On the second order wave equation of fermions II

Nuclear Physics 10 (1959) 468

A.74;~North-Holland PublishingCo., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

O N T H E S E C O N D O R D E R W A V E E Q U A T I O N OF F E R M I O N S II G. M A R X Institute for Theoretical Physics, Roland E6tv6s University and

Central Research Institute for Physics, Budapest Received 2 J a n u a r y 1959 A new space reflection P ' a n d charge conjugation C' can be readily introduced from t h e second order wave equation for fermion doublets. Strong as well as weak interactions are i n v a r i a n t with respect to P ' a n d C' while electromagnetic/nteractions show only P ' C ' invariance. T h e wave operator for t h e nucleon (and for o t h e r fermion doublets) is t h e irreducible representation of the Lorentz group extended b y P" a n d C'. Results obtained b y this t h e o r y a n d b y t h a t of Gell-Mann disagree only as to decay oi strange particles.

Abstract:

1. I n t r o d u c t i o n

The theory Of universal Fermi interactions has increasing evidence in its favour. The basic idea is that vectors have to be constructed for various fermion pairs: G,(A, B) = iV~(~AT,½(l+75)~vB) ~: G,+(A, B)

(1)

(%//is real) and the universal Hamiltonian for weak interactions is given by the scalar product of such vectors: H a = G,(A,

B)G,+(C, D)+G,+(A, B)G,(C, D).

(2)

It is of interest to note, however, that such vectors cannot be built from any fermion pair and only certain pairings are allowed. GeU-Mann, for instance, proposes the following three classes 1): I. Lepton pairs: (e,v), (/zv). II. Strangeness-conserving baryon pairs: (np), (~°2~+), (2~-Z°). (S-S°). (3) III. Strangeness-violating baryon pairs: (Ap), (2:°p), (Z-n), (S-A), Class II is selected according to the unique rule that by the omission of the axial vector part in eq. (1) the current of the isobaric spin component J _ is obtained *). These pairs are products of ~r+-dissociation. The only difference is that the latter selection principle would lead, in addition, to the A°2~+ doublet. Products of K+-meson dissociation are found in class III. Class I contains lepton pairs selected by the rule that in interaction processes an equal number of charged and neutral leptons should take part 3). 468

ON THE

SECOND

ORDER

WAVE

EQUATION

OF FERMIONS

II

469

A closed theory of universal Fermi interaction would require that fermion pairs occurring in eq. (1) should be defined, instead of the specification given in the enumeration (3), by a rule "expressible in a single comprehensive sentence". In a previous paper *) we made the following attempt: it was assumed that the known fermions could be assigned to doublets according to a natural and unambiguous rule (e.g. (np), (ev)) and a fermion doublet could be described by a single field operator X. This could be constructed in an unequivocal way from the Dirac wave operators of the doublet and would satisfy a four component second order wave equation of the Feynman type 5). 2. The Theory of Ogievetsky and Chou If we attempt to describe a fermion doublet, e.g. the proton and neutron, by a single multicomponent X, it will not give an irreducible representation of the Lorentz group. The nucleon will be transformed irreducibly only in the combined Lorentz + isobaric group; a reference to the isobaric group is, however, not permitted for weak interaction since, contrary to strong interactions, they are not charge independent. Recently Ogievetsky and Chou proposed e) to consider the nucleons, from the point of view of strong interactions, as the irreducible representation of the extended Lorentz group. Their hypothesis m a y be reformulated as follows: The usual operators for space reflection, (strong) time reflection and charge conjugation are given by P~v(r, t ) P -1 : 74~v(--r, t),

(4)

Xv(r, t ) X -1 = i~5~4~v(r, --t),

(5)

C~v(r, t)C -1 = 7 # ( r , t) = ~0¢(r, t),

(6)

respectively. (Symbols given in ref. 4) are used.) Phase factors are chosen so that p2=x 2=C2=1. (7) According to eqs. (4), (5), (6) the following relationships m a y be obtained: PX = (--1)NXP,

