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A new majorana field theory
Nuclear Physics 22 (1961) 262--282 ;@North-Holland Publishing Co., Amsterdam permission from the publisher Not to be reproduced by photoprint or microfilm without written
A NEW MAJORANA FIELD THEORY H. UMEZAWA t and A. VISCONTI Faculté des Sciences de Marseille, Prance Received 16 June 1960 Abstract : After tr-, short introduction, we review in sect. 2 the various isospin schemes for strong and weak interactions and we show that they can be reduced to a common scheme by writing down an invariant of a set of 2 x 2 matrices, whose elements are linear combinations of the particle wave functions. Various properties and difficulties of those schemes are discussed . In sect. 3, we propose a scheme where each set of a particle and its antiparticle is described by a quaternion . Several interesting properties of that scheme are pointed out: the rule for constructing quaternions leads directly to charge conservation ; the strangeness conservation for strong interactions and the rule IASI = 1 for the weak interactions are direct consequences of this theory, furthermore all the unwanted electromagnetic interactions are naturally excluded.
1. Introduction
The weak interactions present several important and characteristic features First of all, the rule 4AII = 2 which is a generalization of the rule JA 1 a 1 = I (or in terms of the hypercharge : jAU) = 1) seems to be consistent with all the known weak interactions, except for the decay K+ --> a++a0. Then, the mixture of vector (V) and axial vector interactions (A) seems to fit the known experiments and finally there are, for the time being, no phenomena violating the PC invariance . Furthermore, the V and A interactions belong to the 2nd kind of inte-action, whose coupling constants have the dimensions (L) 11 with 71 > 0 in contradistinction to the first kind of interaction where q <_ 0. The distinction between these two classes could be traced to their renormalizability properties, since it is generally believed that no interactions of the 2nd kind can be renormalized tt . It is also knowr~ that the coupling constants for weak interactions are about (10 -17 )2 cm2 for fermion-fermion interactions and 10-20 cm for fermion-boson interactions . In other words, the weak interactions imply extremely small lengths, much smaller than the Compton wave length of the particles known so far. Therefore, one may reasonably suppose that in the theory of elementary particles, one has to consider 3,4) a fundamental length l. However, it must be noted that this assumption does not necessarily mean the nonlocal character of the interactions . Consider, for instance, the low energy phenomena which t On leave from the Physics Dept. University of Tokyo (Japan) . tt See ref . 1, $) for the details of this classification . 262
A NEW MAJORANA FIELD THEORY
263
involve particles whose wave lengths are well above l ; these interactions, which we will call the strong interactions are independent of l, correspond thus to t = Q and belong to the first kind of interactions. The effects involving the fundamental length l are extremely small, when they appear among the former low energy phenomena. They are consequences of weak interactions and belong to the 2nd kind of interactions . The strong interactions are then scalar (S) or pseudo scalar (PS) (both theories are renormalizable), while the weak fermion-boson interactions re of the types V or A . Finally, there is another remarkable feature of the weak interactions : in contradistinction to the stronger ones, they are usually defined by violations of laws which express the conservation of some quantum numbers (strangeness or hypercharge, for instance!) in the case of the strong interactions . But, from our point of view, the definition of the weak interactions has its ground essentially in the consideration of the fundamental length l. It should be therefore desirable to try to understand their systematics by means of only conservationlaws. We are thus faced with the following problems: 1) How to substitute for the rule JJIJ = 2 an invariance law? 2) Why do we have the conservation of the hypercharge in strong interactions but not in the weak ones? 3) How the PC invariance in strong interactions leads to the P invariance? On the other hand, the answer to these questions must not bring any change in the experimentally well established mass spectra of the particles . In particular, the strong interactions have to be charge independent (in the ordinary meaning of the word), and the theory must allow all the mass differences between particles in different sets of charge multiplets (e.g. between the Z, N, A, X multiplets) t. Our aim in the present paper is to investigate the former problems along the lines we just mentioned : after an extensive study of several isospin schemes (sect. 2) we shall introduce in sect. 3 a new scheme (the scheme of new Majorana fields), where by association of particles and antiparticles we shall build up real quaternions. Therefore, to each particle antiparticle set will correspond a quaternion and the investigation of the symmetry properties of- the interaction hamiltonian will show several important and interesting features of this theory .
1' The problem of the charge independence of the strong K interactions needs some comment : there are not enough experimental data which wouh compel oneto conclude that they are (although the cross section of K- capture by the deuteron agrees with the charge independence hypothesis` . However, it seems reasonable to expect that if these interactions were not charge independent, there would have been a quite big mass difference between, say, proton and neutron. This is a reason to assume - as we shall do in this paper - the charge independence of the strong K interactions .
264
$. VUEZAWA AND A. VISOO14TI
It is, however, worth while to note that we shall not deal with the problems of K+ decay (K+ -> -,r++x°) : this problem will be discussed in a forthcoming paper. Z. Isospin Schemes and the 2 x 2 Matrix Scheme
Let us call "convéntional or scheme (I)" the isospin scheme where the nucleon, the H and the K particles build up isodoublets while the X, x particles build up isotriplets (or isovectors) and the A an isoscalar. The JAI( = j rule for weak interactions has been deduced from this scheme. It has been noticed several tines, however, that the charge independence of strong interactions does not necessarily lead to the former scheme uniquely . For instance, the charge independent strong x-interactions can be described either in terms of °) the former scheme (I) as: ~
p
n)
kH-)
(K+
(-KO*) , K+*
KO)
V2 £- --Z+ iV2 or in terms of s, 7) the baryon doublet scheme (II) , , k, ~, X = ~+ ,
r Yo
with Y° _
b11o-E° ~2
,
Z° --
brio-}-~ X8 , -%/2X8-%/2
Z=
Zo
('r-
,
(2)
b a real constant
Furthermore . the same interaction can also be written in terms of the tripletsinglet scheme (111) :
p -#- a .:--/2
p-a,N= No , . ~~ !1 . x, iV2 i1/2
n-a.^O
A NEW MAJORANA FIELD THEORY
26 5
where a is a real constant. In the same way, we shall assign to the K-meson the following set built up by an isovector and an isoscalar: f K
K+ + K+* V2 K+ - K+*
KO
KO + KO*
KO _... KO*
i,%/2
To each scheme belongs an isospace which is well defined: let us call a theory charge independent in a given scheme when it is invariant with respect to the rotations in the isospace belonging to that scheme t. When a =- 1, d'Espagnat, Prentki and Salam 8 ) showed that the scheme (III) leads to a hamiltonian charge independent in both schemes (III) and (1) . Now, the charge independence in scheme (I), on one hand, does not lead to any mass differences between the particles belonging to each charge multiplet set %, 6, E, 916, K and is, on the other hand, compatible with the experimental evidence that the nucleons, pions, H, X, a particles have different masses, therefore each of the hamiltonians has to be charge independent in scheme (I) . One must notice, too, that a free lagrangian charge independent in scheme (II) (in (III)) is charge independent in scheme (1), only when b --- 1 (a = 1) and when the bare masses of Z, 11 (nucleon and H) are equal tt. In order to have an insight into the connection between these schemes, let us consider the strong interactions with a mesons, i.e. NNn, w;r, XZn, X11n in the schemes (I), (II) and (III) . They are obtained by building up all the possible isoscalars by means of field quantities . Since we assumed the strong interactions to be of first kind, we get in scheme (I) the following interaction hamiltonian: Hn u) = 91 91*07 ' 91~7 +92$ *0 r' ~n * +g4,E* 0 A n+g4A* 0,E' aj, (5a) -igsZ A 0.E " ;r where 0 is 0 --
while one gets H, (II)
=
4
(5b)
Y4 Ys
i
'+g3X* 0a ° Xn+g 3 Z* 0t " Zn (6a) gi9l * 0z ° 9179+g2 e* 0T ' e7
.r°,
Z-t- o, AO, nf, o , t Note that in the three schemes, we shall normalize the field quantities p, n, in such a way that they satisfy the canonical commutation relations. tt One may introduce a modified scheme (II') by replacing k and k by the isovector K and the Z++n- is forbidden and thereisoscalar KO, as given by (4) . But for 5 = 1, the process K +p -> fore the scheme (II') disagrees with the expriments. For b 0 1, the free lagrangian is no longer charge independent in scheme (II') .
H. UMEZAWA AND A. VISCONTI
in scheme (II), and HWWI)
= --ig1[N* A
ON - a+NO* ON - a-N * - ONO X} -ig3X A OE - =+ g4Z* - 0A;r+g4A* OE - =
(6b)
in scheme (III) . = bg3, one finds By taking the real coupling constants g2 = a2gi , g4 3r(I) = Ha(II) = H (III) In other words, one may obtain the same interaction hamiltonian which is charge independent in the three schemes, but, as said before, the free lagrangian is charge independent neither in (II) nor in (III), when the constants a, b are different from 1 and the masses of E, A. (N and H) different. The interaction hamiltonian for charge independent K interactions can be written as follows: HKM
= g',%* Or - kZ+g' 2 ~* Or - kZ+g'3 91* OkA+9V* OkA+h.c .
(7a)
in scheme (I), and HK (II) = ig'1(N* A OK - X+No* OK - X-N* OK O - Z}
-- g'3(N * - OKA+N O*OKO A}+h .c. (7b)
in scheme (III) . By taking the real coupling constants g'z- ag'1 , g'3 = bg'1 and g4 = abg'1, one obtains, as before, the same hamiltonian in the above two schemes. The strong K interactions are not charge independent in scheme (II) : there is indeed no way to construct an isoscalar by means of an odd number of doublets. In this theory of global symmetry, Gell-Mann s) assumed that the strong K interactions are charge independent only in scheme (I) (but not in (II)) . However, recent experiments show that the cross-section a (K -}-N -0- K+ N ) has about the same value as that of the pion-nucleon scattering : it seems, therefore, reasonable to believe that these interactions are not weaker than the a-interactions and that the symmetry properties are about the same in both the interactions . The schemes (II) and (III) have a great advantage : even the weak interactions between baryons and bosons are charge independent. But there is an important drawback too. If the total hamiltonian without (the electromagnetic interaction) is charge independent in scheme (III), there is no way to account for the mass difference between the nucleon and .'E". On the other hand, the weak interactions, which are charge independent in (II), disagree with experiments : the E- decay is forbidden as discussed later on. One may explain in another way the existence of the former three charge ndepen0Qnt schemes, and the method we are going to develop here, will introduce spine of the mathematical concepts, which will be used extensively later on. Suppose that we are considering the vectorial space built up by all 2 X 2
267
A NEW MAJORANA FIELD THEORY
matrices; I, e1 e2 , e3 build up a system of generators of this space. Furthermore, we have: ei ef = -iek , [ei , e;] = -26ij, (8) and i, j, k are a cyclic permutation of the numbers 1, 2, 3. The standard representation of this basis is _
e1 - ZQ1
_ 0 i _ 0 1 ) e2 - '12 = ( i 0 ' -1 0
(i e 3 = iß3 = 0 -0i ,
and if A is any 2 x 2 matrix, we may write A =
1 (a+a " e), -%/2
( 1 0)
where in terms of the former representation a 1 . A12+A 21 _, i 1/2
a 2 - A12 -A21 , a3 1/2 -
A11 --A 22 i .%12
,
a=
1 TrA . -X/2
(11)
It is also worth while noting the following formula which may be of use : Tr (ABC) = 1 (abc-a " be-a - be-ab - c+a A b - c) . .%/2
Now, let us define the two matrices A^ = TrA-A = At = A*
A22 - A12 ,
A ll
( -A,,, A22
--A*12
(12)
(13)
A21 A*11 .
