No knock, the door just opened suddenly.

“Hello, Jeremy.  Rule of Three?”

“Huh?  No, I was down the hall just now when I saw you go into your office so I knew you hadn’t gotten busy with something yet.  Sir.  What’s the Rule of Three?”

“Never mind.  You’re up here about virtual particles, I guess.”

“Yessir.  You said they’re ‘now you might see them, now you probably don’t.’  What’s that about and what do they have to do with abstraction and Einstein’s ‘underlying reality’?”

“What have you heard about Heisenberg’s Uncertainty Principle?”

“Ms Plenum says you can’t know where you are and how fast you’re going.”

“Ms Plenum’s got part of the usual notion but she’s missing the idea of simultaneous precision and a few other things.  Turns out you CAN know approximately where you are AND approximately how fast you’re going at a particular moment, but you can’t know both things precisely.  There’s going to be some imprecision in both measurements.  Think about Coach using a radar gun to track a thrown baseball.  How does radar work?”

“It bounces a light beam off of something and measures the light’s round-trip travel time.  I suppose it multiplies by the speed of light to convert time to distance.”

“Good.  Now how does it get the ball’s speed?”

“Uhh… probably uses two light pulses a certain time apart and calculates the speed as distance difference divided by time difference.”

“Got it in one.  Now, suppose that a second after the ball’s thrown the radar says the ball is 61 feet away from the plate and traveling at 92 mph.  Air resistance acts to slow the ball’s flight so that 92 is really an average.   Maybe it was going 92.1 mph at the first radar pulse and 91.9 mph at the second pulse.  So that reported speed has an 0.2 mph range of uncertainty.”

“Oh, and neither of the two pulses caught the ball at exactly 61 feet so that’s uncertain, too, right?”

“There you go.  We know the two averages, but each of them has a range.  The Uncertainty Principle says that the product of those two ranges has to be greater than Planck’s constant, 10-34 Joule·second.  Plugging that Joule-fraction and the mass of an electron into Einstein’s E=mc², we restate the constant as about 10-21 of an electron-second.  Those are both teeny numbers — but they’re not zero.”

“So speed and location make an uncertainty pair.  Are there others?”Zebras“A few.  The most important for this discussion is energy and time.”

“Wait a minute, those two can’t be linked that way.”

“Why not?”

“Well, because … umm … speed is change of location so those two go together, but energy isn’t change of time.  Time just … goes, and adding energy won’t make it go faster.”

“As a matter of fact, there are situations where adding energy makes time go slower, but that’s a couple of stories for another day.  What we’re talking about here is uncertainty ranges and how they combine.  Quantum theory says that if a given particle has a certain energy, give or take an energy range, and it retains that energy for a certain duration, give or take a time range, then the product of the two ranges has to be larger than that same Planck constant.   Think about a 1-meter cube of empty space out there somewhere.  Got it?


“Suppose a particle appeared and then vanished somewhere in that cube sometime during a 1-second interval.  What’s the longest time that particle could have existed?”

“Easy — one second.”

“How about the shortest time?”

“Zero.  Wait, it’d be the smallest possible non-zero time, wouldn’t it?”

“Good catch.  So what’s the time uncertainty?”

“One second minus that tiniest bit of time.”

“And what’s the corresponding energy range?”

“That constant number that I forget.”

“10-21 electron-second’s worth.  Now let’s pick a shorter interval.  What’s the mass range for a particle that appears and disappears sometime during the 10-19 second it takes a photon to cross a hydrogen atom?”

“That’s 10-21 electron-second divided by 10-19 second, so it’d be, like, 0.01 electron.”

“How about 1% of that 10-19 second?”

“Wow — that’d be a whole electron.”

“A whole electron’s worth of uncertainty.  But is the electron really there?”

“Probably not, huh?”

“Like I said, ‘Now you probably don’t’.”

~~ Rich Olcott

And now for some completely different dimensions

Terry Pratchett wrote that Knowledge = Power = Energy = Matter = Mass.  Physicists don’t agree because the units don’t match up.

Physicists check equations with a powerful technique called “Dimensional Analysis,” but it’s only theoretically related to the “travel in space and time” kinds of dimension we discussed earlier.

Place setting LMTIt all started with Newton’s mechanics, his study of how objects affect the motion of other objects.  His vocabulary list included words like force, momentum, velocity, acceleration, mass, …, all concepts that seem familiar to us but which Newton either originated or fundamentally re-defined. As time went on, other thinkers added more terms like power, energy and action.

They’re all linked mathematically by various equations, but also by three fundamental dimensions: length (L), time (T) and mass (M). (There are a few others, like electric charge and temperature, that apply to problems outside of mechanics proper.)

Velocity, for example.  (Strictly speaking, velocity is speed in a particular direction but here we’re just concerned with its magnitude.)   You can measure it in miles per hour or millimeters per second or parsecs per millennium — in each case it’s length per time.  Velocity’s dimension expression is L/T no matter what units you use.

Momentum is the product of mass and velocity.  A 6,000-lb Escalade SUV doing 60 miles an hour has twice the momentum of a 3,000-lb compact car traveling at the same speed.  (Insurance companies are well aware of that fact and charge accordingly.)  In terms of dimensions, momentum is M*(L/T) = ML/T.

Acceleration is how rapidly velocity changes — a car clocked at “zero to 60 in 6 seconds” accelerated an average of 10 miles per hour per second.  Time’s in the denominator twice (who cares what the units are?), so the dimensional expression for acceleration is L/T2.

Physicists and chemists and engineers pay attention to these dimensional expressions because they have to match up across an equal sign.  Everyone knows Einstein’s equation, E = mc2. The c is the velocity of light.  As a velocity its dimension expression is L/T.  Therefore, the expression for energy must be M*(L/T)2 = ML2/T2.  See how easy?

Now things get more interesting.  Newton’s original Second Law calculated force on an object by how rapidly its momentum changed: (ML/T)/T.  Later on (possibly influenced by his feud with Liebniz about who invented calculus), he changed that to mass times acceleration M*(L/T2).  Conceptually they’re different but dimensionally they’re identical — both expressions for force work out to ML/T2.

Something seductively similar seems to apply to Heisenberg’s Area.  As we’ve seen, it’s the product of uncertainties in position (L) and momentum (ML/T) so the Area’s dimension expression works out to L*(ML/T) = ML2/T.

SeductiveThere is another way to get the same dimension expression but things aren’t not as nice there as they look at first glance.  Action is given by the amount of energy expended in a given time interval, times the length of that interval.  If you take the product of energy and time the dimensions work out as (ML2/T2)*T = ML2/T, just like Heisenberg’s Area.

It’s so tempting to think that energy and time negotiate precision like position and momentum do.  But they don’t.  In quantum mechanics, time is a driver, not a result.  If you tell me when an event happens (the t-coordinate), I can maybe calculate its energy and such.  But if you tell me the energy, I can’t give you a time when it’ll happen.  The situation reminds me of geologists trying to predict an earthquake.  They’ve got lots of statistics on tremor size distribution and can even give you average time between tremors of a certain size, but when will the next one hit?  Lord only knows.

File the detailed reasoning under “Arcane” — in technicalese, there are operators for position, momentum and energy but there’s no operator for time.  If you’re curious, John Baez’s paper has all the details.  Be warned, it contains equations!

Trust me — if you’ve spent a couple of days going through a long derivation, totting up the dimensions on either side of equations along the way is a great technique for reassuring yourself that you probably didn’t do something stupid back at hour 14.  Or maybe to detect that you did.

~~ Rich Olcott