PH YSICA L REVIEW D VOLUME 16, NUMBER 6

Extended Yang-Mills formalism

15 SEPTEMBER 1977

Gerry McKeon~Department of Physics, University of Toronto, Toronto M5S 1A7, Ontario, Canada

(Received 12 November 1975; revised manuscript received 7'March 1977)

In analogy with the extension of electromagnetism by the inclusion of the Dirac monopole, the possibilityof extending the Yang-Mills formalism to include two separate types of self-coupling for the vector field is

examined. A local Lagrangian which lacks manifest rotational invariance is presented.

I. INTRODUCTION p~p+VQ (5b)

In two classic papers, Dirac' has postulated theexistence of magnetic monopoles. The existenceof monopoles would provide a symmetry betweenthe electric and magnetic fields in Maxwell's the-ory by replacing the equations

uFuv @v

The arbitrariness in gauge. of Q and p would ap-pear to give both P and p two transverse degreesof freedom. However, owing to Eqs. (3c) and (3d),only two of these four degrees of freedom survive.To see this, the following expansions of the trans-verse components of the potentials are useful:

and

by

(lb)

(2a)

1 ~ 1 ik ~ x-~z (~„~) ~ ffk

+fee ia'x)(6a)kX

1 ~ 1 ik ~ xP 4 ~ ~ (3~@)s/2 kk kx

k

a,nd

Fuv ~ ~v (2b)

V ~ H=O,

v E=O )

(3a,)

(3b)

Here Fuv denotes the electromagnetic field ten-sor and *Fu„ its dual, defined by *Fuv = —,'cu, ~ F~,.The electric and ma.gnetic currents are given byj„and k„, respectively (the notation used is thatof Lurie').

It should be empha, sized that the introduction ofmagnetic source terms in Maxwell's equationsstill leaves the electromagnetic field with two de-grees of freedom. To illustrate this, .first of allconsider the free-field equations,

fT; =~T.. .f k2=-&Ti ~

(8a}

(6b)

This indicates that only two degrees of freedomexist for the electromagnetic field.

Generalizing this approach to a four-dimen-sional notation means that the solutions to theequations

+ yt e "' ") . (6b)

kX

The quantities ek), are the standard transversepolarization vectors, and satisfy the relation

eg, &&eg, = k j~k~ .Upon substituting Eqs. (6a) and (6b) into Eqs. (3c)and (3d), we obtain

a,E =VxH,

~oH=-VxE .(3c)

(3d} and

'uFu. = o

H=H~=Vxy,

E=E =-VX p .(4a)

(4b)

By examining Eqs. (3a) and (3b), we see that H

and E are transverse vector fields that can be ex-pressed in the form

au*Fu„= O

are written in the form

Fu. = ~u4"- 8. 4'u

and(9a)

(5a)

The electromagnetic field is unaltered by the ad-dition of the gradient of a scalar function to thevector potentials,

+VA )

Fuv upv vpu (9b)

4'u-4u+ ~u A (10a)

The functions Qu and pu are invariant under thegauge transf ormations

16 1836

16 E X TEN DED YANG- MI L LS FOR MA LIS M 1837

alld

P'u-P'v + (101)

The identity

G„„=—([n n (n G)]„„—*[n ~ (n ~G)]„„3 (17)1

By coupling the electric current to the electro-magnetic field in such a way that the gauge in-variance of Eq. (10a) is preserved, ordinary elec-trodynamics is derived. Magnetic currents arecoupled to the electromagnetic field in such a waythat the gauge invariance of Eq. (9b) is maintained.However, Eqs. (9a) and (9b) must then be altered,as Eq. (9a) is clearly not compatible with Eq.(2b) and Eq. (9b} is not compatible with Eq. (2a).This situation can be remedied by defining

$'~ „——8~ P„—8„$„+*G~„with

F„„=—,((n A [n ~ (8 n A) ]3))„1

and

—*(n~[n ~ (8 ~B)L.) (16a)

'P„„=—,(*(nr [n ~ (»A)]3„„1

that holds for any antisymmetric matrix G„„allows us to reexpress Eqs. (15) and (16) in theform

