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Flip Tanedo | USLHC | USA

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Hierarchy problems and why electrons don’t have infinite mass

Saturday, September 5th, 2009

One of the reasons why physics is hard to learn in high school is that sometimes it just seems so fraught with inconsistencies. One such inconsistency that always bothered me was the energy of the electron.

Point charges don’t make sense in classical physics

Classical electromagnetism tell us that the energy of a configuration of charges increases as the distance between them decreases. To put it simply, the energy increases the closer together you bring the charges. For two charges q1 and q2 separated by a distance R, the electrostatic energy of the configuration is given by:

twocharges

The first fraction is just an overall constant number, sometimes it’s just called k. Don’t be put off by the fancy pi and epsilon, this number isn’t so important. It’s the second fraction that actually tells us about physics. For constant charges q1 and q2, what we see is that the energy increases the closer we bring them together.

This should make sense without having to look at any equations: the closer you bring two like charges the harder they push each other apart.

Something is fishy here, though. If I bring two charges arbitrarily close together, does this mean that I end up with a configuration with arbitrarily large energy?  (Could I produce a black hole from the electrostatic energy of trying to force two charged particles together?)

In fact, the fishiness is even more insidious: we are told that electrons are point particles that carry some charge density. If we imagine the electron to be a charged spherical shell (like the skin of an orange) and shrink its size down to zero, doesn’t this mean that we end up with an electron of infinite electrostatic energy?

If you do the calculation (this is a common homework problem for undergraduates), the equation for a spherical shell of charge e (the electron’s charge) and radius R is almost the same as the equation above for just two charges:

untitled1

The net effect of the shell is an additional factor of one half. We’re not going to nitpick about overall numbers, the point is that as we take the radius of the shell to zero (i.e. to a “point particle”) the energy seems to go to infinity! Let us call this the electron’s self-energy, i.e. the energy of the electron wanting to push itself apart. Alternately it’s the energy of the electron trying to escape its own electric field.

Something is seriously wrong with classical electromagnetism in a way that is plain to anyone familiar with introductory physics.

Electron self-energy vs. mass

Now let’s try to pull in some things that we’ve learned before. In a previous post I explained Einstein’s relation between energy and mass. You can review it now if you’ve forgotten it, but the punchline is that mass can be thought of as a kind of potential energy belonging to a particle.

This, however, is exactly what the electron self-energy is: it’s a potential energy associated with the electron’s charge. This can be thought of as a contribution to the electron’s mass. Some of you might object: we said that mass is a kind of energy, but this doesn’t mean that all energy can be thought of as mass. Good, you’re thinking like a scientist! The point here is that gravity feels energy, not mass. Usually this is interchangeable because mass energy is so much larger than other types of energy (by factors of the speed of light), but in this case we’re talking about a potentially infinite electrostatic energy, so this should certainly be included in the mass.

We thus end up with the following equation for the “effective” electron mass:

electronmass

We’ve just written out energy using E=mc2. Here m is the observed “effective” mass, M is the “intrinsic” mass, and we’ve included the contribution from the electrostatic energy. Note that m is the mass that we measure, while M is some “non-electrostatic” contribution to the mass energy that is never directly measured. (This is a “deep idea” in quantum physics that I won’t go into right now.)

We know what the electron mass is because we’ve measured it (via experiments by Thompson and Millikan); the left-hand side of this equation is 511 KeV. Experimental results also limit the electron characteristic size to be less than 10-17 cm, so just for the heck of it let’s use that as a limit for the radius R above.

Now if one crunches the numbers, one ends up with the electrostatic term being something like 10 GeV = 10 000 000 KeV.  This makes our equation look really fishy, we end up with:

511 KeV = Mc2 + 10 000 000 KeV

This seems to imply that the Mc2 must cancel out the 10 000 000 KeV at the part-per-mille level to give 511 KeV. (The fact that it has to give a negative contribution isn’t actually too weird since M is never measured.)

Ridiculous fine-tuning: the hierarchy problem

This is an example of a kind of problem that has been at the back of theorists’ minds for over 30 years, it’s called a hierarchy problem. The problem is that the mass of the electron seems to depend on the cancellation of two numbers that are much, much bigger. This might not seem like a big problem, and indeed it took a long time before physicists identified this as something very undesirable in our theory.

The gist of the problem is that nature appears to be very sensitive to how these two big numbers happen to cancel. We say that the two big numbers have to be finely-tuned. The physical mass of the electron, 511 KeV, depends on physics at much higher scales. But this is like wondering why we’re able to calculate the trajectory of a baseball using high school physics without taking into account quantum corrections. The quantum corrections represent very short-distance physics relative to the macroscopic Newtonian physics of the baseball. As explained in a previous post, the physics at one scale should be insensitive to physics of very different scales. In the same way, it would be very surprising that 511 KeV physics would depend on very fine cancellations between quantities from 10,000,000 KeV physics.

In some sense this is an aesthetic problem. It may well be that nature is “finely-tuned” and the electron mass does come from a miraculous cancellation. But this sensitivity to much-higher-scale physics goes against the intuition that one develops from other physical examples of hierarchies.

Quantum theory: antimatter to the rescue

It turns out that quantum mechanics saves us from this “electron mass hierarchy problem. ” This is not actually so surprising: it appeared that our problem came from using classical physics to probe scales (R ~ 10-17 cm ~ 10 GeV) that are much smaller than the electron mass. We know, however, that quantum mechanics becomes effective at small scales.