(8)

PC =

(9)

xc

=

0o)

(N is the operator of the fermion number.) Geometrically the space reflection and the time reflection commute and therefore a new interpretation of time reflection is required so that the new time reflection operator Y should satisfy the following relations: Y~---- 1,

(11)

G. MARX

470

PY

(12}

= YP,

while the validity of (13)

YC = (--1)NCY

remains unchanged. It is easy to find such an operator Y; it has the concrete form Y = X C e -~'¢j, = T e -~''~*,

(14)

where T = X C is Wigner's weak time reflection, •, is the 2nd component of the isobaric spin operator. Considering that v)

[P, .G] = o,

[c, J , ] = 0,

{x, J , } = 0,

(15)

the validity of eqs. (11), (12) and (13) can be readily proved. Since the charge symmetry operator e -~"~', leads to the interchange of p and n, we obtain b y extending the group of proper Lorentz transformations L with P, Y, C an extended Lorentz group which can be irreducibly represented only b y the eight component nucleon wave operator b u t not b y the proton wave function in itself. The electromagnetic interaction, on the other hand, is not invariant with respect to Y. Nor do weak interactions show isobaric invariance, but some invariance is found under charge symmetry transformation s). Now it seems very likely that a similar procedure m a y be used for the required interpretation of fermion doublets in weak interaction. Of course the Ogievetsky-Chou theory cannot be applied because of the important role played in it b y P and C, for weak interactions are not invariant with respect to P and C.

3. P' and G' Operators In this section our considerations will be restricted to the interactions of the nucleon doublet with 7 and with ev: the study of strong interactions does not lead to new aspects, and the weak interactions of other elementary particles will be discussed in this respect in section 4. The IMrac equation for free nucleons is written (7,a,+K)~p = 0,

(7,~,+K)~. = 0.

(16)

The Haxnfltonian for electromagnetic interaction is given b y H r = ~ i e c ~ p T w p A ~ +charge conjugate.

(17)

From eqs. (1) and (2) the Hamiltonian for fl-interaction is obtained as

Ha = --/[(~nT ,½(I +76)Wp) (~vyil (1+76)~Ve)

+ ( pr,

(is)

It can be seen that all these equations are invariant under the proper Lorentz group L. (The transformation formulas for A~ are always the usual

ON T H E SECOND O R D E R WAVE E ~ U A T t O N OF F E R M I O N S II

47][

expressions.) Formulae (16) and (17) are invariant with respect to t h e transformations P, C given in eqs. (4) and (5), the expression (18), however, shows only PC = K invariance. Usually, this phenomenon is interpreted as the reduced degree of s y m m e t r y of week interactions: they are not invariant under space reflection and charge conjugation. (There is no question of time reflection: the TCP theorem holds now, and from L and K invariance the T invariance follows throughout.) Let us rewrite the above equations according to ref. 4). By introducing the new operator = ~(l+rs)~pp-t-½(1--ys)~n e , (19> one gets from eqs. (16), (17), (18) the following relations: z = o, H 7 :

i e c 2 7 ~ 4(1-f-75)z+charge

(20)

conjugate,

(21)

is the spinor constructed from ~ and ~e similarly to eq. (19). Equations (20), (21), (22) are completely equivalent to (16), (17) and (18) because ~p and ~n can be expressed from X as V'p = [l'4-K-1757,O,]½(l+rs)Z = ½(I+rs)X+~(1--Ys)$ e, =

i[l

(23>

°

now denotes a subsidiary quantity constructed from the derivatives of Z: Since the equations were simply rewritten, the formulae (20) and (21) continue to be invariant with respect to L, P, C and (22) is invariant with respect to L, K. The transformations P and C have, however, a very complex effect on the wave function Z: Pz(r,

t ) P -~

-= r,$C(--r, t) = [tc-lr5ri~/r4Z]r..~_r,

Cx(r, t)C-S =

=

O,x¢(r, t),

(25).

