Richard Feynman to Enrico Fermi

Home , Richard Feynman

CBPF-CS-012/97

Richard Feynman to Enrico Fermi:

a letter from Rio de Janeiro, 1951

J. Leite Lop es

Centro Brasileiro de Pesquisas Fsicas - CBPF

Rua Dr. Xavier Sigaud, 150

22290-180 - Rio de Janeiro-RJ, Brazil

and

Universidade Federal do Rio de Janeiro

Richard Feynman came to Rio de Janeiro for the rst time in 1949. The Centro

Brasileiro de Pesquisas Fsicas | CBPF | had just b een founded by a group of Brazilian

physicists led by Cesar Lattes | who had contributed in 1947 to the discovery of pions

| and mathematicians led by Leop oldo Nachbin and Maurcio Mattos Peixoto, in order

to promote researchworkincontemp orary themes of physics and help in the university

education of physicists.

Feynman liked the atmosphere at the Centro and was attracted by the city and its

b eaches. He came backayear later and stayed during his U.S. Sabbatical year, 1951 -

1952. He gave lectures at the University and at the Centro on electro dynamics and was

working an meson theory after his well-known success in quantum electro dynamics. He

prop osed to me that weinvestigate whether the symmetrical pseudoscalar meson eld

theory could give a description of the deuteron which could b e exp erimentally checked, in

spite of the dicultyofthe -singularity at the origin of the tensor-force in the Yukawa

p otential. Feynman comments this work in his letter to Fermi in the p ort-scriptum P S .

The results of this research are in a pap er published in the Pro ceedings of a Symp osium

whichwas held in Rio and which I presented after Feynman left for Japan in May 1952.

Feynman enjoyed very much his stay in Rio and came back during his summer vaca-

tions several times. In 1953, he had his ideas on sup erlfuidity in our city.Ashesays in

Details of his activities anf feelings in Rio may b e found in the pap er J. Leite Lop es, RichardFeynman

in Brazil: recol lections, QUIPU, Revista Latino-Americano de Historia de las Ciencias y la Tecnologia,

vol. 7 numero 3, Mexico (1990)

J. Leite Lop es and R.P.Feynman, On the pseudoscalr meson theory of the deuteron New Research

Tecniques in Physics, Academia Brasileira de Ciencias, Rio de Janeiro (1954)

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this letter to Fermi: \(I) am going to the Copacabana b each to see if I can get one (ideas)

of myown. I get lots of ideas at the b each."

It was a great joy for me to work and discuss with him. Wewere together in Pasadena,

at Caltech, during the year 1956-1957 and in Mexico City in 1972 where we discussed on

the emerging gauge eld theories and nally in 1977 when I invited him to come to Stras-

b ourg and present a review pap er on the parton mo del to an International Symp osium

on multiparticle dynamics whichwas held in the charming Alsacian town of Kaysersb erg,

birthday of Alb ert Schweitzer.

Here is Feynman's letter to Enrico Fermi:

Rio de Janeiro

Decemb er 19, 1951

Dear Fermi,

Being thousands of miles awayIhave only heard by amateur radio from friends in the

U.S. that you are doing exp eriments in meson scattering from protons. I don't know what

your theoretical friends are saying, so I should liketomakesomecoments at the risk of

only saying what is obvious to everyb o dy in the U.S.

To b egin with I am of the opinion that Yukawa's meson theory with pseudoscalar

mesons gradient coupling, is wrong, (or least useless) in its present form | b ecause at

least p erturbation theory is N.G. and otherwise divergences cloud the issue. But I think

mesons are pseudoscalar, and I think the amplitude that a nucleon emits just one may

b e prop ortional to .Q (where is the nuclear spin, Q the meson momentum) for Q

small. (This is of course in agreement with the Yukawa theory | to all orders in account,

H H HHH

b ecause for low Q one op erator in the series H + + etc. is prop ortional

E E E

to Q and others, involving all the virtual mesons are not (the virtual moments are of

order , the meson mass) so for Q low enough the sum will b e prop ortional to Q, and

further will b e Qtimes the sum with the op erator in place of one of the H's | which

means Q.times a spin ob ject which can only therefore b e prop ortional to ). Let us say

then the coupling is G(Q)( Q)u for emission of one meson amplitude u, momentum

Q, mass where G(Q) is a function of Q (and p ossibly the nucleon moments at higher

Q??) and I exp ect G to have the prop erties of not varying muchfor Q small, just is a

Imake all analyses thinking of the theory non-relativistic in the nucleons.

