I’d like to use some part of this summer to learn some Standard Model physics – my ignorance is growing rather inexcusable. The plan is to first brush up on Nonabelian gauge theories, and then jump to the Standard Model as soon as is convenient. The main text I’m going to follow is Srednicki’s book on QFT. It provides an accessible route through the maze of theoretical machinery. I’ve made a list of sections I’d like to cover in the book, with prerequisite sections in brackets. I certainly won’t get through it all, but any progress will be worth the effort put in. I’m hoping this path will be a good mix of neat theoretical concepts and connections to real world physics.
As kids tend to discover rather early on, it’s fun to throw stones into water. As the stone hits the water, it gives it energy and creates ripples that move outwards in concentric circles. These ripples carry the energy outwards, and as they move further out they eventually die out and the water is smooth again (this is true even if you ignore the effect of friction). The behavior of the water is well described by the wave equation — very likely the first field equation that people are taught about in physics. In one dimension, it describes waves on a string or electric and magnetic waves. In two dimensions, it describes water waves. In three, sound waves. Waves are ubiquitous in our daily lives.
In contrast, when you kick a soccer ball, the energy you give it doesn’t spread out. It stays with the ball as it rolls away. The difference between soccer balls and water waves is that in the former case the energy density is localized, and in the latter it dissipates. So the question arises: are there any wave type phenomena with a localized, non-dissipative energy density?
Solutions of field equations that have this property are called solitary waves. We do hear of such objects from time to time – moving lumps of water that don’t die out like waves do – and usually with catastrophic consequences. These are the tsunamis. The physics of tsunamis is rather involved, but certain kinds of tsunamis involve solitary waves – those that are caused due to sharp localized impulses such as an asteroid collision.
If these solutions have the further property that they retain their shapes after collision with each other, they are known as solitons. These objects show up in theories where non-linear interactions balance out the tendency to disperse, and they give rise to lots of interesting physics!
The discovery of the solitary wave is the kind of story that physicists love to tell. It’s a example of Isaac Asimov’s principle – “The most exciting phrase to hear in science, the one that heralds new discoveries is not ‘Eureka!’ but ‘That’s funny…'”. The story is described in this article on the history and application of solitons:
In August 1834, the naval engineer John Scott Russell was watching a horse-drawn barge on the Union Canal, Hermiston, Edinburgh, as part of his work on hull design. When the cable snapped and the barge suddenly stopped, Russel was impressed by what happened: ‘A mass of water rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some 30 feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel’.
I find it extremely cool that an idea that started off with a seemingly mundane observation of horse drawn boats can predict the existence of magnetic monopoles in certain gauge theories, or can explain flux quantization in superconductors! The important point is that rather than write off his observation as a quirk of nature, John Russell was the first person to follow through and experiment to try to understand this phenomenon.
Q) Why is it that manhole covers are round?
The question achieved notoriety when it started being used by microsoft as an interview question. The answer that I’ve always liked is that a circular manhole cover can never fall into a circular hole. If you were to use a square, for example, it could easily fall through the diagonal.
So here’s another puzzle: is there any other shape for which this property holds – that you can make a manhole cover out of it that will never fall into the manhole?
This is now my second year as a TA, and I’ve been able to see first hand some of the problems with science education in the US pre-college school system. Most of the student I’ve been teaching physics to are not physics majors, but engineering, pre-med or biology students. The students have drastically different levels of preparation, depending on their schooling. On my first time teaching, I was asked by some student ‘three times zero is three, right?’. The university and the teachers work hard to try and bring these students up to speed with the rest, but it’s no easy job. For one thing, people have a real disinclination to learn math. It’s considered hard, painful, boring, and some students even become hostile when forced to think mathematically. I couldn’t even tell you how many times when I’ve asked students some question about finding the x and y components of a vector, I’m told “Oh, this is that SOHCAHTOA shit!”.
It’s not that SOHCAHTOA shit. It’s trigonometry. It’s about as old as christianity. Where were your schools when they should have been teaching you about it? Prior to coming to college, these students were literally living in the scientific dark ages – in a world that’s not just before Newton and Descartes, it’s before Galileo and Euclid!
Many of these students come from low-income school districts, and have learned no math or science to speak of. The college system gives them another chance to get up to speed with their peers – but this is not an easy gap to bridge, and they don’t have a lot of time to do it. Sadly, many do not succeed and end up dropping the class, or not going any further in the sciences. And these students are out of the minority who actually did have the opportunity and inclination to go to college and try to become an engineer or a doctor.
But what amazes me is the resilience and dedication of those that do overcome the odds and make it. Every time I teach, there are a few moments – when they realize they’ve proved something neat and counter intuitive and there’s a glimmer of excitement in their eyes. Those are the moments that make all the hard work and sweat worth it.
