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- Subject: sci.physics Frequently Asked Questions (Part 4 of 4)
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- --------------------------------------------------------------------------------
- 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
-
-