citnum Twelve Tales in Mathematical Physics: An Expanded Heineman Prize
LectureTwelve Tales in Mathematical Physics: An Expanded Heineman Prize
Lecture

# Twelve Tales in Mathematical Physics: An Expanded Heineman Prize Lecture

Barry Simon1,2

###### Abstract.

This is an extended version of my 2018 Heineman prize lecture describing the
work for which I got the prize. The citation is very broad, so this describes
virtually all my work prior to 1995 and some afterwards. It discusses work in
non-relativistic quantum mechanics, constructive quantum field theory and
statistical mechanics.

###### Key words and phrases:

Simon, Schrödinger operators, quantum mechanics, quantum field theory,
statistical mechanics

###### 2010 Mathematics Subject Classification:

Primary: 81Q10, 81T08, 82B24; Secondary: 47A55, 81Q15, 81Q20, 81Q70, 81T25,
81U24

1 Departments of Mathematics and Physics, Mathematics 253-37, California
Institute of Technology, Pasadena, CA 91125. E-mail<EMAIL_ADDRESS>
2 Research supported in part by NSF grants DMS-1265592 and DMS-1665526 and in
part by Israeli BSF Grant No. 2014337.

###### Contents

  1. 0 Introduction
  2. 1 Summability of Divergent Eigenvalue Perturbation Series
  3. 2 Complex Scaling Theory of Resonances
  4. 3 Statistical Mechanical Methods in EQFT
  5. 4 Thomas–Fermi Theory
  6. 5 Infrared Bounds and Continuous Symmetry Breaking
  7. 6 $N$–Body quantum mechanics
  8. 7 Magnetic Fields in NRQM
  9. 8 Quasi-classical and Non–quasi-classical limits
  10. 9 Almost Periodic and Ergodic Schrödinger Operators
  11. 10 Topological Methods in Condensed Matter Physics
  12. 11 Anderson Localization: The Simon-Wolff Criterion
  13. 12 Generic Singular Continuous Spectrum
  14. 13 Further Remarks

##  0\. Introduction

The citation for my 2018 Dannie Heineman prize for Mathematical Physics reads:
_for his fundamental contributions to the mathematical physics of quantum
mechanics, quantum field theory, and statistical mechanics, including spectral
theory, phase transitions, and geometric phases, and his many books and
monographs that have deeply influenced generations of researchers_. This is
very broad so I decided to respond to the invitation to speak at the March
2018 APS which says the talk should be _preferably on the work for which the
Prize is being awarded_ , by discussing the areas of my most important
contributions to mathematical physics. I couldn’t say much in the 30 minutes
allotted to the talk so it seemed to make sense to prepare this expanded Prize
Lecture.

I will discuss 12 areas in theoretical and mathematical physics. The first
seven involve areas where my work was largely done during my Princeton years,
1969–1980 (a kind of golden era in mathematical physics [707]) and the last
four during my Caltech years, 1980–1995 (I’ve remained at Caltech since 1995
but my interests shifted towards the spectral theory of long range potentials
and of orthogonal polynomials whose connection to physics is more remote). The
eighth area is one where I had work both before and after I moved to Caltech.
It’s a pleasure to thank Michael Aizenman, Michael Cwikel, David Damanik, Jan
Derezinski, Rupert Frank, Jürg Fröhlich, Fritz Gesztesy, Leonard Gross, George
Hagedorn, Bernard Helffer, Svetlana Jitomirskaya, Martin Klaus, Elliott Lieb,
John Morgan, Derek Robinson, Israel Sigal, Alan Sokal and Maxim Zinchenko for
feedback on drafts of this article.

Many of the topics I’ll discuss have spawned industries (as shown by my
current Google scholar h-index of 113); any attempt to quote all the related
literature would stretch the number of references far beyond the citnum so
I’ll mainly settle for quoting relevant review articles or books where they
exist or perhaps limit to on a very small number of later papers that shed
light on my earlier work. In particular, I focus very much on my own work and
make no pretense of doing comprehensive reviews or a serious history even of
all the ideas floating around at the time of my work and certainly not all the
work after I essentially left a subject.

##  1\. Summability of Divergent Eigenvalue Perturbation Series

Eigenvalue perturbation theory depends on formal perturbation series (aka RSPT
(or just RS) for Rayleigh-Schrödinger Perturbation Theory) introduced by
Rayleigh [550] and Schrödinger [586]. The core of the rigorous theory about
1970 when I began my research in this area were results of Rellich [560],
extended by Nagy [507] and Kato [373] (see my review [713] of Kato’s work
written on the centenary of his birth) and summarized in Kato’s magnificent
1966 book [386].

The Kato–Rellich theory in its simplest form considers operator families

$A(\beta)=A_{0}+\beta B$ (1.1)

where $A_{0}$ and $B$ are typically unbounded self–adjoint operators ($B$ need
only be symmetric; see [711, Chap. 7] for a presentation of the language of
unbounded self–adjoint operators) on a Hilbert space, ${\mathcal{H}}$. One
demands that there are $a$ and $b$ so that

$D(B)\supset D(A_{0});\qquad\forall\varphi\in D(A_{0}):\,\lVert
B\varphi\rVert\leq a\lVert A_{0}\varphi\rVert+b\lVert\varphi\rVert$ (1.2)

In Kato’s language, one says that $A(\beta)$ is a _type A perturbation_ of
$A_{0}$. The big result of this theory is

######  Theorem 1.1.

If $A(\beta)$ is a family of type A and $E_{0}$ is an isolated eigenvalue of
$A_{0}$ of finite multiplicity, $\ell$, then there exist $\ell$ analytic
functions, $\\{E_{j}(\beta)\\}_{j=1}^{\ell}$, near $\beta=0$ which are all the
eigenvalues of $A(\beta)$ near $E_{0}$ when $\beta$ is small. Moreover, there
exists an analytic choice $\\{\varphi_{j}(\beta)\\}_{j=1}^{\ell}$ of
eigenvectors, orthonormal when $\beta$ is real and small. The Taylor
coefficients of $E_{j}$ and $\varphi_{j}$ are given by the
Rayleigh–Schrödinger perturbation theory.

######  Remarks.

1\. For textbook presentations of this theorem, see Kato [386], Reed–Simon
[556] or Simon [711, Sections 1.4 and 2.3].

2\. The _Kato—Rellich theorem_ assets that for $\beta\in(-a^{-1},a^{-1})$, one
has that $A(\beta)$ is self–adjoint on $D(A_{0})$.

3\. The theory is more general than the self–adjoint case. It suffices that
$A_{0}$ is closed, that $B$ obey (1.2) and $E_{0}$ be a point of the discrete
spectrum (isolated point of finite algebraic multiplicity). In that case, one
has analyticity in the non-degenerate case ($\ell=1$) but, in general, $E_{j}$
may be one or more convergent Puiseux series (fractional powers in $\beta$)).
The is also a generalization that only requires quadratic form estimates.

4\. It is also not necessary that the $\beta$ dependence be linear; a suitable
kind of analyticity suffices.

While this is elegant mathematics, the striking thing is that it doesn’t cover
many cases of interest to physics. Perhaps, the simplest example is

$A(\beta)=-\frac{d^{2}}{dx^{2}}+x^{2}+\beta x^{4}$ (1.3)

the quantum anharmonic oscillator. This is the usual textbook model of RSPT
because the sum over intermediate states is finite and one can compute the
first few terms in the RSPT by hand.

Moreover, it can be regarded as a toy model for a $\varphi^{4}$–field theory.
Indeed, if one specializes a $(\varphi^{4})_{d+1}$ QFT in $d$ space dimensions
to $d=0$, one gets a path integral for the $A(\beta)$ of (1.3) and the RSPT
terms can also be written in terms of Feynman diagrams, at least for the
ground state (see, for example, Simon [627, 693]).

In this regard, a celebrated argument of Dyson [172] is relevant. He noted
that quantum electrodynamics (QED) isn’t stable if $e^{2}<0$ since electrons
then attract. Since there isn’t a sensible theory for such $e$, he argued the
Feynman perturbation series must diverge. Similarly, (1.3) for $\beta<0$ isn’t
bounded below (indeed, even worse, the operator isn’t self–adjoint and a
boundary condition is needed at $\pm\infty$ \- see, for example, [711, Theorem
7.4.21]). In fact, various estimates [346, 65, 613] show that the perturbation
coefficients for the Rayleigh–Schrödinger series of (1.3) grow like $n!$. (see
the further discussion around (2.10) below).

Two other standard models to which RSPT is applied are the Stark effect in
Hydrogen(indeed the title of Schrödinger’s paper [586] where he introduced his
version of RSPT is “Quantization as an Eigenvalue Problem, IV. Perturbation
Theory with Application to the Stark Effect of Balmer Lines”)

$A(\beta)=-\Delta-\frac{1}{r}+\beta z$ (1.4)

and the Zeeman effect in Hydrogen

$A(\beta)=-\frac{1}{2}\Delta-\frac{1}{r}+\frac{\beta^{2}}{8}(x^{2}+y^{2})+\beta
L_{z}$ (1.5)

The $z$ in (1.4) and $x^{2}+y^{2}$ terms in (1.5) are clearly not bounded at
infinity by $A(0)$ (i.e. (1.2) fails) and it is known that both problems have
divergent RSPT. The Stark effect is more singular than the other two examples
in that, as first noted by Oppenheimer [520], its bound states turn into
resonances, an issue that I will discuss in Section 2.

One might think, on the basis of these three examples that convergent RSPT is
irrelevant to physics but that is wrong. First of all, the Kato–Rellich
theorem implies that in the Born–Oppenheimer limit (i.e. infinite nuclear
masses), the electronic energies are real analytic in the nuclear coordinates
(at non-coincident points if the internuclear repulsion is included); see
[467, 502] for some of my work on Born–Oppenheimer curves.

Moreover, we have the following interesting example on
$L^{2}({\mathbb{R}}^{3}\times{\mathbb{R}}^{3})$

$A(\beta)=-\Delta_{1}-\Delta_{2}-\frac{1}{r_{1}}-\frac{1}{r_{2}}+\beta\frac{1}{|\mathbf{r_{1}}-\mathbf{r_{2}}|}$
(1.6)

where the Kato–Rellich theory applies. Up to a scale factor of $Z^{-2}$, when
$\beta=Z^{-1}$, this describes a two electron system moving around a nucleus
of charge $Z$. This $A(0)$ is the sum of two independent hydrogen atoms so it
has continuous spectrum $[-\tfrac{1}{4},\infty)$ and eigenvalues

$E_{n,m}=-\frac{1}{4n^{2}}-\frac{1}{4m^{2}};\qquad n,m=1,2\dots$ (1.7)

For $n$ or $m$ equals $1$, these are below $-\tfrac{1}{4}$, so discrete and
Theorem 1.1 applies.

There is a huge literature on the discrete eigenvalues of this system,
especially the ground state. Some of it is summarized in [713, Example 2.1]. I
have a joint paper [318] on what happens at $\beta_{c}$, the coupling where
the ground state hits the continuous spectrum.

The major theme of this section is that RSPT tells you something about the
eigenvalues, even when the series diverges. Before my work, the standard
connection, where Kato [378] was the pioneer, concerned asymptotic series, a
notion first formalized by Poincaré [540] in 1886. Given a function, $f$,
defined in a region $R$ with $0$ in its closure, we say that $f(\beta)$ has
$\sum_{n=0}^{\infty}a_{n}\beta^{n}$ as an asymptotic series on $R$ if an only
if, for any $N$, we have that

$\lim_{|\beta|\to 0,\,\beta\in
R}(f(\beta)-\sum_{n=0}^{N}a_{n}\beta^{n})\beta^{-N}=0$ (1.8)

Kato’s method allows one to prove that RSPT is asymptotic when $R=(0,B)$ for
any eigenvalue of the anharmonic oscillator, (1.3), and for the Zeeman effect,
(1.5), and the method in his book [386] allows one to take $R$ to be suitable
sectors in the complex plane.

(1.8) shows that $f$ determines $\\{a_{n}\\}_{n=0}^{\infty}$ but since, for
example, when $R=(0,B)$, if (1.8) holds for $f$, it also holds for
$f_{1}(\beta)=f(\beta)+10^{6}\exp(-1/10^{6}\beta)$, if we only know (1.8) and
$\\{a_{n}\\}_{n=0}^{\infty}$, we can’t say anything about the value of
$f(\beta)$ for any particular fixed, non–zero $\beta$. Over the years,
mathematicians have developed a number of methods for recovering a unique
function among the several associated to a given asymptotic series. Hardy
[297] is a discussion of many of them. Two of them – Padé and Borel
summability are relevant to our discussion here.

Truncated Taylor series are polynomial approximations to a formal series
$\sum_{n=0}^{\infty}a_{n}\beta^{n}$. Padé approximation involves rational
approximation (the name is after the thesis of Padé [522]; his advisor,
Hermite, was a great expert on rational approximation). Given a formal series,
$\sum_{n=0}^{\infty}a_{n}\beta^{n}$, we say that $f^{[N,M]}(\beta)$ is the
$[N,M]$ Padé approximant if

$f^{[N,M]}(\beta)=\frac{P^{[N,M]}(\beta)}{Q^{[N,M]}(\beta)};\quad\deg
P^{[N,M]}=M,\quad\deg Q^{[N,M]}=N$ (1.9)
$f^{[N,M]}(\beta)-\sum_{n=0}^{N+M}a_{n}\beta^{n}=\textrm{O}\left(\beta^{N+M+1}\right)$
(1.10)

A formal power series is called a _series of Stieltjes_ if it has the form

$a_{n}=(-1)^{n}\int_{0}^{\infty}x^{n}d\mu(x)$ (1.11)

for some positive measure, $d\mu$ on $[0,\infty)$ with all moments finite.
This is related to the _Stieltjes transform_ of $\mu$ which is defined by

$f(\beta)=\int_{0}^{\infty}\frac{d\mu(x)}{1+x\beta}$ (1.12)

since it is easy to see that $\sum_{n=0}^{\infty}a_{n}\beta^{n}$ is an
asymptotic series for such an $f$ in any region of the form
$\\{\beta\,|\,|\arg\beta|<\pi-\epsilon\\}$. A basic result on convergence of
Padé approximants is

######  Theorem 1.2.

(Stieltjes Convergence Theorem) If $\\{a_{n}\\}_{n=0}^{\infty}$ is a series of
Stieltjes, then for each $j\in{\mathbb{Z}}$, the diagonal Padé approximants,
$f^{[N,N+j]}(z)$, converge as $N\to\infty$ for all
$\beta\in{\mathbb{C}}\setminus[0,\infty)$ to a function $f_{j}(\beta)$ given
by (1.12) with $\mu$ replaced by $\mu_{j}$ which obeys (1.11) (with
$\mu=\mu_{j}$). The $f_{j}$ are either all equal or all different depending on
whether (1.11) has a unique solution, $\mu$, or not.

######  Remarks.

1\. For proofs, see Baker [54, 55, 56] or Simon [711, Section 7.7]

2\. Stieltjes [731] didn’t discuss Padé approximates by name but instead
studied continued fractions which lead to the result for $j=0,1$ from which
one can deduce the general result.

3\. There is a huge literature on convergence of Padé approximants in cases
where they are not series of Stieltjes (see Lubinsky [479]).

In applying Theorem 1.1 it is useful to know when (1.11) has a unique
solution. A sufficient (but certainly not necessary) condition is that

$|a_{n}|\leq AC^{n}(kn)!$ (1.13)

for some $k\leq 2$.

The second summability method relevant to us here may work if (1.13) holds for
$k=1$. One forms the Borel transform

$g(w)=\sum_{n=0}^{\infty}\frac{a_{n}}{n!}w^{n}$ (1.14)

which defines an analytic function in $\\{w\,|\,|w|<C^{-1}\\}$. Under the
assumption that $g$ has an analytic continuation to a neighborhood of
$[0,\infty)$, one defines for $\beta$ real and positive

$f(\beta)=\int_{0}^{\infty}e^{-a}g(a\beta)da$ (1.15)

if the integral converges. Since $\int_{0}^{\infty}e^{-a}a^{n}\,da=n!$,
formally, $f(\beta)=\sum_{n=0}^{\infty}a_{n}\beta^{n}$. Here, one has a
theorem of Watson [758]; see Hardy [297] for a proof:

######  Theorem 1.3.

Let $\Theta\in\left(\tfrac{\pi}{2},\tfrac{3\pi}{2}\right)$ and $B>0$. Define

$\displaystyle\Omega$ $\displaystyle=\\{z\,|\,0<|z|<B,|\arg z|<\Theta\\}$
(1.16) $\displaystyle\widetilde{\Omega}$ $\displaystyle=\\{z\,|\,0<|z|<B,|\arg
z|<\Theta-\tfrac{\pi}{2}\\}$ (1.17) $\displaystyle\Lambda$
$\displaystyle=\\{w\,|\,w\neq 0,|\arg w|<\Theta-\tfrac{\pi}{2}\\}$ (1.18)

Suppose that $\\{a_{n}\\}_{n=0}^{\infty}$ is given and that $f$ is analytic in
$\Omega$ and obeys

$\left|f(z)-\sum_{n=0}^{N}a_{n}z^{n}\right|\leq AC^{N+1}(N+1)!$ (1.19)

on $\Omega$ for all $N$. Define

$g(w)=\sum_{n=0}^{\infty}\frac{a_{n}}{n!}w^{n};\qquad|w|<C^{-1}$ (1.20)

Then $g(w)$ has an analytic continuation to $\Lambda$ and for all
$z\in\widetilde{\Omega}$, we have that

$f(z)=\int_{0}^{\infty}e^{-a}g(az)\,da$ (1.21)

######  Remark.

By writing (1.21) as $f(z)=z^{-1}\int_{0}^{\infty}e^{-a/z}g(a)\,da$, one sees
that the natural regions of analyticity aren’t sectors but regions of the form
$\operatorname{Re}(1/z)>1/R$ which are circles tangent to the imaginary axis.
A few years after Watson, Nevanlinna [516] proved a stronger version of
Theorem 1.3 using such regions; see also Sokal [725].