(26)

(there is no change in L). However, all transformations are such that through them the positive and the negative chirality part of g each is transformed into itself, and therefore, as stated previously, the proton and neutron fields deduced from those according to (23) remain disconnected; Z is thus a reducible representation of group (E) =-- (L, P, T). With L, P we m a y write T instead of K according to the TCP theorem. Now, let us forget for a moment the Dirac equation and consider eqs. (20), (21), (22) in themselves, expressed with g- Obviously they are invariant with respect to L. Neglecting the electromagnetic interaction, the invariance of field equations, of weak (and strong) interaction Hamiltonians with

472

G. MARX

respect to space reflection becomes clearly apparent if the space reflection is interpreted in the most obvious way: P ' z ( r , t)P'-* -- 74z(--r, t).

(27)

The same are also invariant with respect to charge conjugation according to the simple formulation C';~(r, t)C '-1 : ~,e~(r, t) -- ;~e(r, t).

(28)

The electromagnetic interaction, however, shows only invariance with respect to P' C'. It is seen now that X is an irreducible representation of the extended Lorentz group (E') = (L, P', T). A similar irreducible representation of (E') is given b y the lepton wave function. Vector (1) is the only covariant quantity constructed from the irreducible Z field which 1) is not Hermitian, 2) does not contain derivatives, 3) does not violate the law of conservation of fermion number. It can be seen that this treatment leads naturally to a set of fermion doublets and to the law of universal Fermi interaction (2). Of course, other bilinear vectors m a y also be constructed from X, but these are Hermitian. Even scalars and tensors can be built without derivation in violating, however, the principle of fermion conservation, thus of baryon conservation, defined in equation (41) of ref. 4). If nucleons are separated into protons and neutrons according to their charges, we have to use spinors ,pp and Y~ninstead of the comprehensive X. Let us see what will be the effect of the new P', C' inversions on the usual Dirac wave operators. Departing from the inversion formula (23), we obtain ~on

\~,pe/_r

Vn

~P '

(30)

Now, we introduce the charge symmetry operator Z as reflection on the coordinate plane (1.3) of the isobaric space~): Z =

(__{)N e-i~.Js,

Z ("~P'~ Z -1 = --~ ("~On~, \~On/ \~0p/

(31)

which induces the interchange p ~- n. Let us extend itsinterpretation to the leptons by the replacement of p -> v, n -> e. Then, the n e w inversion can be

expressed with the usual operators P , C, Z in the form P' = PCZ,

C' = Z ,

P'C' = PC = K.

(32)

The weak interaction (18) is invariant, in the known way, with respect to K and with respect to the simultaneous interchange of p ~ n, v ~ e.

ON T H E SECOND O R D E R W A V E

E ~ U A T I O N OF F E R M I O N S I I

473

This explains the P' and C' invariance. The same is violated b y (17). From the above it can be stated that weak interactions show not fewer b u t other symmetry properties than electromagnetic interactions. Using electrical analysators, that is, if electromagnetic interactions come into prominence, it is suitable to use a description method based upon the Dirac equation; space reflection is best interpreted b y P which does not mix the states with different charges. Dealing, however, with weak interactions the combined second order wave equation is more convenient because it leads readily to the space reflection operator P' which does not mix chirality states. The (VJn, ~0p)doublet Z is an irreducible representation of the combined Lorentz group (E, E') = (L, P, P', T) fitting to both cases.

