Locally gauge-invariant formulation of parastatistics

Nuclear Physics B219 (1983) 358-366 ~) North-Holland Publishing Company

LOCALLY GAUGE-INVARIANT F O R M U L A T I O N OF PARASTATISTICS* O.W. GREENBERG and K.I. MACRAE

Centerfor Theoretical Physics, Department of Physics and Astronomy, Universi(v of Mao'land, College Park, Maryland 20742, USA Received 20 January 1983

A locally gauge-invariant formulation of parastatistics, which is equivalent to a Yang-Mills gauge theory, is given, using a complex Clifford algebra (case of SU(N)) or a real Clifford algebra (case of SO(N)). In particular, for the SU(3) case, the gauged theory of para-Fermi quarks is equivalent to quantum chromodynamics.

1. Introduction

The color degree of freedom of hadronic physics was first introduced [1] in the context of para-Fermi statistics of order three for quarks. Bose and Fermi combinations of para-Fermi quarks are in one-to-one correspondence with the color singlets of the formulation with explicit color [2], and thus the counting of states, and the explanation of the apparent conflict with the spin-statistics theorem with quarks in the symmetric representation of SU(6) [3] are in agreement with the explicit color formulation, as is the "symmetric quark model" for baryons [1,4], which was first proposed in the context of the para-Fermi formulation. Other predictions of the two formulations which agree include the decay rate for ~r° to two photons, and, at least from the standpoint of naive counting, the ratio of the cross sections of e +e- to hadrons to that to /~+/~-. The gauge theory of color, quantum chromodynamics (QCD) differs, however, from the ungauged parastatistics formulation in predictions involving gluons, such as quark and gluon jets and the existence of glueballs. We show in this article that parastatistics can be gauged and that, when gauged, parastatistics is equivalent to the corresponding Yang-Mills gauge theory, in particular, for the case of para-Fermi quarks of order three, to QCD. To make this article self-contained, we include a brief review of parastatistics here. Motivated by the fact that the commutation relations between the number operators and the creation (and annihilation) operators have the same form for both the Bose * This work was supported in part by the National Science Foundation. 358

O. W. Greenberg, K. 1. Macrae / Parastatistics

359

and Fermi cases [nk,a~]

=6,;a~,

(1)

where

nk=½[a~,ak]++const.,

[A,B]+=AB++_BA,

(2)

and here and throughout this article upper (lower) signs stand for the para-Bose (para-Fermi) case, Green [5] generalized (1) to allow all three indices to differ, and considered the following trilinear commutation relations [[a*k,a,]_+,am] = -23kma;,

(3)

[[a,, a,]+, am]_= 0.

(4)

We will only consider the analog of the Fock representation of the Bose and Fermi cases, where there is a no-particle state which satisfies a,{O> =

o.

(5)

Green found solutions of these commutation relations using "Green's" ansatz: Let P

a, = Y'. a~,") ,

(6)

a=l P

~**= E ~,"(~*,

(7)

a=l

where for equal values of a (the "Green index") the operators obey the usual commutation or anticommutation relations, but for different values of the Green index, the operators have abnormal relative commutation relations, [at "), aS~'*] q:= 3,,,

[a(~', a}B'*]+= O,

(8)

a# =ft.

(9)

Then the expression inside the nested brackets of (3) and (4) has the form

[a**, a,] + = E Z [4-,', a~,] ~ =E[~y,
(,0)

ot

and it is clear that (3) and (4) are satisfied. The number of values over which the

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Green ansatz runs defines the order of the parastatistics. The order can assume any positive integer value. The physical interpretation of parastatistics of order N is that for para-bosons (para-fermions) at most N particles can be in an antisymmetric (symmetric) state, while any number of particles can be in a symmetric (antisymmetric) state. Greenberg and Messiah [6] showed that all solutions of the trilinear commutation relations (3) and (4) are associated with a positive integer order N, so that Green's ansatz exhausts the independent solutions. As we will see below, this does not mean that solutions equivalent to those of Green cannot be represented in a different way. An important application of para-Fermi fields of order three was made by showing that such fields, which contain a new three-valued degree of freedom, color, allow three quarks to be in a state which is symmetric in the other degrees of freedom. Consider

=

4

"/i

"/j

~'/k

(ll)

,

abe, all

different

where the subscripts i, j, k each stand for the space, spin and flavor quantum numbers associated with a single quark. B t is a Fermi creation operator for a baryon, and is symmetric under permutations of i, j, k. The creation operator for mesons is

M,? =

--

2 ~-,qi

qj ,o,'.

