July 2014 - mathfun@mailman.xmission.com

Re: [math-fun] Problem I never solved
by Bill Gosper 05 Jul '14

05 Jul '14

Wouter> nice! prod( k=1..m, A026147(k) ) / prod( k=1..m, A181155(k) ) approaches sqrt( 2 ) as m-> infinity goes like 1, 1/2, 2/3, 7/10, 286/405, 144305/204102, 276620298878/391202754597, ... (* those who abhor Mathematice lines, please look away : *) k=1/2;NestList[#/(#/. i->i+(k=2k))&,x[i],6] %/. x[i_]->i %/. i->1 %//N <Wouter. -----Original Message----- From: David Wilson Sent: Saturday, July 05, 2014 5:34 PM To: math-fun Subject: [math-fun] Problem I never solved I know the answer to this, but not how to derive it: What is the value of (((1/2)/(3/4))/((5/6)/(7/8)))/... <DW More to abhor: NestList[{#,#+Max[#]}&,1,6]/.List->Divide Divide::argrx: Divide called with 7 arguments; 2 arguments are expected. >> Divide[1, 1/2, 2/3, 7/10, 286/405, 144305/204102, 276620298878/391202754597] (-: rwg

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[math-fun] Problem I never solved
by David Wilson 05 Jul '14

05 Jul '14

I know the answer to this, but not how to derive it: What is the value of (((1/2)/(3/4))/((5/6)/(7/8)))/...

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[math-fun] random points on sphere, normals
by Warren D Smith 05 Jul '14

05 Jul '14

Salamin: Random points on the unit n-sphere can be generated without the need for sqrt and trig. Generate n+1 independent Gaussian random numbers, and normalize the vector to unit length. -- Gene --WDS: I agree this is the way that would have occurred to me to generate point on sphere... but... First of all, seems to me "normalizing vector to unit length" requires sqrt. Second, generating random Gaussians in the first place is often done with the aid of trig, log, and/or sqrt (Box-Muller method). There is however is a nice approach due to Chris Wallace for generating certain kinds of randoms, especially Gaussians, without any fancy functions. You have a pool of random standard Gaussian deviates. If we then apply a KxK orthogonal matrix to any K of the pool members, the result is K new Gaussian deviates. In this way you can keep updating the pool. Once the pool is re-randomized enough you output some of its members. This depends inherently on the fact the Gaussian is the maxent distribution with mean=0 and variance=1. The Wallace approach can be implemented in ways not requiring sqrt or trig, and very fast -- this is one of the fastest, and perhaps *the* fastest, random-Gaussian generation method known.. It has some small statistical flaws however, and if you get serious about correcting the flaws, that slows it down somewhat. I happen to think, which is I believe a new remark by me, this same kind of approach would be a good way to generate "stable" random variables. I also have ways to generate "gamma" variates by such an approach. But the trouble is, the standard Gaussian has no parameters, but other things such as stables, gammas, etc do have parameters. And if you generate samples from a parameterized distribution via a Wallacian pool-update procedure, then you've got to stick with just one choice of parameter values for everybody in the pool. Which is very limiting because users might want to keep changing parameters.

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[math-fun] Julian's tympanohedron--publication at last
by Bill Gosper 04 Jul '14

04 Jul '14

Colm Mulcahy is planning to feature it in his Martin Gardner Centennial article in the October Scientific American. But when it rains, it pours. While NeilB and I were preparing materials for Colm, Gary Antonick added finding it (fat chance) as a "bonus puzzle" to this week's http://wordplay.blogs.nytimes.com/2014/06/30/wcup1/ To check out the pix, be sure to click on the little "32 [or whatever] Comments" and then the Enhancer button near the bottom of the main section. Now Wikipedia can no longer damn it as Original Research. --rwg

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[math-fun] Solid angle reminder
by Bill Gosper 04 Jul '14