My own work on the anharmonic oscillator was motivated by my thesis advisor,
Arthur Wightman, who had the idea of exploring this as a way of understanding
QFT perturbation theory. He wanted to exploit an idea of Symanzik to use
scaling. One looks at

$H(\alpha,\beta)=-\frac{d^{2}}{dx^{2}}+\alpha x^{2}+\beta x^{4}$ (1.22)

and notes that if $\lambda$ is positive, then

$(U(\lambda)f){x}=\lambda^{1/4}f(\lambda^{1/2}x)$ (1.23)

is unitary and

$U(\lambda)H(\alpha,\beta)U(\lambda)^{-1}=\lambda^{-1}H(\alpha\lambda^{2},\beta\lambda^{3})$
(1.24)

So for any $\alpha,\beta,\lambda$ real with $\beta,\lambda>0$, one has that

$E_{n}(\alpha,\beta)=\lambda^{-1}E_{n}(\alpha\lambda^{2},\beta\lambda^{3})$
(1.25)

where $E_{n}$ is the $n$th eigenvalue of $H(\alpha,\beta)$.

Wightman gave the problem to another graduate student, Arnie Dicke, but they
came to me with a technical problem they ran into. Then, in early 1968, I was
a second year graduate student in physics but I had been charmed by Kato’s
book [386] and was regarded as a local expert on some of the material. The
problem was that $U(\lambda)$ was only a bounded operator if $\lambda>0$ and
so (1.24) only made sense for such $\lambda$ and they wanted (1.25) for
complex $\lambda$.

I came up with the following argument. In the region
$R=\\{\alpha,\beta\in{\mathbb{C}},\,\beta\notin(-\infty,0]\\}$,
$H(\alpha,\beta)$ is an analytic family of type A (I proved estimates like
(1.2) for $A=H(\alpha,\beta),\,\beta\notin(-\infty,0]$ and $B=x^{2}$ or
$x^{4}$). Thus, as long as the eigenvalue is simple at
$(\alpha_{0},\beta_{0})\in R$, we have that $E_{n}(\alpha,\beta)$ is analytic
near $(\alpha_{0},\beta_{0})$. Since (1.25) holds for $\lambda$ real, it holds
for small complex $\lambda$ by analyticity. (There was an issue of eigenvalue
labelling - there was no guarantee that if one went around a loop starting and
ending on ${\mathbb{R}}\times(0,\infty)$, that $n$ couldn’t change.)

Here, I missed a golden opportunity. I had proven an invariance of discrete
spectrum under complex scaling. It didn’t occur to me to ask about an operator
like $-\tfrac{d^{2}}{dx^{2}}-\beta e^{-\mu x}$ or $-\Delta-\tfrac{1}{r}$ which
like $H(\alpha,\beta)$ has an analytic continuation for
$H(\lambda)=U(\lambda)HU(\lambda)^{-1}$ from real $\lambda$ to complex
$\lambda$. If I had, I might have found Combes great discovery of a year later
(I’ll discuss his work in Section 2).

After I found this, given that Dicke was bogged down in his construction of
solution with the expected WKB asymptotics at infinity (which turned into his
thesis and which he asked me to publish as an appendix to my long paper
[613]), he and Wightman felt that I should explore aspects of this problem
beyond the existence of solutions that Dicke was looking at. I immediately
noticed that (1.25) implies that

$E_{n}(1,\beta)=\beta^{1/3}E_{n}(\beta^{-2/3},1)$ (1.26)

so since $E_{n}(\alpha,1)$ is analytic near $\alpha=0$, $E_{n}(1,\beta)$ has a
convergent series near infinity, not in $\beta^{-1}$, but in $\beta^{-2/3}$,
so that $E_{n}(1,\beta)$ has a kind of three sheeted structure.

In some of my work, I made an assumption that $E_{n}(\alpha,1)$ has no natural
boundaries – this was proven to be true many years later (Eremenko–Gabrielov
[186]) but for $|\arg\alpha|<2\pi/3$, as we’ll see shortly, it was proven
there were no singularities at all in the same time frame as my paper.

In 1968–69, Wightman was on leave in Europe and he thought about and talked to
others about the anharmonic oscillator and wrote me letters. Andre Martin
pointed out to him that the large $\beta$ expansion couldn’t converge for all
$\beta\neq 0$. For, if it were, $E_{n}(\alpha,1)$ would be an entire Herglotz
function and so linear which one can easily see isn’t true for
$E_{n}(\alpha,1)=\alpha^{1/2}E_{n}(1,\alpha^{-3/2})$ shows that

$\lim_{\alpha\to\infty\,\,\alpha\in{\mathbb{R}}}E_{n}(\alpha,1)/\sqrt{\alpha}=E_{n}(1,0)$

I’d never seen the theorem about entire Herglotz functions which I’m sure
Martin got by using the Herglotz representation theorem. While I’d later often
use that representation theorem heavily in my career and even find a useful
extension for meromorphic Herglotz functions on the disk [692], I’d never
heard of it at the time. In those pre–Google days, I couldn’t easily find much
about Herglotz functions which was good because it forced me to find my own
unconventional proof of the entire Herglotz theorem and that allowed me to
prove that $E_{n}(\alpha,1)$ couldn’t have an isolated singularity at
infinity. I’d already proven using Kato’s methods that for any fixed
$\omega=e^{i\theta},\,|\theta|<\pi$, one had that $p^{2}+\omega x^{2}+\beta
x^{4}$ has a RS asymptotic perturbation series as $\beta\downarrow
0,\,\beta>0$ which, by scaling, implied that $E_{n}(1,\beta)$ has an
asymptotic series in
$\\{\beta\,|\,|\arg\beta|<3\pi/2-\epsilon,\,|\beta|<R_{\epsilon}\\}$. This in
turn implied that on the three sheeted Riemann surface, there were an infinity
of singularities with limit point 0 (on the natural three sheeted surface) and
asymptotic phase $\pm 3\pi/2$.

Around this time, I got a hold of a preprint of Bender and Wu [65]. (In those
days, Xeroxing was expensive so preprints were mimeographed and of limited
distribution. While I had known Carl Bender when I was a senior at Harvard and
he a graduate student and we were in Schwinger’s QFT course together, I
certainly didn’t know him well enough to get a preprint, but fortunately, he
and Arnie Dicke were friends and I got it from Arnie). Bender and Wu computed
the first 75 coefficients, $\\{a_{n}\\}_{n=1}^{75}$ for the anharmonic
oscillator ground state RS series and they did a numerical analysis of the
$a_{n}$ leading to a conjecture of the large $n$ asymptotics (I’ll say more
about this subject in the next section). They also did a mathematically
unjustified WKB analysis of the analytic behavior of $E_{n}(1,\beta)$ which
was consistent with what I had found. (I still remember that my first seminar
outside Princeton was a physics talk at Chicago where I made reference to the
“notoriously unreliable WKB approximation”. Afterwards, a kindly older
gentleman came up to me and introduced himself: “I’m the W of WKB”!).

Early in 1969, I got a letter from Arthur Wightman that began “The specter of
Padé is haunting Europe…”. Various theoretical physicists had the idea of
using diagonal Padé approximants on some field theoretic Feynman series and
Wightman suggested that I try it on the anharmonic oscillator. I’d never done
any scientific computing (and haven’t done any since!) but with the first 41
coefficients from the Bender–Wu preprint and explicit determinantal formulae
from Baker’s book [55], it was straightforward.

In those days, one did computer calculations by writing the program in Fortran
on punch cards, submiting the deck and waiting a day to get back the results.
My initial output was nonsense, but I realized I’d left out a $(-1)^{n}$,
fixed it, and the second time was golden! I computed $f^{[N,N]}(\beta)$ for
$N=1,2,\dots,20$ and $\beta=0.1,0.2,\dots,1$ and got rapid convergence to
answers consistent with less accurate variational calculations already in the
literature.

The approximants $f^{[N,N]}(\beta)$ were monotone in $N$ suggesting that the
underlying series was a series of Stieltjes. I realized that with my methods,
to prove this, one needed to show that on $\\{\beta\,|\,|\arg\beta|<\pi\\}$,
the $E_{j}(\beta)$ have no natural boundaries and no eigenvalue crossing,
equivalently the same for $E_{j}(\alpha,1)$ within
$\\{\alpha\,|\,|\arg\alpha|<2\pi/3\\}$. Nick Khuri, a physicist at
Rockefeller, heard of my work and invited me to talk while Martin was visiting
there and I explained the situation to him. Loeffel–Martin [475], using a
clever argument tracking the zeros of eigenfunctions were able to show no
eigenvalue crossing assuming one could make analytic continuation and I could
show, using their results, that one could be sure one could analytically
continue.

The four of us (Loeffel, Martin, Simon and Wightman [476]) then published an
announcement putting everything together. The analyticity results implies that
for a positive measure on $(0,\infty)$, one has that

$E_{j}(\beta)=E_{0}-\beta\int_{0}^{\infty}\frac{d\rho_{j}(x)}{1+x\beta}$
(1.27)

for all $\beta\in{\mathbb{C}}\setminus(-\infty,0]$. My results on the RS
series being asymptotic in the cut plane then implied that the RS series was a
series of Stieltjes, so the diagonal Padé approximants converge. Moreover, I
had shown that (1.13) holds for $k=1$, so the limits are the same and equal
the eigenvalues.

While this Padé result is nice, the known scope where one can prove Padé
summability is very limited. Loeffel et al. [476] note that their methods
imply that for $m=2,3,\dots$, the RS series for the eigenvalues of
$p^{2}+x^{2}+\beta x^{2m}$ are series of Stieltjes so the diagonal Padé
approximants converge. However, (1.13) holds for $k=(m-1)$ so they only knew
uniqueness when $m=2,3$. In fact, several years later, Graffi–Grecchi [268]
proved that for the $x^{8}$ oscillator, the $f^{[N,N+j]}$ converge as
$N\to\infty$ to $j$ dependent limits, none of which is the eigenvalue!!
Moreover, the Loeffel–Martin [475] method tracks zeros and so is limited to
ODEs and there is no rigorous Padé result known for anharmonic oscillators
with more than one degree of freedom.

Borel summability turns out to be much more widely applicable. Shortly after
the four author announcement appeared, I got contacted by Sandro Graffi and
Vincenzo Grecchi whom I hadn’t previously known. They enclosed a Xerox of the
pages of Hardy’s book dealing with Watson’s Theorem and more importantly some
numerical calculations of the Borel sum of the $x^{4}$ ground state (based on
a not rigorously justified use of Padé approximants of the Borel transform,
$g$, of (1.14)) which not only converged but more rapidly than ordinary Padé
approximants. I quickly determined that my techniques showed the hypotheses of
Watson’s theorem held for $x^{4}$ oscillators in any dimension and that a
higher order (i.e. $(kn)!$ instead of $n!$) Borel summability works for the
$x^{2m}$ oscillator so we published a paper with these results [274]. Before
leaving the issue of the perturbation series for the anharmonic oscillator, I
note that using the first 60 terms in the series and the computer power
available in 1978, Seznec and Zinn–Justin [597], using modified Borel
summability and large order expansions, claimed to be able to find the ground
state for all values of $\beta$ to one part in $10^{23}$!

I wrote several papers on applying Borel summability in $\Phi_{2}^{4}$ cutoff
field theory [614, 574] and other contexts [617, 618]. Avron–Herbst–Simon [37]
proved Borel summability of Zeeman Hamiltonians and there have been proofs by
others of Borel summability of various quantum field theoretic perturbation
series (Feynman diagram expansion of Schwinger functions): $P(\phi)_{2}$
[177], $\phi^{4}_{3}$ [481], $Y_{2}$ [565], $Y_{3}$ [482]. As we’ll see in the
next section, there is a sense in which the Stark series is Borel summable.

Before leaving asymptotic perturbation theory, I mention a striking example of
Herbst–Simon [311]

$A(\beta)=-\frac{d^{2}}{dx^{2}}+x^{2}-1+\beta^{2}x^{4}+2\beta x^{3}-2\beta x$

If $E_{0}(\beta)$ is the lowest eigenvalue, we proved that for all small,
non–zero positive $\beta$

$0<E_{0}(\beta)<C\exp(-D\beta^{-2})$

Thus $E_{0}(\beta)$ has $\sum_{n=0}^{\infty}a_{n}\beta^{n}$ as asymptotic
series with $a_{n}\equiv 0$. The asymptotic series converges but, since
$E_{0}$ is strictly positive, it converges to the wrong answer!

##  2\. Complex Scaling Theory of Resonances

Our second tale also concerns eigenvalue perturbation theory, but in
situations where the eigenvalue turns into a resonances. One of the simplest
real physical examples where, at the time of my work, this was expected to
occur involves the $1/Z$ expansion of (1.6). The eigenvalues $E_{m,n}$ of
$A(0)$ given by (1.7) when $m,n\geq 2$ are embedded in the continuous
spectrum. For example $E_{2,2}=-\tfrac{1}{8}>-\tfrac{1}{4}$. For $\beta\neq
0$, one expects the bound state to dissolve into a resonance.

There is a standard physics textbook calculation called time–dependent
perturbation theory (TDPT). The lifetime, $\tau$, is by the Wigner–Weisskopf
formula $\tau=\hbar/\Gamma$ with $\Gamma=2\mbox{Im}\,E(\beta)$. The leading
order for $\Gamma$ is called the _Fermi golden rule_ and is given by
$\Gamma=\Gamma_{2}\beta^{2}+\mbox{O}(\beta^{3})$ where

$\Gamma_{2}=\left.\frac{d}{d\lambda}\langle
B\varphi_{0},\tilde{P}_{(-\infty,\lambda)}(A_{0})B\varphi_{0}\rangle\right|_{\lambda=E_{0}}$
(2.1)

Here $\tilde{P}$ is a spectral projection for $A_{0}$ with the eigenspace at
$E_{0}$ removed and $\varphi_{0}$ is the eigenvector with
$A_{0}\varphi_{0}=E_{0}\varphi_{0}$ for the embedded eigenvalue with
$\lVert\varphi_{0}\rVert=1$ (this is only the correct form if the eigenvalue
is simple). Usually, the right side of (2.1) is written as $\langle
B\varphi_{0},\delta(A_{0}-E_{0})B\varphi_{0}\rangle$. This version is from
Simon [622]. There is a subtlety here that we won’t discuss in detail (but see
[556] or [713, Example 3.2]): $E_{2,2}$ is actually a degenerate eigenvalue
and a subspace of the eigenspace at $-\tfrac{1}{8}$ has a symmetry with a
continuum only beginning at $-\tfrac{1}{16}$ (put differently, the continuum
it is imbedded in has a different symmetry from part of the eigenspace), so
only part of the eigenspace dissolves into resonances. These resonances are
observed in nature and are called autoionizing states or Auger resonances.

A second important model is the Stark Hamiltonian, (1.4). If $\beta\neq 0$, it
is not hard to see that spec($A(\beta))$ is all of ${\mathbb{R}}$ (for
$A_{0}(\beta))$. For the operator without Coulomb term, one can write down
$\exp(itA_{0}(\beta))$ explicitly [32] and show that $-1/r$ doesn’t change the
spectrum, indeed, wave operators exist and are complete [70, 308]. Thus the
discrete eigenvalues are swamped by continuous spectrum. The theoretical
physics literature based on a formal tunnelling calculation studied the
leading asymptotics of the width, which is $\mbox{O}(\beta^{n})$ for all $n$,
and found that the leading order is

$\Gamma(\beta)=(2\beta)^{-1}\exp\left(-\frac{1}{6\beta}\right)\left(1+\textrm{O}\left(\tfrac{1}{n}\right)\right)$
(2.2)

This formula, first found correctly by Lanczos [433], is called the
Oppenheimer formula after [520].

There were fundamental mathematical questions discussed by Friedrichs [207],
who was the first person to look mathematically at issues of eigenvalues
turning into resonances. First, in cases like the Stark effect, where there
are RSPT series but no eigenvalues, what is the meaning of the perturbation
coefficients. Second, what exactly is a resonance? Third, in a case like
autoionizing states, what exactly are the higher order terms of TDPT (the
physics literature was unclear on this point) and is the series ever
convergent?

Before the complex scaling approach, there was the idea of solving the first
problem by connecting the series to the asymptotics of the spectral
projections of the perturbed operators. This notion, called spectral
concentration was pioneered by Titchmarsh [751] and Kato [375, 386] and later
by others [116, 566]. It works well for the Stark effect where the width is
$\mbox{o}(\beta^{n})$ for all $n$ so one can hope to prove spectral
concentration to all orders [557, 559] although it does not seem possible to
fit a result like (2.2) into this framework. But for the case of autoionizing
states, the widths go as $c\beta^{2},c\neq 0$ and there is only spectral
concentration to first order.

Howland wrote several papers [322, 323, 324, 325, 326] that addressed both
kind of models, but they required either the perturbation or some other object
be finite rank so they didn’t cover either physical model mentioned above.

In two remarkable papers, Combes with his collaborators Aguilar and Balslev
[6, 57] developed a framework to study the absence of singular spectrum that I
realized was ideal to study autoionizing states. They called it the theory of
dilation analytic potentials but after the quantum chemists started using it
in calculations, the name shifted to complex scaling.

