A unified field theory of mesons and baryons

A unified field theory of mesons and baryons

Nuclear Physics 31 (1962) 556--569; ~ Not North-Holland Publishing Co., Amsterdam to be reproduced by photoprint or microfilm without written permis...

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Nuclear Physics 31 (1962) 556--569; ~ Not

North-Holland Publishing Co., Amsterdam

to be reproduced by photoprint or microfilm without written permission from the publisher

A U N I F I E D FIELD T H E O R Y OF M E S O N S A N D B A R Y O N S T. H. R. S K Y R M E t

A.E.R.E., Harwdl, England Received 29 S e p t e m b e r 1961 A b s t r a c t : Some aspects of a field theory, similar to b u t m o r e realistic t h a n , t h a t examined in the preceding p a p e r are discussed. The w a y in which a non-linear m e s o n field t h e o r y of this t y p e m a y contain its o w n sources, and h o w these m a y be idealised to p o i n t singularities, as in the conventional field theories of interacting linear systems, is formulated. The s t r u c t u r e of the particle source in the classical t h e o r y is calculated, and some qualitative features of the interactions b e t w e e n these particles and m e s o n s are described.

1. I n t r o d u c t i o n The classical field theory described in the preceding paper 1) has several features similar to that of a conventional field theory describing a system of interacting mesons and their particle sources, and the correspondence can be made close also in the quantized theory ~). It is interesting therefore to see whether a more realistic theory of similar type could provide a more fundamental basis for the usual description of mesons and baryons in terms of linear fields interacting through various types of coupling. The objective is the construction of a theory of self-interacting (boson) meson fields, which will admit states that have the phenomenological properties of (fermion) particles, interacting with mesons. This programme is the obverse of the more fashionable endeavour to reduce the truly elementary particles to a set of spinor fields, out of which everything can be built b y simple conjunction. It is a priori much less reasonable because, in particular, it is more difficult to construct half-integral representations of rotation groups out of integral than conversely; indeed it is patently impossible to do this within the limitations of a polynomial expansion. The hope that remains is that the particle-like states will be of a kind that cannot be reached by perturbation theory, and which cannot necessarily be discounted by general arguments. In the type of theory we are using the particle-like solutions have this character, arising from the fact that periodic, transcendental, functions may have properties essentially different from those of any polynomial approximation to them. There is then also the related difficulty that if such states can be made they ? N o w a t D e p a r t m e n t of Mathematics, University of Malaya, P a n t a i Valley, K u a l a L u m p u r , Malaya. 556

A UNIFIED FIELD THEORY

~57

should occur in pairs, just as when massive particles are made from massless ones there will be states with both positive and negative mass-constants. It has been argued that this duplicity need not imply a doubling of the particles to be seen, only a doubling of their description. A theory of the sort we propose m a y help to clarify this phenomenon b y providing evident classical analogies. In this paper we look at some more details of a possible theory of this kind 8). The principles underlying the separation of the field into meson-like and particlelike parts can be illustrated in a parallel w a y for this theory and for the simpler model considered in the preceding paper 1). In both the fields at any point are characterised b y a unitary symbol U: U U , = U* U = 1.

(1)

In one case U is a complex number (e ~ for the model), in the other a quaternion. The domains of U are respectively the circumference of a unit circle, and the surface of a unit sphere in a four-dimensional Euclidean space. The constant of motion N that is interpreted physically as the number of particles has the geometrical meaning of the number of times that the field distribution U(x) maps (algebraically) the 1- or 3-dimensional configuration space onto this domain. The field U is postulated to satisfy the boundary condition u(oo) 1, =

which ensures that a continuous field generates an integral N. In a state where there is one particle, N = 1, U must then take on the particular value --1 at least once. This suggests a natural definition for the position of particles; the field m a y be said to contain a particle (or antiparticle) at t h e point x -----Xo, whenever

u( 0) = - 1 .

