Lorentz-harmonic (super)fields and (super)particles

NUCLEAR

P H Y S I CS B

Nuclear Physics B 368 (1992) 143—171 North-Holland

_________________

Lorentz-harmonic (super)fields and (super) particles Francois Delduc

Alexander Galperin

~,

2,*

and Emery Sokatchev

~

*

‘LPTHE, T.24, Unit’ersité de Paris VII, 2, p1. Jussieu, 75251, Paris Cddex 05, France 2 The Blackett Laboratopy, Imperial College, London SW7 2BZ, UK Received 14 June 1991 Accepted for publication 5 August 1991 We apply the recently developed concept of a compact Lorentz-harmonic space, which is 2, to the problem of covariant quantization of the superparticle in isomorphic to the sphere S° D = 3, 4, 6, 10. We study the structure of the representation of the Lorentz group realized on harmonic functions on S°2. The crucial difference between compact and non-compact harmonic analysis is explained. The massless harmonic fields depend on one space-time coordinate x~ only, and on D —2 harmonic coordinates. It is shown how ordinary massless fields can be obtained from the harmonic ones by means of covariant integration on S’°2. We construct a Lorentz-harmonic superspace, which involves only one quarter of the usual number of Grassmann coordinates. It closely resembles the light-cone superspace, however it is Lorentz covariant. This framework is used to formulate a superparticle action, in which all the constraints are first class, Lorentz-covariant and allow a straightforward canonical quantization.

1. Introduction The problem of the quantization of the Brink—Schwarz superparticle [1] (and the related problem of the Green—Schwarz superstring [2]) has a long history. The simplest and historically the first approach made use of the light-cone frame. It allowed us to separate the Grassmann constraint Da 0 of the BS superparticle into first- and second-class parts, and thus to perform canonical Dirac quantization. The high price to pay was the loss of Lorentz invariance. Siegel [3] proposed a modification of the BS superparticle, in which the Grassmann constraint was (]~D)~0 instead of D,~ 0. The singular matrix ~ (on-shell P2 0) played the role of a Lorentz-covariant projector which singled out the first-class part of the constraints D~ 0. However, a problem remained, this time it appeared when trying to fix the gauge invariance generated by the constraint (~D)~ 0 (the so-called “kappa symmetry”). This invariance affects only half of the Grassmann coordinates, but that half could not be separated out in a Lorentz-covariant way. *

=

=

=

=

=

=

*

** *

On leave from the Laboratory of Theoretical Physics, JINR-Dubna, USSR. On leave from the Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria. D~,is the conjugate momentum of the Grassmann coordinate 0”; after quantization it becomes the spinor covariant derivative.

0550-3213/92/$0500 © 1992

—

Elsevier Science Publishers WV. All rights reserved

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In 1986 it was suggested [4] to combine the advantages of the light-cone approach with manifest Lorentz invariance, using vector harmonic variables for the Lorentz group. They were supposed to play the role of bridges bringing (but not breaking!) the Lorentz symmetry SO(D 1, 1) down to the light-cone subgroup SO(1, 1) x SO(D 2). This idea was further developed in refs. [5—7],where the necessity to use spinor harmonic variables was underlined. However, all those attempts suffer from a principal problem: the coset SO(D 1, 1)/[SO(1, 1) x SO(D 2)], which the harmonic variables describe, is a non-compact space. This means that the functions on it, which contain finite-dimensional representations of the Lorentz group, cannot be integrated invariantly and do not have a harmonic expansion [8]. Moreover, a number of basic features of the compact-space harmonic analysis [9] have been assumed to apply to the non-compact case as well. Unfortunately, this is not the case. For example, in the compact case of SU(2)/U(1), the harmonic condition D~~I~(u)0 implies ‘I~’(u) 0 if the U(1) charge of the harmonic function is negative. As we shall see below, in the non-compact case of SL(2, ~)/SO(1, 1) this condition has non-trivial solutions for q <0, and precisely those correspond to physical fields. The importance of the compactness problem was first pointed out in a recent paper [101.There a solution to the problem was proposed. It turns out that another coset space of the Lorentz group is a compact space. It is SO1(D 1, 1)/ [SO 1(1, 1) >< SO(D —2) x K], where the divisor is the isotropy group of the light-cone. In fact, this space is isomorphic to the sphere 5D—2~ In ref. [10] it was —

—

—

—

=

=

—

also shown how to properly define spinor Lorentz harmonic variables for 5D-2 In the cases D 3, 4, 6 they form matrices of the corresponding spin groups SL(2, l~), SL(2, C), SL(2, El-fl) SU(4)*. The case D 10 requires special care, since Spin(9, 1) is not isomorphic to any of the classical groups. In the present paper we adopt this new point of view on the Lorentz-harmonic space. In sect. 2 we study in detail the structure of the Lorentz representations realized on harmonic functions ~t1(U) with external SO 1’ (1, 1) weight q. We show that these representations are not completely reducible and contain a finite- and an infinite-dimensional parts. It turns out that the harmonic measure on 5D—2 has positive SO 1(1, 1) weight, so only functions cIx~1(u)with negative weight can be integrated covariantly. The result of the integration is an SO’~(D 1, 1) covariant tensor. Thus, a Lorentz-harmonic function ~D~’(u)gives rise to one Lorentz tensor only. This is in sharp contrast with the case of harmonic functions for a compact group, where the harmonic expansion contains infinitely many tensors of that group. In sect. 3 we consider2 Lorentz-harmonic fields. The idea byis choosing that the 0 can be solved massless in the light-cone frame masslessness condition P ~ 0, P’ 0, P~±0. We can do the same, but this time using Lorentz-covariant light-cone projections of the momentum ~ ±± u~±piL,P’ ~ where (u, u~, u~) SOI(D 1, 1) is the matrix of Lorentz-harmonic van=

=

—

=

=

~,

~

—

=

=

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ables. Correspondingly, we define massless Lorentz-harmonic fields ~‘(x~, u~), which depend on one space-time coordinate x~~= u~x~ only (the rest are replaced by the harmonic coordinates ~ of the sphere S’22). We show that the invariant harmonic integral of such a field defines an ordinary massless field f(x,). We prove that the harmonic fields are in one-to-one correspondence with the solutions of the massless on-shell conditions for fields of arbitrary (integer in this section) spin. This solution of the massless field equations (or rather its spinorial version of sect. 4) is reminiscent of the well-known Penrose formula [11]. In sect. 4 we introduce spinor Lorentz-harmonic variables needed for describing half-integer spin fields. Here we follow closely ref. [10]. In sect. 5 we use the formalism developed to construct an on-shell massless superspace. It strongly resembles the light-cone superspace of ref. [12]. Like the latter, it contains only 1/4 of the original number of Grassmann coordinates 0~. However, this quarter is projected out in a Lorentz-covariant way, with the help of the spinor Lorentz harmonics u~ and of a second set of harmonic variables for the compact space SU(2)/U(1) in the case D 6 or SO(8)/SO(6) x SO(2) [6] in the case D 10. We study the Lorentz-harmonic superfields ~ 0~,u~)for D 3, 4, 6, 10 and show how they give rise to different massless SUSY multiplets in those dimensions. Finally, in sect. 6 we apply all this to solve the problem of covariant quantization of the superparticle. Our superparticle theory has only first-class Lorentz-covaniant constraints. The kappa symmetry of the BS superparticle now simply means that half of the 0’s (their u~projections) can be gauged away. In addition, for D 4, 6, 10 one can eliminate another quarter of the U’s by a “chirality” constraint, thus arriving at the light-cone-like superspace of sect. 5. The wave-functions of our superparticle are Lorentz-harmonic massless superfields, which describe on-shell Lorentz-covaniant supermultiplets, as explained in sect. 5. In sect. 6 we also briefly discuss the relation with the “twistor”-like particle of ref. [13] and with other twistor approaches [14—16]. =

