2 (1953) 1

CZECHOSLOVAK

JOURNAL OF PHYSICS

ON THE ISOTOPIC SPIN OF ELEMENTARY PARTICLES Vhclav Votruba and Milo~ Lokaji~ek, I n s t i t u t e of Theoretical Physics, Charles University, Prague.

It is shown that the electron, the positron and the Ma]orana neutrino can be described jointly as three charge states o/a single particle, the lepton, which in contrast to the nucleon is an elementary particle with isotopic spin 1. Similarly, the mesons ~• and ~o are described jointly as three charge states o / a single particle, the pion, with isotopic spin 1. Interactions o/the lepton field (and also o/the pion field) with the electromagnetic field and with the nucleon field are considered. The whole theory is shown to be independent of any special choice el the representation o] the isotopic spin operators.

I. Introduction. Recent discoveries of a rather large number of different kinds of elementary particles and of their mutual transmutations have emphasized the importance of the problem of the ':families" of related particles and of their unified description. The ultimate aim of theoretical investigations in this branch of physics is, of course, the establishment of a "periodic system" of elementary particles, in which all possible kinds of elementary particles would have their natural places. Needless to say that this ultimate aim is still far from being realized [1]. Nevertheless, the existing theory of elementary particles already knows and can describe some relations between them. Thus for example the pair relation between the electron e- and the positron e + is well known from the very beginning. A unified mathematical description of these two kinds of elementary particles is provided b y Dirac's electron-positron theory, which is now usually formulated (more elegantly than earlier) b y means of the charge conjugate wave functions. Also the pairs P• N~, /z~ are usually considered as related in the same sense a s e =l=.

Another pair-relation is quite naturally assumed between the mesons z* and ~ - which are both described jointly b y a single complex wave field. Besides the above mentioned pair relations there are still further relationships between elementary particles, generally assumed and also described mathematically in a more or less perfect way in the existing theory. Thus, first of all, the proton P+ and the neutron N + (together with their antiparticles P - a N - which will not be specially

mentioned in the following) are generally considered as being merely two different "inner states" of a single kind of elementary particle, the nucleon. This nucleon m a y be characterized briefly as a particle which has two spins, the ordinary and the isotopic. Both these spins are 89 spins, because each of their components has only two eigenvalues. In the existing theory the isotopic spin ~+of the nucleon is represented exclusively b y the special Pauli matrices. In chapter I I I a of the present paper a slightly more generM description of the nucleon is given and shown to be independent of any special choice of the representation of the isotopic spin matrices ~i. A further "family" of elementary particles consists presumably of electrons, positrons and neutrinos, for which the common name leptons has been introduced. The true character of t h e relation s in this family of leptons seems, howeverl to be more problematic than in the family of nucleons. There are, in principle, two possibilities: 1. The antineutrino is an analogue of the antineutron and i s distinguishable from the neutrino. In this ease Dirac's theory of holes applies to neutrinos. 2. The antineutrino does not exist (or is undistinguishable from the neutrino) and the ~Iajorana theory is valid for the neutrinos. 1) In the first case the family of the leptons would contain four members e ~, ~ and would be quite similar to the family of nucleons. The electron and the neutrino (together with their antiparticles, the positron and the antineutrino) could be eonsi1) E x p e r i m e n t a l evidence concerning t h e double fld e c a y [2] does not ye~ p e r m i t of conclusive discrimination b e t w e e n these two possibilities. [3]

2

Vdclav Votruba, Milog Lokayi&k: On the Isotopic Spin o] Elementary Particles

dered as being merely two different inner states of a single kind Of particle, the lepton, having the ordinary spin ~ and the isotopic spin 89too (exactly like the nucleon). From this point of view the family of "light particles" (now leptons) has been considered by WENTZEL [4] who attempted a charge symmetrical theory of the interaction between the nucleons and leptons 15 years ago. Such a description of the lepton would be very attractive if it were possible to treat the nucleon as an excited mass state of t h e lepton. This should be a rather difficult task, however, above all because of the very high stability of the nucleon "mass state". Conversely, if we must give up the description of the nucleon as an excited mass state of the lepton, then the duplicity in nature of the particle with both ordinary and isotopic spin 89 becomes rather improbable. It is therefore worth while to consider the problem of the unified description of leptons also from the point of view of the Majorana theory of the-neutrino. In this case the family of the leptons contains only three members, the neutrino v, t h e electron e- and the positron e +. We will show in chapter I I I b that these three particles may be described as mere three different charge states of a single kind of particle, the lepton, which (like the nucleon) has the ordinary spin 89 b u t (in contrast to the nucleon) the isotopic spin 1. The components of the isotopic spin of the lepton can be represented by an irreducible set of three Hermitian matrices ~i (of rank 3) satisfying the Duffin-Kemmer relations. The three charge states of the lepton e T, v are to be defined as eigenstates of the operator ~a which simultaneously plays the role of the charge operator of the lepton. An appropriate mass operator commutative with the charge operator can be chosen to give equal non-vanishing masses to the electron and positron but mass zero to the neutrino. The interaction of the lepton field with the electromagnetic field can be introduced on quite similar lines as for the nucleon field. Furthermore, a charge symmetrical direct interaction between the lepton field and the nucleon field can be introduced which leads to the #-radioactivity (with vector and tensor coupling) and also to the non-electromagnetic emission of electron-positron pairs (but not of neutrino pairs) from nuclei. The Whole theory is independent of any special

