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Thus,-it seems that'the general picture of the expanding universe with the in-clusion of the turbulent motion in primordial gas gives a reasonably accurate pic-ture of the formation of galaxies. One can ask, of course, Why was the primordialgas in a supersonic turbulent state? The possible answer to this question is thatturbulence may have been the result of some kind of internal instability in a 50-50mixture of matter and thermal radiation. In fact, mechanical and thermodynami-cal properties of such a system have never been investigated, and further investi-gations are badly needed in that field.

1 G. Gatnow, Kong. Dan. Vid. Selsk., 27, No. 10 (1953), and earlier publications.2 H. Shapley, these PROCEEDINGS, 37, 191 (1951), and earlier publications.3 C. D. Shane and C. A. Wirtanen, Proc. Amer. Phil. Soc., 94, 13 (1950), and a new article in

preparation.4Vera Cooper-Rubin, these PROCEEDINGS (in press).6 C. von Weizsicker, private conversation.

ON THE PROGRA.M1 OF A SYSTEMATIZATION OF PARTICLES ANDINTERACTIONS

By A. PAIS

INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY

Communicated by J. R. Oppenheimer, April 19, 1954

1. Recently, the author has attempted' to consider baryons (nucleons andhyperons) as various half-integer representations of the full 3-dimensional rotationgroup 0(3). The classification of states proceeds by assignment of a'total isotopicspin I, its "3"-component 13, and a parity. I3 essentially determines the electriccharge, and electromagnetic phenomena are invariant only with respect to a sub-group of 0(3) of rotations around a preferred axis. The parity quantum numberserves to stabilize a limited number of baryon states against a rapid decay intonucleons under the emission of 7r-mesons or photons. The weak w-baryon decayinteraction has again different properties than the meson-baryon or the electro-magnetic interactions as it disturbs the parity (i.e., the reflexion invariance).As has been pointed out in Paper I, the extension of the underlying marnifold

considered there is not unique. In discussing the conservation of baryons (Paper1, p. 871), it was noted that the framework is actually too limited for an incorpora-tion of this law. Meanwhile, the discovery2 of the cascade particle (Y-) shows in adirect way that 'the model is inadequate: the introduction of a parity can neveraccount for this phenomenon. Thus the "minimum extension" discussed in PaperI (p. 873) must be considered too narrow.These and other attempts are based on an extension of the fundamental role of

c(harge-independence (CI) ih ir-nucleon interactions to a wider system of particles.If the symmetries and stability properties that do not seem to find a place in ourpresent theoretical picture are manifestations of invariance properties with re-spect to, a group G, it is natural first to try and identify G with the "Cl-group"of the ir-nucleon system, which may be considered as 01(3) (3-dimensional rotation

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*group with det. = + 1) or, alternatively,3 as the 2-dimensional unitary un.modulargroup SU(2). The latter is well known as the transformation group4 of 2-componentspinors in 3-space: two complex variables wI = xl + ix2, W2 = X3 + iX4 transformaccording to SU(2) as wI'= awI + w2, W2' = -I*wI + a*w2, yaa* ++ - 1.Note that x12 + x22 + x32 + x42 invariant so that SU(2), must be contained inthe group 01(4) of 4-dimensional real orthogonal transformations with det. = + 1.

01(3), as well as SU(2), is generated by infinitesimal operatorsJI satisfying [4,If,] = iU, a, 6, y = 1, 2, 3 cycl. In Paper I we made a generalization 0 1(3)0(3), i.e., we replaced 01(3) by a group which contains 01(3) as a factor group. Asa result of this we gained the parity operation without losing the CI.