=

=

=

2. Lorentz harmonics (vector case) We begin by introducing harmonic variables for the D-dimensional ortochronous Lorentz group SOt(D 1, 1). They are defined as a matrix H U H E *

**

*

**

—

The name “harmonic” changes its meaning in the cases of compact and non-compact groups. In the case of a compact group, e.g. SU(2) (see ref. [9]) one could write down the harmonic expansion of a function on the coset S2 SU(2)/U(l) in terms of those SU(2) covariant variables. This is not true for a non-compact group like SO1(D —1, 1), as we shall see below. Only after restricting the functions to the compact coset one can define their expansions in terms of SO(D —1) covariant variables. We restrict ourselves to the ortochronous (proper) Lorentz group SOT (D —1, 1), because this is what one obtains when constructing the vector harmonics of this section in terms of the spinor ones of sect. 4 (see ref. 1101).

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SOT (D 1, 1) with one Lorentz index ~ and the other index decomposed into SO 1(1, 1) x SO(D 2) representations *: u,~,u, u~.The fact that this matrix belongs to SO t(D 1, 1) leads to the following restrictions on the harmonics U: —

—

—

u0~~ > 0, uu~

—2,

=

2 = (u~)2

u~u~

=

0,

=

(u~) =

—

~(u~u~

+ u~u~) + u~u~ det MuM =1. =

(1) (2)

~,

They reduce the number of independent variables from D2 to D(D 1)/2, i.e. to the dimension of the group SOt (D 1, 1). The harmonic matrix 1 u H transforms as follows: —

—

Iu~I’=LI~u~R~.

(3)

Here Lu SOI(D —1, 1) and Ru SO1’(1, 1)x SO(D —2). In other words, the left group acts on the Lorentz index ~xand the right one acts on the SO ~(1, 1) X SO(D —2) indices ±±,i. In the group space of SOt(D 1, 1) one can define harmonic derivatives, which are L invariant and R covaniant, and are compatible with the conditions (1) —

a

a

+u~L

Dih=uih~____

——

au~

-

D1’

=

a

a

.

u”~aug’ —

—

D° u~’~

U2t’

au~

a

=

—

a

u~

(4)

.

The derivatives D°, D’2 form the algebra of the right-group SO 1(1, 1) x SO(D 2). Together with D ±±‘theyform the algebra of the right-group SOT (D 1, 1) —

—

*

[D~’

D~J]

[D~’,

D~J]=D” +~°D°

We use the following conventions: ~

1=1

D—2.

=

7J~,,=(+,

[D~1

+

D~J]

~);

=

0 (5)

A±± = A

2 0

±

AD,

A/IA”

(A’)

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The harmonic variables u 1 above parametrize the group space of SOT (D 1, 1), but in fact they are designed to describe the coset SOT(D 1, 1)/ [50 1(1, 1) X SO(D —2)]. To this end one considers fuiictions I~1(u), which are —

—

singlets of the left-group SO I (D 1, 1) and transform homogeneously under the right-group SO 1(1, 1) x SO(D —2). In other words, they carry SO 1(1, 1) weight 2q * and SO(D 2) indices i1 in. Thus, they effectively do not depend on 1 + (D 2)(D 3)/2 of the D(D 1)/2 degrees of freedom in lu ~, so they are functions of the 2(D 2)-dimensional coset SOI(D 1, 1)/ [SO 1(1, 1) x SO(D 2)]. The requirement of homogeneity may be formulated as differential conditions on the harmonic functions —

—

—

. . .

—

—

—

—

—

D°I~’,,(u)

=

2qcJ~d1

Dk/’I~’~’1(u)

=

~leI2k

,,,(u),

(6)

27

1,(u)

—

~i1k~~1i2

.

1~(u)+

...

(7)

The main problem with the coset SOI(D 1, 1)/[SOI(1, 1) x SO(D —2)] is its non-compactness. This severely restricts the functions which can be invariantly integrated and can have an expansion. In particular, the constant is excluded from the class of integrable functions. This circumstance is crucial for us, because it means that the Fourier (harmonic) transform of an ordinary (harmonic-independent) scalan field 4(x) does not exist. The way out of this problem is to restrict the harmonic functions to a compact subspace of the above harmonic space. This subspace is obtained in the following way [10]. The algebra (5) suggests an interpretation of SOI(D 1, 1)2 There as theD’2conformal group D°is for the the dilatation, (D —2)is the rotation, dimensional euclidean space R’~ D~~’ is the conformal boost (K) and D~’ is the translation (P) generator. The —

—

compact space is the coset SOI(D 1, 1)/ [SO 1(1, 1) x SO(D —2) x K]. Our harmonic functions are restricted to this coset by the additional differential constraint —

~

~,(u) =0.

(8)

If the function I2~?(u)is a singlet with respect to SO(D 2), then eq. (8) implies that it depends only on u~. Indeed, only u~ is inert under the action of the K generator D~~’ (see (4)): —

D’u~ *

=

0,

~

=

~

D~~’u~ u~. =

(9)

In this section all the objects have even SO 7(1, 1) weights, because we consider vector harmonics. Odd weights will be introduced in sect. 4, where we consider spinor harmonies.

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In fact, the newly defined coset is the sphere

5D—2

To see this, it is convenient

to parametrize the light-like vector u~ as follows: v~ 1.

2, P2”a)’

u~

=

(10)

=

(p

Here p is a non-vanishing * real number. It is the coordinate corresponding to the homogeneous action of SO 1(1, 1). The S0(D 1) unit-length vector Va defines the sphere —

5D—2

The functions (8) can be expanded on the sphere (for simplicity we first study the case of SO(D 2) singlet functions) —

2~(u~)

=

~2q~(

Va)

E~

2q

=

~

0

...a*V

(11)

a~’

n=0

where 4~a1. . . a~ are

symmetric traceless SO(D 1) tensors. This expansion is an infinite-dimensional representation of SO I (D 1, 1). It is well known [17] that the representations of the Lorentz group realized on homogeneous functions with integer SO 1(1, 1) weights are not completely reducible and consist of finite- and infinite-dimensional parts. We shall demonstrate this explicitly. To this end, let us rewrite the SO I (D 1, 1) algebra —

—

—

~ in SO(D

—

LAP]

LCd]

[Lab,

[La, La

ThLPLPA

+

Th’ALt*p

—

—

~7~~AL~P

(12)

1) notation: [Lab,

Here

=

L~]

~ad”hc

+ ~bcLad

~hc’~a

~acLb,

a =

~

—

~bdLac

—

ôacL~bd,

(13)

Lb] =Lab.

In terms of the harmonics

Lao.

Lab

=

La

we have

a —

~

a La

=

(dab

—

VaVb) ~

Va

a

+ —~-p~---.