One of the consequences of quantum mechanics is that one can violate energy conservation, but only for very short amounts of time. This “energy-time” uncertainty is a direct analogue to the “position-momentum” Heisenberg uncertainty relation that is now part of the popular zeitgeist. Written as an equation,

dtde

At short distance scales, quantum electrodynamics predicts that virtual electron-positron pairs may pop out of the vacuum before disappearing again. Their main effect is to become polarized under the influence of the actual electron’s charge, as illustrated below:

Image from Peskin and Schroeder.

Image from Peskin and Schroeder.

These virtual electron-positron pop in and out of existence and end up smearing out the electron’s charge. As we probe the electron at shorter and shorter distances, it no longer behaves as a hard ball of charge, but rather an infinitesimal point particle with a cloud of charge. (Don’t confuse this with an atom having a quantum ‘electron cloud,’ this is a similar but different phenomenon.) In fact, as you keep probing even shorter distances the electron-positron pairs would “screen” the original electron’s charge and you would think that the original electron has a smaller charge than expected. This is an actual effect that physicists call renormalization. (It’s a fancy word that you can tell your friends.)

Thus our toy model of a charged shell breaks down and we are saved from having to think that the electron should have an infinite mass.

Have we solved our hierarchy problem? We need to figure out at what scale does the quantum picture become effective? Using the uncertainty relation above and converting into distance,

distance

Where we’ve written the energy uncertainty as the amount of energy required to create a virtual electron-positron pair; m is the physical electron mass and the h-bar is Planck’s constant (representing the scale of quantum fluctuations). Putting in actual numbers, we find that quantum mechanics becomes relevant at the scale

R ~ 10-11 cm ~ 10 KeV

So indeed we find that quantum effects cancel out the classically infinite contribution to the electron mass at roughly the scale of the electron mass itself. (To the best of my knowledge this argument was first made clear by Hitoshi Murayama in his ICTP lectures on supersymmetry.)

The Higgs Mass Hierarchy Problem and why we expect something new at the LHC

It turns out that this “electron mass” hierarchy problem is exactly analogous to what is more commonly known as The Hierarchy Problem. This is the question of why the Higgs mass is so small.

Wait a second, we haven’t even discovered the Higgs boson… why do we think its mass is small? If there is a Higgs, then it turns out have a similar self-energy as the electron example above. It would appear that the Higgs mass should be arbitrarily large. From the above example, we know what this means now: the Higgs mass should be roughly at the scale of the new physics which “completes” the previous theory.

In order for the Higgs mass not to be finely-tuned, there should be some new phenomena waiting to be discovered at the TeV scale. One proposal for new physics is called supersymmetry, which predicts a “superpartner” particle for each known particle in much the same way that “charge-parity symmetry” predicts an antimatter particle for each known particle. In the above example the virtual effects of matter-antimatter pairs smeared out the electron charge to cancel the electrostatic contribution to its mass. In exactly the same way the virtual supersymmetric particles cancel out the contribution to the divergent contributions to the Higgs mass.

This, however, is a whole different story that I’d like to tell in another blog post. The take-home message is that the “Hierarchy Problem” that physicists always mention as a motivation for new physics can be understood in terms of the classical problem of an electron’s self-energy: a problem that even high school physics students can identify from their textbooks.

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The secret lives of particles

Tuesday, September 1st, 2009

Up next on your favorite sensationalist news program: What do quantum particles do when we’re not looking? Probably not what you’d expect.

Quantum physics is weird… but most readers of this blog probably already knew that.

While the mathematical formalism ‘behind the scenes’ is perfectly well-defined and the predictions by the theory are completely sensible (and rigorously tested), it is often difficult to interpret the mechanism of quantum theory into ideas that make sense relative to everyday experiences. You’ve probably already heard of several examples of quantum weirdness:

  • At very small scales pop in and out of existence like the bubbles in a quantum root beer
  • Somehow the cat-in-the-box is both dead and alive (who put the cat in such a dangerous box?)
  • Particles behave like waves… or maybe waves behave like particles

The subject of quantum physics is too broad and deep for me to give an complete summary of the nuts and bolts, but the three popular examples above are all related to one very neat feature of quantum theory:

Particles behave very differently (perhaps even misbehave) when nobody is looking*.

A quantum system looks just fine when you observe it, but between observations some seriously funky things can occur. I’m being deliberately vague when I say “observation,” suffice it to say that it could be anything from peeking to look at the cat-in-the-box or looking for the remnants of an exotic particle in a collider. Here’s the point: once we make an observation, everything makes sense. Energy and momentum are conserved, the cat is either dead or alive (but certainly not both), etc.

Quantum Billiards

An analogy is in order. In high school physics people love to talk about billiard balls. They’re the perfect example of Newtonian physics: if you know the positions and the momenta of the cue ball, you can calculate (up to friction) the trajectories of all the other balls. Thus if you only knew the initial an final configurations of the billiard balls, you could work out everything that happened in between. Consider, for example, the following picture of my friend and colleague, Sven:

untitled-1

Between these two photos one can work out all of the kinematics of the billiard balls. As a string theorist, however, Sven knows that nature is quantum mechanical.

Quantum mechanics tells us that when you’re not looking, all sorts of crazy stuff happens. If we “close our eyes” between the initial and final configurations, the balls ought to have taken any of an infinite number of arbitrary paths (as shown above). In fact, to be technically correct, they take a combination of all the paths. In one possible path, the eight ball can hit the side bumper and then fly off to the moon at faster-than-the-speed-of-light and return to the table to innocently come to rest just short of the corner pocket.

quantumbilliard

The “sum” of all of these possible (if non-sensical) histories conspire to give a final configuration of billiard balls which just happens to obey all of our rules about conservation of momentum, etc.

