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See reader questions & answers on this topic! - Help others by sharing your knowledge -------------------------------------------------------------------------------- FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 4/4 -------------------------------------------------------------------------------- Item 24. Special Relativistic Paradoxes - part (a) The Barn and the Pole updated 4-AUG-1992 by SIC --------------------- original by Robert Firth These are the props. You own a barn, 40m long, with automatic doors at either end, that can be opened and closed simultaneously by a switch. You also have a pole, 80m long, which of course won't fit in the barn. Now someone takes the pole and tries to run (at nearly the speed of light) through the barn with the pole horizontal. Special Relativity (SR) says that a moving object is contracted in the direction of motion: this is called the Lorentz Contraction. So, if the pole is set in motion lengthwise, then it will contract in the reference frame of a stationary observer. You are that observer, sitting on the barn roof. You see the pole coming towards you, and it has contracted to a bit less than 40m. So, as the pole passes through the barn, there is an instant when it is completely within the barn. At that instant, you close both doors. Of course, you open them again pretty quickly, but at least momentarily you had the contracted pole shut up in your barn. The runner emerges from the far door unscathed. But consider the problem from the point of view of the runner. She will regard the pole as stationary, and the barn as approaching at high speed. In this reference frame, the pole is still 80m long, and the barn is less than 20 meters long. Surely the runner is in trouble if the doors close while she is inside. The pole is sure to get caught. Well does the pole get caught in the door or doesn't it? You can't have it both ways. This is the "Barn-pole paradox." The answer is buried in the misuse of the word "simultaneously" back in the first sentence of the story. In SR, that events separated in space that appear simultaneous in one frame of reference need not appear simultaneous in another frame of reference. The closing doors are two such separate events. SR explains that the two doors are never closed at the same time in the runner's frame of reference. So there is always room for the pole. In fact, the Lorentz transformation for time is t'=(t-v*x/c^2)/sqrt(1-v^2/c^2). It's the v*x term in the numerator that causes the mischief here. In the runner's frame the further event (larger x) happens earlier. The far door is closed first. It opens before she gets there, and the near door closes behind her. Safe again - either way you look at it, provided you remember that simultaneity is not a constant of physics. References: Taylor and Wheeler's _Spacetime Physics_ is the classic. Feynman's _Lectures_ are interesting as well. ******************************************************************************** Item 24. Special Relativistic Paradoxes - part (b) The Twin Paradox updated 04-MAR-1994 by SIC ---------------- original by Kurt Sonnenmoser A Short Story about Space Travel: Two twins, conveniently named A and B, both know the rules of Special Relativity. One of them, B, decides to travel out into space with a velocity near the speed of light for a time T, after which she returns to Earth. Meanwhile, her boring sister A sits at home posting to Usenet all day. When B finally comes home, what do the two sisters find? Special Relativity (SR) tells A that time was slowed down for the relativistic sister, B, so that upon her return to Earth, she knows that B will be younger than she is, which she suspects was the the ulterior motive of the trip from the start. But B sees things differently. She took the trip just to get away >from the conspiracy theorists on Usenet, knowing full well that from her point of view, sitting in the spaceship, it would be her sister, A, who was travelling ultrarelativistically for the whole time, so that she would arrive home to find that A was much younger than she was. Unfortunate, but worth it just to get away for a while. What are we to conclude? Which twin is really younger? How can SR give two answers to the same question? How do we avoid this apparent paradox? Maybe twinning is not allowed in SR? Read on. Paradox Resolved: Much of the confusion surrounding the so-called Twin Paradox originates from the attempts to put the two twins into different frames --- without the useful concept of the proper time of a moving body. SR offers a conceptually very clear treatment of this problem. First chose _one_ specific inertial frame of reference; let's call it S. Second define the paths that A and B take, their so-called world lines. As an example, take (ct,0,0,0) as representing the world line of A, and (ct,f(t),0,0) as representing the world line of B (assuming that the the rest frame of the Earth was inertial). The meaning of the above notation is that at time t, A is at the spatial location (x1,x2,x3)=(0,0,0) and B is at (x1,x2,x3)=(f(t),0,0) --- always with respect to S. Let us now assume that A and B are at the same place at the time t1 and again at a later time t2, and that they both carry high-quality clocks which indicate zero at time t1. High quality in this context means that the precision of the clock is independent of acceleration. [In principle, a bunch of muons provides such a device (unit of time: half-life of their decay).] The correct expression for the time T such a clock will indicate at time t2 is the following [the second form is slightly less general than the first, but it's the good one for actual calculations]: t2 t2 _______________ / / / 2 | T = | d\tau = | dt \/ 1 - [v(t)/c] (1) / / t1 t1 where d\tau is the so-called proper-time interval, defined by 2 2 2 2 2 (c d\tau) = (c dt) - dx1 - dx2 - dx3 . Furthermore, d d v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t) dt dt is the velocity vector of the moving object. The physical interpretation of the proper-time interval, namely that it is the amount the clock time will advance if the clock moves by dx during dt, arises from considering the inertial frame in which the clock is at rest at time t --- its so-called momentary rest frame (see the literature cited below). [Notice that this argument is only of heuristic value, since one has to assume that the absolute value of the acceleration has no effect. The ultimate justification of this interpretation must come from experiment.] The integral in (1) can be difficult to evaluate, but certain important facts are immediately obvious. If the object is at rest with respect to S, one trivially obtains T = t2-t1. In all other cases, T must be strictly smaller than t2-t1, since the integrand is always less than or equal to unity. Conclusion: the traveling twin is younger. Furthermore, if she moves with constant velocity v most of the time (periods of acceleration short compared to the duration of the whole trip), T will approximately be given by ____________ / 2 | (t2-t1) \/ 1 - [v/c] . (2) The last expression is exact for a round trip (e.g. a circle) with constant velocity v. [At the times t1 and t2, twin B flies past twin A and they compare their clocks.] Now the big deal with SR, in the present context, is that T (or d\tau, respectively) is a so-called Lorentz scalar. In other words, its value does not depend on the choice of S. If we Lorentz transform the coordinates of the world lines of the twins to another inertial frame S', we will get the same result for T in S' as in S. This is a mathematical fact. It shows that the situation of the traveling twins cannot possibly lead to a paradox _within_ the framework of SR. It could at most be in conflict with experimental results, which is also not the case. Of course the situation of the two twins is not symmetric, although one might be tempted by expression (2) to think the opposite. Twin A is at rest in one and the same inertial frame for all times, whereas twin B is not. [Formula (1) does not hold in an accelerated frame.] This breaks the apparent symmetry of the two situations, and provides the clearest nonmathematical hint that one twin will in fact be younger than the other at the end of the trip. To figure out *which* twin is the younger one, use the formulae above in a frame in which they are valid, and you will find that B is in fact younger, despite her expectations. It is sometimes claimed that one has to resort to General Relativity in order to "resolve" the Twin "Paradox". This is not true. In flat, or nearly flat, space-time (no strong gravity), SR is completely sufficient, and it has also no problem with world lines corresponding to accelerated motion. References: Taylor and Wheeler, _Spacetime