On unusual statistics of elementary particles

Nuclear Physics 40 (1963) 518--523; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microffimwithout written permission from the publisher

ON UNUSUAL STATISTICS OF ELEMENTARY PARTICLES t P. R. P A U L A E SILVA and TAKAO TATI-tt

Department of Physics, University of S~o Paulo, Brazil Received 9 August 1962 Abstract: When we introduce a constant vector N z in momentum space (Minkowskian space), we can define a unitary Lorentz-invariant S-matrix for the non-microcausal (non-local) interactions. The existence o f Nimplies the existence of a special reference system in the Minkowskian space. In this scheme, certain unusual quantizations are possible. As an example, we examined the possibility of the case in which the state vector is symmetric for the interchange o f two particles, while nevertheless, the maximum occupation number is one (exclusion principle). We call this kind o f quantum b-particle. The b-particle can interact with photons only in very unusual ways especially in the case of spin ½. The muon is excluded from being a b-particle by its anomalous magnetic moment and the results o f pair creation experiments. But the present experimental data cannot exclude the possibility o f z~ or K being a b-particle.

1. Introduction and Summary The relation between spirt and statistics has been considered to be one of the most important results of field theory. We think it, however, worthwile to examine the possibility of statistics different from the usual one, since we expect that certain modifications of fundamental assumptions of field theory will be made in the future. At present, we know elementary particles belonging to ten categories, V, e, v,/1, N, A, X, 8, n, K. Experimentally, the statistics has been tested directly only for ~, e, N and n 1). In order to establish the statistics by experiment, the observation of a process is necessary in which more than two particles or two anti-particles of the same kind exist in the initial or final state. Drell discussed the possibility of spin - ½ bosons and spin 0-fermions as strange particles and gave examples of possible experiments to test this 2). We note here also that there is no experimental evidence in support of a maximum occupation number for n or K larger than one. The purpose of this note is to show a theoretical possibility of unusual statistics and to emphasize that it is important to establish the statistics for all elementary particles directly by experiment. The statistics is determined by the properties of the creation and annihilation operators (a*, b*), (a, b) in interaction representation: ~b(x) = constant ~,Sr{a(p,

r)ur(p)e~P'~+b*(p,r)vr(p)e-~P'~},

(1.1)

where p and r represent respectively momentum and spin degrees of freedom (we do t The main content of this paper was presented to the 13th annual meeting of the Brazilian Society for the Progress o f Science in July 1961. tt On leave of absence from Kanazawa University, Japan. 518

U N U S U A L STATISTICS OF ELEMENTARY PARTICLES

519

not consider other degrees of freedom, for the sake of simplicity). When we define a or b as operators different from the usual ones, the field operators at x and x', (x - x') being space-like, do not satisfy [~b(x), qS(x')] + = 0 ( + / - for haft-integral/integral spin). Then, in general, for the interaction Hamiltonian, [H(x), H(x')]_ ~ 0,

( x - x ' ) a > 0.

(1.2)

In this case, the descriptions of one event by two observers are not connected by a Lorentz transformation x' = Lx. This fact can be seen, for instance, in the perturbational expansion of the S-matrix (assuming its convergence)

S = y,(-i"/n,) f ... f P(H(xl)...H(x,))d4xl

...d4x,,

(1.3)

where P is the ordering operator with respect to x °. The transformed S-matrix contains the ordering operator P ' with respect to L(0, 0, 0, x °) (not x '° = (Lx)°), so that S is not invariant under L in the ease of (1.2). A possible way to obtain an invariant S-matrix in the case of (1.2) is to introduce a constant vector N in x-space and define

S = y , ( - i " / n , ) f . . . f PN(H(xl). . . H(x,))d4xl . . . d4x,,

(1.4)

where Pu is the ordering operator with respect to N. x = N xx ~ + N 2 x 2 + N 3 x 3 - N Ox °. The expression (1.4) follows from the following assumptions. The state vector • ' is a function of a one-parameter family of planes aN(r) perpendicular to N. The parameter • is the component parallel to N o f the space-time displacement ¢ of the observer (apparatus), i.e. v = N - ¢. The function ~[aN(z)] obeys

~(0/0~)~, = ~U. 0 ~ ' = (N" e)~,,

(1.5)

where the total energy momentum P , is given by Pu

=

-p (, o ) _ p- ~' ul )

,

p~l)

=

N,fo,o,H(x)da,

(1.6)

where do" is the (invariant) surface element of the constant plane o"NO) and p~O) is the free part ofP~. In the case of (1.4) or (1.5), the theory is invariant under Lorentz transformations, but it does not satisfy, in the case of (1.2), the postulate that there exist no special inertial system. The violation occurs in phenomena in which the energy o f the participating particle is higher than a certain critical value. F r o m this violation, it is possible, in principle, to find the direction of N, if it exists. The constant vector N is defined when the theory starts from the definition of the momentum degree o f freedom, as follows. We can define N by assuming that the momentum satisfies p2 = _ m 2 (1.7)

•~ 2 0

P.R.