We shall call the matrix  the "associated matrix" t of A . A matrix is self associated (antiassociated) if  = A ( = -A), and to any given matrix corresponds a self associated part 2 (A -}- Â) and an antiassociated part 2 (A -A) : A = 2(A+Â)+2(A-Â) . One has for instance
et = e, e* = -e, ê = -e.
( 1 5) A matrix A is called a real quaternion if A = A t, and imaginary one if At = -A . If q is a vector of the tridimensional space, the 2 x 2 matrix Q = exp - 26e -
q) Iql
(16)
is on one hand a real quaternion and a unitary matrix (since the e; are antihermitian) and on the other hand, as well known, Q is the operator describing
(A B) t = A t Bt . t Note that ÂB = f?A,
shown terms two this may We an the thus let be that Q the A under in have way isoscalar isodoublets us transformation viewed is therefore in see of the that matrix any meaning consider a to (10 indeed the matrix the that A' real form the Aunitary the baryon as both -~ (isovector) say e'' the quaternion xtwo "left self Q of under --Q (we the represents as (-A21 that sides AQ* matrix (16) a~xek schemes doublet associated follows and shall respectively "right (18) hand the =transformation The real under = is Q-le,Q, call -A',2 A right = transformation" right (Q -A12 hand equivalent scheme the AQ* obtained transformation behaves -~ Q Aj it quaternion 'il =W1, (antiassociated) the hand rotation A'= the and Q*), hand as"both side (II) AO since an both Q side like by we A transformation" 2side to isoscalar (17) Q* and = az the and sides transformation with have A' a sides (Q) transformation 21 of set while -A21 the (17) transformations = a-1 a, = A22 transformation") -A12 the c-number of part QÂ, transformation triplet-singlet = it and Q has it an follows (-A,2 a,kat isodoublets, is which an of isovector therefore = Al transformed defined aa+a',e,, isovector that (18) elements We matrix can (18), Qek is (17) scheme cin Qaand as completely (see be and W2 Q* clear tand behaves follows further written can as = in an iteke (15)) a(III) the can cut (18) isobe set
ut4zzAWA Ago A . VIO
I
NT9
the rotation of angle 0 around the unit axis qjlgi " It is also known that the three generates e1 e., e2 behave like vectors under the rotation (i.e. e! Q*ef Q =- es where e,' is the vector obtained by rotating the e, vector amind the axis q qj by the angle 0) . This remark shows therefore that the quantities ao and a which describe the =
.
matrix
behave
transformation A --- .
= QA Q_"I = Qfa+auex)Q * = a+a,% a , xet
(17) .
where prove written
.
geometrical be
.
like Now
: .
A
(18)
Since A' and in
(19) :
A'22 (-A'?~
A
.
=
A22 -- .4
(20)
A,,)
We equivalent A22 -A'21)
A22)
A12 A'11)
=
We scalar of 1 _-
-_
(21)
In may t
..
A NEW MAjORANA FIELD THEORY
269
applied to a single scineme, where all the particles will be represented b~ 2 ~ 2 matrices build up in the following way t - ayoJ
a~ b~1o-~a ~/2
~2
~+
(22)
1
(23)
~/2 ~=i
K
-
1/"2
~+
-~o -y/2
-Ko K_ à K+* go*~
1
~/2 ~ " e,
- i (Ko-I-K . e) . ,r/2
{24)
( 25 )
All the former matrices have been expressed in terms of the isovectors and isoscalars as defined in scheme (III) by means of the formula (10) . Remembering the relations {26a) ~* _ ~~ K* = Ky Ko* = Ko~ one obtains We may, therefore, characterize the ~ and K matrices as real and imaginary quaternions . On the other hand the doublets ~Xi and ~ 2 given by the formula (21) are s~l -_ -is,~- -i P {27a) n' for the matrix N, and for the matrix ~ (see (24) and {25) ) . Consider now the hamiltonians H and HK deffined as f c llc~~s
t It should be noted thwt K° is not the particle K° but the isoscalr.r attached to the isovectôr K in scheme (III) .
270
H. UMEZAWA AND A, VISCONTI
where 0 is given by (5b) and the trace operation is to be performed on the isospins indices . As well known, a canonical transformation generated by the unitary operator Q in isospace leaves the trace invariant, hence the formula (28) may be as well put into the form H, = i1/20a,# Tr {& QNa* Q* QN eQ* QKQ* +$3QEa* Q* QE8 Q* QnQ*), (29a) HK =
--,V L g' 1 OaR
Tr {QNa*Q*QKQ*QE8Q* ..~ h.c. } .
(29b)
They can be interpreted in three ways : a) One may consider this canonical transformation as just the both sides transformation (17), Nca ->
QNa Q*, Ia -> Q£a Q*, a -> QnQ * , K -> QKQ* .
(30)
Taking into account the text before formula (17), we are thus led directly to the triplet singlet scheme (III). Indeed a straightforward calculation taking (12) into account shows that the hamiltonians (29a) and (29b) are equal to (6b) and (7b) represtively . b) One may consider the hamiltonian (29a) as resulting from the transformation Na -> Na Q * , _a -3. -ra Q* , 9L --. QaQ * , ( 31 ) applied to (28a) and this leads to the doublet scheme (II), if we take into account the formula (27) . We may remark that (29b) cannot be deduced from (28b) by the former transformation (31) together with : K --> KQ*, i.e. HK is not charge independent in schema (II) . c) One may consider finally the transformations K -+ KQ *, a -> QxQ * , (32) of the hamiltonians (29) and this leads, as easily seen, to the scheme (I) the hamiltonians H. and HK are both charge independent in scheme (1) . Let us now consider the weak ;r baryon interactions . It can be shown that the following hamiltonian is invariant under the transformation (31) Na -> N. Q*,
Xac -> Qga Q*,
i(fY4Y#+ .f' Y4Y.,Y5)a.8 Tr {Na*Z,eô.a)-}-h.c. (33) Therefore it is charge independent in scheme (II) . On the other hand it is not invariant under the transformation (32), but it leads us to the rule 14II = 1. Indeed, if one looks at the transformation of the product Ea n by the formula (32), la
a --> QI, zQ* ,
one sees that Ea a behaves as a set of isovector and isoscalar, while Na behaves as two doublets, one may then conclude that Tr {*~?a*E,#a} satisfies the rule mentioned above . One can show that (33) is equivalent to an interaction propos-
27 1
A NEW MAJORANA FIELD THEORY
ed by Takeda 7) . A perturbation treatment in its lowest order shows that the decay is forbidden and therefore this decay may be explained only by an extremely large a mesonic correction. One of the authors (H. U.) s) and independently d'Espagnat and Prentki 1 °) proposed once consideration of the following hamiltonian for the weak baryon-pion interactions t: H(ENa)
= if [jn* Y4Y#Y( 1 +PY5)Z° r
1
~2
a~,Z
P*Y4Y~Y(1-PYs)~+ aw~°-i-n*Y4Y,~~-a~~++Pn~`Y4Y~YYs~+aw~-+h .c. ],
H(ANa)
= -
(34a)
°+ 2P * Y4Ylt Y( 1- PYs) Xoa p n+
Z .f? P * Y4Y,~Y( 1 +Y5)A0,zn+- _
1/2
1 n*Y4Y,,Y(1+Ys)11ô,~°+h .c .
1/2
where
p=±1, y=1
, (34b)
or y,, .
It has been proved by d'Espagnat and Prentki 1 °) that (29) and (30) together with "--a weak interactions can be charge independent in scheme III when we assume p = 1 . This can be seen from the following relation : 1
2
f{ (Y4YwY),,8 Tr[Na
*
e pnEB ] - (Y4YwYsY)aB Tr { ;~Ta * ~B 2~~}}
--- H(EN ;r)+H(ANn)--if 1 - ~~
+ so* Y4Y,AY5YZ ~2
- *Y4YuY(1__Ys)E- 0 ,i :T°
-*Y4Y#Y(1-Ys) X°0~7t - +2'° *Y4Y j Y(1+Ys)X°0~~z°
2
+
1
~
0/43++
ow
0* V4Y#YE+
Y4Y Y(1+Yb A00 n + X/2
(35)
lu
0.
Y4YItY( +Ys
,1 .
with f =bf,p=1 t The lowest order perturbation applied to results :
(34)
is in agreement with the following experimental
-> oc (,JO -> p +n - ) <- -0 .67+0 .13, a(AO p+n) - 2, c (AO -~ n +n") +) ee Q(Ear -~ p+n°) ti n+TC
(,r+
l a(Z+ -> p+no)l > 0.70+0.30, oc(E+ -> n+:z+) , ce(£- -> n+ ,-r -) \ 0, where a is the asymmetry factor . Note that one may conclude from experiments for the E- del=ay that Pa ~ 0, where P is the polarization of the Z-. We assumed here that a ,. 0 (instead of P -- 0) in the X- decay.
272
H . VMBZAWA AND A . VISCONTI
3. New Majorana Field Formalism As we have seen, all the known schemes in isospace can be formulated in terms of 2 x 2 matrices. We have gone even further and tried to pave the way to a theory where all the interactions - including the weak ones -- must be governed by invariance laws only. But, by associating particles of different masses (e.g. nucleon and H) in our fundamental matrices, we were forced to violate some of the invariance laws in order to explain the mass differences (cf. text just before the formula (7)) . Therefore, we have to construct our basic 2 x 2 matrices by means of particles of same mass (electromagnetic mass differences are here disregarded.) This is only possible if we construct, say, the nucleon basic matrix by means of nucleons and antinucleons. In order to do this, we shall be led to use in an extensive way the formalism of the quaternion algebra. In sect. 2, we defined the real (imaginary) qu~rternion as a matrix A, whose elements are c-numbers, and which satisfies the relations A = At - Â*
(A = -At) .