(3)

The particle or antiparticle character is distinguished b y the sign of the Jacobian, symbolically - - i ~ log U/3z at this point. The geometrical interpretation of N implies that it must equal the difference between the numbers of particles and antiparticles so defined. We now t r y to separate U into a 'mesonic' part 0 that never takes the value --1, and a 'particle' part that is significant only near the particle positions. Now the square root of U that reduces to unity at infinity is equal to

(l+U)/{(l+U)[ and is continuous and one-valued if the branch-point at U = --1 is excluded. The branch-point would disappear if 1 were replaced b y (1+~), for arbitrarily small positive e, and this suggests identifying the modified function with 0½; alternatively we have

0 = (l+e+U)/(l+~+U ?).

(4)

558

T . H. R. SKYRME

Evidently 0 :/: --1 for positive e, and approaches U almost everywhere as -+ O. The field described b y 0 must have N -- O, and if we put U = OS

(5)

a natural separation is made. Now S = 1 except near points x 0 where U = --1; near a particle we can write (with the notation introduced in the following section) U =

-- l--i~B~

~ (x--x0) i + . . . .

so that near any particle S = -- ( e + i , ~ B , ~ (X--Xo),}/(e--iv ~ B , ~ (X--Xo),).

(6)

The B, ~ are proportional to the field gradients at x = x0, and characterise the particle-source in addition to the position x o. Applied to the model theory this treatment is essentially the same as that given in ref. 3); the programme is to show that, as was demonstrated in that case, so in the more realistic theory, singularities like S, describing the branchpoints of U½, or certain types or combinations of them behave, in the eventual quantized theory, like Dirac particles coupled to the residual meson field U, Whether or not this theory has any direct connection with the description of the observed strongly interacting particles the principles involved seem important enough for further study. In this paper we consider some implications of the classical field theory, partly because a good understanding of it m a y be necessary as a prelude to consideration of the quantized theory, and partly with the optimism that the theory really is relevant to the physical particles, in which case the classical theory m a y illuminate phenomena at small distances in which many particle quanta are involved and which are beyond the scope of calculation with current theories.

2. Definitions of the Theory The definitions used here are identical with those in ref. 3). The basic field quantity is the unit quaternion U, which may also be written in terms of four real fields ~,: 3

where ,4 are a set of three Pauli matrices. The field gradients are expressed conveniently by writing OU/~x~, = iv ~ Ba~ U, or

Ba ~ = (1/2i)Tr(U*z ~ ~U/~xt, ).

(7)

A UNIFIED

FIELD

THEORY

559

at ° From these we can form the antisymmetric tensor Ca,.

Ct, , = ½ (OB~,at[Ox,-- OB,at/Ox~,) = ( 1 / 2 i ) T r ( O V + / O x ~ v ~ OY/Oxt,)

(s)

= %p~,B~p Bv ~. The space parts of this tensor will also be written as a vector Clat ~ C2~ etc. For reasons that were given before 3) the dynamics will be derived from an assumed Lagrangian density ~=

-(,/s~

2

)c,,,c,,,-(~,, at

at

2

I ~ 2 )B~,at B / ' .

(9)

Here we use units in which c = 1, K is a reciprocal length (about 2 or 3 fm -1 in our interpretation), and e is the constant fixing the energy scale of the classical theory. We conjecture, for reasons one of which is given below that e = ~ c , as in the model theory of ref. 3). In this Lagrangian the independent variables are the fields Ca, strictly only three of them on account of the unitary condition 4

¢ 2 = 1. p=l

The relations (8) show .L~ as a polynomial in the Ba ~, and the canonically conjugate variables can be defined in the usual way. They are equivalent to the fourth components of P/'

= &L~'/~Bt, ~' = - - ( e / 2 ~ ) B T [ ( B ~ T

B~'+K2)~c,~--B,~'B~P],

(I0)

which are proportional to the isobaric spin currents J~at = --½Puat. The equations of motion are identical with the conservation laws

~aC/'l~x~, = o.