Physics_ (An *excellent* discussion) Goldstein, _Classical Mechanics_, 2nd edition, Chap.7 (for a good general discussion of Lorentz transformations and other SR basics.) ******************************************************************************** Item 24. Special Relativistic Paradoxes - part (c) The Superluminal Scissors updated 31-MAR-1993 ------------------------- original by Scott I.Chase A Gedankenexperiment: Imagine a huge pair of scissors, with blades one light-year long. The handle is only about two feet long, creating a huge lever arm, initially open by a few degrees. Then you suddenly close the scissors. This action takes about a tenth of a second. Doesn't the contact point where the two blades touch move down the blades *much* faster than the speed of light? After all, the scissors close in a tenth of a second, but the blades are a light-year long. That seems to mean that the contact point has moved down the blades at the remarkable speed of 10 light-years per second. This is more than 10^8 times the speed of light! But this seems to violate the most important rule of Special Relativity - no signal can travel faster than the speed of light. What's going on here? Explanation: We have mistakenly assumed that the scissors do in fact close when you close the handle. But, in fact, according to Special Relativity, this is not at all what happens. What *does* happen is that the blades of the scissors flex. No matter what material you use for the scissors, SR sets a theoretical upper limit to the rigidity of the material. In short, when you close the scissors, they bend. The point at which the blades bend propagates down the blade at some speed less than the speed of light. On the near side of this point, the scissors are closed. On the far side of this point, the scissors remain open. You have, in fact, sent a kind of wave down the scissors, carrying the information that the scissors have been closed. But this wave does not travel faster than the speed of light. It will take at least one year for the tips of the blades, at the far end of the scissors, to feel any force whatsoever, and, ultimately, to come together to completely close the scissors. As a practical matter, this theoretical upper limit to the rigidity of the metal in the scissors is *far* higher than the rigidity of any real material, so it would, in practice, take much much longer to close a real pair of metal scissors with blades as long as these. One can analyze this problem microscopically as well. The electromagnetic force which binds the atoms of the scissors together propagates at the speeds of light. So if you displace some set of atoms in the scissor (such as the entire handles), the force will not propagate down the scissor instantaneously, This means that a scissor this big *must* cease to act as a rigid body. You can move parts of it without other parts moving at the same time. It takes some finite time for the changing forces on the scissor to propagate from atom to atom, letting the far tip of the blades "know" that the scissors have been closed. Caveat: The contact point where the two blades meet is not a physical object. So there is no fundamental reason why it could not move faster than the speed of light, provided that you arrange your experiment correctly. In fact it can be done with scissors provided that your scissors are short enough and wide open to start, very different conditions than those spelled out in the gedankenexperiment above. In this case it will take you quite a while to bring the blades together - more than enough time for light to travel to the tips of the scissors. When the blades finally come together, if they have the right shape, the contact point can indeed move faster than light. Think about the simpler case of two rulers pinned together at an edge point at the ends. Slam the two rulers together and the contact point will move infinitely fast to the far end of the rulers at the instant they touch. So long as the rulers are short enough that contact does not happen until the signal propagates to the far ends of the rulers, the rulers will indeed be straight when they meet. Only if the rulers are too long will they be bent like our very long scissors, above, when they touch. The contact point can move faster than the speed of light, but the energy (or signal) of the closing force can not. An analogy, equivalent in terms of information content, is, say, a line of strobe lights. You want to light them up one at a time, so that the `bright' spot travels faster than light. To do so, you can send a _luminal_ signal down the line, telling each strobe light to wait a little while before flashing. If you decrease the wait time with each successive strobe light, the apparent bright spot will travel faster than light, since the strobes on the end didn't wait as long after getting the go-ahead, as did the ones at the beginning. But the bright spot can't pass the original signal, because then the strobe lights wouldn't know to flash. ******************************************************************************** Item 25. Can You See the Lorentz-Fitzgerald Contraction? 12-Oct-1995 Or: Penrose-Terrell Rotation by Michael Weiss People sometimes argue over whether the Lorentz-Fitzgerald contraction is "real" or not. That's a topic for another FAQ entry, but here's a short answer: the contraction can be measured, but the measurement is frame-dependent. Whether that makes it "real" or not has more to do with your choice of words than the physics. Here we ask a subtly different question. If you take a snapshot of a rapidly moving object, will it *look* flattened when you develop the film? What is the difference between measuring and photographing? Isn't seeing believing? Not always! When you take a snapshot, you capture the light-rays that hit the *film* at one instant (in the reference frame of the film). These rays may have left the *object* at different instants; if the object is moving with respect to the film, then the photograph may give a distorted picture. (Strictly speaking snapshots aren't instantaneous, but we're idealizing.) Oddly enough, though Einstein published his famous relativity paper in 1905, and Fitzgerald proposed his contraction several years earlier, no one seems to have asked this question until the late '50s. Then Roger Penrose and James Terrell independently discovered that the object will *not* appear flattened [1,2]. People sometimes say that the object appears rotated, so this effect is called the Penrose-Terrell rotation. Calling it a rotation can be a bit confusing though. Rotating an object brings its backside into view, but it's hard to see how a contraction could do that. Among other things, this entry will try to explain in just what sense the Penrose-Terrell effect is a "rotation". It will clarify matters to imagine *two* snapshots of the same object, taken by two cameras moving uniformly with respect to each other. We'll call them *his* camera and *her* camera. The cameras pass through each other at the origin at t=0, when they take their two snapshots. Say that the object is at rest with respect to his camera, and moving with respect to hers. By analysing the process of taking a snapshot, the meaning of "rotation" will become clearer. How should we think of a snapshot? Here's one way: consider a pinhole camera. (Just one camera, for the moment.) The pinhole is located at the origin, and the film occupies a patch on a sphere surrounding the origin. We'll ignore all technical difficulties(!), and pretend that the camera takes full spherical pictures: the film occupies the entire sphere. We need more than just a pinhole and film, though: we also need a shutter. At t=0, the shutter snaps open for an instant to let the light-rays through the pinhole; these spread out in all directions, and at t=1 (in the rest-frame of the camera) paint a picture on the spherical film. Let's call points in the snapshot *pixels*. Each pixel gets its color due to an event, namely a light-ray hitting the sphere at t=1. Now let's consider his & her cameras, as we said before. We'll use t for his time, and t' for hers. At t=t'=0, the two pinholes coincide at the origin, the two shutters snap simultaneously, and the light rays spread out. At t=1 for *his* camera, they paint *his* pixels; at t'=1 for *her* camera, they paint *hers*. So the definition of a snapshot is frame-dependent. But you already knew that. (Pop quiz: what shape does *he* think *her* film has? Not spherical!) (More technical difficulties: the rays have to pass right through one film to hit the other.) So there's a one-one correspondence between pixels in the two snapshots. Two pixels correspond if they are painted by the same light-ray. You can see now that her snapshot is just a distortion of his (and vice