A unified theory of the fermi interaction

Nuclear, Physics 8 (1958) 661--674; ~ ) North-Hotlmtd Publishing Cc~., q ~t~:~e?'d~ Not to be reproduced by phe,toprint or mierrdilm without written l:.~rmissc ~ [!:o~:~,.;.~

e

,4 ,~ ,-......

} :5 :

:

i

A UNIFIED

THEORY

OF TILL' FLRMI~ I N T E R A C T I O N

T{-')SH [ YU t:/.[ "f':)YODA

Received 3I~ Juae 1958 A b s t t ' a c t : A m e t h o d developed bv Infeld and va~ der \Vaer~,ien f~r gener;:dia;.~:~g~f~: ~ , D i r a c field equation into the scheme of general relativity is a~ed to urify gi~e, i-:;t-~':~.~:~,,~';, t y p e s b y a d d i n g some new hvootlaeses., but sti.tl remai.qing, h~. t!~c u s u a MmIa ~,~~- ~:~..... A n e x p l a n a t i o n for a modified Pauii-Gtirsey ~:ransformatiem is given i~ r c } a t i ~ +~ ;~. ~: , c o u p l i n g tw.:-pes. Neither conventional i~ospin.-pace m~r strar~gel~e~:_~ q~ ~ a t , ~ ;a ~_~ } ,,,~ ,~:; used, b u t o n l y the usual three dimensional :;pin space is ne,::e~sa~y, The: r,e~,~lc-:~::*::~ ~, ',.>: s i m i l a r to those of the Feynman---Gell-Mann t heory, except for a ~ew p~t~'~t~ ~ c, ~'~ • ~ ~,;:,~: w i t h their theory is made.

1. Introduction Since Lee and Yang 1) discovered that there exist: parity n::moc,.~i~vrv~:;: interactions in weak inleractions many. hvpotheses~. ~-st, have k~:-:~ ,r~:'~'~,~d for a more or less fundamental principle by which parit3" ,~o~;~t~'~:~.. ~:~';f:,! london-number conservation may be underst,:}od it~ s~mu' ,n~ga~a< w : ; , ~ On the other hand, since Gelt-3Iann 9) and Nishijima "~} "~"~" " i ~ : ? . d u c e d the new quantum number to describe the st+.~ctkm ral~ ,A ..4~:':~:4~'~ particles in a phenomenolo~ical way, many attempts .-.~) l~ave b~: :~, :~:~:::~.~& to interpret the strangeness quantum number. Among these the ~..~-~..~ t ........ t w o kinds of iso spinors ?t introduced by d'Espagnat and Pr.:r~tki ~ } ,:.4;~,:'~:~:~~ t o be hopeful, because the vMation of isc parity in the weak d<:ay ~:,4 s,tr~;~i*{tv' b c t w e e l t t h e t w o ki~'ld~ i~f pI ~:~' ~:~.~c~ particles suggests that •a clo~"e relation ~ ~....................... m a y be obtained from the same point of view° Infeld a t l d m a t h e m a t i c a l structure.. of the ..~elativi,;tie wave ,~.q~:;.auo:~ ~i ~.~:~.. ,,~:~ great detail and gave a ~enera[ relativisl:ic cxten~i~m c~i t ~ ~' i~ *';:~c ........ ~ ' : ~ using the concept of gauge i,uv ~triance mv~'~w~ ~~ general spin space a t eact~ a ~ r l [ p o i v ! ~ [ ~ o i d ;~:.1 ~'~i ,;~ %~;~',,i.,:~:~~,;',

. . . . .

~ f :In B e r g m a n n ' s paper of tel ~} tIae rote cd p a ~ ; ' ~t

.................

,~i~:~!% ':..~i ii'

:iii hhe i in the sense of W e : i s ~nc,ple o f gauge m v a m n ~ . . . . . . . We w~uld l ~ e to point out t ~ t arty sift:nor field must _~ ~ u p l e d with

Therefore, if=there e x i t more than two iields, there will be s o m e change of tra~form~tion property of e~ :t ~'~o .~e~. ~t~hI, ":s it~ ~ result: ol t.h~ coupling of the fundamental v¢.~.tor item with 'the other fiel~., A general e x p r ~ i o n for the Fermi interaction which: ~ invariant with r,~pt~:t t o file generalized transformation will be derived ~ l o w an¢~its physical interpretation will be given.