P A U L A E SILVA A N D T . T A T I

and is distributed with the density, e.g.,

dp = #Q(d3p/po) exp [ - 2 2 ( N . p ) 2 ]

(1.8)

in a Minkowskian space. In (1.8), Q is a universal constant and/z and 2 are characteristic positive constants like the mass m. (This may be interperted as a momentum distribution with momentum dependent quantizafion volume Q(#/po)exp [ - 2 2 (N. p)2].) The density (1.8) coincides with the usual one in the limit 2 ~ 0. We can define N also by assuming various functions for p but a decreasing function o f p such as (1.8) gives concurrently a theory with finite degree of freedom 3). We assume the Lorentz invariance in momentum space and the introduction of the spin degree of freedom. We can define creation and annihilation operators a*(p, r), b*(p, r), a(p, r) and b(p, r) and also observables, like P , , given as Hermitian operators consisting of creation and annihilation operators without referring to a Lagrangian or to field equations. In the theory outlined above, certain unusual quantizations are possible. In the next section, we shall examine an example of an unusual particle, i.e., one that satisfies the exclusion principle (maximum occupation number = 1), but for which, nevertheless, the state vector is symmetric with respect to the interchange of particles. We shall call this kind of particle b-particle. In the case of spin ½, such a bparticle can interact with photons only in very unusual ways, so that the existence of a charged b-particle of spin ½ is unlikely. In particular, the/z-meson is excluded from being a b-particle by its anomalous magnetic moment 4) and pair production 5). But it is not excluded, at present, that n or K be a b-particle. 2. Example of an Unusual Quantization

We shall discuss an example of unusual particles in which creation and annihilation operators satisfy

[a(p, r), a*(p, r)]+ = 1, a(p~ r)a(p, r) = a*(p, r)a*(p, r) = O, [a(p, r), a*(p', r')]_

=

[a(p, r)a(p', r')]_

(2.1)

= [a*(p, r), a*(p', r')]_ = 0, for p ~ p', or r ~ r', and similar relations for b and b*. These operators are defined as linear operators in a Hilbert space spanned by basis vectors 7t[.W"] specified by a set of occupation numbers N(p, r) = 0, 1 for allp and r, = {N(p, r)} in the following way (from (1.7), the degree of freedom (iv, r) can be a finite set, so that the dimension of Hilbert space is countable):

a(p, r)Tt[.A/'] = 0,

when

N(p, r) = 0,

a(p, r)~VV'] = ~gI-JV"], when

N(p, r) = 1,

(2.2)

UNUSUAL STATISTICS OF ELEMENTARY PARTICLES

where ,4:' = {N'(p', r')} is equal to ~ a*(p, r ) T [ ~ r] = 0,

52"t

except that N'(p, r) = N(p, !")--1; when

N(p, r) = 1,

a*(p, r)Tt[J/'] = gt[sU"], when N(p, r) = 0,

(2.2)

where ~4"" = {N"(p", r")} is equal to Jg" except that N"(p, r) = N(p, r ) + l . The operators b and b* are defined similarly. Since, for arbitrary 7t. a*(p, r)a*(p, r)~ = b*(p, r)b*(p, r)7/ = 0, it satisfies the exclusion principle (the maximum occupation number is 1). On the other hand, when p ¢ p' or r ~ r', a*(.p, r)a*(p', r')7 t = a*(p', r')a*(p, r)7~ and b*(p, r)b*(p', r ' ) ~ = b*(p', r')b*(p, r)T, so that the state vector is symmetric with respect to the interchange of two particles of p ~ p" or r ¢ r'. We shall call such a particle a b-particle. By the theory mentioned in § 1, the interaction between b-particles and photons is determined when the creation and annihilation operators c*(k, v) and c(k, v) for photons are introduced, and the momentum operators P~ in (1.5) are given. We denote the momentum of a photon by k, with k 2 = 0, and the spin (polarization) functions by w~(k), v = 1, 2, 3, 0. The operators c and c* are defined by

[c(k, v), c*(k, v)]_ = 1,

(2.3)

while the other commutators vanish. We assume that in (1.6)

p~O) = 2,,~,p,(a*(p, r)a(p, r)+ b*(p, r)b(p, r))+ T.kS, kue*(k, v)e(k, v),

(2.4)

and H(x) is a function of ~b(x) and A(x) defined by

c~p(x) =- [(21r)3#Q]-~Z,,r,,(a(p, r)u#e 'p'x + b*(p, r)v#e-'P'x), A~(x) =-- [2(2~)31z~,Q]-~.Sk.Sv(c(k, v)w,,e ~, ,k.x + c *(k, v)w~, " *" e - u , . , , ,).

