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514 L.Michel
from the observed quadrupole moment of the deuteron as 0.3. Hence
B/a-(ka)2/10. T h e value of K to this approximation i n (ka) and (eba) is a
complicated one. Since, however, we are not interested in the absolute magni-
tudes of the scattering cross section, we have taken a N 11 ( € 6 + i k ) for simplicity,
even though it is not strictly justified.

REFERENCES
J., and Wu, T., 1948,Phys. Rev., 73, 973.
ASHKIN,
BARGMANN, V.,1949,Phys. Rev., 75, 301.
CONDON, E. U.,and SHORTLEY, G. H., 1935, Theory of Atomic Spectra (Cambridge:
University Press), p. 76.
FROBERG,C., 1948 a, A r k . M a t . Astr. Fys., 34, N o . 28; 1948b, Ibid., 36, No. 11.
W., and PEIERLS,
HEPNER, R., 1942,Proc. Roy. Soc. A, 181, 43.
HOLMBERG, B., 1948,KurzgZ Fysiografiska Siillskapets I Lund Handl., 18,No. 7 .
HYLLERAAS,E., 1948, Phys. Rev., 74, 48.
W.,and SCHWIXGER,
RARITA, J., 1941,Phys. Rev., 59, 436.
ROSENFELD,L.,1948,Nuclear Forces (Amsterdam : hTorthHolland Publishing Co.).
SCHWINGER, J., 1948,Phys. Rev., 73, 407.
WIGNER,
E.,1941,Proc. N a t . Acad. Sci., Wash., 27, 282.
WOLFENSTEIN, L.,1949,Phys. Rev., 76, 541.

Interaction between Four Half-Spin Particles and the

Decay of the p-Meson
BY L. M I C H E L
Physics Department, University of Manchester

Comntu?iicated by L . Rosenfeld; M S . receined 28th Septembev 1949

A B S T R A C T . T h e general direct coupling between four fermions is studied. It is a

linear combination of the five invariants used in /%decay theory. Altering the order of the
particles in the Hamiltonian changes only the coefficients of the linear combination. The
formalism of charge conjugation is used with the ordinary theory or the Majorana abbreviated
theory for neutral particles. This is applied to the study of the decay of the p-meson into
an electron and two neutrinos.

§ 1. I K T R O D U C T I O N

T
HE purpose of this work is to study the most general contact interaction
between four half-spin particles. T h e interaction Hamiltonian density
used here consists of the most general !inear combination of scalars which
can be made u p from four wave functions (or the conjugate of some of them).
T h e interaction can be produced physically by an intermediary field; then the
Hamiltonian only gives a phenomenological description.
Such an interaction between four fermions can give rise to 24’ 1 6 kinds of
processes, where i particles (0 $ i $ 4 ) are absorbed (M = - l), and j particles
( j = 4 -i) are emitted (M = + l ) . M is a dichotomic variable having the values -1
a n d + 1 for absorbed and emitted particles respectively. For example :
i = l , /?-decay: N = P f + c - + u with ~ ~ = - Ml , . ,= M , = M , = + ~ ;
i = 2, K-capture ; scattering of fermions by fermions ;
i = 3 , pair annihilation near to an electron without emission of photons;
t = O or i = 4 (cf. Critchfield and Wigner 1 9 4 1 , Critchfield 1 9 4 3 ) .
 Interaction between Half-Spin Particles and Decay of p-Meson 5I5
The necessary energies for these processes can be supplied by external fields
(e.g. P++N+€++U).
The study of these processes will be restricted t G the Schrodinger method of
perturbation theory in which the Hamiltonian of the interaction does not contain
the time .explicitly.

I. I N T E R A C T I O N BETWEEN F O U R H A L F - S P I N P A R T I C L E S
51. F O R M A L I S M OF T H E C H A R G E C O N J U G A T E T H E O R Y
The charge conjugate formalism, which is preferable to Dirac’s hole theory
(see $9), has been developed by Majorana (1937) and later by Racah (1937) a n d
Kramers (1937). Pauli’s (1941) notation ( f i = c = 1) will be followed here.
The most convenient representation of Dirac’s matrices, which does n o t
restrict the formalism, is that proposed by Majorana (1937).
( Y / J w+KM = 0 . . . . . . (1)
is the Dirac equation, where the yJ,satisfy
YwY, +Y,YU =2%. . . . . . . (21
In the Majorana representation ( K = 1, 2, 3)
Y:=Yk=?k=Y;; y!=y4= -74=-y:, . . . . . . (3)
or, with u k = iy4yk and ,B = 1r4,
uE=ulc=Mk=u;; p+=p= - p = -p*. . . . . . . (3’).
A solution of (1) can be split into plane waves normalized i n a volume V :
1
$-- C [u+(k,u)ue(k,U) exp {i(k.x - k,x,)} +ut_ ( k ,u)b,(k,U )
(NW.,
x exp { - i(k . x -kOxO)}] . . . . . . (4)
(N=XN,(k, U ) is the total number of particles of this field, see equation ( l l ) ) ,
and because of the Majorana representation
@ , 0)=a,@, 0); . . . . . . (5)
where k is the momentum, k , = ( k z + ~ 2 ) 6and U is a dichotomic variable for t h e
two states of spin. Also