It is indeed clear that, in general, it is sufficient for the incorporation of CIthat the CI-group be a factor group of G. This is of course a very broad limitationof G. There is, nevertheless, one alternative which commends itself by its simplic-ity and which leads to results that are not without interest: this is the group 01(4).It is the main purpose of this note to indicate the line of argument to which one isXt~in this instance. Before going into the details, the following remarks are in

odder:aJ The possible validity of more conventional arguments for the long life of

hyperons and heavy Bosons (like barrier effects due to high angular momenta)should be clarified before one can fully judge to what extent the stability of theparticles discovered in recent years does in fact require new types of selection rules.However these two approaches are n-ot mutually exclusive.

b) Whatever choice for G is may along the lines indicated above, the electro-magnetic field must necessarily introduce a preferred orientation in the (general-ized) isotopic space. It does not seem possible to avoid this without changingdrastically -the physical contents of our present theories.Thus our present aim cannot be to make all interactions invariant under G. We

shall rather try to extend the invariance of the strong 7r-nucleon interaction underthe CI-group to the invariance of a strong meson-baryon interaction under the groupG. As in Paper I, the physical idea is once more to provide a mechanism for thecopious production of hyperons which yet does not lead to their rapid decay under7r- and/or y-emisson. Again, as in previous models, the weak r-baryon decay in-teraction should have different symmetry properties than the strong' coupling andthe electromagnetic interaction. The hierarchy of interactions with differentsymmetry properties, already commented on in Paper I (p. 885), is indeed a generalfeature for any G.

c) Iri this note we shall, as in Paper I, assume the total frame of description tobe the direct product of space-time and the manifold on which G act+. As al'ready noted in Paper II (see also sec. 5 of this paper), this must be considered as atbest an approximate description. This has been especially emphasized by Pauliand by Yang (in unpublished investigations), who have studied attempts to loosenthe rigid orthogonality of space-time and the "isotopic manifold"; they admit thepossibility of free rotations of the isotopic manifold from space-time point to space-time point.We shall be unavoidably faced with-such questions if it turns out that the notion

of "isotopic manifold" makes good descriptive sense. It seems to me, however,that, in order to ascertain this, one may first endeavor to discern in the phenomena

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under consideration the workings of one particular group. It is in this spirit thatPapers I and II and the present note represent attempts at a "penetration into therorld of the nucleon."5

2. In this section we state the properties of Ol(4) which are needed for the subse-quent discussion. The generators of 01(4) are operators Li, = -Lki, i, k = 1,

4 which satisfy

[Lik, Lim] = -i(Limk5& + Lklim - Lilkm - Lkmil)t (1)

All LIk are hermitian. Define Lj = 1/2* (Li ± La4) a, (3 y = 1, 2, 3 cycl. Thei[Li, Lof] = iLi, [Ln, LT] = 0. (2)

Hence (L±2) - 1 (U' + 1), l - 0 1/2 12. Furthermore, (L3) = 13 X-li <1-3 i.Thus the representations of 01(4) are in general specified by two numbers .(it,

I-) which can range independently over the indicated domain of values. Eqpatiaii(2) implies that 01(4) is essentially6 the product of two groups SU(2). Thus 01(4)indeed contains the Cl-group as a factor. (Incidentally, there are no other-dimens-sional rotation groups than 0(3), 01(4), and the full group 0(4) that contain.eitherSU(2) or 01(3) as factor.)We shall need special representations of 01(4) and introduce, correspondingly,

operators that satisfy specifications additional to equations (1) and (2). For thesake of evoking an analogy, I shall call these operators "orbital angular momentum"(Kik); "intrinsic spin 1/2'.' (sik); "total angular momentum for spin 1/2" (Ik).These are their definitions and properties:

a) Kik. Consider a 3-dimensional Euclidean sphere S3 (described by a set ofthree angular variables 0). Kik are the generators of the rotations of this special

representation space. Kik satisfies equations (1) and (2) and K+2 K-2 = 0. Hence

(K 2) = k (ki + 1), while, moreover, k+ = k-. The representations are (k, k),k = 0, 1/2, 1 .

b) sjk Let As be a set of 4 X 4 Dirac matrices (3. Put (-i/2) [3, Ok] = ask,=5=-(1(2(334- We have [7ik, (5] = 0. A representation in which (5 is diagonalis provided by (3 = plTiy i = 1, 2, 3, 4 = P2, hence 5 = p3. Here pi, Ti are two

sets of Pauli matrices. Accordingly, 7 = 1/2(1 ± p3) T.Si= 1/2?7ik satisfies equations (1) and (2). In addition, (7±2) - S(Si + 1) =

' p3). Thus we get two representations: ('/2, 0) + and (0, '/2), where thesuperscript i refers here and in the following to the eigenvalues 1 of p3 (i.e., (5).

c) Iik. Defined by Iik = Kik + Sik* Iik commutes with (35. It satisfies equations(1) and (2). The representations are (i+, i-)+ and (i+, i-)-, where i+, i- are sub-jedt to the conditioni+-i- = 1/2 It follows that Q = I3 + I3- + '/2 hasonly inger eigenvalues.

