******************************************************************************* * Item 5. TOP QUARK updated: 18-APR-1993 by SIC --------- The top quark is the hypothetical sixth fundamental strongly interacting particle (quark). The known quarks are up (u), down (d), strange (s), charm (c) and bottom (b). The Standard Model requires quarks to come in pairs in order to prevent mathematical inconsistency due to certain "anomalous" Feynman diagrams, which cancel if and only if the quarks are paired. The pairs are (d,u),(s,c) and (b,?). The missing partner of the b is called "top". In addition, there is experimental evidence that the b quark has an "isodoublet" partner, which is so far unseen. The forward-backward asymmetry in the reaction e+ + e- -> b + b-bar and the absence of flavor-changing neutral currents in b decays imply the existence of the isodoublet partner of the b. ("b-bar", pronounced "bee bar", signifies the b antiquark.) The mass of the top quark is restricted by a variety of measurements. Due to radiative corrections which depend on the top quark circulating as a virtual particle inside the loop in the Feynman diagram, a number of experimentally accessible processes depend on the top quark mass. There are about a dozen such measurements which have been made so far, including the width of the Z, b-b-bar mixing (which historically gave the first hints that the top quark was very massive), and certain aspects of muon decay. These results collectively limit the top mass to roughly 140 +/- 30 GeV. This uncertainty is a "1-sigma" error bar. Direct searches for the top quark have been performed, looking for the expected decay products in both p-p-bar and e+e- collisions. The best current limits on the top mass are: (1) From the absence of Z -> t + t-bar, M(t) > M(Z)/2 = 45 GeV. This is a "model independent" result, depending only on the fact that the top quark should be weakly interacting, coupling to the Z with sufficient strength to have been detected at the current resolution of the LEP experiments which have cornered the market on Z physics in the last several years. (2) From the absence of top quark decay products in the reaction p + p-bar -> t + t-bar -> hard leptons + X at Fermilab's Tevatron collider, the CDF (Collider Detector at Fermilab) experiment. Each top quark is expect to decay into a W boson and a b quark. Each W subsequently decays into either a charged lepton and a neutrino or two quarks. The cleanest signature for the production and decay of the t-t-bar pair is the presence of two high-transverse-momentum (high Pt) leptons (electron or muon) in the final state. Other decay modes have higher branching ratios, but have serious experimental backgrounds from W bosons produced in association with jets. The current published lower limit on M(t) from such measurements is 91 GeV (95% confidence), 95 GeV (90% confidence). However, these limits assume that the top quark has the expected decay products in the expected branching ratios, making these limits "model dependent," and consequently not as "hard" as the considerably lower LEP limit of ~45 GeV. Unpublished results from CDF and D0 now claim lower top mass limits of 108 GeV and 103 GeV for the respective detectors, presumably at 95% confidence. These numbers will probably change by the time they make it into print. The future is very bright for detecting the top quark. LEP II, the upgrade of CERN's e+e- collider to E >= 2*Mw = 160 GeV by 1994, will allow a hard lower limit of roughly 90 GeV to be set. Meanwhile, upgrades to CDF, start of a new experiment, D0, and upgrades to the accelerator complex at Fermilab have recently allowed higher event rates and better detector resolution, should allow production of standard model top quarks of mass < 150 GeV in the next two years, and even higher mass further in the future, at high enough event rate to identify the decays and give rough mass measurements. There have already been a few unpublished "candidate" events from CDF and D0, which, if verified, would be the first direct evidence of the top quark, with mass in the vacinity of 130 GeV. References: Phys. Rev. Lett. _68_, 447 (1992) and the references therein. ******************************************************************************* * Item 6. Tachyons updated: 22-MAR-1993 by SIC -------- 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 7. 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 7. Special Relativistic Paradoxes - part (b) The Twin Paradox updated 17-AUG-1992 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 A 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 a 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.) *******************************************************************************