- Email: [email protected]
Consistent Dirac bracket for dynamical systems with infinitely reducible second class constraints
3 October
1996
PHYSICS
LETTERS B
Physics Letters B 386 (1996) 141-145
ELSEVIER
Consistent Dirac bracket for dynamical systems with infinitely reducible second class constraints A.A. Deriglazov ‘, A.V. Galajinsky, Department
S.L. Lyakhovich
ofTheoretical Physics, Tomsk State University, 634050 Tomsk, Russia Received 18 May 1996 Editor: PV. Landshoff
Abstract
The standard Dirac bracket construction is generalized to the case of infinitely reducible second class constraints. Reducibility of constraints is shown to manifest itself in weakening the Jacobi identity for the construction, that now holds on the second class constraint surface only. The proposed scheme is realized for N = 1 superparticle model in ten dimensions. P4CS: 04.60.D~; 11.3O.Pb Keywords: Covariant
quantization;
Mixed constraints;
Dirac bracket; Superparticle
1. A constructing of the Dirac bracket associated with reducible second class constraints is an important ingredient of covariant quantization scheme [ 1,2] for dynamical systems subject to mixed first and second class constraints. This construction to a great extent was designed to resolve covariant quantization problem of the superparticle [ 31 and superstring [4] theories (see Ref. [5] for a review). A complete constraint system of the models includes fermionic constraints which, being a mixture of half first class and half second class constraints, lie in the minimal spinor representation of the Lorentz group. Projectors that realize a covariant (reducible) split of the constraints into first and second class have been constructed in Refs. [ 6,7]. A recipe how to generalize the Dirac bracket to reducible case, i.e. to the case when second class constraints xa TZ0 satisfy the identity xaCaO = 0, where
’ E-mail: [email protected]. 0370-2693/%/$12.00 Copyright PII SO370-2693(96)00905-7
the C,” are (weak) null eigenvectors 2 of the Poisson brackets matrix AUp E {xn, xp}, has been formulated in Refs. [ 5,6,8]. The leading principle was to achieve a weak commutation of any quantity with the second class constraints. Presence of the property allows, by analogy with Dirac’s prescriptions of quantization, one to set the second class constraints strongly equal to zero and pass to the reduced phase space [ 91. It seems to be unnoticed, but the Jacobi identity was not automatically provided for the scheme. In the general case, however, the bracket of such a kind may possess the identity in a strong sense, weakly or has no the property at all. For instance the bracket of Ref. [6] possesses the Jacobi identity on the second
* In the general case the C,” themselves may possess null eigenvectors (the next stage of reducibility [ 1 ] ). The most interesting example is [5,6] Au0 = gu,.p,p, where prp is a projector p2 = p, pT = p and g is a reversible matrix. In this case C = 1 - p has evident null eigenvector pA with A being an arbitrary function.
0 1996 Elsevier Science B.V. All rights reserved.
AA. Derigluzou et al./Physics Letters B 386 (1996) 141-145
142
class constraint surface only [ 71, while certain cycles break the identity for the bracket of Ref. [ lo]. Since, in passing to quantum description, correct correspondence rules imply the Jacobi identity for the classical counterpart of the quantum commutator, the question of how to ensure the property for the scheme acquires a great importance. In this letter we propose the general scheme for constructing the consistent covariant Dirac bracket associated with infinitely reducible second class constraints. It will be shown that building such a bracket is equivalent to solving certain system of algebraic equations in appropriately extended phase space. As an application we realize the scheme for the Brink-Schwarz superparticle model in D = 10.
[ 81. Note as well that rank A 1= n/2 by virtue of Eq. (3), which correctly reflects reducibility of the constraints involved. To construct a bracket under which any quantity weakly 3 commutes with the second class constraints, let us find a (symmetric) matrix zap being inverse to the AaP in the following sense: Ai =p-
(4)
Taking into account consequences p+Kp-
= 0,
+2_ + P -P 1 p+p- = 0, satisfying
-2
P
=p-9
(1)
p++p-=1
the conditions
LY,‘, X,‘> M 0, L&Y;1 {x,3 xi} = Asp = (P-A’P-
= 0, lap,
(5)
+p+&p+,
(6)
where A1 is some matrix consistent with Eq. (4) and & is an arbitrary matrix. As will be shown, only the first term from Eq. (6) is essential in building the Dirac bracket. In what follows we will also suppose that the following conditions {x,ttclA~)
=o,
{,&tcIA2)
(7)
x 0
hold 4 . With the & at hand, generalization of the Dirac bracket to reducible case present no special problem. The suitable ansatz is {A, B}D = {A, B} - {A,x,}~~~{x& - {A,
(CIA~}Q~~~Z{$C~,
B},
B} (8)
where 6 is the inverse matrix to VA2B2= {@AZ,I,~B?}, 00 = 1. For specific models the brackets like Eq. (8) were also considered in Refs. [ 5,6,10]. Several comments on the structure of the bracket are relevant here. First, any quantity A weakly commutes with the second class constraints under the bracket (8)
(2)
{A,$B~}D =O,
where x * = p*x. From Eqs. (2) it follows the determining equation for the projector operators A*p+ M 0.