2. Mathematical Formulation Following Infeld and van der Wae;den's treatment ~), let us assume a spin space t at each world point x ~ and describe the spin quantaties by a set of two complex quantities ~ , where latin indices k's run from I to 4, and greek indices A's run from 1 to 2. For tt~e sake of s~mplici%, we shall confine ourselves to the usual flat blinkowski :~I~tce; the metric t e n o r g~, is then Since we want -to extend the usual transformations then as is shown below, the 1aetric tensor of the spin space can no kruger be taken, as constant with respect to variations of z *. This may- be made plausible by considering the cormection of the metric tensor of the spin space ~fith the fundamental vector field mentioned above. We shall define a generalized transformation of V,~ by a two-by-two matrix w'hose determinant is not rarity but e ' : ~" =~ A p a ¢ , ~fi' =~-~A ~ , A ~=: det IApa~ = C ~, (2) wtiere 0 is an arbitrary real function of ~c*and the usual notational convention is used. As is well known, the metric t~nsor can ~:~ defined in the following way:

The tramformafion formula for 7 ~ is e~sily obtained a s • ~spin space is entirelydifferent from Pais':*~)" ~ e , which is ~ e p e n d e n t of the Minkowski space. O n the contrary, ore- spin.space is intimately rgl~ted ~th-.the Minkowski s~ce:thro~gh its transformation propertT, as will tie shown ~lOwi :i . ~.~:i-:i

A tJNtFIED

THEORY

OF THE

F~$RMI [N'Tf3RAC33tON

t

. t

~:,t }

~:'

::= "--e ~ ' / t 2

C o n c e r n i n g t h e c o n n e c t i o n operator~; betweer~ a general work[ v,,~.cvc~: ~,~ a n d a s p i n t e n s o r ~Pi~, a m i ~ . . ; , , re.scar ,¢)'.~, be derived, sa,J,fv,f:~}~{ {i~# f o l l o w i n g c o m m u t a t i o n reiations: aJ;

~';A# ~.,.

a,~,eC./ aa,cesa{

= ;"0it,* ,

w h e r e ~.~' a n d 6i ~ are K r o n e c k e r ' s deltas. Normalizing the vohm~e, t ,~.>,:,; i:~ t h e s p i n space, i,e. Dutting -, o-,.~ 1, we m a y ~wite d,>w~ c'xb~,~ ~::r ~ " ' metric

tensor

1'a~, as

w h e r e ~0 is a n a r b i t r a r y fnnction of th,~ world point ~cL A s w a s ctearl)7 d e m o n s t r a t e d above, the generalizatio~ {2) ~%c~'~i~,~:c~}"-' c o m p a t i b l e w i t h L o r e n t z cowtriance. However, since we like> to :m.d-;~ T,}'<, s m a l l e s t possible modification of the usual theory, we sha~i c~>;~ri,eV ,,~,~' g e n e r a l i z a t i o n (2) to the ra~her ibnited transformati0~m - 0. 1 #a ,
;{:}

det 0 A f ' l :=: 1, w h e r e 0 is still an a r b i t r a r y functiolt c,f x '~, be.{ o;1/t ~ x , ~,~,,v pav>a~~,:> {,-:>, i n d e p e n d e n t of z a. T h e n our generalized u:aasfc~i:-mado~ i~ >~d~>:,.d, '~,}'?./,~ -~ ,,~>,:,5 f r e e field, we h a v e no roon~ to add scm,ethi~g ~>,w t~ z}~< ~,a~,:d. ~:.~'>:::< H o w e v e r , as we shall disct>:~ later, ~hc Illctric ~ci~::~g i~:~ ~:pb~ >,p.~',, , , ..... m a y be t a k e n to be connec.x* wii:h a, ............. g e n e r a l l y is a m i x t u r e of vect<,r and a x h ! xec~o~ at~..> z,g>a .,;~ ;:,:-~ ~ ~ ~ g a u g e t r a n s f o r m a t i o n . "rherei,>'c it 5,.'<=',~s to u,: r:~.ti~c*,: ?~¢a>k,~ ,,* =--:'~,~:~:: .. p with tlfi~ >,w IbA~d ;~:,. ~ ~{[~:~'q:="{ <;'14}' :¢}:i~{>: t h a t e a c h s p i n o r f i e M c m ~ 'es . ,. • ,~;, ,- ~: ,~ :., ,<{ {,,:~, o f t h e c o u p l i n g consta~t. The d.ifkq't,u, c ,d ta,' .... q , ...... ~ a b l e t o e x p l a i n m a n y omserv,;'~ field in t h e general rdaUVity, ia ttte s,mse tha~ }::*'~}.~,,>t ~:i::,,'* > ' r e l a t e d w i t h t h e corres~o-~ .... 5 . . . . . L e t u s c o n ~ I d ~ , r ,t s } ~ t c t n {.,t ,v,~, . . . . . . .

., ~

:{~,~'>~,

:!: )i ~:~ ii i.:~ii~ i i:i~!

