tag:blogger.com,1999:blog-22973357.post5102245342353378637..comments2015-07-06T19:28:43.587-04:00Comments on Backreaction: PS on The Mathematical UniverseSabine Hossenfelderhttps://plus.google.com/111136225362929878171noreply@blogger.comBlogger39125tag:blogger.com,1999:blog-22973357.post-50723597365196227112007-10-18T10:16:00.000-04:002007-10-18T10:16:00.000-04:00Hi Neil:the platitude that "magnitude in wave func...Hi Neil:<BR/><BR/><I>the platitude that "magnitude in wave function does not matter, only the squared amplitude" is wrong</I> 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'.<BR/><BR/>Hi Arun:<BR/><BR/>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. <BR/><BR/>Best,<BR/><BR/>B.Beehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-61876609729614387622007-10-17T01:40:00.000-04:002007-10-17T01:40:00.000-04:00Bee wrote: Complex numbers are without doubt an us...Bee wrote: <EM>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.</EM><BR/><BR/>Why 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.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>Sorry for the "crackpot theory plug", but this reasoning is elaborated <A HREF="http://cc3d.free.fr/tim.pdf" REL="nofollow">here</A>, specifically in section 4.Christophe de Dinechinhttp://www.blogger.com/profile/15212549796119667462noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-43760492199565533702007-10-16T21:29:00.000-04:002007-10-16T21:29:00.000-04:00Since I see chatting about interference, here's a ...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?)<BR/><BR/>BTW, another reminder about those even more wacky surreal numbers.Neil'http://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-45030252800465422712007-10-16T18:15:00.000-04:002007-10-16T18:15:00.000-04:00Hi Bee,Complex numbers, phase and zitterbewegung o...Hi Bee,<BR/><BR/>Complex numbers, phase and zitterbewegung or "oscillatory motion" may all be synonymous <BR/>terms.<BR/><BR/>David Hestenes, 'The Kinematic Origin of Complex Wave Functions' may be correct.<BR/><BR/>http://modelingnts.la.asu.edu/pdf/Kinematic.pdf<BR/><BR/>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.<BR/><BR/>http://www.iop.org/EJ/article/0305-4470/35/29/104/a229l4.pdf?request-id=eIomSjJ83BGb2YLJ2wi7Kg<BR/><BR/>JH Hannay commented on the Xu paper, but I have not been able to read the comment.<BR/>Journal of Physics. A, Mathematical and General, 2002; 35 (45), p 9699-9700, ISSN: 03054470 <BR/><BR/>http://europa.sim.ucm.es/compludoc/AA?a=Hannay%2c+J+H&donde=otras&zfr=0Doughttp://www.blogger.com/profile/07643919214761722345noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-68274189358928051212007-10-16T17:46:00.000-04:002007-10-16T17:46:00.000-04:00Umm, yes, and no, Bee. Phase info in QM is delicat...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).<BR/><BR/>Does 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.<BR/><BR/>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.Arunhttp://www.blogger.com/profile/03451666670728177970noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-90825675531162722792007-10-16T15:46:00.000-04:002007-10-16T15:46:00.000-04:00Hi Arun:err, apologies. But then I don't get your ...Hi Arun:<BR/><BR/>err, 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,<BR/><BR/>B.Beehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-25369995258532867942007-10-16T15:02:00.000-04:002007-10-16T15:02:00.000-04:00Bee:The mystery of the double slit experiment is n...Bee:<I>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.</I><BR/><BR/>I thought I said that.<BR/><BR/>Arun:<I>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><BR/><BR/>I did.<BR/><BR/>Anyway, the point was not wave-particle duality, but rather observability of phase (differences).Arunhttp://www.blogger.com/profile/03451666670728177970noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-3378847241975682942007-10-16T13:37:00.000-04:002007-10-16T13:37:00.000-04:00Bee wrote: The real numbers are a field as well, b...Bee wrote: <EM>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.</EM><BR/><BR/>This is how you get complex numbers in mathematics, but I don't think this is how you get them in physics.<BR/><BR/>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".<BR/><BR/>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.<BR/><BR/>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)).