04 Jul '14

Here's my little solid angle toolbox: Vertex angles a,b,c. Four versions solidAngle[a_, b_, c_] := {ArcCos[-1 + (1 + Cos[a] + Cos[b] + Cos[c])^2/((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c]))], 2 ArcCos[1/ 4 (1 + Cos[a] + Cos[b] + Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[( Sqrt[-1 - Cos[2 a] - Cos[2 b] + 4 Cos[a] Cos[b] Cos[c] - Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2])/(4 Sqrt[2])], 4 ArcTan[Sqrt[ Tan[1/4 (a + b - c)] Tan[1/4 (a - b + c)] Tan[ 1/4 (-a + b + c)] Tan[1/4 (a + b + c)]]]} Largest possible solid angle with vertex angles a,b,c,d: maxSolidAngle[a_, b_, c_, d_] := 2*ArcCos[((Cos[d] + Cos[c] + Cos[b] + Cos[a])/(4*Cos[a/2]*Cos[b/2]*Cos[c/2]* Cos[d/2])) - Tan[a/2]*Tan[b/2]*Tan[c/2]*Tan[d/2]] Dihedral between faces with vertex angles a and b: dihedral[a_, b_, c_] := ArcCos[Csc[a]*Csc[b]*Cos[c] - Cot[a]*Cot[b]] Apex and base solid angles for regular n-gon base, height h pyramidSolidAngle[n_, circumRadius_, h_] := List[2*Pi - 2*n*ArcTan[((h*Tan[Pi/n])/(Sqrt[circumRadius^2 + h^2]))], 2*ArcCos[((h^2*Sin[Pi/n])/(2*(circumRadius^2*(Cos[Pi/n])^2 + h^2))) + Csc[Pi/n]/2]] Apex and base solid angles for regular n-gon base, cos of angle between edge and altitude pyramidSolidAngle[n_, cos_] := {2*Pi - 2*n*ArcTan[cos*Tan[Pi/n]], 2*ArcCos[(Sin[Pi/n]/(2*(1 + Cos[\[Pi]/n]^2*(cos^-2 - 1)))) + Csc[Pi/n]/2]} coneSolidAngle[generatorAxisAngle_] := 2*(1 - Cos[generatorAxisAngle])*Pi sphericalPolygonArea[r_, n_, anglesum_] := r^2*(anglesum - (n - 2)*Pi) E.g., for the vertex of a regular octahedron, In[454]:= pyramidSolidAngle[4, 1, 1] Out[454]= {2 \[Pi] - 8 ArcTan[1/Sqrt[2]], 2 ArcCos[(2 Sqrt[2])/3]} or In[476]:= pyramidSolidAngle[4, 1/Sqrt[2]] Out[476]= {2 \[Pi] - 8 ArcTan[1/Sqrt[2]], 2 ArcCos[(2 Sqrt[2])/3]} or In[455]:= maxSolidAngle @@ ({\[Pi], \[Pi], \[Pi], \[Pi]}/3) Out[455]= 2 ArcCos[7/9] In[456]:= N[{%, %%}] Out[456]= {1.35935, {1.35935, 0.679674}} or, cutting the four-sided pyramidal vertex into two three-sided ones, In[458]:= 2*solidAngle[\[Pi]/3, \[Pi]/3, \[Pi]/2] Out[458]= {2 ArcCos[7/9], 4 ArcCos[(2 Sqrt[2])/3], 4 ArcSin[1/3], 8 ArcTan[Sqrt[Cot[(5 \[Pi])/24] Tan[\[Pi]/24]] Tan[\[Pi]/8]]} In[459]:= N[%] Out[459]= {1.35935, 1.35935, 1.35935, 1.35935} There doesn't appear to be a nice formula for the sum of the vertex solid angles, analogous to the 2D case: In[460]:= 6*%[[1]]/\[Pi] Out[460]= 2.59616 In[461]:= FullSimplify[%458] Out[461]= {2 ArcCos[7/9], 4 ArcCos[(2 Sqrt[2])/3], 4 ArcCsc[3], 8 ArcTan[3 - 2 Sqrt[2]]} Five different FullSimplifiactions of the same quantity! Repeating a neat result from my previous hype, What is the tetrahedral bond angle? Well, three of them lie at the apex of one those four squashed tetrahedra meeting at the center, so each solid angle is 4π/4: In[482]:= Cos /@ (solidAngle[x, x, x] == 4*\[Pi]/4*{1, 1, 1, 1}) Out[482]= {-1 + (1 + 3 Cos[x])^2/(1 + Cos[x])^3, Cos[2 ArcCos[1/4 (1 + 3 Cos[x]) Sec[x/2]^3]], Cos[2 ArcSin[(Sqrt[-1 + 4 Cos[x]^3 - 3 Cos[2 x]] Sec[x/2]^3)/( 4 Sqrt[2])]], Cos[4 ArcTan[Sqrt[Tan[x/4]^3 Tan[(3 x)/4]]]]} == {-1, -1, -1, -1} In[484]:= Solve /@ Thread[%%] Out[484]= {{{Cos[x] -> -(1/3)}, {Cos[x] -> -(1/3)}}, {{Cos[ 2 ArcCos[1/4 (1 + 3 Cos[x]) Sec[x/2]^3]] -> -1}}, {{Cos[ 2 ArcSin[(Sqrt[-1 + 4 Cos[x]^3 - 3 Cos[2 x]] Sec[x/2]^3)/( 4 Sqrt[2])]] -> -1}}, {{Cos[ 4 ArcTan[Sqrt[Tan[x/4]^3 Tan[(3 x)/4]]]] -> -1}}} In[485]:= FullSimplify[%, 0 < x < \[Pi]] Out[485]= {{{Cos[x] -> -(1/3)}, {Cos[x] -> -(1/3)}}, {{Cos[ 2 ArcCos[1/4 (1 + 3 Cos[x]) Sec[x/2]^3]] -> -1}}, {{1 - 32 (1 + 2 Cos[x]) Csc[x]^6 Sin[x/2]^10 -> -1}}, {{Cos[ 4 ArcTan[Sqrt[Tan[x/4]^3 Tan[(3 x)/4]]]] -> -1}}} So two votes for acos[-1/3) and two abstentions. --rwg