:

In generai, t h e ~ two lieids ~re coupled to each oiher through the: B field, Therefore, the metric :of the spin slmce for the case where: there exist two fields :a and b which: are coupled to each other can be different from t h a t for an isolated field, Before d~ussing the general c&~, we shall consMer a very speciM case, viz., in: the limit where the fie~d b ts couple extremely weakly •,~tlt the B field we may make the following hypothesis: (I) The metric of the spin space is determined only by the existence of the field a, so that the transformation of ~a is just the .same as for the isolated case, Le.

,~(a) -~ et'o,p,(a),

(8)

~,~da) -~ e-~':,,:,(a). On the other hand the metric 71a appropriate to the transformation of ~ is only influenced by the coupling of B with the field a because of our assumption of ve~y weak eoupling of b with Bs i.e. tp~ (b) ~-"~e l ' e # (b),

(!3)

Thus. any physical quantity e~ the syste~a sholfld be invariant under the transformation (8) and (9). The neutrino field is the best example of th,.: field b. If we rewrite the transformations (8) an~: (9) using the familiar notation for the 7-matrices t, we obtain. Touschek's transformation s) t Although our iix~al :-esutts do not depend on the representation of the ~,-matrices, it is useful to note that van der ~,Vaerden's repre~ntation (7) is a.~aciated with the following representation for the 7-matrices:

(--0)

(

.....o')

(0

With the usual definition o~ ~'s this is given in this representation by

7. = 7'7"7"'/' := i (I0 _0I)" However, with our choice of metric in the Minkowski space we sh~tt deiine

S'aclx a definition does not ch&rlge the usual important relatiollt~ except tha~. now F~Z ~ L

A

UN1FIND

I".i-fi'2OR~

c}F. "t?Ht:i l " t ~ R 3 1 , r I..~lhd.L%'' .... ,a~'.,;,>~' " ~

~ff't~

;~{[i2

e:.',,,oa T

A s s u m i n g t h e n o n - d e r i v a t i w : c o u N i n < < then, a,; is wNt kl~ .,,:a~. ~. . . . . . . . ¢oUplin g t y p e w h i c h is invar-iant with req~{:.ct h, !s'~ a<~d ~:,:. > ..... ..... g i v e n by i n t h e g e n e r a l case of a system of tv:~ find,; ,, "* ~ .;~ . . . . . .~s . . . . . m-~*' , ...... ~ k ~:~ i'>~ . . . . " < ~' L transformation law of the system, beca;~se the metrR: :~,d ~:;~: ~,9i:.~ ..q:~::ha :,p.>_.v ........ ~:~. ' ~ * , { ,~: L);,:<:(, t o t h e full t r a n s f o r m a t i o n (4), which woukl o c e m m ~l,:,.. ~ ............,~ :. ~!:: ~:h~. ...... f i e l d b, b u t is onh" subject to the s,vnR'.;vhat ie~:5 e.-,.~.-,'"'c<~ '

;'~e("~) ..... {) <
,-r>" '*

S i m i l a r l y , t h e t r a n s f o r m a t i o n of the fiekl b i~ .~ ~v~'~ i:~y v'a (b) . . e0" ~,,*@)~

{2} < c a < I°

invafiant

u n d e r t h e transfo~mado~*~ ( 1 t ) .... * ~::~},

With 1 H o w e v e r , ia ord,'r to o b t d n tha ....... ~'.... > ' , i > > < t o o s-mp'e.i laWandwetheneedB field.m°reFor the morn ~,ut wc do m~t }>~vc d-::i~-~.~;~t~:~P.,~[: .: :~ <:."<: c h o o s e a s h n p l e f o r m for th* ~ traud,~rt~S:h".',~ #.va,ib , , ~ b /v,:-, ,:

The

For the

ex,lstellC

,

....... ,.. = i!~v,%r},~:~l ~N!b{¢< - - . 4 ' ~' ' ~: , , ; , , < , t p ; ,,,

g}:!" :

(IIY) t h e c h o i c e ot el (me, ,,.~ ~....... ,c~mNm'}~e, ,:~-~.,~ i n g t h a t t h e o n l y p{>Sb!e i e v a r i a a t previous

c a s e (t0) or ~vid~

,,~,

't<¢

;,:,,

,.

......

. . . . .....

~:

the~ ~)f four sDiners, If 'we p~~tuLate t | ~ t Fermi t ~ :mter=~tions are to inv~i;m~. ~raler t ~ above tra.nsformation, then;the only p o ~ b l e kinds of

in) ractio

:the

t:hr

nd ii

where A, B, A" and B' ~ arbitraxy cor,s t ~ t s . The differermes among these tht~: types of i n ~ c t i o n come :from the ~fferent transformation properti~ of the four s#nor fields relative to each other. Nmmely for each of the interaction t y ~ (14)-(I6) the foiiowing conditions must be satisfied (see appendix): *t = ~=,

and

~ ~= e~,

(~4')

~'t '~= %, e~ ~nd e( can have different values,

(15')

~ , %, e~ and e~ can have different values.

