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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 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
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