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Some remarks on the conformal group
Vol.31(1992)
REPORTS
ON MATHEMATICAL
No. 2
PHYSICS
SOME REMARKS ON THE CONFORMAL GROUP* J. G. PERERA Institute de Fisica Tebrica, Universidade Estadual Paulista, Rua Pamplona 145, 01405Sao Paulo-SP, Brazil (Received November 18, 1991)
It is shown that the local isomorphism between the conformal group of Minkowski spacetime and the group SO(4,2) makes Sense only if one eliminates from SO(4,2) one of the Sitter boosts contained in it.
In recent years, there has been a growing interest in the study of conformal invariance in field theories as well as in the study of gauge models for gravitation. As a contribution to those developments, we make here some remarks concerning the structure of the conformal group. In particular, we make an effort to clarify the local isomorphism between the conformal group of Minkowski spacetime and the group SO(4,2). The main result is as follows. Starting from a spacetime conformal transformation exhibiting the SO(4,2) structure, in order to rewrite it as the sum of a Lorentz transformation, a translation, a special conformal and a dilatation, we show here that it is necessary to eliminate one of the de Sitter boost transformations occurring in the SO(4,2) conformal group. Consequently, the translation and the special conformal parameters of the resulting conformal transformation will not be independent. As is well-known, the conformal group can be interpreted [l] as the group of hyperbolic rotations in a flat 6-dimensional space, with coordinates xA (A, B, C... = 1, . .. . 6), leaving invariant the 5-dimensional hypersurface defined by
mEXAXB = l2 with Nan the metric of that space VAB
=
(1)
which here is chosen to be diag(-,
-,
-,
+;
-,
+),
(2)
and l- a parameter (length) related to the (constant) curvature of the hypersurface. infinitesimal conformal transformation of the coordinates xA is given by
The
6xc = $WABJ~B~C, where SWAB = -6WBA
are the parameters of the transformation,
and JAB denote the
* A previous version of this paper has been selected for Honorable Mention in the Gravity Research Foundation Essay Contest, USA, 1989.
186
J. G.
generators written in some appropriate ;[JAB,
PEREIRA
representation.
JCDI= 77 AD J BC-VBDJAC
These generators satisfy [l]
+TIBCJAD-QACJBD.
(4)
In this formulation, the conformal group explicitly exhibits the SO(4,2) structure. Let us consider an arbitrary field d(z). In order to obtain its transformation under a spacetime conformal group exhibiting the SO(4,2) structure, it is necessary first to write the generators and the corresponding parameters in a four-dimensional notation. This can be achieved through a stereographic projection [2], yielding the relations [l] (cr,P,r )... = 1)“‘) 4)
J ap = Lao, Jo5 = l&,-),
J56 = D, Ja6 = ET@.
(5)
The set {&” , Lap} generates the transformations of the de Sitter group SO(3,2), and {ni-,-), Lap} generates th e t ransformations of the de Sitter group SO(4,l). In terms of these generators, the commutation relations (4) can be rewritten in the form f[L+
L-y61 = Sa6JQ,,- L&5JL,+ wy.LG- Sayq36,
f[L,p, D] = 0; f [L,,
f[D, D] = 0,
q’l = gps7&*yf) - g,sq,
(*I1 = w2L2flc, $T,(*)‘TO
(64 (6.b)
(6.4 (6.4
f
$rh+‘, $‘] = K2gapD,
(64
The generators &*) are related to the translation and special conformal generators by [l] 7&? = i(Pa f Z-W,).
(7)
Using these expressions in eqs. (6) we obtain the usual commutation relations of the conformal group [3] in terms of the generators Lap, Pa, K,, D. In the same way we have redefined the generators, we now redefine the group parameters according to 6WY” = SWYa, &WY5= Z%4P[-]
= E-1(&9 - 1%?),
6W76 = z-‘sw7[+]
= z-‘(Sa’ + P&Y),
sws6 = SW,
Pa) (8.b) (8.c) (8.d)
where SwY[-] and SwY[+] are the parameters related, respectively, to the boost generators x$-) and XV) of the de Sitter groups SO(4,l) and SO(3,2). Thus, in 4-dimensional
SOME REMARKS ON THE CONFORMAL GROUP
notation, the SO(4,2) conformal transformation form 64(z) = i[;Gw”pL,p
187
of an arbitrary field 4(z) is written in the
- W-]7&,-)
+ CW[+]7rP
- &JD]~(z).
(9)
An important point is the fact that the terms involving the generators ~2) and 7ri-j appear with different signs. This is a consequence of the property
always presented by the metric v,@. Using eqs. (7) and (8), we obtain
+ l-2Sa"Ka-SwD]q5(z). 64(X) = i[$!kL@L,~ + 12Sc"Pa
(10)
Notice that the translation generator appears multiplied by the special conformal parameter, and vice-versa. In order to obtain the usual expression for a conformal transformation of an arbitrary field 4(z), given by [3] Q(z)
+ &z"P,+ Sc"K, - sbJD]q5(~)
= i[$%LP&/j
(11)
it is necessary to impose the constraint SC” = 1-26a”.