Consider first a two body potential, $V(x),\,x\in{\mathbb{R}}^{\nu}$ and for
$\theta\in{\mathbb{R}}$ define

$(U(\theta)f)(x)=e^{\nu\theta/2}f(e^{\theta}x)$ (2.3)

which is a one parameter semigroup of unitary operators. Define, again for
$\theta\in{\mathbb{R}}$:

$V(\theta)=U(\theta)VU(\theta)^{-1}$ (2.4)

which is, of course, multiplication by $V(e^{\theta}x)$. For general $V$, this
doesn’t make sense for $\mbox{Im}\,\theta\neq 0$, e.g. let $V$ be a square
well. But for some $V$’s, one can analytically continue. Particular examples
are $V(x)=|x|^{-\beta},0<\beta<2$, in particular for $\beta=1$, where
$V(\theta)$ continues to an entire function and $V(x)=e^{-\mu r}/r$ where
$V(\theta)$ can be continued (as a relatively bounded operator) so long as
$|\mbox{Im}\,\theta|<\pi/2$. $V$ is called dilation analytic if
$V(\theta)(-\Delta+1)^{-1}$ has an analytic continuation from ${\mathbb{R}}$
to all $\theta$ with $|\mbox{Im}\,\theta|<\Theta$ for some $\Theta>0$.

Let $H_{0}=-\Delta$ and $H=H_{0}+V$. Then

$H(\theta)=U(\theta)HU(\theta)^{-1}$ (2.5)

will have an analytic continuation to
$\\{\theta\,|\,|\mbox{Im}\,\theta|<\Theta\\}$ as an analytic family of type
(A).

The Kato–Rellich theory is applicable. As in the last section, discrete
eigenvalues are $\theta$–independent at least for $\mbox{Im}\,\theta$ small.
But since

$H(\theta)=-e^{-2\theta}\Delta+V(\theta)$ (2.6)

we know that if $V(-\Delta+1)^{-1}$ is compact, then we have that $H(\theta)$
has continuous spectrum $e^{-2\mbox{Im}\theta}[0,\infty)$, i.e. the continuous
spectrum rotates about the threshold $0$.

Balslev–Combes [57] analyzed the spectrum for $N$–body Hamiltonians and found
a spectrum like that shown in Figure 1

Figure 1. Spectrum of a Complex Scaled Hamiltonian

Instead of continuous spectrum rotating about zero, it rotates about each
scattering threshold. By an induction argument, one can prove that the set of
thresholds is a closed countable set. An important point is that as the
spectrum swings down, it can uncover eigenvalues which then persist until
perhaps hit by another piece of continuous spectrum when they can disappear.
Combes and company interpreted these complex eigenvalues as resonances.

A key use Balslev–Combes made of their theory was to prove the absence of
singular continuous spectrum (see Section 6 below). One of my later results on
complex scaling that I should mention is a quadratic form version [619] which
has some significant technical simplifications, some of them involving work
with Mike Reed on the spectrum of tensor products [551, 552, 621].

In [622], I realized that complex scaling was an ideal tool for understanding
autoionizing states. One can prove that embedded eigenvalues also don’t move
if $\mbox{Im}\,\theta$ is moved away from zero to positive values while
continuous spectrum does move. Thus in studying $H+\beta W$, one can look at
$H(\theta)+\beta W(\theta)$. While $E_{0}$ might be an embedded eigenvalue of
$H$, so long as it is not at a scattering threshold of $H$, it is a discrete
eigenvalue of $H(\theta)$ when $\theta=i\epsilon$ with $\epsilon$ small and
positive. So $E_{0}$ will become a an eigenvalue, $E_{0}(\beta)$, of
$H(\theta)+\beta W(\theta)$ given by a convergent power series in $\beta$ (if
$E_{0}$ is degenerate, there are extra subtleties). In general,
$\mbox{Im}\,E_{0}(\beta)<0$, i.e. embedded eigenvalues turn into resonances.
The Rayleigh–Schrödinger series for $E_{0}(\beta)$ provide an unambiguous
higher order TDPT which is convergent! Moreover, one can manipulate the second
order term to validate the Fermi golden rule and so get a rigorous proof of
it.

For the Stark effect, the conventional wisdom among mathematical physicists
was that complex scaling couldn’t work. Because it was known (see, e.g. [308])
that $-\Delta+F\hat{e}\cdot x$ for $F\neq 0$ and $\hat{e}$ a unit vector has
no scattering thresholds, there was no place for the continuous spectrum of

$H_{0}(\theta)=-e^{-2\theta}\Delta+e^{\theta}F\hat{e}\cdot x$ (2.7)

to go when $\mbox{Im}\,\theta$ is small and non–zero. But W. Reinhardt, a
quantum chemist, was fearless and found [558] calculations gave sensible
answers.

This made I. Herbst reconsider the conventional wisdom [309]. In fact for
$\mbox{Im}\,\theta\in(0,\pi/3)$, $H_{0}(\theta)$ defines a closed operator
with empty spectrum! So, since there is no place for the continuous spectrum
to go, it disappears! It is, of course, known that a bounded operator cannot
have empty spectrum (see, e.g. [711, Theorem 2.2.9]) but $H_{0}(\theta)$ is
not bounded and $H_{0}(\theta)^{-1}$ has a single point, namely $0$ in its
spectrum; in some sense, $H_{0}(\theta)$ has $\infty$ as the only point in its
spectrum. Herbst was able to analyze [309] the Hydrogen Stark Hamiltonian
whose resonance energies he showed have width that are $\mbox{O}(F^{k})$ for
all $k$ and had RS series as asymptotic series.

Herbst and I [312] then extended this work to analyze the Stark Hamiltonian
for general atoms. We also proved a kind of Borel summability. The
Rayleigh–Schrödinger perturbation series is Borel summable to a function
defined about the positive imaginary axis in the $F$ plane whose analytic
continuation back to real $F$ is the resonance. We proved this for atoms.
About the same time, Graffi–Greechi [269, 270, 272] discovered this for the
hydrogen Stark effect using the separability of that problem into 1D problems
(see below). Sigal [603, 604, 605, 606] and Herbst–Møller–Skibsted [310] have
further studied Stark resonances in multi–electron atoms proving that the
widths are strictly positive and exponentially small in $1/F$.

Harrell and I then wrote a paper [298] that was able to analyze the small
coupling behavior of the imaginary part of some resonance energies that are
exponentially small. Essentially, this allowed a rigorous proof of some
results obtained earlier by theoretical physicists using a formal WKB
analysis. First of all we proved the Lanczos–Oppenheimer formula (2.2). As
noted earlier by Herbst–Simon [312] this implies asymptotics of the
perturbation coefficients

$E(F)\sim\sum_{n=0}^{\infty}A_{2n}F^{2n}$ (2.8)
$A_{2n}=-6^{2n+1}(2\pi)^{-1}(2n)!\left(1+\textrm{O}\left(\frac{1}{n}\right)\right)$
(2.9)

since one can write $f(x)=\oint_{\gamma}(2\pi i)^{-1}\tfrac{f(z)}{z-x}\,dz$
where $\gamma$ is a contour that is a small circle with a loop around the
negative axis (in the $-F^{2}$ variable) and a large circle.

In the context of the anharmonic oscillator, the same idea of precise
asymptotics of RS coefficients occurred earlier than the work of Herbst–Simon
and Harrell–Simon. As noted in Section 1, Bender–Wu [65] had computed the
first 75 coefficients for the ground state $E_{0}(1,\beta)$ and they did a
numerical fit and conjectured that

$a_{n}=4\pi^{-3/2}(-1)^{n+1}\left(\tfrac{3}{2}\right)^{n+1/2}\Gamma(n+\tfrac{1}{2})\left(1+\textrm{O}\left(\tfrac{1}{n}\right)\right)$
(2.10)

They had the leading constant to 8 decimal places and guessed its analytic
form. In my anharmonic oscillator paper [613], I noted that (2.10) was
equivalent to leading asymptotics

$\mbox{Im}\,E_{0}(1,\beta)\underset{\beta=-b+i0}{\sim}4\pi^{-1/2}b^{-1/2}e^{-2/3b}$
(2.11)

Without noticing my remark, Bender and Wu noted [66] that (2.11), and so
(2.10), follow from a formal WKB calculation of the tunnelling in a potential
x${}^{2}-bx^{4}$. Harrell-Simon [298] have rigorous proofs of (2.11) and so
(2.10). Helffer-Sjöstrand [303] proved Bender-Wu type formulae for higher
dimensional oscillators.

Harrell–Simon uses ODE (i.e. 1D) techniques. The Stark effect can be separated
into 1D problems in elliptic coordinates (noted by Jacobi [345] in classical
mechanics and then Schwarzschild [587] and Epstein [184] in old quantum theory
and in parabolic coordinates by Schrödinger [586] and Epstein [185]) and this
was later used mathematically by Titchmarsh [751, 752], Harrell–Simon [298]
and by Graffi–Grecchi and collaborators [269, 270, 271, 273, 275, 64, 89,
272].

The Zeeman effect for Hydrogen can be reduced to a two dimensional problem.
Avron [31] used this and an instanton calculation of tunnelling (see section
8) to formally compute the asymptotics for RS coefficients for the ground
state of the Zeeman Hamiltonian (1.5).

$E_{k}=\left(\frac{4}{\pi}\right)^{5/2}(-1)^{k+1}\pi^{-2k}\Gamma\left(2k+\frac{3}{2}\right)\left(1+\textrm{O}\left(\frac{1}{k}\right)\right)$
(2.12)

Helffer-Sjöstrand [303] then gave a rigorous proof of this using PDE
techniques.

Quantum Chemists embraced the complex scaling method to do calculations of
resonance energies in atoms and molecules. I wrote a review of the
mathematical theory [645] for a joint conference. The calculations for
molecular resonance curves were done in a Born–Oppenheimer approximation with
fixed nuclei which lead to potentials which are analytic outside a large ball.
I introduced exterior complex scaling Simon [647] to justify what they did and
wrote a paper with Morgan [503] explaining why exterior scaling did indeed
justify their calculations. A more elegant approach (smooth exterior scaling)
was subsequently developed by Hunziker [337] and Gérard [229].

I should note that I have reason to believe that, at least at one time, Kato
had severe doubts about the physical relevance of the complex scaling approach
to resonances. [298] was rejected by the Annals of Mathematics, the first
journal it was submitted to. The editor told me that the world’s recognized
greatest expert on perturbation theory had recommended rejection so he had no
choice. I had some of the report quoted to me. The referee said that the
complex scaling definition of resonance was arbitrary and physically
unmotivated with limited significance. My review of Kato’s work on non-
relativistic quantum mechanics (henceforth NRQM) [713, Part 1, pg. 154-155]
has a long discussion of why I believe the complex scaling definition is
physically relevant with many references to the literature.

I should mention that I used complex scaling [626] to show $N$–body systems
with local potentials that can be continued to the right half plane (in
particular, with Coulomb potentials) can’t have positive energy bound states
or thresholds.

While I’ve focused on the complex scaling approach to resonances, there are
other methods. One, called distortion analyticity, works sometimes for
potentials which are the sum of a dilation analytic potential and a potential
with exponential decay (but not necessarily any $x$–space analyticity). The
basic papers include Jensen [353], Sigal [602], Cycon [124], and Nakamura
[508, 509]. Some approaches for non-analytic potentials include Gérard-Sigal
[230], Cattaneo–Graf–Hunziker [103], Cancelier–Martinez–Ramond [91] and
Martinez–Ramond–Sjöstrand [489]. There is an enormous literature on the theory
of resonances from many points of view. I should mention a beautiful set of
ideas about counting asymptotics of resonances starting with Zworski [776];
see Sjöstrand [723] for unpublished lectures that include lots of references,
a recent review of Zworski [777] and the book of Dyatlov–Zworski [171] (I have
one paper related to these ideas [691]). The form of the Fermi Golden Rule at
thresholds is discussed in Jensen–Nenciu [355]. A review of the occurrence of
resonances in NR Quantum Electrodynamics and of the smooth Feshbach–Schur map
is Sigal [607] and a book on techniques relevant to some approaches to
resonances is Martinez [488].

Finally, I note that these two sections have dealt with eigenvalue
perturbation theory. I’ll return in Section 8 to a different issue involving
perturbations that give birth to eigenvalues from the edge of continuous
spectrum and to eigenvalues at limiting values of coupling constant, namely
$-\hbar^{2}\Delta+V(x)$ as $\hbar\downarrow 0$.

##  3\. Statistical Mechanical Methods in EQFT

The fifteen years following 1965 saw the development of a subject known as
constructive quantum field theory (CQFT) which successfully constructed
interacting quantum fields in $2$ and $3$ space-time dimensions obeying all
the Wightman axioms [767, 734, 369]. Because of the failure to get to $4$
space-time dimensions (except for some negative results [8, 210, 10]), the
long lasting impact to rigorous quantum physics has been more limited than
initially hoped (extending to the physically relevant $4$ dimensional case is
a million dollar problem [347]). Still, the spinoff to various areas of
mathematics and theoretical physics has been substantial.

My main goal in this section is to focus on my work, much of it jointly with
Francesco Guerra and Lon Rosen, on using methods from classical statistical
physics to study Bose CQFT, but I’ll begin with some of my other work
motivated by CQFT that had important mathematical spinoffs.

CQFT was initially developed by many researchers including J. Fröhlich, F.
Guerra, K. Osterwalder, L. Rosen, R. Schrader, I. Segal, E. Seiler, T.
Spencer, A. Wightman and especially J. Glimm, A. Jaffe and E. Nelson. I refer
the reader to the books of Simon [627] and Glimm-Jaffe [255].

The initial work mainly on $(\varphi^{4})_{2}$ theories focused on the
Hamiltonian viewpoint where controlling spatially cutoff theories is hard
because the operators act on an infinite number of variables and the potential
is not bounded from below (we use $(X)_{d}$ shorthand to describe theories
where $d$ is the number of space-time dimensions and $X$ an abbreviation for
the interaction term). The first breakthrough was by Nelson [510] who realized
that the free Bose Hamiltonian, $H_{0}$, in a periodic box in one space
dimension, viewed as an infinite sum of harmonic oscillators (with different
frequencies), could be realized as a Gaussian process by shifting from $dx$ to
$\varphi_{0}^{2}\,dx$, where $\varphi_{0}$ is the ground state, so that
$H_{0}$ acted on ${\mathbb{R}}^{\infty}$ with a Gaussian probability measure,
$d\mu_{0}$. The operator $H_{0}$ was then realized as a pure Dirichlet form
(i.e. $\langle\psi,H_{0}\psi\rangle=\int\,|\nabla\psi|^{2}\,d\mu_{0}$). For
differential operators, this shift to ground state measure and Dirichlet form
goes back to Jacobi (!) and since, Nelson’s representation has been used many
times in mathematical analysis of quantum theories with finitely many degrees
of freedom, e.g. [205]. In this representation, Nelson proved that

$\lVert e^{-tH_{0}}f\rVert_{p}\leq\lVert f\rVert_{p}$ (3.1)

for all $f\in L^{p}({\mathbb{R}}^{\infty})$ and all $t>0$ and he proved that
for some $T>0$

$\lVert e^{-TH_{0}}f\rVert_{4}\leq C\lVert f\rVert_{2}$ (3.2)

for some fixed $C$ and all $f\in L^{2}$. He also showed that while the
$(\varphi^{4})_{2}$ spatially cutoff interaction, $V$, is not bounded from
below, it obeys

$\int e^{-sV}\,d\mu<\infty\qquad\text{all }s>0$ (3.3)
$V\in\bigcap_{p<\infty}L^{p}({\mathbb{R}}^{\infty},d\mu)$ (3.4)

and most importantly that (3.1)-(3.4) imply that $H_{0}+V$ is bounded from
below.

Two important followups were by Glimm [251], who proved that (3.2) plus a mass
gap imply that by increasing $T$, (3.2) holds with $C=1$ (this yields
dimension independence and allows removing the need for Nelson to restrict to
periodic boundary conditions) and by Federbush [190], who used interpolation
to prove that $\lVert e^{-sH_{0}}f\rVert_{p_{s}}\leq C_{s}\lVert f\rVert_{2}$
with $p_{s}\downarrow 2$ as $s\downarrow 0$ and then took derivatives,
implicitly getting the first Gaussian logarithmic Sobolev inequality but which
was dimension dependent.

The next step is to prove essentially self-adjointness of $H_{0}+V$ on
$D(H_{0})\cap D(V)$ for spatially cutoff $\varphi^{4}_{2}$. This was
accomplished by Glimm-Jaffe [252] who proved it using additional estimates
beyond those of Nelson and subsequently by Segal [590, 591, 592] who only
needed the estimates (3.1)-(3.4).

At this point, my work enters via a widely quoted joint paper with Høegh-Krohn
[715] entitled _Hypercontractive semi-groups and two dimensional self-coupled
Bose fields_. We abstracted and simplified Segal’s self-adjointness result.
One significant aspect was inventing the term “hypercontractive” for groups
obeying (3.1) and (3.2) (Nelson complained to me that since (3.2) has a $C$
which might not be one, we should have used “hyperbounded” but I replied that
hypercontractive sounded better). Other terms that I’ve introduced that have
caught on include Agmon metric, almost Mathieu equation, Berry’s phase,
Birman-Schwinger bound, CLR inequality, CMV matrix, coupling constant
threshold, diamagnetic inequalities, HVZ theorem, Kato class, Kato’s
inequality, ten martini problem, Verblunsky coefficients and
ultracontractivity.

Hypercontractivity and its differential version, logarithmic Sobolev
inequalities (first completely explicated by Gross [284]), have had an
enormous number of applications outside quantum field theory; they are even
used in Perelman’s proof of the Poincaré conjecture. See [710, Section 6.6]
for a discussion of the various sides of the mathematical theory with
historical notes, additional references and presentation of some of the
applications. Several years later, in 1983, Brian Davies and I [136] found a
variant of hypercontractivity called ultracontractivity which has evoked
considerable mathematics.

Before turning to the discussion of statistical mechanical methods in QFT, I
should mention another aspect of my work in CQFT with mathematical spinoff. I
wrote a series of papers with E. Seiler [593, 594, 595, 596] on the Yukawa QFT
in two space-time dimensions, aka $Y_{2}$, that developed some mathematical
tools in the theory of trace ideals that have had many applications including
to quantum information theory.

