles. That brings us back to the wave-particle problem, i.e. quantum mechanics.
ReplyDeleteI thought I said that.
Arun:The phases do show up, as cos() or sin(). The whole mysterious two-slit experiment is about how phases show up, even when only one particle is transiting the interference apparatus.
I did.
Anyway, the point was not wave-particle duality, but rather observability of phase (differences).
Hi Arun:
ReplyDeleteerr, apologies. But then I don't get your point? Weren't you trying to argue that phases in qm are essentially the same as in classical ed ("Phase is very real classically"), while I was trying to say phase differences don't necessarily have something to do with complex numbers, it's just more convenient to handle with them? Best,
B.
Umm, yes, and no, Bee. Phase info in QM is delicate and easily destroyed. That is why it doesn't easily show up in our experiments. On the other hand, if you reply to this over a Wi-fi connection, it is probably using phase shift keying or some such - phase information pervades life (and so more real, somehow).
ReplyDeleteDoes phase have anything to do with complex numbers, or are two real functions sufficient? We could replace the complex plane by the real plane, etc.
Unfortunately, I don't think the behavior of real linear systems lends itself easily to this, because of the second complex plane that is needed to represent the Fourier transform. I don't see any nice way of moving back and forth between the (two real functions world) and the (sine, cosine transform world). It would e like doing arithmetic in Roman numerals, except worse.
Hi Bee,
ReplyDeleteComplex numbers, phase and zitterbewegung or "oscillatory motion" may all be synonymous
terms.
David Hestenes, 'The Kinematic Origin of Complex Wave Functions' may be correct.
http://modelingnts.la.asu.edu/pdf/Kinematic.pdf
There is also Donghui Xu, 'Hannay angle in an LCR circuit with time-dependent inductance, capacity and resistance", that contrasts the mechanical Hannay angle with the Berry quantum geometric phase in EE terms.
http://www.iop.org/EJ/article/0305-4470/35/29/104/a229l4.pdf?request-id=eIomSjJ83BGb2YLJ2wi7Kg
JH Hannay commented on the Xu paper, but I have not been able to read the comment.
Journal of Physics. A, Mathematical and General, 2002; 35 (45), p 9699-9700, ISSN: 03054470
http://europa.sim.ucm.es/compludoc/AA?a=Hannay%2c+J+H&donde=otras&zfr=0
Since I see chatting about interference, here's a reminder, that the platitude that "magnitude in wave function does not matter, only the squared amplitude" is wrong! The amplitude is important if it interferes with the amplitude of another wave or part of the same wave. (Hey, do photons ever interfere with other photons, or only themselves?)
ReplyDeleteBTW, another reminder about those even more wacky surreal numbers.
Bee wrote: Complex numbers are without doubt an useful tool to deal with phase differences. However, in electrodyn you can do without, even if inconvenient. Not so in QM.
ReplyDeleteWhy not? I'm reading de Broglie's 1937 book, and while some of it is dated, it remains clear that for the people of that time, complex numbers only appeared as a way to represent a wave. They were not "magical". Actually, de Broglie makes a repeated point that this was the same mathematical wave Jacobi was using in his formulation of classical mechanics. This is the reason we still use Hamilton and Lagrange formalisms so much.
Let me show another way to illustrate how complex numbers emerge in QM. The existence of a particle is proven by a measurement that gives a binary result: particle found, particle not found. Let's call "p_1" the probability that the particle is found, and "p_0" the probability that it is not found.
Before you do the measurement, such probabilities are the best description of the future measurement result physics can provide in the most general case. So we can reasonably argue that they are a description of the "presence state" of the particle as far as physics is concerned.
Now, since both cases are mutually exclusive, and since by construction it's an "either-or" case, we have 0<=p_i<=1, and the sum of the p_i is 1. We can shorten that by writing p_i=u_i^2, in which case the two conditions are summarized as: "\sum_i{u^i^2}=1", so the u_i are on a unit sphere. In the specific case of a two-state measurement like "do we find a particle", the unit sphere is a unit circle, which can also naturally be represented by a unit complex number.
Going from here to a "field" is then natural if you imagine a grid of individual detectors. But if there is a reason why there would be exactly one particle to be detected, then you need a second normalization condition on the place where you will find the particle.
Sorry for the "crackpot theory plug", but this reasoning is elaborated here, specifically in section 4.
Hi Neil:
ReplyDeletethe platitude that "magnitude in wave function does not matter, only the squared amplitude" is wrong It is correct, wave-function is the complete wave-function including superpositions. If you superpose two pure states then their individual amplitudes do of course matter. I don't know why you think there is a 'wrong' platidute'.
Hi Arun:
Yeah sure. The problem with QM is that the evolution equation mixes both parts, Re and Im - it doesn't matter whether you call that complex numbers, or whether you describe the wave-fct. as a two-comp vector instead. In classical ED, you can work with only the real part. That's essentially what I meant to say.
Best,
B.
I strongly recommend the many Geometric Algebra and Geometric Calculus papers of Dr. David Hestenes regarding complex numbers in both Classical Physics and Quantum Mechanics. I've also seen a paper on the Wick rotation using Geometric Algebra. There is a Geometric Algebra group in Cambridge that has also written quite a bit on Geometric Algebra and Geometric Calculus with lectures, handouts, examples etc. One could start with 'Imaginary Numbers are not Real'. There is also a good Youtube video on Geometric Algebra for Physicists by Dr. Anthony Lasenby.
ReplyDeleteOne can start by looking at bivectors and their 'squares', duality, the 'squares' of pseudoscalars and material regarding spinors and rotors.