. . . . . . (6)
(there is a misprint of sign in Pauli’s formula (1941, equation (86)).
It is assumed (without restricting the theory) that in ( U ) the indices
( U ) + or - denotes for charged particles, the particle of charge +e or - e
respectively (e being the charge of the positron) ;
( b ) for neutral particles which have charge conjugate states, the indices
+ or - correspond to the sign of the magnetic moment (with respect to the spin
direction) ;
34-2
SI6 L.Michel
(c) for neutral particles which have no different charge conjugate states,
U+ U- (this could be written uo). These particles have no interaction with the
electromagnetic field and cannot be described by Dirac's hole theory. They
were first treated by Majorana (1937), and they will be referred to as 'Majorana
neutral particles '.
Let us call
*IC = *, -U if K = - 1
,i*
u(li)-

$Ic=C, u(IC)=ut if ~ = + l
. . . . . . (7)
T h e index L is either + 1 or - 1, and indicates either (U) the sign ofthe charge
or ( b ) the sign of the magnetic moment, or ( c ) for Majorana neutral particles is
arbitrary (L or -L indicating the same state).
+
I t is seen from (4) that #* can be obtained from by changing t into -I,,
so complex conjugation and charge conjugation are equivalent. Equations (4)
a n d (6) can be written :

4; = 1 Z Z uLL")(k,a)ul,."(k,a) exp { - LKi(k. x - koxo)} . . . . . , (8)

(NW'ko L

and &+(k, o)a"(k, a) = D ( K ) . . . , . . . ( 9)

F o r Majorana neutral particles #I<=#-" (# is real). I n formula (8) the upper
index LK plays the part of the index K as defined by (7), its values being the product
of the K and L occurring separately in (8).
T h e anti-commutation rules can be written
[uf;")(k, a), u y ) ( k ,a')]+ = 6,,,6,,*6(k - k)6,,, . . , . . (10)
+

and

T h e operator NJk, a) represents the number of particles in the state k, U, L ; its

eigenvalues are 0 or 1 .
ujf.") is an absorption operator if LK = M = - 1, and
a n emission operator if LK = M = + 1 ; * . (12) . ..
I

for a charged particle, of charge Le, it changes the total charge by an amount Ke.
All the dichotomic variables used here are listed i n Table 3.

52. PARTICLES OF D I F F E R E N T K I N D S
If there are two different kinds S of particles, their amplitudes v(S) commute:
[dK)(S, L), d"')(S',L')]- = O if S+=S'. . . ..
. . (13)
However, it is even possible to consider particles of different rest mass as .
like particles. F o r example, neutrons and protons are considered a s nucleons,
o r electrons and neutrinos as leptons, by the formalism of isotopic spin, which
is nothing other than writing the same d s in a more condensed form :
 Interaction between Half-Spin Particles and Decay of p-Meson 5 17
Another interesting extension which has been made is to consider a Majorana
neutral particle and the two states of a charged particle as three conjugate states
(+,o, -) of the same particle (Noma 1948).
Alternately there is a transformation, due to Klein (1938), which makes it
possible always to have' anti-commutation between the amplitudes of two like
or unlike particles.
If the n's satisfy (13), putting dK)( 1 , L) = dK)( L)= t;( l)u(")(2,L)J
1, L) and d1<)(2,
with {(l)=IS[l - 2 N ( 1 , ~ ,k, U)] (II is taken over all the states ~ , k , u one
) gets
[@)(I, L), ~ ( " " ( 2L',)]+ =O.
This can be extended to any number of kinds S of particles, so that one gets
[U'-K)(S, L, k, U), u'""(S', L', k ,U')]+ = SICK,8S.y SLL,S(k - k )Suo,. . . . . . . (14)
$ 3 . SCALARS F O R M E D F R O M F O U R +K

The method of formation of such scalars is well known. Using only t h e

relations (1) and ( 2 ) , it has been demonstrated by Pauli (1936), corrected by
Racah (1937). With the four y, one can build 16 matrices yA=yLi, ( A= 1 to 16
or i = 1 to 5), viz. :

A z a=l a=2 a=3 a=4 a=5 a=6

1 1 1
2to 5 2 Y1 Y2 Y3 Y4
6 t o 11 3 iY2Y3 iY3Y, iYlY2 il/m iY2Y4 iY3'Sya
12 to 6'2Y3Y4 i'Y3Y1Y4 iY1Y2Y4 -'YlY2Y3
16 YIY2Y3Y4

It is easily, the y , satisfying ( 2 ) and (3), that yA2= 1 and y a t = y a . If

Fi, = ' i u ~ 4 ~ [ 1u1, . . . . . . (151

where qa is such that eL2 = +, - - - +, +++---, +++-, -, then
Fiat = Fi, . . . . . .(16)
and [+s"(x> . ~ o ) l , ( F ~ ~ ) l , " [ *~~0 ,) K ' (#s"Fia#s,"'
I v= ~,
p

is a scalar for i = 1, the four components of a vector for i = 2, a skew tensor for i = 3 '
a pseudovector for i = 4 and a pseudoscalar for i= 5.
The scalars we require are J i ( ~P ,) = t h e scalar product of +11"F~u$2K' and
$3Fi&4"d, i.e.
, = C ~:~(*~'~lFi,*?g~)(*3T'~Fi(~~4'~'), (17)
J i ( ~P) ......
r,

where cia= +, + + + -, + + + - - -, + + + -, +, and ( K ) is a contraction

for (K1,K ~ K, ~K, ~ and
) can have 16 different values ; P indicates that the four *E
inJ,(P) are in a definite order: 1, 2, 3, 4.
From (15), (16) and (3) it is found that
-Fi, = %il.;, ......( 1 8 )
with%,=- I , + I , + I , - I , - I .
j I8 L.Michel
54. T H E M O S T G E N E R A L H A M I L T O N I A N OF T H E I N T E R A C T I O N
W e choose the Schrodinger picture for the operator H which then does not
depend explicitly on the time, and have taken the same axes for the Heisenberg
and Schrodinger pictures at time xo=O. We therefore put xo=O in J j ( K , p ) ,
given by (17).
T h e most general H a i l t o n i a n of the interaction, which must be Hermitian,
can be written :
24 llj 5