(14)

The last term in La is chosen so that the space parametenized by (10) is closed

*

2)=P2Va, La(P2Vb)=~abP2. La(P The vector u~ is defined as a column of an S07(D —1, 1) matrix, so it cannot vanish.

(15)

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The action of

La

v)

2q(p

~

~

. . .

A”L~(D2~(p, v)

=

Va

~

=

n=0

=

149

on the expansion (11) is ~(J)

~2q

(Super)fields and particles

a,,)

Ab~a1.a*L~(p2~V

n=0

p2~~Ab~al...a~[(q

—

fl) VbVai

. . .

+ fl~baiVa

V~

2

where A” is a parameter. One can decompose

A~)4~1.

..

a~ as

(16)

a]’

follows:

{b aj...a

=A4

D +

(D

+

n

—

+

—5

4)(D + 2n

3)

—

(D +n

{ }

—

+

~

2

—

where

2n

+

Ac(~la2ba3a*

4)(D +

. . .

...

(17)

+~a*_Ia~~~la*_2)

2n —3)

denotes the traceless part. Then from eq. (16) it follows =

(q

—

n

(n +

+

+

1)(n +q+D—2) D

+

2n

—

,.~

1

>1

q+D n=0.

(18)

We see that the variation mixes the nth term with the (n + 1)th and the (n 1)th terms, except for the following two cases: (i) n —q D + 2, i.e. q < 2 D <0. The subspace Eq {~~~1a,,: n 0,..., Iq I —D + 2} of the space Dq {~ai...a,, : n 0,... ,~}is invariant, and forms a finite-dimensional irreducible representation of SO I (D 1, 1). The space Dq itself forms a not completely reducible representation, however, the coset space Dq/Eq is an infinite-dimensional irreducible representation. (ii) n q + 1, n * 0, i.e. q ~e 0. The subspace F,, a,, : n q + 1,. , ~) is invariant, and forms an infinite-dimensional irreducible representation of 501 (D 1, 1). The space D,, itself forms a not completely reducible representation, however, the coset space Dq/Fq is a finite-dimensional irreducible representation. —

=

—

—

=

=

=

=

—

=

—

=

.. -

=

. .

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The next important issue is how to obtain ordinary Lorentz-covariant tensors from the harmonic functions cf22~(~±+)This can be done by integrating out the harmonic dependence. Let us first write down the measure on du~ ...du~((u~)2).

fdu22)=fEt*It*Du~

(19)

The s-functions enforces the light-likeness condition (u)2 0. This measure is manifestly Lorentz (left) invariant. It is also SO(D 2) and K (right) invariant, since the harmonics u~ are inert under these subgroups. However, under the action of 50 1(1, 1) it transforms with weight 2(D 2), so the integrand must have negative weight —2(D —2). Then one can map a harmonic function a symmetric traceless Lorentz tensor in the following way: =

—

—

f~

~

=

fdu2D_2u~ 1+ .

. .

(20)

~

Note that the use of u~ to neutralize the positive weight of the measure is not allowed, because this would break K invariance. To get a better insight into this phenomenon, let us make use of the SO(D 1) covariant parameterization (10) of 5D—2 Then (19) reduces to —

f du2~ From SO(D

—

=

fa~i

dVa

. . .

2

dta

1) covariance it follows that the fdV

f

dV

fdVVa

Vai

I

a,,,

=

2~(v~ 1)

fdv

—

~2(D_2)

(21)

p

0

V

integral (21) has the properties

1,

~(a 1a,

(22)

a2,,*I0~

Then it is clear that the integral in (20) picks out the terms n 0,.. k from (11), i.e. the representation Eq. The rest of 1’ (the coset Dq/Eq) does not contribute to the integral. In sect. 3 we shall show that in the case of an on-shell massless harmonic field the infinite-dimensional part is determined by the finite-dimensional one via the field equations. We would like to stress the unusual structure of the SOT (D 1, 1) harmonic functions, compared to the SU(2) ones (see ref. [9]). In the case of SU(2)/U(1) the terms in the harmonic expansion are SU(2) tensors, thus one obtains an infinite set .,

=

—

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of SU(2) representations. In the Lorentz case one harmonic function gives rise to only one Lorentz tensor, as well as to one infinite-dimensional irreducible Lorentz representation. Finally, a few words about harmonic functions with external SO(D 2) indices. For them the constraint (8) allows a restricted dependence on u’~.To see this, we use once again the SO(D 1) parametrization. Using the K gauge freedom (see ref. (9)), we can choose * —

—

(0,

=

Then the

SD_2

t’~),

1aL~

=

0,

L’,~L’~ =

(23)

~

expansion of a function with, e.g. one S0(D

~

u)

p2~

~haj

~

=

n

=

.

—

2) index is (24)

a,,~

~

0

Here 4 is traceless, but not necessarily symmetric in the indices (bak). From (9) it follows Diht~ 1~ ~51v,,,so the condition (8) for the function (24) becomes D~1~7~(u) =

0

~ha1...a,,1,1,

~

n

=

...

~,,

=

0

~(~a,,)

=

0.

(25)

U

Note that the integral transform (20) cannot be applied to such functions (a combination like u~I2~” is not K invariant). However, this will change in sect. 4, where the conformally-invariant spinor Lorentz harmonics will carry S0(D 2) indices. —

3. Massless harmonic fields A massless of theThis Poincaré groupcan is characterized the condi2 0 onrepresentation the momentum. condition be solved bybygoing to a tion p light-cone frame, in which ~ (~, ~ p’), and putting p~=p’ 0. Using Lorentz harmonics, we can do the same without losing Lorentz invariance. We can decompose P~L into p+±=ptLu±± p’ =p~u~,and then demand p~~=p’ 0. In coordinate representation this means to consider harmonic fields L~2’?(x~,, u~) satisfying the constraints ** =

=

=

=

(26)

a1*p~ut~a~cp=o *

**

v~ and

L,’,

form a (D

—

(27)

1)x(D —1) matrix belonging to SO(D —1), so they are the harmonic

variables on the coset SO(D — 1)/SO(D —2)’~SD.2. Note that (26) is actually a consequence of (27), (8) and (9).

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Their solution is c12/ cf2~(~++u~), where x =x~’u,j/.As a consequence, D 0. These massless harmonic fields generate ordinary (harmonic-independent) symmetric traceless fields by means of the integral transform (20) =

=

f~ t*k(~)

=

fdu2

..

.

~

(28)

2~(x~u~,u~).