The work on statistical mechanical methods depends on the second big
breakthrough in CQFT, namely Euclidean Quantum Field Theory (EQFT). The
Wightman axioms show that the Wightman functions (vacuum expectation values of
the product of quantum fields as tempered distributions on Minkowski space) of
any QFT can be analytically continued in time to pure imaginary time
differences and that these continued functions are invariant under the
Euclidean group. Schwinger [588] first emphasized this, so the analytic
continuation to imaginary times are sometimes called Schwinger functions.
Symanzik [737, 738] noted the analogy between classical statistical mechanics
and EQFT focusing on the analog of the Kirkwood-Salzburg equations.

The central development was due to Nelson [512, 513]. He understood that for
Bose QFT, EQFT is essentially an infinite dimensional path integral with the
extra bonus of Euclidean invariance. A key role was played by the extension of
the Feynman-Kac formula that Guerra-Rosen-Simon called the Feynman-Kac-Nelson
formula. This immediately implied a symmetry, later dubbed Nelson’s symmetry:

$\langle\Omega_{0},e^{-tH_{\ell}}\Omega_{0}\rangle=\langle\Omega_{0},e^{-\ell
H_{t}}\Omega_{0}\rangle$ (3.5)

where $H_{\ell}=H_{0}+\int_{0}^{\ell}:P(\varphi):(x)\,dx$ is the spatially
cutoff Hamiltonian and $H_{0}\Omega_{0}=0$. Nelson also realized the key
multidimensional Markov property which allowed one to go from Euclidean fields
to Minkowski fields (later Osterwalder-Schrader [521] found an alternate way
to do this, which, because it extended to Fermi fields and provided necessary
and sufficient conditions, supplanted this part of Nelson’s approach).

Nelson gave a few lectures on this new approach in Princeton early in 1971 and
lectured at a Berkeley summer school that summer attended by many experts on
CQFT (but I was not there). Even though this work eventually rapidly
revolutionized the subject, initially, it had little impact. I think part of
the reason this happened was that the language, especially as presented by
Nelson, was so foreign to the functional analysts working in the field, part
was that Nelson’s lectures seemed obscure and, most importantly, his original
work provided no new technical results in conventional CQFT. Indeed, the only
CQFT technical result was a new proof of a lower bound for $(\varphi^{4})_{2}$
theories

$E_{\ell}\equiv\inf\text{\rm{spec}}(H_{\ell})\geq-c\ell-d$ (3.6)

a result originally proven by Glimm-Jaffe [253]. Nelson’s proof was much
simpler than theirs but its impact was lessened by a simple proof that I found
(Simon [620]) shortly before Nelson.

The work that made Nelson’s theory take off was a remarkable note of Francesco
Guerra [287], then a postdoctoral visitor at Princeton. Guerra was out of town
when Nelson lectured but he got notes from Sergio Albeverio, then another
fellow postdoc. Guerra realized that (3.5) and

$E_{\ell}=\lim_{t\to\infty}-\frac{1}{t}\log\langle\Omega_{0},e^{-tH_{\ell}}\Omega_{0}\rangle$
(3.7)

provided tools to study $E_{\ell}$ and $\Omega_{\ell}$, the vector with
$H_{\ell}\Omega_{\ell}=E_{\ell}\Omega_{\ell}$ (for example, these two
equations immediately imply that $\ell\mapsto E_{\ell}$ is concave). He proved
that $E_{\ell}/\ell$ had a limit, $\alpha_{\infty}$, and that
$|\langle\Omega_{\ell},\Omega_{0}\rangle|=\text{O}(\ell^{-k})$ for all $k$.
This was way beyond anything obtained via the purely operator theory used
previously in CQFT.

Indeed, I have a vivid memory of how I first learned of these results. Guerra
had been visiting Princeton at that point for about 18 months. He was very
quiet - I’d probably exchanged only a few words with him and he’d given no
talks. Wightman told me that Guerra had asked Wightman to set up a meeting
with Lon Rosen (another postdoc and a student of Glimm with several
significant CQFT results) and me and we met in Wightman’s office in early
January, 1972. Guerra began by writing three facts that he was going to prove.
Lon and I later compared notes and we had the same thought “yeah, sure, you’re
going to do that”. These went so far beyond what was known that it was
literally unbelievable. He began by writing (3.5) on the blackboard which we’d
seen since it was part of Nelson proof of (3.6) and ten minutes later, he’d
proven the three facts. We were shell shocked!

After Guerra told us of these results, Lon, Francesco and I began working
together on exploiting these ideas (our work went through two phases - first
we mainly exploited consequences of (3.5) and similar results but later we
fully embraced the Euclidean viewpoint). In short order, we found [289, 290]
improvements on what Guerra had found: first
$E_{\ell}=-\alpha_{\infty}\ell-\beta_{\infty}+\text{o}(1)$ as $\ell\to\infty$
and secondly, for some $c,d>0$ and $\ell\geq 1$, one has that
$e^{-c\ell}\leq|\langle\Omega_{\ell},\Omega_{0}\rangle|\leq e^{-d\ell}$ (see
Lenard-Newman [452] for further developments on these subjects). Moreover, we
found a new and much simpler proof of some bounds of Glimm-Jaffe [254] that
allow one to show that limit points of the cutoff Wightman functions (as the
spatial cutoff in $H_{\ell}$ is removed) are tempered distributions.

The above mentioned work of Guerra and GRS got the attention of experts in
CQFT and virtually all papers in the subject after early 1972 used the EQFT
framework. I recall that a few weeks after GRS started working together, Glimm
came to Princeton to talk about the bounds in [254] and spent the hour seminar
sketching their subtle proof. Afterwards, Francesco, Lon and I waylaid him and
explained in 10 minutes the short proof we had found using an extended version
of Nelson’s symmetry. It was Glimm’s chance to be shell shocked!

The further introduction of techniques from rigorous statistical mechanics
and, in particular, the use of correlation inequalities, the major
accomplishment highlighted in this section, were introduced in two papers, one
by Guerra-Rosen-Simon [292] and one by Griffiths-Simon [283]. GRS [292] was a
long paper, so long that the Annals of Mathematics broke it into two parts so
it spread between two issues. Among other things, it provided a detailed
exposition of EQFT so that it and my book on the $P(\varphi)_{2}$ theory
(Simon [627], based on lectures I gave at the ETH) served as the standard
references on the subject for a time.

The most important set of ideas in GRS [292] involve the lattice
approximation. Our work was announced [291] a year earlier than Wilson’s work
[769] on lattice QCD, which of course went much further by allowing Fermion
and Gauge fields albeit without mathematical rigor. (It appears from Wilson’s
historical note [770] that he didn’t start to think about lattice
approximations to EQFT until early 1974 while we were already working on it in
the spring of 1972; that said, there is no reason to think that Wilson knew of
our work in 1974 or even in 2005!). The free EQFT is a Gaussian random field
with covariance $(-\Delta+m^{2})^{-1}$. One gets the lattice approximation by
replacing $-\Delta$ by a finite difference operator. Since it is the inverse
of the covariance that appears in the exponent of the Gaussian field, the free
lattice field is formally

$Z^{-1}\exp\left(-\tfrac{1}{2}\sum_{|i-j|=1}(s_{i}-s_{j})^{2}\right)\displaystyle\prod_{j\in{\mathbb{Z}}^{2}}e^{-m^{2}s_{j}^{2}}\,ds_{j}$
(3.8)

which is an Ising type ferromagnet with nearest neighbor interactions and
spins lying in ${\mathbb{R}}$ (rather than just $\pm 1$ with single site
distribution $e^{-as^{2}}\,ds$. While (3.8) is a formal infinite product, if
one puts it in a finite box, the spins lie in ${\mathbb{R}}^{k}$ and the
product is a simple finite measure. An analysis of the interaction just
changes $e^{-as^{2}}$ to $e^{-Q(s)}$ for a suitable semibounded even
polynomial.

One powerful tool in the statistical mechanics of spin systems is correlation
inequalities, a method initiated by Griffiths [277, 278, 279] whose
inequalities were extended by Kelly-Sherman [394] (hence GKS inequalities). A
different set of inequalities are due to Fortuin, Kasteleyn and Ginibre [197]
(hence FKG inequality). Relevant to EQFT are versions tailor made for spins
with continuous values due to Ginibre [250] for GKS and Cartier [100] for FKG.
With these extensions, GRS [292] obtained GKS and FKG inequalities for
Euclidean $P(\varphi)_{2}$ theories.

The most important application of these correlation inequalities (namely of
GKS) is to show monotonicity in volume of the so-called half-Dirichlet
Schwinger functions, a suggestion of Nelson [514], exploited by GRS [292] to
obtain $P(\varphi)_{2}$ quantum fields obeying all the Wightman axioms except
perhaps uniqueness of the vacuum (for this last axiom, see below). It should
be mentioned that the earliest construction of $P(\varphi)_{2}$ theories
(indeed the first construction of non-trivial examples of theories obeying all
the Wightman axioms, albeit in 2 space-time dimensions), using cluster
expansions, was by Glimm-Jaffe-Spencer [256] for $\lambda P(\varphi)_{2}$
theories with small $\lambda$ and, then, by Spencer [730] for
$P(\varphi)_{2}+\mu\varphi$ with $|\mu|$ large, The Nelson-GRS work (for
$P(X)=Q(X)+\mu X$ with $Q$ even) was the first results without restrictions on
coupling constant.

The second application that I mention is results by Simon [623] who, following
work of Lebowitz [448] on spin systems, used the FKG inequalities to show that
decay of the truncated two point function dominates the decay of all the
truncated vacuum expectation values. This means to prove uniqueness of the
vacuum (respectively, existence of a mass gap), it is enough to prove that as
$x-y\to\infty$, one has that
$\langle\varphi(x)\varphi(y)\rangle-\langle\varphi(x)\rangle\langle\varphi(y)\rangle$
goes to zero (respectively goes to zero exponentially).

While the work of Ginibre and Cartier nicely proves GKS and FKG inequalities
for fairly general single spin distributions, there are other results for $\pm
1$ spins that don’t extend so generally. In this regard, Griffiths [280]
introduced a simple but powerful tool. Consider two $\pm 1$ spins, $s_{1}$ and
$s_{2}$ and let $t=\tfrac{1}{2}(s_{1}+s_{2})$. Then $t$ takes values $0,\pm 1$
just like a spin $1$ spin but if $s_{1}$ and $s_{2}$ are uncoupled, the
weights are $\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{4}$ rather than equal
weights. If we find a coupling with energy $H$ so that the Gibbs weight
$e^{-H}$ has the values $2,1,2$, then the adjusted weights are all equal. We
thus pick $H=-(\log\,2)t^{2}$ which is ferromagnetic. Thus any correlation
inequality that holds for ferromagnetically coupled spin $1/2$ spins extends
to spin $1$ spins. Using this idea and a second order deMoivre-Laplace limit
theorem, Griffiths-Simon [282, 283] realized a lattice system with real valued
spins with a weight $\exp(-as^{4}-bs^{2})\,ds\quad(a>0,b\in{\mathbb{R}})$ as a
limit of scaled spin $1/2$ ferromagnetic chains.. This allowed one to obtain
GHS (after Griffiths-Hurst-Sherman [281]) and Lebowitz [449] inequalities and
a Lee-Yang [450] theorem for $P(\varphi)$ theories with
$P(X)=aX^{4}+bX^{2}+\mu X,\,a,\mu\geq 0,b\in{\mathbb{R}}$. This in turn can be
used to prove that when $\mu>0$ such theories have a unique vacuum (Simon
[631]) and even a mass gap (GRS [293]). It is known (see Newman [517]) that if
the polynomial $P$ is of (even) degree larger than $4$ then
$\exp(-P(x)dx)\,dx$ may not be approximated by ferromagnetic arrays of $\pm 1$
spins, so the Griffiths-Simon result is restricted to $\varphi^{4}$ theories.

I should remark that correlation inequalities are useful in the study of
Schrödinger operators on ${\mathbb{R}}^{\nu}$. For example, it is known [632,
179], using GHS inequalities, that if $V(x)$ is an even function on
${\mathbb{R}}$ with $V^{\prime\prime\prime}(x)\geq 0$ for $x>0$ and if
$E_{1}<E_{2}<E_{3}$ are the first three eigenvalues of $-d^{2}/dx^{2}+V(x)$,
then $E_{3}-E_{2}\geq E_{2}-E_{1}$. And, in Section 7, we’ll discuss
applications of FKG inequalities to Schrödinger operators in magnetic fields.

While I only worked on CQFT in two space-time dimensions, there is some deep
work by others on the three dimensional case. This, as well as work by others
on two dimensions, is presented in the book by Glimm-Jaffe [255].

I should close the discussion of my work in CQFT by mentioning a paper with
Fröhlich [214] that, among other things, constructs $P(\varphi)_{2}$ theories
obeying all the Wightman axioms for any semibounded polynomial $P$. It relies
on Spencer’s large $\mu$ expansion [730] and FKG inequalities.

##  4\. Thomas–Fermi Theory

In 1972-73, Elliott Lieb and I found results on the Thomas-Fermi (TF) theory
that we announced in 1973 [465] with full details only published in 1977 [466]
due, in part, to a long journal backlog. We first of all established existence
and uniqueness of solutions to the TF equations for neutral (and positive
ionic) atoms and molecules and, more importantly, proved that TF theory was an
exact approximation to quantum theory in suitable $Z\to\infty$ limits. Since
then, an entire industry has been spawned from this work.

TF theory goes back to Thomas [747] and Fermi [193] in 1927 near the dawn of
quantum mechanics as an approximation expected to be valid in regions of high
electron density. Interestingly enough, it approximated a linear equation in
$3N$ variables as $N\to\infty$ by a non-linear equation in $3$ variables! They
originally found their basic equation using a Fermi surface heuristic argument
but we relied on the 1932 approach of Lenz [454] who used energy functionals
giving birth to the density functional method of atomic and molecular physics
that has become such a standard that the 1998 Nobel Prize in Chemistry was
awarded to Walter Kohn “for his development of the density-functional theory”.

In units where

$\hbar^{2}\left(\tfrac{3}{2}\right)^{2/3}(2m)^{-1}=1$ (4.1)

(where $m$ is the electron mass and we assume the electron has $2$ spin states
- state counting is important because one assumes Fermi statistics), the Lenz
functional is

${\mathcal{E}}(\rho;V)=\frac{3}{5}\int\rho^{5/3}(x)\,d^{3}x+\frac{1}{2}\int\frac{\rho(x)\rho(y)}{|x-y|}\,d^{3}xd^{3}y-\int\rho(x)V(x)\,d^{3}x$
(4.2)

Here $\rho(x)$ is the electron density, so, if there are $N$ electrons, we
have that

$\int\rho(x)\,d^{3}x=N$ (4.3)

and $V(x)$ is the one electron potential; for a molecule with nuclear charges
$z_{1},\dots,z_{k}$ at distinct points $R_{1},\dots,R_{k}$, we have that

$V(x)=\sum_{j=1}^{k}\frac{z_{j}}{|x-R_{j}|}$ (4.4)

We set

$Z=\sum_{j=1}^{k}z_{j}$ (4.5)

The last term in (4.2) is just the interaction of the electrons with the
nuclei and is exact, not an approximation. The second term is an electron
repulsion and assumes no electron correlation so that the two point density is

$\rho_{2}(x,y)=\rho(x)\rho(y)$ (4.6)

which cannot be even approximately true unless $N$ is large, since
$\int\rho_{2}(x,y)\,d^{3}xd^{3}y=N(N-1)$, while, by (4.3),
$\int\rho(x)\rho(y)\,d^{3}xd^{3}y=N^{2}$. The first term relies on a quasi-
classical calculation. If one has $N$ particles in a box, $\Omega$, of size
$|\Omega|$ with $\rho=N/|\Omega|$ and fills phase space by putting particles
in $\\{p\,|\,|p|\leq p_{F}\\}$, then
$N=\tfrac{4\pi}{3}p_{F}^{3}|\Omega|/2h^{3}$ ($2$ in the denominator from $2$
spin states) by the rule that each particle takes volume $h^{3}$ in phase
space. The total energy of this is then $C|\Omega|\rho^{5/3}$ with an explicit
$C$ which explains where the first term in (4.2) comes from. The choice (4.1)
leads to $C=3/5$. Of course, the notion of states taking $h^{3}$ in phase
space is an approximation justified in a large $N$ limit by Weyl’s celebrated
eigenvalue counting result (see later and [711, Section 7.5] for exposition
and references).

The Euler-Lagrange equation with Lagrange multiplier to take the condition
(4.3) into account with the restriction $\rho(x)\geq 0$ is that there is
$\varphi_{0}\geq 0$ so that with

$\displaystyle\varphi(x)$
$\displaystyle=V(x)-\int\frac{\rho(x)}{|x-y|}\,d^{3}y$ (4.7)
$\displaystyle\rho^{2/3}(x)$
$\displaystyle=\left\\{\begin{array}[]{ll}\varphi(x)-\varphi_{0},&\hbox{ if
}\varphi(x)\geq\varphi_{0}\\\ 0,&\hbox{ if
}\varphi(x)\leq\varphi_{0}\end{array}\right.$ (4.10)

This is the Thomas-Fermi integral equation which implies the Thomas Fermi PDE

$\Delta\varphi=\left[\max(\varphi-\varphi_{0},0)\right]^{3/2}$ (4.11)

One result that Lieb and I proved [465, 466] is the following:

######  Theorem 4.1.

Let $V$ be given by (4.4) and $N,Z$ given by (4.3)/(4.5). Then
${\mathcal{E}}(\rho;V)$ is well defined if $\rho\geq 0$ lies in $L^{1}\cap
L^{5/3}$. Moreover:

(a) If $N\leq Z$, there is a unique minimizer of ${\mathcal{E}}(\rho;V)$ among
those $\rho$’s obeying (4.3).

