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steuard: (strings)
Thursday, April 22nd, 2010 10:34 am
Apparently, [livejournal.com profile] ukelele's daughter just pointed to my string/physics icon and asked, "What's that?". [Edit: Since I've changed physics icons, you'll need to look at the enlarged version below to see what she was asking about.] It strikes me that she may not be the only one who's wondered that. So here's a stab at a broadly understandable explanation. I'll try to keep it fairly short, but I'll stick it behind a cut anyway (since I'll embed a picture or two). Mind you, this won't make any sense to a 3-year-old, but I already took a stab at that in my reply to the original comment.

( No, it's not a severed aorta... )
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Quantum histories

steuard: (physics)
Thursday, April 22nd, 2010 12:12 am
I've been thinking lately about the many things we usually leave out of intro physics. It's hard to strike the right balance between practical and inspirational, between strong foundations and broad horizons. But I'd like to see that change, at least a little: you won't entice students to love the subject without a glimpse of what makes it beautiful.

Just for fun, let me share an example: the quantum idea that reality is a sum over all possible histories. It goes something like this. When I throw a ball and you catch it, common sense and classical physics agree that the ball follows a specific path through the air.

But quantum mechanics tells a remarkably different story. In this picture, the ball takes every possible path from me to you at once: a straight line, or a high arc, or a zig-zag, or three quick loops around my head before diving between your legs and then up over your head and back down into your hand, or anything else you can imagine. Moreover (and further straining credulity), every single possibility is equally likely! Of course, with so many options, the odds of any given path are basically nil: what we really have to compare are "neighborhoods" of almost-identical paths (those similar enough that we couldn't tell them apart anyway).

Very roughly speaking, quantum mechanics says that every possible path gets to cast an equal vote in favor of its neighborhood. But there's a catch: each path gets a specific "direction" assigned to its vote (think of this like the spinner from a game of Twister). In most neighborhoods those directions go every which way, so when you add up all the votes they pretty much cancel out (just as many lefts as rights, etc.). But there's usually one special neighborhood that votes as a block, with all its vote directions more or less the same. That overwhelmingly probable neighborhood is what we see as the classical path. Common sense emerges from chaos in a truly remarkable way.


[The deeper beauty of this story is that the math of these "vote directions" immediately explains the reason behind the equations of classical physics. The vote direction of a path is precisely its "action" (interpreted as an angle). We learn in calculus that near the minimum of any function its value is nearly constant, so the neighborhood of the path with minimum action will have nearly constant vote direction. And indeed, the "principle of least action" was the basis of the Lagrangian formulation of mechanics long before quantum mechanics was ever imagined, but nobody knew why. Good stuff!]
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Musings on the infinite

steuard: (physics)
Friday, April 2nd, 2010 11:21 am

A few weeks ago, my friend and former colleague Sean Carroll was a guest on the Colbert Report to promote his book about the nature of time. Toward the end of the interview, they discussed the idea of the "multiverse", which Sean uses to refer to the (possibly) infinite number of "universe-sized" regions within the vast web of space and time where we live. The notion is that if we could somehow travel far enough (faster than light) to regions many times more distant than our telescopes can see, we could find countless independent "universes" that can never talk to each other at all. Some of them would be much like our own but others could be very different, maybe even with different laws of physics. Steven Colbert seemed quite interested:

Colbert: Am I in these other universes?

Carroll: There will be people very much like you.

Colbert: In these other universes, is it possible that my show's on at 11 and John Stewart is at 11:30?

Carroll: Maybe more often!

It's a cute exchange, and it's a variant on the old idea that "in an infinitely big universe, everything that could possibly happen must happen somewhere."

Trouble is, I don't know that I buy that argument, for rather subtle reasons. However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer. But many sets (like the real numbers) are uncountably infinite: no matter how you try to label each real number by an integer, you'll miss the vast majority of them. The real numbers are just a much bigger infinity than the integers are. Going on, the set of all possible curves in space is a yet larger infinity. (Assuming space and time are continuous! If they turn out to be discrete, then the set of curves has the same infinite size as the real numbers.)

The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity. So no matter how large our multiverse may be, it's mathematically impossible for every possible history to occur somewhere. Does that mean that our Steven Colbert (on at 11:30) is the only one? Quite possibly so. I'm not convinced that the multiverse idea opens up as many possibilities as people sometimes think.

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Angular "units": a nice perspective

steuard: (physics)
Tuesday, December 1st, 2009 04:19 pm
I had a neat realization during a conversation with my colleague Cameron today, so I thought I'd share. Feel free to tune this out if you're not a math/physics type.

Cameron was pondering the various "units" for angles: the common ones are degrees, radians, and rotations (a.k.a. revolutions). Physics students learn that many common equations (such as v = r ω) only work when angles are in radians: they given wrong answers if you use degrees or rotations. It's a standard story: radians are defined in terms of a ratio of lengths (arc length/radius) so they are automatically dimensionless (since a ratio like "meters/meters" cancels out).