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reasonable function of Q=.For Q =0,callG(O )=G (If p ert theory were OK G is just

0 0

the usual g ). Further this is most reasonale on nearly any theory | for the meson b eing

pseudoscalar the co ef to emit one (even if proton is a p ositron + 18 neutrinos + 4 neutral

mesons) must b e ps.scalar | which , if it do esn't involvethenucleon momenta (and I

can't see how it easily can b e galilean invariant | but Nature's imagination always has

my resp ect) can hardly b e other than Q. (According to Yukawa theory, standard form,

the total series would givea G(Q)which, if g were very small and integrals converged,

would b e nearly constant for all Q andequaltog | but if g is larger, correction terms

set in for Q of order ).

I wish to app eal to exp eriment to try to establish, if p ossible, wheter the ab oveis

correct and the coupling is like Q for one meson absorption. You see tho I mean only

to refer to low energy mesons |for Q or higher I have no arguments ab out what to

exp ect.

Yet it is imp ossible to measure the absorption of one meson bynucleon directly for

the conservation of energy demands that another coupling enter to take out the energy.

If we do it with a -ray, or a collision b etween nucleons new uncertainties arise, but if we

do it by means of another meson (scattering) the situation would app ear to b e as simple

as p ossible.

The \intermediate states" (if they mean anything) have, mayb e, energy of order

so that as long as Q remains small enough (non-rel. mesons) the intermediate states do

not dep end muchon Q. Then, if we assume the coupling for two mesons is essentially like

the double action of the lst order coupling, we see that the matrix element for scattering

ought to b e prop ortional bilinearly to Q and Q .Itmust therefore have the form

1 2

M = X Q Q + X i (Q Q )

1 1 2 2 1 2

of if Q ;Q lie in x; y plane at angle one to other using c.g. system Q = Q = Q

1 2 1 2

M = Q (x cos + i sin X )

1 2

where X ;X are some functions of Q, insentitiveto Q for small Q. But in principle

1 2

knowledge of the coupling of one meson do es not determine that for two. There could

still b e a term with arbitrary co ecient in the Hamiltonian of form u u which is scalar.

1 2

Hence we mightexpect

M = Q (X cos + i sin X )+X

1 z 2 3

(For example, gradient and direct coupling theories agree on Q for one meson, but

for two X is very di erent b eing very small for grad. and very large for direct- in p ert.

theory).

Because, if you like, now in the p ert series one of the H is prop. Q , other to Q and otherwise

1 2

nothing is sensitivetothevalues of Q ;Q .

1 2

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Naturally such a form is completely general | but what I wanttoverify is that

X is very small (mayb e order =M smaller than X ;X); (1)

3 1 2

(could in principle dep end on spin | I will assume it do esn't)

2 2 2

X ;Xare insensitiveot Q for Q well b elow : (2)

1 2

I am not in p osition to calculate X ;X in terms of G, nor to get a relation b etween

1 2

them | for wehave no go o d theory. (One p ossibility of course is that relations of the lst

order p ert theory may b e true, but let us rst nd out if (1), (2) are true and that b eing

established go on from there.)

Coments: (1) is a pure guess | various evidence (suchas emission comp eting

favorably with emission in H capturing ) indicates it is so | all the evidence which

is usually aduced to prefer the grad. to direct coupling is just a question of how big X

is. I assume for no excellent reason that X do es not dep end on spins.

(2) could b e wrong. It would b e very interesting. For it probably would mean there

exist imp ortant\intermediate states" at low (rel. to ) energy | whichwould b e a vital

discovery. Hence I urge you to try to see whether the predictions of (1), (2) are satis ed.

Incidentally since M for the inverse reaction should b e the complex conjugate I con-

clude all X s are real (but I am notoriously punk at such arguments | get a eld theory

or group theory exp ert).

Next, very interesting is the relation of the X s for di erent reactions (I mean mesons

of di erentcharges, neutral etc.). It would b e very interesting if we could verify that the

symmetric theory is valid. Let us lo ok at the predictions of this theory for this problem

and test it later exp erimentally.If~u; ~v are the vectors in isotopic spin space representing

the mesons in and out, and is the op erator for the nucleon M must b e bilinear in u,

and v and invariant in isotopic spin, or of the form

M = A(~u~v )+Bi~ (~u ~v ) (3)

where A; B are matrices involving spin etc. (Whichwe later write in the form

A = A + i A ; (4)

1 z 2

B = B + i B (5)

1 z 2

and we exp ect nearly to write

2 2

A = Q X cos + X , A 2Q X sin

1 1 3 = 2

2 2

B = Q Y cos + Y , B = Q Y sin ; X ;Y

1 1 3 2 2 3 3

small, X; Y nearly constant

small Q . All real?