(b) If $N>Z$, there is no minimizer of ${\mathcal{E}}(\rho;V)$ among $\rho$’s
obeying (4.3)

(c) If $N<Z$, the minimizing $\rho$ has compact support and obeys the TF
integral equation for some $\varphi_{0}>0$ and is real analytic on the open
set $\\{x\,|\,\varphi(x)>\varphi_{0};x\notin\\{R_{j}\\}_{j=1}^{k}\\}$.

(d) If $N=Z$, the minimizer minimizes ${\mathcal{E}}(\rho;V)$ on all $\rho\in
L^{1}\cap L^{5/3},\rho\geq 0$ without any condition (4.3). This minimizing
$\rho$ obeys the TF integral equation with $\varphi_{0}=0$. One has that, for
all $x$, $\varphi(x)>0$ and so $\rho(x)>0$ on all of ${\mathbb{R}}^{3}$.
$\varphi$ is real analytic on ${\mathbb{R}}^{3}\setminus\\{R_{j}\\}_{j=1}^{k}$
and

$\rho(x)\sim 1728|x|^{-6}$ (4.12)

as $x\to\infty$.

######  Remarks.

1\. The only prior results on existence were for the neutral atomic case
($k=1,R_{1}=0,\varphi_{0}=0$) where one looks for spherically symmetric
solutions of (4.11). Since, if $\varphi$ is spherically symmetric, one has
that $\Delta\varphi=r^{-1}(r\varphi)^{\prime\prime}$, we see that if
$Y(r)=r\varphi(r\omega)$, then (4.11) when $r\neq 0$ is equivalent to

$Y^{\prime\prime}(r)=r^{-1/2}Y(r)^{3/2}$ (4.13)

which goes back to the work of Thomas and Fermi. Thomas noticed that
$Y(x)=144x^{-3}$ solves (4.13) (which leads to $\varphi=144x^{-4}$ and
$\rho=\varphi^{3/2}=1728|x|^{-6}$). In 1929, already, Mambriani [483] proved
existence and uniqueness of solutions of (4.13) with $\lim_{r\downarrow
0}Y(x)=a$ and $\lim_{x\to\infty}Y(x)=0$; see Hille [315] for further work. But
Lieb-Simon had the first results on existence and uniqueness going beyond the
spherically symmetric case. We note that uniqueness of spherically symmetric
solutions of the PDE doesn’t prove that the minimizer for ${\mathcal{E}}$ is
spherically symmetric nor that the minimizer is unique.

2\. Sommerfeld [727] suggested that the singular solution $144x^{-4}$ should
control general asymptotics of the TF PDE and that was proven by Hille [315]
in the spherically symmetric case and by Lieb-Simon [466] in the non-central
case. We used subharmonic comparison ideas, a technique we learned from Teller
[746], who used it in a different context.

3\. Uniqueness of minima follows from strict convexity of ${\mathcal{E}}$,
i.e.

$0<\theta<1,\,\rho_{1}\neq\rho_{2}\Rightarrow{\mathcal{E}}(\theta\rho_{1}+(1-\theta)\rho_{2})<\theta{\mathcal{E}}(\rho_{1})+(1-\theta){\mathcal{E}}(\rho_{2})$

since one term in ${\mathcal{E}}$ is linear and the other two are strictly
convex.

4\. Existence uses what has come to be called the direct method of the
calculus of variations (see, for example Dacorogna [126]). Namely, one looks
at $\\{\rho\,|\,\rho\in L^{1}\cap L^{5/3},\,\rho>0,\,\int\rho dx\leq N\\}$
which is compact in a suitable weak topology (if $\leq N$ is replaced by $=N$,
it is not weakly closed, so not compact) and one proves that
$\rho\mapsto{\mathcal{E}}(\rho)$ is weakly lower semicontinuous. A potential
theory argument shows that if $N\leq Z$, the minimizer has $\int\rho=N$ but if
$N>Z$, the minimizer obeys $\int\rho=Z$. These weak compactness, lsc ideas are
now standard analysis but, at the time, while they were used in some areas,
they were not widely known in mathematical physics. While Lieb eventually
became a world expert in subtle extensions of this method, he learned the
necessary functional analysis from me at the time of our work.

With the existence out of the way, we turned to figuring out the connection to
atomic physics. In this regard, there are several reasons that the TF theory
might not have anything to do with true quantum systems. As we saw, in the
neutral case, $\rho_{TF}$ decays as $x\to\infty$ as $C|x|^{-6}$ but true
atomic bound states decay exponentially (see Section 6 below; O’Connor’s work
was done before Lieb and I were working). As we’ll see, scaling shows that in
the atomic case the TF density shrinks as $Z$ grows (at a $Z^{-1/3}$ rate),
while true atoms expand in extent (although it might be that atomic radii,
defined as where $Z-1$ electron live, might be bounded as $Z\to\infty$, they
certainly don’t shrink). Finally, it is a result of Teller [746] that
molecules don’t bind in TF theory while they do exist in nature. For technical
reasons, Teller had a short distance cutoff in the Coulomb potential in his
argument leading some people to question whether his result held in TF theory
without cutoffs, but, it does, as Lieb and I showed. Interestingly enough,
this apparent negative result in TF theory was a key, several years later, in
the elegant Lieb-Thirring proof [471] of the stability of matter!

Early in our work, Lieb understood why none of these issues were problems in
connecting TF theory to quantum mechanical atoms. TF theory describes the
cores of atoms while chemistry involves the outermost electrons so it isn’t
surprising that molecules don’t bind in TF theory - it is an expression of the
repulsion of the cores. The $|x|^{-6}$ Sommerfeld asymptotics describes the
mantle of the core while exponential decay describes the last few electrons. I
still remember the start of our collaboration while we were both visitors at
IHES in the fall of 1972. Lieb had the idea that Weyl type estimates should
show that TF was a proper semiclassical limit of atoms. At the end of a long
day of discussing this idea, I told him of Teller’s result which I’d learned
about in a course taught by Wightman, so since TF theory didn’t bind atoms, it
couldn’t describe physics. The next morning Lieb walked in and said to me:
“Mr. Dalton’s hooks are in the outer shell.” In other words, chemistry had
nothing to with region in which a leading quasi-classical limit is valid. (The
notion behind density functional theory is that chemistry can be connected to
non-leading terms).

One key to the large $Z$ results is scaling. The following is easy to check.
If $Z>0$ and

$V_{Z}(x)=Z^{4/3}V(Z^{1/3}x);\qquad\rho_{Z}(x)=Z^{2}\rho(Z^{1/3}x)$ (4.14)

then

${\mathcal{E}}(\rho_{Z};V_{Z})=Z^{7/3}{\mathcal{E}}(\rho;V);\qquad\int\rho_{Z}(x)\,d^{3}x=Z\int\rho(x)\,d^{3}x$
(4.15)

In particular if

$E_{V}(Z;N)=\inf\\{{\mathcal{E}}(\rho;V_{Z})\,|\,\rho\in L^{1}\cap
L^{5/3},\,\rho\geq 0\,\int\rho(x)\,d^{3}x=N\\}$ (4.16)

then

$E_{V}(Z;nZ)=Z^{7/3}E_{V}(1;n)$ (4.17)

Given $z_{1},\dots,z_{k},R_{1},\dots,R_{k}$, we let
$E_{TF}(N;z_{1},\dots,z_{k};R_{1},\dots,R_{k})$ be the TF energy (i.e. minimum
of ${\mathcal{E}}(\rho;V)$ with $V$ given by (4.4) over $\rho$’s obeying
(4.3)). Then (4.17) says that

$\displaystyle E_{TF}(nZ;z_{1}Z,\dots,z_{k}Z;$ $\displaystyle
Z^{-1/3}R_{1},\dots,Z^{-1/3}R_{k})$
$\displaystyle=Z^{7/3}E_{TF}(n;z_{1},\dots,z_{k};R_{1},\dots,R_{k})$ (4.18)

We next describe quantum atomic energies. Let ${\mathcal{H}}_{phys}$ be those
elements in $L^{2}({\mathbb{R}}^{3N};{\mathbb{C}}^{2N})$ which are functions
of $N$ points $x_{1},\dots,x_{N}$ in ${\mathbb{R}}^{3}$ and spins
$\sigma_{1},\dots,\sigma_{N}$ in ${\mathbb{C}}^{2}$ which are antisymmetric
under permutations of $(x_{j},\sigma_{j})$ (see [711, Section 7.9] for more on
this formalism). On ${\mathcal{H}}_{phys}$ let

$H=-\sum_{j=1}^{N}\frac{\hbar^{2}}{2m}\Delta_{j}+\sum_{i<j}\frac{1}{|x_{i}-x_{j}|}-\sum_{j=1}^{N}\sum_{\ell=1}^{k}\frac{z_{\ell}}{|x_{j}-R_{\ell}|}$
(4.19)

where $\hbar$ is given by (4.1). We set

$E_{Q}(N;z_{1},\dots,z_{k};R_{1},\dots,R_{k})=\inf_{\varphi\in{\mathcal{H}}_{phys}}\langle\varphi,H\varphi\rangle$
(4.20)

If $N\geq Z\equiv\sum_{j=1}^{k}z_{j}$, it is known (see [775, 615]) that $H$
has a discrete ground state, $\psi$; we set

$\displaystyle\rho_{Q}(x;$ $\displaystyle
z_{1},\dots,z_{k};R_{1},\dots,R_{k})$ $\displaystyle=\sum_{\sigma_{j}=\pm
1;j=1,\dots,N}\int|\psi(x,x_{2},\dots,x_{N};\sigma_{1},\dots,\sigma_{N})|^{2}\,d^{3}x_{2}\dots
d^{3}x_{N}$ (4.21)

(the ground state can be degenerate in which case in the theorem below one can
take any ground state eigenfunction).

The main result of Lieb-Simon [466] is

######  Theorem 4.2.

For any distinct $R_{1},\dots,R_{k}$ and positive $z_{1},\dots,z_{k}$, and
$n>0$, we have that

$\displaystyle\lim_{Z\to\infty}Z^{-7/3}E_{Q}(nZ;$ $\displaystyle
z_{1}Z,\dots,z_{k}Z;Z^{-1/3}R_{1},\dots,Z^{-1/3}R_{k})$
$\displaystyle=E_{TF}(n;z_{1},\dots,z_{k};R_{1},\dots,R_{k})$ (4.22)

Moreover, if $n\leq\sum_{j=1}^{k}z_{j}$, then

$\displaystyle\lim_{Z\to\infty}nZ^{-2}\rho_{Q}(Z^{-1/3}x;$ $\displaystyle
z_{1}Z,\dots,z_{k}Z;Z^{-1/3}R_{1},\dots,Z^{-1/3}R_{k})$
$\displaystyle=\rho_{TF}(x;n;z_{1},\dots,z_{k};R_{1},\dots,R_{k})$ (4.23)

in the sense of convergence of the integral over $x$ in any fixed open set.

Our proof of (4.22) uses the method of Dirichlet-Neumann bracketing. This goes
back to Weyl [765] as formalized by Courant-Hilbert [119] (see [16] for the
discrete analog and [711, Section 7.5] for another textbook discussion). They
used it to count eigenvalues of the Laplacian in regions with smooth boundary.
It was later used to prove that when $V\in
C_{0}^{\infty}({\mathbb{R}}^{\nu})$, then as $\lambda\to\infty$, one has that
$N(\lambda V)$, the number of negative eigenvalues of $-\Delta+\lambda V$ on
$L^{2}({\mathbb{R}}^{\nu})$ obeys

$\lim_{\lambda\to\infty}\lambda^{-\nu/2}N(\lambda
V)=(2\pi)^{-\nu}\tau_{\nu}\int\max(-V(x),0)^{\nu/2}\,d^{\nu}x$ (4.24)

(where $\tau_{\nu}$ is the volume of the unit ball in ${\mathbb{R}}^{\nu}$).
This was discovered independently about the same time by Birman-Borzov [75],
Martin [486], Robinson [570] and Tamura [742]. (Lieb and I only knew of
Martin’s work when we looked at Thomas-Fermi, although all but Tamura existed
at the time.) In Section 8, I’ll discuss what happens if $V$ is not
$C_{0}^{\infty}({\mathbb{R}}^{\nu})$.

Using these ideas of dividing space into small boxes, it wasn’t hard to show
that $E_{Q}(V_{Z})/E_{TF}(V_{Z})\to 1$ as $Z\to\infty$ if $V\in
C_{0}^{\infty}({\mathbb{R}}^{3})$. These methods don’t deal with the boxes
around the nuclei at $R_{j}$. Basically, one needs to show that the system
doesn’t collapse on those points, i.e. most of the electrons wind up in the
boxes containing those points. When I left IHES for Marseille at the end of
1972, Lieb and I were at this point and were left with the problem we called
between ourselves “pulling the poison Coulomb tooth”. I spent a long weekend
in Paris in March, 1973 and we figured out how to pull the tooth. With current
technology, one would use Lieb-Thirring inequalities (discussed in Section 8
below; see also [471, 472] for the original papers, [332] for the discrete
case and [710, Section 6.7] for a textbook discussion) but they didn’t exist,
so we used an ad hoc argument exploiting the angular momentum barrier.

The proof of (4.23) isn’t hard. The $\rho$’s are functional derivatives of the
energy under adding an infinitesimal $V_{Z}$ to the Coulomb attraction.
Normally convergence of functions doesn’t imply convergence of derivatives but
it does for concave functions and one can show the energies, as minima of a
set of functions linear in $\lambda$, are concave under $\lambda\to\lambda V$.

I have one other result on large Z ions. As noted above, it is known (see
[775, 615]) that the Hamiltonian, $H(Z,N)$ for a charge $Z$ nucleus and $N$
electrons has infinitely many bound states if $N\leq Z$. What happens if
$N>Z$? It is a result of Ruskai [581] and Sigal [600, 601] that there is a
finite number $N(Z)$, so that $H(Z,N(Z))$ has no discrete spectrum (i.e. there
is a negative ion with nuclear charge $Z$ and total charge $-(N(Z)-Z)$) and so
that for all $N>N(Z)$, we have that $N(Z)$ has no bound states below the
continuum. In [464], Lieb, Sigal, Thirring and I showed that $N(Z)/Z\to 1$ as
$Z\to\infty$. That this is especially subtle is seen by the fact that if one
replace fermionic electrons by bosons (with negative charge), then Benguria-
Lieb [68] have shown that the analogous $\liminf$ is strictly bigger than $1$.
I note that there are no twice negatively charged ions known in nature so it
is possible that $N(Z)$ is always either $Z$ or $Z+1$. In fact, one of the
fifteen open problems in my 2000 list (Simon [690] of which 11 remain open) is
to prove that $N(Z)-Z$ remains bounded as $Z\to\infty$.

Since 1973, there has been a huge literature on large $Z$ atoms and molecules
and on density functional theory. I will not attempt a comprehensive review,
but I should mention the work on non-leading asymptotics beyond (4.22) and
(4.23). For the energy, Hughes [328] and Siedentop-Weikard [598, 599] obtained
the O$(Z^{2})$ term for atoms and Ivrii-Sigal [344] for molecules. Later,
Fefferman-Seco [192] obtained the O$(Z^{5/3})$ term. For the density,
Iantchenko-Lieb-Siedentop [339] found the O$(Z^{2})$ term (recently, I was
coauthor of a paper [201] that proved an analog for a relativistic
Hamiltonian). Since much of the work on higher order corrections to (4.2) was
done by Lieb and collaborators, I refer the reader to the relevant volume of
Lieb’s Selecta [463] for references. The reader can also look at two somewhat
dated review articles by Lieb [462] and Hundertmark [329].

Before leaving this subject, I should mention that Lieb and I [468] used
methods similar to those we used to prove existence of solutions to the TF
equation to prove existence of solutions to the Hartree and Hartree-Fock
equations for neutral (and positive) atoms and molecules. Later works on
existence of solutions of Hartree-Fock include Lions [474] and Lewin [456].

##  5\. Infrared Bounds and Continuous Symmetry Breaking

A fundamental problem in statistical physics concerns the following. The Gibbs
states of statistical mechanics are clearly analytic in all parameters, yet
nature is full of discontinuities, for example the direction of a magnet as a
magnetic field is slowly varied through zero field. We now realize that the
way to understand this is by looking at the thermodynamic limit, i.e. infinite
volume, where states can become non-analytic in parameters. That this is far
from evident can be seen by a story told in Pais’ book [523, pp 432-433] that
as late as 1937, at the van der Waals Centenary conference, a vote of the
physicists present was taken on whether this view was correct and the vote was
close (although, given that Peierls’ work mentioned below was in 1936, it
shouldn’t have been close; that means that Peierls’ work was not well known,
or at least not understood, at the time).

The simplest models on which this can be explored are the lattice gases whose
formalism is described in the books of Ruelle [580] and Simon [678]. Two of
the simplest examples both have spins on a lattice, say ${\mathbb{Z}}^{\nu}$,
$\boldsymbol{\sigma}_{\alpha}$, at points $\alpha\in{\mathbb{Z}}^{\nu}$. In a
finite box $\Lambda\subset{\mathbb{Z}}^{\nu}$, the Hamiltonian (energy
functional) is

$H_{\Lambda}=-J\sum_{\alpha,\gamma\in\Lambda,\,|\alpha-\gamma|=1}\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{\gamma}$
(5.1)

The sign is there so that when $J>0$, energies are lowest when spins are
parallel, i.e. the model is ferromagnetic. $J<0$ describes the
antiferromagnet. We will normally take $J=\pm 1$ which is no loss since we can
vary the inverse temperature, $\beta$ in $e^{-\beta H}$ defining the Gibbs
measure. We have not been careful about boundary conditions; we will most
often take periodic BC although sometimes free or plus BC.