One can extend without difficulty this definition to commuting fields ; we shall define the Bt corresponding to a Bose field as follows : where the hermitic conjugation * is to be worked out both in Hilbert space and in isospace, and the operation ^ in isospace only. Let us now consider a fermion field F with components Fa and define P = F* y4 . Since the hermitic conjugation for a boson field is equivalent to the charge conjugation, we are led to define Ft as follows: Fat = (CP)a,
where C is the well known charge conjugation matrix If j and t are isospin indices running from 1 to 2,
(36a) (C'I y.C
=
)
(36b) ap(Fp* ) iz , where the * operation is to be worked out both in Hilbert space and isospace . A fermion field will be called a real or an imaginary quaternion if we have (Fat)sa -
On the other hand,
(Cy4
Ft = F or
Ft = -F .
.P0T +CFT6 *) . (C -v/2 Therefore if the quaternion F is real, we obtain CP0T = FQ , CFT= F
(37)
A NEW MAJORANA FIELD THEORY
and if it is imaginary,
C~o _-_ --F° ,
CFT --
27 3
-FT.
One may conclude that the decomposition of a real or an imaginary quaternion into an isoscalar Fo and an isovector F defined two Majorana fields Fo and F. As well known, there is no conservation law for the fermion number in Majarana's theory. In order to avoid this drawback, we shall assign to the pa~~ticles Fo and T a definite handedness . V6Te may fiat remark that by means of the well known handedness projection operators one may prove that the field
hf
= 2(I~YS)~
tas)
(with F' i CS T) is again a Majorana field. We have indeed for the field h+ F, for instance t Since the particle and its antiparticle are of opposite handedness, we can define tire fermion number as shown later. Let us, then, define the basic quaternions by associating, in. a same matrix, particles and antiparticles, each of them having a definite handedness . ~~e define first the followïng four 2 x 2 matrices in isospace :
and the four matrices :
N_, ~_, ~_, ~_,
(~2)
which are deduced from the matrices (41) by exchanging; l~+ for la_ and vice versa. The ~ and K matrices are given by (2~) and (25) and iii all the above definitions the pxime denotes the antiparticle . We note further that ~~, ll t , ~z, are real quaternions and Nom , fit , IL, are imaginary quaternions . We assign, therefore, to each field a real or a~i imaginary * t Another formula which will be of use later on is ~'* Ocp' = 4~ 0~~, where O is y4 or iy~ ys .
214
19 . UMBZAWA AND A . VISCONTI
quaternion as follows: to fields of hypercharget 0, i.e. to X, A. and x particles we will assign real quaternions, to fields of hypercharge different from 0 we will assign imaginary quaternions . Let us note further that .4 = --x, , = -- , 11 =11 . Following (10), isoscalars and isovectors corresponding to the above quaternions are given by N+ = V2 1 (N+o+N+ - e) . . . etc.,
where N+i +]L =
N +3 = .
P+h-P, V2
h+n+h-n' V2
N+a - +
~
~
N +o =
h+ P - h-P, i .%/2
h+ n n~ -~ . . . etc. i~2
(43)
(44)
It is remarkable that all the components of the isoscalars and isovectors are Majorana fields . Let us now consider. the structure of the interaction hamiltonian. To obtain conservation of thebaryon number, we shall introduce a generalized gauge transformation F+ -> L (h+, h-)F+L(h +, h-) = h+ elf+h-e -"m, ( 45)
where T is a real c-number. If the quaternion F+ is real (imaginary), the transformation (45) keeps its real (imaginary) character. The transformation for F_ has to be
F_ -> L (h-, h+ ) F-,
(46)
because F_ is obtained by exchanging h+ and h- in F+ . Our generalized gauge transformation is built up of two transformations (45), (46). On the other hand, it is easy to verify the following formulas
L* (h+, h_)L (h+, h-) = L* (h-, h+)L (h-, h+) = 1,
L (h+ , h_)Y4 = Y4L (h-, h+ ),
L (h+, h_)Y4Ys = Y4YsL (h-, h+ ),
(47 ) (48)
and tha. L (h+ , h_) commutes with the matrices y4yw, Y4YJAYs Since we assumed in sect. 1 that the strong interactions are of the y4 or Y4Ys type (first kind interactions) and the weak interactions are of the Y4Y,. or Y4Y,.Ys the (second kind interactions), the generalized gauge invariance together with the formulas (46) shows that in strong interactions, the two bary ons must be F+ and G_ respectively, while in weak interactions, the two baryons are F+ and G+ respectively, or F_ and G_ . It is also worth while noting that the following baryon products are identically zero : t The hypercharge of a field is twice the average value of the charges of the fields which build up the charge multiplet of the given field in scheme (I).
A NEW MAJORANA FIELD THEORY'
+ YQ N- _ e_* Y4 Ys N+ _ e+* Y4Yâi N- - 0, +* Y4 Y,. N+ _ _* Y4 Y o N- = e+* Y4Y aYsN+ = , e-* Y4Ye YsN- = 0.
E- * Y4N+ °_
275 (49)
We shall now give four theorems which will lead us to selection rules for the choice of hamiltonians : let us consider two fermion fields F and G and a boson field B and, as previously, define the matrices 0+
Y
m
ZY4Ys
014
= iY4Yl .
The trace operation in the following theorems is to be taken in isospace . THEOREM I
The operators 0,.p Tr {F,* G,B B} and 0,,.8 Tr {F.* BG,O}, are hermitic when there is an even of zero number of imaginary quaternions among the three quaternions F, G and B. THEOREM II
The equation
Oa# Tr {F.* Gf B+h.c.} = Oa,# Tr {F.* BG,g--Fh .c.} = 0 holds when there is an odd number of imaginary guaternions among the three quaternions F, G and B. THEOREM III
The operators O,ag Tr {Fa* Gq B} and O,,a,# Tr IF,,* BG,#} are hemitic cVhen there is an odd number of imaginary quaternions F, G and B. THEOREM IV
The equation
O,,a,o Tr{Fa*G,6 B+h.c.} = 0,,f Tr{Fa*BG,B +h .c.} = 0 holds when there is an even number of imaginary quaternions among F; G and B. The reader may find the proof of these theorems in the appendix. In the following we shall assume that the Dirac matrices in the interactions are hermitic and that coupling constants are real. Then, the theorem II excludes the following combinations among the stroii~f interactions (N,1, a), (N, A., a), (e,1, n), (e, A., a), (N, N, K), (50) (~, $, K), (N, $, K), (X, Z, K), (Z, A, K), (A, A, K) . These results added to the ones obtained in (49) show an extremely remarkable property of our scheme : almost all the strong interactions which violate the hypercharge conservation (or the strangeness conservation) are excluded. There
276
H. UMEZAWA AND A. VISCONTI
are two exceptions, and both are characterized by the fact that the pion quaternion appears between the two baryons field quantities. But, as we shall see later, there is a reason to exclude these interactions . Let us now consider the weak interactions : we shall exclude later the case 0,, = iy4y,.yr, by means of a space reflection invariance. We are therefore left with 0,, = iY4Yj., and then the theorem IV excludes the follo-vNing combinations for weak interactions : (N, N,
(lv,
)r
(A, A, z),
e, x), (e, e, Z), (,r, 1, x),
(N, X, K),
(N,11, K),
(~, A, K),
(~, E, K)-
(52)
This remark gives the well known hypercharge r» ie for weak interactions A U = I (or I A S -- 1 for strangeness) . To discuss he problem of the space reflection, we shall briefly discuss the free field eque tions of the quaternions. From their definitions (41) and (42) it is easy to see that they satisfy 4
yx 0x F+ =
-"&F F-,
yx ô,h F_ = -mF F+ ,
(53)
where F is one of the baryons and mF its mass. These equations are invariant under the following space inversion t: x -} -- x,
t --> t,
F* -> paY4
Ff,
(54a)
with N 1, Vii. «'e shall choose p, = 1 for all the baryons. One further assumes that the boson quaternions transform as =
1,
z(x)
and
K(x) -+ K(x'),
(54b)
under the space reflection and that the interactions are invariant under this space reflection. Then we are led to choice 0 = y4 for the strong interactions and 0,. = 1 '4 'f for the weak ones . Collecting the strong z interactions, we may write Tr
Hz = 9174,P Tr {N+*
+gy4. Tr{,E,*. +g,Y4,,,,q
19, 1`,,
111*'r " Y47; ~ a
3
1*
A :,
7rs 'E
r}
(55)
.
_
94 til2
13 0*
z-h.c.
are invariant under the following space-inversion toy : Fs --3- pa7,gFY . 7i -, -x, t --i- i (pa = ±1, ±_). This corresponds to the well known P-transformation. We do not however, assume that the theory invariant under this kind of space reflection . equations
in
+ EA
192 .? eT ' Y4 Ys y 7 g
The
Tr
(53)
A NEW MA]ORANA FIELD THEORY
277
where the particles T, ~, T, AO and a0 have been defined in scheme (1) . is is equal to ±1 and y = 2(1+E)+2(1-E)ys(56) The hamiltonian (55) is the most general hamiltonian under the following assumptions: (a) the invariance of the interaction under space reflection (54a) and (54b), (b) the Dirac matrix 0 is hermitic and the coupling constants gb are real, and (c) the quaternion n never appears between two baryon quaternions. The (55) shows evidently that H is charge independent in scheme (1). This can be seen also from the fact that (55) is invariant under the right hand side transformation for N, $ and K (which are equivalent to isodoublets) and under the both sides transformations for A, Z and a (which are equivalent to isoscalars and isovectors) N -> NQ*, ~ -> ~Q *, K -> KQ*, Z -~ Q-Q* , ;r -> QnQ *, A ->. QAQ * . (75) Note that this invariance is entirely due to the assumption (c) . However, this transformation does not leave invariant the real or imaginary character of the quaternions ; only the "both sides transformation" preserves this invariance. Since we assume that the real or imaginary character of the quaternions is intrinsic, we requi: e only that the interactions must be invariant under the "both sides transformation" A -> QAQ*, where A is one of the quaternions N, ~, . . . and Q is a unitary real quaternion . One may see from (55) that the a-nucleon, a-H and n-Z interactions naturally appear to be P-invariant t. On the other hand, the AXa can be P-invariant only when E = -~- 1 since we have then y = 1 or V5 . The space reflection (54) does not lea3 to , these values of E as a direct consequence. The value E = -j- 1 is then to be considered as .1 supplementary hypothesis . Moreover, it is a remarl - able result that the nucleonpion interaction appears to be pseudoscalar in (55) . As is shown in (55), the type of the AEI interact ;-~)n depends on the constant E i.e. (scalar for E = 1 and pseudoscalar for E = -1) . Consider now th .~ K strong interactions, making use of the assumptions (c) and (b) used for the z strong interactions and replacing (c) by (c') ; "the quate :.nion K never appears between two baryon quaternions", we may write do-,` n the following hamiltonian: HK = b'~10a~ Tr(N+aKI ,6+el(+ E-->-)) +9'20pTr(N* .KA+f --E2 ({-)? +g' 30,,,6 Tr(~+ .KE,#+E3(+ -)) +9'4 0,,,6 Tr ;~*aKA+,q+-E4(+ H-) .N/2
g'
191* 0y (1) T . U+
(!, S)
1 g ' 291* py (2) kA
~2
g 3e* py (3)T
, k'-
vL g~4e* py (4), A j-h . ., r
t This P-invariance is due to the following relation : Utt# Tr { N~, N_p z) = - Op Tr {N*-, N+,0=r}, etc .