versa). You could take his snapshot, scan it into a computer, run an algorithm to move the pixels around, and print out hers. So what does the pixel mapping look like? Simple: if we put the usual latitude/longitude grid on the spheres, chosen so that the relative motion is along the north-south axis, then each pixel slides up towards the north pole along a line of longitude. (Or down towards the south pole, depending on various choices I haven't specified.) This should ring a bell if you know about the aberration of light: if our snapshots portray the night-sky, then the stars are white pixels, and aberration changes their apparent positions. Now let's consider the object--- let's say a galaxy. In passing from his snapshot to hers, the image of the galaxy slides up the sphere, keeping the same face to us. In this sense, it has rotated. Its apparent size will also change, but not its shape (to a first approximation). The mathematical details are beautiful, but best left to the textbooks [3,4]. Just to entice you if you have the background: if we regard the two spheres as Riemann spheres, then the pixel mapping is given by a fractional linear transformation. Well-known facts from complex analysis now tell us two things. First, circles go to circles under the pixel mapping, so a sphere will *always* photograph as a sphere. Second, shapes of objects are preserved in the infinitesimally small limit. (If you know about the double-covering of SL(2), that also comes into play. [3] is a good reference.) References: [1] and [2] are the original articles. [3] and [4] are textbook treatments. [5] has beautiful computer-generated pictures of the Penrose-Terrell rotation. The authors of [5] later made a video [6] of this and other effects of "SR photography". [1] Penrose, R.,"The Apparent Shape of a Relativistically Moving Sphere", Proc. Camb. Phil. Soc., vol 55 Jul 1958. [2] Terrell, J., "Invisibility of the Lorentz Contraction", Phys. Rev. vol 116 no. 4 pp. 1041-1045 (1959). [3] Penrose, R., and W. Rindler, "Spinors and Space-Time", vol I chapter 1. [4] Marion, "Classical Dynamics", Section 10.5. [5] Hsiung, Ping-Kang, Robert H. Thibadeau, and Robert H. P. Dunn, "Ray-Tracing Relativity", Pixel, vol 1 no. 1 (Jan/Feb 1990). [6] Hsiung, Ping-Kang, and Robert H. Thibadeau, "Spacetime Visualizations," a video, Imaging Systems Lab, Robotics Institute, Carnegie Mellon University. ******************************************************************************** Item 26. Tachyons updated: 22-MAR-1993 by SIC -------- original by Scott I. Chase There was a young lady named Bright, Whose speed was far faster than light. She went out one day, In a relative way, And returned the previous night! -Reginald Buller It is a well known fact that nothing can travel faster than the speed of light. At best, a massless particle travels at the speed of light. But is this really true? In 1962, Bilaniuk, Deshpande, and Sudarshan, Am. J. Phys. _30_, 718 (1962), said "no". A very readable paper is Bilaniuk and Sudarshan, Phys. Today _22_,43 (1969). I give here a brief overview. Draw a graph, with momentum (p) on the x-axis, and energy (E) on the y-axis. Then draw the "light cone", two lines with the equations E = +/- p. This divides our 1+1 dimensional space-time into two regions. Above and below are the "timelike" quadrants, and to the left and right are the "spacelike" quadrants. Now the fundamental fact of relativity is that E^2 - p^2 = m^2. (Let's take c=1 for the rest of the discussion.) For any non-zero value of m (mass), this is an hyperbola with branches in the timelike regions. It passes through the point (p,E) = (0,m), where the particle is at rest. Any particle with mass m is constrained to move on the upper branch of this hyperbola. (Otherwise, it is "off-shell", a term you hear in association with virtual particles - but that's another topic.) For massless particles, E^2 = p^2, and the particle moves on the light-cone. These two cases are given the names tardyon (or bradyon in more modern usage) and luxon, for "slow particle" and "light particle". Tachyon is the name given to the supposed "fast particle" which would move with v>c. Now another familiar relativistic equation is E = m*[1-(v/c)^2]^(-.5). Tachyons (if they exist) have v > c. This means that E is imaginary! Well, what if we take the rest mass m, and take it to be imaginary? Then E is negative real, and E^2 - p^2 = m^2 < 0. Or, p^2 - E^2 = M^2, where M is real. This is a hyperbola with branches in the spacelike region of spacetime. The energy and momentum of a tachyon must satisfy this relation. You can now deduce many interesting properties of tachyons. For example, they accelerate (p goes up) if they lose energy (E goes down). Futhermore, a zero-energy tachyon is "transcendent," or infinitely fast. This has profound consequences. For example, let's say that there were electrically charged tachyons. Since they would move faster than the speed of light in the vacuum, they should produce Cerenkov radiation. This would *lower* their energy, causing them to accelerate more! In other words, charged tachyons would probably lead to a runaway reaction releasing an arbitrarily large amount of energy. This suggests that coming up with a sensible theory of anything except free (noninteracting) tachyons is likely to be difficult. Heuristically, the problem is that we can get spontaneous creation of tachyon-antitachyon pairs, then do a runaway reaction, making the vacuum unstable. To treat this precisely requires quantum field theory, which gets complicated. It is not easy to summarize results here. However, one reasonably modern reference is _Tachyons, Monopoles, and Related Topics_, E. Recami, ed. (North-Holland, Amsterdam, 1978). However, tachyons are not entirely invisible. You can imagine that you might produce them in some exotic nuclear reaction. If they are charged, you could "see" them by detecting the Cerenkov light they produce as they speed away faster and faster. Such experiments have been done. So far, no tachyons have been found. Even neutral tachyons can scatter off normal matter with experimentally observable consequences. Again, no such tachyons have been found. How about using tachyons to transmit information faster than the speed of light, in violation of Special Relativity? It's worth noting that when one considers the relativistic quantum mechanics of tachyons, the question of whether they "really" go faster than the speed of light becomes much more touchy! In this framework, tachyons are *waves* that satisfy a wave equation. Let's treat free tachyons of spin zero, for simplicity. We'll set c = 1 to keep things less messy. The wavefunction of a single such tachyon can be expected to satisfy the usual equation for spin-zero particles, the Klein-Gordon equation: (BOX + m^2)phi = 0 where BOX is the D'Alembertian, which in 3+1 dimensions is just BOX = (d/dt)^2 - (d/dx)^2 - (d/dy)^2 - (d/dz)^2. The difference with tachyons is that m^2 is *negative*, and m is imaginary. To simplify the math a bit, let's work in 1+1 dimensions, with coordinates x and t, so that BOX = (d/dt)^2 - (d/dx)^2 Everything we'll say generalizes to the real-world 3+1-dimensional case. Now - regardless of m, any solution is a linear combination, or superposition, of solutions of the form phi(t,x) = exp(-iEt + ipx) where E^2 - p^2 = m^2. When m^2 is negative there are two essentially different cases. Either |p| >= |E|, in which case E is real and we get solutions that look like waves whose crests move along at the rate |p|/|E| >= 1, i.e., no slower than the speed of light. Or |p| < |E|, in which case E is imaginary and we get solutions that look waves that amplify exponentially as time passes! We can decide as we please whether or not we want to consider the second sort of solutions. They seem weird, but then the whole business is weird, after all. 1) If we *do* permit the second sort of solution, we can solve the Klein-Gordon equation with any reasonable initial data - that is, any reasonable values of phi and its first time derivative at t = 0. (For the precise definition of "reasonable," consult your local mathematician.) This is typical of wave equations. And, also typical of wave equations, we can prove the following thing: If the solution phi and its time derivative are zero outside the interval [-L,L] when t = 0, they will be zero outside the interval [-L-|t|, L+|t|] at any time t. In other words, localized disturbances do not spread with speed faster than the speed of light! This seems to go against our notion that tachyons move faster than the speed of light, but it's a mathematical fact, known as "unit propagation velocity". 