(12)

~t

Further references about parastatistics appear in [7]. There are two apparent problems which impede making Green's formulation of parastatistics into a local gauge theory: (i) neither the commutator nor the anticommutator of the para-Fermi fields has the correct number of components to couple to an S U ( N ) or SO(N) gauge field [8], and (ii) a unitary transformation of the Green components is not again a Green component, because of the scrambling of the commutation and relative commutation relations. The fact that there is an exception to (i) for SO(3) was pointed out by one of us (OWG) and was reported in [8]. The physically interesting case of SU(3) was gauged in the context of octonionic field theory in ref. [9].

o. w. Greenberg, K. I. Macrae / Parastatistics

361

2. Local gauge theory Usually, every index summed over occurs twice (one up, one down) rather than once. This suggests the introduction of basis elements, so that the parafield has the form =

(13)

In later formulas we will not exhibit the summation sign; we will use the convention that repeated indices are summed over, but with one exception: when we write the formula ~p(~)= e%p~(x) we do not sum. Here, we will take the basis elements to be independent of space-time. Below, we will discuss a formulation, motivated by ideas of differential geometry, in which the basis elements depend on space-time. The ~b(~) will be Green components provided the basis elements anticommute for unequal values of their index. For the parastatistics version of S U ( N ) gauge theory, we choose the basis elements to form a complex Clifford algebra, with the anticommutation relations [e~,e*B]+=26#I,

[ e ~ , e # ] + = 0;

(14)

and for the S O ( N ) gauge theory, we choose a real Clifford algebra, with the anticommutation relations [e ~, e#]+ = 26~BI.

(15)

The unit elements on the right-hand sides of (14) and (15) are the n × n unit matrices in the representation space of the basis elements. For our detailed discussion, we will consider the S U ( N ) case. (The changes necessary for the S O ( N ) case are straightforward.) We take all the fields 6 , to be Fermi fields with normal (Fermi) relative anticommutation relations. (For para-Bose fields, we take all the corresponding fields to be Bose fields with normal (Bose) relative commutation relations.) We also take the fields to commute with the basis elements. Now a unitary transformation of the 6,, is again a Fermi field, and a unitary transformation of the basis elements is again a Clifford algebra, so that we have overcome objection (ii) above. With our construction, the Green trilinear commutation relations must be altered slightly. The analog of (3) becomes

[[Vfl(x),V~(y)]-([Vfl(x),VJ(y)]->o,~P(z)] =-26'(x-z)+(y), (17) at equal times. The symbol ( )0 stands for the vacuum expectation value. The analog of (4), again taken at equal times, has the same form

[[+(x), ,(y)]

, ,(z)]

= 0.

(18)

To generate the gauge interaction, we follow the procedure of Yang and Mills, and

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O. W. Greenberg, K. 1. Macrae / Parastatistics

consider space-time dependent gauge transformations U( x ) + (x) U*( x ), where U(x) = exp{/4nnltrf [ ~ p * ( x ) ' [ t ( x ) ' + ( x ) ] - ]

- d3x}

=exp( if~t(x)tJ3(x)~a(x)d3x},

t( x ) = e~t,t~( x )e~,

ta l~ = ,l~~* ,

(19)

t, ~ = O.

(20)

? = +U(x) = e ~ ( V ~ ) , , ( x ) ,

(21)

We find

V(x)+(x)U(x) where

( U + ) ~ ( x ) = UJ3(x)~13(x),

UJ3(x) = ( e x p i t ) ~ ( x ) .