(16')

The condition (1,;~') is ve:~, stror~ and we shM1 discuss h t e r in detail the c~se owh,ne it ~s~s~dsfied~ The coupling (16). is identical ~t;h that of Fe)m~ai;lli attd Getl*Mann ~) and corresfumds to the V - A combination in the two-com~nent neutrino ~h~)D,. However, in the cv~se of (15) we have freedom in the choice of the mixing ratio of V--A and V + A in the sense o~i' the two-component neutrino theoD~ but this freedom is gained at the expense of the zather strong condition (lo). In the p r i n t paper we are ,oncerned with l¢cal and non-derivative couplings. However. in order to ~ how our metric spin space works, it may be wo~h whim to ~view ~ m e equations ~*) concerning the eovariant derivative. ~o define a covariant derivative of a spinor we need the soc~filed affine tensor I a r

6

=

07)

In order to determine I~*, ~two postulates are essentmL (i} ~ must be covariant under I~:)rentz transfo~ations, (~) a volume element of spinor space must be i n v ~ a n t under a parallel displacement in the Minkowski space (local conservation law) L

A U N [ F H ~ D TH~O~IY O~i' T]{E F E ~ M ~ I N T E R A C T I O N

t~!,~

Since w e confine ourselves to the usual 31iakc.,wski s;)a~>h :.m ;~w" :.i# ' r e p r e s e n t a t i o n can be give:a for /7;%:

w h e r e ~ e is an arbitral T re,ai flmctio;a of ,t a', g a~ arbkra~ T g.>:'.>~:~v'< :x>:<; K r o n e c k e r s delta. If there is only o.~ spinet ":.... Q , , ~ .........a.~ a n d is subject t o the second kind gauge tmnstormatS,~a # 0

T h e n # ~ could be interpreted as the eiectro-~magnetic fi>ur po~(~,~':i~i_t ~i<:.,:~ ever, it should be stressed that this ~,,ee,~:YaI vector field which m~:~ h< .~::~,.~,,,d a t e d w i t h a n y spinor field need not satisfy the l[axw~ il equaU~:)~z, f.,.:~~.~.;<. if w e b o r r o w Stueckelberg's method 2s} which e~abbs ~,s v) tv*:,~ a ~>:~<~:) vecto:r field similarly to the electromagnetic fiel:L the {nwr:~c'tk~:s be::~!,c':'-"<~ , s p i n e t fields a n d a vector field whict~ has a firfite re:at ~i.tass ca~; i>~"b~;>.~:i.~-<~4 b y a f o r m a l i s m which is completely analogous t o that d~T<~p~.l f~:~ ~Y:<'~:{:>,~ electr, x t y n a m i c s . A V.~ssibility is then to assume an ext:.'{u~::% ~.~:......... f o r this n e u t r a l vector field so that the region of nowloe~J:i~ )' a~s<,<,}.~w.4::<.,~f~t h i s is v e r y small. It n:fight be possible to tw:at~,, this ];,<:~lln:;<~ "~: ~.: w i t h t h a t proposed b y Lee and Yang :~! for the ~:>xptamati<~ ,:~ ,r~: >,~,.:,~;=;' n o n - l o c a l i t y in ,u-e decay. ~. I n t h e case of interacting spinet fields, corr,.-q:,.,:vadh~.gto, the:' ~:~;~:¢~:"~':-,:~-~ t i o n ( l a ) , ¢~ should undergo the trm~sformati.~

T h e n ¢ ~ no longer has a definite sp~.ce p ~ w ~

~. . . . . . . . .

T h e idea of sucI~ a . a t , . ~ . . . . . . . a r b i t r a r y w a y to explain the .~ebcti.. ,~..... T h e r e f o r e , followiRa" ~ a - s and ,~E[!s. ,.v~' h a y , ~,.+:rr~'~{ ['~ ~"+'~ field as the B field. In the prese.~,'..t -'~'"" a~':*-!d,, explicit behaviour of dw l~ ficM but .~,{',~.~!hc ,.~ ,a~. '..... ' t ~r'x,\~ 4 : a ,"*,<'"~ effects on the form of the mu:racd{,{~ ' * ' .... ~>.:,,>,~ .,.vv:x. A s is well known, {he usual M " £ c < ~ I ';>t>ea~ ~..t{: a;.,:~ :'::,, v a r i a n t w i t h req~ect to a U,aasbrma~i~'~ i~-vr~&i>;: 7, a,w,c},,~ ~,:"~) : ~=:~:' '~; ?~, v a n i s h i n g mass. In particub, r it i.~ u~,': ~......... •

:

H o w e v e r , as Giirsev ~'~ hack. ,,b'w,~: ' ' c '~m{'~ i ~'~: i~,:{~'': >':1 ~:.v:.} f o u r - c o m p o n e n t spir>r ~,."~............... !~,' ,=: <

%':..*.,= ' > ::: T ~

t h e n we can co~struct t!w .,qua~th.m 1£ ' *

{"

which ~ in~'ari.ant*if :W,~ translorm :according to {2!) and (22)resp~tiveIy. VCe may ~ h t l y ~ t e tl~ds eq~tion in the following form:

where

r*

.....