278
H . UMEZAWA AND A . VISCONTI
where k and k have been defined in (1) and y o) =
(Yr,
for
Et =
( 59)
-1
and the symbol (-}--) means the replacement of N+ , ~ +, . . . by N_, , . . . . This interaction is charge independent in the ordinary scheme (I) . This fact is due to the assumption (c') . (58) is also invariant under the both sides transformation . It is P-invariant only when leil = 1. Let us now discuss the weak interactions. It can be proved that H(ENn)
=
Z.fl(Y4Y,)ap Tr[{N,*.a(a~~~+,~ -}-PE+~ ben)
H(AN=) _ i,f2(Y4Y e )a,g Tr [N+a{0e aA+P+A+,q 0ex} +eN*_a{ôexA +,. -A+,e 00x}]
= 2if2(Y4Ye) «,# Tr[N+.A+#â~x]. (61) Here H(ENx) and H(ANx) are defined by (34a) and (34b), where now y corresponds to s as follows : (62) These interactions are invariant under the both sides transformation as one may easily check. We now come to the electromagnetic interactions. We introduced at the beginning of sect . 2 the associated part and the antiassociated part of a matrix A by writing A l = 2(A - ;-Â), A2
=2(A
A = A 1 +A 2 .
-Â),
Under the transformation A --->- QAQ*, one has  ->. QAQ* = QAQ*,
(6) (64)
since from Qt = Q one deduces Q* = Q. Therefore, A 1 and A 2 are transformed as A under the both sides transformation : the separation of A in .4 al1d A 2 is an invariant one . One may prove also that if A is real (imaginary), p 1 and A2 are real (imaginary) . We define the electromagnetic quaternion corresponding to the four-potentiel A e , as follows: (A ." 0 ) = - 2iA~e3 . 0 -A# One has
%+=
To A e, corresponds therefore a fixed direction
(65) (66)
(e3)
in isospace .
A NEW MAJORANA FIELD THEORY
219
We shall assume that only the antiassociated part A 2 interacts with the electromagnetic field t. Further, let us not the following relations : ~(Y4Yp)aR Tr[N.*2a$+2,,9f..] = ~(Y4Y~)aR Tr [N+2aW~
+2/9 .J
Tr [N2a _2jW,] = Z (Y4Y~c)a~ Tr [N*2a 0, (67) Taking into account the theorem IV, we are left with the following combing tions for the electromagnetic interactions : (N2 , N2 , 91j.),
%,
~2,
(£+, F,
%I),
(-To
no WJJ,
(K2, K2, %p)'
It must be noted that nl = 0, n2 = n, 'E1 --. 0, X2= E, A1 A2 -- 0 . The position of W.. is of no importance because we have Tr (A 2 B2W..) = Tr (A2BAp) =-c Tr (W,, B2A 2 )
=
=
A,
(61.'i) arms
-Tr(A2W,, B2), (69)
where A . and B2 are given anti-adjoint quaternions and c is + 1 (or -1) when A 2 and B2 are bosons (or fermions) respectively. Thus, we are left with the following electromagnetic interaction hamiltonian : ie(Y4Y,),R Tr [N2+ *N2+9f1, +( +H -)~ = -ep*Ov pA .« ,
ie Tr [aâ.9 n%] = --lie (a+ bo n--n- b .. n+)A r , ie Tr (K2 â., K2 W.) = -- jie(K+* ô.,K+-K-ôl,K-)Ae .
It is remarkable that the theory automatically leads us to the right combinations for the electromagnetic interactions . Let us note chat H(XNa) ziid H(ANx) (eqs. (60), (61)) can be written in terms of N±1 and N:,,2 as follarvs : H(XNn) = 2ifi(Y4Y~)aR Tr [N+ia~+R ôp n-e~T*2« "~- âEl 'al,
or
for p = 1, (i 1)
H(XNn) = 2i8f1(74YEc)c~~ Tr FN *la -P-,6 0 1.n-EN+2«E+ ,6 ô~n ], for p -- 1, (72)
and
H(ANn) = 2if2(Y4~ ~)x~ Tr
CN+2a /1+.8 a,un1-
( 73 )
We shall close this section by the study of the charge conservation law. "; .the reader may have noticed that we always used the following rule called the charge rule in constructing the quaternions: for any quaternion A, ale4Po and t This assumption excludes the E°--A°-y interaction as a direct one ; the decay Z° -> A"-}-y can appear through several steps involving strong and electromagnetic interactions .
280
8 . VMLZAWA AND A . VISCONTI
describe a positive an_.d negative charge state, respectively, and al,0o and an0o, neutral states only in the representation (2). Here 00 is the vacuum state. We shall now prove that the charge conservation law is based on this rule . Indeed making a product C = AB out of two quaternions A, B, we may see that C satisfies the charge rule when A and B do : Ci10a a neutral state, etc. . . . Therefore, when A, B, . . . are constructed according to the charge rule, the state a2IL0o
Tr [AB . . . ]0u is neutral and this property corresponds to the conservation of charge.
4. Concluding Remarks In the last section, we have formulated a new Majorana theory under the fundamental assumptions that any field must be described by a real or imaginary quaternion and that the theory is invariant under the both sides transformation . The X, A and x have been assumed to be real and N, ~ and K imaginary and the basic quaternions have been constructed according to the charge rule. We have further assumed the invariance of the theory under the space refïection (.54). Taking the coupling constants and Dirac matrices to be real and hermitic respectively, we are then naturally led to the conservation of charge, the conservation of hypercharge, (there are two exceptions (51)) the P-invariance of th, nucleon-pion, ~-=, E-a interactions and the pseudoscalar character of latter interactions . Assuming that the electromagnetic field is represented by an imaginary quaternion and that it interacts only with the antiassociated parts of all fields, all the unwanted electromagnetic interactions appear to be zero . The invariance under the both sides transformation governs all the interactions including the electromagnetic interactions . The strong interactions appear to be charge independent in the scheme (1) when we assume a) that 7F never appears between two baryon quaternions and b) that K stands between them . This assumption holds only in the case of strong interactions and excludes the unwanted interactions (51) . The weak baryon-pion interactions lead us to the rule in scheme (1) and will explain the known interactions . In many JAI( cases, we have found terms of the form e(+ H -) . Nothing has been said, in this article about the four fermion weak interactions : this will be discussed in a separate paper. As was shown in sect. 2, the invariance under both sides transformations leads us to an isoscalar and isovector scheme. It is furthermore, important to remark that the Lorentz space is not, in the formalism of the new Majorana fields theory, independent of the isospace . Some of the components of the quaternions are left handed states, and some are right handed ones. This seems to be very characteristic of this theory.
=I
A NEW MAJORANA FIELD THEORY
281
The authors . would like to acknowledge extremely helpful discussions with Professors D. Kastler and J. M. Souriau who introduced them to the quaternion algebra and who took a continuous interest in this work in all its stages. One of us (H. U.), would like togratefully acknowledge a fellowship of invited Professor at the University of Marseille. Appendix Consider two fermion fields Fi , F2 and a boson field B and suppose that Flt _-_ p l F Flt .= P2 F2 , Bt .-- pB, (A .1)
where pl , P2 and p are c-numbers. We want to show that if 0 = ys or iy4 ys , then --1 0ap Tr {Fi F2pB} = (P1P2P) Opa Tr{B* F* F a} .
(A.2)
One may first show that
Fla 0ap F'29 =
FuOap F1ß,
where F',.a = Ca,& F,,j and taking into account the anticommutation properties of the Fermi fields . Then by using (A .1) one may write the left hand side of (A .2) as follows Bt} Oap Tr{F« F** B} = Oap (P1P2P) -1 Tr {Fi t F2at _ -Oap(PIP2P) -1 Tr {B*(Cp2)ß(C-p1)a *} Oap (P1 P2P) -1 Tr {B* (C.F 2 ) p (Fl* C-1 )a} r
since the association operation does not change the trace of a given matrix . Finally, if one notes that --C-1 y4 0Y4 C = Ot, where "t" means the transpose in rpinspace, a few algebraical steps are enough to prove (A.2) . In a similar way we can prove that B} = - (p1P2P)-1'y4y ap Tr {B* F2*ß Fl ap Tr {F, . } i'74y
F2pa~
To prove this, use is made of the following relation : 2F1* y4yp F' 2 = -ZF2* y4 y,, Fl .
The formulae (A.2) and (A .3) are at the basis of the four theorems of sect . 3. References '.) W. Heisenberg, Solvay Berichte report (1939) Theor. Phys . 17 (1952) 377 2 ) S. Sakata, H. Umezawa and S. Kamefuchi, Prog . 3) H. Umezawa, Prog. Theor. Phys . Suppl. 1 7 (1959) 67 4) H. Umezawa, b'1. Konuma and K. Nakagawa, Nuclear Physics 7 (1958) 169
282
8. VMEZAWA AND A. VISCONTI
5) hi. GellMann, Phys. Rev. 92 (1953) 833; K. Nishijima, Frog. Theor. Phys. 13 (1955) 285; K. Nishijima and T. Nakano, Prog. Theor. Phys. 10 (1953) 581 6) Id. Cell-Mann, Phys. Rev. 106 (1957) 1296 7) G. Takeda, Prog Theor. Phys. 19 (1958) 63 8) B. d'Fspagnat, J. Prentki and A. Salam, Nuclear Physics 5 (1958) 447 9) H. Umexawa and K. Nakagawa, Nuovo Cimento (to be published) 10) B. d'Espagnat and J. Prentki, Nuclear Physics
A NEW MAJORANA FIELD THEORY H. UMEZAWA t and A. VISCONTI Faculté des Sciences de Marseille, Prance Received 16 June 1960 Abstract : After tr-, short introduction, we review in sect. 2 the various isospin schemes for strong and weak interactions and we show that they can be reduced to a common scheme by writing down an invariant of a set of 2 x 2 matrices, whose elements are linear combinations of the particle wave functions. Various properties and difficulties of those schemes are discussed . In sect. 3, we propose a scheme where each set of a particle and its antiparticle is described by a quaternion . Several interesting properties of that scheme are pointed out: the rule for constructing quaternions leads directly to charge conservation ; the strangeness conservation for strong interactions and the rule IASI = 1 for the weak interactions are direct consequences of this theory, furthermore all the unwanted electromagnetic interactions are naturally excluded.