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[math-fun] testing - ignore it
by rcs＠xmission.com 03 Jul '14

03 Jul '14

Test message. Ignore it. --Rich

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Re: [math-fun] Quantum Mechanical Pilot waves: they're baaaack !
by Henry Baker 02 Jul '14

02 Jul '14

Cramer's "transactional" interpretation of QM is very computer sciencey, at least to my eyes. Consider a thought experiment in which you're going to construct a parallel computer simulation of the double slit experiment, and you compute the probability density of photons at a uniform sampling of points in space. The problem comes when you have to flip a probability-weighted coin at each point of the photon absorber in order to decide if a photon will be absorbed at that point. The problem is that you can't decide _locally_ and _independently_ at each absorbing point whether to absorb a photon at that point, because you have to guarantee the conservation of energy. The problem is that energy isn't just conserved _probabilistically_, but _exactly_, so if you choose badly, you can have more photons being absorbed than were emitted -- a clear violation of the conservation of energy. What a computer simulation program would do would be to utilize _transactions_, which would guarantee that energy was exactly conserved on a global basis; such transactions would certainly slow down the parallel simulation program due to contention of the locking mechanisms on the shared resource which keeps the energy in exact balance. For a computer scientist, who expects the universe to be "embarrassingly parallel", this transaction/locking mechanism is extremely inelegant. Yet the possibility of _entanglement_ virtually guarantees that some sort of transaction mechanism will be required to guarantee consistency of the simulation. There are several ways to look at this problem. One is to assume that it is a wart, and attempt to remove it with Bohm-like and pilot wave-like models. The other is to turn this bug into a feature, and attempt to utilize QM itself to do the dirty work in computer simulations by having QM entanglement handle whatever transactional guarantees are required (I'm not sure exactly how this might be done, but there are lots of computer scientists looking into the usefulness of quantum computers). BTW, there are several different flavors of computer transaction implementations, which are also mirrored in discussions about the philosophy of QM. "Conservative" transaction implementations refuse to do any work until a process has _exclusive_ access to a shared resource, while "speculative" transaction implementations are willing to do quite a lot of work, so long as they are also willing to throw away work that proves not to be consistent when the transaction is "closed". QM seems to be quite content to do its thing obliviously until a "measurement" is made, at which point all transactions must be closed, so that the observer sees a consistent picture. At 02:12 PM 6/30/2014, Jeff Caldwell wrote: >I hope to better understand the curious case of the photon. > >It has no frame of reference and, were it conscious, would perceive itself to be emitted and absorbed simultaneously, i.e. no time passing between the events. > >With zero time between emission and absorption, whimsy allows me to think of emitter and absorber as in some sense touching, albeit one is an ancient star and the other a cone in my living eye. > >Zero time means zero distance, in my book, although applying that rule to the no-frame-of-reference photon is probably a category error. > >Cramer's transactional interpretation, inspired by Wheeler-Feynman time-symmetric theory, has both forward and backward-in-time waves between emitter and absorber, agreeing upon the transaction before (as? timey-wimey words ...) it takes place, which leaves precious little room for the free will electrons have if you or I do, John Conway and Simon Kocken say, and leaving no room at all for deciding whether or not to slide a detector into a photon's path after the photon has been emitted. > >Emitters, detectors and photons have united, and their agreements will be kept!

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[math-fun] Quantum Mechanical Pilot waves: they're baaaack !
by Henry Baker 01 Jul '14