Obviously, they satisfy the on-shell conditions (29) Such an integral representation of the solutions of the massless field equations in four dimensions (or rather its spinor version, see sect. 4) is known by the name of “Penrose’s formula” [11]. Note that the same formula for the solutions of the three-dimensional Laplace equation appears in ref. [18]. Now we shall show that eq. (28) establishes a one-to-one correspondence between the solutions of eq. (29) and the harmonic fields I2~(x~,u~). First, it is easy to see that in this way one obtains the general solution of eq. (29). Indeed, the Taylor expansion of a field satisfying eq. (29) is

f

Y’ ~

1x~ =

2t-~l--t-~k~ ~

f

2t’l---/~k”l--”,,

x”’

(30

xC~‘

n=0

where the coefficients are symmetric traceless Lorentz tensors. Now consider the expansion of ~_2(k+D_2)(x++, u~) in powers of x~ ~_2(k±D_2)( 1±±

u~)

=

~

(31)

~_2(k+n+D_2)(u++)(x+±)fl

n=0

It is clear that the component 4,—2(k+n+D_2)(u++) can produce any given component f,,~ 1~k~I by means of eq. (20). Another way to study the correspondence between 2 harmonic and ordinary massless fields is to look at the expansion of cP on S~ ~_2(k±D_2)(x++,

u++)

~_2(k+D_2)

~

faI...a~(x)V

(32)

a~

n=0

As we explained in sect. 2, the integral in (28) picks out the terms n 0,.. k from (32), i.e. the E,, part of the Lorentz representation realized on cP. There arises the legitimate question: What is the role of the infinite-dimensional representation Dq/Eq contained in 1? Does it remain arbitrary, once the finite-dimensional part has been fixed? We shall show that it is actually determined by the finite-dimensional part (28). To this end we consider a field ~_2(k±D_2)(x++, ~ =

.,

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such that f~, ~(x) 0 (i.e. that faI...a~, 0 for n 0,..., k). We shall demonstrate that the rest of it must vanish as well. The point is that there are strong relations among the component fields fa1(l*(x,~) in (32) following from the fact that ‘1 satisfies the constraints (26), (27). Eq. (26) can be rewritten as follows: .

=

=

2( —a (u~°a() + u~”a~)t1~ 0 =

~ [~a0fa~v~1

+

=

V~a~)cIJ=o

+ a~fa1.--

a,,

a~VhVa

a,,]

0.

=

(33)

n= 0

Now, assume that f~ .a,,

=

0 for

n

=

0

...aj,~

k. Then eq. (33) implies

a~f”~

(34)

(see eq. (17)). So, we shall show that ahft*l 0 as a consequence of eq. (27). Using the parametrization (23), one can rewrite eq. (27) as follows: =

~ abfal ...a,,L:~,V

L’~

=

0.

(35)

n=0

With the help of eqs. (17) and (23) this becomes ~

a,,

a{”fa...a~v~v

n’=0

n(n—1) —

2(D+n—4)(D+2n—3)

a fCba1...a~~2VlL, b

c

~

L’

=

0

(36)

a,,2

If fal...a,, 0 for n 0, k, then the term acf~akv~Va a5 is the only one of this type, so 0. Then eq. (34) implies a~f ~+ 0. Now, it is known that for the irreps Dq/Eq there exists a positive-definite norm (i.e., they are unitary, see ref [17]) =

=

=

(~‘

~)

=

=

E

anfdDx I

fa!...a,,

12,

an >

0.

(37)

n=k±I

The requirement that this norm be finite implies Iim~,,,,f~~1a,,(x)0, ,~ k+ 1 ~, so we conclude that f~h1~lk+1 0. The argument goes on by induction to show that all the components of 1 vanish, if its finite-dimensional part vanishes. This completes the proof of the one-to-one correspondence between Lorentz-harmonic fields ~_2(k+D—2)(x++, u~) and solutions of the on-shell conditions (29). =

=

=

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4. Lorentz harmonics (spinor case) In order to describe massless fields with half-integer spin, we shall introduce spinor Lorentz harmonics. The vector ones discussed in sect. 3 will then be bilinear combinations of the spinor ones. The main idea can be traced back to the “twistor method” [19] of solving the light-likeness condition p2 0 ~p5, u”u’3(y,)~0, =

=

where u~’is a commuting spinor (“twistor variable”). This is possible only for t’) D 3, 4, 6, 10, where the identity (y5,)~(0(y 5~)0 holds. We shall complete u” to a full matrix II u I belonging to the spinor (Majorana—Weyl) representation of SOI(D 1, 1). In each dimension we encounter some specific features, therefore we treat them case by case. In this section we follow closely ref. [10]. =

=

—

4.1. D=3

The D lows:

=

3 Lorentz group SO 1(2, 1)

lull uSL(2, R);

lull

=

=

SL(2, R)/~2,so we write

(ut, u:),

u~”u: I ~-*det =

lull

lull as fol-

=

1.

(38)

Here u~are real two-component spinors carrying SO 1(1, 1) weight ±1 [SO 1(1, 1) acts from the right]. Note the completeness relation u~u~ u:u~ —

(39)

~

=

The variables u,,~ are harmonic coordinates on the coset SL(2, R)/ SO 1(1, 1). To get a better feeling of the structure of this coset, it is instructive to look at the following parametrization of SL(2, R): p cos U

llull= p sin U .

(1/p)(—sin 0+y cos 0) (1/p)(cos O+y sin 0) .

,

0
The first column in (40) corresponds to u~ and the second one to u. The functions defined on SL(2, R) and homogeneous under the right action of SO 1(1, 1) have the form f”(p, 0, y) p”f(O, y). The harmonic derivative =

a

a

a

D°=u~”-~--—~ —u”—---~—-=p—

(41)

F. Delduc et al.

/ (Super)fields and particles

qf”. The

is the weight operator, D°f” derivative

=

155

conformal boost operator is the harmonic

a

a

au

ay

D~~= tt~~ D~u~ 0,

D~u

=

u~.

=

(42)

It is clear that the conformal condition (43) implies f~=f”(u~)=p”f(0).

(44)

The function f(0) is defined on the circle 51 S01(2, 1)/[SOI(1, 1) x K] and 0(this is the analog of expansion has a harmonic expansion f(0) ~ ~ e~~z (11)). One sees that the situation with the non-compact coset SO 1(2, 1)/SO 1(1, 1) is radically different from the compact coset SU(2)/U(1), although the two formalisms look very similar. In the case of SU(2) one has harmonics u,~—, II u I u SU(2), which can be parametrized as follows: =

llull=

1 /

e

-

V1+zz

-~

—ze

Ze

0
,

(45)

‘~

The first column in eq. (45) corresponds to u~ and the second one to u~.The phase i,1i is a coordinate for the subgroup U(1). The harmonic functions are by definition homogeneous in e’4 f”(z, 2, iii) thus they are equivalent to functions on ~ is

=e’~’f(z, 52

2),

(46)

SU(2)/U(1). The harmonic derivative

a ~z a D~~=e2”—(1+z2)-—+—-— a2

2a

(47)

.

1~, Then the analog of eq. (43) is 2)1~

D~f~=

~(~±

~

~

qz

=0.

(48)

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(Super)fie/ds and particles

It has the general solution f~(z, 2)

=

(1 +z2)~”2f0(z),

(49)

where f0(z) is an arbitrary holomorphic function. Now comes the crucial difference between the compact and non-compact cases. In the compact case of 2 SU(2)/U(1) ~2 one looks for solutions (49) which are square integrable on S and thus have a harmonic expansion. This restricts f’~to a polynomial of degree q if q ~ 0 (i.e. to the product u 1 u, of the harmonics), or to 0 if q <0. In the non-compact case one cannot require square-integrability (otherwise one would rule out such functions as the constant), so the functions of SL(2, R)/ SO 1(1, 1) do not have harmonic expansions. Only after restricting them to the circle S’ one may define their harmonic expansions. Note the striking difference: In the SU(2) case eq. (48) has no non-trivial solutions if q <0, whereas in the SL(2, R) case not only do there exist non-trivial solutions to eq. (43) for q <0, but just those are the useful ones. To make contact with sect. 2 we show how the vector harmonics u,L(yt*),,4 are expressed in terms of the spinor ones ..