Okay, let’s stop right here. You should be angrily jumping up and down saying,

But that’s not science, that’s mumbo-jumbo! Where is the experimental evidence? You’re just making up stories about what happens when nobody is looking, then you’re saying that everything magically works out when someone is looking! Isn’t this just another version of the, “if a tree falls in the woods and there’s nobody there to hear it” paradox?

To this I would say, “good job!” You’re thinking like a scientist. Indeed, up to now what I’ve been telling you is a story and you have no reason to believe it. It turns out, however, that one can set up situations where the effect of this quantum “sum over histories” can actually be made manifest. Let’s start by being a little more concrete about what I’m trying to sell you.

“Hacked” Quantum Mechanics

Here’s a somewhat idiosyncratic introduction to quantum mechanics. It’s not rigorous, but should give a flavor of what’s actually being done. (And I’ll refer back to this in future posts.) Suppose you have a particle that you’ve observed at point A. We would like to know the probability of finding the particle at point B at some later time.

  • For each path between A and B, no matter how funky, we associate a complex number called the amplitude of that path.
  • If we sum together all of these numbers, we get a complex number called the amplitude for the particle to go from A to B.
  • The probability of finding the particle at B is given by the squared absolute value of the amplitude for it to go from A to B.

When we say that a particle takes “all possible paths” between A and B we are really referring to this sum over amplitudes. For those who like fancy words, this is known as Feynman’s “path integral formulation” of quantum mechanics.

Here’s the interesting thing about these amplitudes: complex numbers have relative phases (i.e. angles) and they can “interfere,” i.e. they can sum in a way that partially or completely cancel each other out. This is exactly the same interference when we say quantum mechanics displays wave-particle duality. What usually happens is that the amplitudes for very exotic paths end up canceling each other out, leaving only the contribution from more mundane paths. In particular, this usually leaves behind the classical path as the dominant contribution.

We can set up situations, however, where there are two different classical paths that should both give “dominant contributions.” In this case, one can observe that these two paths exhibit quantum interference. The most well-known example of this is the famous double-slit experiment, where the distribution of electrons on the other end of a barrier with two slits shows a pattern characteristic of quantum interference. In fact, one can “read” the distribution of electrons as a way of hinting at the existence of two slits in the barrier. This experiment is usually presented as proof that electrons behave like waves. In this context I present it as evidence that electrons obey a “sum over histories.” The formalism of quantum mechanics tells us that these are actually the same thing. Thus all this quantum “sum over histories” is not just mumbo-jumbo and really does make testable predictions.

Not-so-secret lives of particles at the LHC

So far I’ve given a very hand-wavy discussion of what is meant when physicists say “a particle takes all possible paths between two points.” This has been rather abstract and it’s not clear why it’s so useful since, as we’ve mentioned, most of the time the punchline is that the result of the quantum effects is to reproduce something classical.

The point is this: one way to describe the program at the LHC is that we’re trying to reveal the “secret lives of particles.” When we smash particles together at the LHC we end up with a lot of stuff coming out. We know that the stuff that we’re colliding are protons. We also know what most of the stuff we’re detecting: light baryons and leptons. None of these things are directly interesting to me: these are all rather boring Standard Model junk that we’ve known about for a long time. We’re not going to directly observe the Higgs boson, or supersymmetry, or extra dimensions in our detectors. Those things would all decay into the boring junk before we have any chance of observing them. (This is looking grim…)

What we can observe, however, is the distribution of the “boring junk.” In between the collision point and the detector, the quantum particles take all possible “paths.” This includes some ‘paths’ that involve creating a Higgs boson (or other exotic particle) which then promptly decays into boring junk. Even though we don’t detect the Higgs directly, we can see its imprint on the distribution of boring junk in the same way that one could “read” the distribution of electrons in the double-slit experiment.

Further Reading**

I’ve been unfortunately terse in my description here. After getting to know some of the readers from their comments, I’m sure many will want to learn about these ideas in somewhat more detail. Allow me to suggest two excellent reference by Richard Feynman, both of which are accessible to a lay audience,

-Flip

* — For Doctor Who fans, the best analogy to a quantum system I can think of is the award-winning episode Blink. Here statues of weeping angels (actually aliens) stalk about when nobody is looking andcreep up on people to attack them. (Supremely creepy.)

** — I recently found out that my favorite childhood television show, Reading Rainbow, has unfortunately completed its last show and will not be continuing due to lack of funding. At the end of each episode LeVar Burton would suggest some books that kids might be interested in picking up in their local library.

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Sick days

Sunday, August 30th, 2009

Ah, the beginning of fall semester. All the undergrads come back, the weather changes, and everyone gets caught up in the hustle and bustle of a new term. In other words, the perfect storm for viruses.

On Thursday a rather hectic week caught up with me and I found myself feeling rather ill. I ran up a bit of a fever before eventually being able to head back to my apartment to hibernate for the better part of the weekend. Thankfully I’m much better now and can catch up on some paper reading for next week.

What I’ve come to realize is a double edged sword in grad student life. While we have a tremendous amount of freedom in how we spend our time, it’s really tough when one has to take some sick days off. It’s not that people aren’t sympathetic or willing to give you some time to recuperate (especially these days with H1N1 in the back of everyone’s minds), but rather that research doesn’t stop when a grad student gets sick. The more time one takes off the more there is to catch up on. I’m responsible to keep up with collaborators and to provide meaningful input, so I have to make sure that I keep up with my project even when I’m out-of-commission.