In what follows, 4V(x, Q) shall denote a field that is a spinor both with respect tothe Lorentz group7 and with respect to 01(4). Thus it has 16 components. Wecan write #(x, Q) = #+ (x, Q) + xt- (x, Q)yp,± = '/2 (1 i (85)3. The #,6+ are semi-spinors with respect to 01(4), having 8 components each. One has I++ = I+- 6 +K.++ Ia-p = K-7A+ + I4,A-, hence:

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6+(x, Q) transforms like a e field with respect to the group SU(2) generatedby I+, and it transforms like a spinor field with respect to the group SU(2) gener-ated by Ij. 6+(x, Q) is in general a superposition of all states labeled by (i+, i-)+.When we talk in the following of an a-dependent field like Jt(x, &2), this should be

considered as a convenient shorthand for taking together all representations of01(4) that satisfy certain constraints, like| i+-i- = 1/2, above. It is possible touse a more abstract terminology (the same could have been done in Paper I).However this may be, the emphasis lies for the present on the group and its rep-resentations rather than on the manifold on which the group acts (see also theend of sec. 4).

3. Let us identify the 8-component nucleon wave function with the constant(a-independent) semispinor b+(x), corresponding to the representations ('/2, 0) +.

Then CI uniquely fixes the representation of the 7r-meson field +0(x): it is (1, 0),

i.e., a (constant) self-dual tensor. The CI ir-nucleon interaction is now9' .

{+ (x) I +0t+ (x) = Ad+ (x) r 0t'+ (x). Thus CI emerges as invariance with respect tothe group generated by I+.

We can now ask for the most general interaction which contains +(x), 4'(x, Q),and {(x, P.) only and which is invariant under 01(4). This is now (apart from in-variant weight factors that do not affect the argument)

4+(x) f {(x, Q) n+#(x, l) d1 = +(x) f {+(x, Q)?)T++(x, Q) dQ. (3)Thus there clearly exists a fundamental dissymmetry between 4'+ (x, 11) and *-(x,12) due to the characteristic transformation properties of the w-mesons.We shall now attempt to identify the baryon field with the semispinor i&+ (x, Q),

which is strongly coupled to 4 by relation (3). A+ (x, Q) is a superposition of stateslabeled by (i+, i-) + among which we have to identify the hyperons. Whicheverthey are, the experimentally indicated strong bonding of hyperons to nucleons5 isguaranteed by relation (3) through the intermediary of the Tr-mesons.

Define eQ as the charge operator for the baryons and e(I+3 + 1-3) as the corre-sponding quantity for Bosons. We are then ready to discuss the stability of thebaryon states by means of the following rules: (1) In any transition 2ji3+ andVi3- are conserved. (2) Two baryon levels combine under single r-emission ifAi+ = ±1, 0(0O - 0 forbidden), Ahi- = 0, if energetically possible; (3) Theycombine under single Remission if Ai+-= 1, 0, Ai- = 0 or Ai+ = 0, Ai- =4 1, 0. Thus the stability considerations are based on the conservation of Zi3,together with the impossibility for baryon states to combine under 'y- or 7r- emissionif Ai+ or Ai- = half-integer. Note that there is no parity argument as in PaperI.

It was Gell-Mann'0 who first drew attention to the fact that the forbiddennessof half-jumps by strong interactions may be a cause for hyperon stability.His model has two basic features: (1) baryons are described by an isotopic spin Icapable of both integer and half-integer eigenvalues; (2) the charge need not be13 + '/2 for all multiplets. One will note a certain correspondence with the presentnotions by reading I+, I+ for I, 13.