(11)

The energy momentum tensor has the density

and the density of angular momentum is just the orbital angular momentum associated with f , i.e. •"~'a,c,v ~- xa'~av--xv'~"a,u •

The other important conserved current is that interpreted as the particle number-density, defined by ~a

= (i/2z2)eal,~v B t, l B ~ B v 3.

The conservation law ~/e~

= 0

is satisfied identically, independently of the equations of motion.

(12)

560

3. a. a. SKYRM~-

A different form for these currents arises when, in £0, C ~ is regarded as the curl of B~ ~, instead of its explicit form as a product of B. The differences arising are in the form of divergence terms that do not alter the integrated constants of motion, b u t the new forms are suggestive in relation to our programme of interpretation. In this view we regard the Ba ~ as first a set of vector meson fields, and add the terms that arise from the fact that they are really gradients of the basic fields ¢p. Therefore, we define =

R,,

= =

)C.,,

aLflaB p-- a(J , lax .

Then P is replaced b y P P~ = R~ " + 2e~p~ ~,,,,~ B~

= t' p+

(13)

and the additional term does not contribute to the equations of motion nor to the total isobaric spin constant. Likewise we get !

= •

t

t

(14)



= "~',~av+ (e/4:rr2)e [Xa B va C~p-- x vB l,~ C~,a]/ ~x a" The last terms in (13) and (14) are the contributions of B as vector fields to the spin currents. The latter is interesting as it suggests that there m a y be associated with the particle current Mf~ a parallel spin of magnitude ~. In conjunction with the expectation, b y analogy from the analysis of ref. 2), that the elementary particle structures m a y be Pauli spinors this supports the conjecture that ~ m a y have the value ½hc. The interpretation of the current ~Y'z as a particle current was first suggested because it is conserved absolutely. This interpretation is reinforced b y the definition of particle position, equation (3); for if U has there a constant value, the velocity 4-vector v~ must satisfy

vt,~gU[~z~, =:- 0, or with the definition (7)

vj, Bt,~

= 0,

at

x -- x o.

(15)

These three equations (15) determine va proportional to vC'~(xo), provided that ~ 0 cc det (B~~) ~ 0. This condition is just that equation (3) should not have coincident roots, when there would be a pair of particles at x 0. Corresponding to the separation of U into particle and meson parts (eq. (5)),

. u m r x E , rZSLD THEORY

561

the field gradients B separate into a mesonic part/~, defined from 0 as B from U, and a source part which vanishes except in the neighbourhood of particles. This is ~,~ = Bt,~ + [eB~,a/(e2+ (B,~'(x--xo),)2)]Tr(S-½vaS½za), (16) where S was defined in eq. (6), and in the source term the B~~ are taken at the point x = x o. The source contribution provides something like a 0-function multiplied by a rotation induced by S½. It is this rotation which makes the discussion of the present theory much more difficult than that of the model, where S was a scalar number. 3.

Simple

Solutions

It the fields have a wave-like character in which U is a function only of the combination (kt, xt, ), for some fixed wave-number k~, then evidently the B~~ are of the form k , b~(kx), and the C~ will vanish identically. In this case the equations of motion reduce to =

0

(17)

which is satisfied if k~,k~, = 0, i.e. for any waves progating with the velocity of light in a fixed direction. A similar simple solution occurs when the fields ¢~ lie in a fixed direction, when we can put ¢~ ---- t ~ sin *¢(x),

¢4 -----cos ~(x).

Eq. (17) again applies and reduces to [ ~ = 0. These are interpreted as meson waves, and indeed eq. (17) applies to all small amplitude waves. A relation with the conventional description b y linear fields is suggested b y the structure of U, eq. (4); if we define

/ L = 2QSd(l+¢4),

(is)

where Q is the scaling constant given by Q 2 = (eK2/2u2),

(19)

then the variables//= extend over the range (-- 0% oo) as usual, and the terms of lowest order i n / / ~ in are just =

(20)

These 'mesons' have zero mass, ultimately on account of the full rotational s y m m e t r y of the Lagrangian (9). This s y m m e t r y is, however, destroyed by the boundary condition (2), and we believe that the mass m a y arise as a selfconsistent quantal effect. This point will not be followed here, but when, for

~6~

T.H.R.