ttexoca. ~8. ~yp~. z (1958)

choice of the representation of the matrices z~ and ~/as well as of the Dirac matrices r~. There is still another family of three particles, viz., the mesons :r~ and n~ which are generally assumed to be related elementary particles. The relationship of these particles was anticipated in the well known Kemmer's charge symmetrical meson theory of nuclear forces. We will show in chapter IIIc that these three particles can be considered as mere different charge states of a single kind of particle, the pion, which can be characterized briefly as a particle with ordinary spin 0 and isotopic spin 1. The description of this pion can be given on quite similar lines as for the lepton using the Duffin-Kemmer fl-formalism instead of the Dirac y-formalism. One-column matrices, quite similar to the well known HarrishChandra's ~r-matrices, can be introduced in the charge space and prove useful for the formulation of the laws of interaction between the pion field and the nucleon field. In chapter II a review is given of the properties of the most important quantities defined in the three-dimensional charge space. It is well known that there also exists an irreducible representation of rank 4 of the three abstract operators ~i satisfying the Duffin-Kemmer relations. This representation can be used to describe a family of four particles (two oppositely charged ones and two different neutral ones). All these four particles would be inner states of a particle the isotopic spin of which is able to have two values 1 and 0. Thus we would have a particle with an inner structure beyond the isotopic spin. A more detailed physical discussion of this case will be given in a future work by one of the authors

(M. L.). II. Pauli, Duffin-Kemmer and Harrish-Chandra Operators in Charge Space. 1. The operators ~ i ( i : 1, 2, 3) satisfying the relations T~. § ~T~ = 26~j, (1) w h e r e i , ] = 1,2,3;~i~ : l ( i = ] ) , 6 ~ j = 0 ( i # ]), can be represented by Hermitian matrices. We are only interested in irreducible representations, i t is well known that all such representations are of degree 2, and that only two of them are inequivalent. One special irreducible representation is provided by the Pauli matrices

Vdclav Votruba, Milo~ LolcajiSelc : On the Isotopic S p i n o/Elementary Particles

Czechosl. Journ. Phys. (1953)

~

, ~2~

0'

~3~

__

.

(2)

r -~

(2)

Another such representation ~ is obtained b y (2)

(1)

(1)

(1)

m a t r i x u satisfying the three conditions ~ = uv~u +. Thus an arbitrary irreducible representation of the (1)

abstract operators ~i is equivalent either to v~ or to (~)

Ti.

Now let v be a (2 • 1)-matrix with elements v~ (a = 1, 2). If the u n i t a r y t r a n s f o r m a t i o n v--~ v ~ = Uv is performed, t h e n the three expressions F i ~- v+v~v (i = 1, 2, 3) undergo the following real orthogonal transformation

where X~ = ~SpU+viU~:~, because Sp~d:~ = 2~i~ by virtue of (1). Thus F~ are components of a vector in charge space. Similarly I ~ v+v is a scalar. On the other h a n d a simultaneous u n i t a r y substitu~ion v --~ ~ = u v a n d Ti --)- T UTiU + leaves the expressions F~ as well as I unchanged. (~) Using the representation ~ one verifies t h a t ~-

9~ T ~ - ~,v~ = 2T~2 = 2i~3, (cyel. l, 2, 3),

(3)

which relations obviously remain valid for all (1)

representations equivalent to ~. 2. F o r the operators ~i (i = 1, 2, 3) satisfying the relations ~/~

+ ~G~

= ~

+ ~,

(4)

there exist four m u t u a l l y inequivalent irreducible representations [5], The first one is trivial, of degree 1. There are further two representations of degree 3 a n d one of degree 4. Let ~r be an a r b i t r a r y irreducible representation of degree 3 o r 4 a n d let v be a (3 • 1)- or (4 • 1)m a t r i x with elements v~ (a = 1, 2, 3 or 1, 2, 3, 4~. Xf now a u n i t a r y t r a n s f o r m a t i o n v - + v ~ = Uv is performed, t h e n the expressions F i = v +~iv undergo the real orthogonal transformation F~ - > F~. =

v~

0 i

,

r ~

0 0

--

,

(2)

p u t t i n g ~ = - v~. B o t h representations v~ a n d ~ are inequivalent, because there exists no u n i t a r y (2)

3

~v ~ = X~F~

where X~- = 8 9 because, as we shall see later, S p ~ = 2 ~ . T h u s F i are components of a vector in charge space. Similarly I = v+v is a scalar. (i)

A special irreducible representation ~ of degree 3 is given b y the H e r m i t i a n matrices

~'~

=

o

.