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In assigning states we shall assume that electromagnetic mass splittings withina given multiplet are relatively small. Theni this picture of metastable particle'sdevelops: (a) ('/2, 0)+ + nucleon; (b) (0, '/2) + +-+ A', A +. Thus it is essentialto verify whether or not A0 is a member of a charge doublet, also because this con-stitutes a distinguishing feature between the Gell-Mann and the present model.There- is little evidence" for the existence of a charged hyperon with m''&40iev.12(c) There is evidence'I for charged hyperons with charge -e and +e and Q - 130mev. These can be contained in (1, 1/2)+, with Ii3+,+i3- = {I-1-/2} and { 1,-'/21, respectively. But then there should also exist a second A0 with higher Q:-1, 1/21, as well as'4 a A++: { 1, '/21. Note that it is essential to the argument

that the mass difference between (1, '/2)+ and (0, '/2)+ is <m,. (a) ('/2, 1) + callnow contain the cascade particle Y-: {-'/2, - }, but again one shows that thismultiplet should then contain similar particles with charge (0, 1, 2)*e.Thus it is clear that the present considerations, however qualitative, lead to vari-

ous verifiable statements die to the group properties that underlie them. Weshall not go into further properties of the spectrum, such as the degeneracy forgiven charge of some of the levels or the stability of higher multiplets. Concerningthe latter, we may only notX that for baryon levels 'l1000m, above the nucleon,heavy Boson instability has' also to be envisaged. Due to the combined effectsof heavy Boson-, 7r-, and y-decays, the "hyperon region" can be seen not to extendbeyond -2850m, if the spedi~um is widening sufficiently.A provisional classification- of heavy Bosons can be given on the same basis as

Gell-Mann's ordering'0 byi equivalence considerations with baryon-antibaryonpairs. Considering in particular the reaction X+ nucleon -- A + B, it follows fromthe above assignments that _l is a Boson with (i+, i-) both half-integer, e.g., ('/2,'/2), which can therefore n,. be a 7r-meson. This exemplifies a genra feature ofthis scheme: again, V-particles are produced in pairs"' in nucleon-nucleoon or 7r-nucleon collisions; note, however, that heavy Boson + nucleon - A + X is by nomeans excluded. -To couple Bosons other than the 7r-meson invariantly to baryons, one wvill now

introduce a-dependent Boson fields. Example: Replacing O(x) by +(x, i2), onegets a superposition of states for which Ii+-i = 0 or 1. It is then.possible todiscuss the metastable properties of Boson states by the same rules as those for thehyperons. Thus, e.g. ('/2, 2'/2) cannot combine with 7r-mesons and photons.(Note: in these considerations no arguments are provide4that linkthe space-timetransformation properties ofany Boson field with its isotopic assignment.)

Gell-Mann has pointed out'I that -the reaction 2 nucleons 2A's is forbiddenif one can distinguish between heavy Bosons and anti-Bosons. This important re-mark is valid for any model.Ak Example: Let B0 be equivalent to A0 + antineu-tron and B0' equivalent to ahnti-A0 + neutron. Then, if Bo is not identical withB°', 2 neutrons -- 2A0's cannot be brought about through the intermediary of B0or B01. The reaction P + N -- A0 + A+ can be discussed along similar lines.Thus further restrictions may be imposed on which specific combinations of V-particles are actually allowed- in pair production. In any case the Bosons whichin this pi ture are equivalenltto a particle-antiparticle pair, one belonging to ('/2,0)+, the other to (0, '/2)+, are' all metastable with respect to decays into- 7r's and -y's.

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The charge operator Q contains 13+ + Is- = I12 (cf. sec. 2), i.e., the generator ofrotations "in" the plaiie (12), "around" the plane (34). Thus electromagneticphenomena introduce, geometrically speaking, a preferred plane in the space of S3.Also, the weak decay interactions introduce an element of preference: Let us con-sider the slow decay interactions AO P + r-, Y- A0 + r-, which are so farrigorously forbidden. -This means that the representation of the particle on theleft-hand side is notocontained in the produ t of the representations of the particleson the right. We can next ask: What representation'has to he multiplied furtherinto this product in order to produce the initial state?. From the assignment ofstates it is seen that this is the representation ('/2, '/2) :- a vector. Thus, geometri-cally speaking, the decay interactions may be characterized by the introduction of apreferred axis in the space of S3. From the above-mentioned equivalence proper-ties of the heavy Bosons, it follows that the same characterization holds true for theheavy Boson decay.