(12)
But, according to eq. (8.b), this is equivalent to fW[-]
= 0,
(13)
which means that the boost transformation of the de Sitter group SO(4,l) is eliminated. Alternatively, we could have imposed the condition &P[+]
= 0,
(14)
or equivalently SC* = -I-%“,
(15) which would eliminated the boost transformation of the de Sitter group SO(3,2). In this case, we would obtain a conformal transformation that differs from eq. (11) in the signs of ba* and SC*. In either case, one of the de Sitter boosts will be absent in the resulting conformal transformation, and consequently the group parameters will not be completely independent. Of course, it is possible to define a transformation in the basis {L, P,K, D} with a completely independent set of parameters, but this transformation will not present the pseudo-orthogonal structure of the SO(4,2) group. Summing up: we have succeeded in obtaining the usual expression for the conformal transformation of an arbitrary field 4(z) starting from a transformation that exhibits the SO(4,2) structure. However, in order to do that, the group parameters must satisfy one of the conditions Sw”[f] = 0.
(16) The resulting conformal transformation, therefore, is such that one of the de Sitter boosts occurring in the SO(4,2) group must be absent. Only in this case, the {L,P,K, D} conforma1 transformation, eq. (1 l), will be a pseudo-orthogonal transformation with the SO(4,2)
188
J. G. PEREIRA
structure. These remarks, aside from their natural interest for the understanding of the conformal transformations, may be particularly important in the study of gauge models for the conformal group as alternative theories for gravitation [4]. In this context, conditions (16) turn out to be related to the inverse Higgs effect, [5] a useful device of theories with nonlinear realizations and spontaneous breakdown of gauge symmetry. Acknowledgements
I would like to thank the hospitality at the Center for Relativity, The University of Texas at Austin, where this work was done. This work was partially supported by Conselho National de Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil. REFERENCES [l] [2] [3] [4]
[5]
G. Mack and A. Salam: Ann Phys. 53 (1969), 174. F. Giirsey: in Lectures of the Istanbul Summer School of Theoretical Physics, ed. F. Giirsey, Gordon Breach, New York, 1962. E. S. Fradkin and M. Ya. Palchik: Phys. Rep. 44 (1978), 249. See for example: A. B. Borisov and V. I. Ogievetskii: Theor. Math. Phys. 21 (1973), 1179. M. Kaku, P. K. Towmsend and P. V. Nieuwenheuizen: Phys. Lett. B 69 (1977), 304. C. Fronsdal: Phys. Rev. D 30 (1984), 2081. E. A. Ivanov and V. I. Ogievetskii: Theor. Math. Phys. 25 (1975), 1050.
and
REPORTS
ON MATHEMATICAL
No. 2
PHYSICS
SOME REMARKS ON THE CONFORMAL GROUP* J. G. PERERA Institute de Fisica Tebrica, Universidade Estadual Paulista, Rua Pamplona 145, 01405Sao Paulo-SP, Brazil (Received November 18, 1991)
It is shown that the local isomorphism between the conformal group of Minkowski spacetime and the group SO(4,2) makes Sense only if one eliminates from SO(4,2) one of the Sitter boosts contained in it.
In recent years, there has been a growing interest in the study of conformal invariance in field theories as well as in the study of gauge models for gravitation. As a contribution to those developments, we make here some remarks concerning the structure of the conformal group. In particular, we make an effort to clarify the local isomorphism between the conformal group of Minkowski spacetime and the group SO(4,2). The main result is as follows. Starting from a spacetime conformal transformation exhibiting the SO(4,2) structure, in order to rewrite it as the sum of a Lorentz transformation, a translation, a special conformal and a dilatation, we show here that it is necessary to eliminate one of the de Sitter boost transformations occurring in the SO(4,2) conformal group. Consequently, the translation and the special conformal parameters of the resulting conformal transformation will not be independent. As is well-known, the conformal group can be interpreted [l] as the group of hyperbolic rotations in a flat 6-dimensional space, with coordinates xA (A, B, C... = 1, . .. . 6), leaving invariant the 5-dimensional hypersurface defined by
mEXAXB = l2 with Nan the metric of that space VAB
=
(1)
which here is chosen to be diag(-,
-,
-,
+;
-,
+),
(2)
and l- a parameter (length) related to the (constant) curvature of the hypersurface. infinitesimal conformal transformation of the coordinates xA is given by
The
6xc = $WABJ~B~C, where SWAB = -6WBA
are the parameters of the transformation,
and JAB denote the
* A previous version of this paper has been selected for Honorable Mention in the Gravity Research Foundation Essay Contest, USA, 1989.
186
J. G.
generators written in some appropriate ;[JAB,
PEREIRA
representation.
JCDI= 77 AD J BC-VBDJAC
These generators satisfy [l]
+TIBCJAD-QACJBD.
(4)
In this formulation, the conformal group explicitly exhibits the SO(4,2) structure. Let us consider an arbitrary field d(z). In order to obtain its transformation under a spacetime conformal group exhibiting the SO(4,2) structure, it is necessary first to write the generators and the corresponding parameters in a four-dimensional notation. This can be achieved through a stereographic projection [2], yielding the relations [l] (cr,P,r )... = 1)“‘) 4)
J ap = Lao, Jo5 = l&,-),
J56 = D, Ja6 = ET@.
