Monday, March 24, 2008

Joseph Liouville: an anniversary

Joseph Liouville, a 19th century string theorist and politician, was born 199 years ago, on March 24th, 1809.

He lived with his uncle for a few years, showed his talents as a kid, attended all kinds of schools, and met important French mathematicians and physicists. Throughout his life, he wrote about 400 technical papers.

He was also a science official. At some moments of his life, Liouville was defeated by Count Guglielmo Libri Carucci dalla Sommaja, a guy who escaped France during the 1848 revolution to avoid prison sentence for stealing precious books and manuscripts. ;-)

As a moderate republican politician, Liouville was depressed for years after 1849 when he lost an election because working-class socialist radicals and similar political foam became "hot" at that time. He recovered and 1856 and 1857 were among his most scientifically fruitful years.

Science

There have been many different disciplines of mathematical sciences where Liouville left us valuable insights. For example, he was the first man to prove the existence of transcendental numbers (that are not solutions to any algebraic, polynomial equation with rational coefficients). The simplest (but later) example he found is
0.1100010000000000000000010000...
that only has "1" in place "n!". It is so close to the rational numbers that approximate it yet so different that he could show it was not algebraic. An older example he found used continued fractions. Liouville also investigated number theory.

In theoretical physics, he is known for Liouville's theorem, the local continuity equation for a probability distribution function on the phase space (with the current explicitly expressed in terms of derivatives of the probability density, as dictated by the Hamiltonian equations of motion).

He studied various differential equations involving eigenvalues. See, for example, the Sturm-Liouville problem. Many of the problems he studied resemble the tasks one must routinely solve in quantum mechanics - even though he lived 100 years earlier.

Liouville and string theory

But string theorists will surely know him for the Liouville action, also known as the Liouville theory. How the hell could a 19th century mathematician write down an equation for two-dimensional non-critical string theory including the linear dilaton and the exponentially increasing tachyonic wall? Something that is a frequent ingredient of many perturbative string-theoretical constructions? And believe me, he did so. ;-)

The answer is that Liouville's equation is truly natural and someone who studies how to solve partial differential equations and what non-trivial pieces they are made out of will inevitably run into such an equation.

The string-theoretical terminology for Liouville's equation arises from a more general type of a differential equation that Liouville studied, namely the equation requiring that the Laplacian (or d'Alembertian) of the function "u" is equal to the exponential of "u". When "u" is interpreted as the coordinate "X_1(sigma,tau)" in spacetime along which the dilaton is linear, a coordinate that is mixed with the exponent encoding a Weyl scaling and a coordinate treated as a function of the worldsheet variables, Liouville's 19th century equation becomes the same thing as the equation of motion for "X_1(sigma,tau)" in non-critical string theory.

In fact, Liouville didn't just encounter the same equation: he studied it in the context of very similar mathematical procedures that string theorists do before they end up with Liouville's equation, namely an analysis of conformal transformations.

More generally, all kinds of similar, generalized, or nearly-equivalent equations of this kind are referred to as Liouville's equation, for example
y'' + g(y) y'2 + f(x)y' = 0
You shouldn't confuse non-linear Liouville's equations with the linear equations that appear in Liouville's theorem or the Sturm-Liouville problem discussed above.

I have emphasized this point many times but let me say it once again. String theory naturally incorporates, explains, interprets, and unifies most of the deep mathematical ideas, concepts, and equations inspired by and/or indirectly or directly connected with the laws governing the physical Universe. There is really no way for a real 21st century mathematical or theoretical physicist to "cut string theory off" without amputating his whole brain.

And that's the memo.

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