1. Introduction
The weak interactions present several important and characteristic features First of all, the rule 4AII = 2 which is a generalization of the rule JA 1 a 1 = I (or in terms of the hypercharge : jAU) = 1) seems to be consistent with all the known weak interactions, except for the decay K+ --> a++a0. Then, the mixture of vector (V) and axial vector interactions (A) seems to fit the known experiments and finally there are, for the time being, no phenomena violating the PC invariance . Furthermore, the V and A interactions belong to the 2nd kind of inte-action, whose coupling constants have the dimensions (L) 11 with 71 > 0 in contradistinction to the first kind of interaction where q <_ 0. The distinction between these two classes could be traced to their renormalizability properties, since it is generally believed that no interactions of the 2nd kind can be renormalized tt . It is also knowr~ that the coupling constants for weak interactions are about (10 -17 )2 cm2 for fermion-fermion interactions and 10-20 cm for fermion-boson interactions . In other words, the weak interactions imply extremely small lengths, much smaller than the Compton wave length of the particles known so far. Therefore, one may reasonably suppose that in the theory of elementary particles, one has to consider 3,4) a fundamental length l. However, it must be noted that this assumption does not necessarily mean the nonlocal character of the interactions . Consider, for instance, the low energy phenomena which t On leave from the Physics Dept. University of Tokyo (Japan) . tt See ref . 1, $) for the details of this classification . 262
A NEW MAJORANA FIELD THEORY
263
involve particles whose wave lengths are well above l ; these interactions, which we will call the strong interactions are independent of l, correspond thus to t = Q and belong to the first kind of interactions. The effects involving the fundamental length l are extremely small, when they appear among the former low energy phenomena. They are consequences of weak interactions and belong to the 2nd kind of interactions . The strong interactions are then scalar (S) or pseudo scalar (PS) (both theories are renormalizable), while the weak fermion-boson interactions re of the types V or A . Finally, there is another remarkable feature of the weak interactions : in contradistinction to the stronger ones, they are usually defined by violations of laws which express the conservation of some quantum numbers (strangeness or hypercharge, for instance!) in the case of the strong interactions . But, from our point of view, the definition of the weak interactions has its ground essentially in the consideration of the fundamental length l. It should be therefore desirable to try to understand their systematics by means of only conservationlaws. We are thus faced with the following problems: 1) How to substitute for the rule JJIJ = 2 an invariance law? 2) Why do we have the conservation of the hypercharge in strong interactions but not in the weak ones? 3) How the PC invariance in strong interactions leads to the P invariance? On the other hand, the answer to these questions must not bring any change in the experimentally well established mass spectra of the particles . In particular, the strong interactions have to be charge independent (in the ordinary meaning of the word), and the theory must allow all the mass differences between particles in different sets of charge multiplets (e.g. between the Z, N, A, X multiplets) t. Our aim in the present paper is to investigate the former problems along the lines we just mentioned : after an extensive study of several isospin schemes (sect. 2) we shall introduce in sect. 3 a new scheme (the scheme of new Majorana fields), where by association of particles and antiparticles we shall build up real quaternions. Therefore, to each particle antiparticle set will correspond a quaternion and the investigation of the symmetry properties of- the interaction hamiltonian will show several important and interesting features of this theory .
1' The problem of the charge independence of the strong K interactions needs some comment : there are not enough experimental data which wouh compel oneto conclude that they are (although the cross section of K- capture by the deuteron agrees with the charge independence hypothesis` . However, it seems reasonable to expect that if these interactions were not charge independent, there would have been a quite big mass difference between, say, proton and neutron. This is a reason to assume - as we shall do in this paper - the charge independence of the strong K interactions .
264
$. VUEZAWA AND A. VISOO14TI
It is, however, worth while to note that we shall not deal with the problems of K+ decay (K+ -> -,r++x°) : this problem will be discussed in a forthcoming paper. Z. Isospin Schemes and the 2 x 2 Matrix Scheme
Let us call "convéntional or scheme (I)" the isospin scheme where the nucleon, the H and the K particles build up isodoublets while the X, x particles build up isotriplets (or isovectors) and the A an isoscalar. The JAI( = j rule for weak interactions has been deduced from this scheme. It has been noticed several tines, however, that the charge independence of strong interactions does not necessarily lead to the former scheme uniquely . For instance, the charge independent strong x-interactions can be described either in terms of °) the former scheme (I) as: ~
p
n)
kH-)
(K+
(-KO*) , K+*
KO)
V2 £- --Z+ iV2 or in terms of s, 7) the baryon doublet scheme (II) , , k, ~, X = ~+ ,
r Yo
with Y° _
b11o-E° ~2
,
Z° --
brio-}-~ X8 , -%/2X8-%/2
Z=
Zo
('r-
,
(2)
b a real constant
Furthermore . the same interaction can also be written in terms of the tripletsinglet scheme (111) :
p -#- a .:--/2
p-a,N= No , . ~~ !1 . x, iV2 i1/2
n-a.^O
A NEW MAJORANA FIELD THEORY
26 5
where a is a real constant. In the same way, we shall assign to the K-meson the following set built up by an isovector and an isoscalar: f K
K+ + K+* V2 K+ - K+*
KO
KO + KO*
KO _... KO*
i,%/2
To each scheme belongs an isospace which is well defined: let us call a theory charge independent in a given scheme when it is invariant with respect to the rotations in the isospace belonging to that scheme t. When a =- 1, d'Espagnat, Prentki and Salam 8 ) showed that the scheme (III) leads to a hamiltonian charge independent in both schemes (III) and (1) . Now, the charge independence in scheme (I), on one hand, does not lead to any mass differences between the particles belonging to each charge multiplet set %, 6, E, 916, K and is, on the other hand, compatible with the experimental evidence that the nucleons, pions, H, X, a particles have different masses, therefore each of the hamiltonians has to be charge independent in scheme (I) . One must notice, too, that a free lagrangian charge independent in scheme (II) (in (III)) is charge independent in scheme (1), only when b --- 1 (a = 1) and when the bare masses of Z, 11 (nucleon and H) are equal tt. In order to have an insight into the connection between these schemes, let us consider the strong interactions with a mesons, i.e. NNn, w;r, XZn, X11n in the schemes (I), (II) and (III) . They are obtained by building up all the possible isoscalars by means of field quantities . Since we assumed the strong interactions to be of first kind, we get in scheme (I) the following interaction hamiltonian: Hn u) = 91 91*07 ' 91~7 +92$ *0 r' ~n * +g4,E* 0 A n+g4A* 0,E' aj, (5a) -igsZ A 0.E " ;r where 0 is 0 --
while one gets H, (II)
=
4
(5b)
Y4 Ys
i
'+g3X* 0a ° Xn+g 3 Z* 0t " Zn (6a) gi9l * 0z ° 9179+g2 e* 0T ' e7
.r°,
Z-t- o, AO, nf, o , t Note that in the three schemes, we shall normalize the field quantities p, n, in such a way that they satisfy the canonical commutation relations. tt One may introduce a modified scheme (II') by replacing k and k by the isovector K and the Z++n- is forbidden and thereisoscalar KO, as given by (4) . But for 5 = 1, the process K +p -> fore the scheme (II') disagrees with the expriments. For b 0 1, the free lagrangian is no longer charge independent in scheme (II') .
H. UMEZAWA AND A. VISCONTI
in scheme (II), and HWWI)
= --ig1[N* A
ON - a+NO* ON - a-N * - ONO X} -ig3X A OE - =+ g4Z* - 0A;r+g4A* OE - =
(6b)
in scheme (III) . = bg3, one finds By taking the real coupling constants g2 = a2gi , g4 3r(I) = Ha(II) = H (III) In other words, one may obtain the same interaction hamiltonian which is charge independent in the three schemes, but, as said before, the free lagrangian is charge independent neither in (II) nor in (III), when the constants a, b are different from 1 and the masses of E, A. (N and H) different. The interaction hamiltonian for charge independent K interactions can be written as follows: HKM
= g',%* Or - kZ+g' 2 ~* Or - kZ+g'3 91* OkA+9V* OkA+h.c .
(7a)
in scheme (I), and HK (II) = ig'1(N* A OK - X+No* OK - X-N* OK O - Z}
-- g'3(N * - OKA+N O*OKO A}+h .c. (7b)
in scheme (III) . By taking the real coupling constants g'z- ag'1 , g'3 = bg'1 and g4 = abg'1, one obtains, as before, the same hamiltonian in the above two schemes. The strong K interactions are not charge independent in scheme (II) : there is indeed no way to construct an isoscalar by means of an odd number of doublets. In this theory of global symmetry, Gell-Mann s) assumed that the strong K interactions are charge independent only in scheme (I) (but not in (II)) . However, recent experiments show that the cross-section a (K -}-N -0- K+ N ) has about the same value as that of the pion-nucleon scattering : it seems, therefore, reasonable to believe that these interactions are not weaker than the a-interactions and that the symmetry properties are about the same in both the interactions . The schemes (II) and (III) have a great advantage : even the weak interactions between baryons and bosons are charge independent. But there is an important drawback too. If the total hamiltonian without (the electromagnetic interaction) is charge independent in scheme (III), there is no way to account for the mass difference between the nucleon and .'E". On the other hand, the weak interactions, which are charge independent in (II), disagree with experiments : the E- decay is forbidden as discussed later on. One may explain in another way the existence of the former three charge ndepen0Qnt schemes, and the method we are going to develop here, will introduce spine of the mathematical concepts, which will be used extensively later on. Suppose that we are considering the vectorial space built up by all 2 X 2
267
A NEW MAJORANA FIELD THEORY
matrices; I, e1 e2 , e3 build up a system of generators of this space. Furthermore, we have: ei ef = -iek , [ei , e;] = -26ij, (8) and i, j, k are a cyclic permutation of the numbers 1, 2, 3. The standard representation of this basis is _
e1 - ZQ1
_ 0 i _ 0 1 ) e2 - '12 = ( i 0 ' -1 0
(i e 3 = iß3 = 0 -0i ,
and if A is any 2 x 2 matrix, we may write A =
1 (a+a " e), -%/2
( 1 0)
where in terms of the former representation a 1 . A12+A 21 _, i 1/2
a 2 - A12 -A21 , a3 1/2 -
A11 --A 22 i .%12
,
a=
1 TrA . -X/2
(11)
It is also worth while noting the following formula which may be of use : Tr (ABC) = 1 (abc-a " be-a - be-ab - c+a A b - c) . .%/2
Now, let us define the two matrices A^ = TrA-A = At = A*
A22 - A12 ,
A ll
( -A,,, A22
--A*12
(12)
(13)
A21 A*11 .