H= Jv h ( x ) d x with h = I nC= lKC C

= l <;;I
!I{(K, P)[Ji(K,P ) + J ~ ( K , P ) ] .
T h e s u m C indicates a summation over all the twenty-four permutations of the
. . . . . . (19)
P
four indices S = 1 , 2, 3 , 4 ; the s u m C indicates a summation over the 16 possible
K
, ; and O{(K, P ) are real numbers called
sign combinations of K = ( K ~ K, ~ K, ~ KJ
interaction constants.
For the following considerations we must exclude the transition in which
two particles are absorbed and emitted again i n their initial states: this would
correspond to the self-energies of these particles due to this interaction.
We can then say that the #Ic anti-commute, and from (16) it follows that
(#SR Fju#s?c)'t= - +s;Ic F,#,-".
Therefore, if Pl is the permutation (2143)
J;(K, P ) =Jj( - K , P I P ) . . . . . . . (20)
From (14) and ( 1 8 ) one gets
Jl(K, P I P )=( - 1)' 6j2Ji(K,P ) =Ji(K, P ) . . . . . . . (21)
Therefore (20) can be written
J?(K, .P) =Ji( - K , P )
and (19) becomes
h = C C C g i ( K , P)[Ji(K,P ) +Ji(- K, P ) ] = C C[&(K, P ) + S i ( -K, p)]J((K,p )
P K i I' K i
or h= Cgi(K, p)J$(K, P) . . . . . . (22)
1' K i
with !li'(K, p ) = g i ( K , p ) Of( - K , p ) ;
therefore one has g i ( K , P ) = si( - K, P ) . . . . . . .(23)
$ 5 . E L I M I N A T I O N O F T H E S U M M A T I O N OVER T H E
TWENTY-FOUR PERMUTATIONS
I t is well known (see for instance Speiser 1945) that the S, group (four object
permutations) is soluble. All the permutations are given by the twenty-four
terms of the symbolic expression: (1 +P3)(1+ P 3 P 4 + P 4 P 3 ) (+P,)(11 +PI)
with P I =(2143), Pz =(3412), P3=(2134), P4=(1432). Each P is a product
of the form
P=P,P, P,, .... (24) ......
where a,b, ....
k = 0 t o 4 and Po 1. Let US write ( - 1 ) ' = + 1 or - 1 according
as P i s an even or an odd permutation.
W e will now show that, for every P ,
Jj(K, PPo) =E( - 1)'c:Jj(K, Po), ...... (25)
j
 Interaction between Half-Spin Particles and Decay of W-Meson 5 19
which means that under any permutation P of the four indices S , the Jiundergo
a linear substitution; later on J i ( ~P)
, will simply be written J:(K).
Indeed, for P = PI,P, or P3 we very easily find : CG1= 6 , (see (21)), C: = 6,
(evident), C? = S& (see (18)). T h e case of P = P4has been studied by Fierz (1937),
and the same method can be extended to the present choice of F,, ; it is found that

*
cp=

of the four permutations PI,

I
cp =
-1

-$
Q
1
t -&
-4
- 12

-t
0 -4
-&
k
0

0 - 3
-$
0

B
‘Therefore for every permutation P,which is always a product o f the form (24)
P,,P3, P4,we get
E C;CL
c,:b..
-B
-1
-8

. . Czk.
1
t I
83 h,.?.
Then (22) can be written h =xP K C g;(K, ij
P)( - 1)‘ CGJ,(K).

By putting g j ( 4= 9’(K, PI( - 1 v ;

2, I-’

.one gets h=Ex&(K)+(K) . . . . . I (26)

K j

and (23) gives g j ( K ) =gj( - K ) * (27) ......

This last relation will be studied in §7.
Changing the order of the particles in H only means ‘changing the name’ of
the interaction constants g, ; thus H remains invariant.

$6. A R E M A R K A B L E I N T E R A C T I O N
Can we define an interaction which is independent of the order of the four
tL” in H ?
A permutation leaves H invariant :
h= cgj(K)J,(K)
K J
=x x g ; ( K ) J , ‘ ( K )
K L
with J,(K, P)=Jj‘(K),

but changes the ‘five vector’ j’(~)into “ g ’ ( ~ )according t o g = ( - l)‘?. “g‘ because
equation (25) gives J,’(K) =x(
-1)z’CtjpJ3(K),so that one must have
J

h=xEg,(K)J,(K)=E~gl’(K)( - l)pcljpJj(K),
K J K 73

which gives Xg,’( - l)pCj92’

7
=g, or ( - 3’. Cz’ =zor ( - l)”Ep.-j’=z.
The above question is thus reduced to this : are there common ‘eigenvectors’
to the twenty-four (-l)‘C? matrices? T o answer this it is sufficient to look
for the common ‘eigenvectors’ of - and - 61’4, since the other ( - 1)‘z“ are
$8

products of 1, - e ‘ s and -ep&.

