**
H. Kleinert: Collaborating with Feynman
at Caltech
**

Richard Feynman was one of
the most fascinating characters of the 20^{th} century,

both as a physicist and as a person.
My friendship with him
had a rather slow start,

mostly due to the respect he inspired to me and the other
young people around him.

I met him first
in 1973 when spending a sabbatical winter semester

at Caltech. I was invited by
Murray Gell-Mann
after he had heard a lecture of mine on Current
Algebra at a

winter school in Schladming, Austria

(reporting on the results of a paper I had written at CERN, Geneva)

Caltech was an extremely exciting place.

The theory department met every Wednesday for a

luncheon seminar where everyone had a chance of presenting
his problems
and solutions.

At that time, experimentalists had discovered
point-like

constituents in hadrons with the help of

deep inelastic scattering of electrons.
The point-like structure

had been explained phenomenologically
by Feynman with his
*parton**
model.*

Gell-Mann was trying
to go further

by constructing a fundamental quantum field theory

which would combine
the point-like structure with

well-known results of his Current
Algebra.

So he gave
partons

the
quantum numbers of *
quarks* and described them

by fundamental fermion
fields.

Together with Harald
Harald Fritzsch he had just shown that

currents constructed from free quark fields

would explain most of the data.

What was missing at that time was an explanation

why free fields worked so well although

nobody was able to detect quarks as particles in the
laboratory!

The model had, however, an important weakness, as was immediately noticed by Feynman:

It did not account for the fact that high-energy collisions produced jets of particles ---

the free-quark model would lead to uniform distributions.

It was Gell-Mann who
realized that local color gauge fields

could provide
them with the necessary glue

to keep them forever inside the hadrons.

The point-like structure was assured by what is
called

*
asymptotic freedom* of
color gauge theory, which ensures

that quarks inside hadrons
would behave almost like free particles.

This property had just been discovered independently by 't Hooft,
Politzer, and by
Gross and Wilczek.

Unfortunately, I did not participate in this
fundamental
endeavor

since
I was working
on a field-theoretic derivation of

the Algebra of
Regge Residues (see also a href="http://www.physik.fu-berlin.de/~kleinert/kleiner_re45/45.pdf

which had been postulated seven years earlier
on phenomenological grounds

by Cabibbo, Horwitz, and Ne'emann [Phys. Lett. *166*, 1786 (1968)].

I found a derivation by generalizing Current Algebra

to an algebra of bilocal quark charges of free quark fields.

In fact,
Yuval Ne'eman

[who had discovered in 1961, simultanously with Gell-Mann,
the fact

that mesons and nucleons occur in multiplets
representing the symmetry group
SU(3)
and who became later

Israel's Minister of Science and development (1982--1984)
and of Energy (1990--1992)]

was my office mate at Caltech in 1980
and was very happy about my derivation.

We became close friends,
and whenever he appeared on a ministerial visit in Berlin,

he always found some spare time
to meet me and discuss physics

(protected by several body guards
sitting at the tables around us).

Gell-Mann, however, convinced me that according to my
derivation, the algebra

was not exact but only
approximately true
due to logarithmic corrections.

This made it uninteresting to
him. He always said

"Don’t waste your time
with theories which do not have a chance of being true."

He taught
me that
theories which
are only approximate
from the beginning

are not worth pursuing.

This was quite different with
Feynman

who loved simple models which can explain things
approximately.

Feynman
appeared regularly at the
seminars, and it was an experience to witness the
pointed discussions

evolving between him and Gell-Mann.
There was always a

tension between them, a certain rivalry,

which led to interesting exchanges. Some of them were
plainly silly:

for instance, a speaker said on the blackboard:
"I am now using Feynman's

parton
model" and was interrupted by Gell-Mann with

the provocative question "What's that?"
(whose answer he knew, of course).

Feynman
responded with a boyish smile: "It's published"!

Upon which Gell-Mann said
"You mean that phenomenological model

which I superceded with my
quark field theory?".

The students thoroughly enjoyed seeing the greatest
physicists

of that time fighting like kids.

In this environment I had the opportunity

of participating in a series of seminars Feynman
gave to graduate students and

postdocs
on path integrals with applications to quantum electrodynamics.

He told us that when he had first come to Caltech

as a young professor he had

used path integrals quite extensively in his course on
quantum mechanics.