.

=

±±_ ± ± Uaj3 — U,~,,14~3 ,

0 UaO

— —

+ U— U(~

Their properties (1) can be checked using (36). The covariant measure (19) on 5’ becomes fdu+2=fu+adu~

fp2de

(51)

The analog of the integral transform (20) now allows to obtain half-integer-spin representations of SL(2, R) f~1..akf~

(52)

~

Further, one can define massless fields (see eq. (28))

f du~2

u

fa,

ak(xs,)

=

2(x~, ut), 1

. . .

(53)

u~5~

where x~~=x”0u~u. They satisfy the on-shell conditions

a”~f$aak=0 (k~1), (the latter follows from the former).

Df=0 (k~0)

(54)

F. Dc/due et al. 4.2. D

=

/ (Super)fields and particles

157

4

In this case SO 1(3, 1)

lull uSL(2, C):

SL(2, C)/7L2 and we write

=

Ilull

=

(up, uk’),

~

=

1 -*det

llutl

=

1. (55)

Here u~ are complex two-component spinors carrying SO 1(1, 1) weight ±1 and S0(2) U(1) charge ±1, so they are harmonic coordinates on the coset SOI(3, 1)/[SO 1(1, 1) x S0(2)]. The conjugate matrix is II u I (ut’, uç’~).The conformal boost generators =

D+±,++=u+,+~

a a

_____

(56)

—

leave the harmonics ~

ü~’ inert. Thus, the conditions

D’~f””°(u ~) ~ =

~

~)

=

0

~f’1P

_f~P(u+,

i~~)

(57)

restrict the functions on SOI(3, 1)/[SOI(1, 1) x SO(2)] with SOI(l, 1) weight q and S0(2) charge p * to functions 52 50 1(3, 1)/[SOI(1, 1)x SO(2)X K]. A 2 isongiven by convenient parametrization of S U~’+=pV~, u~’ =pV,, (58) where V~-~are SU(2)/U(1) harmonic variables. Then the functions in (57) have a harmonic expansion on S2 fqP

=

p~fP(~±) p”Ef’1~~v~ =

. ..

V~V~

..

.

V 1.

The vector harmonics of sect. 2 un,,, ++ Uaá

a,,

=

=

+,+—+,—

a

,~

=

(59)

e~i)aau~’ are ——

U,~

U+,+i7_,+ a a

—

~Ua =

U,,

u,~+ a u,,,

2 is The measure on S fdu~4 f(u~’~du~)(F~’~’ di~) =

*

=

fp4(v±’dv~)(V~dvl).

(61)

This SO(2) charge is an example of an external SO(D —2) index of a harmonic function. In sect. 2 we were not able to have such functions without the help of the eonformally non-invariant harmonics ut,. Here the eonformally invariant harmonics u5~, ~ carry SO(2) charges themselves.

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This time the on-shell Lorentz-harmonic fields can carry both SO 1(1, 1) weight and S0(2) charges. Thus, for q —4, p 0 I~4(x~,u~)defines a scalar field =

=

f(x) =fdu~4~4(x~, ut),

for q

=

—5, p

=

—1 =

Df=0,

(62)

5’(x~, u~)defines a spinor field fdu~4 u,,5’(x~,

ut),

~

=

0,

etc..

(63)

4.3. D=6

In D

=

6 the Lorentz group is SO 1(5, 1)

=

SU(4)*/7L

2. It has two inequivalent fundamental representations corresponding to Weyl and anti-Weyl spinors 4. (64) ~J,,,4”, a=l,2,3, They are complex, four-dimensional and transform with inverse Lorentz matrices. These representations are pseudoreal, i.e. transform like i~i,,

C7,~/i 0,

(65)

where Cf~is a charge conjugation matrix. It 2is useful work in the representation on the to diagonal. where C/~is a block-diagonal matrix with iTcan be unified into the so-called SU(2) A Weyl spinor and its complex conjugate Majorana—Weyl spinors

i~i,,,

i

=

1, 2 with the following reality condition (66)

~=C~u’~çli~.

The matrix of the Lorentz group in the fundamental, spinor representation, i.e. SL(2, I-El) has the form u~,where a and f3 are a Weyl and an anti-Weyl indices. It

satisfies the following (pseudo)reality condition (67) as well as the unimodularity condition

detllull=1. Now, when we consider the coset space SO 1(5, 1)/[SOI(l, 1) index f3 decomposes into a pair of SU(2) x SU(2) S0(4) indices: 13

llull=(u~A,u:A),

llull=(u~,u~).

(68) X =

S0(4)} the (A, A), so (69)

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159

The entries are SU(2) Majorana—Weyl spinors with the corresponding reality conditions. They should satisfy the condition of unimodularity, which reads (70)

E”U,i~iAUj~BuyAU,y~j= ~AB~AB~

There are no other conditions, which is clear from counting the independent parameters. The functions on SOI(5, I)/[SOI(1, 1)xSO(4)] are ~ ~ They carry SO 1(1, 1) weight q, SO(4) SU(2) x SU(2) indices (A1... A,,,A, A,,) ...

and indices (i, ik) of the spinor automorphism SU(2). 4 SO 1(5, 1)/group [SO 1(1, 1) xThese SO(4)functions K] by are the restricted to the compact coset S conformal condition . ..

X

a

D~

- ~

a

u~—~

±~

=

~

+U~”A,

fq~...i 5

AAJAI...A,,4I...A,,kU

The vector harmonics

~

u~, are

...

+(

f’I’i---’s •( JAJ..A,,4I...A,,\U

±\ ~

~71 k

—U,,

±

\“OAB

~

C U0,~, 0E~U~,

u~ =UaA(Y~) ±A~ \(‘P U AA 1,

=

given by

±±

Ut,

.

AA1A

—À

~~Y~*) U0

Their reality, 4as iswell as the properties (1) follow from conditions (67) and (70). The measure on S

fdu~

=

f

(u+~~Adu~)(u~0du~A)(u~du±Th~)(u~B du~”).

(73)

The on-shell fields are given by S4 integrals, e.g. ~(x)

F~(x)

=

fdu~ ~8(x~,

=

fdu~ u~A~9(x~,u~)~

=

fdu~ u~Au~0~’°(x~, u~) FJ~’ (y)F’~”, =

u~)

U

=

a~F~ =

C

0,

=

AP,,,~aF

=

0,

0.

The field F describes the on-shell field-strength of an abelian gauge field.

(74)

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(Super)fields and particles

4.4. D=10

In this case the definition of the Lorentz harmonics is more complicated, because the group Spin(9, 1) is not isomorphic to any of the classical groups. We define them as 16 x 16 matrices + — Ilull =(u,,A, u,,A),

lull

—,

—a

=(uA

,

+C,

U~

(75)

),

where

a is a 16-component Majorana—Weyl spinor index of SO 1(9, 1), ± are SO 1(1, 1) weights and A, A are indices belonging to the spinor representations (s), (c) of SO(8). The defining constraints on I u II (which mean that I H Spin(9, 1)) are as follows [10]. Denote the elements of I I by u~, where a (+A, —A). One can define the 10 x 10 matrix U

U

=

(76)

0(T”)a~

L~=U~U~(F~)

The first condition is that

I u ll~II u II’ —s L L’. It implies .