One finds that the only common ‘eigenvectors’ are given by
g, =g3 = O,? g, = -g4 =g,. - (28a) ..... .
 520 L.Michel
They correspond to the ‘ eigenvalue ’ + 1. Therefore h is symmetrical for
the $lC (which anti-commute) ; this is easy to verify, for, starting from
h= ~ ~ ~ K ) F JP), (=K , E ( K ) ~ ( ? (-
K a I K z 3 I
1)pc,,p)4(K) ==ExZ(K)Z$,J,(K),
K 73

with 2 =(1 - C’.)(l + Cp3Cr’++ Cp4Cp~)(1 + Cre)(1 + Cpl), one finds

h=zz(g1-4g4 +gs)(J,-J4 fJ5) 01 h=zg&K)[Ji(K)-Jq(K) +J5(K)].
K K
......(28 4
It is also easy t o verify that there is no h completely antisymmetrical with respect
to the $IC (which anti-commute) because
(1 i;CP3)(1 + c p c p 4 + C P * C P 3 ) ( 1 + CP,)(1 + Cl>$)
= 0.

Has this remarkable interaction a physical meaning ? This question has na

meaning if H is only phenomenological (see introduction). However, the field
theory allows us to assume a physically direct interaction between four particles
(that would be unthinkable with only the classical notion of forces), and we have
seen that the only direct interaction symmetrical with respect to the four particles
is the remarkable one go (see equation (28)).
Now, in a Dirac equation, it is well known that F, and F3 correspond to
electromagnetic interaction and the sign of the interaction constant (electric
charge or magnetic moment) is changed by charge conjugation ; F,, F4, F5 would
correspond to a non-electromagnetic interaction and the sign of the interaction
constant (mesic charge (Okayama 1949) for instance) is not changed by charge
conjugation; but the J,(K), which are quadratic with respect to the F,,are
invariant by charge conjugation and a(.) =g,(- K). However, there is only one
interaction built either with F,, F3 or F,, F4, F5 independently of the order
chosen for the $K in writing h : it is the remarkable interaction go built with
the three non-magnetic operators.
T h e interaction go has already been proposed by Critchfield (1943) and can
be written as the determinant of the four components of the four $I<.

97. C L A S S O F R E A C T I O N S W I T H S A M E g(K)

A reaction between four particles is defined by (see Table 3) the values of

~ each particle (S= 1 t o 4).
the M~ and of the L , of
T h e conservation of the electric charge requires
MsLs=O, ...... (291,
(S)l

where C means the sum over the charged particles only.

W e have already seen (12) that
MSLS = K S , ......(30)
so (29) can be written
Ky=O. ......(31)
(SIC

T h i s relation indicates the possible different values which can be taken for
, ~ and
K (K1,KZ, K ~ K ) for Z(K)=g‘( - K) in H .
 Interaction between Half-Spin Particles and Decay of p - M e s m 521
For example, in P-radioactivity the magnetic moment of the neutron is of
negative sign ; let m be the sign of the magnetic moment of the neutrino emitted
in p--decay. T h e n the reaction
N P E Y
(a) N--+P+ corresponds to + + +
+E- +U,
+ - m
and (28) gives for K K + + - m
With four Dirac neutral particles there are of course Z4 x Z4 =256 distinct
possible reactions M = (Mlr MZ, M,, M ~ ) L, = ( L ~L, ~ L,,
, L ~ ) . T h i s number is different
for other kinds of particles since the charged particles have to fulfil (29) and the
Majorana particles have only one state L. T h e result is given in the fifth line
of Table 1.
Table 1
No. of charged particles 0 0 0 0 0 2 2 2 4
No. of Dirac neutral particles 4 3 2 1 0 2 1 0 0
No. of Majorana neutral particles 0 1 2 3 4 0 1 2 0
No. of different? or classes 8 4 2 1 1 4 2 1 3
No. of different reactions 256 128 64 32 16 128 64 32 96
No. of reactions per class 32 32 32 32 16 32 32 32 32
This table refers only to the cases of four distinguishable particles.

When a reaction is realized in nature, a Z(K) -K) =z(

must be given for
describing it, and all the reactions which have the same f K = ( K ~ ,K ~ K,,
, K~ or
- K ~ , -K*, -K,, -KJ can be associated with this f. T h e set of all reactions
which can be described by the same g' will be called a ' class '. Therefore the
interaction Hamiltonian density which describes all the interactions of a class is
(like (26), but without summation over K)

For example, the reaction

(b) N-+v-,,+P+ +E- belongs to the same class of reactions as ( a ) :
K=( +, +, -,
m), but the reaction
(c) N- +u+~~,+P+ +E- belongs to another class: K( +, +, -, -m), unless
the neutrino is a Majorana particle, when the two reactions would be identical,
v-111= VI,,.
This very simple remark will be useful later on.
For given kinds (charged, Dirac or Majorana neutral) of the four fermions
the fourth line of Table 1 gives the number of different classes of reactions and the
sixth line the number of reactions per class.