SKYRME

calculation purposes, we want to allow phenomenologically for a finite mass this will be done by adding to £v a term

(21) This adds to .W0 the usual mass term --{#2H=2, for a meson of mass ~c#. Solutions with N---- ~ 1 must evidently have non zero C, and cannot be reached by perturbation theory from the small amplitude solutions, all of which have N ---- 0. An exhaustive study even of static solutions of the equations of motion is much more difficult than for the model. The only form that we have found which leads to a solution has

$= : (e,=x,/r)sin ~(r),

64 = cos ~(r),

(22)

where the 'centre' of the particle has been put arbitrarily at the origin x, = 0. The quantity e~= is an arbitrary fixed orthogonal matrix, with determinant equal to --N. Substitution of this form (22) into the equations of motion (11) leads to an ordinary non-linear second order differential equation for the shape-function x(r). This equation is that which determines the minimum of the total field energy: E = (~/rr)f: {(d,t/dr)2(K2r~+2 sin 2 c¢) + (sin2 o~/r~) (2K2r~+ sin ~ ~+/z2K2r4)}dr,

(23)

where the contribution of the phenomenological mass term (21) has also been included. A n estimate of the solution was m a d e in ref. 3); we have n o w solved the differential equation numerically on the Harwell Mercury Computer. Evidently st must tend to a multiple of ~ both at infinity and at the origin; choosing that branch which goes to zero at infinity it appears that there is just one solution for each possible value of ,c = N ~ at the origin, and N is equal to the particle number for the corresponding fields (22). For N -----0 the solution is trivially zero everywhere. For N = ± 1 the numerical solution gives a total energy of about 4,K; identifying this with the nucleon mass M and taking ~ = ½ requires (ILK) ~ 0.4 fro. For N = 2 the total energy is about 12eK; in a naive interpretation this means that stable particles with particle number two would not exist, since the energy is m u c h more than twice that of a particle with N = I; alternatively it suggests that, at least in one special case, there is a short-range repulsion between particles equal in order of magnitude to a particle mass. These numerical solutions have been found both for the (original) case of zero meson mass, and with # = 0.3K, which is of the correct order of magnitude to give the observed pion/nucleon mass ratio. There is very little difference between the total energies in the two cases, or between the forms of

A UNIFIED FIELD THEORY

56~

,t(r) for Kr < 2. In the former case at large distances one has ¢¢ ~ 2.2/K2r 2, while in the case of finite mass, one has 2.2e-~r(l+/xr)/K2r 2.

~

(24)

The coefficient 2.2 is a measure of the effective coupling constant. The asymptotic form (24) is the same as would be obtained from a linear theory with a point derivative interaction, having, in the notation of eq. (20), =

(25)

The corresponding numerical value of F is F = 4.4(2e)½/~. If we put F = g/2M in analogy with the conventional linear theory, we get g2/4~C

~

25(2e/JiC) a.

The quantity a(r) begins to deviate appreciably from its asymptotic form (analogous to the one-pion contribution of linear theory) at distances <2/K. The r.m.s, radius of the energy distribution is 1.25/K. These particles have no charge in the static classical limit, but the charge distribution of a slowly rotating particle can be estimated with eq. (41) of ref. a), giving

p(r) o¢ (sin s ~/r*) [K~r*+ (rda/dr)" +sin* o~. For # = 0.3K this distribution has an r.m.s, radius 1.93/K. The proper interpretation of these particle solutions is not clear. It is plausible that the classical particle represents a primitive, highly degenerate, form of baryon that is the classical limit of the various observed baryons and possibly also of their excited states. In the quantised theory, which cannot admit a fixed matrix e~=, this basic particle should split into a number of different 'rotational' states with various eigenvalues of spin, isospin and hypercharge, and this would conform with the idea of 'global symmetry' within the system of baryons and pions. The Eulerian representation of the orthogonal matrix e~= interprets it in terms of a spin rotation b y a unit quaternion u, with T=