(5)

0 (2)

(])

Putting ~ = --~

we o b t a i n another represents(i) tion of degree 3 which is inequivalent to r Thus each irreducible representation of degree 3 of the abstract operators satisfying (4) is equivalent (1)

(2)

either to r or to ~i. As the sign of the matrices ~i is immaterial for our purposes, we will confine ourselves in the following to the consideration of those representations of degree 3 which are equi(1) valent to $i. I n the representation (5) the special relations are satisfied where X is the (real, orthogonal) m a t r i x of the coefficients X ~ ~hemselves. Thus we can p u t U = X a n d c a n restrict the u n i t a r y transformations U of the triple of quantities v ~ ( a = 1, 2, 3) to the real o r t h o g o n a l t r a n s f o r m a t i o n s X of the triple of components (of the vector) F i (i = 1, 2, 3). I n other (equivalent) representations such a restriction is n o t possible. Xi~ - - - ~ - S p X + r

A special irreducible representation of degree 4 is given b y the H e r m i t i a n matrices

(o00)00 00)00 (0000) 00 00

,

00

$2 ~

'

--/0

~3 =--

0

o

"

(6)

0--/ I n this representation the relations Xii = ~SpU+~iU~ ~

m

oro oro first ~hree components Vl, v~, v3 of v form a vector, whereas v4 is a scalar in the charge space. I n other (equivalent) representations this is n o longer true. I n an a r b i t r a r y representation (equivalent either to (5) or to (6)) we can, of course, always form three linear combinations T [ v (i = 1, 2, 3) of the quantities v~, which are components of a vector

tIexoea, a)~a. m~p~. ~ (lg53)

V deiav V otruba, M ~log Loka]i&k : On the Isotopic Spin o/Elementary Particie~

4

in charge space. It is easy to verify b y means of (5) or of (6) that the (3 X l/- or (4 X 1)-matrices T~ satisfy the following sets of relations: In case (5): T~T + = 1, T+T~ = 5~, T + ~ T ~ = ie~r (7) $~$~T~ =- (~i~T~ - - ~ T ~ , $~T~ -= - - $r where e ~ = 4- 1 if i~k isan even (odd) permutation of the numbers 1, 2, 3 but otherwise ~r = 0. In case (6): T~T + = h, h$i § ~3 = $i, T + T j = ~ir

In those representations (of degree 3) which are equivalent to (5), we have especially hi = $~. In the irreducible representations of degree 4 is 2~ # $~. In the special representation (6) the h-matrices are (1)

just reduced in the form

, where $~ are given

by (5).

III.

The

Ordinary Spin and the Isotopic Spin

a) o / t h e nucleon, b) o / t h e lepton, c) o[ the pion.

a) Let us have two independent Dirac fields ~ (a = 1, 2) satisfying the equations [Y,(~-

(8)

ie ~-~ A t ) +

gel ~'[1= 0,

(12)

E~,2. + ~=] w~ = 0.

These relations (7) and (8) are the analogues of the well known relations for the Harrish-Chandra ~r-matrices [6] and remain valid in an arbitrary representation equivalent to (5) or (6). The transformation which leads to an equivalent set of matrices ~, Ti and $i is, indeed, given b y v-+ 9 = uv, T i ~ T~ = uT~, $~-+ "$~= u$~u +,

In these equations ~ ~ ~/~x~, where x~ (# =- 1, 2, 3, 4) are the space-time coordinates @1 = x, x2----y, x3 = z, xa = let). The four-vector A~ is the four-potential of the electromagnetic field, A~ (A~, A~, A~, iV) and e is the elementary electric charge. The sign h means the Planek constant, divided b y 2~. The coefficients y~ are Hermitian (4 • 4)-matrices satisfying the relation

(9)

where u is any unitary (3 X 3)- or (4 • 4)-matrix. From (9) we see that T~) = T ~ v and that the relations (7) and (8) are valid also for T~ and ~'~. Further, it is easy to prove that T~+$~v~ = T~$iv, (v~ Uv). Thus the expression T,+.$~v is a scalar in charge space. This quantity is zero in any representation of degree 3. In the special representation (6) it equals 3iv~(i =- 1,/-~l). From (7) we have for any representation of rank 3 (equivalent to (5))

Similarly from (8) we have for any representation of rank 4 (equivalent to (6))