(zs)

/'

and sat is~, the commutation vain (26)

F* FI + F~ F* : 2g ~t,

It is then quite e~ident that eq. (24,)is a l)ir~c equation in an eight-component spinor representafiorx, Such an eight-component representation is of course reducible ! but not tri~4_a/. Th~ is because this change of representa~on is nece~h~, in order that the Dirac equation is invariant ~ t h respect to the transformation (21), (2:.~) and because it can now also have another important p r ~ r t y : ~q. (24) is invariant with respect to Pauli's transformation { I ) ~ # = ae+~,s~c,

#¢, = ~ . ~,,_b../s~t, "

(27)

Since tM~ g ~ u p of tra~formations is i~morphic to the unimodular group and also maintains the invariance of the generalized Dirac equation (24), ~e may interpret this group as the usual isospin group. Therefore to a solution of eq. (24) we can attribute an isospin quantum number and an isosf&,~ parity, but not a defimte space parity. It is 1~ther straightfor~-ard to apply the above discussion to our theory when ~ is not a real parameter but a real function such as ½(I--e)0 where 0 is a :function of ~ : if we replace 0, in eq. (23) by ~** and 0~, so that nOW

--i.~r,

y~'Ovd( ~ )

=0'

(28)

this equation is covariant with ~ s ~ t to our generalized transformation :if ~t~ and ~la tramform according to ....

(-29)

A '(TN1FtED

However,

THEORy

OF

.'.'}~

w e n o w lose t h e fi~variaac~

t )e

:: y e n i n t h e c a s e w h e r e ,~ a.)~c~ b :m:

F£~.'~I.

72:T::~!&CT~:~N'

~:~,i',i:

: . . . . . . :.,>>: ,-. . . . .

~.

. ........

~-.-~.,,.......~,.

o t h e r w o r d s , o n l y i n a c o r t : d n ,..~p0rox~.,s~a.t~o~,' " ',~I,,,,.:~'" . w. - . electric approximation", i.e. (rely -,v,~>:~ O ,* k : ~

can

we define

oniy

t h e i s o s p i ~ a ~ 4 isopad~.y,

I n f a c t , il w~ :: l~3>,ittlte . 5

identical

with

" $ 7~:~V ; ;' " : :," ' r[

'

i~ ~.~ . . . .

t h e c,:>,av~.:~tiomd iour-c(>~v,p<,~>,~t [.,~,,~.~ , ~ , . ~ '

We shall now construct of ~

;,, ::¢~

:5....... %

t h e d i f f e r e n c e of tr~msf..~-;>a~,.,.~'~ , , , ' , .... ; ,

disappears.

" " - " "

'

m~.,.,~:.:,:~-~,~,

; a ~ . , . , , :~:

a s w e l l a s q~,

Defining

and

'~" ~ i k :-at. & Cg,

i n t h e c a s e of i n t e r : : c t i n g q~in'or sy~r., ~ . F - 'r ............... :~,.,.,, ..... ~,.~

e ~--- 1 a n d

. ,.: ,~.~i

noting

~ and

the

& by

followi"~g q m p l c ~2

we can These

readily

r,.k~>ic,i,s:

:-:: 2 ,

:

wr:ite d o w n ti~c' i ~ w a d a ' ~

{

a ~,

{},

m ~hc c~,.,~" ~,~ ~,~

,

e,

,,¢

are g

e,

~ ~ii!~¸

Ia =

~F: ~.'F.e #>*a""~g~'

]::. .... ~/: ~g>.//~ "~¢'a,

;q

K a .... ¥'~ !I', g a where

~

a~d

'"

. a:,.' :,i ~'* d ~ ~-,.,=.:,~ . . . . . . . .

E a c h o f t h e a b o v e " ~"- ~ : ! : ~ : : ' ! : ; ~ > :~:i ,: g e n e r a l i z e d tran~form~,.',i,~,. { 1:!.. ~:.~'~'~ ~=. i ~'.':.~-: >:, "~: {~ " ' '

familiar

Garsey i

'

way,

i t is c ,~:;'c.ai~s~ >> r ~v~.v. *{:'.,."~ :i ~: "

~ transformatic>:~

. ' ,:~ { '~= ~ ,.-..:i} ~.

.: ,:<:,}e:!