=

eiaTi = UTigt ,

so that

e,= = ½ Wr(utz~ur~). The matrix u m a y also be written as a column 4-vector v, labelled by isospin z3 and ordinary spin aa, with the 'reality condition' Cr2Z'2V ~

--vt,

so that e~ = ~

--lvtg=aiV.

564

T.H.R.

SKYRME

This last form shows the matrix as an 'expectation value' of the operator V=a~. Formula (6) indicates that the particle-source should be characterised by the B{'(xo), or at least b y their ratios, which amounts to eight numbers instead of the three that fix e~~. Thus in general the classical particle-source has a more complex structure than the static solution, presumably connected with the possibilities of rotation. In the model theory, ref. ~), the classical velocity of the source appears in the quantised theory as an expectation value shared between the two possibilities ± 1 of the quantized states. In the same way, perhaps, the B,~ m a y be related to an expectation value of several orthogonal matrices, defined by the eigenvalue equation ~ a , B [ ' v = by. 4. Interactions

Interactions in this theory can, at present, only be discussed in approximate or perturbative ways; it is interesting to see, however, the extent to which they resemble physical interactions. We note first that for the state described by a field U that is a product U (x) ~- U '1) (x) U c2)(x),

(26)

the particle number is additive, N = N~I~+N ¢~.

(27)

This result, which is obvious in the model theory, requires more formal proof here. From the definition (12), we find MZ~ = (i]2~)~a~,~p(e~pr]6)B ~,~B~B B 7 , and the definition (7) gives in the case of the form (26), B~,~= B;(1)+ga~(1)B~,a(2),

(28)

where ga~ is the orthogonal matrix gp~, = ½Tr(U~aUtz~). We substitute (28) into JV'z and use the relations (8) to simplify the crossterm; the result is W~ -----JV~ (1) + Jg'~ (2) + (i]4~*)e~a~p~[B S (2)ga~ B p~(1) J/ ~x~. Since the last term is a divergence it contributes nothing to the integrated particle number N, establishing (27). In particular for the separation (5) we have 2~ = 0 by definition and for the 'source' S, the particle number m a y be calculated by formula (16). This gives N ----- -- (1/2z ~) f [2e] (e~+ (Si ~ (X--Xo)i)2)~det B, ~d3x,

A UNIFIED

FIELD

THEORY

565

which tends in the limit ~ -+ 0 to the value N = --sgn det Bi ~(x0). This justifies the definition of particle/antiparticle character. Two particles far apart might be described approximately by the product field (26) with U ( i ) equal to the separate free-particle fields. We substitute (28) into the energy density 8 = ~'44, which gives 8 = d'(1) +va(2)--2B,a(1)gp~ J,~(2) 2B, p (2)ga~ J , ~ (1) -- (8K2/2~z2)B, a (2)gB~Bi ~ (1) -

+ terms quadratic in At large separations negligible, and to the with the equations of

-

both B(1) and B(2). the last terms not written explicitly will be relatively same approximation ( B i ~ - - ~ $ ~ / ~ x i ) is zero. Together motion

~ J , ~ / ~ x , = o,

a ( g ~ J , ~ ) l ~ x , = o,

this implies that the integrals of the third and fourth terms are also negligible. This leaves for the asymptotic interaction energy V = -- (sK2/27~~) f (~¢~(1)/~x,)(O¢~(2)/~x,)dax. To evaluate this we integrate by parts, remembering that in this theory the $~ are everywhere finite and continuous. Since V2¢~ vanishes except in the neighbourhood of particle 2, we can expand ¢~(1) in a Taylor's series about that point using the asymptotic form ¢~ :

