IC/82/113

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON TEE CLASS OF SIMPLE SOLUTIONS OF THE SU(2) YANG-MILLS EQUATIONS

Piotr Raczka Jr.

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1982MIRAMARE-TRIESTE

IC/82/113

1, Introduction

International Atomic Energy Agency There has been growing interest in the last years in and the classical Yang-Hills /YM/ theory, the hope being that United Nations Educational Scientific and Cultural Organization the understanding of classical theory will shed some light- 1 IHTERNATIOMAL CENTRE FOR THEORETICAL PHYSICS on the nonperturbative aspects of quantum gauge theory ( ).. However, despite the fundamentals role played by the YM equations, the number of discovered exact solutions of theae equations ia rather small. /(2Jj( )and references given there/. It also proved more difficult to solve these aquations in OH THE CIAES OF SIMPLE SOIUTIOKS OF THE SU(2) YAMG-MIUB EQUATIONS • ttlnkowski space than in the Euclidean space, and only very special solutions on Minkowski space are known. In this paper we want to discuss the class of solutions of ~iU(2j Piotr Raczka Jr. •• ¥M equations on Minkowski space, for which the potentials International Centre for Theoretical Physics, Trieste, Italy. depend on spacetime variables through the lorentz. scalar

H m Avx?t where A is a constant four—vector. The SU(2) YM equations in the "'matrix'" notation have the form:: ' ABSTRACT

= 0 We investigate the solutions of SU(2) Yang-Mills equations of the form A (i) = A (R), R = XvxV, Xv = constant four- vector. We prove, that for null I only embedded maxwelllan solutions exist. For spacelike and timelike X we reduce the problem to the solution of the system of two ordinary 1 1 where x^* = [x°,x jX ^'] r ^,v = O,L,2,3. Our metric differential equations for two unknown functions. Some is f+ ). Ay and P^v are functions on Minkowski explicit solutions are given. space with values in the Lie algebra of SUf2j , and £•, -J denotes the cxumnutator for this Lie algebra. Equations MIRAMAEE - TRIESTE (l.l) are invariant under the conforraal group and the July 1982 group of gauge transformations:

* To be submitted for publication. ** Permanent address: Department of Physics, Institute of Geophysics, Warsaw University, 02-093 Warsaw, ul. Pasteur T, Poland. where wfx) ia SU (2) valued function on Minkowski space.. -2- Taking in the Lie algebra at 6U(2) the basis of Pauli (1.3) matrices <£, =< = 1,2,3, we can d«fine the "Vector" notation = 0 for YM fields? The energy-moment-um tensor for Tnl iiUfz) field hag the form: i C

where

where E- := FD; , H; :=£ *;jk Fjk , i,j,k, = 1,2,3. V == solution of the equations fl.l) is called real, if all A^, are real, and nonabelian, when for some /<>*•,--ii?) ^v *• C : .Ve are looking for real, nonabelian solutions of (1.1/ of the form:

Such "triples A^ ,Pi,v can be treated as vectors in ordi- v nary three-dimensional euclidean "izospin space" with = A, r R =Avx , Xv = constant scalar product a*b = a b^ and vector product (S x b/W'^a b^. . Since we have Inserting (i.6) inta (l.3) we obtain:

d2A d A dH

then

The paper i3 organised as follows: in Section 2 we reduce potentials 0—&) *a one &£ three "canonical"1 forms, in = and the equation (l.l) may be rewritten as: Sections 3,4,5 we discuss separately the cases when -1 O X^^.KO • oection 6 contains some comments on the results. 1**9^ = o> In the Section 2 the '"matrix" notation is used,.and in Sections 3,4,5 we use the "vector" notation, but the '"hats" denoting vectors in the izospin space are omitted

for convenience. In Sections 3,4,5 we also denote A^ r= —^f or as a second order equation on A.

-it- -3- 2. Reduction of the problem "" - 2~~ w"

'A, - 'A3 = W (A,, - A3y ite shall prove, that by Lorentz. transformation, dilatation and gauge transformation any solution of the If W(R) satisfies 2~ = w (A„ + A.3 _) , then form (l*6) can be reduced to one of three following forms; 'A0 + 'Aij = 0-. Such W(R) always exists fM/at least a) when A1 =0, to the form for nonsingular AB, A^/. For R = x* we have: b) when A1 y 0, to the form c A^.(R) = A^(x ) , Aj, 31 0 (2.3-b) -A dw -1 -^ = 1,2,3 c) when A? K. 0, to the form A-(RJ = A (x3) , A- a 0 (2.1.c) f Taking W(R) auch, that ~ = wA0 , we obtain X^ SO. For R = x3we have: First we note, that every solution of the type (i.6.) we can transform by a Lorentz transformation to such, a 'A3 = wAjW~ - -- w" A . = wA_ w coordinate system, in which -i> would have one of the following forms: (a,0,0,a) , ^a,0,0,o) , ^0,0,0,a) Taking W(R) such, that -- = wA-j , we obtain 'ij2 0 a = const. Jince equations (l.l) are invariant under the Thus we completed the proof. It is easy to see, that in transformation Xp, = ax*. , A^ = —• A.^. ,, we can always the cases (2.1.a-c) the gauge is fixed up to the constant reduce.-A.vta: (l,OrO,V or (l,O,O,,aj) or (a,Qi,0,>l) ; gauge transformations /i.e. such that —=0/. then we have E = x° + x or R = x or R = x . dxt Further simplification comes from the choice of gauge.