Finally the wave functions T~ and T2 are (4 • 1)matrices with elements TI~ and T ~ (a = 1, 2, 3, 4), namely the Dirac four-component spinors. cm~ cm~ If we p u t ~ t h and ~2 -- -~ ", then the firs~ equation (12) describes the wave field of the protons (in a given electromagnetic field) and the second equation (12)describes the wave field of the neutrons (which do not interact with the electromagnetic field directly). It is well known that both equations (12) can be written in a compact form 9,

~-~ . 89 + ~a)A, + ~ T = 0, (12a)

where

Thus we always have

= If we form the matrices h~ ~-- -- i($~$a-- $a$~), (cycl. 1, 2, 3),

(10)

these h-matrices satisfy the relations 2 ~ h ~ - h~2~ = iXa, (cycl. 1, 2, 3) which are valid in all representations.

(11)

( 1 + Ta) Jr -~ ( 1

-

-

ra)

and va is the third of the Pauli matrices (2). (The index (~) over Ta has been dropped because, as we shalI show in the following, the choice of the special Pauli representation is immaterial). In (12a) the wave function T has eight components T,~ (a = 1, 2, a = 1, 2, 3, 4). The matrices

Czechosl.;lourn.Phys. (1953)

Vdclav Votruba, Milo$ Lokaii~ek : On the Isotopic Spin o] Elementary Particles

?g operate on the index ~ only and the matrix ~a on the index a only. Thus we have e. g. (?gT)a~ = = (?Z)~pTap, (TaT)a~ = (~a)abTb~. Also in the following we always use operators which only operate on one of the two sorts of indices of the components of the wave function. The equation (12a), as well as the equation s) +

89

+

--

o, (12b)

for the adjoint wave function ~ = ~P+?4 can be deduced 'from the variation principle with the Lagrangian

5

special choice of the representation of the matrices Ti .3) In an arbitrary representation, the wave functions corresponding to the proton or neutron state of the nucleon are to be defined as eigenfunctions belonging to the eigenvalues • 1 of the operator ~a (or to the eigenvalues e, 0 of the operator 89 . 9 (1 +.%)). Developing the field k~ in a series of these eigenfunctions we easily find t h a t the total energy, momentum and charge of the free nucleon field are the sums of energies, momenta and charges of the protons and neutrons. (see Appendix A.) b) We start with three independent Dirae fields ~a Ca = 1, 2, 3) which satisfy the equations

(13)

ie

The Hamiltonian H (energy density), the momentum density G and the current four-vector s z =-- (], ice) of the nucleon field are given by the well known canonical formulas

a) ] ~

ie

= O,

1~2 = 0,

[ r , ~ + oJ.] ~3 = 0. c =

OL

(14)

Let ~ be the Schwinger charge conjugate of ~a, defined by the equation

=

~'~ = r

(20)

where C is a matrix satisfying the relations 8/~=C

_~ iec~?~, 89 A- ~3)~.

~L

(16)

In the quantum theory of the nucleon field the following commutation relations are satisfied at any given time t: -'~

- - -->t

--~

)}=

"~t

?~ = - - C-X?l,C, C T =

'

Equations (12a) also follow from the Hamiltonian equations of motion i

T---> ~-~ ~- uYT, v i - + ~ -~ uviu+,

(18)

'

'

(22)

In order to write the equations (19), (20) and (22) in a compact form, we introduce first of all the matrices ~1 :

The equations (12a, b) and (17) as well as the expressions (14)--(16) are invariant with respect to the substitution

(21)

Should the equations (19) describe the wave fields of the electrons, the positrons and of the Majorana neutrinos the following conditions must be satisfied:

)}= 0.

The curled brackets denote anticommutators.

- - C, C + = C -1.

2-}

0

r

--

~-

,

~2 :

1

2-t

0

,

(23)

0

where u is an arbitrary, constant, unitary (2 X 2)matrix. Thus the theory is idependent of any

which constitute an irreducible representation, equivalent to (5), of the three operators ~ satisfying (4). Further we introduce the matrix

T =) In (12b) ?/,, vTa and are transposed matrices y~, *a and ~ respectively.The matrices ~, gt+, ~are considered either as column-matrices or as row-matrices. We can therefore write ?gg~ = gry~, etc.

a) The matrices 31 and ~ do not occur in the equations and expressions mentioned so far. They wi!l appear, however, in the ttamiltonian describing the direct interaction between the nucleon field and the lepton field or pion field.

~Iexoc=. q~g8.H{~-prt. ~ (1953)

Vdclav Votruba, Milog Loka]i~ek: On the Isotopic Spin o] Elementary Particles

6

D =

(01 ) 1'0 00

(24)

which fulfils the relations ~T = __ D ~ D _ ~ , D T = D, D + = D -~,

(25)

quite similar to (21). Finally, we define the mass operator

=-

+

0(1 - -

We can then write equations (19) in the form

ie~~A~) -t-. ~gly; = 0

_

[7~(~

(19a)

which is the analogue of (12a). Equations (20) and (22) can be collected in the equation 9' ~

C~=

By2.