(22)

The kinetic term in the lagrangian for the Dirac field + is ½(i[t~, 0~b] i([~7, 0+]_ )0). Under a gauge transformation the kinetic term acquires the following non-invariant term due to the space-time dependence of U,

i:

auo

This non-invariant term can be canceled, as usual, by the introduction of the gauge field associated with changing the derivative to the gauge-covariant derivative

iO~aS--+ iO,aJ - gA,.S,

where A ~ J

~xa#a~

(23)

and requiring A ~ # to transform as

1 A,,J 3 ~ U~YA~( U* ) 8~ + -~g ( O~U ) ~Uy~.

(24)

The trilinear commutation relations for the gauge field are,

[tr2[A,(x),A~(y)]+-(same)o,

Ax(z)}=2ig~x*3(x-z)A~(y),

(25)

at equal times; the analogous trilinear commutators with only fields or with only time derivatives of fields vanish. What remains to be shown is that the interaction term can be written in terms of parafields. Let

A~ = ½e~A~e~,

A~J~*= A ~ ~ ,

A ~ ~ = 0.

(26)

O. W. Greenberg, K. I. Macrae / Parastatistics

363

Then the interaction term is ~ntr([¢,[Au,~b]_]

- ( s a m e ) o ) ='r'~.4'54"~:.~b/~

(27)

The pure gauge term in the lagrangian is -

1 4-n tr

G.~G~,

(28)

where

c,,= a,A,- o.A, +;g[A,. a.]_ = le"[Ot, A~,B- O.A~,al~+ ig(Aj,,,aA~al~-A.~,aA~,a~)Je~ -- ~ e

(G..)~

e B.

(29)

We have now shown that the entire lagrangian of an SU(N) Yang-Mills field interacting with a spinor field can be expressed in terms of parafields, as follows: £=

tr[ f , ib~P ] - Tnntr(G~G uÈ),

(30)

where

D =½e~( 61~8 +igA~B)e~.

(31)

A similar construction can be given if scalar fields are also present. Parastatistics can also be gauged in the functional integral approach. Replace the fermionic coefficients ~Pa by standard Grassmann elements, and the bosons by standard c-numbers. This implicitly defines a para-Grassmann algebra and a parascalar algebra. The brackets of these algebras are the same as those in (8) except that the 6k~ should be replaced by zero. Now we point out a way in which the parafield description of Yang-Mills theory differs from the usual description. Since the basis elements are nilpotent, the ( N + 1)st power of the spinor field vanishes identically. This problem can be avoided as far as SU(N) singlet states are concerned as follows: introduce operators for the "baryons" and "mesons" b ( x l , x2 . . . . XN ) = [~P(XN), [~P(XN

I),[---[tP(X2), ~P(XI)]+-'-]+]+]+

= [e"N,[... [e'~2, e"']_]+...](),,+,~p<,.(XN)...~pa2(X2)~p,,(X,) = 2N

leaN...,,a2e'~,e~u-'''~,a~,r, ¢~. ~ - v,~,,~U)... ~p,~,(Xz)~p~,(X~),

(32)

where the ( - ) N + ! in the second line indicates that the last bracket is a commutator

364

O. W. Greenberg, K. I. Macrae / Parastatistics

for N even and an anticommutator for N odd. Let

(33)

E = e'e2.., e N ,

be the element analogous to 3'5 in the usual Dirac algebra. Then ....

1

2 N- 'n t r ( E b ( x , ,

x 2 .... x u ) ) (34)

where the e tensor is antisymmetric and 8N 2,~= 1, is a field for a color-singlet baryon, and is free of basis elements. For mesons, the analogous construction is simpler. The operator . . . . .

M ( y , x) = ~ n tr[q~ ( y ) , ~b( x ) ] _ = " f ~ ( y ) ~ ( x ) :

(35)

serves as a field for a color-singlet meson. From the point of view of differential geometry, the elements e are a basis for the S U ( N ) space at each point, and, in general, will be space-time dependent. Then the gauge (or connection) field is the matrix which gives the change of the basis elements in a given direction. In general the basis elements at finitely separated points are related in a way which depends on the path taken in going between the two points. This is expressed by writing d e ~ ( x ) = iget~( x ) A~,/fl( x ) d x ~.