$8. S T U D Y O F O N E REACTION. DETERMINATION OF TRANSITION

P R O B A B I L I T I E S : SPIiV A V E R A G E S
For this study it will be assumed that the transition can be studied by the
Born approximation, i.e. the particles are described by plane waves.
The transition is given by M = ( M ~ M, ~ M,,
, M ~ and
) L = ( L ~L, ~ ~, 3 ~, 4;) therefore-
K is known since K~ = M ~ L ~ If. H , is the element of H which leads to the transition
T, the probability per unit time of the transition is proportional to I HTI'.
.5= L. Michel
First case : the four particles are distinguishable.
From (8), (12), (17) and (26) one gets: H T = S v h T ( x ) d x , i.e.
.HT = ~ g ~ ( K ) E I 1 ’ ~ l , , ( l ~ , l ~ , ) ~ ~(, L~ LS ~Y KS ~~( )L~uK 4 )
1.3 L4
In

V--1)4 [exp { - i(XKSLSkS).

x (alLiKi~~ua2La1C,)(a3L~K~~iuu4L~K4)( x}] dx j V S

where XLsKsks = &TSk = 0 from the conservation of momentum. Therefore

S S

t h e integral is equal to V and

1 HTI2= H T + H T = X X ~ - 2 g i g i ~ i u ’ q b ’ U . A j a i ( , . . . . . , (32)
ia j b
with U = 1 if the transition is allowed, i.e. the particles to be absorbed are present
.and the states of the particles to be created are not occupied initially (see (11)).
Afa,b(ks,
us) = ( a 2 - ” ~ F i , u l - ” ~ ) ( a l ” ~ ~ , a 2 ” ~ ) (Laaa,3a-‘’- ~ ~)(*a,”3F
F. j b a4’I *)
. . . . . . (33)
where LK has been replaced by M because of (30).
However, experimental physicists do not usually distinguish between the
states of spin of the particles. By summing over the different states of spin
-of the emitted particles and taking the average over the 11 = 4 x 2-*z’fs initial
.states, which are not distinguished experimentally, one calculates :
1 V-2
-XI HTI2= - X X & g b ~ j ~ ‘ ~ j b ‘ Z A f a j b ...... (34)
I1 .ss ‘I ia j b OS
.and (9) gives:
I;A =Trace F i a D 1 ( ~ 1 ) q b D-2M
-3
( ~ x) - M4)
Trace F & ( M ~ ) ~ $ ~( . . . . . . (35)
Because of the symmetry between 1 , 2 and 3 , 4 one finds that X1HTI2is quadratic
OS
i n the M~ and does not change when one changes all the M~ into - M ~ so , that the
reversibility principle of microphysics is satisfied.
One sees that the probability of the reaction M , L depends only on the Ms;
it is independent of the L~ : the formalism is completely symmetrical with respect
t o the charge conjugate states of the particles ; therefore it is also independent
of K, the class of the interaction.
This shows that for the so-called ‘ allowed ’ transitions of the ,&radioactivity,
.
the life-time, the electron spectrum, etc. . . are the same whether the neutrino
is a’Majorana or a Dirac particle.
Second case : some particles indistinguishable.
Firstly, let us say two particles are indistinguishable, and let us suppose them
t o be 0’ and 0 , i.e. K” = K ~ , L~
, = L ~ , M~
, = M ~ , . So we get
,
$O,Ko = $“,o = $olco = (2V)-* X o ) ( k (,k, R ,uR)exp { - MOikR.X}
~ , ~ ( ~ ~ an)a3ro
R=l
. . . . . . (36)
Jihas two identical ; therefore H is invariant with respect to their permutation,
a n d it follows that H is a function of only three independent constants gi.
 524 L. Michel
For the remarkable interaction go =g, = -g4 =g, one finds
172
- C 1 H,lz =6 +(m12 +mz2 +m3z +m42)-(ml +m2 +m3 +m4)2+6m,mzm3m4
go2 a.q-
+ PI. pZm3m4 +p3 p4m1m2) + 2 ( p 1 .p3mzm4+ p 2 .pam,m3)
+2(pl.p4m2m3 fPZ.p3mlm4) f ( P 1 2 fpZ2 +P3'+p4')

- ( P 1 +PZ +P3 +P4l2 + 2 [ ( P l' PZ)(P3 P4) f (P1* P3)(pZ * p4)

+ ( P 1 ' P4)(PZ * P3)1,
that is, completely symmetrical in the four particles.
F o r another set of values of M = ( M ~ M, ~ M, ~ M, ~ one
) can use Table 2, replacing.
ms by M S m s .
T h i s table depends of course o n the Fju chosen. Ft, is defined in (15) as a
function of yi,. Below, the Fi, are given as a function of the U , ~ ,,B. T h e above
calculations are only possible in the Majorana representation, unless
M~ = - M ~= M = ~ - M ~ , when they are also possible i n the Dirac representation.
F o r the latter case the F used here are given as functions of Dirac's pk and uk.
F1= P =P3

Fz= u1 a2 013 PI L1l

F3 = - ~ P u Z U ~ - i p c ~ 3 0 1 1 - i / 3 ~ l ~ [z- iP.11 [ - i / 3 ~ 2 ] [ - i / 3 ~ 3 ] = p a u k [pzukl
F4 -iu,a3 -i~30ri - i ~ r , ~ z [-i~10r2~3] ==IC [fll
F, = , - P u ~ u ~ u ~ =Pz
(time components are given i n brackets; the sign of the Fiu is immaterial).
I n Dirac's representation one must choose Fjuinstead of Fi, in I,hs"Fi&s."' with
C=p2u2and
S i u Fiu for ~ = + l~ ' = - l
F~~ = E F ~ ~ ~ for c - ~ K = - 1 K' = + 1
.F<u = F,,C-l for K = + 1 K' = + 1
I