4. Weak Interactions of Strange Particles Some consideration is due to the weak interactions of strange particles. If we want to apply consistently the above considerations, it has to be assumed that all fermions appear in doublets and that, contrary to (3), a fermion is a member of only one doublet. H o w is this compatible with the singlet A and triplet 2:-hyperons? There are two possibilities: (a) ,In weak interactions (as in the strongest ~-interactions 9)) A and 27 are members of two doublets rather than of a singlet and of triplet. (b) The only elementary baryon is the nucleon. The hyperons are bound states of N and K 10,n). Case (b) m a y seem more natural. Group III, as written in (3), however, has to be left out in both cases; thus, the universal Fermi interaction is found to be useful only for the description of weak interactions occurring without change in strangeness. Consequently, the sum of vectors G, is defined as the current of the J _ component of the isobaric spin - - b y generalizing its concept also for lepton doublets - - with the modification that 7~ has to be replaced b y ~,i(1+75). In order to represent weak interactions involving change in strangeness AS = 4-1, it is the most convenient to assume a direct weak interaction between charged K mesons and the vectors G~: H~ = l •

(0, ~bK+G,(A, B)+0,~KG,+(A, B)).

(AB)

This represents, in first approximation, the transitions in question quite correctly 1). To decide which one of the two possibilities is the correct one, would require some further calculations and measurements like, for instance, the comparison of the [t values in the t-decays of n and A. According to Gell-Mann's theory a) the renormalized coupling constant of the (ev) interaction with the classe~ II and I I I of (3) m a y then be different, for group

474

G. MARX

III is no longer an isobaric spin component. Therefore, experimental tests become more difficult. The role of the "strangest" particle, the muon, is still problematical. In the two-lepton doublets usually the same ~ is taken to be involved and this is contradictory to our scheme. We suggest two possibilities: (a) For some reason the electron in the doublet (e)) m a y occur in two states: e- and p-, and this phenomenon, violating the P'-invariance, does not occur for neutral components. Thus interaction (e~)(p)) is one of the consequences of (e~)(e)) interaction. 03) We m a y assume 12), besides the ~ of --1 helicity, the existence of a particle ~7of ~ 1 helicity, also of rest mass 0 and spin ½, so that lepton doublets will be listed according to (e-)), (~p+). This means that instead of lepton conservation we m a y speak separately of the conservation of e ~ ) and the conservation of P ~ 7 - The theory leads, for all parity experiments and energy spectrum measurements, to the same results as those obtained b y the two-component )-theory. Finally, we have to point out that the P'-invariance here introduced is spoiled b y electromagnetic fields and electromagnetic mass splittings, whereas P is a constant of motion, the conservation of which is only violated b y interactions which are weaker b y several orders of magnitude. Therefore, the description of space reflection b y P is from a practical point of view undoubtedly more important. In principle, however, it is of interest to attempt to find such symmetries of strong interactions which m a y be violated b y electromagnetic interactions and hold for weaker interactions, because these symmetries m a y lead us to the recognition of the laws of weak interactions. The author is indebted to Dr. Chou Kuang-Chao for valuable remarks. References 1 2 3 4 5 6 7 8 9 10 11 12

M. M. N. G. 1t. V. G. L. M. M. G.

Cell-Mann a n d A. Rosenfeld, Ann. Rev. Nucl. Sci. 7 (1957) 407 Gell-Mann, Phys. Rev. 111 (1958) 362 Menyh~rd, Symposion on Elementary Particles, BalatonvilAgos, H u n g a r y (1958) Marx, Nuclear Physics 9 (1958) 337 P. Feynman, M. Cell-Mann, Phys. Rev. 1 0 9 (1958) 193 J. Ogievetsky and Chou Kuang-Chao, J I N R preprint (Dubna) (1958) Marx and G. GySrgyi, Nuovo Cimento Suppl. 5 (1957) 159 Michel, Progr. Cosmic R a y Phys. 1 (1952) 144 Cell-Mann, Rochester Conference (1957) Goldhaber, Phys. Rev. 92 (1953) 1279 Gy6rgyi, J E T P 32 (1957) 152 j. Kawakami, Progr. Theor. Phys. 19 (1958) 459