(22a)

The commutation relations of the components of the quantized lepton field ~ can be written in either of the following three mutually equivalent forms: {~)aa(X), (~)(X )~4)bfl} (~ab(~afl(~(z --X ), .-+! {yJa~(x), ~v~(x )} = - - D h ( 7 t C ) ~ r

where u is an arbitrary, constant, unitary (3 • 3)matrix. The whole description of the lepton is therefore independent of any special choice of the representation of the matrices ~i- If we choose u=

1( 1, 0)

~-~

---4 i

0 ,

001/ we pass over by means of (31) from the represen(1) ration (23) to the representation (5), i. e. ~i = $i. We find at the same time that /) = 1, so that from (22a) it follows that ~ ' = ~. Thus each of the three Dirae fields ~ is self charge conjugate in Sehwinger's sense.5) The Sehwinger charg e conjugate lepton field ~v' has not the physical significance of the true charge conjugate lepton field in all representations. In an arbitrary representation of the matrices ~, the true charge conjugate field yJ~, for which ss(~v~) = --sl~(~), is given b y ~pw: v~tp, (22b)

=

(26)

{~aJS), ~ ( 5 ' ) } = + Da~(C+r~)~r The equation (19a) can be deduced from the generalized Majorana variation principle [7] with the Lagrangian

where vq is a I-Iermitian and unitary (3 x 3)-matrix satisfying the relations v~+~iO :

( - - 1)~$~, (i :

1, 2, 3).

(25a)

(31)

The matrix v~ transforms under (31) in the same manner as the matrices ~i (and thus unlike the matrix D). Only in representation (23) is = D and ~ = ~o'. It is easily found that the general expression for the matrix v~ is given b y v~ : 1 - - ~ - } - ~ - - ~ ' a2 : 2 ~ - - 1 . In an arbitrary representation of the ~'s, the wave functions corresponding to the electron, positron and neutrino state of the lepton are to be defined as eigenfunetions belonging to the eigenvalues :F 1,0 of the operator ~a (or to the eigenvalues :F e, 0 of the operator eta). Developing the field ~ in a series of these eigenfunctions we easily find that the total energy of the free lepton field equals the sum of the positive energies of the electrons, positrons and neutrinos. The total momentum and charge are the sums (vectorial and algebraic respectively) of the momenta and charges of these particles. (See Appendix B.) c) Experimental evidence shows that all the three mesons z• and u~ are pseudoscalar. The existence of vector mesons (other then u-mesons)

a) The matrices ~z and ~ do not occur in the equations and expressions mentioned so far. They will appear, however, in the Hamiltonian describing the direct interaction between the lepton field and the nucleon field.

a) In this case the three D irac functions v~~,~ ~,y~a are also identical with the expressions ~ , 5~, ~. This means that the three wave functions ~a (a = 1, 2, 3) form a vector in charge space.

[(

The quantities H, G a n d st belonging to the lepton field ~ are given by ~) he

-C

H = -~ ~ r . ~ + ~9)~ - -

~

~A.,

(2s)

h s~ =

ie~ ~ , ~ a ? .

(30)

Equation (19a) can now be also deduced from the Hamiltonian equations of motion ~ = b " i [fH(dS),vA~]" Equations (19a), (22a), (25) and (26) as well as the expressions (27)--(30) remain unchanged if we perform the substitutions

~=u~,

~'-+~'=u+~

',

~ --> (~ = u ~ u +, D ---> D = u+ZDu +,

Czechosl.Journ.Phys.

Vdclav Votruba, Milo~ Lo]ca]t~ek: On the Isotopic Spin o] Elementary Particles

0953)

is not excluded. We will use, therefore, the DuffinKemmer B-formalism which can be applied to t h e description of mesons with spin 0 as well as of mesons with spin 1. In the case of s-mesons the four Hermitian matrices $~ constitute an irreducible representation of degree 5 of the abstract operators satisfying the well known Duffin-Kemmer [6,8] relations Considering the three particles s ~, S ~ as three charge states of the pion, a pseudoscalar particle with isotopic spin 1, we can write the fundamental equations of the pion field, in analogy to t h e corresponding equations of the hapton field, as follows [fl~'(~' - - i eq~ 3JA ~h) + Kc ]

:

O,

(32)

K = ,~'a2 ~- go(1 --~'g), ~' ~ B ~ ~-- D ~ .

(33)

In these equations the pion wave function ~ has, of course, fifteen components ~ ( a - 1, 2, 3; ~ = = 1, 2 . . . . ,5). The matrices flz and B operate only on the index a, the matrices ~ and D again only on the index a. For the matrices $~ and D representations (23) and (24) or an arbitrary equivalent representation can be chosen. The matrix B has now the following properties [9]:

fl~ = ~ B-lfl~B, B T = B, B+ ~__ B-1. The function ~ in (33) is given b y ~ = ~§ v4 =

2~

--

1

(34) where

[9].