2) If we *don't* permit the second sort of solution, we can't solve the Klein-Gordon equation for all reasonable initial data, but only for initial data whose Fourier transforms vanish in the interval [-|m|,|m|]. By the Paley-Wiener theorem this has an odd consequence: it becomes impossible to solve the equation for initial data that vanish outside some interval [-L,L]! In other words, we can no longer "localize" our tachyon in any bounded region in the first place, so it becomes impossible to decide whether or not there is "unit propagation velocity" in the precise sense of part 1). Of course, the crests of the waves exp(-iEt + ipx) move faster than the speed of light, but these waves were never localized in the first place! The bottom line is that you can't use tachyons to send information faster than the speed of light from one place to another. Doing so would require creating a message encoded some way in a localized tachyon field, and sending it off at superluminal speed toward the intended receiver. But as we have seen you can't have it both ways - localized tachyon disturbances are subluminal and superluminal disturbances are nonlocal. ******************************************************************************** Item 27. The Particle Zoo updated 4-JUL-1995 by MCW ---------------- original by Matt Austern If you look in the Particle Data Book, you will find more than 150 particles listed there. It isn't quite as bad as that, though... The (observed) particles are divided into two major classes: the material particles, and the gauge bosons. We'll discuss the gauge bosons further down. The material particles in turn fall into three categories: leptons, mesons, and baryons. Leptons are particles that are like the electron: they have spin 1/2, and they do not undergo the strong interaction. There are three charged leptons, the electron, muon, and tau, and three corresponding neutral leptons, or neutrinos. (The muon and the tau are both short-lived.) Mesons and baryons both undergo strong interactions. The difference is that mesons have integral spin (0, 1,...), while baryons have half-integral spin (1/2, 3/2,...). The most familiar baryons are the proton and the neutron; all others are short-lived. The most familiar meson is the pion; its lifetime is 26 nanoseconds, and all other mesons decay even faster. Most of those 150+ particles are mesons and baryons, or, collectively, hadrons. The situation was enormously simplified in the 1960s by the "quark model," which says that hadrons are made out of spin-1/2 particles called quarks. A meson, in this model, is made out of a quark and an anti-quark, and a baryon is made out of three quarks. We don't see free quarks, but only hadrons; nevertheless, the evidence for quarks is compelling. Quark masses are not very well defined, since they are not free particles, but we can give estimates. The masses below are in GeV; the first is current mass and the second constituent mass (which includes some of the effects of the binding energy): Generation: 1 2 3 U-like: u=.006/.311 c=1.50/1.65 t=91-200/91-200 D-like: d=.010/.315 s=.200/.500 b=5.10/5.10 In the quark model, there are only 12 elementary particles, which appear in three "generations." The first generation consists of the up quark, the down quark, the electron, and the electron neutrino. (Each of these also has an associated antiparticle.) These particles make up all of the ordinary matter we see around us. There are two other generations, which are essentially the same, but with heavier particles. The second consists of the charm quark, the strange quark, the muon, and the muon neutrino; and the third consists of the top quark, the bottom quark, the tau, and the tau neutrino. These three generations are sometimes called the "electron family", the "muon family", and the "tau family." Finally, according to quantum field theory, particles interact by exchanging "gauge bosons," which are also particles. The most familiar on is the photon, which is responsible for electromagnetic interactions. There are also eight gluons, which are responsible for strong interactions, and the W+, W-, and Z, which are responsible for weak interactions. The picture, then, is this: FUNDAMENTAL PARTICLES OF MATTER Charge ------------------------- -1 | e | mu | tau | 0 | nu(e) |nu(mu) |nu(tau)| ------------------------- + antiparticles -1/3 | down |strange|bottom | 2/3 | up | charm | top | ------------------------- GAUGE BOSONS Charge Force 0 photon electromagnetism 0 gluons (8 of them) strong force +-1 W+ and W- weak force 0 Z weak force The Standard Model of particle physics also predicts the existence of a "Higgs boson," which has to do with breaking a symmetry involving these forces, and which is responsible for the masses of all the other particles. It has not yet been found. More complicated theories predict additional particles, including, for example, gauginos and sleptons and squarks (from supersymmetry), W' and Z' (additional weak bosons), X and Y bosons (from GUT theories), Majorons, familons, axions, paraleptons, ortholeptons, technipions (from technicolor models), B' (hadrons with fourth generation quarks), magnetic monopoles, e* (excited leptons), etc. None of these "exotica" have yet been seen. The search is on! REFERENCES: The best reference for information on which particles exist, their masses, etc., is the Particle Data Book. It is published every two years; the most recent edition is Physical Review D vol.50 No.3 part 1 August 1994. The Web version can be accessed through http://pdg.lbl.gov/. There are several good books that discuss particle physics on a level accessible to anyone who knows a bit of quantum mechanics. One is _Introduction to High Energy Physics_, by Perkins. Another, which takes a more historical approach and includes many original papers, is _Experimental Foundations of Particle Physics_, by Cahn and Goldhaber. For a book that is accessible to non-physicists, you could try _The Particle Explosion_ by Close, Sutton, and Marten. This book has fantastic photography. For a Web introduction by the folks at Fermilab, take a look at http://fnnews.fnal.gov/hep_overview.html . ******************************************************************************** Item 28. original by Scott I. Chase Does Antimatter Fall Up or Down? -------------------------------- This question has never been subject to a successful direct experiment. In other words, nobody has ever directly measured the gravititational acceleration of antimatter. So the bottom line is that we don't know yet. However, there is a lot more to say than just that, with regard to both theory and experiment. Here is a summary of the current state of affairs. (1) Is is even theoretically possible for antimatter to fall up? Answer: According to GR, antimatter falls down. If you believe that General Relativity is the exact true theory of gravity, then there is only one possible conclusion - by the equivalence principle, antiparticles must fall down with the same acceleration as normal matter. On the other hand: there are other models of gravity which are not ruled out by direct experiment which are distinct from GR in that antiparticles can fall down at different rates than normal matter, or even fall up, due to additional forces which couple to the mass of the particle in ways which are different than GR. Some people don't like to call these new couplings 'gravity.' They call them, generically, the 'fifth force,' defining gravity to be only the GR part of the force. But this is mostly a semantic distinction. The bottom line is that antiparticles won't fall like normal particles if one of these models is correct. There are also a variety of arguments, based upon different aspects of physics, against the possibility of antigravity. These include constraints imposed by conservation of energy (the "Morrison argument"), the detectable effects of virtual antiparticles (the "Schiff argument"), and the absense of gravitational effect in kaon regeneration experiments. Each of these does in fact rule out *some* models of antigravity. But none of them absolutely excludes all possible models of antigravity. See the reference below for all the details on these issues. (2) Haven't people done experiments to study this question? There are no valid *direct* experimental tests of whether antiparticles fall up or down. There was one well-known experiment by Fairbank at Stanford in which he tried to measure the fall of positrons. He found that they fell normally, but later analyses of his experiment revealed that he had not accounted for all the sources of stray electromagnetic fields. Because gravity is so much weaker than EM, this is a difficult experimental problem. A modern assessment of the Fairbank experiment is that it was inconclusive. In order to reduce the effect of gravity, it would be nice to repeat the Fairbank experiment using objects with the same magnitude of electric charge as