(ei~xi/r) sin ~(r) ~

--ei ~ ~(½F/Kr(2c)½/~xi

in the notation of eq. (25). The first significant term, giving the dominant interaction at large distances, is V ,'~ ( F 2 / 4 ~ ) e , ~ ( 1 ) ~ / ~ x l • e T ( 2 ) ~ / ~ x j • (l/r). (29) This result is just what we should expect from the nature of the asymptotic particle fields which are the same as if there were a point derivative coupling (eq. (25)). It is difficult to determine the potential at smaller distances, because of the algebraic complexity and also on account of the ambiguities in defining a potential and in specifying the spin character of the particles when they are interacting. It is likely that there will be repulsive forces between particles at sufficiently small distances for the same reason as in the model theory (ref. 1), sect. 4), that it is difficult to match the fields of two adiacent particles without having large field gradients. If two configurations of the type (22) are brought together there will be

566

T.H.

R, S K Y R M E

some degree of incompatibility unless the matrices e of the two particles are mirror images in the plane normal to the vector connecting the particles, i.e.

This annihilation state of a pair does not contain any of the singlet-singlet state (e (1) = -- e (2)), which m a y therefore be unfavoured. Another indication of particle-particle repulsion is the high energy found for the static solution with N = 2. Finally repulsion at short distances on the average can be proved b y the following argument for 'particle matter'. Suppose that there are N particles in a box of volume V, i.e. N = (1[2~ 9) t A d ' x , where detB, ~ has been replaced by --A for brevity. The energy density is certainly greater than that of the static terms which are (e/4~t*)[(C,')z+ K2(B,~)2]. Now, by Hadamard's inequality, one has A 2 < (B,1)*(B,*)*(B,Z) * < [(B,~)z/3] z, so that (B?) ~ > 31AI~. Likewise, since detC,~ = A2, we have (C,~) 2 > 3[AI~. Therefore we have certainly E > (3e/4st*) f ([A I~+ x~[AI])d3z. We can establish two results from this inequality. First the integrand is greater than 2x[AI, so that E > (3eK/2zt2) f IAldSx > (3eK)N.

(30)

Thus the energy per particle must be greater than (3tK), the same limit as we found by a different, more special, argument in ref. 3). The present argument shows therefore that the average binding energy per particle could not exceed e~ approximately. Secondly we can apply Holder's inequality to the first term only:

flAl'd3x > (f

[Ald3x)' (fd3z) -½,

which gives E > (3e/4st*)(2rrZN)*V-½.

(31)

A UNIFIED

FIELD THEORY

567

Thus the energy must rise with to½ for small mean separation r 0. The numerical limit is weak as the inequalities used have been very crude. In the discussion of mesons it is natural to use the description (18) in terms of fields H~ that have a linear domain. Then we we get

/

(32)

Lax, ~._I I

To lowest order the system is described by the usual free field Lagrangian (20). The next terms, quartic in the fields/7~, give the interaction Hamiltonian

~Pint

=

--

(1/4Q 9) (II~)2(~II~/~3x~) 2

t\3xj \3x, l

\~x~l\~x~/~Sx, l\3x~l!

In an eventual quantal treatment these terms would describe a mesonmeson interaction, and it is interesting therefore to analyse them in terms of the usual representation of the quantized fields //~. The Hamiltonian #/atnt describes the annihilation of four mesons with 4-momenta k r and isobaric spin directions t, for which the matrix element is equal to 0(tl,

t2) a(ta,

t4)

4Q' (~1 a~,oJa~,l)½ (33) {(k x • k2) (k a • k,) + (1/~2) [2(k,. k2) (k a • k , ) - - (k,. ka) (k 2. k , ) - - (k 1. k4) (k,. kz)]} + two permuted terms. In the centre-of-mass system for the scattering of two mesons we have

k 1, k z = ( ± k , i~o),

k a, k, = (q:k', --ico).