I said two models because I haven’t described the set of single spins and
their distributions. If each $\boldsymbol{\sigma}_{\alpha}=\pm 1$ with equal
apriori weights, we have the nearest neighbor _Ising model_. If instead each
$\boldsymbol{\sigma}_{\alpha}\in S^{2}$, the unit sphere is ${\mathbb{R}}^{3}$
with the rotation invariant apriori weight, we have the _classical Heisenberg
model_. More generally if each $\boldsymbol{\sigma}_{\alpha}\in S^{d-1}$, the
unit sphere is ${\mathbb{R}}^{d}$, we have the _$d$ -rotor model_. Significant
here is the global symmetry of the system: discrete,
$\boldsymbol{\sigma}_{\alpha}\to-\boldsymbol{\sigma}_{\alpha}$ for the Ising
model and continuous, $\boldsymbol{\sigma}_{\alpha}\to
R\boldsymbol{\sigma}_{\alpha}$ with $R\in SO(3)$, the rotation group for the
Heisenberg model.

In 1936, Peierls [534] found a simple argument proving that when $\nu\geq 2$,
the nearest neighbor Ising model on ${\mathbb{Z}}^{\nu}$ has a phase
transition at low temperature. However, the argument depends on the sharp
difference between spin up and spin down and fails for the classical
Heisenberg model where the spins vary continuously. Indeed, in 1966, Mermin-
Wagner [495] (Hohenberg [321] had asimilar result on a related model in the
same time frame) proved that in $2D$, the classical Heisenberg model has no
broken symmetry states at positive temperature (see also [412]). Their
argument relies on the fact that if the spin wave energy (when $J=1$) is given
by

$E_{p}=\frac{1}{2}\sum_{|\alpha|=1}(1-e^{ip\cdot\alpha})=\sum_{j=1}^{\nu}(1-\cos(p_{j}))$
(5.2)

(the Fourier transform of the nearest neighbor coupling with a constant added
so that $E_{p}\geq 0$ with minimum value $0$), then $E_{p}\sim p^{2}$ for $p$
small so that

$\int_{|p_{j}|<\pi}\frac{d^{\nu}p}{E_{p}}=\infty\text{ if }\nu=2$ (5.3)

In 1976, Jürg Fröhlich, Tom Spencer and I (henceforth FSS) [215, 216] proved
that

######  Theorem 5.1 ([215, 216]).

The classical $d$-vector model ($d\geq 1)$ with nearest neighbor interactions
on ${\mathbb{Z}}^{\nu}$ with $\nu\geq 3$ has multiple phases (with broken
symmetry) if $\beta\geq\beta_{c}$ where

$\beta_{c}\leq\beta_{c}^{FSS}\equiv\frac{d}{2}I(\nu)$ (5.4)

with

$I(\nu)=\frac{1}{(2\pi)^{\nu}}\int_{|p_{j}|<\pi}\frac{d^{\nu}p}{E_{p}}$ (5.5)

so $I(\nu)<\infty$ when $\nu\geq 3$

######  Remark.

While this was the first result on the existence of phase transitions in the
isotropic model, there were earlier results on the (in the quantum case,
highly) anisotropic case for the classical (see Malyshev [480] and Kunz et al.
[425]) and quantum (see Ginibre [249] and Robinson [569]) Heisenberg models.
These works used variants of the Peierls method. Of course, only the isotropic
model has a continuous symmetry to break.

$I(3)$ can be computed exactly in terms of elliptic integrals so one finds
(with “errors” computed by comparing with high temperature expansions in
parentheses)

$T_{c}(\nu=3,d=3)\geq 1.3189\,(1.44;\,9\%\text{ error})$ (5.6)

and

$T_{c}(\nu=3,d=1)\geq 3.9567\,(4.5108;\,14\%\text{ error})$ (5.7)

The method of FSS which I sketch below is basically the only method known for
rigorously proving spontaneous continuous symmetry breaking with a nonabelian
symmetry group. We note that such continuous symmetry breaking is not only
central to statistical mechanics but also to models of particle physics.

A basic notion is reflection positivity. This is one of the Osterwalder
Schrader axioms mentioned in Section 3. FSS realized it also played an
important role in statistical mechanical models.

Consider spins in a box, $\Lambda$, with even sides with periodic boundary
conditions and slice the box across bonds into two halves, $\Lambda_{+}$ and
$\Lambda_{-}$. There is a natural reflection $\Theta$ of spins in
$\Lambda_{+}$ onto spins in $\Lambda_{-}$ that extends to a map of polynomials
in the spins. A state $\langle\cdot\rangle$ is called _reflection positive_
(RP) if and only if for any polynomial, $A$, in the spins of $\Lambda_{+}$ we
have that

$\langle\Theta(A)A\rangle\,\geq 0$ (5.8)

First, suppose that we consider uncoupled spins (i.e. a product measure over
sites). Then $\langle\Theta(A)A\rangle=|\langle A\rangle|^{2}\geq 0$ so the
measure is RP. Now suppose we have a Hamiltonian of the form

$-H=A+\Theta(A)+\sum_{j}\Theta(B_{j})B_{j}$ (5.9)

We claim that if $\langle\cdot\rangle_{0}$ is RP, so is
$\langle\cdot\rangle=\langle\cdot e^{-H}\rangle_{0}/\langle e^{-H}\rangle_{0}$
for $\exp(A+\Theta(A))=\exp(A)\Theta(\exp(A))$ and we can expand
$\exp(\sum_{j}\Theta(B_{j})B_{j})$ into a Taylor series.

Consider a box, $\Lambda$, with periodic boundary conditions and state
$\langle\cdot\rangle_{\Lambda}$. Define the magnetization (here and below we
notationally suppress a $\Lambda$ dependence).

$\mathbf{m}=\frac{1}{|\Lambda|}\sum_{\alpha\in\Lambda}\boldsymbol{\sigma}_{\alpha}$
(5.10)

Our sign that there is a phase transition will be that

$\liminf\langle\mathbf{m}^{2}\rangle_{\Lambda}\equiv M^{2}>0$ (5.11)

This implies many other notions of phase transition. For example, one can show
that the derivative of the free energy per unit volume with respect to an
external magnetic field has a discontinuity of at least $2M$. Also, there are
multiple equilibrium states in the sense of Dobrushin [163, 164] and Lanford-
Ruelle [435] (see [678, Section III.2]).

We let $\Lambda^{*}$ be the dual lattice to $\Lambda$ so that
$\\{\alpha\mapsto|\Lambda|^{-1/2}e^{i\mathbf{p}\cdot\boldsymbol{\alpha}}\\}_{\mathbf{p}\in\Lambda^{*}}$
is an orthonormal basis for the $\ell^{2}(\Lambda)$. Define the Fourier spin
wave variables

$\widehat{\boldsymbol{\sigma}}_{p}=\frac{1}{\sqrt{|\Lambda|}}\sum_{\alpha\in\Lambda}e^{-ip\cdot\alpha}\boldsymbol{\sigma}_{\alpha}$
(5.12)

and define the spin wave expectation function

$\displaystyle g_{\Lambda}(\mathbf{p})$
$\displaystyle=\langle\widehat{\boldsymbol{\sigma}}_{p}\cdot\widehat{\boldsymbol{\sigma}}_{-p}\rangle$
(5.13)
$\displaystyle=\frac{1}{|\Lambda|}\sum_{\alpha,\beta\in\Lambda}e^{-i\mathbf{p}\cdot(\boldsymbol{\alpha-\beta})}\langle\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{\beta}\rangle$
$\displaystyle=\sum_{\alpha}e^{-i\mathbf{p}\cdot\boldsymbol{\alpha}}\langle\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{0}\rangle$
(5.14)

Note that $\mathbf{m}=|\Lambda|^{-1/2}\widehat{\boldsymbol{\sigma}}_{p=0}$ so
that

$\langle\mathbf{m}^{2}\rangle_{\Lambda}=|\Lambda|^{-1}g_{\Lambda}(p=0)$ (5.15)

Since $\widehat{\boldsymbol{\sigma}}_{p}$ are components of the functions
$\alpha\mapsto\boldsymbol{\sigma}_{\alpha}$ in an ON basis, the Plancherel
theorem implies that

$\sum_{p\in\Lambda^{*}}g_{\Lambda}(p)=\sum_{\alpha}\langle|\boldsymbol{\sigma}_{\alpha}|^{2}\rangle=|\Lambda|$
(5.16)

Since there are $|\Lambda|$ values of $p$ in $\Lambda^{*}$, this says that
normally each $g_{\Lambda}(p)$ should be of size $1$ while the condition of
there being a phase transition is that $g_{\Lambda}(p=0)$ is of order
$\Lambda$. This allows one to interpret the phase transition as due to a Bose
condensation of spin waves.

The key to the proof will be what FSS dubbed an _infrared bound_ (IRB), that
for $p\neq 0$, one has that:

$g_{\Lambda}(p)\leq\frac{d}{2\beta E_{p}}$ (5.17)

where $E_{p}$ is the spin wave energy.

By (5.15)-(5.17)

$\liminf_{|\Lambda|\to\infty}\langle\mathbf{m}^{2}\rangle_{\Lambda}\geq
1-\frac{dI(\nu)}{2\beta}$ (5.18)

where $I(\nu)$ is given by (5.5) and we use the fact that $\Lambda^{*}$ fills
out the $\nu$-fold product of $[-\pi,\pi]$ as $|\Lambda|\to\infty$. Thus
infrared bounds imply Theorem 5.1.

The first step in the proof of infrared bounds from reflection positivity is
to use RP to prove something called Gaussian domination, namely if we define,
for arbitrary $\\{\mathbf{h}_{\alpha}\\}\in{\mathbb{R}}^{d|\Lambda|}$

$Z(\\{\mathbf{h}_{\alpha}\\})=\int_{S^{(d-1)|\Lambda|}}\exp\left(-\beta\frac{1}{2}\sum_{|\alpha-\gamma|=1}(\boldsymbol{\sigma_{\alpha}-\sigma_{\gamma}}-\mathbf{h_{\alpha}-h_{\beta}})^{2}\right)$
(5.19)

then one has that

$Z(\\{\mathbf{h}_{\alpha}\\})\leq Z(\\{\mathbf{h}_{\alpha}\\}\equiv 0)$ (5.20)

One first proves that if $\Lambda$ is split into two halves $\Lambda_{+}$ and
$\Lambda_{-}$ and, given $\mathbf{h}$, we let $\mathbf{h}_{+}$ be the
$\mathbf{H}$ obtaining by restricting $\mathbf{h}$ to $\Lambda_{+}$ and
reflecting it and similarly for $\mathbf{h}_{-}$, then

$Z(\\{\mathbf{h}\\})\leq
Z(\\{\mathbf{h}_{+}\\})^{1/2}\,Z(\\{\mathbf{h}_{-}\\})^{1/2}$ (5.21)

The details of the proof of (5.20) can be found in [216] or [212].

Once one has Gaussian domination, fix $\mathbf{h}$ all real and use the fact
that $Z(\\{\lambda\mathbf{h}\\})$ is maximized at $\lambda=0$ so the second
derivative is negative. This implies that

$\left\langle\sum_{|\alpha-\gamma|=1}|(\boldsymbol{\sigma}_{\alpha}-\boldsymbol{\sigma}_{\gamma})\cdot(\mathbf{h}_{\alpha}-\mathbf{h}_{\gamma})|^{2}\right\rangle_{\Lambda}\leq\frac{1}{2\beta}\sum_{|\alpha-\gamma|=1}|\mathbf{h}_{\alpha}-\mathbf{h}_{\gamma}|^{2}$
(5.22)

Adding the results for the real and imaginary part extends this inequality to
complex $\mathbf{h}$. Taking $\mathbf{h}$ to be a plane wave with a single
component (and summing over possible components) proves the infrared bound and
completes the proof that RP$\Rightarrow$Phase Transition.

A little about the history of this work with Fröhlich and Spencer. In October,
1975 I heard indirectly that Jürg and Tom had found that there was
spontaneously symmetry breaking in the multi-component $\Phi^{4}_{3}$ EQFT.
The analog of infrared bounds for this case was easy. There is a Källen-
Lehmann representation for the two point function

$\langle\boldsymbol{\varphi(x)}\cdot\boldsymbol{\varphi(x)}\rangle=\alpha+\int
d\rho(m)\int\frac{d^{3}k}{(2\pi)^{3}}e^{ik\cdot(x-y)}(k^{2}+m^{2})^{-1}$
(5.23)

where if $\boldsymbol{\varphi}$ has $N$ components, one has that

$\int d\rho(m)=N$ (5.24)

The infrared bound then just needs that $(k^{2}+m^{2})^{-1}\leq k^{-2}$. Since
I’d heard of this indirectly and they were looking at the field theory, I
felt, perhaps unfairly, that I could think about the statistical mechanical
analog. I realized that the key was (5.24) which followed from canonical
commutation relations and I found a commutation inequality for the transfer
matrix and could use that to push through a phase transition (this is the
argument that appears in [215]). The three of us met at the AMS meeting in San
Antonio in Jan 1976 and agreed to publish our results jointly. We found the
Gaussian domination approach during the writeup of the full paper.

It is not surprising that the work on infrared bounds generated considerable
further work (479 Google scholar citations). I will not try to describe all of
it but will focus on two further developments in which I played a role. The
first concerns phase transitions in spin systems with long range interactions.
It was known for many years that finite range $1D$ systems could not have
phase transitions (see, for example [678, Theorem II.5.3]). Ruelle [579]
proved that this remains true for infinite range interactions with not too
slow decay. In particular, for the pair interacting ferromagnetic model with
$J(n)=(1+|n|)^{-\alpha}$, he proved there were no phase transitions if
$\alpha>2$.

On the other hand, Dyson [174], exploiting correlation inequalities, proved
there are phase transitions if $1<\alpha<2$. For $2D$, as we’ve seen Mermin-
Wagner [495] proved that plane rotors had no symmetry breaking for nearest
neighbor interactions but Kunz-Pfister [424] used Dyson’s method to prove that
2D plane rotors with a similar pair interaction has a broken symmetry phase
transition if $2<\alpha<4$. Because these results depend on correlation
inequalities which fail for classical Heisenberg models, the proofs do not
extend to such Heisenberg models. Indeed, Dyson conjectured but could not
prove that his result was still true in the Heisenberg case.

With Fröhlich, Israel and Lieb, I showed [211] infrared bounds could be proven
for such long range models and, in particular, we proved Dyson’s conjecture
about long range order in the $1D$ classical Heisenberg model with slowly
decaying pair interaction. We called a function $J$ on ${\mathbb{Z}}_{+}$, the
strictly positive integers, RP if and only if for all positive integers, $n$
and $\mathbf{z}\in{\mathbb{C}}^{n}$, one has that

$\sum_{i,j\geq 1}\bar{z}_{i}z_{j}J(i+j-1)\geq 0$ (5.25)

Then FILS show that 1D ferromagnets with
$-H={\sum_{i<j}J(j-i)\boldsymbol{\sigma}_{j}\cdot\boldsymbol{\sigma}_{i}}$
yield periodic BC RP states and infrared bounds. (5.25) comes up already in
the study of the Hamburger moment problem [708, Section 4.17] and, using this,
[211] easily proves that $J(n)=(1+|n|)^{-\alpha},\,\alpha>0$ is RP. The
corresponding $E_{p}$ then has $\int E_{p}^{-1}\,dp<\infty$ precisely if
$\alpha<2$ recovering Dyson’s result and extending it to the $d$-vector model.
[211] is also able to recover the Kunz-Pfister result and extend it to the
classical Heisenberg model.

The second extension concerns quantum lattice gases. In that case the spins,
$\boldsymbol{\sigma}_{\alpha}$ of (5.1) are non-commuting matrices, indeed
quantum spins. The basic formalism (see [678, Section II.3]) has a vector
space ${\mathbb{C}}^{2s+1}$ for quantum spins with total spin $s$ (so
$\boldsymbol{\sigma}\cdot\boldsymbol{\sigma}=s(s+1){\boldsymbol{1}}$) for each
site and the Hilbert space associated with a box $\Lambda$ is
$\otimes_{\alpha\in\Lambda}{\mathbb{C}}^{2s+1}$ so a space of total dimension
$(2s+1)|\Lambda|$. Statistical mechanical states are defined in terms of
traces. For classical Heisenberg models on ${\mathbb{Z}}^{\nu}$ (or any
bipartite lattice) the Heisenberg ferro- and antiferromagnet are equivalent
under flipping every other spin. It is critical to realize this is not true in
the quantum case! It is not even true for two spins. If
$\boldsymbol{\sigma}_{j};\,j=1,2$ are two spin $\tfrac{1}{2}$ quantum spins,
the lowest energy of $-\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$ is
$-\tfrac{1}{4}$ (with multiplicity $3$) while the lowest energy of
$\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$ is $-\tfrac{3}{4}$ (with
multiplicity $1$)! In fact the ground state energy density of the ferromagnet
is explicit while for the antiferromagnet, it is not, so before the work of
Dyson, Lieb and Simon [175, 176] (henceforth DLS), to be discussed below, the
quantum anti-ferromagnet was invariably regarded as harder than the quantum
ferromagnet. But DLS could prove

######  Theorem 5.2 ([176]).

The nearest neighbor quantum Heisenberg antiferromagnet for $s\geq 1$ and
$\nu\geq 3$ and for spin $1/2$ and sufficiently large $\nu$ has a phase
transition with Néel order.

######  Remarks.

1\. Néel order means that for any $\Lambda$ with even sides, one has that
$\liminf\left\langle\left(\frac{1}{|\Lambda|}\sum_{\alpha\in\Lambda}(-1)^{|\alpha|}\boldsymbol{\sigma}_{\alpha}\right)^{2}\right\rangle_{\Lambda}>0$.

2\. Kennedy, Lieb and Shastry [395] in 1988 using a more subtle analysis
filled in the missing spin $1/2$ cases when $\nu=3$.

3\. While the bounds on $\beta_{c}$ are concrete, they involve an implicit
equation which includes the ground state energy of the antiferromagnet which
is not known in closed form.