01 Jul '14

FYI -- Follow the link to also see the pix. http://www.wired.com/2014/06/the-new-quantum-reality/ Have We Been Interpreting Quantum Mechanics Wrong This Whole Time? By Natalie Wolchover, Quanta Magazine 06.30.14 6:30 am For nearly a century, ÂrealityÂ has been a murky concept. The laws of quantum physics seem to suggest that particles spend much of their time in a ghostly state, lacking even basic properties such as a definite location and instead existing everywhere and nowhere at once. Only when a particle is measured does it suddenly materialize, appearing to pick its position as if by a roll of the dice. This idea that nature is inherently probabilistic Â that particles have no hard properties, only likelihoods, until they are observed Â is directly implied by the standard equations of quantum mechanics. But now a set of surprising experiments with fluids has revived old skepticism about that worldview. The bizarre results are fueling interest in an almost forgotten version of quantum mechanics, one that never gave up the idea of a single, concrete reality. The experiments involve an oil droplet that bounces along the surface of a liquid. The droplet gently sloshes the liquid with every bounce. At the same time, ripples from past bounces affect its course. The dropletÂs interaction with its own ripples, which form whatÂs known as a pilot wave, causes it to exhibit behaviors previously thought to be peculiar to elementary particles Â including behaviors seen as evidence that these particles are spread through space like waves, without any specific location, until they are measured. Particles at the quantum scale seem to do things that human-scale objects do not do. They can tunnel through barriers, spontaneously arise or annihilate, and occupy discrete energy levels. This new body of research reveals that oil droplets, when guided by pilot waves, also exhibit these quantum-like features. To some researchers, the experiments suggest that quantum objects are as definite as droplets, and that they too are guided by pilot waves Â in this case, fluid-like undulations in space and time. These arguments have injected new life into a deterministic (as opposed to probabilistic) theory of the microscopic world first proposed, and rejected, at the birth of quantum mechanics. ÂThis is a classical system that exhibits behavior that people previously thought was exclusive to the quantum realm, and we can say why,Â said John Bush, a professor of applied mathematics at the Massachusetts Institute of Technology who has led several recent bouncing-droplet experiments. ÂThe more things we understand and can provide a physical rationale for, the more difficult it will be to defend the Âquantum mechanics is magicÂ perspective.Â Magical Measurements The orthodox view of quantum mechanics, known as the ÂCopenhagen interpretationÂ after the home city of Danish physicist Niels Bohr, one of its architects, holds that particles play out all possible realities simultaneously. Each particle is represented by a Âprobability waveÂ weighting these various possibilities, and the wave collapses to a definite state only when the particle is measured. The equations of quantum mechanics do not address how a particleÂs properties solidify at the moment of measurement, or how, at such moments, reality picks which form to take. But the calculations work. As Seth Lloyd, a quantum physicist at MIT, put it, ÂQuantum mechanics is just counterintuitive and we just have to suck it up.Â A classic experiment in quantum mechanics that seems to demonstrate the probabilistic nature of reality involves a beam of particles (such as electrons) propelled one by one toward a pair of slits in a screen. When no one keeps track of each electronÂs trajectory, it seems to pass through both slits simultaneously. In time, the electron beam creates a wavelike interference pattern of bright and dark stripes on the other side of the screen. But when a detector is placed in front of one of the slits, its measurement causes the particles to lose their wavelike omnipresence, collapse into definite states, and travel through one slit or the other. The interference pattern vanishes. The great 20th-century physicist Richard Feynman said that this double-slit experiment Âhas in it the heart of quantum mechanics,Â and Âis impossible, absolutely impossible, to explain in any classical way.Â Some physicists now disagree. ÂQuantum mechanics is very successful; nobodyÂs claiming that itÂs wrong,Â said Paul Milewski, a professor of mathematics at the University of Bath in England who has devised computer models of bouncing-droplet dynamics. ÂWhat we believe is that there may be, in fact, some more fundamental reason why [quantum mechanics] looks the way it does.