<BR/><BR/>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.<BR/><BR/>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 :-)Christophe de Dinechinhttp://www.blogger.com/profile/15212549796119667462noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-11135548108964013012007-10-16T13:24:00.000-04:002007-10-16T13:24:00.000-04:00Plato wrote: That's a cop out if I ever heard one....Plato wrote: <EM>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."</EM><BR/><BR/>No, 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.<BR/><BR/>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.Christophe de Dinechinhttp://www.blogger.com/profile/15212549796119667462noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-48042153492639093312007-10-16T11:33:00.000-04:002007-10-16T11:33:00.000-04:00Hi Stefan,Sorry that I was so sloppy with mentioni...Hi Stefan,<BR/><BR/>Sorry 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.<BR/><BR/>Now to come to the quaternions, the multiplication in C is commutative. if you drop this requirement you can get more general structures. <BR/><BR/>Dear Arun:<BR/><BR/>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.<BR/><BR/>Best,<BR/><BR/>BBeehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-25081953218272417762007-10-15T22:30:00.000-04:002007-10-15T22:30:00.000-04:00Bee,About those pesky imaginariesI think that the ...Bee,<BR/>About those pesky imaginaries<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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 HREF="http://motls.blogspot.com/2005/02/wick-rotation.html" REL="nofollow">a very nice article on the subject </A> back in the days when he wrote more about physics. The comments are also very informative.CapitalistImperialistPighttp://www.blogger.com/profile/17523405806602731435noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-83598012350786331442007-10-15T21:53:00.000-04:002007-10-15T21:53:00.000-04:00In his work "An Imaginary Tale", Paul Nahin says a...In 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:<BR/><BR/>"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."Cynthianoreply@blogger.comtag:blogger.com,1999:blog-22973357.post-64156517798885715052007-10-15T21:03:00.000-04:002007-10-15T21:03:00.000-04:00If you think the complex numbers are weird (and I ...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.Neil'http://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-9370371343763931812007-10-15T20:15:00.000-04:002007-10-15T20:15:00.000-04:00Hi Bee,The Arun comment of 7:16 PM, October 14, 20...Hi Bee,<BR/><BR/>The 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.<BR/><BR/>“Invisible numbers” is probably a more appropriate term since these numbers are detectable but not visible.<BR/><BR/>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]:<BR/><BR/>a - with #2 dealing with EE Fourier transforms / pairs and ideal unobtainable curves:<BR/><BR/>1 - An Imaginary Tale: The Story of "i" [the square root of minus one]<BR/><BR/>2 - Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills <BR/><BR/>b - and on extrema and game theory:<BR/><BR/>3 - When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible<BR/><BR/>4 - Chases and Escapes: The Mathematics of Pursuit and Evasion<BR/><BR/>Caspar Wessel in 1797 demonstrated the existence of “imaginary” numbers.<BR/><BR/>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.<BR/><BR/>Historically EE separated from physics at MIT about 1882.<BR/><BR/>David Hestenes has knowledge of Wessel’s work.<BR/>http://modelingnts.la.asu.edu/<BR/><BR/>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.Doughttp://www.blogger.com/profile/07643919214761722345noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-61437716895827889032007-10-15T18:42:00.000-04:002007-10-15T18:42:00.000-04:00The mystery is rather why phases disappear in obse...<I>The mystery is rather why phases disappear in observable quantities.</I><BR/><BR/>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.Arunhttp://www.blogger.com/profile/03451666670728177970noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-91326349147957694972007-10-15T17:18:00.000-04:002007-10-15T17:18:00.000-04:00Dear Bee, Well, you can talk about im and re part ...Dear Bee,<BR/><BR/><I> Well, you can talk about im and re part instead of course, and understand C as vector space with a funny multiplication law</I><BR/><BR/>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?" <BR/><BR/>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?