Then in a state of isobaric spin T = 0, the matrix element equals

(1/2Q2o) 2) [-- (¢o2+ 3k 2) + ( 4co2k2+ 2k'-- 2 (k'. k) 2) / Kq, which is negative, corresponding to an attraction, for low momenta changing to repulsion at energies of the order of K. In a T = 1 state the matrix element equals

- - ( k ' . k)/(Q2eo2) (1 +o,21~,), representing an atraction in the J = T = 1 state at all energies. Meson-particle interactions can be studied by perturbation theory applied to the free-particle state. We write, as in equation (26),

U(x)

=

Umes(x)Up(x,),

where Up is the static free-particle solution, and expand [/mes in powers of the field; as in equation (28), we get

568

T.H.R.

SKYRME

B l? = b j,~-i-gp=B ~fl , where b and g refer to We substitute this order in the meson effective interaction ~lnt

~--- --

the meson fields, and B is the particle field. into the Lagrangian, and retain terms up to the second fields. The negative of the interaction terms gives the Hamiltonian

P t,~g~pb t, + (e/4•2) (b t,c'b~P) { ( B f f B f f )6 ~,~8~

--Bp~'B~6~p+2Bt,~BvP--B~BxB~I,~--B~B~,P}, (34) where Pp~, defined by eq. (I0), is proportional to the isobaric spin current of the source. To second order we have g=pb~~ = (~H=/a~) +~=p~(aJT/~x~)/Z~. The linear term gives no contribution on account of the equations of motion OP~,"/Oxt, = 0; the second term contributes

2 J ~%:p~ Hp( OH~/ Ox ~,), an interaction between the isospin current of source and meson field which is familiar in the usual theories of nucleon-pion coupling derived from an assumed symmetrical pseudoscalar interaction. The second term in the interaction (34) is a P-wave meson-particle interaction, repulsive on the average. There is no indication of the strong attraction observed in the pion-nucleon resonant state, but this would hardly be expected in a static classical treatment where the rotational splitting of the particle states has been ignored. In so far as this analysis can be compared with experiment it should represent an average effect of pion-baryon interactions at energies above the resonance regions.

4. Prospects In this paper we have developed a classical field theory somewhat further from its statement in ref. 3) and have shown that there are a number of encouraging resemblances between its consequences and the phenomena of the strongly interacting particles; quantitative comparison is hardly significant however, until the quantized theory is understood. The challenging problem which is unsolved is to understand the nature of the states in the quantized theory which corresponds to the classical solutions with N = I, etc. The usual apparatus of canonical quantization can be applied to the original Lagrangian, but the only obvious starting point for a perturbation theory is the meson Lagrangian (eq. (i0)) ; in this way, since all the operators of the theory commute with N, only states with N = 0 can be built. A different starting-point is needed to describe the particle states. We conjecture that this will involve the definition of singular operators that in-

A UNIFIED FIELD THEORY

569

troduce branch-points, such as described b y S (eq. (6)). We believe also that these operators m a y have many of the properties of Fermion field operators in conventional theories. This was demonstrated for the model theory in ref. 2), but the proper extension to the theory of this paper has not been found. The chief difficulty arises, as might be expected, in the introduction of spins. The simplest operators that change the particle number are characterised b y an orthogonal matrix e, ~, whereas the proper operators that we seek must be states of definite angular momentum. Isobaric spin is contained implicitly in the structure of the basic field quantity U, ordinary spin apparently arises from a strong-coupling between the spin and isospin directions of the source, but its mathematical formulation is elusive. We are indebted to Mr. A. J. Leggatt for carrying out the calculations reported in sect. 3, while a vacation student at A.E.R.E. References 1) T. H. R. Skyrme, Nuclear Physics 31 (1962) 550 2) T. H. R. Skyrme, Proc. Roy. Soc. 262 (1961) 237 3) T. H. R. Skyrme, Proc. Roy. Soc. 260 (1961) 127