Â Riding Waves The idea that pilot waves might explain the peculiarities of particles dates back to the early days of quantum mechanics. The French physicist Louis de Broglie presented the earliest version of pilot-wave theory at the 1927 Solvay Conference in Brussels, a famous gathering of the founders of the field. As de Broglie explained that day to Bohr, Albert Einstein, Erwin SchrÃ¶dinger, Werner Heisenberg and two dozen other celebrated physicists, pilot-wave theory made all the same predictions as the probabilistic formulation of quantum mechanics (which wouldnÂt be referred to as the ÂCopenhagenÂ interpretation until the 1950s), but without the ghostliness or mysterious collapse. The probabilistic version, championed by Bohr, involves a single equation that represents likely and unlikely locations of particles as peaks and troughs of a wave. Bohr interpreted this probability-wave equation as a complete definition of the particle. But de Broglie urged his colleagues to use two equations: one describing a real, physical wave, and another tying the trajectory of an actual, concrete particle to the variables in that wave equation, as if the particle interacts with and is propelled by the wave rather than being defined by it. For example, consider the double-slit experiment. In de BroglieÂs pilot-wave picture, each electron passes through just one of the two slits, but is influenced by a pilot wave that splits and travels through both slits. Like flotsam in a current, the particle is drawn to the places where the two wavefronts cooperate, and does not go where they cancel out. De Broglie could not predict the exact place where an individual particle would end up Â just like BohrÂs version of events, pilot-wave theory predicts only the statistical distribution of outcomes, or the bright and dark stripes Â but the two men interpreted this shortcoming differently. Bohr claimed that particles donÂt have definite trajectories; de Broglie argued that they do, but that we canÂt measure each particleÂs initial position well enough to deduce its exact path. In principle, however, the pilot-wave theory is deterministic: The future evolves dynamically from the past, so that, if the exact state of all the particles in the universe were known at a given instant, their states at all future times could be calculated. At the Solvay conference, Einstein objected to a probabilistic universe, quipping, ÂGod does not play dice,Â but he seemed ambivalent about de BroglieÂs alternative. Bohr told Einstein to Âstop telling God what to do,Â and (for reasons that remain in dispute) he won the day. By 1932, when the Hungarian-American mathematician John von Neumann claimed to have proven that the probabilistic wave equation in quantum mechanics could have no Âhidden variablesÂ (that is, missing components, such as de BroglieÂs particle with its well-defined trajectory), pilot-wave theory was so poorly regarded that most physicists believed von NeumannÂs proof without even reading a translation. More than 30 years would pass before von NeumannÂs proof was shown to be false, but by then the damage was done. The physicist David Bohm resurrected pilot-wave theory in a modified form in 1952, with EinsteinÂs encouragement, and made clear that it did work, but it never caught on. (The theory is also known as de Broglie-Bohm theory, or Bohmian mechanics.) Later, the Northern Irish physicist John Stewart Bell went on to prove a seminal theorem that many physicists today misinterpret as rendering hidden variables impossible. But Bell supported pilot-wave theory. He was the one who pointed out the flaws in von NeumannÂs original proof. And in 1986 he wrote that pilot-wave theory Âseems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.Â The neglect continues. A century down the line, the standard, probabilistic formulation of quantum mechanics has been combined with EinsteinÂs theory of special relativity and developed into the Standard Model, an elaborate and precise description of most of the particles and forces in the universe. Acclimating to the weirdness of quantum mechanics has become a physicistsÂ rite of passage. The old, deterministic alternative is not mentioned in most textbooks; most people in the field havenÂt heard of it. Sheldon Goldstein, a professor of mathematics, physics and philosophy at Rutgers University and a supporter of pilot-wave theory, blames the ÂpreposterousÂ neglect of the theory on Âdecades of indoctrination.