There are two issues involving the quantum case viz a viz the classical case
that should be mentioned. First infrared bounds cannot hold in the form of
(5.17) because they imply that as $\beta\to\infty$ that
$\langle\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{\gamma}\rangle$
goes to a constant (i.e. independent of $\alpha$ and $\gamma$). In the
classical case, this quantity goes to $1$ uniformly in the sites (for
$\Lambda$ fixed). But in the quantum case with spin $S$, it goes to $S(S+1)$
for $\alpha=\gamma$ but only to $S^{2}$ for $\alpha=\gamma$ (for the maximum
spin value of $\boldsymbol{\sigma}_{\alpha}+\boldsymbol{\sigma}_{\gamma}$ is
$2S$ so the maximum value of
$\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{\gamma}=\tfrac{1}{2}[(\boldsymbol{\sigma}_{\alpha}+\boldsymbol{\sigma}_{\gamma})^{2}-\boldsymbol{\sigma}_{\alpha}^{2}-\boldsymbol{\sigma}_{\gamma}^{2}]=\tfrac{1}{2}[2S(2S+1)-2S(S+1)]=S^{2}$).
The solution is to get an initial inequality not on the thermal expectation
$\langle AB\rangle=\text{\rm{Tr}}(ABe^{-\beta H})/\text{\rm{Tr}}(e^{-\beta
H})$ but what DLS call the Duhamel two point function
$(A,B)=\int_{0}^{1}\text{\rm{Tr}}(e^{-x\beta H}Ae^{-(1-x)\beta
H}B)/\text{\rm{Tr}}(e^{-\beta H})$. Since DLS prove that $(A^{*},A)\geq
g(A)f(c(A)/4g(A)$ where $g(A)=\tfrac{1}{2}\langle A^{*}A+AA^{*}\rangle$,
$c(A)=\langle[A^{*},[H,A]]\rangle$ and $f$ is the function given implicitly by
$f(x\tanh x)=x^{-1}\tanh x$ and this implies a direct bound with $\coth$.
Various formula involving $\coth$ occur, indeed, DLS conjecture (but do not
prove) that the correct analog of (5.17) is
$g_{\Lambda}(p)\leq\sqrt{\tfrac{3}{2}}S\coth\left(\sqrt{\tfrac{2}{3}}S\beta
E_{p}\right)$ where $g_{\Lambda}(p)$ is thermal expectation of
$\widehat{\boldsymbol{\sigma}}_{p}\cdot\widehat{\boldsymbol{\sigma}}_{-p}$.

Secondly, to get infrared bounds on the Duhamel functions, one needs that the
algebra of matrices on which the reflection acts in non-commutative RP to be
real matrices. Of course, the usual representation of Pauli spins is not real.
$\sigma_{1}$ and $\sigma_{3}$ are but
$\sigma_{2}=\bigl{(}\begin{smallmatrix}0&i\\\ -i&0\end{smallmatrix}\bigr{)}$
is not! Indeed, because of the commutation relations
$[\sigma_{1},\sigma_{2}]=2i\sigma_{3}$, there is no representation in which
all spins can be real. For the antiferromagnet, one can take
$s_{1}=\sigma_{1}$, $s_{2}=i\sigma_{2}$, $s_{3}=\sigma_{3}$ and let the
reflection be

$\Theta((s_{\alpha})_{j})=\left\\{\begin{array}[]{ll}-(s_{R\alpha})_{j}&\hbox{
, }j=1,3\\\ (s_{R\alpha})_{j}&\hbox{ , }j=2\end{array}\right.$ (5.26)

so that
$-\boldsymbol{\sigma}_{\alpha}\cdot\boldsymbol{\sigma}_{R\alpha}=s_{\alpha}\cdot\Theta(s_{\alpha})$
and so get positivity under a reflection on a real algebra for the
antiferromagnet. The corresponding infrared bound on the Duhamel two point
function then reads

$\left((\widehat{\boldsymbol{\sigma}}_{p})^{j},(\widehat{\boldsymbol{\sigma}}_{-p})^{j}\right)\leq\frac{1}{2\widetilde{E}_{p}}\qquad\widetilde{E}_{p}=\nu+\sum_{1}^{\nu}\cos(p_{j})$
(5.27)

(the sign in $\widetilde{E}_{p}$ is such that it vanishes at $p_{j}=\pi$
consistent with Néel order). This bound leads to Theorem 5.2.

Dyson, Lieb and I initially thought that we had a trick for getting real
matrices for the ferromagnet. One can double dimension and replace
multiplication by $i$ by $\bigl{(}\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\bigr{)}$ and thereby homomorphically map $n\times n$
complex matrices to a subset of all $2n\times 2n$ real matrices. However it
turns out when you do this at each site for Pauli matrices, the natural
reflection no longer has reflection positivity. Our announcement [175] focused
on the ferromagnet and so did the preprint of [176]. However, Fröhlich was
giving a course at Princeton on the work of FSS and DLS and didn’t understand
one step in our preprint. He found this and came to us on the same day we’d
finishing correcting the galley proofs for the longer article; indeed, after
we placed the envelope with them in the outgoing departmental mailbox. We
immediately realized that there was a problem and retrieved and then fixed the
galley proofs so that the published version of [176] is correct. Later Speer
[729] proved that reflection positivity must fail for this model. I note that
now, almost 45 years later, there is no rigorous proof of the existence of a
phase transition in the quantum Heisenberg ferromagnet! It is fortunate that
none of us was a young unknown when this work was done for while there is a
correct very important result, the wrong result was embarrassing. The paper
[176] does have results on the quantum $xy$ ferromagnet where the coupling
drops the $\boldsymbol{\sigma}_{\alpha,z}\boldsymbol{\sigma}_{\beta,z}$ (the
$xy$ model has an abelian continuous symmetry); this is possible because there
is a representation in which two of the Pauli matrices are both real.

There has been considerable literature on the quantum Heisenberg model since.
There is a lovely online bibliography on this subject posted by Kennedy-
Nachtergale [396].

Another application of RP methods involves what is called the _Chessboard
Peierls Method_. Fröhlich, Israel, Lieb and I wrote two papers [211, 212] that
systematized both infrared estimates and this method and, in particular,
applied the Chessboard Peierls method to a number of models. The key is what
is called _Chessboard Estimates_. The name was introduced by Fröhlich-Simon
[214] in a paper in the Annal of Mathematics on the structure of states in
general $P(\varphi)_{2}$ quantum field theories. They could not use the less
fancy term “checkerboard estimate” because that had already been used by GRS
[292] for a different bound.

While FS systematized the estimates and introduced the name, the idea had
appeared earlier in works of Glimm-Jaffe Spencer [257], Guerra[288], Seiler-
Simon [595] and Park [525, 526]. Fröhlich-Lieb (henceforth FL) [213] following
up on their use in QFT by Glimm-Jaffe-Spencer and exploited these estimates
with the Peierls argument to prove phase transitions in spin models and this
was pushed further by FILS.

We consider a box $\Lambda$, typically with periodic BC, that can be
partitioned by hyperplanes into boxes, $\\{\Delta_{\alpha}\\}_{\alpha\in Q}$,
so that there are an even number of boxes in each direction and so that there
is RP in each hyperplane. Given a function $F$ of the spins in box
$\Delta_{\alpha}$, we cover $\Lambda$ by continually reflecting $F$ in
hyperplanes and let $\gamma(F,\Delta_{\alpha})$, be the $|\Lambda|^{th}$ root
of the $\langle\cdot\rangle_{\Lambda}$ expectation of the product of these
reflected copies of $F$.

The chessboard estimate says that given functions, $F_{\alpha}$, of the spins
in $\Delta_{\alpha}$, one has that

$\left|\left\langle\prod_{\alpha\in
Q}\,F_{\alpha}\right\rangle_{\Lambda}\right|\leq\prod_{\alpha\in
Q}\gamma(F_{\alpha},\Delta_{\alpha})$ (5.28)

If the number of edges in each direction is a power of $2$, it is easy to
prove the estimate directly by multiple use of the Schwarz inequality. In
general, one uses an argument reminiscent of the proof of Gaussian domination
(in fact, one can prove Gaussian domination from Chessboard Estimates). One
considers the ratio of the two sides of the Chessboard estimate as each
$F_{\alpha}$ runs through the various $F$’s and their reflections, considers
the one that maximizes it and then uses RP to prove among the maximizers is
one where the ratio is $1$.

The Peierls strategy sums on contours that separate various states of the
system. A key part of the strategy is the estimation of the probability of
large contours. Those can typically be thought of expectations of products of
bad events, typically one for each link in the contour. One can use a
checkerboard estimate to get upper bounds on these probabilities in terms of
thermodynamic quantities and this is the Chessboard Peierls method.

In particular, FL showed this approach was effective in studying anisotropic
classical Heisenberg models; they succeeded in proving a phase transition in
$2D$ for arbitrarily small anisotropy. FILS used this technique in a wide
variety of models including ones with no symmetry. In particular, FILS
recovered results of the Piragov Sinai [538, 539] approach in different way.

Since this section is the only one on statistical mechanics, per se, I end it
with a brief discussion of some of my other work in the subject. First a paper
[651] (and a brief report in [649]) on the classical limit of quantum spin
models. This was motivated by a wonderful paper of Lieb [458] who considered a
classical Hamiltonian,
$H_{\Lambda}(\\{\boldsymbol{\sigma}_{\alpha}\\}_{\alpha\in\Lambda})$, which is
affine in the spins $\boldsymbol{\sigma}_{\alpha}\in S^{3},\alpha\in\Lambda$.
The classical partition function is

$Z_{cl}(\gamma)=\int\prod_{\alpha\in\Lambda}\left[d\Omega(\boldsymbol{\sigma}_{\alpha})/4\pi\right]\,\exp(-H_{\Lambda}[\\{\boldsymbol{\sigma}_{\alpha}\\}])$
(5.29)

where $d\Omega$ is the usual unnormalized measure on the unit sphere, $S^{2}$,
in ${\mathbb{R}}^{3}$. For $\ell=\tfrac{1}{2},1.\tfrac{3}{2},\dots$, define

$Z^{\ell}_{Q}(\gamma)=(2\ell+1)^{-|\Lambda|}\text{\rm{Tr}}(\exp[-H(\gamma
L_{\alpha}/\ell)])$ (5.30)

where $L_{\alpha}$ is an independent spin $\ell$ quantum spin at each site
$\alpha\in\Lambda$.

Then Lieb proved that

$Z_{cl}(\gamma)\leq Z_{Q}^{\ell}(\gamma)\leq Z_{cl}(\gamma+\ell^{-1}\gamma)$
(5.31)

Among other things, this immediately implies convergence of
$Z_{Q}^{\ell}(\gamma)$ to $Z_{cl}(\gamma)$ as $\ell\to\infty$ because
$Z_{cl}(\gamma)$ is continuous in $\gamma$ (indeed it is analytic). Moreover,
in situations where one knows that the infinite volume limit object (the
pressure),
$p_{\cdot}(\gamma)=\lim\tfrac{1}{|\Lambda|}Z_{\cdot,\Lambda}(\gamma)$ exists,
it implies convergence of the limit objects, since while they might lose
analyticity in the limit, they are convex and so continuous.

I had begun teaching a course on group representations which eventually turned
into a book [685] and it occurred to me to wonder what the analog of (5.31)
was if the representations of $SU(2)$ or $SO(3)$ were replaced by a more
general compact Lie group. In particular, what classical limit space replaces
$S^{2}$.

While it was mathematical elegance that attracted me, I had another
motivation. Dunlop-Newman [170] had proven a Lee-Yang zero theorem for $S^{2}$
spins by using the fact that Asano [23] had proven one for spin $\tfrac{1}{2}$
quantum spins, the Griffiths trick [280] then gets it for spin $\ell$ quantum
spins and the limit theorem implied by (5.31) then implies one for $S^{2}$
spins. It was natural to worry about spins on $S^{d-1}$ (i.e. $d$-component
rotors).

Since I eventually showed the classical limit spaces are symplectic manifolds,
$S^{N}$ is never a classical limit if $N\geq 3$ but it turns out it is a
quotient of one. I reduced proving Lee-Yang for $S^{d-1}$ for all $d$ to
proving a conjectured Asano type result for spin $\tfrac{1}{2}\,SO(2k)$
spinors. To this day, not only is that conjecture still open but so is whether
Lee-Yang holds for $S^{d-1}$ spins with $d\geq 4$!

To avoid going too far afield, I’ll describe the main results of [651]
assuming a knowledge of the theory of representations of compact Lie groups as
described in Simon [685], Adams [2] or Fulton-Harris [221]. One fixes a
fundamental weight, $\lambda$, on a compact Lie group, $G$, and for each
$L=1,2,\dots$, one considers the irreducible representation, $U_{L}$, on
${\mathcal{H}}_{L}$, with maximal weight $L\lambda$. By picking a basis in the
Lie algebra, ${\mathfrak{g}}$, of $G$, one considers Hamiltonians multilinear
in the basis vectors at the various sites in a box $\Lambda$. If
$d_{L}=\dim({\mathcal{H}}_{L})$, then

$Z_{Q}^{L}(\gamma)=d_{L}^{-|\Lambda|}\text{\rm{Tr}}(\exp[-H(\gamma
S_{\alpha}/L)])$ (5.32)

Extend $\lambda$ to ${\mathfrak{g}}^{*}$, the dual of ${\mathfrak{g}}$ by
setting it to zero on the orthogonal complement of the Cartan subalgebra.
Under the dual of the adjoint action of $G$ on ${\mathfrak{g}}$, one gets a
manifold by looking at the orbit of this extension of $\lambda$. These
coadjoint orbits, $\Gamma_{\lambda}$, which also play a role in the Kirilov
[398] theory of representations of nilpotent Lie groups and in the closely
related Kostant [415]-Souriau [728] method of geometric quantization, are the
classical limits. Haar measure on $G$ induces a measure on $\Gamma_{\lambda}$
and one uses this to define a suitable classical limit partition function
$Z_{cl}(\gamma)$. By using coherent vectors based on the maximal weight
vectors and the same Berezin-Lieb inequalities that Lieb did, I could extend
his result to this case.

There is a magic weight, $\delta$, which is the sum of all the fundamental
weights. Let $a=2\langle\lambda,\delta\rangle/\langle\lambda,\lambda\rangle$
where $\langle\cdot,\cdot\rangle$ is the Killing inner product on the weight
space. Then I extend (5.31)to

$Z_{cl}(\gamma)\leq Z_{Q}^{L}(\gamma)\leq Z_{cl}(\gamma+aL^{-1}\gamma)$ (5.33)

$SU(2)$ is rank 1, so there is a single fundamental weight and
$\delta=\lambda$ so $a=2$. Moreover, $L=1$ corresponds to $\ell=\tfrac{1}{2}$,
so $\ell=\tfrac{1}{2}L$. Thus, (5.33) in this case is just (5.31). One
surprise of this analysis is that there are several distinct classical limit
spaces if the rank is $2$ or more. For example, for $SO(4)$, the space for the
limit of spherical harmonics is the $4$-d space $S^{2}\times S^{2}$ while for
the spinor representations, it is the $2$-d space $S^{2}\cup S^{2}$.

At Princeton, I ran a “brown bag seminar” which included brief presentations
about current research both on one’s own work and work of others. There were
typically about 25 participants that often included all the senior math
physics faculty at Princeton (Lieb, Nelson, Wightman and me) and Dyson from
the Institute as well as our wonderful group of postdocs/junior faculty/grad
students (see [707] for a complete list but included were Aizenman, Avron,
Fröhlich, Deift and Sigal). In the fall of 1979, Michael Aizenman came back
from a conference in Hungary and, at a brown bag, reported on some work of
Dobrushin-Pecherski (a small part of [165]) that showed sufficiently fast
power decay of correlations in spin systems implied exponential decay. In
trying to understand why this might be, I proved the following:

######  Theorem 5.3 ([653, 652]).

Let $\langle\sigma_{\alpha}\sigma_{\gamma}\rangle$ denote the two point
function of a spin $\tfrac{1}{2}$ nearest neighbor (infinite volume, free
boundary condition) Ising ferromagnet at some fixed temperature. Fix
$\alpha,\gamma$ and $B$, a set of spins whose removal breaks the lattice in
such a way that $\alpha$ and $\gamma$ lie in distinct components. Then:

$\langle\sigma_{\alpha}\sigma_{\gamma}\rangle\leq\sum_{\delta\in
B}\langle\sigma_{\alpha}\sigma_{\delta}\rangle\langle\sigma_{\delta}\sigma_{\gamma}\rangle$
(5.34)

######  Remarks.

1\. One consequence of this is that if the lattice is ${\mathbb{Z}}^{\nu}$,
then if $\langle\sigma_{\alpha}\sigma_{\gamma}\rangle\leq
C|\alpha-\gamma|^{-\mu}$ with $\mu>\nu-1$, then for some $C_{1}$ and $m>0$,
one has that $\langle\sigma_{\alpha}\sigma_{\gamma}\rangle\leq
C_{1}e^{-m|\alpha-\gamma|}$.

2\. I talked about this at a later brown bag which stimulated additional work:
Lieb found an improvement and Aizenman and I found a version for
multicomponent models. We arranged for these three papers and one by Rivasseau
[567] to appear successively in CMP. Lieb’s improved result [461] involved the
component $\Lambda$ of ${\mathbb{Z}}^{\nu}\setminus B$ with $\alpha\in\Lambda$
and allowed $\langle\sigma_{\alpha}\sigma_{\delta}\rangle$ in (5.34) to be
replaced by $\langle\sigma_{\alpha}\sigma_{\delta}\rangle_{B\cup\Lambda}$, the
expectation with interactions outside $B\cup\Lambda$ dropped. This included
Griffiths third inequality [279]. These geometric correlation inequalities are
sometimes called Lieb-Simon inequalities as a result.

3\. As mentioned, Aizenman and I [13] proved a version for $d$-vector models.
Rivasseau [567] extended Lieb’s improved inequality to $d=2$ models.