Later, however, he had given up on this since

he was tired of confessing to the students

that he was unable solve the path integral of the

most fundamental atomic system,

the hydrogen atom, whose solution is so easy

in Schrödinger wave
mechanics.
Feynman
knew

that I had developed the group theory of the hydrogen
atom

in my 1967 Ph. D. Thesis, so he
challenged me to try
my luck

with the path integral. We tried a number of things on the blackboard

which, however, did not lead to anything.

When I returned home to Berlin in spring 1974

the problem stuck in my head
while I was working out my

1976 Erice Lecture "On the Hadronization of Quark Theories"

[in which I calculated the mass differences between current and constituent quarks

and various current algebra relations].

The hydrogen atom sufaced
again in 1978

when a Turkish postdoc, H. Duru, came

to me as a Humboldt fellow. He

was familiar with my thesis,

so I told him Feynman's challenge,

and we began searching for the solution.

It turned out that Feynman's
definition of path integrals

as a product of a large but finite number
of ordinary
integrals

was *in principle* unable to describe the
hydrogen atom.

The situation is similar to ordinary integrals:

if a function is too singular, these

can *not* be approximated by finite sums.

We published the
solution in 1979, and it became the basis

for solving all path integrals whose Schrödinger equation can be solved analytically

(see my textbook on this subject).

This success encouraged me to

loose my shyness when meeting Feynman

again on another sabbatical which I spent mostly

in Santa Barbara in 1984. I frequently

drove to Pasadena, and met with him to

discuss physics, and he always had time for me.

We talked for a few hours and usually went to a diner

to have a soup together.

He was sometimes very funny.

Once he said conspiratively to
me: "Hagen, I know you have a loose mouth.

I shall show you a secret, but only if you promise to
tell

it to everybody." I did, and
he pulled out a photo

which showed him in a huge bathtub with three(!)
beautiful long-haired

California beauties in his arms.

His office had a wall full of tightly filled notebooks

which he
occasionally
pulled one to show
me some calculations

he had done earlier
on the topic we were discussing.

He showed me, in particular, long calculations which had
not produced

any interesting
results so far.

I still possess a copy of such notes

where he calculated the properties of

an analog of a polaron, where

the role of the particle coupled to phonons is played
by a spin.

He always wanted me to find a nice physical system

where his model could be applied, but I did not
succeed.

He put a message to his secretary Helen Tuck

to send me these notes on his blackboard, which remained there

until he died in February 1989. It is

visible on the picture "Feynman's
last blackboard" in

the February issue of Physics Today in 1989,

page 88. The relevant section of the blackboard is pictured in Fig. 1.

Fig.
1
Feynman’s last
blackboard in Physics Today, February 1989 issue, p. 88.

Right bottom shows note to his secretary Helen Tuck
asking her to send his calculations on self-coupled

spin system to Kleinert.

Feynman's
notes were always filled with lots of numerical calculations,

with long columns of numbers

obtained with the help of a pocket calculator.

It reminded me of Riemann's
notes, which are filled with such columns.

Feynman
told me that he liked to get numerical results. This was necessary

back in the forties when working in Los Alamos on the
Manhattan Project.

At that time the machines were quite primitive
by today's standards.

Students often believe that great theoreticians design
only abstract

formulas and leave the calculations to others. But
this is not so.

Feynman
always insisted in going to the end to get
numbers,

And when these did not fit the data, this was an
important source of

discoveries.

One of the
set of notes which he showed me to in 1982

was easier to turn into physical results.

These contained the variational
calculations

which are published in his 1972 Benjamin book on "Statistical
Mechanics". Feynman

suggested to me to work out a simple generalization,
leading to what we called the

*
effective classical potential*.

Fortunately, a simple cheap computer

(the Sinclair ZX81 see Fig. 2)
had
become available
a year earlier,

Fig.
2
The Sinclair Computer bought at Woolworth for 15 US$.

and was just becoming very cheap in the supermarkets
(15$ at Woolworth).

It connected to a television set
and worked at a frequency of
3.25 MHz with a memory of

1 Kilobyte.

and wrote up the manuscript in Spring 1983.

When I told him I had written up the paper

Feynman
came to visit me at the ITP in Santa Barbara

to discuss the results. When he arrived
everybody wanted to talk to him

and he was pressed to
give a seminar.