.F,=CFiu for K = -1 K' = -1

99. D I R A C ' S H O L E T H E O R Y
It is not possible to describe the Majorana neutral particles by Dirac's hole
theory. If no such particles are considered, this theory is of course entirely
equivalent to the formalism of charge conjugation b u t not so convenient. In
this section we shall follow up in some detail this equivalence, which shows in
particular that it does not matter which charge we attribute to the particle or
to t h e hole.
W e distinguish between the physical particle and the theoretical ' corpuscle '.
Let us write the I,hK(x)of one ' corpuscle' of electric charge ee (with E = i 1) or
(for a neutral particle) of magnetic moment etit, of momentum tt, of energy
tio= 7(ti2 + K ~ ) * with 7 = 1, as
I,h,"(x) = V-W<)(tt, 0,?)a,"(k, U, 7 ) exp ( - - i ~ t t . x). . . .
. . . (37)
N ( &U, , 7 ) = u t @ , U, ? ) U / ( & , U: 7 ) is the operator number of ' corpuscles' in the
state it, U,?. Physically this corresponds to
 Interaction between Half-Spin Particles and Decay of p-Meson 525
-N:,(k,U) is the operator number of particles of charge vee (or magnetic moment
1 1 ~ ~of~ ~momentum
), k=+t, in the state U, since the vacuum is defined by
N(t, 0917)= U-17).
This leads to two difficulties :
(i) T h e operator Z' =II[l-ZN(ii, u,q)] corresponding to 5 of $ 2 is not so
easily defined because now a n infinite number of states is occupied. I n a diagonal
representation 5' is of the form( - l)-, which gives rise to additional mathematical
difficulties.

Table 3. Table of the Dichotomic Variables used i n this Paper

Letters
X 1 -1 Physical significance used Definition

M Emission Absorption of the particle §-I

,(d . ut U Hermitian conjugation K, KL=M (7)
The two charge conjugate states
112 11; U- (sign of the charge or of the (8)
magnetic moment)
z', # I,?
Y a*, 4* a, * Complex Conjugation K,KL=M (7)

Sign of the energy of the

corpuscle
Sign of the charge (or the
magnetic moment) of the
corpuscle
U The two states of spin

(-l)p For even For odd permutation P §5

EtaZ I n Fia=qay4y[ia1
foi Fiat=Fio (15)
Eta( Space Time components of the invariant (17)
*SKFi,*S,K'

oj Symmetrical Antisymmetrical Fia;+-l, +1, +1, -1, -1 (18)

(ii) T h e +'<
used in h cannot be the sum of all the plane waves describing the
'corpuscles' (they are infinite in number), but contain only u(")a,"exp ( -K&. x)
for the 'corpuscles' which will be emitted or absorbed. T h e n the exclusion
principle is not automatically satisfied. If there are indistinguishable particles-
they correspond to indistinguishable corpuscles-one must symmetrize h with
respect to the corresponding +Ic (which anti-commute). This symmetrization
for an expression like +s"Fjn+s,~is automatically carried out in the Heisenberg
notation (Heisenberg 1934). One writes &(+s"F,,+,."' - Oi+rK'Fi&s")instead
of ~s"Fj,+s"'.
Otherwise we form h as in $4 and get the same formula for IHTI2 and A
(see (32) and (33)).
 5 26 L. Michel
For the study of a transition given by M and L (with 2 MSLS = 0 ) we have the
two relations : (S)r

MS = VsKs . . . . . . (38)
and =s =?s's. (39) ......
Here a new situation arises in that it is not clear how the eS (sign of the electric
charge or the magnetic moment of the corpuscles) are determined. T h e question
arises whether the eS have a physical meaning or are arbitrary ; it will be shown
that they are arbitrary.
Let us therefore choose the eS arbitrarily: for the study of the M, L reaction,
(39) gives us the vs and (38) the K~ It is well known that the summations over
t h e spin states are made by the projection operator defined by Casimir (1933):
(
a(x)=g 1 + c r . t +h,X P K ) = f ( l + a.k+x7Px
) . . . . . . . (40)
ko
One easily sees, i n the Majorana representation, that
%)Ic(.) ='a(- K X ) =%(KX) . . . . . . (41)
and from (9) D(x) =D(vx). . . . . . . (42)
With the well-known properties Dl'(v)u"(ti, U,7') = G,,,a"(k, U, 7') one gets

Ca,-K(ft, U, 7)a,"(k U,?) = [%Ii( +)IALv= [D(K)],,, (cf. (41)).

T h i s formula is equivalent to (9). Therefore, by the same method, one gets,,

instead of (35),
C A = Trace F,,D(K,)F,<C( - K ~ x) Trace F&(K&D( - KJ . . . . . . (43)
OS
and from (42) and (38) D(K)= D ( ~ K =D ) ( M ) ; hence the formulae (35) and (43)
are identical.
T h u s XI H T ! 2depends only o n M ~ and
, is independent of L~ (sign of the electric
charge or of the magnetic moment of the particles), of vs (sign of the energy of the
' corpuscles ') and of eS (sign of the electric charge or of the magnetic moment
of the ' corpuscles ')
T h u s the eRare completely arbitrary* and have n o physical meaning. That
shows the artificial aspect of the hole theory, which, however, is more customary
and, therefore, more intuitive.
Moreover, this formulation does not contain the conservation of the electric
charge. From (38), (39) and (29)

but it has been seen that the eS are arbitrary.