Fortunately my illness passed rather quickly and I have the weekend to properly recover and catch up with what I’ve missed.

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Comments welcome!

Monday, August 17th, 2009

Hi everyone! I just wanted to say that there have been some really great comments that have motivated on some fun discussions and new posts. I’d like to encourage people to continue to leave comments with questions and thoughts.

We unfortunately can’t always address all of them, but we do our best and it’s always great to get feedback and have a real discussion with our readers.

Two caveats:

  • Facebook viewers: it helps if you to comment on the original posts at http://blogs.uslhc.us/ rather than on the Facebook feed. (This way authors are notified when there’s a new comment.)
  • Comments are moderated by the WordPress spam filter and the US LHC admins who are different people than the actual bloggers. This is why your comments might not show up immediately. Since this is a family-friendly blog, inappropriate comments don’t show up at all.

Let us know what you’d like to read more about!

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c=1 (and how to count calories)

Friday, August 14th, 2009

US LHC reader David left an excellent question on my previous post on E=mc2 that touches on another important physics topic. To give a proper response I’d like to dedicate another post to the matter. Here’s his question:

What I can’t understand is the constant assertion that mass and energy amount to the same thing in the e=mc^2 equation. My (albeit basic) education tells me that e/m = c^2, so how are they the ’same’?

He goes on to express that it’s not clear what kind of units one is supposed to use to make the famous equation make sense. Great questions, David. This will leads us to another equation that sounds really weird: c=1. (Where c is the speed of light.) But let’s start with the basics.

In order for our equations to make sense, then they’d better be consistent to matter what units we use. Nature doesn’t care whether we use inches or centimeters. What nature knows, however (and what we had to discover), is that there are constants of proportionality that allow one to measure one quantity in terms of the units of another.

Okay. That last sentence just got really abstract. Let’s start with simple examples from everyday life. (We’ll get to the speed of light at the end, when hopefully things will be crystal clear.)

What is a unit? E.g. counting calories

The past few weeks I’ve been doing a lot of my work at Starbucks. I know that one grande iced coffee has something like 80 calories. This sets up a natural conversion between units of food energy (calorie) and units of  grande iced coffee:

1 grande iced coffee = 80 calories.

Now I can count calories in terms of grande iced coffees. Let’s say bananas have something like 160 calories. Then I can say that the amount of food energy in a banana is two times the amount of food enregy in iced coffee. Or in short-hand notation,

1 banana = 2 grande iced coffees

Note that in this short-hand notation it has to be clear that the quantities we’re equating are measuring a previously understood quantity; in order for this to make sense we have to say “the food energy in…” before each side of the equals sign. The point is that we can now use grande iced coffees to measure the food energy of other things, like a banana.

This is exactly what we mean by a unit: it’s a conversion between counting numbers and ‘dimensionful’ quantities. For example, it would make sense to say a banana has 160 food energy or 2 food energy. It only makes sense to say that a banana has 160 calories of food energy or a banana ias the food energy of 2 grande iced coffees. But each of these latter expressions is equivalent, they convey the exact same information.

[Technical note: I’m writing ‘food energy’ explicitly here because this represents the energy that can be released by the chemical reactions of digestion. I’m not including things like the ‘matter potential energy’ of the atoms in the food which stays tied up in matter.]

Algebra of units: converting into useful quantities

In our banana = 2 iced coffees equation, one nice feature is that it no longer matters how we actually measured the food energy. We originally used calories because this is what you find on nutrition labels. But this unit doesn’t make any sense to me, I don’t know what it means to ‘spend one calorie.’

My favorite recreational activity is playing basketball. I can look up that for a person of my weight, playing one hour of basketball burns about 500 calories. And just like that we just did another unit conversion:

1 hour basketball = 500 calories.

Since I understand what it means to play an hour of basketball, the natural units to measure the food energy of a banana is in hours of basketball played. You may have already done the calculation in your head.

Now let’s make this a little more formal and do what I call the algebra of units. Let’s see how it works:

The trick is to multiply by 1. That’s right. Okay, I guess the real trick is to write the number 1 in a clever way. Note that one doesn’t have units. Here’s what we do:

basketbaanana

Note that each of the quantities in paretheses is just the number 1. I’ve just written 1 in terms of the ratio of two dimensionful things. Where did I get these expressions for 1? Well I took the equation

1 iced coffee = 80 calories

and I divided both sides by “1 iced coffee” to get 1 = (80 food cal)/(1 iced coffee). Then I did the same thing for the equation of basketball hours to calories. It is critically important that we explicitly wrote out the units of each quantity, because now we can simplify the expression on the right hand side.

This is just simplifying fractions. We have “iced coffee” in the numerator and “iced coffee” in the denominator. So we can cancel out these units. Note that we have to leave the numbers, we’re just cancelling the unit “iced coffee.” Similarly, we can cancel the units of “calories” from the numerator and denominator. What we are left with is

1 banana = (2 x 80 / 500) hours of basketball

Doing the arithmetic we find that in order to burn off the food energy of one banana I have to play .32 hours of basketball, or about 19 minutes. (Until roughly half-time… which would be a good time to snack on another banana.)

Along the way we note that we’ve made a conversion from banana to hours. But I know that bananas are different from hours… so is this statement crazy? No — as long as we know that we mean “the enegy in a banana” and “the energy expended playing an hours of basketball.” This is at the heart of understanding the units in E=mc2.