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but form (3) do es not dep end on assumptions (4) (5) of course, just invariance.).

That is, getting down to cases, the matrix element for each pro cess is given in the

following table. Pro cesses lab eled with the same \TYPE" letter have equal probabilities

|aswould b e exp ected from either reaction-inverse or the most naive use of the charge

+ 0

symmetry idea: is to p as is to n and is impartial.

Now let us lo ok at the X -sect for various cases. In complete generality. A can b e

written in the form A = A + i AA where A is scalar AA is 3 quantities (complex)

1 0 2 1 2

(vector) and B = B + i IB . Summing over all spin directions of the nucleon then we

1 2

obtain that the cross section is prop ortional in each case resp ectively to,

2 2

PROCESS ELEMENT TYPE (a) jA + B j + jAA + IB j

1 1 2 2

+ + 2 2

+ p ! + p A + B (a) (b) 2(jB j + jIB j )

1 2

0 +

+ p ! + n 2 B (b)

0 2 2

! + p A (c) (c) jA j + jAA j

1 1

+ p ! + n + 2 B (b)

2 2

! + p A B (d) (d)jA B j + jAA IB j

1 1 2 2

+ +

+ n ! + n A B (d)

! + p 2 B (b)

0 0 2 2

+ n ! + n A (c) (where jAj means A A; jAAj = A A).A

2 B (b) ! + p +

+ n ! + n A + B (a) Hence the symmetric theory predicts

+ =2 + whichwould b e a wonderful thing to verify for its do es involve the idea

a b c b

that neutral mesons have1= 2 times the coupling of charged. However, unfortunately

is unmeasurable exp erimentally. If (somedayweknow sym. theory is OK we can use this

to get which someb o dy mightwanttointerpret pro duction and subsequent escap e

in heavy nuclei). So so far no test of subtle parts of sym. theory.

But now let us substitute (4), (5). I call X zero for simplicity| you can put it in

and see e ects. Assume X; Y real | I hop e it's true.

4 2 2 2

(a) = Q [cos (X + Y ) +sin (X + Y ) ]

1 1 2 2

4 2 2 2

(b) = Q [cos (2Y ) +sin (2Y ) ]

1 2

4 2 2 2

(d) = Q [cos (X Y ) +sin (X Y ) ]

1 1 2 2

4 2 2

Hence (A) Cross sections should go as Q (up until Q )

2 2

(B) in angle should b e of from a + b cos ,orsays cos + sin . E ect of X will

b e seen as a small residual constant x-sect. as vs. Q is extrap olated to zero, | or more

{6{ CBPF-CS-012/97

sensitively (?) a term in cos in the angular distribution (lack of front-back c.g.) for low

(90 ) (0)

relations

q q q

(a) (d)= (b) (these are not valid

q q q

or (a)+ (d)= (b) if my argument X; Y real is

q q q

or (d) (a)= (b) is faulty)

whichmay serve as a test of that theory.

Could you tell me to what extent these foredictions (A), (B), (C) are veri ed by ex-

p eriments? May I urge the imp ortance of low energy meson exp eriments in establishing

beyond doubt (if they agree) some of our basic premises to day? Higher energy are inter-

esting but in our ignorance wedonotknowhowtointerpret them | so it is well to study

low energy well. In particular there is hop e to check the 2 of the symmetric theory with

low energy data.

Sincerely,

/s/ DickFeynman

P.S. I have already heard that x-sect rises rapidly with energy | stops rising ab out Q = ,

so I am not entirely in the dark in Brazil.

P.P.S. Between us theorists (I imagined you as an exp erimenterabove|hencethelow

remark ab out seeing a eld theory exp ert to see if X ;X must b e real) I'd liketo

1 2

make some remarks. I think now non-relativistically ab out nucleons, so errors of order

(P =Mass proton) (c = 1) can come in. A coupling of one meson Q is not Galilean

nuc

invariant, for at additional velo city V; Q = Q + ! V where ! = frequ. of meson. But

nucleon changes mom by M hence the Galilean invariant coupling must b e (error now

2 2

order V =c , not V=c), i.e. (Q IP is invariant)

1 @u @u

r + IP + IP (6)

2M @t @t

where P is the op erator nucleon (b) momentum.Mayb e you should use this in the x-sect

analysis but it only makes factors of 1 + !=2M or 1 + =2M to the accuracy we exp ect,

so is just an unknown constantanyway.

The ps. theory grad coupling agrees, making for the non-rel. hamiltonian approx.

{7{ CBPF-CS-012/97

! !