UaaU~(F~~ 1)$(FC)ah

=

0.

(77)

It removes 210 degrees of freedom [10]. The second condition is that the matrix (76) belongs to SOI(D —1, 1): L~(LT)~=&~=~sdet L=1.

(78)

It turns out [10] that this condition removes only one degree of freedom, after which II u I has 16 x 16 1 210 45 degrees of freedom left. This is precisely the dimension of 50 1(9, 1), so II u I parametrizes the group space of 50 1(9, 1). The harmonic functions f.4~ A,~u±)are homogeneous under the right-action of SO 1(1, 1) SO(8), so they depend on 45 1 —28 16 variables, which is the dimension of the coset 50 1(9, 1)/ [SO 1(1, 1) x S0(8)]. Finally, the restriction to the sphere S~ SOI(9, 1)/[SOI(1, 1) x S0(8) x K] is achieved by imposing the conformal condition —

—

=

~

—

X

D’f~AAA(u~)=O~fi

=

A,,41...A,,_fA1...A,,,A...A,,(U),

(79)

where D~’

=

(

l)AA1±

aU:A

—

U~au~’~]

(80)

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(Super)fields and particles

161

is the conformal boost harmonic derivative (y’ is an SO(8) y-matrix). The vector harmonics are given by 0

U,~, ~ ++

——

frY’

+

+ UOA,

,,l3

—

U,,,~j(F~,) UOA,

U~,

U~_UaA(F/,)(Y)U~A.

(81)

Their properties can be established with the help of conditions (77) and (78) and their corollaries u~A(F~)uPB

=

8~ABUaC(T~)UOC,

uaA(F~)upA

=

~

(82)

etc.. The covariant measure on S8 is

(83)

fdu+16=fEAl.484l40(u,,+Adu)...(u~~du~j0).

The on-shell fields are, e.g. =

fdu+16 u~”~~7(x~, ut),

F,~(x) fdu+’6 u~

0

=

18AA(x++

a~(F~),,

=

00

0~

ut),

F,~

=

4u~

a~F=E~~PaCFA,,=o.

(84)

5. Lorentz-harmonic supersymmetry We begin by recalling how one constructs the representations of the D-dimensional supersymmetry algebra {Q,,, Q 0}

=

(1,),,0P”

(85)

in the massless case. One goes to the light-cone frame in which P~, (P, 0,. and decomposes (85) into =

{Q~, Q} =0,

{Q~,Q~)=0,

{Q~,Q~) (1),,0P. =

. .

,

0),

(86)

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(Super)fIelds and particles

Here Q,~ ~(T ~F are the two light-cone projections of the D-dimensional spinor Q,,. Clearly, half of the SUSY generators (Q~)anticommute with everything, so they do not take part in the construction of the Fock space. At this stage the original symmetry SOI(D 1, 1) is broken down to SO 1(1, 1) x SO(D —2). If D > 3, one can go one step further by dividing the remaining half (Q~)into pairs of creation and annihilation operators which satisfy the algebra =

—

{q,,, , q,,}

=

~

,

i~,,} 0,

(q,,, , q~,,)= ~

=

(87)

.

Then one chooses a vacuum state 10), qL,, 10) 0, and uses the generators q,,, to build up the Fock space 10), q,,, 0), q,,,q,, 10),... ,(q)N 10). Note that the number N of the creation operators is 1/4 of the original number of components of Q,,. At the Fock-space stage the SO(D 2) symmetry is broken down to SO(D 4) x SO(2). We shall follow the spirit of the above procedure in order to construct a massless on-shell Lorentz-harmonic superspace. The resulting superspace will look very similar to the so-called light-cone superspace [12] (see the comments at the end of this section). The principal difference will be that we shall use Lorentz-covariant light-cone-like projections of the supersymmetry generators, obtained with the help of the spinor Lorentz harmonics. Take, for example, the case D 3. The generator Q,, is decomposed into Q u “Q,,, after which (85) becomes =

—

—

=

±=

{Q~Q~}

=

0

±

{Q~Q-}

=

0

{Q

Q-}

=

P.

(88)

(recall that for Lorentz-harmonic on-shell fields P~=P° 0). The essential part of this algebra consists of the generators Q~and P~, has the form of N SUSY in one-dimensional space, and can he realized in the Lorentz-harmonic superspace =

=

{~ ~X~=

0~,u~}, iu+a0+,

~0+~ U~r,,, =

0.

(89)

A superfield cI+”(x~,0~,u~)in this superspace has the expansion cP”(x~,

0~,u~) ~°(x~, =

u,~)+ 0~(x~,

u,,~).

(90)

If we take, e.g. q —2, then, according to eq. (53) the components in eq. (90) define scalar 4(x) and spinor ifr,,,(x) fields, which form a multiplet of on-shell D =3 SUSY. =

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(Super)fields and particles

163

ln the case D 4 the projections of the SUSY generators and Q~.The non-trivial part of their algebra is =

Q~,~

{Q., Q}

p~

~-,+}

~

It can be realized in the Lorentz-harmonic N (xil, 0++, ôx~=

=

are

~-~} =0.

~

(91)

2 one-dimensional superspace ux),

~+.—

jé+.O+.+ ~0~=

=

Q,,, Q,,

iE~O~’,

~

8ö~~-= ~ 3u~=0.

(92)

However, this time there is a smaller superspace, where the same algebra may be realized. It is the chiral subspace of (92) {x~ =x

+i0~~O~, ~

u~},

=

~0+,+=E+.±, ~u~=0.

(93)

The chiral superfields are: cJ~~.P(x±± ~

u~)

=

~

x~, u~)+

~

1,p—

‘(xt~, ut).

(94)

They describe multiplets of on-shell D 4 SUSY. For instance, for q —4, p 0, we obtain a scalar multiplet (4 ~ 4’(x), ~i~’—s i/i,/x)), for q —5, p —1 we F,, get a vector multiplet(4Y’5—s iji,,(x), ~j,—6’—2 0(x), F,,0 F0,,, a°”F,,0 0 =

=

=

=

=

—~

=

=

~-‘

is an on-shell field-strength). Note that the chiral superfield (94) resembles very much the Fock-space construction (87). This is the superspace of “1/4” SUSY, where all the objects are SO 1(1, 1) x SO(2) covariant. However, unlike the usual light-cone superspace, we have not broken the original Lorentz invariance. The harmonic dependence allows us to go back to manifestly Lorentz-covariant fields. In the case D 6, the SUSY algebra is =

{Q,,~,Q0~}

=

11(F”),,0P~,.

(95)

164

Its

F. Dc/due ct al.

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(Supcr)fields and particles

(—) projection becomes ~

Q~}

=

(96)

EiJEABP~.

The corresponding superspace is {x~, 0~,u~},i.e. that of N 4 one-dimensional SUSY. Once again, it has a subspace containing only half of the 0~,(a quarter of the 0,,). One way to obtain it would be to form complex combinations like those for the Fock algebra (87). However, this procedure would break at least one of the SU(2) symmetries of (96). Instead, we can use SU(2)/U(1) harmonics w,~ for the automorphism group SU(2). Thus we obtain the double-harmonic analytic superspace =

{X++=x+++j0~~+0+,_A, ~ where ~

=

u~,w~},

(97)

w± ‘0~,.The corresponding superfields are

LIY”P(x~~

u~,w) =4”(X~, u~,w)

+

0~’”~(X,

u~,w)

u~,w).