It may be noted that this characterization of the very weak interactions forbidsreactions like Y -> N + 7r-. Whatever the explanation for the cascading processmay be, it is certainly remarkable that the particle Y- should have a preference fordecaying via two very weak interactions rather than combine directly with thenucleon in a one-step process.

These, in outline, are the physical consequences of -this particular choice of thegroup G. in so far as one considers the system of baryons, mesons (of integral spin),and photons.'7 Of course, if one confines all considelations to this subsystem ofparticles, there is no sense in asking for an incorporation of the law of conservationof baryons in the theory. This question becomes meaningful only if one alsointroduces the light particles (electron, neutrino, A, . . ~). Surely the existence oftwo disjoint groups of fermions constitutes an essential clue to the theory of matter.

4. While for the heavy fermions both the symmetry properties embodied in CIand the peculiar stability properties of the hyperons are striking qualitative features,there is no evidence for invariance properties concerning isotopic spin, and littleindication for unusual stability so far as the light fermions are concerned. How-ever, the apparent e-stability of the si-meson (1- e + 'y has not been observed) isnoteworthy; in fact, the absence of any strong meson interaction leaves photonlecay as the main source of information concerning stability. In addition to this,even the very weak interactions may produce information as regards stability.There is, for example, the peculiar circumstance that so far no electrons have teenobserved in Boson decay. However, it is still marginal from the e)per'mentalpoint of view whether the absence of w-electron decay truly poses a problem. '8Thus the light fermions do not give us much information about the central ques-

tion one is led to ask: Can all fermions be labeled by means of representations of G,and can their various interactions be classed in such a way that the law of conserva-tion of baryons emerges?

While it is true that the baryons and the light fermions show an essential dissym-metry so far as the presence of strong meson interactions is concerned, they sharethe property of having electromagnetic interaction, and also as regards the weakdecay interactions there are parallels: Note that the probability ratio for A' -* P+ ir- versus or -- ,u + -. is, within a factor of 2, equal to the corresponding phase-

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space ratio. Thus there are indications that, manifest disparities notwithstanding,a simultaneous discussion of baryons and light fermions may be called for evenapart from the question of the baryon conservation lav.We have seen in section 3 that the requirement of invariance of the strong 7r-

meson interaction with *t'(x, i2) led to attributing to 44+ (x, Q) a role quite distinctfrom V- (x, U) so far as the strong coupling to 7r-mesons is concerned. It seems,therefore, worth while to inquire whether {I-(x, Q) could correspond to the lightfermions in such a way that conservation of baryons is guaranteed.

Let us then assume that all interactions commute with ,I5. By a reasoning similarto that given in Paper I (s& is the full spinor), it follows that

-fh.T { dR = a ffl7,6dQ =O,

which amounts to

f:,h'v,#+ do = 0, (4)

fX I- - dQ = 0. (5)

Equation (4) states that the numbpr of baryons -the number of antibaryons re-mains conserved; equation (5) refers in the same way to light fermions.'9 Ofcourse, processes like /3-decay are not at variance with the conservation due to/3s. It should be noted that the definition Q = I3+ + I3- + '/2 of the charge operatorreferring to 4,+ can be extended to Q = I3+ + I3- + '/2 /35 in applying it tothe full spinor field. Such a definition of electric charge forces one to the conserva-tion of baryons by virtue of the fact that the baryons are then differently gaugedcompared to the light particles. Of course, it remains a matter of conventionwhether one writes I/2Af or - V/A35. The second choice would effectively corre-spond to a simultaneous interchange for heavy and light fermions of (particle,antiparticle). Compare the analogous situation in using T3 + 1/2 or T3 - 1/2 fornucleons in the standard isotopic spin formalism.