We shall call the matrix  the "associated matrix" t of A . A matrix is self associated (antiassociated) if  = A ( = -A), and to any given matrix corresponds a self associated part 2 (A -}- Â) and an antiassociated part 2 (A -A) : A = 2(A+Â)+2(A-Â) . One has for instance
et = e, e* = -e, ê = -e.
( 1 5) A matrix A is called a real quaternion if A = A t, and imaginary one if At = -A . If q is a vector of the tridimensional space, the 2 x 2 matrix Q = exp - 26e -
q) Iql
(16)
is on one hand a real quaternion and a unitary matrix (since the e; are antihermitian) and on the other hand, as well known, Q is the operator describing
(A B) t = A t Bt . t Note that ÂB = f?A,
shown terms two this may We an the thus let be that Q the A under in have way isoscalar isodoublets us transformation viewed is therefore in see of the that matrix any meaning consider a to (10 indeed the matrix the that A' real form the Aunitary the baryon as both -~ (isovector) say e'' the quaternion xtwo "left self Q of under --Q (we the represents as (-A21 that sides AQ* matrix (16) a~xek schemes doublet associated follows and shall respectively "right (18) hand the =transformation The real under = is Q-le,Q, call -A',2 A right = transformation" right (Q -A12 hand equivalent scheme the AQ* obtained transformation behaves -~ Q Aj it quaternion 'il =W1, (antiassociated) the hand rotation A'= the and Q*), hand as"both side (II) AO since an both Q side like by we A transformation" 2side to isoscalar (17) Q* and = az the and sides transformation with have A' a sides (Q) transformation 21 of set while -A21 the (17) transformations = a-1 a, = A22 transformation") -A12 the c-number of part QÂ, transformation triplet-singlet = it and Q has it an follows (-A,2 a,kat isodoublets, is which an of isovector therefore = Al transformed defined aa+a',e,, isovector that (18) elements We matrix can (18), Qek is (17) scheme cin Qaand as completely (see be and W2 Q* clear tand behaves follows further written can as = in an iteke (15)) a(III) the can cut (18) isobe set
ut4zzAWA Ago A . VIO
I
NT9
the rotation of angle 0 around the unit axis qjlgi " It is also known that the three generates e1 e., e2 behave like vectors under the rotation (i.e. e! Q*ef Q =- es where e,' is the vector obtained by rotating the e, vector amind the axis q qj by the angle 0) . This remark shows therefore that the quantities ao and a which describe the =
.
matrix
behave
transformation A --- .
= QA Q_"I = Qfa+auex)Q * = a+a,% a , xet
(17) .
where prove written
.
geometrical be
.
like Now
: .
A
(18)
Since A' and in
(19) :
A'22 (-A'?~
A
.
=
A22 -- .4
(20)
A,,)
We equivalent A22 -A'21)
A22)
A12 A'11)
=
We scalar of 1 _-
-_
(21)
In may t
..
A NEW MAjORANA FIELD THEORY
269
applied to a single scineme, where all the particles will be represented b~ 2 ~ 2 matrices build up in the following way t - ayoJ
a~ b~1o-~a ~/2
~2
~+
(22)
1
(23)
~/2 ~=i
K
-
1/"2
~+
-~o -y/2
-Ko K_ à K+* go*~
1
~/2 ~ " e,
- i (Ko-I-K . e) . ,r/2
{24)
( 25 )
All the former matrices have been expressed in terms of the isovectors and isoscalars as defined in scheme (III) by means of the formula (10) . Remembering the relations {26a) ~* _ ~~ K* = Ky Ko* = Ko~ one obtains We may, therefore, characterize the ~ and K matrices as real and imaginary quaternions . On the other hand the doublets ~Xi and ~ 2 given by the formula (21) are s~l -_ -is,~- -i P {27a) n' for the matrix N, and for the matrix ~ (see (24) and {25) ) . Consider now the hamiltonians H and HK deffined as f c llc~~s
t It should be noted thwt K° is not the particle K° but the isoscalr.r attached to the isovectôr K in scheme (III) .
270
H. UMEZAWA AND A, VISCONTI
where 0 is given by (5b) and the trace operation is to be performed on the isospins indices . As well known, a canonical transformation generated by the unitary operator Q in isospace leaves the trace invariant, hence the formula (28) may be as well put into the form H, = i1/20a,# Tr {& QNa* Q* QN eQ* QKQ* +$3QEa* Q* QE8 Q* QnQ*), (29a) HK =
--,V L g' 1 OaR
Tr {QNa*Q*QKQ*QE8Q* ..~ h.c. } .
(29b)
They can be interpreted in three ways : a) One may consider this canonical transformation as just the both sides transformation (17), Nca ->
QNa Q*, Ia -> Q£a Q*, a -> QnQ * , K -> QKQ* .
(30)
Taking into account the text before formula (17), we are thus led directly to the triplet singlet scheme (III). Indeed a straightforward calculation taking (12) into account shows that the hamiltonians (29a) and (29b) are equal to (6b) and (7b) represtively . b) One may consider the hamiltonian (29a) as resulting from the transformation Na -> Na Q * , _a -3. -ra Q* , 9L --. QaQ * , ( 31 ) applied to (28a) and this leads to the doublet scheme (II), if we take into account the formula (27) . We may remark that (29b) cannot be deduced from (28b) by the former transformation (31) together with : K --> KQ*, i.e. HK is not charge independent in schema (II) . c) One may consider finally the transformations K -+ KQ *, a -> QxQ * , (32) of the hamiltonians (29) and this leads, as easily seen, to the scheme (I) the hamiltonians H. and HK are both charge independent in scheme (1) . Let us now consider the weak ;r baryon interactions . It can be shown that the following hamiltonian is invariant under the transformation (31) Na -> N. Q*,
Xac -> Qga Q*,
i(fY4Y#+ .f' Y4Y.,Y5)a.8 Tr {Na*Z,eô.a)-}-h.c. (33) Therefore it is charge independent in scheme (II) . On the other hand it is not invariant under the transformation (32), but it leads us to the rule 14II = 1. Indeed, if one looks at the transformation of the product Ea n by the formula (32), la
a --> QI, zQ* ,
one sees that Ea a behaves as a set of isovector and isoscalar, while Na behaves as two doublets, one may then conclude that Tr {*~?a*E,#a} satisfies the rule mentioned above . One can show that (33) is equivalent to an interaction propos-
27 1
A NEW MAJORANA FIELD THEORY
ed by Takeda 7) . A perturbation treatment in its lowest order shows that the decay is forbidden and therefore this decay may be explained only by an extremely large a mesonic correction. One of the authors (H. U.) s) and independently d'Espagnat and Prentki 1 °) proposed once consideration of the following hamiltonian for the weak baryon-pion interactions t: H(ENa)
= if [jn* Y4Y#Y( 1 +PY5)Z° r
1
~2
a~,Z
P*Y4Y~Y(1-PYs)~+ aw~°-i-n*Y4Y,~~-a~~++Pn~`Y4Y~YYs~+aw~-+h .c. ],
H(ANa)
= -
(34a)
°+ 2P * Y4Ylt Y( 1- PYs) Xoa p n+
Z .f? P * Y4Y,~Y( 1 +Y5)A0,zn+- _
1/2
1 n*Y4Y,,Y(1+Ys)11ô,~°+h .c .
1/2
where
p=±1, y=1
, (34b)
or y,, .
It has been proved by d'Espagnat and Prentki 1 °) that (29) and (30) together with "--a weak interactions can be charge independent in scheme III when we assume p = 1 . This can be seen from the following relation : 1
2
f{ (Y4YwY),,8 Tr[Na
*
e pnEB ] - (Y4YwYsY)aB Tr { ;~Ta * ~B 2~~}}
--- H(EN ;r)+H(ANn)--if 1 - ~~
+ so* Y4Y,AY5YZ ~2
- *Y4YuY(1__Ys)E- 0 ,i :T°
-*Y4Y#Y(1-Ys) X°0~7t - +2'° *Y4Y j Y(1+Ys)X°0~~z°
2
+
1
~
0/43++
ow
0* V4Y#YE+
Y4Y Y(1+Yb A00 n + X/2
(35)
lu
0.
Y4YItY( +Ys
,1 .
with f =bf,p=1 t The lowest order perturbation applied to results :
(34)
is in agreement with the following experimental
-> oc (,JO -> p +n - ) <- -0 .67+0 .13, a(AO p+n) - 2, c (AO -~ n +n") +) ee Q(Ear -~ p+n°) ti n+TC
(,r+
l a(Z+ -> p+no)l > 0.70+0.30, oc(E+ -> n+:z+) , ce(£- -> n+ ,-r -) \ 0, where a is the asymmetry factor . Note that one may conclude from experiments for the E- del=ay that Pa ~ 0, where P is the polarization of the Z-. We assumed here that a ,. 0 (instead of P -- 0) in the X- decay.
272
H . VMBZAWA AND A . VISCONTI
3. New Majorana Field Formalism As we have seen, all the known schemes in isospace can be formulated in terms of 2 x 2 matrices. We have gone even further and tried to pave the way to a theory where all the interactions - including the weak ones -- must be governed by invariance laws only. But, by associating particles of different masses (e.g. nucleon and H) in our fundamental matrices, we were forced to violate some of the invariance laws in order to explain the mass differences (cf. text just before the formula (7)) . Therefore, we have to construct our basic 2 x 2 matrices by means of particles of same mass (electromagnetic mass differences are here disregarded.) This is only possible if we construct, say, the nucleon basic matrix by means of nucleons and antinucleons. In order to do this, we shall be led to use in an extensive way the formalism of the quaternion algebra. In sect. 2, we defined the real (imaginary) qu~rternion as a matrix A, whose elements are c-numbers, and which satisfies the relations A = At - Â*
(A = -At) .