(5)
The set {&” , Lap} generates the transformations of the de Sitter group SO(3,2), and {ni-,-), Lap} generates th e t ransformations of the de Sitter group SO(4,l). In terms of these generators, the commutation relations (4) can be rewritten in the form f[L+
L-y61 = Sa6JQ,,- L&5JL,+ wy.LG- Sayq36,
f[L,p, D] = 0; f [L,,
f[D, D] = 0,
q’l = gps7&*yf) - g,sq,
(*I1 = w2L2flc, $T,(*)‘TO
(64 (6.b)
(6.4 (6.4
f
$rh+‘, $‘] = K2gapD,
(64
The generators &*) are related to the translation and special conformal generators by [l] 7&? = i(Pa f Z-W,).
(7)
Using these expressions in eqs. (6) we obtain the usual commutation relations of the conformal group [3] in terms of the generators Lap, Pa, K,, D. In the same way we have redefined the generators, we now redefine the group parameters according to 6WY” = SWYa, &WY5= Z%4P[-]
= E-1(&9 - 1%?),
6W76 = z-‘sw7[+]
= z-‘(Sa’ + P&Y),
sws6 = SW,
Pa) (8.b) (8.c) (8.d)
where SwY[-] and SwY[+] are the parameters related, respectively, to the boost generators x$-) and XV) of the de Sitter groups SO(4,l) and SO(3,2). Thus, in 4-dimensional
SOME REMARKS ON THE CONFORMAL GROUP
notation, the SO(4,2) conformal transformation form 64(z) = i[;Gw”pL,p
187
of an arbitrary field 4(z) is written in the
- W-]7&,-)
+ CW[+]7rP
- &JD]~(z).
(9)
An important point is the fact that the terms involving the generators ~2) and 7ri-j appear with different signs. This is a consequence of the property
always presented by the metric v,@. Using eqs. (7) and (8), we obtain
+ l-2Sa"Ka-SwD]q5(z). 64(X) = i[$!kL@L,~ + 12Sc"Pa
(10)
Notice that the translation generator appears multiplied by the special conformal parameter, and vice-versa. In order to obtain the usual expression for a conformal transformation of an arbitrary field 4(z), given by [3] Q(z)
+ &z"P,+ Sc"K, - sbJD]q5(~)
= i[$%LP&/j
(11)
it is necessary to impose the constraint SC” = 1-26a”.
(12)
But, according to eq. (8.b), this is equivalent to fW[-]
= 0,
(13)
which means that the boost transformation of the de Sitter group SO(4,l) is eliminated. Alternatively, we could have imposed the condition &P[+]
= 0,
(14)
or equivalently SC* = -I-%“,
(15) which would eliminated the boost transformation of the de Sitter group SO(3,2). In this case, we would obtain a conformal transformation that differs from eq. (11) in the signs of ba* and SC*. In either case, one of the de Sitter boosts will be absent in the resulting conformal transformation, and consequently the group parameters will not be completely independent. Of course, it is possible to define a transformation in the basis {L, P,K, D} with a completely independent set of parameters, but this transformation will not present the pseudo-orthogonal structure of the SO(4,2) group. Summing up: we have succeeded in obtaining the usual expression for the conformal transformation of an arbitrary field 4(z) starting from a transformation that exhibits the SO(4,2) structure. However, in order to do that, the group parameters must satisfy one of the conditions Sw”[f] = 0.
(16) The resulting conformal transformation, therefore, is such that one of the de Sitter boosts occurring in the SO(4,2) group must be absent. Only in this case, the {L,P,K, D} conforma1 transformation, eq. (1 l), will be a pseudo-orthogonal transformation with the SO(4,2)
188
J. G. PEREIRA
structure. These remarks, aside from their natural interest for the understanding of the conformal transformations, may be particularly important in the study of gauge models for the conformal group as alternative theories for gravitation [4]. In this context, conditions (16) turn out to be related to the inverse Higgs effect, [5] a useful device of theories with nonlinear realizations and spontaneous breakdown of gauge symmetry. Acknowledgements
I would like to thank the hospitality at the Center for Relativity, The University of Texas at Austin, where this work was done. This work was partially supported by Conselho National de Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil. REFERENCES [l] [2] [3] [4]
[5]
G. Mack and A. Salam: Ann Phys. 53 (1969), 174. F. Giirsey: in Lectures of the Istanbul Summer School of Theoretical Physics, ed. F. Giirsey, Gordon Breach, New York, 1962. E. S. Fradkin and M. Ya. Palchik: Phys. Rep. 44 (1978), 249. See for example: A. B. Borisov and V. I. Ogievetskii: Theor. Math. Phys. 21 (1973), 1179. M. Kaku, P. K. Towmsend and P. V. Nieuwenheuizen: Phys. Lett. B 69 (1977), 304. C. Fronsdal: Phys. Rev. D 30 (1984), 2081. E. A. Ivanov and V. I. Ogievetskii: Theor. Math. Phys. 25 (1975), 1050.
and