The commutation relations of the components of the wave function ~ can be written in either of the following three mutually equivalent forms:

[(fl~(~))o~, (~(x')~D~] [(~42~(X))aa,

(~2(Xt)ofl)] -~

~o~(/;~)~(

--

),

+ --+ .-->, ), (35) = Dad?4B)~;,~(x--x

The Lagrangian and the expressions for the quantities H, G and s t are now quite analogous to the expressions (27)--(30). (It is sufficient to write ~, K and/3 instead of F, ~ and ~ respectively.) Theequations (32)--(35) and the expressions for H, G and s~ are again invariant with respect to the substitution of the form (31) (with ~ instead of ~p). The wave functions corresponding to the three charge states s ~, ~~ of the pion are defined quite similarly as for the charge states e ~:, ~ of the lepton. (See also Appendix C.) The theory of this pion with isotopic spin 1

7

can further be joined with the theory of the pure neutral meson field (s' mesons with isotopic spin 0). 6) IV. Hamfltonians for N u c l e o n - P i o n and Nucleon-Lepton Interactions.

A collection of various schemes of direct couplings between the known kinds of elementary particles has been proposed so far [10] with the aim of explaining all the observed transmutation processes of elementary particles and the nuclear forces. None of the proposed coupling schemes is in quantitative agreement with experimental data. The main troubles come from the processes of higher order. Moreover, all these coupling schemes are necessarily incomplete, because we do n o t yet know all the kinds of elementary particles and the relations between them. In the following we will confine ourselves to t h e consideration of the direct couplings nucleonpion and nucleon-lepton. These interactions are defined by means of additive perturbation terms to the ttamiltonian of free particles. 1. The coupling nucleon-pion leads in the first approximation to the emission and absorption of pions b y nucleons9 The most general interaction Hamiltonian (with pseudovector coupling), compatible with conservation of the electric charge, can be written in the form

H ' = i~,5~,,7,,(c~'+(1, + il~r + (I3 + I,~DT~}. 7~q~. In this expression the /'s are real coupling parameters and --~25 = Yl~2Y3Y4" Further 5rs = ffs+~]4 (# = 1, 2, 3), 5r4 = - - 74+U4 and ~'/l, (# --~ 1, 2, 3, 4) are the Harrish-Chandra matrices belonging to our system of fl-matrices. The Hamiltonian H' is a Hermitian operator and the volume integral of H ' is commutative with the sum of total charges of the nucleon and the meson field. To prove the first property, one has to use the Harrish-Chandra relation 7~ = 7zB and its analogue T[D -= T~ together with (25) and (33)9 To prove the second property of H', the following formulas (besides the relations (17), (35), (33), (25) and (34)) are to be applied:

= (1

-

-

K-~)fl,~

-

i_ hc Trsr,K-1[(11 -- ~/,,$a)vT +

+ (1~+ A~)T~] (1 - - ~ ) 7 ~ 6) See the dosing paragraph of the inVroduction.

(37)

8

Vdclav Votruba, Milog Lol~a]i~elc: On the Isotopic Spin o] Elementary Particles

and The latter relation follows from the third and first row of (7). The relation (37) is the well known Kemmer formula expressing the dependent components of ~ in terms of the independent ones. The term with /1 in (36) corresponds to Kemmer's charge symmetrical coupling. The term with/~ couples only charged mesons with nucleons whereas through the terms with /3 and t4 only the mesons u ~ are coupled with nucleons. The term with [a describes the "pure neutral coupling" (with equal signs of the P~~ and N~ ~ couplings) and the term with [4 is the "Kemmer neutral coupling" (with opposite signs of the Pz~ and N~ ~ couplings). i t is possible to deduce from the Hamiltonian (36) a scattering matrix for the scattering of pions on nucleons. The isotopic factor of this scattering matrix is almost identical in its form with the isotopic scattering matrix recently proposed by NAMBII and YAMA(~UCHI [11].7) 2. The direct interaction between nucleons and leptons is usually supposed to be responsible for the fl-radioactivity of the atomic nuclei and, in particular, of the neutron (N-~ P + e- + r). Recent experiments [12] are likely to support the existence of isomere states of nuclei from which the energy cannot be liberated by means of an electromagnetic process and is, instead, liberated through the non-electromagnetic emission of electron-positron pairs (P* -+ P § e- § e + or N* -~ -+ N + e- + e+). I t seems desirable, therefore, to complete the ordinary Fermi term in the interaction Hamiltonian by further terms, which would allow the required processes to occur. The most general interaction Hamiltonian compatible with the conservation of the electric charge should be given by the following expression _

_

<.)

