ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY

"SQUARE ROOT" OF THE DIRAC EQUATION IN EXTENDED SUPERSYMMETRY

S. V. Ketov and Ya. S. Prager UDC 530.12

We discuss the possibility of extracting the "square root" from the Dirac equation in N-extended supersymmetry, with the aim of constructing a more fundamental dynamical theory. Although a "square root" of the Dirac oper- ator can be defined in N-extended superspace for N ~ 2, it is not possible to construct with its help a new dynamical model that meets the standard re- quirements imposed on the theory.

As is well-known, Dirac's original method, which he used ~o obtain the famous equa- tion for the electron wave function, is based on the idea of factorizing the Klein-Gordon equation [i]. As a result, a dynamical first-order equation is obtained from the dynamical second-order equation. The first-order equation is more fundamental in the sense that an arbitrary wave function that satisfies the (free) Dirac equation automatically also satis- fies the Klein-Gordon equation.

The remarkable simplicity and elegance of Dirac's method have stimulated further at- tempts to extract a "square root," this time from the Dirac equation itself. In particular, in a recent work [2], it has been proposed to construct a new theory by extracting a "square root" from the Dirac equation in superspace of simple supersymmetry.

In this work, we generalize the discussion from [2] to the case of N = 2 extended sup- ersymmetry. It is interesting that an analogous generalization to the case N > 2 is appar- ently not possible. In the N = 2 extended superspace, there exists, in the same way as in the N = 1 superspace, an operator A, constructed from covariant derivatives with dimension i/2 (in the units of mass), that can be considered as a "square root" of the Dirac operator D. However, use of this operator A for obtaining equations of the free fields leads to re- lations that do not contain dynamical information. Treating the operator A as an auxiliary operator, used for writing the dynamical system of equations for (free) superfields in a first-order formalism, returns us, in fact, to the Dirac equation for superfields. Such a "horizontal" supersymmetrization of the Dirac equation D~(x) = 0 by way of the naive re- placement of the space-time argument xP by the full set of the (extended) superspace coordi- nates z M = (xP, @i s, @ i), is noncanonical and incompatible with the usual principles of the construction of a field theory. This is revealed, for example, by the appearance of nonphysical states and by violation of the usual relation between spin and statistics. The correct supersymmetrization is "vertical," i.e., the spinor field ~(x) is included in a supermultiplet with other fields, for example, scalar ones whose (free) equations of motion are the second-order Klein-Gordon equations; while the equations of motion for ~(x) (the Dirac equations) are first order [3]. The remaining part of this paper is devoted to the discussion of details.

We consider the N-extended Poincare superalgebra in 4-dimensional space-time, defined by the following (anti)commutation relations [4] (the commutators with the Lorentz gener- ators are omitted):

[Q~, B~] = (b~)'j Q~, [ ~ , B , I = - Q~j(b~lJ~; ( i ) B , ] = = Ip , = o ,

p , ] = o;

Power Electronics Institute, Siberian Branch, Academy of Sciences of the USSR. Trans- lated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 5-9, March, 1990. Original article submitted October 19, 1987.

0038-5697/90/3303-0207512.50 �9 1990 Plenum Publishing Corporation 207

tcrsBt. IB,, BA = "

The first letters of the Greek alphabet are always used to denote spinor indices (~ = I, 2; ~ = i, "2), letters from the middle of the Greek alphabet are used for vector indices (U = 0, i, 2, 3), and letters from the middle of the Latin alphabet are used for indices of the internal symmetry U(N) (i = i, 2 ..... N).

Here, B r are Hermitian generators of the internal symmetry U(N) with the structure constants Crs t. The rule of the Hermitian conjugation for spinor charges Q has the form

(Oh + = Q;,. (2)

In this way, if Qi transforms according to a certain (usually fundamental) representation U(N), then Qi transforms according to a complex-conjugated representation. We note that the fundamental representation of U(N) is real only if N ~ 2.

Since supersymmetry is a "quadratic root" of the space-time Poincare symmetry in the sense of Eq. (i), it is natural to consider the construction in general, in an extended sup- erspace with coordinates z M = (x~, 0i~, ~J) of a linear operator A with dimension i/2 that would be a "square root" of the Dirac operator (with dimension i).

In the 2-component notation for spinors used here, the (free) Dirac equation is writ- ten in the form

We now require that the full set of the N-extended superspace coordinates be the argument of the wave functions ~ and ~ ("horizontal" supersymmetrization). However, in this case, the operator D does not affect in any way the anticommuting superspace coordinates. At the same time, the superspace contains derivatives with the lowest dimension. These are the well-known spinor covariant derivatives [4]. They form the following algebra

{D~, D{} = {D;,, D~/} = 0 ; (4)

and admit the following realization in the superspace:

(5) " ' ~L g . D a =

It is known [2] that in the simple N = l superspace there exists a suitable operatorA:

since

where a Hermitian operator

A = ~-~ _ D~ b ; 1/--~2 -D ~ D~ '

A A + = \ Mtl~ i=2.d~ '

(6)

(7)

I (D~, -D ~ § D'D~) ( 8 )

plays the role of the mass. Acting on a chiral superfield, it gives M(1) 2 = P~P~.

