Superfields with arbitrary spin spectral representation of the dilatational invariant two-point function

Superfields with arbitrary spin spectral representation of the dilatational invariant two-point function

Vol. 9 (1976) REPORTS ON MATHEMATICAL No. 3 PHYSICS ” SUPERFIELDS SPECTRAL WITH ARBITRARY SPIN REPRESENTATION OF THE DILATATIONAL TWO-POINT ...

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Vol. 9 (1976)

REPORTS

ON

MATHEMATICAL

No. 3

PHYSICS



SUPERFIELDS SPECTRAL

WITH ARBITRARY

SPIN

REPRESENTATION OF THE DILATATIONAL TWO-POINT FUNCTION

INVARIANT

R. P. ZAIKOV Joint Institute for Nuclear Research, Laboratory (Received

November

of Theoretical

Physics, Dubna,

U.S. S. R.

26, 1976)

The superfields with arbitrary spin are given as homogeneous functions of one twocomponent complex commuting spinor and one anticommuting spinor. The spectral representation of the invariant two-point function is found.

1. According to Salam and Strathdee [2] the scalar superfield in the Majorana anticommuting spinor 8, i.e. qx;

e) =

is written

as a polynomial

~(x-)+~~(x)e,+~~~~~~(x)e,e,+~~~~~~~I(X)e,e,e,+~~=~~~~~~1(X)e,e,e,e, (ct,B )...

= l)...,

4).

(1.1)

The fields with arbitrary spin may be written in a similar way [4], i.e. as one homogeneous function of the two-component complex (commuting) spinor Yqx; AZ) = ;1y#qx;

z),

(1.2)

where x = [Ye, vz] give irreducible representations of SL(2, C) [4]. We introduce the superfields with arbitrary spin in the following qx;

;lz,

e) = AY?%D(~,

way

z, e).

(1.3)

With respect to f3, the fields U(x; z, 0) are polynomials of degree 2. Here z and r9 are twocomponent complex spinors and 8 are elements of the Grassmann algebra, i.e.

4

[e,, e,]+ = [e,, The transformation

laws of superfield qa,

rl)~(~;

z,

ep-ya,

= b,, zbi- = 0

(a,b = 1,2).

(1.3) are:

A) = ~(il~+~;

z~-l,

eA-y, (1.4)

u(qqx;

z,

e)u-qc()

= Y x++~pB-ea,,a); [

z, e+a

1

where 11 E SO(3, l), A E SL(2, C), a E T4, and cc are two-component complex anticommuting spinor parameters of supertransformations. In the basis in which ys is diagonal we have correspondence between (1 .l) and (1.3) for the scalar case. 13871

388

R. P. ZAIKOV

For the tensor in the following Lr(c+D(x,, where

Q

are given

fields (yl = yz) the supertransformation

law (1.4) may be generalized

way

[ .

&,e)u(a)-’

= @ x,,+.-y

and b are two arbitrary

(ao,,e-ea,cr),

parameters.

&&Q-o~Pz),

e+cr

In the infinitesimal

form

1, (1.5)

(1.4) and

(1.5)

by [CC, @] =

(1.6)

[G’Y@I= We have the transformation 2.

Consider

the two-point

law (1.4) if

a

=

I

and b = 0.

function

F(X1,21,e1,xz,Z2,e2)

=
(2.1)

where the fields YO(x,; z,) are transformed according to some irreducible representations x = [v~, vz] of SL(2, C). The superinvariance conditions for the two-point function (2.1) are (2.3) F

Xi+;-

(a~,~6b-8bU,E);

zb,

=

8b+a

F(X;;

Zb, 6”).

I

[

To satisfy (2.2) and (2.3) it is convenient to pass to the momentum account the translational invariance and spectrality, we write down qx,

-x2

; za, 0s) = S dyiyp)

c~P(xI-x~)&

; Z=, eq

space. Taking

into

(2.4)

where 6(p) = 0(p”)O(p2) is the characteristic function of the future cone. Consider first the relativistic invariance condition (2.2). This condition is satisfied

if

the kernel k(
Using

E;b = Z%@Zb,

r;b =

eyjb,

c;” = Z%Jb,

<;b =

ea~,zb.

(2.5)

the identity gqTLo(l$q,)C~ = 2Eq$

one may prove

the following

identities

x;‘Y,2*+x;*Yyzl

= +2g,,x'&y2x1&j2,

(,,b)2

(fll”)z =

(rab)’ = -

= (pb)2 =

0,

2e”Ee”8be6b (Q, b = i , 3,

(2.6)

SUPERFIELDS

WITH AN ARBITRARY

389

SPIN

where E = ia, and the sign+ 1 is for the case [x, u]- = 0 and sign- 1 when [x, y], = 0. Following paper [4], one can prove that the kernel i is a function of the following 14 independent relativistic invariants

; za,P) = F(jP, Z1&Z2, 2&2, ZIP?, z’pZ’, z2pz2,e1d2, e”pab,elszl, &il, &J

The irreducibility

condition

elpZ1) (p = p@a,J.

