H. Kleinert: Collaborating with Feynman at Caltech


Richard Feynman was one of the most fascinating characters of the 20th 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



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,

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. With this little machine, I easily did the necessary calculations

and wrote up the manuscript in Spring 1983.

When I told him I had written up the paper
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, and developed

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 or J. Schwinger. I guess  the reason must be that
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.