In discussing baryon states we used the intuitive argument of approximate massdegeneracy in a multiplet. There is no ground to argue in this way for the lightfermions, nor are there indications for any other satisfactory procedure in this case.We shall therefore leave the subject of light versus heavy fermions with the remarkthat in this scheme one has room for two fermion families for each of which onehas a conservation law for their total number. As regards the study of specificparticles, it may for the present suffice to state that the concept of the baryon as asemispinor with respect to 01(4) may be of heuristic value.The above remarks are intended to illustrate how the 4-dimensional rotation

group constitutes a simple realization of the group G defined in greater generality at,the beginning, and how, in terms of 01(4), one will go about the ordering of the par-ticles and their interactions. The main aim of this paper has been to state a pos-sible program ;20 it need hardly be stressed that the present suggestion must beconsidered as tentative. Some reasons for this have already been given in section 1.Of course, the group 01(4) is richer in representations than the model 0(3) stud-

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itd in P'ap)er I. One may well ask whether the arbitrariness in the choice of therepresentations ifi+ - = '/2 is not too great. Here it should be noted that wecould have given exactly the same analysis as in the foregoing by using all repre-sentations for which ji+ - i-| = half-integer without necessarily introducing ad-ditional metastable baryon states. Thus, as in Paper I, the entire discussion can begiven in terms of all half-integer spin representations without enriching the results.We have excluded the case ji+ -i- = integer for the baryons, as one has thenno obvious definition of a charge operator for the baryon family as a whole. Thedescription of the electric charge seems one of the most crucial and least under-stood aspects of the problem.

5. Finally, I would like to comment on the connection between the stabilityproblems of fundamental particles and their mass values. No simple connectionnecessarily exists. Consider by way of analogy the situation in atomic physics:If an atomic system has metastable levels, this is due to symmetry properties ofthe system in isolation and to symmetry properties of its interaction with externalagents. The actual position of such levels depends also on other properties of thesystem which have no direct bearing on selection rules.

Likewise, if there exists something like a "manifold of structure" of fundamentalparticles, it may be a rational first step to ascertain by trial and error whether anygroup properties can be attached to this manifold by analyzing all symmetry andstability problems involved. As metastable states have a tendency to be low-lyingand limited in number, this kind of question may be much more easily within reachof solution than the one of the actual mass spectrum of the particles.Over a period of many years attempts have been made to envisage a structure

manifold as a finite extension in space-time. It is not clear that such nonlocaltheories can consistently be formulated; however this may be, none of these en-deavors has so far brought us any new insight into the physical properties ofparticles. Yet, if we should have to admit the existence of a structure manifold,it seems hard to believe that this would not affect the spatiotemporal descriptionof events. However, the underlying idea of the general program here outlinedconsists in an ordering of particles and interactions without touching on space-timeproperties. Hence it seems to me that in this way one may at best uncover some ofthe elements that should be contained in a more complete theory; it should be therole of such a theory to shed light on (1) the meaning of the divergences encounteredin our present formalism; (2) the significance of the "preferred orientations" thatare introduced through the electromagnetic (and possibly through the very weak)interactions. A more intimate connection between space-time and a manifold ofstructure seems difficult to establish as long as we require the unlimited applicabilityof our present space-time concepts.

In conclusion it may be recalled2' that there exists a simple algebraic connectionbetween O1(4) and the (proper) Lorentz group L1(4): there is a one-to-one cor-respondence between the representations of O1(4) and the (finite-dimensional)representations of L,(4). However, the former are unitary, whereas the latter arenot. It is this unitarity property which has enabled us to define the internalparticle states in a consistent way.

I want to express my gratitude to Dr. R. Jost, with whom I have had many

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discussions on these and related topics. I am also greatly indebted to-Dr. M. Gell-Mann for communicating his results before publication and for a stimulating cor-respondence.

1 A. Pais, Physica, 19, 869, 1953 (herein referred to as "Paper I"); Prog. Theor. Phys., 10,457, 1953 (herein referred to as "Paper II").

2R. Armenteros, et al., Phil. Mag., 43, 597, 1952; G(>D. Anderson, et al., Phys. Rev., 92, 1089,1953; E. W. Cowan, Phys. Rev. 94, 161, 1954.

