Tuesday, October 27, 2009

The Cauchy–Schwarz Inequality in a Nutshell (or a "Quarkshell" if you're into that whole Number Theory "Integer" thing)

There are few equations as important in Quantum Mechanics (and therefore Physics in General) as this one from Mathematics. I will present a few bits from its entry in Wikipedia, here, then I'll have a few comments, following:

In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy–Bunyakovsky–Schwarz inequality), is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The general formulation of the Heisenberg uncertainty principle is derived using the Cauchy–Schwarz inequality in the Hilbert space of pure quantum states.

The inequality for sums was published by Augustin Cauchy (1821), while the corresponding inequality for integrals was first stated by Viktor Yakovlevich Bunyakovsky (1859) and rediscovered by Hermann Amandus Schwarz (1888) (often misspelled "Schwartz").

The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space,
where < . , . > is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as Moreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).

... OK, I'm back. If that looks like mumbo jumbo to you, then let me be the first to welcome you back to the wonderful world of Mathematics and Physics from the money fields of Finance, Accounting, and Law, and assure you that although you have a lot of catching up to do, this would be a good place to start.

The reason many of us left Maths + Physics in the first place was due the weirdness of Quantum Mechanics, specifically Indeterminacy also known as Heisenberg's Uncertainty Principle. The Cauchy–Schwarz inequality is key to understanding it.

Uncertainty is taught SO badly to undergrads by Physics professors it is mind-boggling. It got me to quit my boyhood love of Math and Physics, and I'm not the only one. It seems unbelievable that such an important and fundamental thing in its field would be taught so badly, I know, but it's very much taught in a most horrible fashion. Compounding the problem is another unbelievability which I feel is best expressed by Lee Smolin's honest comment that half of all PhD's in Physics aren't sure they believe it, and every crackpot scientific theory also slams it and calls it flat out wrong.

But it is true. It's true, experiment backs it up.

Good luck, and may the Schwartz be with you!
A photograph of actor Rick Moranis as Darth Helmet in Mel Brooks' film, "Spaceballs", reacting to his Physics professor's comment "Shut up and calculate!" in response to Darth's probing question re Uncertainty. The Schwartz was not with him that day.

Augustin-Louis Cauchy (1789-1857)

Victor Bunyakovsky (1804-1889)

Hermann Schwarz (1843-1921)

Werner Heisenberg ( 1901-1976)