The formal statement of what we’re doing

Now that we’ve given a tangible example, let’s explain once again what we’ve been doing using high-falutin’ fancy-pants language.

We’ve used equations that relate fixed numbers of one unit to fixed numbers of another unit. In particular, we’ve defined conversion factors. In the above example these conversion factors were just the number 1 written in fancy ways that combine units. The point is that these conversion factors are constants. If they weren’t constants, then they don’t make sense. For example, maybe I don’t just order a grande iced coffee. Maybe depending on how I feel I’ll order a smaller or larger sized up, or maybe I’ll have it with milk. In this case the number of calories in what I called an ‘iced coffee’ is not constant because there are more parameters. One would have to be more specific when defining the conversion  factor so that it really is a constant.

The lesson to take home is this: dimensionful constants allow us to convert between units.

The speed of light is constant

One of the great experimental discoveries in all of science is the fact that the speed of light [in vacuum] is constant. This is the basis for special relativity. For our present discussion however, the point is that now we have a dimensionful constant which we can use to convert units.

In units that I remember, the speed of light is given by

c = 300 000 000 meters / second.

This tells us that we can write out an equality

[the distance travelled by light in] 1 second = 300 000 000 meters.

Now this looks like our silly “1 banana = x hours of basketball” statement, but it does have a clear meaning. We can change units. In fact, this gives us a natural definition for lightsecond:

1 lightsecond =the distance travelled by light in 1 second = 300 000 meters.

In this way a lightsecond (or lightyear, etc.) is both a measurement of time and distance since we’re using the speed of light (a constant) as a conversion. In these units physicists like to say that

the speed of light, c = 1.

This seems like a weird statement, but it’s really just saying that light in vacuum travels at the speed “1 lightsecond per second.” In any real particle physics calculation we always write things in units where the speed of light is 1 since this makes our equations much simpler (just look at the original post and see how even those equations simplify.) If we want to convert back into useful units we can always insert the appropriate factors of 1 = c = 300 000 000 meters / second, just like we did using the ‘algebra of units’ above.

The meaning of E=mc2, redux

So hopefully this makes the meaning of E=mc2 a little more transparent. In fact, I would write this as E=m. The factors of c are just there to convert into normal units. I think David wanted me to write something out explicitly as an example, so let me consider the energy associated with the mass of the proton. I can look up

proton mass, m = 1.7 x 10-27 kg

speed of light, c = 300 000 000 meters/second.

Then the right-hand side of E=mc2 tells us

mc2 = 1.7 x 10-27 kg x (300 000 000 meters/second)2

= 5.02 x 10-19 kg(m/s)2

I’d like to write this into something like joules, so I’d better look up the appropriate conversion from kilograms, meters, seconds into joules:

1 J = 1 kg (m/s)2

So the conversion of units (‘unit algebra’) is very easy — we can just swap the kg(m/s)2 for J using 1 = J/[kg(m/s)2]. And Voila: we discover that the energy associated with the proton mass is about 5 x 10-19 Joules. (I know this as “approximately 1 GeV.”)

Hope that helps! Thanks for the great question.

Flip

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E = mc^2

Thursday, August 13th, 2009

Last month when I gave a talk about Angels & Demons to a group of high school teachers one of the big discussion topics was the nature of Einstein’s famous equation. Since E=mc2 is at the heart of the entire program of collider physics, I thought it’d be a good thing to go over with everyone.

E=mc2 explained in one sentence

In one line, E=mc2 is the statement that energy E and mass m are somehow the same thing where c is the speed of light, which is a fundamental constant and allows us to convert units of mass into units of energy.

How do I use it?

In the context of the LHC, this equation tells us roughly how much energy is needed to create a particle of a certain mass. In the same way, it also tells us how much energy is contained in some lump of matter. For example, if we annihilated a lump matter with mass m with the exact same amount of antimatter, we would expect to cause an explosion of photons with energy E=2mc2 .

Okay. Is that really all there is to it?

Actually, the common form E=mc2 isn’t the whole story. The famous version of Einstein’s relation is actually just an approximation for the full expression, which is:

emcp

The new letter p is momentum. These are all familiar concepts from high school physics: energy is the ability to do ‘work’ (e.g. move stuff around), mass is some concept of how heavy something is, and momentum characterizes an object’s motion. This equation is telling us that these are all somehow the ‘same’ thing, up to factors of the speed of light.

The first thing you should do is check that this reproduces E=mc2 . Certainly if p=0 we get the old relation. More generally, if the p term is much smaller than the m term then we’re valid in using the old equation as an approximation. So this equation is at least consistent with what all the popular science books tell you.

Potential and Kinetic Energy

The reason why I wanted to write this out is that this explicitly separates energy into kinetic and potential parts, just as we’re used to from basic science. Before I explain this, you should be a little surprised: there is no gravitational or electrical background causing a potential, how do we get potential energy for a free particle drifting through empty space?

It’s all in the mass! The m2c4 term is a kind of potential energy for the particle: it’s the amount of energy borrowed from the universe that is bundled up and stored in the particle. When we annihilate matter and antimatter, we are really releasing this mass potential energy back into the universe.

But wait! Then the p2c2 term is supposed to be some kind of kinetic energy. But you already know what kinetic energy looks like for a particle; it’s 1/2mv2. I know that p=mv, and if I plug this in it doesn’t look right at all!