71~en we obtain t a e following: f a m i I ~ expressions: r

r

+ ,~+"

I~':

-~;

-

e

~ ~s~o~e~ ! , ++++l+,+ l ':t ++I '+ I+" f + I + + 1 ' t + + "" /z

K,,+ K m+ + ~ + + : E , ,.

. . .

..

~

+

7~M:,

(36)

¢ "+ +"

'~ r+++, += P+iF+V+~9'mT,+~+++'.+,

! +='+ (~+r++'++,'+++'+'h +l++z)(¢,+,+++,r,+lW+Pm+1+m), (,~: ++ + ++++++ , + .. ++

,++++K++++K+++-IG, +++ ++,+,,,,,++.++ +,.+ +

(37)

(++,+

If, :+r example, we make the identifications ~++ = p,

~+,':= n,

it may be+~ e n directly that ~he following combination is charge independent:

t

(3s)

Furthermore, it is quite obvious that (38) con~rves the space parity. On t h e other hand, if ~l = ¢~ ~ e3 ==- e4, we cannot construct a charge independent coupling such as (38) now only the last term of (38), corresponding t o the third eompor.ent o;f an isovector, is a ~ssiMe interaction. Thus, in th: e a ~ of ,t = % m F3 = e4 we may have tht~ charge independent coupling in the first appro~mation and the Ts-intcraction in the second approximation u+~)+ 3. Discussion

As has been emphasized in the preceding ~ction, our generalized transformation, under which the physical description must be invariant, Only affec~ s~,ste~ of two or more coupled spinor fields, but does not in any way m o l l y the properties of a single isolated Dirac fidd. In the ~++++-,of two or more fields there is a modification of the transformation properties of each field. This modification enables us to classify various types of interactions, which have previously been propo~d, phenomenologieal!y, and to associate each of the types with a corresponding c o ~ r v a t i o n i law, : : One of the e s s e n t ~ p o i n t s of our generalization :is Lto i postulate affine

W a e r d e n , w e w o u k t like to con~ider t h a t ~a...., ~~ ~, fi~?.i consists c.4 tw~:~ :~" ,~o~.~> •. . . . .,~,~ . :fields, i.e, t h e e l e c t r o m a g n e t i c f M d a n d the JJ r i d & T i > ,~:,~:~~"'->~.~.~.., ~):, ~., f i n i t e r e s t m a s s of t h e B 'hel'<] ea.ik be a v o i d e d b y ' ~ ~-,~i. 4~.~,,> '/l<'>:<.!i:':~ --~~ ~ s a m e cow,piing c o n s t a n t as usual, we readily <:,btaia tt~e charade .'-~ ,.~-...... l a w . I t is e v i d e n t t h a t invariance under our generaii:~ed ~:}~ g u a r a n t e e t h e i n v a r i a n c e trader ti~e special ..... " ~ ~ ;. . . . . . charge conservatiom S i n c e w e a r e i n t e r e s t e d in the m t e r a c i : n ~ lr,.x~.:,, we m a y ck~.:~sif~."[:, ,!~:~:'~o.?:~> f i e l d s i n t o t w o c a t e g o r i e s according, to th~ n-~v~~ii~ude,.. . . . . . . . . .
transformation

function: (i)

lepton for ~" %~ l.

,::~*~

(ii) b a r y o n f-~r ~-, ~ i.

~..

" I I a b a r y o n p a i r a p p e.a. . .r. s :n the l"n t k r , ,. t . t-l ~, . " w ,~ l"'l e t t |': ~ , - a I::,ary~mc ~:'u~'~'~ ,:~.:~:.~ ' w h i c h s a t i s f i e s t h e c o n t i n u i t y equation can be d~.~fi~wd i~',}, ~~- ~ 'o.•~:~', b e c a u s e t h e i n t e r a c t i o n between barvop, fields am{ ~l~e ¢-~' #-" '~ c o u p l i n g a n d direct couplings between baryon fieht~ ar~.' a!:,> i ~v;a~"~,,e~>.~,:~.. ~}~

r e s p e c t t o t h e generalized transformatio:~ (lS). R e w r i t i n g t h e t r a n s f o m ~ a t i o n (la; ;,nr V',~ a n d v,:,<

"~

b.:,-:~. ~.,,

A s w e h a v e m e n t i o a e d at the be,,zinmng of the ~recedi~i:: ~;~'~~'~> ~'>' ' ~'~~': e =. 0 m a y b e i n t e r p r e t e d as t h a t ol a n c a t r m o t~ ~,~. " - ~ ~~ :........ ' M a u s i b l e t o i n t e r p r e t V',, aa i ~'~,~ as ~a'~ia,a fi>: s:m:> l>r~va:i< ,:~::~'.