(2.5)

(From (1.8), ~p and ~k can be replaced by (d.tQ)S(dap/po)exp [ - 2 2 ( N " p)2] and (Yr Q)S(dak/ko) exp [ - 22(N • k)2], respectively.) The subsidiary conditions

2,,k~'c*(k, v)w**(k)~ = O,

2,~U'c(k, v)w;(k)k~ = 0

(2.6)

are imposed. We shah first consider the b-particle with spin ½. Then, u and v in (2.5) are solutions of (i7 .p+m)u = 0 and (/y . p - m ) v = 0, respectively. When H(x) is given by

H(x) = j"A,.

j" = ie~?&p.

(~ = q~,yo).

(2.7)

we have

f~

j°dtr = eZ,,Zt(a*(p, r)a(p, r)-b*(p, r)b(p, r ) + 1}

(2.8)

•(0)

in the limit 2 ~ 0, because of [b(p, r), b*(p, r)]+ = 1. But the particle and antiparticle would be found, in deflection experiments in an electromagnetic field, to have charges of the same sign, because of [b(p, r), b*(p', r')]_ = 0 for p ¢ p'. Then, in

~

P. R. PAULA E SILVA AND T. TAT[

this case, b-particles with different charges should belong to different kinds, even though the mass is the same. The electromagnetic anomalous magnetic moment of this particle is very different from that of the usual spin ½ particle. From (1.4) with N = (0, 0, 0, 1), we get (-0.6)(~/2n)/to(Ct = e2/4rt,/'o = e/2m). When we assume

jr, = (ie/2){~o~A4) + (A~j~,l,~b},

A = iy. p/m,

(2.9)

(A~ = [(2rc)a #Q]-}~,~,(a(p, r)u'e ' ' ~ - b*(p, r)v" e - " "~)), the particle and anti-particle are found to have opposite charges. But we have no terms for pair creation or annihilation (a'b*' or ab'). (Also, in this case, S,~(0)j"da cannot be interpreted as the total charge-current.) Thus, the existence of a charged b-particle with spin ½ is unlikely. In particular, the/z-meson cannot be a b-particle, owing to its anomalous magnetic moment 4) and pair creation 5). lm the case of spin 0, u and v in (2.5) are equal to 1. We assume that

H(x) = j~A~ + e2~*c~A z,

j" = ie{49"~'4~ - (a~b*)ff}.

(2.10)

The commutation relation of q~(x) is

~" (x-*')) -2SpS,(a*(p, r)a(p, r)e'~'~x-")-b*(p, r)h(p, r)e-'P ~-x'~)}. (2.1,)

[~b(x), ~b*(x')] _ = [(2rt) a/tQ] - l{27p(e ,p-(x- x,) _ e-

Although the expectation value ([¢(x), ~b*(x')]_) is equal to the usual A-function in the limit Q -+ m and 2 -+ 0, [¢(x), ¢*(x')]_ and [H(x), H(x')]_ are not c-numbers. And, S~N(0)jt'da is not the usual current. The S-matrix in (1.4) is, however, defined uniquely. The ftmction (01PN(~(x),

~*(x'))10>

i(2~)

, £d,s[s, + ms]-I s = (p,, P2, Pa, So),

exp [ - 22(N • p)2 + is(x

-

x')],

(2.12)

coincides with the usual Ap-fUnction in the limit 2-+ 0. Thus, when we take 2 sufficiently small ( < 0.1 fro), the S-matrix (1.4) with (2.10) can explain the electromagnetic interaction of ~ or K. The other available experimental data cannot exclude either the possibility of ~ or K being a b-particle. This possibility seems interesting from the view point of a compound model in which n and K consist of two fermions 6). We can also define a particle whose maximum occupation number Armis an arbitrary positive integer. 111 the scheme of sect. 1, a photon with finite Nm is also possible. The experiments on black-body radiation imply Arm >> 1 but not necessarily N= = m. We are grateful to Professors Mario SchSnberg, Mituo Taketani and Daisuke It8 for their useful discussions.

UNUSUAL STATISTICS OF ELEMENTARY PARTICLES

References 1) B. S. Thomas and W. G. HoUaday, Phys. Rev. 110 (1958) 981 2) S. D. Drell, Communication for the Kiev Conferonco (1959) 3) T. Tati, Prog. Th¢or. Phys..?4 (1960); R. Koeberl¢ and T. Tati, to be published 4) G. Charpak et al, Phys. Rev. 6 (1961) 6 5) G. E. Masek, A. J. Lazarus and W. K. H. Panofsky, Phys. Rev. 103 (1956) 374 6) E. Fermi and C. N. Yang, Phys. Rev. 76 (1949) 1739; S. Sakata, Prog. Theor. Phys. 16 (1956) 686

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