Attempts to construct an analogous operator in an N-extended superspace encounter dif- ficulties since D i and Dj transform according to nonequivalent representations for N ~ 3. For N = 2 we can raise and lower indices of the internal symmetry using an antisymmetric invariant metric eij, el0, since [5]

(D~)+ = D~i; (O~.)+ = _ D ~ i . ( 9 )

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The operator A in N = 2 superspace has the form

We h a v e

AA+ t,s,,>,j i,ia, ) where now the operator

i 0 )

ii)

I 1 (~u § Dr J) ~l,Ii~, = 4 (D' D ~j + D~D~) =-=- --

plays the role of the mass.

Once we have available a linear differential operator A, it is natural to consider a

"new" set of linear equations:

A B = 0 ; A + F : 0 . (13)

In N = 1 s u p e r s p a c e , t he s u p e r f i e t d s B and F have t h e form [2]

V ; ; f = \ ;A/ , ( 1 4 )

Consequently, (13) can be written in the following way [2]:

D~q~ + D~I/~ = 0, - f)~fl~ ~- D~ V= i =0; (15a)

D',~ - ~ ;~ = o, D ~ ,,~, + Dob-= 0. (15b)

In N = 2 s u p e r s p a c e , t h e r e a r e s e v e r a l p o s s i b i l i t i e s f o r c h o o s i n g B

i2)

and, correspondingly, F

~l~.(V)i~). I~.)=/(D2i j ) ((])3i~ �9 = ' _ V a ; B 3 = . t 2= ' V J s )

(16)

_ . .

The s u b s t i t u t i o n A +-~ A + does no t g i v e a n y t h i n g e s s e n t i a l l y new. have used t h e min imal number o f s p i n e t i n d i c e s .

At f i r s t g l a n c e , Eqs. (13) seem to be more f u n d a m e n t a l t han t he D i r a c e q u a t i o n , s i n c e f o r example i f F s a t i s f i e s ( 1 3 ) , t h e n

A

D F = A A + F = O , (18)

where M takes on one of its eigenvalues. N = 1 superspace

(17)

In Eqs. (14)-(17), we

It is easy to write down a suitable action in

or N = 2 superspace

S, = -- (d 4xd*O [F+AB+ h.c.] (19a)

3.2 .-= - - Sd *xds@ [ F+AB "+- h. c] (19b)

and even to introduce a gauge interaction in a supersymmetric way by further covariantiza- tion of the covariant derivatives with respect to the N = I or N = 2 Yang-Mills connection

Dil) ~ 9~ i) = mi i) + iA~i); D~-~ 9<, = D~o - zAu) (20)

using the standard set of connections for anticommutators of the gauge-covariant deriva- tives in superspace [6, 7].

However, Eqs. (13) are, in fact, relations that express only certain component fields in terms of others and do not lead to dynamical equations of motion. We have verified this assertion by explicitly writing the superfield equations (15) in terms of components and solving the linear dependencies that have arisen. We do not present the proof here since

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it is, on the one hand, extremely cumbersome, and on the other, quite elementary. An anal- ogous statement also takes place for N = 2.

A dynamical equation can be obtained by relating, for example, B and F. Based on dim- ensional considerations, a unique variant of such a relation has the form

B------A+F. (21)

Th i s c o r r e s p o n d s t o t h e f o l l o w i n g L a g r a n g i a n d e n s i t y in s u p e r s p a c e

L =--(F+AB-i-h. c,)q-B+B. (22)

The t h e o r y (22) i s a f o r m u l a t i o n o f t h e t h e o r y w i t h

L = - -F+ (AA +) F ( 23 )

in t h e f i r s t - o r d e r f o r m a l i s m and t h i s r e t u r n s us t o t h e " h o r i z o n t a l " s u p e r s y m m e t r i z a t i o n o f t h e D i r a c e q u a t i o n .

To summarize our discussion, only two supersymmetrizations of the Dirac equation are possible: "horizontal" (supersymmetrization of coordinates) and "vertical" (supersymmet- rization of fields), although the two may also be used together (this can be done on grad- ed supermanifolds, considered recently in [8]). The use of the "horizontal" supersymmetri- zation is always accompanied by the appearance of "extraneous" fields with noncanonical dimensions and "ghosts." This significantly complicates the use of such theories in physi- cal applications.

LITERATURE CITED

i. P. A. M. Dirac, Proc. R. Soc. (London), All T, 610 (1928); AIIS, 351 (1928). 2. J. Szwed, Phys. Lett., 181B, 305 (1986). 3. J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press,

Princeton (1983). 4. P. Fayet and S. Ferrara, Phys. Rep., 32C, 249 (1977). 5. S. V. Ketov and I. V. Tyutin, Teor. Mat. Fiz., 61, 254 (1984). 6. R. Grimm, M. Sohnius, and J. Wess, Nucl. Phys., B133, 275 (1978). 7. S. V. Ketov and B. B. Lokhvitsky, Class. Quantum Grav., i, 431 (1987). 8. J. Kowalski-Glikman and J. W. van Holten, Nucl. Phys., B283, 305 (1987).

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