positrons, but with much more mass, to increase the relative effect of gravity on the motion of the particle. Antiprotons are 1836 times more massive than positrons, so give you three orders of magnitude more sensitivity. Unfortunately, making many slow antiprotons which you can watch fall is very difficult. An experiment is under development at CERN right now to do just that, and within the next couple of years the results should be known. Most people expect that antiprotons *will* fall. But it is important to keep an open mind - we have never directly observed the effect of gravity on antiparticles. This experiment, if successful, will definitely be "one for the textbooks." Reference: Nieto and Goldman, "The Arguments Against 'Antigravity' and the Gravitational Acceleration of Antimatter," Physics Reports, v.205, No. 5, p.221. ******************************************************************************** Item 29. What is the Mass of a Photon? updated 24-JUL-1992 by SIC original by Matt Austern Or, "Does the mass of an object depend on its velocity?" This question usually comes up in the context of wondering whether photons are really "massless," since, after all, they have nonzero energy. The problem is simply that people are using two different definitions of mass. The overwhelming consensus among physicists today is to say that photons are massless. However, it is possible to assign a "relativistic mass" to a photon which depends upon its wavelength. This is based upon an old usage of the word "mass" which, though not strictly wrong, is not used much today. The old definition of mass, called "relativistic mass," assigns a mass to a particle proportional to its total energy E, and involved the speed of light, c, in the proportionality constant: m = E / c^2. (1) This definition gives every object a velocity-dependent mass. The modern definition assigns every object just one mass, an invariant quantity that does not depend on velocity. This is given by m = E_0 / c^2, (2) where E_0 is the total energy of that object at rest. The first definition is often used in popularizations, and in some elementary textbooks. It was once used by practicing physicists, but for the last few decades, the vast majority of physicists have instead used the second definition. Sometimes people will use the phrase "rest mass," or "invariant mass," but this is just for emphasis: mass is mass. The "relativistic mass" is never used at all. (If you see "relativistic mass" in your first-year physics textbook, complain! There is no reason for books to teach obsolete terminology.) Note, by the way, that using the standard definition of mass, the one given by Eq. (2), the equation "E = m c^2" is *not* correct. Using the standard definition, the relation between the mass and energy of an object can be written as E = m c^2 / sqrt(1 -v^2/c^2), (3) or as E^2 = m^2 c^4 + p^2 c^2, (4) where v is the object's velocity, and p is its momentum. In one sense, any definition is just a matter of convention. In practice, though, physicists now use this definition because it is much more convenient. The "relativistic mass" of an object is really just the same as its energy, and there isn't any reason to have another word for energy: "energy" is a perfectly good word. The mass of an object, though, is a fundamental and invariant property, and one for which we do need a word. The "relativistic mass" is also sometimes confusing because it mistakenly leads people to think that they can just use it in the Newtonian relations F = m a (5) and F = G m1 m2 / r^2. (6) In fact, though, there is no definition of mass for which these equations are true relativistically: they must be generalized. The generalizations are more straightforward using the standard definition of mass than using "relativistic mass." Oh, and back to photons: people sometimes wonder whether it makes sense to talk about the "rest mass" of a particle that can never be at rest. The answer, again, is that "rest mass" is really a misnomer, and it is not necessary for a particle to be at rest for the concept of mass to make sense. Technically, it is the invariant length of the particle's four-momentum. (You can see this from Eq. (4).) For all photons this is zero. On the other hand, the "relativistic mass" of photons is frequency dependent. UV photons are more energetic than visible photons, and so are more "massive" in this sense, a statement which obscures more than it elucidates. Reference: Lev Okun wrote a nice article on this subject in the June 1989 issue of Physics Today, which includes a historical discussion of the concept of mass in relativistic physics. ******************************************************************************** Item 30. original by David Brahm Baryogenesis - Why Are There More Protons Than Antiprotons? ----------------------------------------------------------- (I) How do we really *know* that the universe is not matter-antimatter symmetric? (a) The Moon: Neil Armstrong did not annihilate, therefore the moon is made of matter. (b) The Sun: Solar cosmic rays are matter, not antimatter. (c) The other Planets: We have sent probes to almost all. Their survival demonstrates that the solar system is made of matter. (d) The Milky Way: Cosmic rays sample material from the entire galaxy. In cosmic rays, protons outnumber antiprotons 10^4 to 1. (e) The Universe at large: This is tougher. If there were antimatter galaxies then we should see gamma emissions from annihilation. Its absence is strong evidence that at least the nearby clusters of galaxies (e.g., Virgo) are matter-dominated. At larger scales there is little proof. However, there is a problem, called the "annihilation catastrophe" which probably eliminates the possibility of a matter-antimatter symmetric universe. Essentially, causality prevents the separation of large chucks of antimatter from matter fast enough to prevent their mutual annihilation in in the early universe. So the Universe is most likely matter dominated. (II) How did it get that way? Annihilation has made the asymmetry much greater today than in the early universe. At the high temperature of the first microsecond, there were large numbers of thermal quark-antiquark pairs. K&T estimate 30 million antiquarks for every 30 million and 1 quarks during this epoch. That's a tiny asymmetry. Over time most of the antimatter has annihilated with matter, leaving the very small initial excess of matter to dominate the Universe. Here are a few possibilities for why we are matter dominated today: a) The Universe just started that way. Not only is this a rather sterile hypothesis, but it doesn't work under the popular "inflation" theories, which dilute any initial abundances. b) Baryogenesis occurred around the Grand Unified (GUT) scale (very early). Long thought to be the only viable candidate, GUT's generically have baryon-violating reactions, such as proton decay (not yet observed). c) Baryogenesis occurred at the Electroweak Phase Transition (EWPT). This is the era when the Higgs first acquired a vacuum expectation value (vev), so other particles acquired masses. Pure Standard Model physics. Sakharov enumerated 3 necessary conditions for baryogenesis: (1) Baryon number violation. If baryon number is conserved in all reactions, then the present baryon asymmetry can only reflect asymmetric initial conditions, and we are back to case (a), above. (2) C and CP violation. Even in the presence of B-violating reactions, without a preference for matter over antimatter the B-violation will take place at the same rate in both directions, leaving no excess. (3) Thermodynamic Nonequilibrium. Because CPT guarantees equal masses for baryons and antibaryons, chemical equilibrium would drive the necessary reactions to correct for any developing asymmetry. It turns out the Standard Model satisfies all 3 conditions: (1) Though the Standard Model conserves B classically (no terms in the Lagrangian violate B), quantum effects allow the universe to tunnel between vacua with different values of B. This tunneling is _very_ suppressed at energies/temperatures below 10 TeV (the "sphaleron mass"), _may_ occur at e.g. SSC energies (controversial), and _certainly_ occurs at higher temperatures. (2) C-violation is commonplace. CP-violation (that's "charge conjugation" and "parity") has been experimentally observed in kaon decays, though strictly speaking the Standard Model probably has insufficient CP-violation to give the observed baryon asymmetry. (3) Thermal nonequilibrium is achieved during first-order phase transitions in the cooling early universe, such as the EWPT (at T = 100 GeV or so). As bubbles of the "true vacuum" (with a nonzero Higgs vev) percolate and grow, baryogenesis can occur at or near the bubble walls. A major theoretical problem, in fact, is that