G (*) "uf »-»' I'«. ».(»') - «.».(»')Ic

+(n~[n (a~B)L„) .The equations of motion now become

(19b)

x 6'(» - »' - ~.)and taking the Lagrangian to be'

1—~matt ~+gv+P +O'PJP ~

Alternatively, we could take

+Pv )f~v ~vt P+ +Pv

with

)»„„(») »f»'»' =(«,j.l»') —& .j,(*'))c

x 6'(»-»' —~,)

(12) —(n an aA —n aa n A n. n-aa A1n'

+n" 8'n A —n ae""" n"a"B )= ej" -(19a}

—,(n ~ an aB"—n aa n B—nn 88 B1n'

+n" a'n B+n ~ ae""" n"8"A )=-p, k'. (19b)

A suitable Lagrangian from which these equationscan be derived is

g, =g „,, + —Tr[(8 w A) ~ (8 wA)+ (8 AB) (8 &B}]

1—(e„p, - e Ii „}= IV,4n

(14)

where N is any integer. In deriving Eq. (14), Cwas taken to have both ends at infinity and to passthrough the origin. Originally, Dirac obtained Eq.(14) by considering C to have one end at the originand the other at infinity; this allowed X to take onboth integral and half-integral values.

Zwanziger has examined magnetic monopoles byuse of a local Lagrangian. ' The solution to Eq.(2a) is written [we define (P & Q)„„=P„Q„-P„Q„]-

"E„„=(8 n B)„„—e(n 8) '*(nl(j)„„, (15)

where n is an arbitrary fixed vector. Similarly,the solution to Eq. (2b) is written

E„„=( A8)„„+(iin )8' (n ~k.)„„. (16)

1~matter 4 +P v ~P v+ PPP ~P

The line integral C in Eq. (12) destroys the mani-fest rotational invariance of the theory, but uponquantization full rotational invariance is restoredto the theory by restricting the charges on twobodies m and n to the values"

, (n [(8 ~A)+*(a h B)]3'4n'

, (n [(a ~B) —*(8nA}]3'+ej A+ p, k B.4n'

(20)

Despite the appearance of the vector n" in theLagrangian, full rotational invariance can be re-stored by the charge quantization condition of Eq.(14).

The Yang-Mills field has the properties of avector field that itself possesses charge. The pur-pose of this note is to extend the Yang-Millsformalism so that there will be two self-couplingsfor the vector field, analogous to the electric andmagnetic couplings of the electric field. This isto be contrasted to the approach of 't Hooft,"whointroduced magnetic monopoles into gauge theoriesby exhibiting special solutions to the classicalfield equations for a spontaneously broken gaugetheory that had the features of genuine magneticmonopoles, upon identifying the electromagneticgroup as a U(1) subgroup of a larger compactcovering group.

1838 GERR Y McKEON 16

II. YANG-MILLS FORMALISM

The standard Yang-Mills' formalism has a Lag-rangian Aa + caPbAP gb (2'la)

dex a. If we now require the Lagrangian to be in-variant under the transformations

~ = ~matter af ttvf ttv+ gjtt4 tt (21) and

where

(22)

with c"'being the structure functions of the groupof which A'„ forms a representation. The equationof motion resulting from this Lagrangian is

(5aba +gcabb yb )f b —Vabf b

and8(avAb)

(28a)

Bu Bc+c'~ B (27b)

with (9 an infinitesimal constant, then the currentson the right-hand side of Eqs. (26a} and (26b) ac-quire contributions

gjv ~

~ a (23) k= c"a(a„Bo) ~ . (28b)

The entire formalism is invariant under the in-finitesimal gauge transformation By use of the Lagrangian of Eq. (20}, we see that

pa pa ~ Vab8b

Finally, the following equation holds,

vabaf b 0

(24}

(25)

j'„=,c' a(2{ntt[n (8&Ab)]„—n„[n (a ~A')]„j1

+{nv [n '*(8nBb}]„—n, [n '*(a~ Bb)]tt]+ e„„&„[n (8 n Bb)] ~n )A„'

g~ 11

Fg & vk&

(26a)

(26b)

with Fv „and *F'„„define das in Eqs. (17) and (18),except the fields A„and B„now have a group in-

The analogy between Eqs. (la), (1b) and Eqs. (23),(25) is obvious. To extend the Yang-Mills formal-ism to a set of equations corresponding to Eqs.(2a) and (2b) we shall initially exploit the approachof Zwanziger.

