5. E. Peleg and Z. Zinamon, Phys. Fluids, 24, No. 8, 1527 (1981). 6. D. W. Rule and H. Grawford Oakley, Phys. Rev. Lett., 52, No. I, 934 (1984). 7. O. F. Nemets and Yu. V. Gofman, Handbook of Nuclear Physics [in Russian], Naukova Dumka,

Kiev (1975). 8. N. M. Belyaev and A. A. Ryadno, Methods in Thermal Conductivity Theory, Vol. 2 [in Russian]

Vysshaya Shkola, Moscow (1982). 9. B.M. Budak, A. A. Samarskii, and A. N. Tikhonov, Handbook of Problems in Mathematical

Physics [in Russian], Nauka, Moscow (1972). i0. A. A. Samarskii, Theory of Difference Methods [in Russian], Nauka, Moscow (1983).

LINEARIZED N = 2 SUPERGRAVITY IN TERMS OF SU(2)-EXTENDED SUPERFIELDS

S. V. Ketov UDC 530.12:531.51

The linearized theory of N = 2 conformal supergravity is expressed in terms of a single irreducible SU(2)-extended superfield. In particular, in this way the or- igin of the N = 2 superconformal multiplet is established.

One of the basic problems in the theories of N-extended supergravity consists of finding the auxiliary fields that close the supersynnnetry algebra off the mass shell. The solutions are known for N = i [i, 2] and N = 2 [3-5], and in the latter case of gauge fields of the superconformal algebra SU(2,2 ]2~ it turns out to be insufficient to close the algebra after finding the constraints [5]. a result, the N = 2 superconformal multiplet has a complex

structure in the component approach.

On the other hand, a systematic and complete description of the multiplets is given by the superfield approach. Thus the linearized N = 2 supergravity can be formulated by using SO(2)-extended superfields [6]. However, the analysis in [6] is incomplete because the N = 2 theories of supergravity have a SU(2) symmetry, which means that a formulation must exist in terms of the SU(2)-extended superfields.

The arguments in this paper are based on the results [7, 8], where the expansion of a general N = 2 scalar superfield in terms of irreducible SU(2) extended superfields was ob- tained. One of these superfields ~<1,o,a>, which corresponds to a superspin i and a super- isospin 0, has the following content of independent components (which are irreducible with re- spect to SO(3, i) | SU(2)

(O, '~̂ ~, ~:~, I~ ~>, L~, ~L, g~), (1)

where according to [7, 8] the fields are redefined as follows

A ~ D ~ D , 2i(~ ~-?~)-+~j,

Va--+ 1~ ), D i~a---"Ima, (2) A A

~ . ~ . _ _ . . A - i �9

The characteristics of the multiplet, the explicit form of the superfield, and the trans- formation laws of the components (i) are given in [7, 8]. Using the explicit form of the superfield we can calculate the supersymmetric lagrangian

(3)

V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, No. 5, pp. 93-95, May, 1986. Original article submitted October 16, 1984.

416 0038-5697/86/2905-0416512.50 �9 1986 Plenum Publishing Corporation

Here, however, the supersymmetry transformation laws for the redefined components (2) become nonlocal. The nonlocality can be eliminated by introducing gauge fields in place of the ir- reducible fields that correspond to a given spin by means of introducing the corresponding projectors~ Then by directly checking, one can be convinced that all the nonloeality in the supersymmetry transformation laws can be absorbed into the gauge transformation

~g~ = ~ + 0 ~ + O~, ~ - O~ ~ + 7~= ~, ~Ima = OaAm, a i ~ ) = a a A " (4)

In order to write down the final result it is convenient to return to the four-component form by setting

4

The reality condition in the SU(2)-invariant form is

A f t e r a l l t he se r e d e f i n i t i o n s the l a g r a n g i a n (3) can be r e w r i t t e n as

(6)

l ( t R 3 3 A

X i7~,7oD~ Rd -~-TdT'R' F~,~ (I(~)) - F~o ([ , . ) -- '4(0: 2L 3

(7)

where

Rab and ~a,bc are the linearized Ricci tensor and connection, respectively.