Â At this stage, Goldstein and several others noted, researchers risk their careers by questioning quantum orthodoxy. A Quantum Drop Now at last, pilot-wave theory may be experiencing a minor comeback Â at least, among fluid dynamicists. ÂI wish that the people who were developing quantum mechanics at the beginning of last century had access to these experiments,Â Milewski said. ÂBecause then the whole history of quantum mechanics might be different.Â The experiments began a decade ago, when Yves Couder and colleagues at Paris Diderot University discovered that vibrating a silicon oil bath up and down at a particular frequency can induce a droplet to bounce along the surface. The dropletÂs path, they found, was guided by the slanted contours of the liquidÂs surface generated from the dropletÂs own bounces Â a mutual particle-wave interaction analogous to de BroglieÂs pilot-wave concept. In a groundbreaking experiment, the Paris researchers used the droplet setup to demonstrate single- and double-slit interference. They discovered that when a droplet bounces toward a pair of openings in a damlike barrier, it passes through only one slit or the other, while the pilot wave passes through both. Repeated trials show that the overlapping wavefronts of the pilot wave steer the droplets to certain places and never to locations in between Â an apparent replication of the interference pattern in the quantum double-slit experiment that Feynman described as Âimpossible Â to explain in any classical way.Â And just as measuring the trajectories of particles seems to ÂcollapseÂ their simultaneous realities, disturbing the pilot wave in the bouncing-droplet experiment destroys the interference pattern. Droplets can also seem to ÂtunnelÂ through barriers, orbit each other in stable Âbound states,Â and exhibit properties analogous to quantum spin and electromagnetic attraction. When confined to circular areas called corrals, they form concentric rings analogous to the standing waves generated by electrons in quantum corrals. They even annihilate with subsurface bubbles, an effect reminiscent of the mutual destruction of matter and antimatter particles. In each test, the droplet wends a chaotic path that, over time, builds up the same statistical distribution in the fluid system as that expected of particles at the quantum scale. But rather than resulting from indefiniteness or a lack of reality, these quantum-like effects are driven, according to the researchers, by Âpath memory.Â Every bounce of the droplet leaves a mark in the form of ripples, and these ripples chaotically but deterministically influence the dropletÂs future bounces and lead to quantum-like statistical outcomes. The more path memory a given fluid exhibits Â that is, the less its ripples dissipate Â the crisper and more quantum-like the statistics become. ÂMemory generates chaos, which we need to get the right probabilities,Â Couder explained. ÂWe see path memory clearly in our system. It doesnÂt necessarily mean it exists in quantum objects, it just suggests it would be possible.Â The quantum statistics are apparent even when the droplets are subjected to external forces. In one recent test, Couder and his colleagues placed a magnet at the center of their oil bath and observed a magnetic ferrofluid droplet. Like an electron occupying fixed energy levels around a nucleus, the bouncing droplet adopted a discrete set of stable orbits around the magnet, each characterized by a set energy level and angular momentum. The ÂquantizationÂ of these properties into discrete packets is usually understood as a defining feature of the quantum realm. If space and time behave like a superfluid, or a fluid that experiences no dissipation at all, then path memory could conceivably give rise to the strange quantum phenomenon of entanglement Â what Einstein referred to as Âspooky action at a distance.Â When two particles become entangled, a measurement of the state of one instantly affects that of the other. The entanglement holds even if the two particles are light-years apart. In standard quantum mechanics, the effect is rationalized as the instantaneous collapse of the particlesÂ joint probability wave. But in the pilot-wave version of events, an interaction between two particles in a superfluid universe sets them on paths that stay correlated forever because the interaction permanently affects the contours of the superfluid. ÂAs the particles move along, they feel the wave field generated by them in the past and all other particles in the past,Â Bush explained. In other words, the ubiquity of the pilot wave Âprovides a mechanism for accounting for these nonlocal correlations.