Lies they taught you at school…

Good. Now we can discuss another ‘lie’ they taught you at school: the usual expression for kinetic energy is also just an approximation! Look back at our main equation: the mass term is multiplied by c4, while the momentum term is only multiplied by c2. Since the speed of light is a big number compared to the usual velocities that we’re used to, we can see that the mass term is much, much bigger than the kinetic energy term.

The reason why we never talk about the mass potential energy in high school physics is that usually it’s not possible to convert mass energy into energy useful for work; a particle’s mass doesn’t change. The first time we use such a conversion is in chemistry when we look at nuclear beta decay. (So E=mc2 is part of the explanation for “why does the sun shine?”)

In order to recover the usual form of the kinetic energy, we can make an approximation. Mathematically this means we do a Taylor expansion. (For those unfamiliar with calculus: this is just a natural way of expanding a function in terms of smaller and smaller corrections.) A good chunk of physics has to do with making clever Taylor expansions. 🙂 In order to do an expansion we need an expansion parameter which is small and dimensionless (it doesn’t make sense to call a dimensionful quantity ‘small’ without a reference point). In this problem we are saying p2 is much smaller than m2c2, so we can write the expression for the energy as:

peke

Voila! We’ve explicitly written out the energy as a mass potential term plus the usual kinetic energy form. Here the dots mean terms which are smaller by factors of (p2/m2c2), which is indeed a very small number for everyday velocities much smaller than the speed of light.

Another short summary

Our conclusion is that the Einstein relation tell us that a particle’s energy is given by a [quadratic] sum of its mass and kinetic energy. Momentum, energy, and mass are all the same thing in different forms. A particle’s mass is energy that stored up in making that particle heavy while a particle’s momentum is energy that is used to make that particle move.

A hint of more advanced stuff

That’s it for the main idea of this post. While we’ve done some work, however, I wanted to share something to entice any future physicists (or recreational physicists) out there. We can compare our ‘complete’ energy-mass-momentum with the equation for a circle from high school algebra to motivate a mathematical understanding of Einstein’s so-called ‘special relativity’.

mep

In the first line is the equation for a circle of radius r. In the second line we’ve rewritten our energy-mass-momentum relation in a suggestive way. The left-hand sides of both equations are constants.

The first equation tells us that a point (x,y) is part of a circle of radius r if the sum of the squares of its coordinates is equal to r2. The actual point (x,y) can change, but in order for the point to stay on the circle it has to change in such a way that the relation is maintained. If x increases, y has to decreases; and neither x nor y can increase/decrease too much or else it’s impossible to satisfy the equation. (e.g. the point (2,y) doesn’t live on the circle of radius 1 for any y.)

Let’s understand the second equation in the same way. Now I’m telling you that a particle’s mass is constant. It’s a fundamental property of the particle. (There’s an old notion of ‘relativistic mass’ which has been discarded in the modern way of looking at this.) The particle’s energy and momentum can change (e.g. through elastic collisions), but they must change in such a way that the above relation is satisfied. If the momentum increases, then the energy increases. Well that makes sense from our intuitive understanding of momentum. This also tells us that there is a minimum energy given by p2=0, E2=m2c4. I.e. when the particle is at rest the energy is just the mass potential energy.

Great. This all seems like I’m stating the obvious in an overly complicated way. The point is this: the equation for a circle is also a way of defining length. The distance from the point (0,0) to (x,y) is given by r, the length of the ‘radius’ to the point. A definition of length (called a ‘metric’) defines a particular kind of geometry. The symmetries of the metric are symmetries of the geometry: for example, the rotational symmetry of the circle manifests itself in the rotational symmetry of the two-dimensional Euclidean plane.

In the same way the rewritten mass-energy-momentum equation is also a definition of ‘length’ in this energy-momentum space. It has a funny minus sign. The relation can be written in terms of space-time (the combined coordinates of space and time) as

plength2

where t is time, x is distance in space, and s is some constant (like r or m2c4) called the proper distance.. In special relativity the trajectory of a free particle must obey this equation of constraint. The symmetries of this ‘metric’ (this thing which defines a preserved length) are called Lorentz transformations. The space defined by these symmetries is called Minkowski space (versus Euclidean space that we’re used to). Just as the rotations caused a point to move around in a circle of constant radius, Lorentz transformations are a rotation in spacetime that preserve the proper distance, s.

In particular, what that means is this remarkable fact:

Space and time are in some sense the same thing.

Of course this statement needs to be understood in a mathematical context. Of course space and time are different: we can move back and forth in space but only in one direction in time, etc. But mathematically one can do rotations between space and time. This is precisely the origin of the magnificent results of special relativity: length contraction and time dilation!

It turns out that the analog of a rotation in Euclidean space by some angle is a boost in Minkowski space. The name is chosen specifically to make clear the relation to picking a reference frame.

Anyway, this opens up the rabbit hole to the fantastic story of special relativity which one can find in any number of excellent books or online references. For those who really want to pursue the mathematical story at a basic level, I cannot recommend enough Sander Bais’ book Very Special Relativity. Those with a high-school physics background can read the relevant chapter of The Feynman Lectures on Physics.

Flip

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Forget black holes… ninjas at the LHC?

Friday, August 7th, 2009

There might be a more immediate threat than black holes at the LHC… ninjas working for shady arms dealers intent on a new world order!

A ninja at the LHC. No... really. Image from the GI Joe Movie fankit.

A ninja at the LHC. No... really. Image from the GI Joe Movie fankit.