2_2~2(X~,

+(O~

(98)

In order to obtain ordinary on-shell fields (independent of w one has to impose a constraint on the w-harmonic dependence. This is done with the help of the w-covariant derivative. Its form in the analytic basis (97) is ±),

a D++ w —w +1 —

Take, for instance, the case q condition *

=

a

aw~ —8, p

‘A

=

ax~~

_____

~‘

1. Then it is not hard to see that the

+

(100) implies 8’~=4~8’(x~, u~)w,~ —s~’(x), ,p_9A(x++, u~) —

=

l0,~..

a 4~8’w~. ‘ax~

(101)

The fields 4’(x), ~/i,,(x) constitute the D 6 matter multiplet (hypermultiplet). The D 6 gauge multiplet is obtained from an odd superfield carrying an external spinor index, 9’~(X~,~ u~,w). Then the components and ~ =

=

~

*

Note the similarity with the description of the q5 hypermultiplet in SU(2) harmonic superspaee

19].

F. Dc/due et a/.

/ (Super)fie/ds and particles

165

generate the spinor field and the field-strength in (74), and is auxiliary. Note 9~~ should satisfy a reality condition that in this case the superfield cJ~ ~“

—

‘~A

-

~AB

where means combined conjugation on the sphere S2 SU(2)/U(1) [9]. Finally, in the case D 10 the (—) projected algebra (85) becomes -

—~

=

{Q.~Q~}~

(103)

It can be realized in the N 8 one-dimensional harmonic superspace {x~, 0~,u~}. As in the case D 6, the definition of an analytic subspace containing only half of the U’s (i.e. one fourth of the initial number of U’s) requires the use of a second set of harmonic variables. This time they reduce the SO(8) automorphism of (103) to SO(6) x SO(2) There are two types of such harmonics =

=

~.

V. 4~’,VAi and

(104)

~

Here ± are S0(2) U(1) charges and i are SO(6) SU(4) indices in the two fundamental representations. They satisfy a number of algebraic conditions, among them —~

V~’VAJ =

V(y)AAW,~)

~,

=

0,

etc.

(105)

(here r is an S0(8) vector index). With the help of V ± the SO(8) spinor 0~ is decomposed into 0~’~’~ 0,~’ ~ Then one can define the analytic double-harmonic superspace =

{

X

=

=x~+

i0±.+l0/~,

0+~+~,

u+,

The harmonic derivative for the coset SO(8)/[SO(6) 1’~1 V~1’ =

D~

a

a

+ ~

~ A]]

awA

The superfield which can describe the D ~~b.++(X++, ~ =

~“’‘~(X

=

X

V,

w}.

(106)

SO(2)] is

+ i(0~’~)21’~’ .a

ax

++

.

(107)

10 Maxwell on-shell supermultiplet is

u~,v, w) U V

w)

+

~

+ (0~)2hu2]p~ 1~8

+~ *

(108)

+ (0~)4~_20__.

The use of a second set of harmonies was for the first time proposed in ref.

151.

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The condition that i be independent of the SO(8)/[SO(6) and w is

x S0(2)] harmonics v

0.

~

(109)

One can check that its solution is p~8 p~8(x~, U ~)(t~~w~AJ] =

7+

—

•-f’i -&

—

l7

—

~‘A

I(~.±+ (

k

+ EiIk/LkWAl),

--i WA,

,~—

/‘x

\

—1.8

JPAA

,+i ± LA WAt,

ia;p8t’~w~,

=

(110) The fields p~8 and ~ generate the members of the D 10 on-shell Maxwell supermultiplet (see (84)). The other fields are auxiliary (note that 16 is obtained by dividing by P, which is a typical light-cone procedure). The reality condition for the superfield (108) is non-trivial and involves spinor derivatives ~

=

~—

(D

)~~TI~ 16.++ =

~ijk/

=

E,jkl( D ~

)2[k/1

~—

i~.-t--t-

(111)

~i8

Note that the superfield f is anti-analytic, i.e. it depends on 07 only. The reader may have noticed the resemblance between the D 10 Maxwell superfield (108) and the D 4 N 4 light-cone superfield of ref. [12]. The 0-structure is exactly the same. The principal difference is that 1 (108) depends on one space-time variable (x~) only. The rest of them have been replaced by the 8 harmonic coordinates u of 58~ This allows us to recover manifestly Lorentz-covariant fields from the components of P (108) by means of the integral transform (28) (see eqs. (84)). Also, in the light-cone superspace of ref. [12] only half of the SUSY algebra is realized linearly. In our case, after removing the harmonic dependence, we find supermultiplets of the full (SOI(D 1, 1) covariant) SUSY. =

=

=

+

—

6. Particles and superparticles The on-shell Lorentz-harmonic (super)fields can be regarded as wave functions for quantized (super)particles. We shall show this in the simplest case D 3. The =

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(Super)fields and particles

167

particle action consists of two parts. The kinetic part S0

=

fdr[P,,0~’~0

+

D(t~u~)

+

D(t~u~”) +D°(r~u”)], (112)

introduces conjugate momenta for (y~’),,0x1,and for the Lorentz harmonic variables u,~.The latter satisfy the defining condition u~”u~ 1, therefore there are 3 independent momenta for them (after quantization they become the harmonic covariant derivatives [4]). The second part of the action introduces the constraints which the wave function ‘~P(P,u) is supposed to satisfy. Those are 0~I2 0, p°i P,, 0cP 0, p++çp P,,0u’u~ 0ti~~u =

=

=

D°e12 qb,

D~I2

=

=

=

0.

(113)

It is easy to check that these are first-class constraints. The first two of them can be replaced by an equivalent pair of constraints p+±~j~(j p2~j~ij

(114)

Indeed, P2 _PilP~+(Po)2, so (114) implies (p°)2 0 —~P° 0 (the same will be true in higher dimensions, where p° will be replaced by the transverse momentum P~, (P 2 0 —s P~ 0). The third constraint means that the wave function carries an1)S01(1, 1) weight, so it is defined on the coset SOI(2, 1)/ SOI(1, 1). The last constraint restricts cP to the compact harmonic space S’ SO 1(2, 1)/[SOI(l, 1) ~ K]. All of those constraints are introduced into the action by lagrange multipliers =

=

=

Sconsir=

=

=

fdr[~eP2 + ~

P,,

0 + A~D~~+ ~(D°

—

q)].

(115)

0u~’u~

The quantization is straightforward. The momenta P,, 0, ~ D~, D° become space-time and Lorentz-harmonic derivatives. After that the constraints (113) imply that the wave function is a Lorentz-harmonic field ~“(x~, ui), which describes a massless field, if the weight q’ is an integer less than or equal to —2 (see eq. (53)). Note that the classical value of q in (115) does not necessarily coincide with the quantum weight q’ of the wave function, due to ordering-ambiguities in the operator D° u~~a/au~” u”a/au”. The action S0 + S~can be rewritten in a different form. First by of all, the classical 2 0, P,, 50 0 can be solved presenting the equations of motion P 0u~”u momentum P,, 0 in the “twistor” form =

=

—

=

P,,0=eu~u~

(116)

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/ (Supcr)fie/ds and particles

(e~ is an arbitrary scalar field). Putting this back into the action, together with the equations of motion D~~= 0, Dt~ q 0, and making the change of variables —

=

x,,

~

0 —~x,,0— (1/e

one obtains S

=

fdT[e

u,,~u~i~O + q(à~u-”)].