Â Yet an experimental test of droplet entanglement remains a distant goal. Subatomic Realities Many of the fluid dynamicists involved in or familiar with the new research have become convinced that there is a classical, fluid explanation of quantum mechanics. ÂI think itÂs all too much of a coincidence,Â said Bush, who led a June workshop on the topic in Rio de Janeiro and is writing a review paper on the experiments for the Annual Review of Fluid Mechanics. Quantum physicists tend to consider the findings less significant. After all, the fluid research does not provide direct evidence that pilot waves propel particles at the quantum scale. And a surprising analogy between electrons and oil droplets does not yield new and better calculations. ÂPersonally, I think it has little to do with quantum mechanics,Â said Gerard Ât Hooft, a Nobel Prize-winning particle physicist at Utrecht University in the Netherlands. He believes quantum theory is incomplete but dislikes pilot-wave theory. Many working quantum physicists question the value of rebuilding their highly successful Standard Model from scratch. ÂI think the experiments are very clever and mind-expanding,Â said Frank Wilczek, a professor of physics at MIT and a Nobel laureate, Âbut they take you only a few steps along what would have to be a very long road, going from a hypothetical classical underlying theory to the successful use of quantum mechanics as we know it.Â ÂThis really is a very striking and visible manifestation of the pilot-wave phenomenon,Â Lloyd said. ÂItÂs mind-blowing Â but itÂs not going to replace actual quantum mechanics anytime soon.Â In its current, immature state, the pilot-wave formulation of quantum mechanics only describes simple interactions between matter and electromagnetic fields, according toDavid Wallace, a philosopher of physics at the University of Oxford in England, and cannot even capture the physics of an ordinary light bulb. ÂIt is not by itself capable of representing very much physics,Â Wallace said. ÂIn my own view, this is the most severe problem for the theory, though, to be fair, it remains an active research area.Â Pilot-wave theory has the reputation of being more cumbersome than standard quantum mechanics. Some researchers said that the theory has trouble dealing with identical particles, and that it becomes unwieldy when describing multiparticle interactions. They also claimed that it combines less elegantly with special relativity. But other specialists in quantum mechanics disagreed or said the approach is simply under-researched. It may just be a matter of effort to recast the predictions of quantum mechanics in the pilot-wave language, said Anthony Leggett, a professor of physics at the University of Illinois, Urbana-Champaign, and a Nobel laureate. ÂWhether one thinks this is worth a lot of time and effort is a matter of personal taste,Â he added. ÂPersonally, I donÂt.Â On the other hand, as Bohm argued in his 1952 paper, an alternative formulation of quantum mechanics might make the same predictions as the standard version at the quantum scale, but differ when it comes to smaller scales of nature. In the search for a unified theory of physics at all scales, Âwe could easily be kept on the wrong track for a long time by restricting ourselves to the usual interpretation of quantum theory,Â Bohm wrote. Some enthusiasts think the fluid approach could indeed be the key to resolving the long-standing conflict between quantum mechanics and EinsteinÂs theory of gravity, which clash at infinitesimal scales. ÂThe possibility exists that we can look for a unified theory of the Standard Model and gravity in terms of an underlying, superfluid substrate of reality,Â said Ross Anderson, a computer scientist and mathematician at the University of Cambridge in England, and the co-author of a recent paper on the fluid-quantum analogy. In the future, Anderson and his collaborators plan to study the behavior of ÂrotonsÂ (particle-like excitations) in superfluid helium as an even closer analog of this possible Âsuperfluid model of reality.Â But at present, these connections with quantum gravity are speculative, and for young researchers, risky ideas. Bush, Couder and the other fluid dynamicists hope that their demonstrations of a growing number of quantum-like phenomena will make a deterministic, fluid picture of quantum mechanics increasingly convincing. ÂWith physicists itÂs such a controversial thing, and people are pretty noncommittal at this stage,Â Bush said. ÂWeÂre just forging ahead, and time will tell. The truth wins out in the end.Â