__> --->

(.)(.)

<,)

(.) 9

_~

(,)~,

9

~

g(5.)

+

~

+

,

~

-~

+

_~

(3s) 7)The Hamiltonian method permits the ~[ambu-Yamaguchi parameters A, ..., L to be expressed in terms of the four ]'s occurring in (36). These questions will be discussed in a paper by one of the authors (M. L.) on a more general basis of the unified description of the fourmember family of mesons ~z• ~o, ~,.

tIexoc~,o~a. ~pH. ~ (1953)

Here ~ denotes the sum over the five covariant (.) Dirac operators A ( ' ) = 1, 7/, as~, 757~, 75. The operators AI~) operate on the lepton wave function 9, whereas AI~) operate on the nucleon wave function 9. The square brackets denote vector products 9 The g's are real coupling parameters. Expression (38) can be considerably restricted. Using relations (26) and (22a), we can write yJ(x)~Aw(x ) = ~p(x)(D~D-1)~(C-IAC) ~f(x) +

--'-~

-->

---->

T

"~

+ 6(o)Sp(r4A)Sp~.

Now, because we have

Sp~t =-0

and

D~iD -1 = - - ~ ,

-~A~, = -- ~(C-1AC)'~.

On

calculating (C-1AC)'=

(C-1AC) ~" one

finds

§ A

for A = 1,7~, 7a7,,

(39a)

--A

for A = 7 ~ , % ,

(39b)

Thus we see t h a t the terms with g['), ..., g~') in (38) can be effective only in combination with vector or tensor coupling. Similarly, it is easily found, t h a t the terms with 9~), ..., g(10 ) can contribute only when combined with either scalar or pseudoscalar or pseudovector coupling. After these restrictions it is not difficult to prove that H" is a Hermitian operator and t h a t its volume integral is commutative with the sum of the charges of nucleons and leptons. If the condition of conservation of the total isotopic spin is imposed on the interaction, theft only the terms with g~) (with V and T coupling) and the terms with g~') (with S, P S and P V coupling) remain in (38). There is, however, another restriction which is likely to be imposed on H". The terms with g~') and g~') would lead not only to the emission of electron-positron pairs but also to the emission of neutrino-pairs from nuclei. Now, if the nonelectromagnetic emission of electron-positron pairs should occur at all, then the terms with g~') and g(e") must be absent from H", because otherwise the emission of neutrino-pairs would compete successfully with the emission of electron-positron pairs. Thus after all these restrictions only the terms with g~') which proved most convenient for the interpretation of the fl-radioactivity data would be left in H".

Vdclav Votruba, M i l o ~ Lel~a]ige]c: On the Isotopic S p i n o] E l e m e n t a r y Particles

Czechosl. a o u r n . Phys.

(1953)

Finally let r (A ~ 1, 2, 3, 4) be the four eigenfunctions satisfying the equations

V. Conclusion.

The formal analogy between the ordinary and the isotopic spin of the nucleon has been considered until quite recently as a rather accidental circumstance. The components of the isotopic spin of the nucleon have been represented exclu"sively b y the special Pauli matrices, with va diagonal. The existence in nature of elementary particles (plea, lepton) which can be considered and described as particles with higher value of the isotopic spin and the fact that this description can be made independent of any special choice of the representation of the isotopic spin matrices (as well as of the ordinary spin matrices), all this in our opinion supports the view that the analogy between the ordinary and isotopic spin is not accidental. The abstract three-dimensional "charge space" (isotopic spin space) is, of course, physically anisotropic. The electric charge and the mass of the particle depend on the orientation of its isotopic spin to the third axis. In the ease of the particle with isotopic spin 89 or 1, the operators of the charge and mass can be expressed in terms of the third component of the isotopic spin. Some recent experiments on the scattering of pions on nucleons are likely to give further support to the conservation of the total isotopic spin in these processes [13]. Thus, one can expect that the ide~ of the isotopic spin and a more detailed study of the properties of the wave functions and of the Hamiltonian in the "charge space" will allow the elucidation of further, up to now, puzzling relations between various elementary particles. Appendices.

Each of the three operators ~i has only two eigenvalues =k 1. Let v~ (J = 1, 2) be the two eigenfunct;ons satisfying the equations q-v~ ( J ~

(~ + ~3)v~ = s,v~,

(A 3)

e1 =

(1 + h) + - ' ~ ( 1 - - ~ 3 ) vj=~jV.,,

e, 8 2 =

0.

(A4)

(A 6)

where E"4j(p) = = ~ c V ~ q - h 2 u ~ ( q f o r A : 1,2; - - for A : 3,4). The wave function g*(~'), periodic in the cube 153, can be developed in the series

~(5) = L

~ c~(~) r

J(~'~).