(117)

For q 0 * this is the “twistor” action of STVZ [13]. Their twistor field A,, ~ Note that to prove the classical equivalence of the STVZ particle to the usual one, one must assume that A * 0. In our case this is a natural consequence of the definition of the Lorentz harmonics. The advantage of the action (112), (115) is the transparent and simple structure of the hamiltonian constraints. They can be read off directly from (115) and form a first-class system. On the other hand, the quantization of the STVZ action is not straightforward and cannot be done covariantly. Here one should mention other twistor-like approaches to the (super)particle [14—16].A common point2 for0. all of them is the twistor variables solution (116) After that the space-time x,, of the light-likeness condition P 0,ut). This is easily done 0in are D re3, placed by pairs of spinors (“twistors”) (x,,0u~ 4. However, in higher dimensions the constraints generalizing (116) are not functionally independent, which leads to infinite reducibility [15]. Attempts have been made to use the notion of quaternionic (for D 6) or octonionic (for D 10) valued spinors [14], but analysis in such spaces is practically impossible. In our approach we do not use the twistor-like solution (116). Instead, we impose the space-time constraints from (113), which are independent. As a result we have harmonic functions of (x,, 0, ut). Note that this is closer to the very old 0u~”u~ idea of ref. [18] than to Penrose’s twistors [11,19]. The supersymmetrization of the action (112), (115) is very easy. One replaces ja0 in (112) by the supersymmetric one-form ~ iO~”80~ and introduces a kinetic term —iD,,O~for 0~(after quantization D,, becomes the spinor covariant derivative D,, =a/a0~ Finally, one adds the Grassmann constraint ~u~”D,, (meaning that 1 is a function of U ~ ±ao only) to (115). The resulting superpartide action is =

=

=

=

=

=

—

—P,,

0).

00

S

=

f dr[P,,0(.k”O

+ =

—

iU~~ 0 13))

—

iD,,0~

+D~(içu~) (à~u~”)+D°(t~u~) 2+ P,, +0 wu~~D,, + A~D~~+ ff(D° —q)]. (118) +~eP 0u~~u~ +D

*

The ease q ±0 has been considered in ref. 1201, although we cannot agree with the interpretation of the harmonic space given there.

F. Delduc ct al.

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(Supcr)fields and particles

169

Once again, the constraint system is first class. In particular, (u~”D,,,U~0D0}

0P,,

=U~”U~

(119)

0.

Quantization gives a Lorentz-harmonic wave function P”(x~, 0~,uk), which describes an on-shell supermultiplet (for specific values of q). Some important comments are due at this point. In the Brink—Schwarz superparticle [1] the Grassmann constraints are 13,, 2 0. 0,Because of the algebraic these constraints form a relation {D,,, D0} (yt~~),,0P~, and the constraint P mixed first—second-class system. Their covariant separation is not possible in the =

=

=

framework of BS. To solve this problem Siegel [3] replaced the constraints 13,, 0 by P, 0 0. The latter are first class ({(PD),,, (PD) 2 0), but this is 0D 0) P,,0P not the end of the story. Those constraints generate a fermionic gauge invariance (“kappa symmetry”), ~0,, P,, 0. Because of the degeneracy of the matrix P,,, 13,~ 0 these transformations affect only half of the components of 0,,, but that half cannot be singled out in a covariant way. This is the origin of specific problems (infinite reducibility [21], or “ghosts for ghosts” [22]) in the quantization of the system. It is clear that in the Lorentz-harmonic case we have actually covariantly projected out the first-class part of the BS constraint D,, 0. This is the harmonic projection In aone sense, this constraint equivalent to that of Siegel, 0 0, u~”D,, assuming0.that has first solved the ismasslessness constraint P2 0 P,,0D in the form (116). Note also that the closure of the algebra (119) of our Grassmann constraints requires that we add the constraint P~= 0 to the usual one P2 0. This in turn leads to the vanishing of the transverse part of Pt,, and one ends up with the Lorentz-harmonic superspace Mx~,0~,u~}. The generalization to higher dimensions is, in principle, straightforward. The constraints now involve the SO(D 2) generator D’2 as well (see eq. (7)). In D 6, 10 one has to introduce new sets of harmonic variables, which help to define the on-shell superspace containing only ~- of the initial number of U’s. Correspondingly, one adds new harmonic constraints to the action (see (100), (109)). In addition to the Grassmann constraint D~cI 0 (it eliminates the 0 dependence), one has to impose the chirality condition D’~P 0 in 13 4 and the similar ones D~’~I 0 in 13 6 and D~’cI 0 in D 10. The D 4 and D 10 ones are not real (the one for D 6 is real in the sense of (102)). Therefore special care is needed when including them into the superparticle action. This can be done by combining the chiral wave-function i and its conjugate anti-chiral one f in a doublet (1, D) and introducing (2 X 2) y-matrix variables into the action (just like one describes spin- ~ in particle quantum mechanics). Then the pair of chirality—antichirality conditions DD) can be introduced by means of the projectors y~y and yy~. Thus the action can be kept hermitian. Finally, the reality condition (111) for the 13 10 Maxwell supermultiplet can be introduced by another combination of those projectors. The details will be given in a future publication. =

=

=

=

=

=

=

=

=

=

—

=

=

=

=

=

=

=

=

(DcP,

=

=

=

=

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(Super)ficlds and particles

7. Conclusions In this paper we have presented a study of the Lorentz-harmonic spaces and superspaces based on the new compact coset of the Lorentz group. We have applied that formalism to the old problem of covariant quantization of the superparticle. We believe we have found a convincing, sufficiently simple and straightforward scheme, which does not suffer from the usual problems (mixed first- and second-class constraints, functionally dependent or infinitely reducible constraints, etc.). Our formulation is close in spirit to the twistor formulations, although we have taken special care to clarify the meaning and role of the commuting spinors as Lorentz-harmonic variables. There are two major problems to which one should try to apply the new formalism. One is to covariantly quantize the Green—Schwarz superstring. We hope that this will be possible, although some important modifications may be needed (for instance, in string theory there are two light-like vectors). The other, perhaps even more important issue is the second quantization of the superparticle. An eventual success in this direction would solve the old problem of finding off-shell formulations of the D 10 supersymmetric theories, and might open up the way to a new superstring field theory. =

The authors are grateful to F. Bergshoeff, J. McCabe, Y. Eisenberg, P. Howe, R. Kallosh, V. Ogievetsky, W. Siegel, K. Stelle and J. Yamron for useful discussions. AG. would like to thank the Theoretical Group at IC, London and E.S. would like to thank LPTHE, Paris for warm hospitality.

References Ill L. Brink and J. Schwarz, Phys. Lett. Bl00 (1981) 310 121 M. Green and J. Schwarz, Phys. Lett. B136 (1984) 367; NucI. Phys. B243 (1984) 285 [31W. Siegel, Phys. Lett. B128 (1983) 397; B203 (1988) 79; Class. Quantum Gray. 2 (1985)

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