If we want the transformation (1.2.) ta preserve the form 3. The case A2 = 0 (i.6) of the solution, we must have w(x) = W(R). For

R = x + x the potentials A^,(R) transform under the Inserting {2.Lay into (1.7/ we obtain: transformation fl.2) with wfx) = w (R) aa:

-A/ = 0 (3. l.a)

dw = 0

A, - (3.

-5- -6- ( - 2Ao - 2A^X A^ + A,, X ij + AZK A^ [2f,, f2 + 4fe f/J s * q + A A (3.3.c) + ^ * C , * o) + A2 < (A2 K Ae ) = 0 = 0

4 2 Adding A«l. a) to(3»l.

f„ (k) = C( R + p , where <%tp are arbitrary constants

f4fR) , f^r^ are arbitrary functions af R a = p = q = constant unit izaapin vector / Potentials A can be always represented as: Ta prove that, we take the scalar product of s and (R) , A, ^3.3-d) and, since s.fa JCS')= 0, we get: |[l- fs.qfj =Q where ta , f^, f^ are scalar functions of E and a,p,q. are unit vectors in the izospin space. Then we haves If © ia the angle between a and p, and «> is the angle between s and n, we have

= 0 (3.5) In this notation the equations f3.2.a) , ^3*1» f3.1.c) , (3.2.d) take the form: Equation (3-5) can be satisfied only when:

a) fos a To) f,a 0, q =f s , £ = 11 + fos" + 2fo's' -if^ = 0 C) faS 0, p =f S , £ =±1- (3-6)

[2 fj d) fHS 0, fa3 0

4f0 = 0 (3.3.b) e) P =£ s, q =ij a, £,*j = + 1

-T-

•;*• .-l^|i. -• Inserting each of these conditions into the system ($.$) (4.1. a) after simple algebraic manipulations we come to special cases of C3-4) . for illustration we give the details for (3.6.a) . Substituting (3. 6.a) into ("3. 3) we find:

f* pxp' q( = a p) = 0 A^' = 0 (4.1.0) (3.7) Por components of energy - momentum tensor we obtain;-

The last two relations can be satisfied only when fAs 0. or fjSO or pxq = 0. If f^= 0, then (3-7) reduces to

fjquq' = 0, so if we want fa^ 0, then must be q * q' =0, (4.2. a) i.e. q = const. iVe obtained:

fo= 0. , f,i 0 , fz arbitrary , q. = const. (4-2. b) If f,3 0 we obtain:

fo= 0 , fA arbitrary , p = const. (4.2. c) If pxq =0, then p =£q , s. = ±1. Substituting that in (3-7) we get only one equation: T« = ^'' (A3x A0+ Ai " (A J* A0 (4*2- d) (f? + .#)dxq' = 0, Since we want f^ Q) , f^ 0 ,, then must be q.x,qr = 0, The system (4-lJ has some interestins properties. First so q = const. Thus we have: let us prove, that by virtue of (4.1.d) we have;

fe= 0 , f,j and f2 arbitrary functions of R

p = i q , q = const. To(< = 0 k = 1.2.1 (4.3) which is a special case of (3.4^) . In the cases A.6.b-e) the calculations are identical. Indeed, taking scalar product of HA and (4.1.d) we have:

4. The case A2 >& By virtue of the vector identity Inserting (2.1.b) inta (1-?) we obtain: k ' fB X G ) = (A x. B j • C

we have: -9- -10- jystem ^4.l) can be formally regarded as a system describing the motion in three dimensional euclidean space of three material points at locations A., A , ij, U(A.) 'aeing the and thus T,,., = 0. The same is true for Ttf2 and T^j , 2 potential of their interactions, and subjected to the con- Next, we find that T#£) is a constant of motion for the system (4-l) : straints (4-1-d) /that the total "angular momentum" ia zero/. The simplest way to find the solutions of f4.l) is 5£2? o to look for solutions satisfying identically constraints dR =

It follows trivially from continiuty equation fox energy - momentum tensor:

where pfq.,u are constant unit vectors in the izospin space

and f., f2 , f-j are scalar functions of R. oulostituting Uince here everything depends on R = x° only, we have:: this into (4.1/ we obtain:

(4.7-a)

q = Another property of the system (4-l) is that /using (4.7.10 the language of classical mechanics/ the "forces1" on the f^ u = f*f3p«(p< uj + (4.7.c) right side of the equations ^4.1.a—c) are derivable from the potential, namely: Assuming that p, q, u are mutually orthogonal, we obtaint

d (4.5) lc = 1,2,3 where:

{A, - A,/ For •£.»! to:

Proof by straightforward differentiation , £"= - P

-li- -12- which ha3 "the solution: One more configuration can be given, for which (A- reduces to f4.ll) , namely: enf M"5R,1 (4.10) where cnf',-)is a Jacobi elliptic function with parameter A, = 0 l (4.12) — and M is positive integration constant. (4.1Ojis a where b,d are orthogonal constant unit vectors in izospin regular and periodic function of H. space and tA , f^ satisfy ^4.11). To demonstrate, how we For fj= 0 (4.8)reduces to:- arrive at (4.12), let us assume , that in f4.l) we have A = A, and A-= 0. ,/e obtain: - tXt. ft. 11) (4.13-a) Some properties of this system are investigated in the

Appendix. If fA = f^ it obviously reduces to (4.9) . A* = C4.13.bJ However, the solution of the system (4.l) obtained in this way is different than the previous one since here fj=O. A, k.'2 = 0 (4.13.c) Another way to satisfy (4-7^ is to take: Adding and subtracting by sides we obtain:

and to assume, that p fq ,u are eoplanar vectors forming (4.14) with each other an angle of 2"/3- Then we have for (4.7.a)

ufp-u) - 2pJ We now introduce new variables: J = A, - A but here q.(vi) + ufp'u) = - p, so we obtain: B-D = Q, i ^ .3 - f' Equations (4-14) in variables 13, j take the form:-

Thus we found three distinct types of solutions af J = - ± BX in the form (4.6y . If me assume, that:

-13- where b,d are orthogonal, constant unit vectors in the jome properties of this system are investigated in the izoapin space, then (4.15) reduces ta (4.1lJ . Moreover, Appendix. In the spacial case f. = f_ = f we obtain: it follows that: B « B1 =0 = a«3' = f' so we have: which has the solutions: and thus

A-< A/+ A^ A/ = 0.. f(R) = so f4.13.c^is also satisfied. where ncf* ,^) 13 a Jacobi elliptic function with parameter — and is positive integration constant. The function 5. The case X2

A" = - A,* .,* A,)- (5.1.a) Components of energy-momentum tenaor are: 2 = f' - + I f« Te.= 0 i = 1,2,3

(AO x A,)- Jince for (5-3J we have f'1 = - f^ + const f A x A ) - z e t then ^oo = f + const. Thus energy density .carried by this solution is singular at the points where f is singular. .Vhen A s 0 13 assumed, some solutions of (5.l) can be D Another configuration, for which (5-1/ reduces to obtained in complete analogy with the results in section 4, 6-2) is: And thus, taking: A, = f; fR)p .^ = f (ll)q. 2 = iffjR)b-f2(Hjd] where p,q are orthogonal constant unit vectors in izospin apace, we reduce (5.l) to: = 0.

2 2 "= f f f ' - f f where b,d are orthogonal constant unit vectors in the izospin space.

-15- -16- 6, Comments Gase Ar K 0 Theae solutions; represent static field confi;juration3 depending only on one opace dimenjion or, Case I2- - Q a'he system (3.l) is at the first glance more generally, they de^ciibe disturbances with plane quite complicated and it is a surprise that we found the symmetry propagating with the speed leas than speed of general solution (3-4) . It came out trivial, since all light. It would be interesting if solutions of thiu the components of the potentials are proportional to the type existed, having nonsin^ular energy density - explicit same constant vector in the izospin space. Juch solutions solutions found here have singularities which make their are embedded maxwellian solutions and of course they are physical interpretation obscure. abelian. for such solutions the nonlinear terms in YM A.11 the solutions discussed in .lections 4 and 5 are equations vanish identically, so they are in fact solutions real. It is easy to check, that they are also nonabelian to the linear equation. and gauge inequivalent to each other. Absence of nontrivial solutions with explicit plane The equation f" = 1 f appeared in the context of symmetry moving with the speed of lijht is not unusual for YM equations several times. it wia obtained in (*) and f J a nonlinear equation - the same thing happens for example but was used in ansatze which lead to complex till potentials. with nonlinear Klein-Oordon equation. On the contrary, It was also discussed in ( ) , but the conclusions of thia YM equations have solutions moving with the speed of light paper do not seem to be true. Liolutions depending on but without full plane symmetry [ J. the x only were reported in [ ) , but I was not able to Case -X^y 0 It is a remarkable property of this class recast them into the form comparable with those in thia of solutions, that even though tho field has nontrivial paper. time dependence /even periodic!/, the energy density remains constant. One might wonder, whether physical interpretation might be given to this fact. Maybe such solutions describe "vacuum fluctuations" in the classical theory, generating uniform energy density in the whole space, or maybe they are some kind of an "aether" - a medium, on which "physical"' disturbances propagate?

