Infinitesimal variables

I was recently at the "Euler Festival" in St. Petersburg, for the 300-th anniversary of the birth of Euler.
The wonderful hospitality of the great scientists there, such as Ludwig Faddeev, created a perfect atmosphere for the event. With the long evenings, added to the charm of the city, there was plenty of time to try and get in the right mood to get more familiar with some of Euler's thoughts.

I spent time reading the little booklet that was distributed to participants. It contained the talk given by A. N. Krylov in October 1933 on the occasion of the 150-th of Euler's death, at a special session of the USSR academy of sciences. One of my down to earth motivation was to find some connection between the topic of the talk I had planned to give and the works of Euler, but that was easy since as we all know he is "everywhere dense" in mathematics.

In volume 1 of his book "Introductio in Analysis Infinitorum", he gives his computation of zeta values at even integers plus a lot of other things, like partition number generating functions etc. He also gives the numerical zeta values (which he found first) in 23 decimal digits. He also gives the value of pi in 127 decimal digits!

The second volume is devoted to analytic geometry. According to the article of A. N. Krylov, "it is marked by astonishing simplicity and clarity, and Euler uses only tools from elementary algebra and trigonometry"; "the aim of the second volume, as Euler understands it, consists not in analysing properties of curves given geometrically, but, on the opposite, in using curves and their properties for visual presentation of functions given by equations of the first, second, third, fourth, and greater degrees".

In fact I got most intrigued by the discussion which follows in Krylov's talk, of the relation of Euler with the calculus of infinitesimals. In the preface of the above book, Euler wrote:

"Many times I saw the difficulties of those who start to study analysis of infinitesimals, stem from the fact that they want to acquire the knowledge in this higher branch of analysis having insufficient prerequisites in elementary algebra. This not only causes obstacles they encounter from the beginning, but also gives a false idea of the infinity, whereas a true treatment of this notion must lead in studies"

The article of A. N. Krylov then gives a detailed comparison of the points of view followed by Newton and Leibniz on the calculus of infinitesimals. One striking point of the discussion is the role that "variables" play in Newton's approach, while Leibniz introduced the term "infinitesimal" but did not use variables. According to Newton:

"In a certain problem, a variable is the quantity that takes an infinite number of values which are quite determined by this problem and are arranged in a definite order"

"A variable is called infinitesimal if among its particular values one can be found such that this value itself and all following it are smaller in absolute value than an arbitrary given number".

In the classical formulation of variables as maps from a set X to the real numbers R, the set X has to be uncountable if some variable has continuous range. But then for any other variable with countable range some of the multiplicities are infinite. This means that discrete and continuous variables cannot coexist in this modern formalism.

Fortunately everything is fine and this problem of treating continuous and discrete variables on the same footing is completely solved using the formalism of quantum mechanics.

The first basic change of paradigm has indeed to do with the classical notion of a "real variable" which one would classically describe as a real valued function on a set X, ie as a map from this set X to real numbers. In fact quantum mechanics provides a very convenient substitute. It is given by a self-adjoint operator in Hilbert space. Note that the choice of Hilbert space is irrelevant here since all separable infinite dimensional Hilbert spaces are isomorphic.

All the usual attributes of real variables such as their range, the number of times a real number is reached as a value of the variable etc... have a perfect analogue in the quantum mechanical setting.

The range is the spectrum of the operator, and the spectral multiplicity gives the number of times a real number is reached. In the early times of quantum mechanics, physicists had a clear intuition of this analogy between operators in Hilbert space (which they called q-numbers) and variables.

What is surprising is that the new set-up immediately provides a natural home for the "infinitesimal variables" and here the distinction between "variables" and numbers (in many ways this is where the point of view of Newton is more efficient than that of Leibniz) is essential.

Indeed it is perfectly possible for an operator to be "smaller than epsilon for any epsilon" without being zero. This happens when the norm of the restriction of the operator to subspaces of finite codimension tends to zero when these subspaces decrease (under the natural filtration by inclusion). The corresponding operators are called "compact" and they share with naive infinitesimals all the expected algebraic properties. Indeed they form a two-sided ideal of the algebra of bounded operators in Hilbert space and the only property of the naive infinitesimal calculus that needs to be dropped is the commutativity. It is only because one drops commutativity that variables with continuous range can coexist with variables with countable range.

Thus it is the uniqueness of the separable infinite dimensional Hilbert space that cures the above problem, L^2[0,1] is the same as l^2(N), and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute, and the real subtlety is in their algebraic relations. For instance it is the lack of commutation of the line element ds with the coordinates that allows one to measure distances in a noncommutative space given as a spectral triple.