4\. Related inequalities appeared earlier in work of Kasteleyn-Boel [372].

Among some of my other results on lattice gases are

1\. A work with Sokal [716] which made rigorous an argument of Thouless [748]
exploiting energy-entropy estimates, that, for example, provided another proof
of the result of Ruelle [579] that a pair of spin $\tfrac{1}{2}$ Ising
ferromagnets whose coupling obeys $\sum n|J(n)|<\infty$ has zero spontaneous
magnetization.

2\. A paper [657] on the one dimensional $d$-rotor model with critical
$J(n)=|n|^{-2}$ (for $n\geq 1$). As mentioned above, for $J(n)=|n|^{-\alpha}$
Dyson showed phase transitions for Ising spins if $1<\alpha<2$, FILS proved
phase transitions for $d$-rotor models with $1<\alpha<2$ and Ruelle proved no
phase transition if $\alpha>2$. The case $\alpha=2$ is borderline. Fröhlich-
Spencer [217] proved Ising models with this borderline $\alpha$ have discrete
symmetry breaking. In [657], I proved $d$-rotor models with $d\geq 2$ and this
critical coupling do _not_ have continuous symmetry breaking.

3\. A note with Aizenman [14] comparing Ising and plane rotors that gives a
lower bound on the Berezinskii-Kosterlitz-Thouless transition temperature.

4\. A note [656] showing that the rather complicated directional dependence of
the high temperature decay in the Ising model is explained by leading order
perturbation theory.

5\. A note [654] showing that for $d$-rotor models, mean field theory provides
upper bounds on transition temperatures.

6\. A book [678] that discusses the lattice models that have been at the
center of this section. It focuses on formalism and does not discuss
correlation inequalities, Lee-Yang, the Peierls argument and infrared bounds,
some of the most fascinating aspects of the subject. I have a book on those
aspects of this subject in preparation [714].

##  6\. $N$–Body quantum mechanics

The previous sections have focused on one or two problems, all (but the last
section) within a limited area of mathematical physics. This section is much
more diffuse dealing with general $N$-body NRQM, so I will leave more
background to references and only briefly discuss a lot of work. In
particular, by thinking of $2$ as a possible value of $N$, I’ll throw in some
subjects that are not usually considered $N$-body QM like some inverse
potential scattering and even a little bit of general $1D$ Schrödinger
operators.

A full $N$-body Hamiltonian acts on $L^{2}({\mathbb{R}}^{\nu N})$ where
$\boldsymbol{x}\in{\mathbb{R}}^{\nu N}$ is written
$\boldsymbol{x}=(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N})$ with
$\boldsymbol{r}_{j}\in{\mathbb{R}}^{\nu}$. We write

$\widetilde{H}_{0}=-\sum_{j=1}^{N}(2m_{j})^{-1}\Delta_{\boldsymbol{r}_{j}},\quad\widetilde{V}=\sum_{1\leq
i<j\leq
N}V_{ij}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j}),\quad\widetilde{H}=\widetilde{H}_{0}+\widetilde{V}$
(6.1)

Then a basic preliminary is

######  Theorem 6.1.

In any coordinate system, $\boldsymbol{\rho}_{1},\dots,\boldsymbol{\rho}_{N}$
where $\boldsymbol{\rho}_{j},\,j=1,\dots,N-1$ is a linear combination of
$\boldsymbol{r}_{k}-\boldsymbol{r}_{\ell}$ and (with $M=\sum_{j=1}^{N}m_{j}$)

$\boldsymbol{\rho}_{N}=\frac{1}{M}\sum_{j=1}^{N}m_{j}\boldsymbol{r}_{j}$ (6.2)

we have that, realizing $\widetilde{{\mathcal{H}}}\equiv
L^{2}({\mathbb{R}}^{\nu N})=L^{2}({\mathbb{R}}^{\nu})\otimes
L^{2}({\mathbb{R}}^{\nu(N-1)})\equiv{\mathcal{H}}_{CM}\otimes{\mathcal{H}}$,
where the first factor is functions of $\rho_{N}$ and the second functions of
$\\{\rho_{j}\\}_{j=1}^{N-1}$,

$\widetilde{H}_{0}=h_{0}\otimes{\boldsymbol{1}}+{\boldsymbol{1}}\otimes H_{0}$
(6.3) $\widetilde{H}=h_{0}\otimes{\boldsymbol{1}}+{\boldsymbol{1}}\otimes H$
(6.4)

where $h_{0}=-(2M)^{-1}\Delta_{\boldsymbol{\rho}_{N}}$, $H_{0}$ is a positive
quadratic form in
$-i\boldsymbol{\nabla}_{\boldsymbol{\rho}_{j}},\,j=1,\dots,N-1$ and
$H=H_{0}+V$.

I refer the reader to [713, Section 11] for a discussion of various coordinate
systems and the formalism of Sigalov–Sigal [610] (see also Hunziker–Sigal
[338]). In that formalism, a major role is played by the inner product

$\langle
r^{(1)},r^{(2)}\rangle=\sum_{j=1}^{N}m_{j}\boldsymbol{r^{(1)}}_{j}\cdot\boldsymbol{r^{(2)}}_{j}$
(6.5)

In this inner product, $\widetilde{H}_{0}$ is the Laplace-Beltrami operator
and the reason that (6.3) holds is that $\rho_{N}$ is orthogonal to the other
$\rho_{j}$’s. One coordinate system that we’ll need soon is atomic coordinates
where

$\boldsymbol{\rho}_{j}=\boldsymbol{r}_{j}-\boldsymbol{r}_{N},\,j=1,\dots,N-1;\qquad\boldsymbol{\rho}_{N}=\frac{1}{M}\sum_{j=1}^{N}m_{j}\boldsymbol{r}_{j}$
(6.6)

In this coordinate system when $m_{1}=m_{2}=\dots=m_{N-1}=m$ and
$\frac{1}{m}+\frac{1}{m_{N}}\equiv\frac{1}{\mu}$, one has that (see [713,
(11.48)] for the calculation)

$H_{0}=-\sum_{j=1}^{N-1}\frac{1}{2\mu}\Delta_{j}-\frac{1}{m_{N}}\sum_{j<k}\boldsymbol{\nabla}_{j}\cdot\boldsymbol{\nabla}_{k}$
(6.7)

One normally studies $H$, _the $N$-body Hamiltonian with center of mass
removed_, aka _reduced $N$-body Hamiltonian_. If one takes $m_{N}$ to
infinity, the annoying extra last term in (6.7) known as the _Hughes-Eckert_
term is gone. This extra term is present because in the inner product (6.5),
the $\rho_{j}$ of (6.6) are not mutually orthogonal. For this reason, it is
often convenient to use coordinate systems like Jacobi coordinates [713,
Example 11.6].

One last piece of kinematics we need is the notion of a _cluster
decomposition_ or clustering, ${\mathcal{C}}=\\{C_{\ell}\\}_{\ell=1}^{k}$,
which is a partition, i.e. a family of disjoint subsets whose union is
$\\{1,\dots,N\\}$. We use ${\mathbb{P}}$ for the set of all non-trivial
clusterings, i.e. those with $\ell\geq 2$. We set $\\#(C_{\ell})$ to be the
number of particles in $C_{\ell}$. A coordinate, $\boldsymbol{\rho}$, is said
to be internal to $C_{\ell}$ if it is a function only of
$\\{\boldsymbol{r}_{m}\\}_{m\in C_{\ell}}$ which is invariant under
$\boldsymbol{r}_{m}\to\boldsymbol{r}_{m}+\boldsymbol{a}$ (all $m\in
C_{\ell}$), equivalently, it is a linear combination of
$\\{\boldsymbol{r}_{m}-\boldsymbol{r}_{q}\\}_{m,q\in C_{\ell}}$. A _clustered
Jacobi coordinate system_ is a set of $\\#(C_{\ell})-1$ independent internal
coordinates for each cluster together with $\boldsymbol{R}_{\ell}=(\sum_{q\in
C_{\ell}}m_{q}\boldsymbol{r}_{q})/(\sum_{q\in C_{\ell}}m_{q})$, If we write
${\mathcal{H}}(C_{\ell})$ to be $L^{2}$ of the internal coordinates of cluster
$C_{\ell}$ and ${\mathcal{H}}^{({\mathcal{C}})}$ to be $L^{2}$ of all the
centers of mass of the clusters then

$\widetilde{{\mathcal{H}}}=\widetilde{{\mathcal{H}}}^{({\mathcal{C}})}\otimes\bigotimes_{\ell=1}^{k}{\mathcal{H}}(C_{\ell})$
(6.8)
$\widetilde{H}_{0}=\widetilde{H}_{0}^{({\mathcal{C}})}\otimes{\boldsymbol{1}}\dots\otimes{\boldsymbol{1}}+\sum_{\ell=1}^{k}{\boldsymbol{1}}\otimes\dots\otimes
H_{0}(C_{\ell})\otimes\dots\otimes{\boldsymbol{1}}$ (6.9)

where
$\widetilde{H}_{0}^{({\mathcal{C}})}=-\sum_{\ell=1}^{k}(2M(C_{\ell}))^{-1}\Delta_{\boldsymbol{R}_{\ell}}$
and $H_{0}(C_{\ell})$ is a quadratic form in the derivatives of the internal
coordinates.

In (6.9), the operator $\widetilde{H}_{0}^{({\mathcal{C}})}$ has a
decomposition like (6.3) where ${\mathcal{H}}$ is replaced by
${\mathcal{H}}^{({\mathcal{C}})}$, the functions of the differences of the
centers of mass of the $C_{j}$. We write

$\widetilde{H}_{0}^{({\mathcal{C}})}=h_{0}\otimes{\boldsymbol{1}}+{\boldsymbol{1}}\otimes
H_{0}^{({\mathcal{C}})}$ (6.10)

Given a cluster decomposition, ${\mathcal{C}}=\\{C_{\ell}\\}_{\ell=1}^{k}$, we
write $(jq)\subset{\mathcal{C}}$ if $j$ and $q$ are in the same cluster of
${\mathcal{C}}$ and $(jq)\not\subset{\mathcal{C}}$ if they are in different
clusters. We define

$\displaystyle V(C_{\ell})$ $\displaystyle=\sum_{\begin{subarray}{c}j,q\in
C_{\ell}\\\ j<q\end{subarray}}V_{jq}$ (6.11) $\displaystyle V({\mathcal{C}})$
$\displaystyle=\sum_{\ell=1}^{k}V(C_{\ell})=\sum_{\begin{subarray}{c}(jq)\subset{\mathcal{C}}\\\
j<q\end{subarray}}V_{jq}$ (6.12) $\displaystyle I({\mathcal{C}})$
$\displaystyle=\sum_{j<q}V_{jq}-V({\mathcal{C}})=\sum_{\begin{subarray}{c}(jq)\not\subset{\mathcal{C}}\\\
j<q\end{subarray}}V_{jq}$ (6.13)

$V({\mathcal{C}})$ is the _intracluster interaction_ and $I({\mathcal{C}})$
_the intercluster interaction_. We define on ${\mathcal{H}}(C_{\ell})$

$H(C_{\ell})=H_{0}(C_{\ell})+V(C_{\ell})$ (6.14) $\displaystyle
H({\mathcal{C}})$
$\displaystyle=H_{0}^{({\mathcal{C}})}\otimes{\boldsymbol{1}}\dots\otimes{\boldsymbol{1}}+\sum_{\ell=1}^{k}{\boldsymbol{1}}\otimes\dots\otimes
H(C_{\ell})\otimes\dots\otimes{\boldsymbol{1}}$ (6.15)
$\displaystyle=H-I({\mathcal{C}})$

$\Sigma({\mathcal{C}})=\sum_{\ell=1}^{k}\inf\sigma(H(C_{\ell}))$ (6.16)

We note that

${\mathcal{C}}\in{\mathbb{P}}\Rightarrow\sigma(H_{0}^{({\mathcal{C}})})=[0,\infty)$
(6.17)

By (6.15), we have that (where $\sigma(\cdot)$ is the spectrum)
$\sigma(H({\mathcal{C}}))=\sigma(H_{0}^{({\mathcal{C}})})+\sigma(H(C_{1}))+\dots+\sigma(H(C_{k}))$.
By (6.17)

${\mathcal{C}}\in{\mathbb{P}}\Rightarrow\sigma(H({\mathcal{C}}))=[\Sigma({\mathcal{C}}),\infty)$
(6.18)

fir some $\Sigma({\mathcal{C}})$. When I discuss $N$–body spectral and
scattering theory below, I’ll be interested in thresholds. A _threshold_ ,
$t$, is a decomposition
${\mathcal{C}}=\\{C_{\ell}\\}_{\ell=1}^{k}\in{\mathbb{P}}$ and an eigenvalue,
$E_{\ell}$ of $H(C_{\ell})$ for each $\ell=1.\dots,k$. The _threshold energy_
is $E(t)=\sum_{\ell=1}^{k}E_{\ell}$. Of course,
$E(t)\geq\Sigma({\mathcal{C}})$.

With these preliminaries in hand, I can describe the central mathematical
questions in the analysis of $N$-body NRQM. I assume that the reader is
familiar with the basic notions of self-adjointness of unbounded operators,
the spectral theorem for them [711, Chapters 5 and 7] and the spectral
decompositions into discrete and essential spectrum and into absolutely
continuous, singular continuous and point spectrum (see [711, Theorem
5.1.12]). One always supposes that the two body potentials, $V_{ij}$, go to
zero at infinity, usually faster than $r^{-1-\varepsilon}$.

(1) The self-adjointness of $H$.

(2) The determination of the essential spectrum of $H$. This is solved by the
celebrated (see [125, Section 3.3] for a proof and references to the original
papers of Hunziker, van Winter and Zhislin whose proofs are very different
from the proof in [125]).

######  Theorem 6.2 (HVZ Theorem).

For reduced $N$ body Hamiltonians with two body potentials vanishing at
infinity, one has that

$\sigma_{ess}(H)=[\Sigma,\infty)\qquad\Sigma=\inf_{{\mathcal{C}}\in{\mathbb{P}}}\Sigma({\mathcal{C}})$
(6.19)

(3) Absence of singular continuous spectrum for $H$. My advisor, Arthur
Wightman had a colorful name for this: “the no goo hypothesis”. His point was
that a.c. spectrum had an interpretation as scattering states and point
spectrum as bound states. If there were singular continuous spectrum, it would
have to be goo. Connected to this is that point spectrum should only have
limit points at thresholds; one might expect no embedded point spectrum but
the examples discussed at the start of Section 2 show that is too simple
minded although one might like to prove the absence of positive energy
eigenvalues and thresholds.

(4) Asymptotic Completeness. To describe this, we need some additional
preliminaries. Let $t$ be a threshold and ${\mathcal{C}}$ the associated
cluster decomposition. Under the decomposition (6.8), we let
${\mathcal{H}}_{t}$ be all states of the form $\varphi\otimes\eta$ where
$\varphi$ is an arbitrary vector in
$\widetilde{{\mathcal{H}}}^{({\mathcal{C}})}$ (i.e. function of the
differences of centers of mass of the clusters) and $\eta$ a sum of products
of eigenvectors of $H(C_{\ell})$ with eigenvalue $E_{\ell}$. Let $P_{t}$ be
the projection onto ${\mathcal{H}}_{t}$. In 1959, Hack [295] proved that, for
each threshold the limits (the funny convention that has $\Omega^{\pm}$
associated to limits as $t\to\mp\infty$ comes from the physics literature
where $\Omega^{\pm}$ defined this way is connected in time independent
scattering to $\lim_{\pm\varepsilon\downarrow 0}(H-E-i\varepsilon)^{-1}$).

$\Omega^{\pm}_{t}=s-\lim_{t\to\mp\infty}e^{itH}e^{-itH({\mathcal{C}})}P_{t}$
(6.20)

exists so long as the two body potentials decay faster that
$r^{-1-\varepsilon}$ (for longer range, including the physically important
Coulomb case, following Dollard [166], one needs to use modified wave
operators - see, for example [555, Section XI.9] \- we’ll refer to this case
below without further technical detail). These are the _cluster wave
operators_. If $\psi=\Omega^{-}_{t}\gamma$, then as $t\to\infty$, we have that
$e^{-itH}\psi$ looks like bound clusters of $C_{\ell}$ in eigenstates with
energy $E_{\ell}$ moving freely relative to each other, i.e. intuitively
scattering states. One can show that for distinct thresholds, $t\neq s$, one
has that $\text{\rm{ran}}\,\Omega^{-}_{t}$ is orthogonal to
$\text{\rm{ran}}\,\Omega^{-}_{s}$. Asymptotic completeness is the assertion
that

$\bigoplus_{\text{all thresholds
}t}\text{\rm{ran}}\,\Omega^{+}_{t}=\bigoplus_{\text{all thresholds
}t}\text{\rm{ran}}\,\Omega^{-}_{t}={\mathcal{H}}_{ac}(H)$ (6.21)

where ${\mathcal{H}}_{ac}(H)$ is the space of all vectors whose spectral
measures for $H$ is purely absolutely continuous.

For each of these four, I made significant, albeit not the definitive,
contributions as I’ll describe soon. Kato (see [713, Sections 7-10]) was both
the pioneer and continuing master of the self-adjointness problem but I made a
basic discovery on allowed local positive singularities and followed up on
Kato’s work on what I called Kato’s Inequality. I not only named the HVZ
theorem (where Hunziker, van Winter and Zhislin were the initiators) but
reworked and extended it twice, including my work with Last on the ultimate
HVZ theorem. Perry, Sigal and I were the first to prove the absence of
singular continuous spectrum for fairly general $N$-body operators although we
relied heavily on ideas of Mourre (and Balslev-Combes had earlier handled
suitable analytic potentials including the important Coulomb case). Sigal-
Soffer were the first to establish $N$-body asymptotic completeness but they
(and later, others) relied in part on my work with Deift which reduced the