there may be _too_ _much_ B-violation in the Standard Model, so that after the EWPT is complete (and condition 3 above is no longer satisfied) any previously generated baryon asymmetry would be washed out. References: Kolb and Turner, _The Early Universe_; Dine, Huet, Singleton & Susskind, Phys.Lett.B257:351 (1991); Dine, Leigh, Huet, Linde & Linde, Phys.Rev.D46:550 (1992). ******************************************************************************** Item 31. The EPR Paradox and Bell's Inequality Principle updated 31-AUG-1993 by SIC ----------------------------------------------- original by John Blanton In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. This so-called "EPR paradox" has led to much subsequent, and still on-going, research. This article is an introduction to EPR, Bell's inequality, and the real experiments which have attempted to address the interesting issues raised by this discussion. One of the principal features of quantum mechanics is that not all the classical physical observables of a system can be simultaneously known, either in practice or in principle. Instead, there may be several sets of observables which give qualitatively different, but nonetheless complete (maximal possible) descriptions of a quantum mechanical system. These sets are sets of "good quantum numbers," and are also known as "maximal sets of commuting observables." Observables from different sets are "noncommuting observables." A well known example of noncommuting observables is position and momentum. You can put a subatomic particle into a state of well-defined momentum, but then you cannot know where it is - it is, in fact, everywhere at once. It's not just a matter of your inability to measure, but rather, an intrinsic property of the particle. Conversely, you can put a particle in a definite position, but then its momentum is completely ill-defined. You can also create states of intermediate knowledge of both observables: If you confine the particle to some arbitrarily large region of space, you can define the momentum more and more precisely. But you can never know both, exactly, at the same time. Position and momentum are continuous observables. But the same situation can arise for discrete observables such as spin. The quantum mechanical spin of a particle along each of the three space axes is a set of mutually noncommuting observables. You can only know the spin along one axis at a time. A proton with spin "up" along the x-axis has undefined spin along the y and z axes. You cannot simultaneously measure the x and y spin projections of a proton. EPR sought to demonstrate that this phenomenon could be exploited to construct an experiment which would demonstrate a paradox which they believed was inherent in the quantum-mechanical description of the world. They imagined two physical systems that are allowed to interact initially so that they subsequently will be defined by a single Schrodinger wave equation (SWE). [For simplicity, imagine a simple physical realization of this idea - a neutral pion at rest in your lab, which decays into a pair of back-to-back photons. The pair of photons is described by a single two-particle wave function.] Once separated, the two systems [read: photons] are still described by the same SWE, and a measurement of one observable of the first system will determine the measurement of the corresponding observable of the second system. [Example: The neutral pion is a scalar particle - it has zero angular momentum. So the two photons must speed off in opposite directions with opposite spin. If photon 1 is found to have spin up along the x-axis, then photon 2 *must* have spin down along the x-axis, since the total angular momentum of the final-state, two-photon, system must be the same as the angular momentum of the intial state, a single neutral pion. You know the spin of photon 2 even without measuring it.] Likewise, the measurement of another observable of the first system will determine the measurement of the corresponding observable of the second system, even though the systems are no longer physically linked in the traditional sense of local coupling. However, QM prohibits the simultaneous knowledge of more than one mutually noncommuting observable of either system. The paradox of EPR is the following contradiction: For our coupled systems, we can measure observable A of system I [for example, photon 1 has spin up along the x-axis; photon 2 must therefore have x-spin down.] and observable B of system II [for example, photon 2 has spin down along the y-axis; therefore the y-spin of photon 1 must be up.] thereby revealing both observables for both systems, contrary to QM. QM dictates that this should be impossible, creating the paradoxical implication that measuring one system should "poison" any measurement of the other system, no matter what the distance between them. [In one commonly studied interpretation, the mechanism by which this proceeds is 'instantaneous collapse of the wavefunction'. But the rules of QM do not require this interpretation, and several other perfectly valid interpretations exist.] The second system would instantaneously be put into a state of well-defined observable A, and, consequently, ill-defined observable B, spoiling the measurement. Yet, one could imagine the two measurements were so far apart in space that special relativity would prohibit any influence of one measurement over the other. [After the neutral-pion decay, we can wait until the two photons are a light-year apart, and then "simultaneously" measure the x-spin of photon 1 and the y-spin of photon 2. QM suggests that if, for example, the measurement of the photon 1 x-spin happens first, this measurement must instantaneously force photon 2 into a state of ill-defined y-spin, even though it is light-years away from photon 1. How do we reconcile the fact that photon 2 "knows" that the x-spin of photon 1 has been measured, even though they are separated by light-years of space and far too little time has passed for information to have travelled to it according to the rules of Special Relativity? There are basically two choices. You can accept the postulates of QM as a fact of life, in spite of its seemingly uncomfortable coexistence with special relativity, or you can postulate that QM is not complete, that there *was* more information available for the description of the two-particle system at the time it was created, carried away by both photons, and that you just didn't know it because QM does not properly account for it. So, EPR postulated that the existence of hidden variables, some so-far unknown properties, of the systems should account for the discrepancy. Their claim was that QM theory is incomplete; it does not completely describe the physical reality. System II knows all about System I long before the scientist measures any of the observables, thereby supposedly consigning the other noncommuting observables to obscurity. No instantaneous action-at-a-distance is necessary in this picture, which postulates that each System has more parameters than are accounted by QM. Niels Bohr, one of the founders of QM, held the opposite view and defended a strict interpretation, the Copenhagen Interpretation, of QM. In 1964 John S. Bell proposed a mechanism to test for the existence of these hidden parameters, and he developed his inequality principle as the basis for such a test. Use the example of two photons configured in the singlet state, consider this: After separation, each photon will have spin values for each of the three axes of space, and each spin can have one of two values; call them up and down. Call the axes A, B and C and call the spin in the A axis A+ if it is up in that axis, otherwise call it A-. Use similar definitions for the other two axes. Now perform the experiment. Measure the spin in one axis of one particle and the spin in another axis of the other photon. If EPR were correct, each photon will simultaneously have properties for spin in each of axes A, B and C. Look at the statistics. Perform the measurements with a number of sets of photons. Use the symbol N(A+, B-) to designate the words "the number of photons with A+ and B-." Similarly for N(A+, B+), N(B-, C+), etc. Also use the designation N(A+, B-, C+) to mean "the number of photons with A+, B- and C+," and so on. It's easy to demonstrate that for a set of photons (1) N(A+, B-) = N(A+, B-, C+) + N(A+, B-, C-) because all of the (A+, B-, C+) and all of the (A+, B-, C-) photons are included in the designation (A+, B-), and nothing else is included in N(A+, B-). You can make this claim if these measurements are connected to some real properties of the photons. Let n[A+, B+] be the designation for "the number of measurements of pairs of photons in which the first photon measured A+, and the second