There are so many systemic problems with science education in this country that it’s a can of worms I’d rather not open in this post. It’s certainly depressing. But happily, there are teachers who, with the help of the internets and generous folk like yourself, are taking matters into their own hands.
Donorschoose is a website based on a marvelous idea – teachers pitch their cases for the classroom projects that need funding, and you can go there and fund the ones of your choice. The students win. The website has an annual blogger challenge to raise money from blogs. This year our friends at Cosmic Variance have taken them up and put together a list of worthy causes in math and science that are in desperate need of funding. Their goal is to raise $10,000 and are offering prizes for large donations! Take a look at their list, and do what you can. They’re competing with scienceblogs, who have also put together their list.
All kids are born with a natural interest in science. Their curiosity drives them to acquire language, and to explore and try to understand the world. A poor education only dulls this spark. We must do what we can to try and reignite it.
Looks like the LHC will miss its run this year.
Every once in a while, one of the 1232 superconducting dipole magnets that bend the beam can heat up to above the critical temperature, causing it go from being superconducting to normal conducting. This rise in temperature is usually caused by some small misalignment in the beam, causing a few stray particles to hit a magnet. Once this happens, the current passing through the magnet’s now significant resistance causes it to heat up like crazy – parts of the magnet can be heated up from -271 to 700 deg C in under a second! This is known as quenching, and it comes as no surprise to particle physicists, who have been dealing with it since superconducting magnets where first used in the Tevatron. They’ve come up with effective ways of dumping the heat and dealing with the problem – the LHC has failsafes built in to deal with quenches effectively, and they should normally be back online after a quench in a matter of hours.
However, in this case it seems that the quench was caused due to a faulty electrical connection between two magnets that likely melted at high current, resulting in mechanical failure and a leakage of liquid helium into the tunnel. The sector (number 23) of the ring will have to be warmed up for repairs, and the cooled down after. The process, which would take a few days to fix in a normally conducting machine, will delay the LHC by two months!
This news comes as a bit of a disappointment, as the experimentalists were hoping to use this years run with a lower energy beam to calibrate the detectors. It looks like they’ll now probably have to wait until the LHC runs at full energy in 2009. However, as long as next years run is not delayed, this should hopefully not be a setback to physics.
For more thorough coverage:
The US LHC Blog has very informative coverage
Last december’s issue of symmetry has an excellent fact-filled article on quenching
CERN has put up a press release
And you can see the incident and the progress of the warmup for yourself here
George Jones put up a neat little puzzle on physicsforums. Following up on this taught me quite a bit about some of the mathematical subtleties of QM that physicists tend to gloss over.
“Suppose A is an observable, i.e., a self-adjoint operator, with real eigenvalue a and normalized eigenket . In other words,
Suppose further that A and B are canonically conjugate observables, so
where I is the identity operator. Compute, with respect to , the matrix elements of this equation divided by :
In the first term, let A act on the bra; in the second, let A act on the ket:
0 = 1.
This is my favourite “proof” of the well-known equation 0 = 1.
When you’re done racking your brains, you can take a look here, here and on the arxiv for solutions to this and more such QM puzzlers. The last link is to a paper by F. Gieres called ‘Mathematical surprises and Dirac’s formalism in Quantum Mechanics’, which I found pretty fascinating. Apparently, when we’re taught as undergrads that state vectors in QM live in a Hilbert space, that’s one of those white lies that teachers tell.
One of the things that I love about science is that it frees us from our parochial, human-centered world view. It allows us to extend our range of experiences far beyond our limited innate faculties. After all, there is no good reason why the sensory apparatus on an east-african-plains ape should come close to sampling what’s really out there.
In Kurt Vonnegut’s Slaughterhouse 5, the Tralfamadorians try to describe what human experience must be like to each other. They imagine it must be something like spending your entire life with a steel ball on your head, with only one eyehole that was welded to six feet of pipe.
The analogy is fairly accurate.
Hope I passed! It’s a pretty painful ordeal. I wasn’t happy with some of the problems, they seemed to test diligence and memory over skill. In any case, for the moment I can stop letting my schooling interfere with my education.
Most hideous experience – the first question on my quantum exam was to write out the full spectroscopic notation (1s2 2s2 blah blah blah) for Europium — that’s atomic number 63! Needless to say, those who didn’t remember their high school chemistry (myself very much included) were floored.
Posting here has been slower than its usual trickle. This is probably mostly because of the following
- I’ve started studying for my PhD. qualifiers in late August. Am going through my old undergrad textbooks. The physics community collectively owes David J. Griffiths a giant hug.
Also, Sean Carroll has been staking his bets on LHC discoveries over here at Cosmic Variance.