T h e r e are no rules like (31) for the choice of the K ~ . T h e K~ are completely
arbitrary (a transition between particles can be described by 16 different transitions
between ' corpuscles ' ; but after their choice the E~ are determined, by (38) and
(39), and give rise to a charge interpretation of the representation.
* In Tiomno and Wheeler's (1949) paper on the interaction p, 6 , v, v , the authors have had to
choose a sign of the charge of the p and E corpuscles. This is only due to the fact that they have
limited their choice of the Hamiltonian to the kind h=gpJi(+, -, +, -)=gl(+*Fi+). (+*Fi#),which
is the simplest when one dces not use the Majorana representation of Dirac's matrices (see P. 52').
 Interactian between Half-Spin Particles and Decay of p-Meson 5 2 7
11. T H E D E C A Y O F T H E p - M E S O N
Some experimenta1,results published in 1948 pointed to the decay of t h e
p-meson into more than two particles (Steinberger 1948, Hincks and Pontecomo.
1948). The similarity of this decay to a P-decay was immediately noticed
(Klein 1948, Horowitz et d.1948). The suggested scheme is p*+pO + E * f V ,
where po is a neutral meson already known (Lattes, Occhialini and Powell 1947)
from the m-meson decay: z-f?p* +A, where h is written for po.
From the work of Part I It 1s easy to study this phenomenon.
510. D E C A Y ~ + A - E + v
Here the Born approximation is used and the electromagnetic interactions-
are neglected (cf. Feer (1949) for photon production by charge acceleration).
From the results of Part I one sees that it is not necessary to know whether
and v are, independently, ' Dirac' or ' Majorana' neutral particles. The
results are the same in any case.
Let us call the rest-masses, in energy units, of the corresponding particles
p, A, E , v (with v = 0). The frame of reference will be used in which the p-meson
is at rest. Let us denote the momenta and the energies (in energy units) of the
particles by p;'=O, p, = p , p;, p. ; E,, = p , E, = E , E,, E, =p,. The conservation
law gives: p + p i +pv= 0, ,u = E + Ei +E, = E , (total energy). Table 1 immediately
gives C [HT12 for this decay; we know t h a t ~ = , - 1, M ~ = M ~ = M , =1 ; the order
chosen is A, v,
E , Y (corresponding to that usually chosen for P-decay : P,N , e, v)-
Then (33) gives* (the gi have the dimensions of energy. volume)

We want to calculate the energy spectrum of the emitted electron. Let us.
denote the angle between p and pv by 0 and let P ( E ) d E be the probability per
unit of time of observing the electron from a disintegration with a total energy
between E and E + dE. We have E < E < W =(p2 e 2 - h2)/2p, +

n=2 (see (34)); p d p = E d E ;

(p- E ) ( p - E + p COS 0) - p ( W - E )
( p- E + p cos e ) z
* This formula contains as special cases the expressions (8-12) of Tiomno and Wheeler (1949).
One would therefore expect that our formulae (45) and (46)would include the corresponding formulae
in that paper. unfortunately a closer comparison reveals various discrepancies, which, however,
has been cle&d up in th: course of a correspondence with Prof. Wheeler and Dr. Tiomno.
It appears that many formulae as given in their excellent paper are marred by errors of tran-
SCnptlon or by slight'slips in calculations. They intend to publish a full list of these errata.
 X

During the last year the existence of the po-meson has become doubtft
and some recent experimental data (Leighton, Anderson and Seriff 1949) show
that W = 5 5 MeV. i.e. about 8 p. Therefore the mass h must be very small and
the decay of the p-meson into an electron and two neutrinos is very probable.
We must consider two cases.:
(i) All neutrinos are Dirac neutral particles, and the two emitted neutrinos are
distinguishable (their magnetic moments are of opposite sign). We can then
obtain CI HTI2 and P(E)by putting h =O i n (44) and (45).
0
We get a simpler formula by choosing the order p, E , Y, Y (of course now thegj
are not the same as in ( 4 5 ) ):
2- 2 4
c) )2 3 p [ 3 E ( W - E ) K l + 2 ( E 2 - e 2 ) K 2 + 3 e ( W - E ) K 3 ] . . . .
P ( E ) = 3(5E( 2 ~ 5 . . . (46)
+
with Kl =g12 2(gZ2+g32+g42)+gj2,K , =gZ2+2g32 +g42,K 3 =g12- 2g,2 + 2g42--g,2.
I n the preliminary note (Michel 1949) that formula was incorrect (put K4= O and
correct K 3 ) , but the errors do not change t h e spectral distribution by an
appreciable amount.
(ii) Either all the neutrinos are Majorana neutral particles or all the neutrinos
are Dirac neutral particles, and the two emitted neutrinos have the same charge
conjugate state (their magnetic moments have t h e same sign); in both cases
the two emitted neutrinos are indistinguishable.
From Part I (5 8, second case) we know that then we obtain the same result
as above excep; that there is no interaction in g , and g3 (in the formula (46) put
g , =g3 =O).

