Category Archives: Physics

E = m c-Squared: The Simple Dimensions of a Discovery

In my last post I introduced you to dimensional analysis, an essential trick for theoretical physicists, and showed you how you could address and sometimes solve interesting and important problems with it while hardly doing any work. Today we’ll look at it differently, to see its historical role in Einstein’s relativity.

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Dimensional Analysis: A Secret Weapon in Physics

It’s not widely appreciated how often physicists can guess the answer to a problem before they even start calculating. By combining a basic consistency requirement with scientific reasoning, they can often use a heuristic approach to solving problems that allows them to derive most of a formula without doing any work at all. This week I want to introduce this to you, and show you some of its power.

The trick, called “dimensional analysis” or “unit analysis” or “dimensional reasoning”, involves requiring consistency among units, sometimes called “dimensions.” For instance, the distance from the Earth to the Sun is, obviously, a length. We can state the length in kilometers, or in miles, or in inches; each is a unit of length. But for today’s purposes, it’s irrelevant which one we use. What’s important is this: the Earth-Sun distance has to be expressed in some unit of length, because, well, it’s a length! Or in physics-speak, it has the “dimensions of length.”

For any equation in physics of the form X = Y, the two sides of the equation have to be consistent with one another. If X has dimensions of length, then Y must also have dimensions of length. If X has dimensions of mass, then Y must also. Just as you can’t meaningfully say “I weigh twelve meters” or “I am seventy kilograms old”, physics equations have to make sense, relating weights to weights, or lengths to lengths, or energies to energies. If you see an equation X=Y where X is in meters and Y is in Joules (a measure of energy), then you know there’s a typo or a conceptual mistake in the equation.

In fact, looking for this type of inconsistency is a powerful tool, used by students and professionals alike, in checking calculations for errors. I use it both in my own research and when trying to figure out, when grading, where a student went wrong.

That’s nice, but why is it useful beyond checking for mistakes?

Sometimes, when you have a problem to solve involving a few physical quantities, there might be only one consistent equation relating them — only one way to set an X equal to a Y. And you can guess that equation without doing any work.

Well, that’s pretty abstract; let’s see how it works in a couple of examples.

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Physics is Broken!!!

Last Thursday, an experiment reported that the magnetic properties of the muon, the electron’s middleweight cousin, are a tiny bit different from what particle physics equations say they should be. All around the world, the headlines screamed: PHYSICS IS BROKEN!!! And indeed, it’s been pretty shocking to physicists everywhere. For instance, my equations are working erratically; many of the calculations I tried this weekend came out upside-down or backwards. Even worse, my stove froze my coffee instead of heating it, I just barely prevented my car from floating out of my garage into the trees, and my desk clock broke and spilled time all over the floor. What a mess!

Broken, eh? When we say a coffee machine or a computer is broken, it means it doesn’t work. It’s unavailable until it’s fixed. When a glass is broken, it’s shattered into pieces. We need a new one. I know it’s cute to say that so-and-so’s video “broke the internet.” But aren’t we going a little too far now? Nothing’s broken about physics; it works just as well today as it did a month ago.

More reasonable headlines have suggested that “the laws of physics have been broken”. That’s better; I know what it means to break a law. (Though the metaphor is imperfect, since if I were to break a state law, I’d be punished, whereas if an object were to break a fundamental law of physics, that law would have to be revised!) But as is true in the legal system, not all physics laws, and not all violations of law, are equally significant.

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Did BICEP2 Detect Gravitational Waves Directly or Indirectly?

A few weeks ago there was (justified) hullabaloo following the release of results from the BICEP2 experiment, which (if correct as an experiment, and if correctly interpreted) may indicate the detection of gravitational waves that were generated at an extremely early stage in the universe (or at least in its current phase)… during a (still hypothetical but increasingly plausible) stage known as cosmic inflation.  (Here’s my description of the history of the early universe as we currently understand it, and my cautionary tale on which parts of the history are well understood (and why) and which parts are not.)

During that wild day or two following the announcement, a number of scientists stated that this was “the first direct observation of gravitational waves”.  Others, including me, emphasized that this was an “indirect observation of gravitational waves.”  I’m sure many readers noticed this discrepancy.  Who was right?

No one was wrong, not on this point anyway.  It was a matter of perspective. Since I think some readers would be interested to understand this point, here’s the story, and you can make your own judgment. Continue reading

The Amazing Feat of Quantum Tunneling

Our quantum world has many odd and counter-intuitive features.  One of these is “tunneling” — the ability of objects to pass through walls, escape from traps, and slip under mountains into the next valley.   We don’t encounter this effect in daily life; objects we’re used to are so incredibly unlikely to tunnel from one place to another that we will never hear of one doing the apparently impossible.   But in the atomic and subatomic realms, even in various types of modern technology, tunneling is an essential and commonplace feature of the quantum reality in which we live.

I’ve written a short article about this phenomenon, which you can read here, emphasizing the central role that tunneling plays in the world’s most powerful microscopes.  It should be suitable for anyone who has read a little about atoms.

This article lays the groundwork for a discussion of how tunneling could someday, in the distant future, end the universe as we know it.  It also prepares the way for a more advanced post about how a single physics theory (i.e., a set of equations designed to describe some aspect of nature) may have multiple `vacua’ (i.e. multiple solutions that each represent different ways that the universe could be configured — what empty space could be like, and what types of fields, forces and particles could be found in the universe — over long periods of time.)  If that’s confusing, stay tuned for a few days; I’ll soon explain it.