-17- -18- i'rom (A.3^ , (A. 3) , (A. 4] we get: APPENDIX

Let us consider the system:

(A. ; X* = f^x y" =£ x*y £ il (A.l) 5 Third power of firat derivative makes this equation where x = x(t) and y = y (t) . difficult to solve. When F= + l , fA.5j can be reduced (A.l)has the first integral: to. the first order equation in the case E=0 /for £ = — 1 there are no nontrivial solutions with S - 0/. .Ye x' + y' - g x2 ij* = B = const. CA.2J have then:

The simplest solutions of (A.l) are: (g/j - •

a) linear functions: x = 0 , y=a( R +/J =<;pconstants After the change of variables b) elliptic functions: x = y , y* =f y' y = x^f^) ^ = In x we obtain jystera (A. l) can be reduced to one ordinary differential equation of the second order on the function y = y(xj . We have: Taking S5 aa a function of^ , i.e. dx x we obtain the first order equation on namely the Prom we have: Abel equation of the second kind.

x = »e 4? =

30

dx dfi V r; (|F

-19- -20- IC/82/23 SUN KUN OH - Mass splitting between B+ and B mesons. INT.REP.*

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IC/82/32 W. MECKIJUIDURO - Hiorarchial spontancoas compactifieation. INT.REP.*

IC/82/33 C. PAHAGI0TAK0F0UL03 - Infinity subtraction in a. quimtum field theory of chargeH and moncjioJ.(.-::.

IC/82/3'l M.W. ICAL1KOWGKI, M. GEWERYHEKI und L. HZUl-lABOWSKI - On the F equation. INT.REP.*

IC/82/35 H.C. LEE, LI BING-AU, EII1SM QI-XIH(J, ZIIAMG MEI-HAN and+iU HONG -+ IMT.KEP.* Electrovr^ak interference effects in the hip;h energy e + e —> e + c + hadrons process.

IC/82/36 G.A. ClIhlSTOS - Some aspects of the 0(1) problem, and the paeudoacalar masa spectrum.

IC/82/37 C. MUKKU - Gauce theories in hot environments: Fermion contributions to one-loop.

IC/82/38 W. KOTARSKI and A. KOHALEWSKI - Optimal control of distributed parameter INT,REP.* system with incomplete information about the initial condition.

IC/82/39 M.I. TOUSEF - Diffraction model analysis of polarized triton and He INT,REP,* elastic scattering.

IC/82/UO S. SELZEB and H. MAJLIS - Effects of 3urfa.ee exchange atlisotropy INT.REP.* in Heiaenberg ferro.iiagnetic insulators.

IC/82A1 H.R. HAHOOM - Subcritical assemblies, use and their feasibility IHT.REP.* assessment.

IC/82A2 W. ANDREONI and M.P. TOSI - Why is AgBr not a superionic conductor?

THESE PREPRINTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586, 1-31*100 TRIESTE, ITALY. -21- -i- -IC/82/U3 ' N.S. CHAIGIE and J. ETERH - What can we learn from sum rules for vertex functions in QCD?

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IC/82/U8 B.R. BULKA - Electron density of states in a one-dimensional distorted INT.REP.* system with impurities: Coherent potential approximation.

IC/02/1+9 J. GORECKI - On the resistivity of metal-tellurium alloys for lov INT.REP.* concentrations of tellurium.

IC/02/50 S. RAHDJBAR-DAEMI and R. PERCACCI - Spontaneous compactification of a (lt+d)-dimensiontil Kaluza-Klein theory into M^ * G/I[ for arbitrary G and H. p IC/82/51 P.S, CURE - On the extension of 11 -functions in polydiaca. INT.REP.*

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IC/82/60 Y. FUJIMOTO and Z1LA.0 ZHIYOHG - U-ff oscillation in S0(10) and SU(6) supersymmetric grand unified models.

IC/82/61 J. GORECKI and J. POPIELAWSKI - On the application of the long mean free path approximation to the theory of electron transport properties in liquid noble metals.

IC/82/62 I.M. REDA, J. HAFNER, P. PONGRATZ, A. WAGENDKISTEL, H. BANGERT and P.K. BHAT - Amorphous Cu-Ag films vith high stability. IC/PD/82/1 PHYSICS AND DEVELOPMENT (winter College on Nuclear Physics and Reactors, INT.REP. 25 January - 19 March 1982).

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