One can extend without difficulty this definition to commuting fields ; we shall define the Bt corresponding to a Bose field as follows : where the hermitic conjugation * is to be worked out both in Hilbert space and in isospace, and the operation ^ in isospace only. Let us now consider a fermion field F with components Fa and define P = F* y4 . Since the hermitic conjugation for a boson field is equivalent to the charge conjugation, we are led to define Ft as follows: Fat = (CP)a,
where C is the well known charge conjugation matrix If j and t are isospin indices running from 1 to 2,
(36a) (C'I y.C
=
)
(36b) ap(Fp* ) iz , where the * operation is to be worked out both in Hilbert space and isospace . A fermion field will be called a real or an imaginary quaternion if we have (Fat)sa -
On the other hand,
(Cy4
Ft = F or
Ft = -F .
.P0T +CFT6 *) . (C -v/2 Therefore if the quaternion F is real, we obtain CP0T = FQ , CFT= F
(37)
A NEW MAJORANA FIELD THEORY
and if it is imaginary,
C~o _-_ --F° ,
CFT --
27 3
-FT.
One may conclude that the decomposition of a real or an imaginary quaternion into an isoscalar Fo and an isovector F defined two Majorana fields Fo and F. As well known, there is no conservation law for the fermion number in Majarana's theory. In order to avoid this drawback, we shall assign to the pa~~ticles Fo and T a definite handedness . V6Te may fiat remark that by means of the well known handedness projection operators one may prove that the field
hf
= 2(I~YS)~
tas)
(with F' i CS T) is again a Majorana field. We have indeed for the field h+ F, for instance t Since the particle and its antiparticle are of opposite handedness, we can define tire fermion number as shown later. Let us, then, define the basic quaternions by associating, in. a same matrix, particles and antiparticles, each of them having a definite handedness . ~~e define first the followïng four 2 x 2 matrices in isospace :
and the four matrices :
N_, ~_, ~_, ~_,
(~2)
which are deduced from the matrices (41) by exchanging; l~+ for la_ and vice versa. The ~ and K matrices are given by (2~) and (25) and iii all the above definitions the pxime denotes the antiparticle . We note further that ~~, ll t , ~z, are real quaternions and Nom , fit , IL, are imaginary quaternions . We assign, therefore, to each field a real or a~i imaginary * t Another formula which will be of use later on is ~'* Ocp' = 4~ 0~~, where O is y4 or iy~ ys .
214
19 . UMBZAWA AND A . VISCONTI
quaternion as follows: to fields of hypercharget 0, i.e. to X, A. and x particles we will assign real quaternions, to fields of hypercharge different from 0 we will assign imaginary quaternions . Let us note further that .4 = --x, , = -- , 11 =11 . Following (10), isoscalars and isovectors corresponding to the above quaternions are given by N+ = V2 1 (N+o+N+ - e) . . . etc.,
where N+i +]L =
N +3 = .
P+h-P, V2
h+n+h-n' V2
N+a - +
~
~
N +o =
h+ P - h-P, i .%/2
h+ n n~ -~ . . . etc. i~2
(43)
(44)
It is remarkable that all the components of the isoscalars and isovectors are Majorana fields . Let us now consider. the structure of the interaction hamiltonian. To obtain conservation of thebaryon number, we shall introduce a generalized gauge transformation F+ -> L (h+, h-)F+L(h +, h-) = h+ elf+h-e -"m, ( 45)
where T is a real c-number. If the quaternion F+ is real (imaginary), the transformation (45) keeps its real (imaginary) character. The transformation for F_ has to be
F_ -> L (h-, h+ ) F-,
(46)
because F_ is obtained by exchanging h+ and h- in F+ . Our generalized gauge transformation is built up of two transformations (45), (46). On the other hand, it is easy to verify the following formulas
L* (h+, h_)L (h+, h-) = L* (h-, h+)L (h-, h+) = 1,
L (h+ , h_)Y4 = Y4L (h-, h+ ),
L (h+, h_)Y4Ys = Y4YsL (h-, h+ ),
(47 ) (48)
and tha. L (h+ , h_) commutes with the matrices y4yw, Y4YJAYs Since we assumed in sect. 1 that the strong interactions are of the y4 or Y4Ys type (first kind interactions) and the weak interactions are of the Y4Y,. or Y4Y,.Ys the (second kind interactions), the generalized gauge invariance together with the formulas (46) shows that in strong interactions, the two bary ons must be F+ and G_ respectively, while in weak interactions, the two baryons are F+ and G+ respectively, or F_ and G_ . It is also worth while noting that the following baryon products are identically zero : t The hypercharge of a field is twice the average value of the charges of the fields which build up the charge multiplet of the given field in scheme (I).
A NEW MAJORANA FIELD THEORY'
+ YQ N- _ e_* Y4 Ys N+ _ e+* Y4Yâi N- - 0, +* Y4 Y,. N+ _ _* Y4 Y o N- = e+* Y4Y aYsN+ = , e-* Y4Ye YsN- = 0.
E- * Y4N+ °_
275 (49)
We shall now give four theorems which will lead us to selection rules for the choice of hamiltonians : let us consider two fermion fields F and G and a boson field B and, as previously, define the matrices 0+
Y
m
ZY4Ys
014
= iY4Yl .
The trace operation in the following theorems is to be taken in isospace . THEOREM I
The operators 0,.p Tr {F,* G,B B} and 0,,.8 Tr {F.* BG,O}, are hermitic when there is an even of zero number of imaginary quaternions among the three quaternions F, G and B. THEOREM II
The equation
Oa# Tr {F.* Gf B+h.c.} = Oa,# Tr {F.* BG,g--Fh .c.} = 0 holds when there is an odd number of imaginary guaternions among the three quaternions F, G and B. THEOREM III
The operators O,ag Tr {Fa* Gq B} and O,,a,# Tr IF,,* BG,#} are hemitic cVhen there is an odd number of imaginary quaternions F, G and B. THEOREM IV
The equation
O,,a,o Tr{Fa*G,6 B+h.c.} = 0,,f Tr{Fa*BG,B +h .c.} = 0 holds when there is an even number of imaginary quaternions among F; G and B. The reader may find the proof of these theorems in the appendix. In the following we shall assume that the Dirac matrices in the interactions are hermitic and that coupling constants are real. Then, the theorem II excludes the following combinations among the stroii~f interactions (N,1, a), (N, A., a), (e,1, n), (e, A., a), (N, N, K), (50) (~, $, K), (N, $, K), (X, Z, K), (Z, A, K), (A, A, K) . These results added to the ones obtained in (49) show an extremely remarkable property of our scheme : almost all the strong interactions which violate the hypercharge conservation (or the strangeness conservation) are excluded. There
276
H. UMEZAWA AND A. VISCONTI
are two exceptions, and both are characterized by the fact that the pion quaternion appears between the two baryons field quantities. But, as we shall see later, there is a reason to exclude these interactions . Let us now consider the weak interactions : we shall exclude later the case 0,, = iy4y,.yr, by means of a space reflection invariance. We are therefore left with 0,, = iY4Yj., and then the theorem IV excludes the follo-vNing combinations for weak interactions : (N, N,
(lv,
)r
(A, A, z),
e, x), (e, e, Z), (,r, 1, x),
(N, X, K),
(N,11, K),
(~, A, K),
(~, E, K)-
(52)
This remark gives the well known hypercharge r» ie for weak interactions A U = I (or I A S -- 1 for strangeness) . To discuss he problem of the space reflection, we shall briefly discuss the free field eque tions of the quaternions. From their definitions (41) and (42) it is easy to see that they satisfy 4
yx 0x F+ =
-"&F F-,
yx ô,h F_ = -mF F+ ,
(53)
where F is one of the baryons and mF its mass. These equations are invariant under the following space inversion t: x -} -- x,
t --> t,
F* -> paY4
Ff,
(54a)
with N 1, Vii. «'e shall choose p, = 1 for all the baryons. One further assumes that the boson quaternions transform as =
1,
z(x)
and
K(x) -+ K(x'),
(54b)
under the space reflection and that the interactions are invariant under this space reflection. Then we are led to choice 0 = y4 for the strong interactions and 0,. = 1 '4 'f for the weak ones . Collecting the strong z interactions, we may write Tr
Hz = 9174,P Tr {N+*
+gy4. Tr{,E,*. +g,Y4,,,,q
19, 1`,,
111*'r " Y47; ~ a
3
1*
A :,
7rs 'E
r}
(55)
.
_
94 til2
13 0*
z-h.c.
are invariant under the following space-inversion toy : Fs --3- pa7,gFY . 7i -, -x, t --i- i (pa = ±1, ±_). This corresponds to the well known P-transformation. We do not however, assume that the theory invariant under this kind of space reflection . equations
in
+ EA
192 .? eT ' Y4 Ys y 7 g
The
Tr
(53)
A NEW MA]ORANA FIELD THEORY
277
where the particles T, ~, T, AO and a0 have been defined in scheme (1) . is is equal to ±1 and y = 2(1+E)+2(1-E)ys(56) The hamiltonian (55) is the most general hamiltonian under the following assumptions: (a) the invariance of the interaction under space reflection (54a) and (54b), (b) the Dirac matrix 0 is hermitic and the coupling constants gb are real, and (c) the quaternion n never appears between two baryon quaternions. The (55) shows evidently that H is charge independent in scheme (1). This can be seen also from the fact that (55) is invariant under the right hand side transformation for N, $ and K (which are equivalent to isodoublets) and under the both sides transformations for A, Z and a (which are equivalent to isoscalars and isovectors) N -> NQ*, ~ -> ~Q *, K -> KQ*, Z -~ Q-Q* , ;r -> QnQ *, A ->. QAQ * . (75) Note that this invariance is entirely due to the assumption (c) . However, this transformation does not leave invariant the real or imaginary character of the quaternions ; only the "both sides transformation" preserves this invariance. Since we assume that the real or imaginary character of the quaternions is intrinsic, we requi: e only that the interactions must be invariant under the "both sides transformation" A -> QAQ*, where A is one of the quaternions N, ~, . . . and Q is a unitary real quaternion . One may see from (55) that the a-nucleon, a-H and n-Z interactions naturally appear to be P-invariant t. On the other hand, the AXa can be P-invariant only when E = -~- 1 since we have then y = 1 or V5 . The space reflection (54) does not lea3 to , these values of E as a direct consequence. The value E = -j- 1 is then to be considered as .1 supplementary hypothesis . Moreover, it is a remarl - able result that the nucleonpion interaction appears to be pseudoscalar in (55) . As is shown in (55), the type of the AEI interact ;-~)n depends on the constant E i.e. (scalar for E = 1 and pseudoscalar for E = -1) . Consider now th .~ K strong interactions, making use of the assumptions (c) and (b) used for the z strong interactions and replacing (c) by (c') ; "the quate :.nion K never appears between two baryon quaternions", we may write do-,` n the following hamiltonian: HK = b'~10a~ Tr(N+aKI ,6+el(+ E-->-)) +9'20pTr(N* .KA+f --E2 ({-)? +g' 30,,,6 Tr(~+ .KE,#+E3(+ -)) +9'4 0,,,6 Tr ;~*aKA+,q+-E4(+ H-) .N/2
g'
191* 0y (1) T . U+
(!, S)
1 g ' 291* py (2) kA
~2
g 3e* py (3)T
, k'-
vL g~4e* py (4), A j-h . ., r
t This P-invariance is due to the following relation : Utt# Tr { N~, N_p z) = - Op Tr {N*-, N+,0=r}, etc .