3See, e.g., B. L. van der Waerden, Die gruppentheoretische Methode in der Quantenmechanik(Berlin: J. Springer Verlag, 1932), sec. 16.

4Ibid. Cf. also E. Cartan, Lefons sur la theorie des spineurs. (Paris: Hermann & Cie, 1938),Vol. 1.

6 Cf. C. F. Powell, Nature, 173, 469, 1954.6 Cf. B. L. van der Waerden, Gruppen von linearen Transformationen (Berlin: J. Springer

Verlag, 1935), esp. sec. 7. To be precise, we have 01(4) = SU(2) X SU(2)/H, where H is the dis-crete group with elements (1, 1) and (-1, -1).

7This implies that in the following we consider all baryons as spin-1/2 particles. More generalassumptions on the spin of the various baryon states are not necessarily at variance with thepresent ideas, however.

8 One other choice is the constant semispinor (0, 1/2). This alternative in no way changes thetrend of the subsequent argument.

We shall not write down explicitly any detailed structure of the interactions in s8 far as space-time is concerned (e.g., a y5 in the above case), as this does not affect the argument presentedhere.

10 M. Gell-Mann, Phys. Rev., 92, 833, 1953; and a sequel, "On the Classification of Particles"(in press). The same ideas have independently been arrived at by T. Nakano and K. Nishijima,Prog. Theor. Phys., 10, 581, 1954.

1"A. M. York et al., Phys. Rev., 90, 167, 1953 (A+); W. B. Fretter et al., Bagnrres de Bigorrereport, July, 1953, p. 101 (A+); P. Barrett, Phys. Rev. (in press) (A-).

12 At this stage it may be observed that we could Give replaced i + in equation (3) by K+.

But then nucleons would not be coupled to ir-mesons. Similarly, using I+, we would have nosuch interaction with the A', A +. Note the exceptional position of the level (0, 1/2) +: it can onlybe coupled to nucleons by exchange of an even number of r-mesons,,due to the forbiddenness of0 - transitions.

13 M. Ceccarelli and M. Merlin, Nuov. Cim., 10, 1207, 1953 (A ); D. Lal, Y. Pal, and B. Peters,Phys. 'Rev., 92, 438, 1953 (A ); H. S. Bridge and M. Annis, Phys. Rev., 82, 445, 1951 (Ak); D. T.King et al., Phys. Rev., 92, 838, 1953 (A+); W. B. Fowler et al., Phys. Rev., 93, 961, 1954 (A-); A.Bonetti et al., Nuov. Cim., 10, 345, 1953 (A+).

14 A cloud-chamber picture recently obtained by G. Ascoli (Phys. Rev., 90, 1079, 1953) givessome evidence for a doubly charged particle with transprotonic mass.

15 A. Pais, Phys. Rev., 86, 663, 1952.16Formally this expresses itself in terms of reality conditions to be imposed on a Boson field.17 For possible causes of "anomalous Q-values" see H. PXimakoff and W. Cheston, Phys. Rev., 92,

1537, 1963; 93, 908, 1954. Also the possibility should be borne in mind of radiative transitionsA°-P + 7- + -Y, -,'/137 less probable than ordinary.A0-decay.

'8 Cf. H. Friedman and J. Rainwater, Phys. Rev., 84, 684, 1951.19 On the possible consequences of a conservation law of light fermions cf. E. Konopinski and H.

Mahmoud, Phys. Rev., 92, 1045, 1953.*' After-the completion of this work, I received a preprint of a manuscript by Drs. Nakano and

Utiyama. Using the language of the present paper, thei authors propose to describe the nucleonby (1/2, 0) + and the (A 0, A +) by (0, '/2)-. They extend the 7r-meson field to a tensor of the secondrank, containing two charge triplets. One of these is supposed to be the wr-meson, the other a T-meson. This approach has no direct connection with the present one. I would like to thankthe authors for communicating their results before publication.

21 Cf. van der Waerden, Die gruppentheoretieche Methode in der Quantenmerhanik, sec. 20, andE. Cartan, Legons sur la theorie des spineurs, 2, 75, 1938.

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