<BR/><BR/><BR/>Best, Stefanstefanhttp://www.blogger.com/profile/09495628046446378453noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-77880575097031250602007-10-15T17:15:00.000-04:002007-10-15T17:15:00.000-04:00Hi Thomas,polarized light is crying out to be desc...Hi Thomas,<BR/><BR/><I>polarized light is crying out to be described with complex numbers.</I><BR/><BR/>that reminds me... there is a way to describe <A HREF="http://en.wikipedia.org/wiki/Polarization#Parameterizing_polarization" REL="nofollow">polarised light by complex two-dimensional vectors</A>, related to the the Stokes parameters, and there is this connection with the Poincaré sphere... <BR/><BR/>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.<BR/><BR/><BR/><BR/>Hi Dr. Who,<BR/><BR/><I>"The arrow of time is imo an illusion"<BR/><BR/>Do you do requests? Please write a blog entry about this!</I><BR/><BR/>I guess we should wait at least until after the <A HREF="http://www.arrowoftime.org/" REL="nofollow">Workshop on the Arrow of Time</A> ;-)<BR/><BR/>Best, Stefanstefanhttp://www.blogger.com/profile/09495628046446378453noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-84100114152264630052007-10-15T15:18:00.000-04:002007-10-15T15:18:00.000-04:00Regarding the Cambridge intro to geometric algebra...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:<BR/><BR/>(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.<BR/><BR/>(2) There is a <A HREF="http://modelingnts.la.asu.edu/pdf-preadobe8/SymmetryGroups.pdf" REL="nofollow">fascinating paper</A> 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.<BR/><BR/>(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.<BR/><BR/>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.CarlBrannenhttp://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-76054643573183053572007-10-15T13:33:00.000-04:002007-10-15T13:33:00.000-04:00Arun,You're saying classical phase implies complex...Arun,<BR/><BR/>You're saying classical phase implies complex numbers? Get real!Kris Kroghhttp://www.psych.ucsb.edu/people/researchers/krogh/index.phpnoreply@blogger.comtag:blogger.com,1999:blog-22973357.post-42690565438700173622007-10-15T13:17:00.000-04:002007-10-15T13:17:00.000-04:00Hi Who: got me ;-)Hi Who: got me ;-)Beehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-46075541634997398422007-10-15T13:16:00.000-04:002007-10-15T13:16:00.000-04:00Dear Arun:A vector space comes by definition with ...Dear Arun:<BR/><BR/>A 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. <BR/><BR/>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. <BR/>Best,<BR/><BR/>B.Beehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-39845393441596647992007-10-15T12:51:00.000-04:002007-10-15T12:51:00.000-04:00Dear Bee,Similarly we don't need multiplication, r...Dear Bee,<BR/><BR/>Similarly we don't need multiplication, repeated addition will do :)<BR/><BR/>Phase is very real classically, listen to your FM radio and think about it :)Arunhttp://www.blogger.com/profile/03451666670728177970noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-8060437429723494652007-10-15T09:33:00.000-04:002007-10-15T09:33:00.000-04:00Not just electrical circuits - polarized light is ...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.<BR/><BR/>As 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, ...<BR/><BR/>The mystery is rather why phases disappear in observable quantities. Wave function collapse, decoherence and all that?Thomas Dnoreply@blogger.comtag:blogger.com,1999:blog-22973357.post-45213971104717430062007-10-15T08:04:00.000-04:002007-10-15T08:04:00.000-04:00"So, it might take some more while, but I'll keep ..."So, it might take some more while, but I'll keep it in mind. Best,<BR/>"<BR/><BR/>Thanks!<BR/><BR/>ps I note the care with which you avoided the word "time" in that sentence....Dr Whonoreply@blogger.comtag:blogger.com,1999:blog-22973357.post-2706850158388701512007-10-15T07:44:00.000-04:002007-10-15T07:44:00.000-04:00Dear Arun:Complex numbers are without doubt an use...Dear Arun:<BR/><BR/>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. 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.<BR/><BR/>Hi Dr. Who,<BR/><BR/>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,<BR/><BR/>B.Beehttp://www.blogger.com/profile/06151209308084588985noreply@blogger.com