But in what sense is "rotations" not dimensionless? After all, you're just counting things, so what you end up with ought to be a pure number (which has no units by, definition). For that matter, "degrees" is just counting things too (albeit more finely spaced things). Given that multiplying by a pure number can't possibly change the units of your answer, it's hard to understand on a conceptual level why angles measured in degrees or rotations shouldn't work in the equations. (The equations themselves are entirely clear about what works and what doesn't, mind you! It's just understanding them that's subtle.)

The resolution that I've come to is that "degrees" and "rotations" are not units in the sense that "seconds" or "meters" are units. They can't be, as noted. Instead, "degrees" and "rotations" have the same status as SI prefixes like "kilo" and "micro". After all, "kilo" effectively translates as "times 1000", a pure number, but you wouldn't expect your equation to give the right result if you removed a "kilo" that was supposed to be there! So I would assert that one can think of "degrees" as an SI "prefix" meaning "times 2π/360" and "rotations" as an SI prefix meaning "times 2&pi". Explicitly plugging in these definitions then just gives your result in terms of the base unit, the most natural one: radians.

It may not actually be all that helpful in practice (my advice is still "just use radians"), but I think this gives a cute new perspective on what these angular "units" actually mean.

Update: Another useful note: The choice of "base unit" occurs when we define the angle in radians: θ = s/r. That choice then carries through the rest of the equations that we use for angular motion. If we had instead defined the angle as θ = s/(2πr), the "base unit" would have been rotations, and that factor of 2π would show up in every other rotational equation. Radians are clearly the simplest choice, but there isn't anything privileged about it. (Just imagine the ugliness of equations throughout rotational motion if we decided to choose our fundamental definition of angle to be θ = 360s/(2πr)!)
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Life after physics

steuard: (physics)
Saturday, September 13th, 2008 01:32 am
All the talk of LHC doomsday scenarios can get you thinking (even though it may be more likely that the LHC will produce dragons than Earth-swallowing black holes or strange matter), and the HTML source of hasthelhcdestroyedtheearth.com has reminded me of an incredible paragraph from a paper by Sidney Coleman and Frank De Luccia ("Gravitational effects on and of vacuum decay". Physical Review D21 (1980) p. 3305).

In the paper, Coleman and De Luccia studied implications of "vacuum decay", the suggestion that the laws of nature that we know have not settled into their final form. Instead, the idea goes, perhaps the laws of physics got "stuck" as the universe cooled down from the Big Bang much a rock falling down a cliff might get caught on a ledge partway down. The rock is likely to be knocked loose in the next big storm or sooner, perhaps with no apparent cause at all. The same thing could happen to physics if we really are stuck in a "false vacuum": whether caused by a specific event or just bad quantum luck, our seemingly eternal laws of physics could come loose ("decay") and fall into a totally different state. Coleman and De Luccia analyzed what might be left after such an event happened, and had this to say about their results:
This is disheartening. The possibility that we are living in a false vacuum has never been a cheering one to contemplate. Vacuum decay is the ultimate ecological catastrophe; in the new vacuum there are new constants of nature; after vacuum decay, not only is life as we know it impossible, so is chemistry as we know it. However, one could always draw stoic comfort from the possibility that perhaps in the course of time the new vacuum would sustain, if not life as we know it, at least some structures capable of knowing joy. This possibility has now been eliminated.

With that quote in mind, it may be disquieting to realize that a fair number of physicists have come to believe that string theory predicts that the universe has something like 10500 different choices of "vacuum" and that we might be living in any of them. I don't think we know anywhere near enough to say how likely they are to decay.
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Baby universes = awesome

steuard: (physics)
Friday, June 13th, 2008 10:53 am
[livejournal.com profile] ricevermicelli recently posted an awesome quote by Sean Carroll: "...a universe could form inside this room and we’d never know." Sean is a cosmologist at Caltech whom I knew when we were both at Chicago, and if you ever have the chance to see him speak, do it: he's fantastic. (He's got a few talks online, some with video.) I offered to give a capsule summary of what he was talking about, but I figure it might be of broad enough interest to give it its own entry rather than burying it as a comment.

Because this is turning out to be less "a capsule" and more "three capsules and a blender", I'll put the details behind a cut.

( I'll try to keep it short, I promise! )
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Future of physics decided by a deck of cards?

steuard: (physics)
Thursday, July 19th, 2007 09:07 am
I'm guessing that almost nobody who reads my little blog keeps up with the high energy physics preprint server at arXiv.org, but an article just appeared there that I have to share. The title is "Search for Future Influence from L.H.C." The LHC is the upcoming "Large Hadron Collider", the biggest particle physics experiment in history; high energy physicists like me are exceedingly excited about what it might reveal about our universe. This paper seems to be a serious attempt to test some of our most fundamental ideas about physics by using (or perhaps not using!) the LHC, but that very seriousness makes it one of the funniest things I've read in a long time.

( The physics, and the hilarity... )
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