Dog Brains and Fishing Line: 2 Fun Articles

Nothing about quantum physics today, but … wait, everything is made using quantum physics…

Could you imagine getting a dog to sit absolutely still, while fully awake and listening to voices, for as much as 8 minutes? Researchers trained dogs to do it, then put them in an MRI [Magnetic Resonance Imaging] machine to obtain remarkable studies of how dogs’ brains react to human voices and other emotional forms of human expression. [MRI is all about magnetic fields, protons, spin, and resonance; particle physics!! more on that another time, perhaps.]   The authors claim this is the first study of its type to compare human brains to those of a non-primate species. Here’s something from the scientific article’s abstract:

We presented dogs and humans with the same set of vocal and non-vocal stimuli to search for functionally analogous voice-sensitive cortical regions. We demonstrate that voice areas exist in dogs and that they show a similar pattern to anterior temporal voice areas in humans. Our findings also reveal that sensitivity to vocal emotional valence cues engages similarly located non-primary auditory regions in dogs and humans… 

So it seems, as dog owners have long suspected, that we’re not just imagining that our best friends are aware of our moods; they really are similar to us in some important ways.

Here’s a BBC article:

Once you’re done with that, would you like to build up your muscles?  No exercise needed, just call the University of Texas at Dallas.  They’ve found that “ordinary fishing line and sewing thread can be cheaply converted to powerful artificial muscles.  The new muscles can lift 100 times more weight and generate 100 times higher mechanical power than a human muscle of the same length and weight… The muscles are powered thermally by temperature changes, which can be produced electrically, by the absorption of light or by the chemical reaction of fuels.”  [Quantum Physics = cool!!] The quotation above is from an interesting press release from the university, reporting the research which was just published in the journal Science.  I recommend the press release because it mentions several interesting possible applications, including robotics technology  and clothing that adjusts to temperature.  Here’s also a nice article by Anna Kuchment (who’s on Twitter here):

Though evolution left us with many wonderful abilities, it does seem that, year by year, humans are becoming less and less practically useful.   But at least our dogs will comfort us in our obsolescence.


Happy (Chilly) New Year

Welcome 2014! And quite a start to the year, with a cold snap that rivals anything we’ve seen in two decades. I don’t remember cold like this since the horrid winter of 1994, when the Northeastern U.S. saw snowstorms and extreme cold that alternated back and forth for weeks. Of course, when I was a child in the 1970s, such chills happened a lot more often; I remember a number of New England mornings where I awoke to a thermometer reading of -20ºFahrenheit (-29ºCelsius) [244 Kelvin].

The scariest negative temperature numbers that one hears about from the media are associated with the “wind chill”, which is a number that is supposed to measure how cold the air “feels” to your skin.  But “wind chill” is a rather subjective and controversial measure — there’s no unique way to define it, since you’ll feel differently depending on how much exposed skin you have, on your body weight, on your age and conditioning, etc.  By contrast, the temperature measured by a thermometer is defined independent of how humans feel, and experts agree on what it is and means. Oh sure, people use different scales to measure it: Fahrenheit (F), Centigrade or Celsius (C), and Kelvin (K).  But the differences are no more than the distinction between meters and feet, or between kilograms and pounds; it’s straightforward, if a bit annoying, to convert from one to the other.

So everyone agrees the temperature is and feels extremely cold, But is it, from the point of nature, really that much colder than usual? To say it another way: it was 84ºF (29ºC) in southern Florida yesterday.  How much warmer is that than the -40ºF (-40ºC) that was registered in the cold Minnesota morning?

Well, you might first think: wow, it’s a difference of 124ºF (69ºC), which sounds like a huge difference.  But is it really so huge? Continue reading

Quantum Field Theory, String Theory, and Predictions (Part 6)

For More Advanced Non-Experts

[This is part 6 of a series, which begins here.]

I’ve explained in earlier posts how we can calculate many things in the quantum field theory that is known as the “Standard Model” of particle physics, itself an amalgam of three, simpler quantum field theories.

When forces are “weak”, in the technical sense, calculations can generally be done by a method of successive approximation (called “perturbation theory”).  When forces are very “strong”, however, this method doesn’t work. Specifically, for processes involving the strong nuclear force, in which the distances involved are larger than a proton and the energies smaller than the mass-energy of a proton, some other method is needed.  (See Figure 1 of Part 5.)

One class of methods involves directly simulating, using a computer, the behavior of the quantum field theory equations for the strong nuclear force. More precisely, we simulate in a simplified version of the real world, the imaginary world shown in Figure 1 below, where

  • the weak nuclear force and the electromagnetic force are turned off,
  • the electron, muon, tau, neutrinos, W, Z and Higgs particles are ignored
  • the three heavier types of quarks are also ignored

(See Figure 4 of Part 4 for more details.)  This makes the calculations a lot simpler.  And their results allow us, for instance, to understand why quarks and anti-quarks and gluons form the more complex particles called hadrons, of which protons and neutrons are just a couple of examples. Unfortunately, computer simulations still are nowhere near powerful enough for the calculation of some of the most interesting processes in nature… and won’t be for a long time.

Fig 1:

Fig 1: The idealized, imaginary world whose quantum field theory is used to make computer simulations of the real-world strong-nuclear force.

Another method I mentioned involves the use of an effective quantum field theory which describes the “objects” that the original theory produces at low energy. But that only works if you know what those objects are; in the real world [and the similar imaginary world of Figure 1] we know from experiment that those objects are pions and other low-mass hadrons, but generally we don’t know what they are.

This brings us to today’s story.  Our success with the Standard Model might give you the impression that we basically understand quantum field theory and how to make predictions using it, with a few exceptions. But this would be far, far from the truth. As far as we can tell, much (if not most) of quantum field theory remains deeply mysterious. Continue reading