(36)

Then, the gradient of the spinor field q~= e"q~ contains two terms. One is the gauge (or connection) field and the other is the change of the components ~ , d~ = d(e'k),~) = e~'(0~,+,~) d x ~' + ige~A~.~t),~ d x ~

= e " ( D ~ , + ) , d x ~,

(37)

where (DJ~)~ is the usual gauge-covariant derivative. This motivates the notation dJ~ = e ~ ( D , } ) ~ . The covariantly coupled kinetic term for the spinor field is

tr[,L

_.

(38)

In a similar way, the commutator of the d r s generates the gauge field tensor (or curvature),

[d~,, d.] e '~ = ige~(G~,~)fl",

(39)

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365

and the gauge field term in the lagrangian is _

1

4ng2

t r [ d ,d~]

t ~

(40)

e~'[,

This completes our demonstration that parastatistics can be gauged to give S U ( N ) gauge theory. One of us (OWG) acknowledges helpful discussions with S.J. Brodsky, P.G.O. Freund, F. Gtirsey, K. Johnson, M. Ogilvie and C.H. Woo.

Note added After this work was completed, we received three D u b n a preprints, see ref. [10], dealing with the gauging of parastatistics. Although there is some similarity between the work of Govorkov and our present article, Govorkov concludes that it is not possible to gauge parastatistics in such a way as to get a theory equivalent to QCD. As shown above, we conclude the opposite: the gauged theory of order three para-Fermi fields is equivalent to QCD. We thank Dr. Govorkov for making his results available to us prior to publication.

Appendix CONSTRUCTION OF BASIS ELEMENTS For the S U ( N ) case, let

e'=~(o~+iov)XlXlXlx...

X1X1,

e 2 = a z X ~ (1 o ,:+ i o v ) X 1 X I X . - .

X1X1,

e3=o~Xo:X~f22(o~+ir~r)XlX...X l X l , e N - ' = . z x oz x o. x ~ x . - .

x f~ (ox + i ~ y ) x 1,

eN=o=×oz×o,xox×-..×o.×~(~x+io,), ( N factors),

where the × 's stand for tensor products. The dimension is n = 2 x.

(A. 1)

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O. W. Greenberg, K.I. Macrae / Parastatistics

For the SO(N) case, let el=ox×lX1X..-Xl, e2=oyX

1X 1 × -.- Xl,

e3=ozXOx

X 1 X ...

X 1,

1 X ..-

X 1,

e4=ozXOyX •

.

o

e 2m-1 = O z X O z X ~ z

x ...

XOx,

e2m = oz X Oz X oz X . . .

X ov,

(m

factors).

(A.2)

F o r e v e n N , N = 2 m ; for o d d N , o m i t e 2" a n d t a k e N = 2 m - 1. T h e d i m e n s i o n n = 2 m.

is

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

O.W. Greenberg, Phys. Rev. Lett. 13 (1964) 598 M.Y. Han and Y. Nambu, Phys. Rev. 139 (1965) B1006 F. Giirsey and L.A. Radicati, Phys. Rev. Lett. 13 (1964) 173 O.W. Greenberg and M. Resnikoff, Phys. Rev. 163 (1967) 1844 H.S. Green, Phys. Rev. 90 (1953) 270 O.W. Greenberg and A.M.L Messiah, Phys. Rev. 138 (1965) B1155 O.W. Greenberg and C.A. Nelson, Phys. Reports 32 (1977) 69 P.G.O. Freund, Phys. Rev. D13 (1976) 2322 M. Giinaydin and F. Giirsey, Phys. Rev. D9 (1974) 3387, F. Gi.irsey, in Proc. of the Johns Hopkins Workshop on current problems in high energy particle theory, 1974, ed. by G. Domokos and S. Kgvesi-Domokos (Johns Hopkins Univ. Press, Baltimore, 1974) p. 15 [10] A.B. Govorkov, preprints Dubna P2-81-749: P2-82-296; E-2-82-470