IP IP 1 q @u g @u g g @u

H = + ( IP ) + IP + + ru

2M 2M @t @t 2M @t

(b) (c) (a)

Now I argued ab ove for the term (a) with a g renormalized to G as e ect for absorption of 1

meson. Hence the galilean argumentshows the g in (b) is the same as that in (a). Nowthe

AA . (c) lo oks like the typ e of e AA AA term that comes in electro dynamics from IP

In that case renormalizing the charge must change the e in e(IP A+A AA P ) and in e AA AA

by the same amountby an argumentofgaugeinvariance. Now is there some reason for

a particular size (c)? Or, is there some principle whichshows the renormalized g in (a)

and (c) must b e equal? Do es anyone in U.S. know ab out this? It is very interesting

b ecause (c) of course is the origin of the X term | so if X is known in size it maytell

3 3

us something.

Also when electric p otential is presentthe ru gets another AAu. So one way

G might b e got is from the cross sect for p ! p + capture from at rest. We

know this comp etes sucessfully with p+ ! p + and the latter absolute x-sect can

b e got extrap olating scattering cross sections down. (The latter is small either b ecause

it uses the Y term, or if this is zero (as in lst order p ert. theory) by Q of the out

followed by the (b) term for the in ). Can we argue that the emission comes just

from ( A)A u I think yes. If you imagine the p ert. series again and try to get in by

Q it can only go by (b) term and hence is so small that could complete. Hence

must go in via G( A)A u. Next should the G be G(O )? I am not clear on this. Probably

not, for if the nucleon has a structure it would dep end on the raywave length | or

otherwise put, in principle we cannot exclude additional terms of the kind ( (AA IK ))u

etc. Anywayitmaybeinteresting when enough data is available to put in numb ers ans

see how comparable are the G s obtained from X , from this reaction, from an attempt

to get X ;X from p ert. theory, etc. Also interesting is to see if any electromagnetic

1 2

prop erties can b e got from the scattering based on the principle that any function of

momentum for charged particles go es to Q e=c A.

P S. I'm sorry to have to write by hand but secy's here have language trouble, and are

slow, and are nowonX -mas vacation, and I'vedelayed to o long. If anything herein lo oks

interesting enough to tell any other meson labs please tell them, I am not writing this to

anyone else. (If you make copies please send me one.)

P S. Leite Lop es and I nished that test of the Yuk. theory p otential I said I mighttry.

The idea was take lst order in g potential from ps. grad. sym. meson theory. Assume

{8{ CBPF-CS-012/97

OK large r but not for small. Integrate from outside in, but don't assume r go es

exactly to O at origin (b ecause there is wrong). Starting with singlet, scattering length

and e ective range determine g 18 hc. r ! 0at0:1=. But using these for triplet

entirely to o much D state results and no accord is got to exp eriment no matter what

phase is chosen at origin. It is so bad that we can say the p otential must b e wrong by its

own order of magnitude even as far out as at r =0:7. The next order (g ) p otential of

Yuk theory makes changes of 200% at r =1:0, even for g as large as 0.2 (the co ecients

in the series are so large!) in directions whichdo not seem righttostraighten things out.

Hence wehave no idea of what the p otential should b e even if the meson theory were OK.

5 2

| p erturbation expansions are inconsistent for such large g , and even if all is dropp ed

arbitrarily but the rst term the p otential disagrees exp eriment. The terms (b) in (6)

pro duce in 4th order (g ) a quite strong spin orbit force b etween nucleons and a closed

shell as well. It seems to b e of the right sign and order of magnitude (it actually is to o

NUC

big, but ) for Mayer (order main force). I am writing all particulars to Bethe,

in detail. It is hard to b elieveinanything from the ps. theory b ecause the p erturbations

are inconsistent. I have tried for 6 months and 100 closely written pages of formulas to

work out intermediate coupling problems. I think I could succeed but the grad. meson

theory diverges everywhere so I am disheartened to pick out any false mo del (without

divergencies) and push it through, b ecause it's so muchwork. IthinkIcoulddoany

sp ecial problem which didin't havedivergencies (e.g. a cut-o theory) but I don't want

to waste mytime.

So I am, with this letter to you and one to Bethe, giving up Yuk. idea 1934 and am

going to the Copacabana b each to see if I can get one of myown. I get lots of ideas at

the b each.

Marry X-mas.

Don't b elieveany calculations in meson theory whichusesaFeynman diagram! Eg. p erturbation

values of X; Y are

X =0;X =+1;X = u =2M ) Simple but

1 2 3

Y = 1;Y =0;Y = 0) false. How false!

1 2 3