Ok, I’m being facetious. I just got back from watching the new action movie, G.I. Joe: The Rise of Cobra. As someone who grew up watching the G.I. Joe cartoon in the 80s and reading the many iterations of G.I. Joe comic books, I couldn’t help but go out to watch the film despite my low (i.e. nonexistent) expectations for anything academy award-worthy.

What I felt like sharing with the US LHC blogosphere, however, was an oblique mention of the LHC in the middle of the movie. There’s technically some small spoilers ahead… but look, I don’t think anybody is going to watch this movie for its plot, so I’ll spill the beans.

One of the movie bad-girls, the Baroness — because she’s married to a French baron — forces her physicist husband to activate some nano-bot missiles using a “particle accelerator in France.”  Did I mention that the French baron is also a particle physicist? Yeah, it’s that kind of movie. Anyway, the scene involves the ninja assassin Storm Shadow (pictured above) slicing up some innocent particle physicists (experimentalists, no doubt*). In the background of all this is a big machine that looks very similar to the ATLAS detector.

Those of you who have attended one of the “Science of Angels & Demons” talks will be familiar with the misrepresentations of particle physics labs. (Lab coats, windows to the interaction point, …) But like the many liberties in science, technology, and plausibility that the movie takes, one is just asked to take these in stride; this isn’t meant to be a `cerebral’ film, it’s a movie about action figures.

By the way, how should you know that the premise of “activating” nano-bots using the LHC is silly? The energy scales at the LHC can be converted into length scale**. If one does a back-of-the-envelope calculation, one finds that the TeV-scale energies probed by the LHC corresponds to length of  roughly 10^{-19} meters. This is way smaller than a “nanobot” or anything that would be built out of atoms. For a concrete comparison, Cornell’s `nanoguitar‘ is only on the order of 10^{-5} meters.

So trying to use the LHC to ‘activate’ nanobots would be like trying to use a toothbrush to wash your kitchen floor… only your toothbrush would be the width of a DNA molecule and your kitchen floor would be the size of Jupiter.

While we’re at it, here are a few other priceless particle physics ‘moments’ in recent blockbusters:

  • Star Trek (2009): Vulcans in the future work with ‘red matter,’ a fictional substance that can warp space and is presumably named to mimic ‘dark matter.’
  • Spiderman 3 (2007): While fleeing the police, Flint Marko accidentally falls into a particle accelerator (apparently a chain-link fence is a sufficient barrier to synchrotron radiation in the Marvel universe), turning him into the supervillain The Sandman.
  • The World is Not Enough (1999): Okay, this movie is now a decade old, but I’ll never forget the ridiculous scene where Denise Richardson pulls herself out of some large service pipe, peels off a tight-fitting jumpsuit, and introduces herself as “Christmas Jones, I’m a theoretical physicist.” (At least I remember her saying that, it was so long ago.)

Anyway, to all my experimental colleagues: keep an eye out for ninjas!

-Flip (“… I’m a theoretical physicist.”)

* — I say this because it’s a well-known fact that most theorists posses crazy ninja abilities.

** — The idea of converting energy scales to length scales comes from the observation that nature appears to have ‘fundamental’ dimensionful constants. For example, the speed (=length / time) of light is constant which sets an `equality’ between length and time. Similarly, Planck’s constant (h-bar) sets an equality between energy and rate (inverse time). Thus one can convert an energy (TeV) into a length scale by dividing by the speed of light, dividing by the Planck constant and then taking the inverse of the result. This isn’t an exact relation since sometimes the constants are defined with factors of pi floating around, but it gives an order of magnitude estimate for the relation between length and energy scales.

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(Micro-)collaborations across continents

Tuesday, August 4th, 2009

Particle physicists are known for having large collaborations that incorporate groups from universities and labs around the world. Experimentalists have developed fantastic tools for managing these collaborations including a range of video-conferencing tools, wiki-based resources, and `starter kits‘ to get new collaborators up-and-running.

Theorists, on the other hand, tend to use more mundane methods to manage much smaller collaborations of around three to five researchers from different places around world. A particular favorite is to organize workshops where a large group of researchers chat and develop new ideas in-person. After that most work is done via e-mails and conference calls (e.g. Skype) and is relatively independent.

Being good netizens of the 21st century, however, younger physicists have started thinking about how to use technology to help facilitate smaller-scale scientific collaborations. Web 2.0 staples like blogs and wikis can are simple ways to build a common set of knowledge in a research collaboration. For example, they can help

  • organize an annotated paper-trail of past literature on a subject
  • provide a truly collaborative (almost real-time) writing environment
  • let other collaborators know what you’re reading/thinking about.

For the past two weeks I’ve been collaborating with other graduate students at different institutions and playing with some of these tools to see how we can best make up for the lack of face-to-face-to-chalkboard interactions.

While there are lots of useful tools out there, we’ve yet to find a true `killer app’ that combines collaborative knowledge-management (wiki), threaded sub-discussions (forums, blogs), multimedia (VoIP, LaTeX), and real-time (instant messaging) + off-line (e-mail) communications.

There is one particularly promising tool in the near future, however: Google Wave. In the web search company’s May product preview, they demonstrated the system’s unique ability to facilitate mundane tasks like sharing photos or arranging a movie outing among friends. That’s all well and good, but my colleagues and I have especially high hopes for what it may do for scientific (and mathematical) collaboration.