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Re: [math-fun] Robert Smith's vector problem...
by Henry Baker 01 Jul '14

01 Jul '14

Sorry, Robert, but I still don't understand. What is the meaning of "(v . <i, j, k>)" ? Do you simply mean a vector (= pure (imaginary) quaternion; i.e., S(v)=0) ? At 01:50 PM 6/26/2014, Robert Smith wrote: >Most formally speaking, for a quaternion Q and a vector v in R^3, and >for * meaning quaternion multiplication, > >Q # v = Q * (v . <i, j, k>) * Q^{-1} > >where i, j, and k are the pure unit quaternions. Originally I said >conj(Q) instead of Q^{-1} which is only correct some of the time. > >Robert > >On Thu, Jun 26, 2014 at 11:23 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I didn't understand your "#" operation. Could you please try again?

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[math-fun] pilot waves in QM
by Warren D Smith 01 Jul '14

01 Jul '14

Au Contraire, I believe Bell realized and admitted pilot waves could not handle QFTs (relativistic QM)? Re Meeker "trouble is the pilot wave must change faster than light"... let me amplify/clarify that thought. First of all "pilot wave" and "wavefunction" were in Bohmian nonrelativistic QM anyhow, the same thing. Second, already in the Dirac equation for the electron wavefunction that wavefunction can (and according to Dirac equation must) change "faster than light" i.e. outside the lightcone. This fact caused great angst in the early days. However, the solution of this "paradox" is stunningly beautiful and was originally missed by Dirac himself. It is: An electron traveling forward in time, can be equivalently regarded as a positron traveling backward in time. "Faster than light" travel is paradoxical because for some observer the electron "gets there before it started." However, any observer who feels that way can regard it as positron going forward in time. And you cannot measure the wavefunction. All things you CAN measure, always obey causality (sublightspeed propagation of cause & effect). So anyhow THE EXISTENCE OF ANTIMATTER IS LOGICALLY FORCED BY RELATIVISTIC CAUSALITY is the brilliant conclusion. Now another issue with Bohmian pilot wave jive, is the "superparticle" encodes locations of all particles and is guided by pilot wave. But in relativistic QM, there is no such thing as "the vector of locations of all the particles" because the very number of particles itself is not fixed. Oops. So then you are already in immense trouble trying to construct Bohmism. He could not.

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