(2.7)

(1.3) gives ([4])

k. m, n=O

x H,&

p2(z1a,Z1)(z2aPZ2)

p2, w = l -

(z’pZ’) (z”pZ’)

, 01&02,#&, =

eap# ,

(2.8).

where vr = lI+lo-I, v2 = 1,-1,-l and Hklm are arbitrary functions of p2 and w and polynomials in 19~and (_I2of degree 2. Equation (2.8) gives the general form of the relativistic invariant two-point kernel for the superfields with arbitrary spin. The superinvariance condition (2.3) in the infinitesimal form is written as follows: G"k(p; Zb,eb)= o, G"F(p;zqeb) = 0

where G and G are the generators of the supertransformations acting From (1.10), (1.11) (for a = 1 and b = 0), and (2.4) we have G” =

(2.9

(a,b = 1,2),

on the kernel

&+3;2-+; (p?@“+;(p32)“, a (I

(2.10)

G” = +~ + 2& + + (elp); - + (ezp)i Substituting

i.

(I 0 (2.10) in (2.9), we have

,

+

(ZIP)“‘--cc

+ (Z’&&

iTz'p3'

= 0.

(2.11)

(2.12)

390

R. P. ZAlKOv

The solution

of this system

is given

by

x

dxf($ s

where_f($,

x) is an arbitrary

F’ = 6(0, -0,)

function

f

X)ln(ull+u*l)+(:-x)Y**-(~+~)~~~

of p2 and x .There is a second

(z’EZ2)la+lb(t’pZ2)I,-I~(~l~~l)J,-l,-’(Z2pZ2)1;-I,--lg(P2,

When lo, 1; # 0 there y(x-, z) is transformed

solution ~I,oI),

where g(p2, 8~8) is an arbitrary function. The function (2.13) is not invariant under the space reflections. = 0 the function (2.13) possesses such an invariance if ([3])

f(P2, x>= 4P’)

3

(2.13) given by (2.14)

In the case lo = /A

w .

is invariance with respect to space reflections, provided the field according to representations [lo, I,]@ [-lo, /,I of SL(2,C).

3. The two-point function (2.13) describes the propagation of particles with spin max(O, ]l0l - 1) Q s < I, for finite-component fields and max(O, ]l0] - 1) < .c < co when both the fields are infinite-component ([I]). Let us decompose the superinvariant kernel (2.13) into the sum over the spin variable, i.e. over the eigenvalues of the second Casimir operators of the Poincare subgroup

&;

z’, e’) =

c F&I;

Z', 0”).

(3.1)

ey = 0,

(3.2)

s Here

2, satisfy

the following

equation [+-.s(s+

l)]F,(p;

za,

where

is the second Casimir operator of the Poincare group. Consider first the scalar superfields (these fields contain case the generators of the Lorentz group are given by

spins .Y= 0, l/2,

1). In this

(3.4)

SUPERFIELDS

Substituting

(3.4) in (3.3),

WITH AN ARBITRARY

we write

equation

SPIN

391

(3.2) in the relativistically

invariant

variables

The solutions

of this equation

are given

by (3.6j

where f(p2) t:,

=

is an arbitrary

1.

t;

=

function t,”

ll,,U22.

=

and t,” are given t,3 =

112 12,

112 21,

by t;

=

112 12 112 21,

(3.7) t$,

t:

=

=

tl’,/2

1112,

u,,ul_+2u 7

=

u2,,

t:

2 =

u121i:,,

$2

=

N2111f2,

12Ii2,.

The coefficients c,, may be found (2.13) and (3.6) and (3.7) we get

from

the power

decomposition

of (2.13). From

C; = 8C; = SC,” = SC; = 64C,s = 2C;;l

= -2Cf2

= l6Cz,2 = -16C;(;,

In the case of superfields with spin, the decomposition of complicated. From (2.15) it follows that we may decompose with over the variables 0 and z separately. The decomposition variables z is given in [4]. Combining these decompositions,

= -SC:

= 1.

(3.8)

the kernel (2.14) is more the kernel into the sum respect to the commuting we get

(3.9) where (T&is an arbitrary function, C:,l are given by (3.8) and P~“l’(s) are the Jacobi polynomials. Any term in (3.9) describes propagation of particles with spin 1.~~-x2] d .\ < s,+s,. The kernel (3.8) is local for the finite-component fields ([4]). 4. The kernel equation ([5]j:

(2.15) is invariant

with respect

D&I,

to dilatations

@I, zl, 02, z2) = 0.

if it satisfies

the following (3.1)

392

R. P. ZAIKOV

where

D is the generator

acting

on the fields,

This operator symmetric

of dilatations,

given

which

is the sum of corresponding

operators

by

obeys the followin g commutation

relations

with generators

of the super-

algebra [D,

M,,,,.l= 0,

[D, Pul = -if,,. [D,G"]= ; (j".

[D,G"]= -;G", Substituting

(1.3)

(3.2) in (3.1). \\c obtain

(3.1) From (2.13) and (4.4) WC find that the general symmetric two-point kernel is given by

form of the dilatationnlly

invariant

supcr-

Acknowledgement The author is indebted to Dr Tz. Stojanov for valuable discussions and to Professor5 D. 1. Blokhintsev and V. A. Meshcherjakov for hospitality kindly extended to him at the Laboratory of Theoretical Physics of JINR Dubna. where the present paper \\a\ completed. REFEKENCES [I] O’Raifeartaigh,

L.:

I21 Salam,

J. Strathdee:

A., and

131 Stojanov,

D.Tr.,

141 Todorov.

I. T., and

iS] Zaikov,

R. P.:

M/ci,ollt &ugrtm~

and Bulg.

I.T.

for

s~~prfi~/~/.s, Preprint

;h;ucI. Phys. Todorov:

R. P. Zaikov: J. Phys.

B76 (1’)74),

J. Math. ibid. 9

2 (1975),

89.

Phys.

DI AS, Dubl’n.

477. 8 (1969).

(1969). X13.

2146.

lY7-l