>~::':

opposite leptonic charge . . . . S u c h a n i d e a was first proposed by" l',.>.>chek s1 .;_rod ~ t .... ~w,,,',;~,~> ,~:<: extensively

b y K o n u m a a~;i m e ' ~- ~'- . . . . . . . . .

"

number . . . . . . L e t u s n o w discuss briefly tlw corr,,,pt~,~'~'~ f o r m s a n d p r e v i o u s l y propo..;ed F,,,r~m " interactions,

~ .....

°'

,,<~g

~'~ '~ "

' "~ . . . . . . . . . .

e.g, ,.e,-e oeca,..

,}~

~: ~. . . . . . . . . .

a *

i ;~":e

b e e m l s e y , e a n d ,' m:~.5 a!l ' , v .a;tt,''<~t v,.~Ka"~" t h e s e a r e n e a r l y zero, "l"hi~ i-- v>_~.c!;v .~h~ ..,,~m, .,~.~.,d~. ~- .~" '!~:.:~' : ' :~:~ a n d G e l l - M a n n , w h o have" i,c:ves~ i~:at~',i * !~> <>"~....... ., ~ ,.,.,:~r. ......~.~:i"" :::~:~ !:[ ~:~-~..":": :i:: t h e p o s s i b l e interacti, m~ ~'~ o m v ~his ~5! w- .\~,.~.5,.'r ~..........' ~ ; :

:;~::~)

,p ._.~.~>1~,e,+4,

2

e::~|

~ e ~ :of mtx~::o~ t ~ a!ternat|ve ¢ o n i ~ t ~ g-~ A,: V @A (m? ~ : ~ of two~mponent:: ~ t ~ t~)ry-); ~ the Contrary, in Feymnan :and GeltM ~ n ' s t ' h ~ R ~ m s t h a t ot>e can one" Choose: oaeo~ t ~ ~t~tiv~,

~erefore, It the ex~fir~mnto, e.n the a n g ~ a r d!strtbutum of ~ t r o n s from lx~!ari~M n cutron~ do not agr,, ~ h h ot~e of the Mter:tmtiv~ l ~ i c t M by Feyaman and GelLMan:rt:~ t:hs w~mid ~:~eni MvouraMe for our theory. : Finally, keeping in mitM a ~pinor m ~ e l in which all bosons truly ~. regarded ~ comf~site ~,~.. ~"s"t:~n~ ~ s o[ iermi~ms, e ~ e p t the eku:tromagtietic: fidd and the B fieM, we ~mR consider interactions of Fermi interaction type. In the e a ~ of interacting tmM.~ it is u:,c~try to introduce the eight¢omt~nent ~pinor relpr~ntation o5 the Dirac equat:itm in order to describe each spinor field in a covariam way. WRhin this expanded framework the concept of ~ p i n appears naturally in as~)ciation with a particuFar interaction ty!~. In fact, in the ca~: of q ~= t~ ~ ,~ := et we flare charge-inde~ n d t , n t interaction:~,~; or

If we take into account the I:mssibitity of introducing two kinds of isospinor n) the strangeness (corresponding to the ~ p a r i t y ) is obviously conserved in t~th the above ca:~es, a_swell as the space parity. Therefore it seems plausible to take the above two interaction forms rLs typical of strong interactions. Furthermore it is inter(~ting I.;.~at the case q ,~ ~ ~./=ea ~ *a gives us only the la.~t~s'term (*a part) in the a~ove* • ~two interactions (45}, (46).~ Therefore, if ,,i ~:~ ~ ~ ~.a ~o ~ we i~ve the: strong interaction (45) or (46) in the first approximation and %-interactions in the second apprdx4anation. This corrt..'sv~:nds to the modification of Gell-Mann~Nishi;ima's. . j selection rule otm:~d by Takeda an) a:nd also by d'Espagnat, Prentki and SMam an). "rile modification of the spin metric as~mlated ' ..... • . . . . . fields has .... with interacting some analogy to the mlxtificatmn of the Riemamnian metric ~ i a t e d with the interaction of two or .more bodies :in general relati~ty. Another analogy

~:;~:' ,: :

A :UN1F[ED THEoRy

O~~ THI~ FEi?.~-[[ I N T E ~ C T 1 O N

~

b y ! n theofpresenta generalizedPaper we have tried to unify all known eoasem, at i , . , . laws ~ ..'ILse transformation group, which !is a:;soc~ ~to I a i t h tb,;? t r a n s f o r m a t i o n of the metric in t h e , ' .......... ~ s u a ~ . spin sic,ace. Fho a~.,t~,::~~,~.i:~l~.il.ik~ t o e x p r e s s Iris sincere thanks to P rofcs~ or B. Ferretti f .)r wfl~ abk, dis.<:~:¢~.-i,~:v.,s a n d P r o f e s s o r L. Ro~emeld': " for the' Innd" intere.~t.- Fie wi,,hea to ' ...... p-¢c. J. Bakker for the hosI'&aiity. extended to him ~:.t uEt(N,~: ' :