5 12. C O M P A R I S O N W I T H E X P E R I M E N T A L R E S U L T S
T h e term proportional to K3does not appreciably change the curve P(E), and
it is neglected in the Figure.
If T is the mean life of the p-meson (2.15 microseconds)
 Interaction between Half-Spin Particles and Decay of p-Meson 529
It is already well known (Tiomno and Wheeler 1949, Horowitz et al. 1948) that
theg, are of the order of the Fermi constant ( erg. cm3). T h e condition (47)
gives us the scale for drawing the different curves which we want to compare.
Let us denote the curve .?'(E) by P,when only one g,#O (P, =PE and P, = P4).
All the possible curves P have (for 0 < p < 1) the shape
case (i) P,(E) +p[P,(E) - P,(E)] and sweep the whole areas A and B
case (3) P,(E) +p[P2(E)- P,(E)] and sweep only the whole area B.
All the curves pass through the same point M.
The experimental curve and the expected statistical spread given by Leighton,
Anderson and Seriff (1949, Figure 5 ) are plotted together. T h e curve Po
corresponding to the remarkable interaction go (see (28)) is also shown.

. .

0 5 IO 15 20 25 30 35 40 45 50 55
Energy E (MeK)

_- Theoretical curves P,-Pp,, P,-Pp,, P,, boundaries of the A and B areas.

_ _ _ - Theoretical
- curve Po, remarkable interaction.
., . , . . Experimental curve of Anderson et al.

The agreement is quite satisfactory and the experimental curve also passes through
the point M.
There is thus strong evidence for the decay of the p-meson into one electron
and two neutrinos.* But, since the experimental curve falls in the B area,
nothing can be said about the nature of the neutrinos ; furthermore, the good fit
of the theoretical curves with experimental results does not prove that a ' direct
interaction ' necessaiily exists between pmesons, electrons and neutrinos.
* Of course this conclusion rests on the assumption that the p-meson has a spin 0: one-half. It
\\.auld be possible, as pointed out by J . Tiomno (Phys. Rez.., 1949, 76, 856), to reproduce the experi-
mental curve on the assumption of an integral spin for the p-meson and one of :he neutral particles.
PROC. PHYS. SOC. L X I I I , 5-A 3s
530 L. Michel
$13. P R O P E R T I E S O F T H E N E U T R I N O
( a ) Its magnetic moment. From the theoretical point of view it is more exact
to ask : is the neutrino a Dirac or a Majorana neutral particle ?
F r o m the decay of the p-meson it would have been possible to say that the
neutrino is a Dirac particle if the experimental curve lay in the A area, but this
is not the case. However, there is one phenomenon rather favourable to the
hypothesis : the neutrino is a Majorana particle. This, as was shown by Furry
(1939), is a double ,El-radioactivity without the emission of neutrinos. Recently
experimental evidence for this phenomenon was published (Fireman 1949) ; it
is the spontaneous decay
Sn +l;$Te +2e -.
No neutrinos are emitted, otherwise the lifetime would be lolo times larger
than the observed one.
Theoretically this is explained by the reactions : ( U ) N--+P+ +e- +vnl and
( c ) N- +u,,+P+ +e- i n the following scheme:

intermediary
initial state + + final state
virtual state
N-1 +N-z-+Pi+ +el- +v,, +N-Z+P,+ +el- +Pz+ +EZ-
but we have seen ($8) that the ( a ) and ( c ) reactions do not belong to the same
‘ class ’ unless the neutrino is a Majorana particle.
T h e simplest course is, therefore, to assume that the interaction Hamiltonian
is reduced to terms corresponding to the single class embodying (a)and ( c ) when
the neutrino is treated as a Majorana particle. However, one cannot exclude the
possibility that the Hamiltonian would consist of two distinct f ( ~ )corresponding
t o t h e two classes involved in which neutrinos are Dirac particles. This
possibility has been discussed by Touschek (1948); for ordinary P-decay this.
leads for each process to two competing transitions which differ only by the sign
of t h e magnetic moment of the neutrino, of course without interference between
them for such phenomena of first order in t h e z . I t may be observed, in particular,
that if g’(Kl) =?(KJ (the same g’for both classes) this theory, which also maintains.
distinction between neutrinos and anti-neutrinos, is formally identical for all
,El-processes with that involving Majorana neutrinos.
T h u s if the neutrino is a Majorana particle it has no electromagnetic interaction.
T h e neutrino appears only in the spontaneous decay of elementary particles, and
it is only required to preserve the conservation laws of energy, momentum and
angular momentum. I t therefore seems to be connected with the gravitational
field ; but perhaps not so simply as might be thought (Gamow and Teller 1937).
T h e interaction between p-meson, electron and neutrinos gives, in the second
order, a very small interaction of infinite range between a p-meson and an electron
of the same charge. According to the calculation of Noma (1948), the static
potential interaction is proportional to y-5.
( b ) Its mass. From the formulae (45) and (46) it appears that there is a large
difference between the two cases X # O and X=O. This is due to the fact that,
in (45), P(E) is proportional to ( W - E ) Z , but I his factor, which disappears if =o,
would also have disappeared if v+O when X f O , and even if v = h # O , as in the
well-known case of @-spectra the curve P(E, u)+P(E, 0 ) when u+O, although t?
same is not true for the derivative dPldE. One cannot hope evidence of this
 Interaction bc'tween Half-Spin Particles and Decay of p-Meson 5 3 I
kind will give an answer to the question: is the mass of the neutrino finite?
Equation (48)gives a good example for the mass E of the electron. T h e tangent
of P(E) at the point E = E is vertical when e+O and horizontal when E = 0, but
P(E, e)+ P(E,0) continuously as E+O.
AC K N 0 W L E D G M E N T S
T h e author wishes to thank Professor Rosenfeld for helpful advice and
discussions. H e is indebted t o the French Service des Poudres for making
possible his stay at Manchester University.
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