(A 7)

~, "4, J

The energy W ~ fH~re~ (d~), the m o m e n t u m P = .+

i

-~

: fG (dx), and the charge Q = - - c fs4 (dx) are then given b y

W=

2

E"4Ap)N~Ap),

p~ .4, J

P 9 "4AP),

(A 8)

p~ "4, J

Q = ~

ejN"4j(p) = e A, N"41(p ).

Here N"4.r(p ) : C ~ ( p )C Aj(P ) are operators with eigenvalues 1, 0, because the c's fulfil the relations

{c, c'} = {c*, e'*} = 0. as follows from (17). The terms with A : 3, 4 in (A 8) must be interpreted, of course, according to the Dirac "theory of holes". B. L e p t o n s . Each of the three operators ~i has three eigenvalues -[= 1,0. Now let vI (J ~ 1, 2, 3) fulfil the equations

(A lo) (A 11)

e~ev J -~ %v j,

wheree~ = - - e , e ~ = e , e a : 0 . Then we also have YJvj = wjvj, 0)1 ~

0)2 ~

0 ) , 0)3 ~

(A 12)

0)0.

Further let ~ ] ( ~ ) fulfil again the equations CA5) and (A 6) with 0)j instead of ~]. From (A 10) and (A 5), by using (25) and (21) respectively, we obtain the relations vj : Dvj, Cezj(--p)

(A 1)

"4j(p), (A 5)

~J(P) = )'4. E"4j(p)

~"4~(~)r,~,~(~) = ~"4"4,,

= r

where J : 2 , 1 , 3 for J ~ 1, 2,3 a n d ] A -- 1 , 2 , 3 , 4 . Since 0)1 ~ 0)2, we also have Es

e

where

( ~ . p ) -t- uJ

1),

( J = 2),

Then we also have

~vj =--

hc.

where

A. N u c l e o n s .

~3vr = - - v j

9

= - - E~j(p), C~)i] ( - ~) :

(A 13) :

3, 4 , 1 , 2 for

~"4,(~). (A 13')

The lepton wave function ~o(~) can again be doveloped in the form (A 7). Using CA 13, 13') we deduce from (22a) that the coefficients c"4~(~) satisfy the relations

Vdclav Votruba, Milo~ Loka]i~ek: On the Isotopic Spin o] Elementary Particles

10

The c o m m u t a t i o n relations in the first row of (A 9) r e m a i n valid also for the lepton coefficients c~](~). The relations in the Second row of (A 9) m u s t be replaced, however, b y those relations which result f r o m t h e relations in t h e first row b y m e a n s of (A 14). I t is now easy to prove t h a t the t o t a l energy, m o m e n t u m , and charge of the l e p t o n field become (after dropping the v a c u u m values)

qexoc~. (D~s. }lfyp~. ~ (1953)

Using (34) we can deduce f r o m (A 17) t h a t Bq~zr ( - ~) = q5 (~)

where -4 ~ 2,1 for A = 1,2. Since k~ = /%, we also h a v e

B q ~ ] ( - - ~) = q~j({), E2](p) = -- E~r

~(p),

(A 15)

e~(~~).

(A 22)

p A~1,2

a n d from t h e c o m m u t a t i o n relations in the first row of (35) the following c o m m u t a t i o n relations

C. P i o n s . L e t t h e matrices v~ ( J = 1, 2, 3) satisfy again the equations (A 10, 11) and the first one of t h e equations (A 13). T h e n also the e q u a t i o n s

Kvr ~ kjvj,

(A 16)

where k~ = k~ = k, k a = k 0, are satisfied.

hc [ h ( f " ~) A- kj] cI).,~.i(p) = fl,E~j(p) q~aJ(P),

be normalized b y means of t h e relations

-~

E,~j(p) ~ ,

(A 23)

The t o t a l energy, m o m e n t u m , and charge of the free pion field become (after dropping the v a c u u m values)

=

--

/--~ p 9

1J(p),

(A

24)

(A17)

where E~,(p) = • cV~-4- h2k~ (d- for A = 1, - - f o r A = 2). As is well known, the two matrices ~ can q~

[b~j(p), ~,~,(p H = ~ , ~ , ~ .

w =

F u r t h e r let ~ar (A = 1, 2) be two (5 X 1)-matrices satisfying t h e equations

~.

(A 21)

F r o m (33) we can now deduce the relations

b]~(-- ~) = b~(~), pJ All,2

(A 20)

The w a v e function ~(~), periodic in the cube L a, can be developed in the series

~(~) = L - } ~ b~(~) ~ ( ~ ) ~ p

(A 19)

(A 18)

The eigenvalues of t h e operators Nxj(p ) = b~j(p )b~j(p ) are the n u m b e r s 0, 1, 2 . . . . R e c e i v e d 16.2.52, F i r s t Published in Czech 27. 11.52

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