Appendix F o r calculations which include ;.% the foilovdng identities; are us<:,ta_~: ( l + y s ) e ° 5 " = (t-SZ~)e'",

(1--;.,s)e':rs'~' =: t~1- - . 5, , et ..... " w h e r e u is real and exp (i;,~u) = cos u+-i),~ sin u. If we transform tv, c ~,~:a o c o m p o n e n t spinets as ~'$

'

It-

the couphng I = !t~1(.-t--~.',, ~"~T., is t r a n s f o r m e d in the following way:

"q- "~ 1 "

gi"

--

"/~"---zS]z

~ 2"

T h e r e f o r e I is i:,variant for the following two ca~o~,< (i) A~ B are arbitrary, b u t 81 == ,~Q, ~'1 :~: ~"~" (ii) .4 . . . . . !¢,

b,.it u,-! vi .....,~-{ c~.

T h e case of u x - v ~ = (u.2--z'2) is omitted because i~', our th~.o~y i~ i'~ ~ ~,.;'~.~.~~:~:~

in (i). N o t e a d d e d i n p r o o / : Rocent ,' " " ' ..... ' ¢hiral boson, tield to derive th=, I'crmi ipA~r,~c'~:~;m ri:,a ' S. W a t a n a b e for sendin~ him the pr~ p~iz'~t,and ab~,~ for i~:e,."e~m~ d~,~ ~:~ ';~:~~=-' "~:~:'~ bet~veen their theory :rod ~m:'s.

References 1) 2) 3) 4) 5~ 6) 7)

/

T, D. Lee and (7. N. Y~mg. Phy~. R~.v. 104 (~!):~0 ,'.:~ A . Salam, Nuow:, ( m w n t o 5 (t957i 2!~9 L, L a n d a u , Ntxciear pi~vsh:s 3 (p357 127 T. D, L e e aml C. N'. Y a n ~ Phys. Roy. 105 ~i~[,"7 iU'7I F . Touschek, Nuovo (qmci:al 5 {t9.;7?~ I:2~! K . Nislfijima, Nuovo qia'.c~to 5 {U)57i 13t9 E . J. Schremt-L Phys. Rev. 108 (!957; 1o76

:

ieM,

ithi Fiz. Nau~ Sl' :(1953~ ~17

19} J, Rav~ki, :Nt~ovoCjm¢ot~ | 0 {195!] :1729; |;# (1951)8t5 22) L, latdd and B. L~ va~ der 'VCaerdcn. Sit~ung~¢iehte der Preusslsehen Akad. der Wis~. (Jga3) 3 ~ ; revt,wed in W~ L~ B~de and H~ Jehle, Revs. Mod. ]~rtys, 2~ (1953) 714; ~IsO inP,. G, Bergmann; Phys, R~,, 107 {1957} 624 23) H, Weyl, Sitzungs~cht~ dec K0nlg!ich Prcu~ai~chen ; ~ d e m i e der W ~ h a f t c n 26 (1918) 465; See Nar.htrag yon W'eyl; Setecta Hcrmamt Weyt (BirkbJiuser Verlag, Basel und

Stuttgart, 1956)

24) A, Paisl Physica i9 (i9~3) 869 25) ~ ior review L. Infeld, Revs. Mod~ Phys. 29 0957) 398 26) R, P, Feynman and M~ Cell-Mann, Phys. Rev. 109 (19~8) 193; J. J. Sak:urai, Nuovo Cimento 7 (1958)649; E. C. G. Sud~rshanand R. E. Marshak, Padua-Venice International Conference on Mesons and Recently ~ v e ~ e d Partide~ (19~7); Physo Rcv. 109 (1958) 1860; S . B . Freiman andH~ W, WyiG J r , t~vs. Rev. 106 (1957) 1320 27) B. I~MrrettL Nuovo Cimento 6 (1957) 997" 28) E. C. G. $tueckelberg, tielv. Phys. Acta 11 (1938) 225, 299 29) T. D, Lee and Co NLI Yang, Phy~ Rcw t08 (1957} 1611 30) C:N~ Yang and R, L~ Mi}ls. Phys. Rev. 96 {1954) 191 31) F, G~lr~y, Nimvo Cimen~o 7 (1958) ~ll 32) L Biedenhara, Phys. Rev. 82 (1951) I(~) 33) W. l~,uli, Nuovo Ci:mento 6 (t957) 204 34) G. Takeda, Prog, Theor, Phys. i9 (1958) 631 35) B. d'Espagrtat, J. Prentki and A. Salam, Nuclear Physics 5 (1958) 447 36) M. Konuma, Nuclear Physics 5 (1958) 504