Most of you web-saavy readers will already know a thing or two about Wave. For those that don’t, I’d recommend taking a look at their product preview. At the very least it’s a neat idea. As a researcher managing my own small collaborations, however, a few features stick out:

  1. As reported by Terence Tao, one of the early extensions appears to be a LaTeX package which would really help physics discussions. (The IM client Adium has had a similar feature for some time.)
  2. “Waves” combine the discussion threads of a blog/forum the collaborative document preparation of a wiki. In this way one could, for example, write up a section of a paper while having a thread of questions and answers to work out tricky calculations. (We’d never have to worry about a modern-day Fermat saying “there’s not enough room in this margin.”)
  3. The wiki and IM nature of a ‘wave’ automatically builds in real-time version control. One would never have to worry about e-mailing back and forth several versions of a paper and then losing track of which revisions are most recent. (This is a real annoyance when writing a paper.)
  4. I suspect it would not be difficult to write an API to send a document written in Google Wave to one’s local LaTeX compiler. This way one can really work collaboratively on a LaTeX file without all the overhead and limitations of having a central LaTex compiler (e.g. ScribTeX) or having to manually send LaTeX files back and forth via e-mail.

The current status is that Google already has a Wave beta in operation for developrs and plans on opening a limited public beta in September. It sounds very exciting!

Further reading

Flip
(I’ve been informed that I should sign my posts to help the Facebook crowd identify authors.)

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Some summer reading…

Tuesday, July 28th, 2009

Some time ago I had a discussion originally got interested in particle physics and I traced it all back to a book I read in 7th grade, The Physics of Star Trek by Lawrence Krauss. (The book not only turned me into a physicist, but also a trekkie.)

It’s a little late to put out a summer reading list, but for those of you who are still looking around for something to tickle your particle physics fancy, here are a few other personal recommendations.

First of all, let’s start with just about anything that Richard Feynman has written. Perhaps the most enjoyable and playful volume for a general audience is his auto-biographical Surely You’re Joking, Mr. Feynman, a collection of anecdotes about his life. There isn’t much physics in this book, but you’ll see why Feynman is a hero to generation after generation of physicists and laypeople alike. If you’re looking for something with more physics intuition (accessible to ambitious high school students), check out QED, a surprisingly accurate description of the work for which Feynman was awarded the Nobel prize in physics.

To the best of my knowledge, Brian Greene’s The Elegant Universe is still one of the most successful popular introductions to theoretical physics, focusing on string theory. The NOVA documentary is also available free online. While I haven’t read it, I’ve heard that Lisa Randall’s Warped Passages is also an excellent read with more of a particle physics emphasis.

For those interested in extra dimensions (one of my current projects), I would also strongly recommend Edwin Abbott’s 19th century novel, Flatland. It’s available free online, but I’d recommend an annotated version to fill in some of the background. The book is, in some sense, a social commentary on Victorian life; but more importantly it is a remarkable lesson in how to think about higher dimensions by analogy to the two-dimensional flat land. (And it was written more than a century ago!)

These are all great popular-level books, but what about something for people who want to get their hands dirty? I remember the summer before starting college I was itching to start learning ‘real physics.’ For the autodidactically inclined, I would recommend starting with any fairly recent college physics textbook. Ugh! So dry, right? Don’t worry, I suggest skipping to the last few chapters that discuss modern physics. Most textbooks I’ve seen have some neat things to say about special relativity, quantum physics, and even the Standard Model. So to all you eager college freshmen, I suggest taking a skim through these chapters now because it’s unlikely your intro-level courses will ever get to the ‘good stuff.’

For more specific suggestions, I’ve recently found a wonderful exposition on special relativity called Very Special Relativity by Sander Bais. The book is accessible to anyone with a high-school physics background and a good grasp of algebra. It is remarkably adept at conveying a real working understanding of special relativity through illustrations. I would even recommend this book as a model of excellent pedagogy to graduate students who will be teaching courses in modern physics.

For advanced undergrads or beginning grad students who are interested in particle physics, an excellent first book in quantum field theory is Zee’s Quantum Field Theory in a Nutshell, which illuminates the concepts behind the equations with expert clarity. Don’t rush out to buy it just yet, though, as I’ve been informed that a new expanded edition is right around the corner. For those rising graduate students who want a lighter summer, get acquainted with PhD Comics… you’ll laugh now, and then realize it’s all spot-on. 🙂

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Positrons from Bananas

Tuesday, July 21st, 2009

I was recently preparing a “Physics of Angels & Demons” talk for a group of high school physics teachers who were visiting Cornell for a “Contemporary Physics for Teachers” workshop. While researching ‘natural sources of antimatter,’ I discovered a curious article about a naturally occurring potassium isotope that, some fraction of the time, decays via positron emission. The conclusion was that:

Tthe average banana (rich in potassium) produces a positron roughly once every 75 minutes.

Now any time you find something like this you have to remember that not everything on the Internet is true — not even Wikipedia, but I checked it out (e.g. the LBNL/Lunds table of isotopes) and indeed this seems to be correct!

Potassium-40 (40K) is a naturally occurring isotope that is unstable and decays, but it has a huge half life, about a billion years. These days only a small fraction (100 parts per million) of potassium atoms are actually 40K, but objects that are dense in potassium — such as bananas — are likely to have tens of micrograms of the stuff. If one crunches the numbers (as they do in the original article), it turns out that bananas pop out a positron every 75 minutes or so.

These positrons quickly annihilate with ambient electrons, perhaps undergoing some other interactions and releasing some photons beforehand. I’m sure the bloggers here who work on LHC calorimetry would have a better description of what happens to it! Advanced readers can read the “Passage of particles through matter” section of the PDG.

Potassium plays a necessary role in our biology, so yes, even you produce positrons every once in a while.

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