He finally agreed, although he was not in

good shape at that time.

He began his talk with
the words: “Sorry, that I am unprepared but I came
here

only to
finish a paper with my friend Hagen”.

His talk dealt
with
the proliferation of vortex lines

explaining the critical properties of
superfluid helium.

He had proposed this mechanism in a lecture in 1955,

probably inspired by a similar proposal
by Shockley in 1952

that the proliferation of defect lines
should drive the melting transition.

Both have probably known
the pioneering 1949 paper by Onsager

who
had explained the phase transition
of the
two-dimensional Ising model

by the
proliferation
of the line-like domain walls between up and down spins,

and conjectured a similar mechanism
for the vortex lines in superfluid helium.

To his surprise, Feynman encountered strong but unjustified
criticism from the young people

in the audience who wanted to show that they were
smart.

They had just learned to calculate critical properties
by applying the renormalization group

to complex scalar field theory, and did not believe
that vortex lines were relevant.

At that time,
the disorder field description of superfluid helium
was not yet common

knowledge. These describe ensembles of vortex lines by
fields

whose Feynman diagrams are
direct pictures of these lines.

I had inroduced these fields
in 1982 and published a full theory in a textbook in 1989

(*
Gauge Fields in Condensed Matter, Vols. I
and II *).

After his talk and the fights, Feynman
was extremely exhausted and

disappointed by
the agressiveness of the audience.

He lay down in my office and sighed "I should not
have talked!

Why am I doing this to myself?". When he returned
to Caltech

he felt quite ill and I was very worried. Of course, I
did not dare

to press him to go through our paper

and permit me to send it off to a journal.

In February 1984 I returned to Berlin

to my regular teaching duties, and forgot all
about our

manuscript, when

in a night of
May 1986 the telephone rang

and Helen Tuck, Feynman's
secretary, told me that

Feynman
found the paper OK as it was

and wanted me to submit it to Physical Review A.

It was interesting to observe the reaction of the referrees

to a paper with Feynman as
an author: One wrote:

“The manuscript shows the

clarity and conciseness typical for Feynman's
writing”.

The paper appeared in
December 1986.

Feynman
never expected the method would be applicable to quantum field theory.

In recent years, however, I have found a way of doing
this,

what I have named *
Field-Theoretic Variational
Perturbation Theory*
which

has led to the
most reliable
calculations of critical

properties of systems near phase transitions so far.

In particular, the most-accurately known singularity
of the specific heat of superfluid helium

which has been measured with great effort in a
satellite experiment by J. Lipa and collaborators

was precisely predicted, and I have written a textbook
on this subject

(*Critical Properties of phi^4 Theories*).

Inspired by the collaboration with Feynman
I published in 1990 the

first edition of my textbook

on *
Path Integrals in Quantum Mechanics, Statistics,
and Polymer Physics*

which has become a real best seller.

The
third edition appearing now contains also
applications to financial markets.

Without him, this book would have never appeared

and many problems would have remained unsolved.

Another system we discussed many times

was a stack of biomembranes. If they are contained between two walls

they exert a pressure obeying a law very similar to the ideal gas law

(Helfrich's law: p d^3=c T^2/k,
where p=pressure, d=distance between the walls, T=temperature,

k=stiffness of
membranes). The challenge was
to find the proportionality constant c

which was only known from
Monte-Carlo simulations.
Unfortunately we could not find
a solution at that time.

The problem was not forgotten, however, and many years later I succeeded

in solving it with the help of field-theoretic variational perturbation theory.

Of all the physicists I have known, Feynman impressed me most

with the simplicity and elegance with which he
articulates and solves

complicated problems. His course I attended was
extremely clear.

He never used camouflaging mathematical language to
express physical

facts. If someone did, he always stopped him with the
question “what’s that?”,

and insisted on a down-to-earth explanation. He never made the student feel stupid

(unless he really was) and always explained his ideas
to exhaustion.

He was a perfect teacher. It is somewhat amazing that
he has produced

only a relatively small number of excellent postdocs of his own, in comparison with

J.A. Wheeler

too few
students had the courage to approach him.

Feynman has certainly shaped the thinking of all Caltech students

who went through his
classes even if they graduated with other thesis advisors.

And, of course, the many
students and colleagues around the world

who have studied his fascinating textbooks on physics.