Thus the free iagrangian for the superfield {<~,o~z> is none other than the linearized theory of the N = 2 conformal supergravity:

the gauge fields gab, ~ia, Ima, I(~)a,

the matter fields Tab , ki,

the auxiliary field Do

By using the results of [7, 8] the transformation laws for the supersymmetry can be written

~D = i - ^ ----siOk i, 2

A | " " S}~ Dsi q_ 2i%%bO=jTao _ i]j~Fab (l(S)) si -- -- %~F~b (Ira) r

= ' 2

1 - ,~ 1 1 - .

= T § ,

- 1 -~" - ~ ~i~[5~a ~ ,

( __1 ) I 7i ].__~ -Ta'miAT' 8faro 2 ~ mJ

i - ~ i - - ,~

~,, i i �9 " - ] ( S ) e / ~ ;~i o'?,, - - 2D.~ - - z%%j~.s 'T~c + 7~ ,~ T- ~- ( '%)/ Ia~,

(8)

417

where D~ is the covariant derivative for the spinors

I (9)

io 2. 3o 4. 5. 6. 7~ 8.

LITERATURE CITED

P. Van Nieuwenhuizen, Phys. Rep., 68C, 189 (1981). T. Kugo and Sh. Uehara, Nucl. Phys., B226, 49 (1983). B. DeWitt et al., Nucl. Phys., B155, 530 (1979). E. S. Fradkin and M. A. Vasiliev, Phys. Lett., 85B, 47 (1979). B~ DeWitt et al., Nucl. Phys., B167~ 186 (1980). J. Kim, Irreducible S0(2) Extended Superfields, Preprint UCLA 83/TEP/I, Los Angeles (1983). S. V. Ketov and io V. Tyutin, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 9, 124 (1984). S~ V~ Ketov and I. V. Tyutin, Teor. Mat. Fiz., 61, 254 (1984).

PROPERTIES OF EXACT SOLUTIONS FOR A MASSIVE CLASSICAL YANG--MILLS FIELD

Ao S. Vshivtsev, V. K. Perez-Fernandez, and A. V. Tatarintsev UDC 539.12:530.145

Exact solutions for a massive Yang--Mills field are found and solutions of class- ical Wong equations and quantum Dirac equations are discussed for the field con- figurations obtained. A procedure for constructing constant fields is given and transition to solutions of the Yang--Mills equations in the case of a massless field is discussed.

INTRODUCTION

There has been considerable interest in studies of exact classical solutions of the Yang-- Mills equations, because they can help us understand the vacuum structure and its nontrivial properties that result from the nonlinearity. At present, only a few special solutions are known: kinks, string, monopoles, and localized nonsingular solutions (in imaginary time) i.e., instantons [1-4]. These solutions are classically stable due to the presence of topo- logically conserved charges and are at the center of interest of contemporary theoretical physics. Besides the above mentioned solutions, the nonabelian, plane-wave solutions of the Yang--Mills equations, first obtained by Coleman [5], have also been extensively discussed in the literature. Somewhat more recently a number of authors [6-8] have studied the possibility of obtaining plane~wave solutions of the Yang--Mills equations on the basis of substitution proposed, for Euclidean metric, by t'Hooft [9] and Corrigan and Fairlie [i0]. Shortcomings of solutions obtained for the Minkowski space have been discussed in [iI]~

In this note we study exact solutions of the classical Yang--Mills field with mass. Pre- viously (using another ansatz) such a problem was considered by Actor [8]; however, his tran- sition to the massless field is incorrect and also the conditions for connecting solutions in various regions of the values of the parameter~ characterizing the field mass, have not been explained. Simultaneously, we consider the behavior of the classical and quantum isospin particles in an external field defined by potentials that are given by solutions of the Yang-- Mills equation.

!o Solution of Equations for a Massive Yang--Mills Field

The equation describing the classical massive Yang-~Mills field can be written in the fol- lowing form:

D~Fg=K~, (1)

M. V. Lomonosov State University, Moscow. Translated from Izvest!ya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 96-i00 , May, 1986. Original article sub~itt~ d June 15, 1984o

418 0038-5697/86/2905-0418512.50 �9 1986 Plenum Publishing Corporation