photon measured B+." Use a similar designation for the other possible results. This is necessary because this is all it is possible to measure. You can't measure both A and B of the same photon. Bell demonstrated that in an actual experiment, if (1) is true (indicating real properties), then the following must be true: (2) n[A+, B+] <= n[A+, C+] + n[B+, C-]. Additional inequality relations can be written by just making the appropriate permutations of the letters A, B and C and the two signs. This is Bell's inequality principle, and it is proved to be true if there are real (perhaps hidden) parameters to account for the measurements. At the time Bell's result first became known, the experimental record was reviewed to see if any known results provided evidence against locality. None did. Thus an effort began to develop tests of Bell's inequality. A series of experiments was conducted by Aspect ending with one in which polarizer angles were changed while the photons were `in flight'. This was widely regarded at the time as being a reasonably conclusive experiment confirming the predictions of QM. Three years later Franson published a paper showing that the timing constraints in this experiment were not adequate to confirm that locality was violated. Aspect measured the time delays between detections of photon pairs. The critical time delay is that between when a polarizer angle is changed and when this affects the statistics of detecting photon pairs. Aspect estimated this time based on the speed of a photon and the distance between the polarizers and the detectors. Quantum mechanics does not allow making assumptions about *where* a particle is between detections. We cannot know *when* a particle traverses a polarizer unless we detect the particle *at* the polarizer. Experimental tests of Bell's inequality are ongoing but none has yet fully addressed the issue raised by Franson. In addition there is an issue of detector efficiency. By postulating new laws of physics one can get the expected correlations without any nonlocal effects unless the detectors are close to 90% efficient. The importance of these issues is a matter of judgement. The subject is alive theoretically as well. In the 1970's Eberhard derived Bell's result without reference to local hidden variable theories; it applies to all local theories. Eberhard also showed that the nonlocal effects that QM predicts cannot be used for superluminal communication. The subject is not yet closed, and may yet provide more interesting insights into the subtleties of quantum mechanics. REFERENCES: 1. A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review 41, 777 (15 May 1935). (The original EPR paper) 2. D. Bohm: Quantum Theory, Dover, New York (1957). (Bohm discusses some of his ideas concerning hidden variables.) 3. N. Herbert: Quantum Reality, Doubleday. (A very good popular treatment of EPR and related issues) 4. M. Gardner: Science - Good, Bad and Bogus, Prometheus Books. (Martin Gardner gives a skeptics view of the fringe science associated with EPR.) 5. J. Gribbin: In Search of Schrodinger's Cat, Bantam Books. (A popular treatment of EPR and the paradox of "Schrodinger's cat" that results from the Copenhagen interpretation) 6. N. Bohr: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review 48, 696 (15 Oct 1935). (Niels Bohr's response to EPR) 7. J. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1 #3, 195 (1964). 8. J. Bell: "On the problem of hidden variables in quantum mechanics" Reviews of Modern Physics 38 #3, 447 (July 1966). 9. D. Bohm, J. Bub: "A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory" Reviews of Modern Physics 38 #3, 453 (July 1966). 10. B. DeWitt: "Quantum mechanics and reality" Physics Today p. 30 (Sept 1970). 11. J. Clauser, A. Shimony: "Bell's theorem: experimental tests and implications" Rep. Prog. Phys. 41, 1881 (1978). 12. A. Aspect, Dalibard, Roger: "Experimental test of Bell's inequalities using time- varying analyzers" Physical Review Letters 49 #25, 1804 (20 Dec 1982). 13. A. Aspect, P. Grangier, G. Roger: "Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment; a new violation of Bell's inequalities" Physical Review Letters 49 #2, 91 (12 July 1982). 14. A. Robinson: "Loophole closed in quantum mechanics test" Science 219, 40 (7 Jan 1983). 15. B. d'Espagnat: "The quantum theory and reality" Scientific American 241 #5 (November 1979). 16. "Bell's Theorem and Delayed Determinism", Franson, Physical Review D, pgs. 2529-2532, Vol. 31, No. 10, May 1985. 17. "Bell's Theorem without Hidden Variables", P. H. Eberhard, Il Nuovo Cimento, 38 B 1, pgs. 75-80, (1977). 18. "Bell's Theorem and the Different Concepts of Locality", P. H. Eberhard, Il Nuovo Cimento 46 B, pgs. 392-419, (1978). ******************************************************************************** Item 32. Some Frequently Asked Questions About Virtual Particles ------------------------------------------------------- original By Matt McIrvin Contents: 1. What are virtual particles? 2. How can they be responsible for attractive forces? 3. Do they violate energy conservation? 4. Do they go faster than light? Do virtual particles contradict relativity or causality? 5. I hear physicists saying that the "quantum of the gravitational force" is something called a graviton. Doesn't general relativity say that gravity isn't a force at all? 1. What are virtual particles? One of the first steps in the development of quantum mechanics was Max Planck's idea that a harmonic oscillator (classically, anything that wiggles like a mass bobbing on the end of an ideal spring) cannot have just any energy. Its possible energies come in a discrete set of equally spaced levels. An electromagnetic field wiggles in the same way when it possesses waves. Applying quantum mechanics to this oscillator reveals that it must also have discrete, evenly spaced energy levels. These energy levels are what we usually identify as different numbers of photons. The higher the energy level of a vibrational mode, the more photons there are. In this way, an electromagnetic wave acts as if it were made of particles. The electromagnetic field is a quantum field. Electromagnetic fields can do things other than vibration. For instance, the electric field produces an attractive or repulsive force between charged objects, which varies as the inverse square of distance. The force can change the momenta of the objects. Can this be understood in terms of photons as well? It turns out that, in a sense, it can. We can say that the particles exchange "virtual photons" which carry the transferred momentum. Here is a picture (a "Feynman diagram") of the exchange of one virtual photon. \ / \ <- p / >~~~ / ^ time / ~~~~ / | / ~~~< | / \ ---> space / \ The lines on the left and right represent two charged particles, and the wavy line (jagged because of the limitations of ASCII) is a virtual photon, which transfers momentum from one to the other. The particle that emits the virtual photon loses momentum p in the recoil, and the other particle gets the momentum. This is a seemingly tidy explanation. Forces don't happen because of any sort of action at a distance, they happen because of virtual particles that spew out of things and hit other things, knocking them around. However, this is misleading. Virtual particles are really not just like classical bullets. 2. How can they be responsible for attractive forces? The most obvious problem with a simple, classical picture of virtual particles is that this sort of behavior can't possibly result in attractive forces. If I throw a ball at you, the recoil pushes me back; when you catch the ball, you are pushed away from me. How can this attract us to each other? The answer lies in Heisenberg's uncertainty principle. Suppose that we are trying to calculate the probability (or, actually, the probability amplitude) that some amount of momentum, p, gets transferred between a couple of particles that are fairly well- localized. The uncertainty principle says that definite momentum is associated with a huge uncertainty in position. A virtual particle with momentum p corresponds to a plane wave filling all of space, with no definite position at all. It doesn't matter which way the momentum points; that just determines how the wavefronts are oriented. Since the wave is everywhere, the photon can be created by one particle and absorbed by the other, no matter where they are. If the momentum transferred by the wave points in the direction from the receiving particle to the emitting one, the effect is that of an attractive force. The moral is that the lines in a Feynman diagram are not to be interpreted literally as the paths of classical particles. Usually, in fact, this interpretation applies to an even lesser extent than in my example, since in most