=0
one comes to the decomposition a = p-alp-
2. Consider a model constraint system of which includes (among other) fermionic constraints xn z o,Ly= l,... , n, and suppose half of them to be first class and another half to be second, i.e {xa, xp} -_ A& and rank A*1 = n/2. Since in the original phase space r, covariant separation of the constraints may present a nontrivial task [ 5, 111, we extend this space to r* = (r, radd ) , where r&d is some set of additional variables. It is implied that nondynamical character of the variables is provided by some new constraints. Denote first and second class constraints of this type as ‘PA,(r*)MOand~IA2(r*)M0,Al=l ,..., nl,Al= 1,. . , n2, respectively and we suppose the latter to be linearly independent. To split the constraints xa z 0 in reducible and equivalent manner, we suppose the existence in the space r* of a pair of (strict) projectors
p-hp+
of this equality
{A, X,}D = {A, P,~}x~ In the geometrical brackets matrix
= 0.
(9)
terms it means that the Poisson
(3)
Since the second class constraints have been extracted by means of a strict projector, they are infinitely reducible and automatically satisfy the criterium of Ref.
’ Here and up to second 4 Equations represent the
later by the term “weak equality” we mean equality class constraints. (7) are only technical restrictions that allow one to bracket (8) in the simplest form.
AA. Deriglazov et al./Physics
-
‘dik;kt+bA2f?A2B2 a ~~~B2dj
is degenerate (corankp’i null eigenvectors I)ija,j$A? = 0,
pijajx;
= 122+ n/2)
M0
(10) and (weak)
- 0. Jxp -
143
M ,+, is reversible Map@* = aay and consequently &A = -fi(&M)A?.For the case concerned, however, we deal with Eq. (4), from which it immediately follows the another relation
(11)
are normals to the second class constraint surface. It is worth mentioning that the latter condition can be strengthened to a strict equality #jp-Pa. a
Letters B 386 (1996) 141-145
(12)
The first term in r.h.s. of Eq. (16) reproduces the situation of the standard Dirac bracket. The remaining terms are manifestation of reducibility of the constraints involved and it is their contribution into the cycle ( 15) which breaks the identity. To suppress the contribution let us impose the following restrictions
Second, there is a natural arbitrariness in definition of the bracket being related to the possibility to add arbitrary combination of second class constraints to the ansatz (8). By this reason, the contribution of the second term from Eq. (6) into the bracket (8) can always be neglected, if one takes into account the identity
Considering now Eq. ( 15) with p’j being presented in the form ( 10) and making use of Eq. ( 16) and ( 17) one can get, after tedious calculations, the equality
p;“&x,
( -l)‘j”pk’Ylpij
= -(Z”p;“)x,.
(13)
{PZ
XY)
(‘kA2493
= 0,
+ cycle(ijk)
-(-1)
I’+& = 0.
+ cycle( ijk) .
Third, by construction the bracket (8) possesses the graded symmetry, linearity and acts in a functional space as differentiation. Thus, the only conventional property to be discussed is the Jacobi identity. It is straightforward to check that for the bracket ( 8) the identity does not hold. We will show, however, that imposing further restrictions on the structure of the projectors p,fp and the constraints @AZM 0 one can provide the property on the second class constraint surface. For this purpose consider the graded cycle
+ (-_l)~~~~k~a~,ii
(15)
which identically vanishes for the ordinary Dirac bracket and ensures the Jacobi identity for the construction. The proof of the identity in the irreducible case is, actually, based on the fact that the matrix of Poisson brackets of second class constraints, say
=
l,Q+fitE,+%,) [WkL-Wk”;;,~Xp~PyapX;““I]
The latter circumstance means that one can search for the A in the form h = p-&lp- and consider that (14)
(17)
=o,
x wiGnx,
[ ps’” &P
+ ;,iPp;q
&/y$LPj ( 18)
Taking into account now the identity ( 13) one concludes that, under the restrictions ( 17)) the Jacobi identity for the bracket (8) holds on the second class constraint surface. Thus, Eqs. (l), (3), (4), (7) and (17) represent the sufficient conditions for the Dirac bracket associated with infinitely reducible second class constraints to exist. Several remarks seem to be relevant. First, although some properties of the bracket (8) (a commutation of any quantity with second class constraints and the Jacobi identity) hold on second class constraint surface only, this bracket is consistent with setting all second class constraints strongly equal to zero. Consequently, the proposed construction do generalize the Dirac bracket to reducible case. Second, in passing to quantum description (when second class constraints operators are treated as strong equations) the standard correspondence rules (A, B}D + [A, sl may be imposed. Third, for some models the suggested scheme
AA. Deriglazov et d/Physics
144
can be realized without extension of the original phase space. The example is D = 9 massive superparticle with the Wess-Zumino term [ 6,7]. Forth, making use of the natural arbitrariness one can always continue the bracket (8) up to the one with the strong Jacobi identity. It should be noted, however, that for certain models such a continuation may conflict with manifest Poincare covariance [ 71. 3. As an example of application of the scheme proposed we consider D = 10 superparticle model with the action [ 31 (we use the notations from Refs. [ 6,7] )
Letters B 386 (1996) 141-145
to the Hamiltonian formalism one can find the following constraints for the model (22) Pe M 0,
7r2 = 0;
(24a)
ps - if!F7r,,
Xn : (PA,
:
$A?
