Calculus without limits

# Stability of the Vacuum

Explanations of the Casimir effect using common physics intuition like “polarization” (it originally was studied in the context of van der Waals forces) or “pressure” do not work. The reason in the case of the Casimir effect is that in the case of two planes or two cylinders the Casimir force is attractive while in the case of spherical shells the force is repulsive. The mathematical calculations involve divergent series which must be Zeta regularized. But when doing analytic continuations, one loses intuition. One has to use some ingenuity to argue that $\sum_{n=1}^{\infty} n = \zeta(-1) = -1/12$. That adding up all positive integers gives a negative value is something far from intuitive. As one can see in slides like this, the zeta function appears. One has to look at the difference of zero point energies with discrete energy values and continuum leading to the difference $(\sum_{n=0}^{\infty}n) - \int_0^{\infty} t \; dt$, which is of the form $\infty-\infty$, when regularized gives $-1/12$. About intuition: how can one see intuitively that $\sum_n n^3 = 1/120$ and $\sum_n n = -1/12$ and $\sum_n n^5=-1/252$.

I had been asked during dinner on Wednesday about the zero point energy. As a mathematician we can ignore the philosophical and even conceptional difficulties of a problem and look at simpler related problems. The zero point energy can be defined the infimum of the spectrum of some operator. It is also called the “ground state energy” but this is a bit misleading since if we look at operators without eigenvalues, there is no “state” = “unit vector” in the associated Hilbert space. For the free discrete Laplacian operator Lu(n) = u(n+1)+u(n-1)-2u(n) on $l^2(\mathbb{Z})$ for example, the spectrum is [0,4] with explicit density of states given by the probability density function of the arcsin distribution on [0,4]. It is the equilibrium measure of the Julia set of the quadratic map f(x)=x(4-x) which is conjugated explicitly to the tent map and so Bernoulli (as the induced dynamics on a Julia set always is). The bounded operator is conjugated by Fourier theory to the multiplication operator $2 \cos(x)-2 = 4 \sin^2(x/2)$ the inverse of which is the integrated density of states (CDF of the arcsin distribution in stats) which when differentiated gives the density of states $dk=1/(\pi(\sqrt{x(4-x)}))dx$. As a senior in college I already got interested in the topic, especially about the almost Mathieu operator $Lu(n) = u(n+1)+u(n-1)-2u(n)+c \cos(n a) u(n). I had read in high school the Douglas Hofstadter cult book Goedel-Escher-Bach'' and got interested then already in fractals, recursion and especially the Hofstadter butterfly which illustrates the spectrum of the almost Mathieu operator for c=2 where a goes from 0 to$latex 2\pi\$. I would spend the next 13 years with trying to prove that the operator $Lu(n) = u(n+1)+u(n-1))+c \cos(n x_n) u(n)$ where $x_n$ is the orbit sequence of the Standard map (=equilibrium state of the Frenkel-Kontorova wave model) has determinant larger than 1 if c is large enough. The operator L as a random operator over a dynamical system is an element of a non-commutative type $II_1$ von-Neumann algebra and so has a natural determinant. By the Thouless formula $\log(\det(L)) = {\rm tr} \log(L) = \int \log|x|dk(x)$ and Pesin formula, a $\det(L)>1$ rephrases that the Kolmogorov-Sinai entropy of the Standard map is positive for large enough c. (The trace in tr(log(A)) for an operator is an average $\lim_{n \to \infty} \sum_{k=-n}^n A_{kk}/(2n)$ which makes sense as a Birkhoff average because $A_{kk}$ are random variables. One calls such operators random operators even if there is nothing random associated with it; it is just a non-commuative random variable. Also in probability theory, a random variable is not a priori random at all. It is just a measurable function on a probability space.)

The Frenkel-Kontorova model is at the heart of many things I got exposed to in college. Juergen Moser gave a special one semester course on “Selected chapters in the calculus of variations” (to which I wrote the lecture notes in 1987 and retyped it around 2000 in LaTeX) in which he looked especially at Aubry-Mather theory, a beautiful (relatively trivial) modification of the KAM approach of finding smooth solutions of the nonlinear problem (*) $q(x+\alpha) + q(x-\alpha) - 2 q(x) =c \sin(q(x))$ for small $c$. Linearization at $c=0$ gives in the Fourier picture the multiplication with $2 \cos(\theta + n \alpha)-2$ which famously when inverted produces small divisors. If $\alpha$ is Diophantine, meaning $|\alpha - p/q| \geq C/q^2$ for all integers $p \neq 0, q$, one can invert the linearization operator on a subspace of smooth functions. But the inverted function is less smooth. If one uses a naive Newton iteration to find roots of the nonlinear problem (*) one loses smoothness in each step. Moser (similarly than Nash) found a way to modify the Newton step, adding some smoothing and show that one still can have convergence of that now called Nash-Moser iteration. In Aubry-Mather theory, one drops the requirement that q is smooth near q(x)=x but is monotone. The solutions produce Aubry-Mather sets. They exist for all $\alpha$ and are minima of a variational principle but in general are Cantor sets. Often, the corresponding linearization operator $L$ is hyperbolic. I looked in my thesis also at the more general variational problem, where we take instead of an irrational rotation any ergodic dynamical system. Using the Aubry anti-integrable limit one can easily show (using the standard soft implicit function theorem) that for large c, a given dynamical system of finite entropy can be found within the standard map meaning that there are invariant measures of the Standard map for which the corresponding metric dynamical system is isomorphic to the given system. I called this “embedding of a metric dynamical system” in a topological dynamical system. Here is a paper of mine from August 1996. It shows also that there are arbitrary large c for which there are periodic points that are parabolic and so points which are elliptic (A result of Douarte which appeared around the same time). Typically near elliptic points, the twist theorem in KAM theory gives stable invariant islands. This also shows that it can not be true that for c large enough the Standard map is ergodic. It is not impossible however (even so unlikely) that there are values of c for which the Standard map is ergodic. I saw at UCLA (during a conference called “Dynamical systems Quo Vadis”) a talk of Sinai in which he claimed to have a proof ergodicity for some values of c. I also saw at Caltech (in a special seminar), Michael Herman give a proof that for some large values of c, the Standard map has positive entropy. I myself gave once a proof (maybe encouraged by such claims). All these claims have fallen. There is no value c known for which the Standard map has positive entropy. There is no value c known for which the standard map is ergodic. I’m still interested in the problem but do not work on it any more. It might be futile. What if the claim is false and the determinant is zero for all c? I also used it in teaching, like in this Unit 24: Chaos, of a course Math 22b given in Spring 2019 at Harvard. I have in 1999 written an online javascript calculator for the entropy of the Standard map. There is also a C program which produces (arbitrary large resolution) pictures of the standard map.

In the talk, I look mention again connection calculus. I talked about this several times already.

About the talk. There is a small typo on the right hand side of the board. where it is mentioned that the Fourier transform is $2 cos(x)-2$, there is a $\hat{u}$ missing. I should have mentioned that the anti-integrable limit easily shows that for larger c there are many solutions of $L u = c \sin(u)$ as it is just a case for the standard (not-hard) implicit function theorem for nonlinear smooth maps in Banach spaces. The problem $Lu=c \sin(u)$ can be written as $\epsilon L u = \sin(u)$ with $\epsilon=1/c$. Now $\epsilon=0$ produces many measurable solutions like taking values ${0,\pi\}$ each on sets of positive measure. These solutions continue to exist for $\epsilon>0$ by the standard implicit function theorem because for $\epsilon=0$ the linearization is invertible. This is Aubry’s argument. It is described also here.