First of all, consider the equations of motion

and

jbV—

a Cava(2{n ~ (8 ~ B')]„-n„[n ~ (a ~ B')]„j1

—{n„[n~ *(a &Ab)]„—n„[n *(a~A')]„j

+e„„b„[n (8~A )]),n, )B„a.

Inclusion of these terms in the equations ofmotion leads to the following additional terms inthe Lagrangian,

c"'i,"=, [gA.'Att(2{n [n (8&A )]tt —ntt[n (8&A )] j+{n [n *(8&B)]tt—ntt[n *( &8B ]b}j

f ay [n (»t B')]~n, )+oB' Btt(2{n [n (8 «B')]tt —ntt[n (8 +B )] j—{n [n a~(» A')]8—ntt[n *(8~A')] j+e a~, [n ~ (8~Ab)]~n, )] . (28)

These terms also contain derivatives of the fields, and so, by Eqs. (28a) and (28b), there will be addi-tional contributions from 2' to j'„and k'„. Upon inclusion of these terms into the equations of motion by aprocedure analogous to the one above, we find that the total Lagrangian is given by

', Tr[(a r A+gA -xA) (8 itA+gA xA~)+ (8 B+cBxB) ~ (8~B+oBxB)]1 1, {n [(8 &A + gA xA) + "(a~B+&rBx B)]j ' ——,{n ~ [(8 r B+tJBx B) —*(a i A + gA xA) ]j '

4n''

4n

+gj.A+ok. 8 (30)

where A xA denotes c'~'A~~A„'. There are no addi-tional terms in this Lagrangian. It is invariantunder the infinitesimal transformations

A'„(x) -A '„(x)+ [8„8"+gc'bbA bv (x)]8'(x)/g

=-A '„(x)+ „v '„'8'(x)/g (31a)

and

B'„(x)-B'„(x)+[8„8"+(rc"bBbv(x)) 8b(x)/g

= B„(x)+,V „8 (x}b/ob. (31b)

From Eq. (30) we can derive the equations of

EXTENDED YANG-MILLS FORMA LISM

motion

A+lf +Ps g~v

upon varying A~ (x) and 8'„(x). Here we havetaken

F = —,({n~[n (s~x+gWxW)]}

-*{nrfn (s wg+g+XQ)]}) . (33)

III. DISCUSSION

The program outlined here is obviously incom-plete. First of all, it is not clear whether thereare conditions analogous to Eq. (14) under whichthe theory can have rotational invariance in spite

of the appearance of the fixed vector n in the I ag-rangian of Eg. (30). Secondly, it is not evidenthow this theory is related to the standard Yang-Mills formalism; even in the limit cr 0 the I ag-rangian of Eq. (30) does not reduce to the standardYang-Mills formahsm. A third problem that re-mains is that there does not seem to be any mean-ingful way of making even a perturbation theorycalculation of matrix elements.

Despite the considerable problems that remainto be solved, it could be worthwhile to entertainsome speculations about use of this theory. Forexample, it might be feasible to relate one of thethoro couplings of the vector field to matter witheither the weak or strong interaction. It may thenbe possible to explain CP violation in the K~ —K~complex by use of the fact that the vector fieldsA.„and B„have different transformation propertiesunder CI.

*Present address: College of Physical Science, De-partment of Math and Statistics, University of Guelph,Guelph, Ontaxio, Canada.P. A. M. Dirac, Proc. B. Soc. London A133, 60 (1931);Phys. Bev. 74, 817 g.948).

~D. Lurie, Particles and Ilields gnterscience, NewYork, 1968).

3J. Schwinger, Phys. Bev. 144, 1087 (1966); 173, 1536g.968) .

D. Zwanziger, Phys. Bev. D 3, 880 (1971).G. 't Hooft, Nucl. Phys. 979, 276 (1974).

68. Julia and A. Zee, Phys. Bev. D11, 2227 (1975).YC. N. Yang and B. L. Mills, Phys. Bev. 96, 191 (1954).