278
H . UMEZAWA AND A . VISCONTI
where k and k have been defined in (1) and y o) =
(Yr,
for
Et =
( 59)
-1
and the symbol (-}--) means the replacement of N+ , ~ +, . . . by N_, , . . . . This interaction is charge independent in the ordinary scheme (I) . This fact is due to the assumption (c') . (58) is also invariant under the both sides transformation . It is P-invariant only when leil = 1. Let us now discuss the weak interactions. It can be proved that H(ENn)
=
Z.fl(Y4Y,)ap Tr[{N,*.a(a~~~+,~ -}-PE+~ ben)
H(AN=) _ i,f2(Y4Y e )a,g Tr [N+a{0e aA+P+A+,q 0ex} +eN*_a{ôexA +,. -A+,e 00x}]
= 2if2(Y4Ye) «,# Tr[N+.A+#â~x]. (61) Here H(ENx) and H(ANx) are defined by (34a) and (34b), where now y corresponds to s as follows : (62) These interactions are invariant under the both sides transformation as one may easily check. We now come to the electromagnetic interactions. We introduced at the beginning of sect . 2 the associated part and the antiassociated part of a matrix A by writing A l = 2(A - ;-Â), A2
=2(A
A = A 1 +A 2 .
-Â),
Under the transformation A --->- QAQ*, one has  ->. QAQ* = QAQ*,
(6) (64)
since from Qt = Q one deduces Q* = Q. Therefore, A 1 and A 2 are transformed as A under the both sides transformation : the separation of A in .4 al1d A 2 is an invariant one . One may prove also that if A is real (imaginary), p 1 and A2 are real (imaginary) . We define the electromagnetic quaternion corresponding to the four-potentiel A e , as follows: (A ." 0 ) = - 2iA~e3 . 0 -A# One has
%+=
To A e, corresponds therefore a fixed direction
(65) (66)
(e3)
in isospace .
A NEW MAJORANA FIELD THEORY
219
We shall assume that only the antiassociated part A 2 interacts with the electromagnetic field t. Further, let us not the following relations : ~(Y4Yp)aR Tr[N.*2a$+2,,9f..] = ~(Y4Y~)aR Tr [N+2aW~
+2/9 .J
Tr [N2a _2jW,] = Z (Y4Y~c)a~ Tr [N*2a 0, (67) Taking into account the theorem IV, we are left with the following combing tions for the electromagnetic interactions : (N2 , N2 , 91j.),
%,
~2,
(£+, F,
%I),
(-To
no WJJ,
(K2, K2, %p)'
It must be noted that nl = 0, n2 = n, 'E1 --. 0, X2= E, A1 A2 -- 0 . The position of W.. is of no importance because we have Tr (A 2 B2W..) = Tr (A2BAp) =-c Tr (W,, B2A 2 )
=
=
A,
(61.'i) arms
-Tr(A2W,, B2), (69)
where A . and B2 are given anti-adjoint quaternions and c is + 1 (or -1) when A 2 and B2 are bosons (or fermions) respectively. Thus, we are left with the following electromagnetic interaction hamiltonian : ie(Y4Y,),R Tr [N2+ *N2+9f1, +( +H -)~ = -ep*Ov pA .« ,
ie Tr [aâ.9 n%] = --lie (a+ bo n--n- b .. n+)A r , ie Tr (K2 â., K2 W.) = -- jie(K+* ô.,K+-K-ôl,K-)Ae .
It is remarkable that the theory automatically leads us to the right combinations for the electromagnetic interactions . Let us note chat H(XNa) ziid H(ANx) (eqs. (60), (61)) can be written in terms of N±1 and N:,,2 as follarvs : H(XNn) = 2ifi(Y4Y~)aR Tr [N+ia~+R ôp n-e~T*2« "~- âEl 'al,
or
for p = 1, (i 1)
H(XNn) = 2i8f1(74YEc)c~~ Tr FN *la -P-,6 0 1.n-EN+2«E+ ,6 ô~n ], for p -- 1, (72)
and
H(ANn) = 2if2(Y4~ ~)x~ Tr
CN+2a /1+.8 a,un1-
( 73 )
We shall close this section by the study of the charge conservation law. "; .the reader may have noticed that we always used the following rule called the charge rule in constructing the quaternions: for any quaternion A, ale4Po and t This assumption excludes the E°--A°-y interaction as a direct one ; the decay Z° -> A"-}-y can appear through several steps involving strong and electromagnetic interactions .
280
8 . VMLZAWA AND A . VISCONTI
describe a positive an_.d negative charge state, respectively, and al,0o and an0o, neutral states only in the representation (2). Here 00 is the vacuum state. We shall now prove that the charge conservation law is based on this rule . Indeed making a product C = AB out of two quaternions A, B, we may see that C satisfies the charge rule when A and B do : Ci10a a neutral state, etc. . . . Therefore, when A, B, . . . are constructed according to the charge rule, the state a2IL0o
Tr [AB . . . ]0u is neutral and this property corresponds to the conservation of charge.
4. Concluding Remarks In the last section, we have formulated a new Majorana theory under the fundamental assumptions that any field must be described by a real or imaginary quaternion and that the theory is invariant under the both sides transformation . The X, A and x have been assumed to be real and N, ~ and K imaginary and the basic quaternions have been constructed according to the charge rule. We have further assumed the invariance of the theory under the space refïection (.54). Taking the coupling constants and Dirac matrices to be real and hermitic respectively, we are then naturally led to the conservation of charge, the conservation of hypercharge, (there are two exceptions (51)) the P-invariance of th, nucleon-pion, ~-=, E-a interactions and the pseudoscalar character of latter interactions . Assuming that the electromagnetic field is represented by an imaginary quaternion and that it interacts only with the antiassociated parts of all fields, all the unwanted electromagnetic interactions appear to be zero . The invariance under the both sides transformation governs all the interactions including the electromagnetic interactions . The strong interactions appear to be charge independent in the scheme (1) when we assume a) that 7F never appears between two baryon quaternions and b) that K stands between them . This assumption holds only in the case of strong interactions and excludes the unwanted interactions (51) . The weak baryon-pion interactions lead us to the rule in scheme (1) and will explain the known interactions . In many JAI( cases, we have found terms of the form e(+ H -) . Nothing has been said, in this article about the four fermion weak interactions : this will be discussed in a separate paper. As was shown in sect. 2, the invariance under both sides transformations leads us to an isoscalar and isovector scheme. It is furthermore, important to remark that the Lorentz space is not, in the formalism of the new Majorana fields theory, independent of the isospace . Some of the components of the quaternions are left handed states, and some are right handed ones. This seems to be very characteristic of this theory.
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A NEW MAJORANA FIELD THEORY
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The authors . would like to acknowledge extremely helpful discussions with Professors D. Kastler and J. M. Souriau who introduced them to the quaternion algebra and who took a continuous interest in this work in all its stages. One of us (H. U.), would like togratefully acknowledge a fellowship of invited Professor at the University of Marseille. Appendix Consider two fermion fields Fi , F2 and a boson field B and suppose that Flt _-_ p l F Flt .= P2 F2 , Bt .-- pB, (A .1)
where pl , P2 and p are c-numbers. We want to show that if 0 = ys or iy4 ys , then --1 0ap Tr {Fi F2pB} = (P1P2P) Opa Tr{B* F* F a} .
(A.2)
One may first show that
Fla 0ap F'29 =
FuOap F1ß,
where F',.a = Ca,& F,,j and taking into account the anticommutation properties of the Fermi fields . Then by using (A .1) one may write the left hand side of (A .2) as follows Bt} Oap Tr{F« F** B} = Oap (P1P2P) -1 Tr {Fi t F2at _ -Oap(PIP2P) -1 Tr {B*(Cp2)ß(C-p1)a *} Oap (P1 P2P) -1 Tr {B* (C.F 2 ) p (Fl* C-1 )a} r
since the association operation does not change the trace of a given matrix . Finally, if one notes that --C-1 y4 0Y4 C = Ot, where "t" means the transpose in rpinspace, a few algebraical steps are enough to prove (A.2) . In a similar way we can prove that B} = - (p1P2P)-1'y4y ap Tr {B* F2*ß Fl ap Tr {F, . } i'74y
F2pa~
To prove this, use is made of the following relation : 2F1* y4yp F' 2 = -ZF2* y4 y,, Fl .
The formulae (A.2) and (A .3) are at the basis of the four theorems of sect . 3. References '.) W. Heisenberg, Solvay Berichte report (1939) Theor. Phys . 17 (1952) 377 2 ) S. Sakata, H. Umezawa and S. Kamefuchi, Prog . 3) H. Umezawa, Prog. Theor. Phys . Suppl. 1 7 (1959) 67 4) H. Umezawa, b'1. Konuma and K. Nakagawa, Nuclear Physics 7 (1958) 169
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8. VMEZAWA AND A. VISCONTI
5) hi. GellMann, Phys. Rev. 92 (1953) 833; K. Nishijima, Frog. Theor. Phys. 13 (1955) 285; K. Nishijima and T. Nakano, Prog. Theor. Phys. 10 (1953) 581 6) Id. Cell-Mann, Phys. Rev. 106 (1957) 1296 7) G. Takeda, Prog Theor. Phys. 19 (1958) 63 8) B. d'Fspagnat, J. Prentki and A. Salam, Nuclear Physics 5 (1958) 447 9) H. Umexawa and K. Nakagawa, Nuovo Cimento (to be published) 10) B. d'Espagnat and J. Prentki, Nuclear Physics