A web site about physics and other things
vttoth — November 29th, 2008
I am sure this is a fine CBC journalist and her report about OPEC was interesting, but I do wonder: why did she have a dead Christmas tree (looks like leftover from last year) to her left in the background?
OPEC and last year's Christmas tree
Categories: Economy, Media | No Comments
vttoth — November 28th, 2008
Looks like Stephen Hawking is coming to Waterloo. I may not be an adoring fan, but I am certainly an admirer: being able to overcome such a debilitating disease and live a creative life is no small accomplishment even when you don’t become a world class theoretical physicist in the process.
Categories: Physics | No Comments
vttoth — November 28th, 2008
Are we going to have a coalition government in Canada? Perhaps not, now that the Conservatives backed off on their idea to drop federal financing of political parties. I’d have liked to see a coalition government. Sure, multi-party politics are inherently messier than a neat two-party or one-party system, but so long as we don’t end up like Italy or Israel, the result may very well be a more representative, more responsive government.
Anyhow, you just gotta love Chretien’s “Je ne comprends pas anglais” comment…
Categories: Politics | No Comments
vttoth — November 27th, 2008
Like other software, this Web logging software, WordPress, also needs to be updated from time to time. It appears that my attempt to update it just now to version 2.6.5 was successful.
Categories: Computers, Internet | No Comments
vttoth — November 27th, 2008
Watching the pictures from Mumbai, I cannot help but wonder: when WW3 inevitably arrives, will we also be seeing live pictures and breathless news media coverage as major cities around the world turn into radioactive mushroom clouds and millions of lives are reduced to ash and smoke? Will there be a new Wikipedia article about the nuking of London, Paris, and New York City moments after they occurred, just as there is already an extensive article in Wikipedia on the 26 November 2008 Mumbai attacks?
Categories: Media, Politics | No Comments
vttoth — November 27th, 2008
A few hours ago, I became rather alarmed, as suddenly, my outgoing network connection was saturated. “What the…?” asked I, as it took a little bit of frantic searching in the log files before I had my answer: Somehow, my old Web page about the 4-bit processor I built many years ago became rather popular, as apparently, it was featured on reddit.com. Cool! Now if only those visitors actually clicked on the Google ads that I hastily placed on these pages…
Categories: Computers, Internet | No Comments
vttoth — November 26th, 2008
This sounds almost like a rallying cry for white supremacists and their crazy claims of reverse discrimination, but did Carleton U. really cancel Shinerama in an act of political correctness gone rampant because supposedly, cystic fibrosis is a “white male’s disease”?
Categories: Politics | No Comments
vttoth — November 25th, 2008
A new paper by Sean Carroll asks this question in its title “What if time really exists?” I feel reassurred that Carroll thinks it does (hmmm, let me check my watch… yup, I think it exists, too) but the fact that a paper with this title appears in an archive of theoretical physics papers perhaps illustrates what is so wrong with physics today. To quote Carroll, “when something is so obvious and important, declaring that it isn’t real is sure to win points for boldness”, but have physicists really become this shallow?
Categories: Physics | No Comments
vttoth — November 23rd, 2008
I’m listening to Mitt Romney. He’s not the only one suggesting that the big problem with Detroit is that it is burdened by its unions: that excessive benefits like generous pension plans are the reason why Detroit cannot compete with others, and that the solution is a restructuring that helps the automakers get rid of these undue burdens.
I don’t want to sound like a grumpy socialist (which I am not, or at least I sure hope I am not) but is the rolling back of worker benefits really the right solution in this time of crisis? I am certainly not advocating an isolationist economic policy that protects an inefficient industry from foreign competition, but how about requiring that other automakers who either manufacture cars in, or export cars to, the United States, play by the same rules as the “big three”?
Either Romney is wrong, and the unions can take solace in the fact that a Democratic president with a large Democratic majority in both houses is about to be unaugurated. Or, Romney is right and Obama and the Democrats are about to make some colossal economic mistakes. Time will tell.
Categories: Economy, Politics | No Comments
vttoth — November 23rd, 2008
I’ve been reading a lot about gauge theories lately. I once wrote what I thought was a fine and concise description of the principle of gauge invariance, but I needed Schrödinger’s equation for it, which made my explanation both non-classical and non-relativistic.
Finally, with the help of a Wikipedia article no less, I think I managed to understand how a gauge theory can come into being without involving any quantum physics. It’s simple, really, surprisingly so.
What you need is a (classical) field theory. A field theory is specified by its Lagrangian, which, crudely speaking, is just the difference between the kinetic and potential energy. For a scalar field φ, the kinetic energy of the field is the square of its gradient, ∂μφ∂μφ. The potential term can be nothing (massless particle in empty space) or it can contain a mass term in the form m2φ2.
Where things begin to get interesting is when we allow φ to be a complex field. In this case, rather than writing φ2, we must now write φφ*, where φ* is the complex conjugate of φ. The same thing happens in the kinetic term. So now the Lagrangian reads,
L = ∂μφ∂μφ* – m2φφ*.
The reason why this is so interesting is that if we change the phase of φ by a set amount (i.e., multiply φ by eiψ) the conjugate’s phase changes by the opposite amount (i.e., φ* it gets multiplied by e–iψ). Their product, therefore, multiplied by eiψe–iψ = 1, remains unchanged. In other words, our Lagrangian is invariant under a global rotation in the complex plane. Right there, this has an important implication: as per Noether’s theorem, a global symmetry implies the existence of a conserved current.
But what if the symmetry is not global but local? Meaning that we rotate φ in the complex plane as before, but the angle of rotation is not the same everywhere? Clearly, φφ* still remains unchaged just as before, but the same is not true for ∂μφ∂μφ*; the derivative operator brings new terms into the Lagrangian.
These new terms are best dealt with by changing the derivative operator into a covariant derivative: ∂μ → Dμ = ∂μ + Aμ, where Aμ is an arbitrary vector field.
Or maybe not so arbitrary. We can make Aμ anything we want, of course, but that also means that we can demand that Aμ satisfy a field equation. Perhaps the field equations of electromagnetism… why not? (After all, every vector field satisfies the field equations of electromagnetism.)
The difference between the original Lagrangian (written using the ordinary derivative ∂μ) and the new Lagrangian (written using the covariant derivative operator Dμ) is the interaction Lagrangian that describes how the φ field interacts with itself through a vector field Aμ. By making the complex φ field locally gauge invariant, we have, in effect, invented the electromagnetic vector potential Aμ.
This is, after all, what gauge theories do: they turn a local symmetry into a force. The local symmetry can be geometric in nature (e.g., a rotation) or it can be an internal symmetry of a field that is not described by simple real numbers. In the present example, the field was made up of complex numbers, and the symmetry was that of the complex plane. This symmetry group is U(1), which is an Abelian group: two rotations in the complex plane, executed one after the other, produce the same result regardless of the order in which they are executed.
In many physically important cases, the symmetry is non-Abelian. The most profound consequence of this is that in place of the gauge field Aμ, which is “inert”, we get gauge field(s) that interact with themselves. In practical terms, when the theory is Abelian, like electromagnetism, the gauge field Aμ represents photons, which are uncharged; but when the theory is non-Abelian, like electroweak theory, the gauge fields are non-Abelian, carry charge, and interact with each other.
Categories: Physics | No Comments