Feynman diagrams the incoming and outgoing particles are not very well localized; they're supposed to be plane waves too. 3. Do they violate energy conservation? We are really using the quantum-mechanical approximation method known as perturbation theory. In perturbation theory, systems can go through intermediate "virtual states" that normally have energies different >from that of the initial and final states. This is because of another uncertainty principle, which relates time and energy. In the pictured example, we consider an intermediate state with a virtual photon in it. It isn't classically possible for a charged particle to just emit a photon and remain unchanged (except for recoil) itself. The state with the photon in it has too much energy, assuming conservation of momentum. However, since the intermediate state lasts only a short time, the state's energy becomes uncertain, and it can actually have the same energy as the initial and final states. This allows the system to pass through this state with some probability without violating energy conservation. Some descriptions of this phenomenon instead say that the energy of the *system* becomes uncertain for a short period of time, that energy is somehow "borrowed" for a brief interval. This is just another way of talking about the same mathematics. However, it obscures the fact that all this talk of virtual states is just an approximation to quantum mechanics, in which energy is conserved at all times. The way I've described it also corresponds to the usual way of talking about Feynman diagrams, in which energy is conserved, but virtual particles can carry amounts of energy not normally allowed by the laws of motion. (General relativity creates a different set of problems for energy conservation; that's described elsewhere in the sci.physics FAQ.) 4. Do they go faster than light? Do virtual particles contradict relativity or causality? In section 2, the virtual photon's plane wave is seemingly created everywhere in space at once, and destroyed all at once. Therefore, the interaction can happen no matter how far the interacting particles are from each other. Quantum field theory is supposed to properly apply special relativity to quantum mechanics. Yet here we have something that, at least at first glance, isn't supposed to be possible in special relativity: the virtual photon can go from one interacting particle to the other faster than light! It turns out, if we sum up all possible momenta, that the amplitude for transmission drops as the virtual particle's final position gets further and further outside the light cone, but that's small consolation. This "superluminal" propagation had better not transmit any information if we are to retain the principle of causality. I'll give a plausibility argument that it doesn't in the context of a thought experiment. Let's try to send information faster than light with a virtual particle. Suppose that you and I make repeated measurements of a quantum field at distant locations. The electromagnetic field is sort of a complicated thing, so I'll use the example of a field with just one component, and call it F. To make things even simpler, we'll assume that there are no "charged" sources of the F field or real F particles initially. This means that our F measurements should fluctuate quantum- mechanically around an average value of zero. You measure F (really, an average value of F over some small region) at one place, and I measure it a little while later at a place far away. We do this over and over, and wait a long time between the repetitions, just to be safe. . . . ------X ------ X------ ^ time ------X me | ------ | you X------ ---> space After a large number of repeated field measurements we compare notes. We discover that our results are not independent; the F values are correlated with each other-- even though each individual set of measurements just fluctuates around zero, the fluctuations are not completely independent. This is because of the propagation of virtual quanta of the F field, represented by the diagonal lines. It happens even if the virtual particle has to go faster than light. However, this correlation transmits no information. Neither of us has any control over the results we get, and each set of results looks completely random until we compare notes (this is just like the resolution of the famous EPR "paradox"). You can do things to fields other than measure them. Might you still be able to send a signal? Suppose that you attempt, by some series of actions, to send information to me by means of the virtual particle. If we look at this from the perspective of someone moving to the right at a high enough speed, special relativity says that in that reference frame, the effect is going the other way: . . . X------ ------ ------X you X------ ^ time ------ | ------X me | ---> space Now it seems as if I'm affecting what happens to you rather than the other way around. (If the quanta of the F field are not the same as their antiparticles, then the transmission of a virtual F particle >from you to me now looks like the transmission of its antiparticle >from me to you.) If all this is to fit properly into special relativity, then it shouldn't matter which of these processes "really" happened; the two descriptions should be equally valid. We know that all of this was derived from quantum mechanics, using perturbation theory. In quantum mechanics, the future quantum state of a system can be derived by applying the rules for time evolution to its present quantum state. No measurement I make when I "receive" the particle can tell me whether you've "sent" it or not, because in one frame that hasn't happened yet! Since my present state must be derivable from past events, if I have your message, I must have gotten it by other means. The virtual particle didn't "transmit" any information that I didn't have already; it is useless as a means of faster-than-light communication. The order of events does *not* vary in different frames if the transmission is at the speed of light or slower. Then, the use of virtual particles as a communication channel is completely consistent with quantum mechanics and relativity. That's fortunate: since all particle interactions occur over a finite time interval, in a sense *all* particles are virtual to some extent. 5. I hear physicists saying that the "quantum of the gravitational force" is something called a graviton. Doesn't general relativity say that gravity isn't a force at all? You don't have to accept that gravity is a "force" in order to believe that gravitons might exist. According to QM, anything that behaves like a harmonic oscillator has discrete energy levels, as I said in part 1. General relativity allows gravitational waves, ripples in the geometry of spacetime which travel at the speed of light. Under a certain definition of gravitational energy (a tricky subject), the wave can be said to carry energy. If QM is ever successfully applied to GR, it seems sensible to expect that these oscillations will also possess discrete "gravitational energies," corresponding to different numbers of gravitons. Quantum gravity is not yet a complete, established theory, so gravitons are still speculative. It is also unlikely that individual gravitons will be detected anytime in the near future. Furthermore, it is not at all clear that it will be useful to think of gravitational "forces," such as the one that sticks you to the earth's surface, as mediated by virtual gravitons. The notion of virtual particles mediating static forces comes from perturbation theory, and if there is one thing we know about quantum gravity, it's that the usual way of doing perturbation theory doesn't work. Quantum field theory is plagued with infinities, which show up in diagrams in which virtual particles go in closed loops. Normally these infinities can be gotten rid of by "renormalization," in which infinite "counterterms" cancel the infinite parts of the diagrams, leaving finite results for experimentally observable quantities. Renormalization works for QED and the other field theories used to describe particle interactions, but it fails when applied to gravity. Graviton loops generate an infinite family of counterterms. The theory ends up with an infinite number of free parameters, and it's no theory at all. Other approaches to quantum gravity are needed, and they might not describe static fields with virtual gravitons. ******************************************************************************** END OF FAQ ## User Contributions:1 DavidLeags Sep 18, 2019 @ 8:20 pm Invest $ 5,000 in Bitcoin mining once and get $ 70,000 passive income per month: https://hec.su/lY2i?&iriji=rckmifpJ8I5De1 ## Comment about this article, ask questions, or add new information about this topic: |