:
=
TA,l
Pnr
-
M 0;
(24b)
0;
(24~)
OTT,, =
0,
Pnt + &I =O,
mpn
=
0,
(24d)
%-B,, %O
where we denoted momenta conjugate to the variables (e,x”*,p”‘,8”,A”*,Bm)
s=
I d+12,
Constraint
(19)
system of the theory is of the form: 7rX) 2
Pe = 0,
0
x z pH - iW”l77,, M 0,
(20)
where (pe, T,,,, porn) are momenta conjugate to the variables (e, xrn, 0(l) respectively. Evaluating the Poisson bracket of the spinor constraints ( {x”, rm} = a”,,,
as %-B,) respectively. The constraints (24a), (24~) are first class. Among the fermionic constraints (24b) half are first class and another half are second class. Constraint system (24d) is second class and implies that the pairs (p”‘, rpm ) , ( B” , TTB,,) are trivial. To split the constraints xa M 0 into first and second class in covariant manner, one has to find a projector pf being a (weak) null eigenvector of the matrix A* = 2i( F’r,,, ) . The result proves to be of the form (Pr 9~lll, 7TI,l?I 7PBn 7Th,
pf = ;(l+K),
p-+(1-K),
{O”* Psp) = -Fp) 1
K= {xa,
xp}
= 2i(r”*~,,),p,
one concludes that there are eight first class and eight second class constraints among xu M 0. To construct the Dirac bracket for the case concerned, let us rewrite the theory (19) in the equivalent form S=
2
(21)
dT{p,(An’-i/T”&)
+B,(A”‘-k”‘)
- $1,
flnrnllT
A n I?*,
(TTA)~ - IT~A~
(25)
K2=l.
and the constraint
algebra is
{x,‘, x,‘,
= 0,
{XZ, xp}
{x;,/@
= 20-I”*r,,p-),p
{,&VA,}
=
= 0, - Asp,
./ (22)
where the new variables p”, A”, B”’ have been introduced. Taking into account the equations of motion for the variables 6S
~=Pm+B,,=O,
-
6S
Sp”l
gq= - iW,b)
{,&+Az}
=
{,&A}
=o
Inverting Fq. (4)
further the matrix A,,
0,
(26)
in accordance
with
A,, - &,, = 0, h”P =
= ep,,, - (A,
0,
= 0,
2i(J(n-A)2
(23)
one can easily show that the theory (22) is on-shell equivalent to the original superparticle ( 19). Passing
p-m’s(pA,)““p-Pn - m2A2 + (TA))
’
iA=pand inserting the result into Eq. (8) one gets
(27)
AA. Deriglazov et ok/Physics
{A,C}D = {AX}
-{A x,1
Letters B 386 (1996) 141-145
145
References (p+Anp-)@
2i(-/(n-A)*
- m2A2 + (?rA))
txp1 Cl
III LA. Batalin and ES. Fradkin, Lebedev Inst. Preprint No. 259 (1982);
Lebedev Inst. Preprint No. 165 ( 1983).
t21 I.A. Batalin and I.V. Tyutin, Nucl. Phys. B 381 (1992) 619;
Since the conditions (7), (17) hold, one concludes that Eqs. (25), (27) and (28) represent the consistent covariant Dirac bracket for the superparticle model in ten dimensions. To quantize the theory it is suffice to realize the algebra of quantum observables with respect to the bracket (28) and set the first class constraint operators to vanish on the physical states. The explicit representation for the algebra in a certain Hilbert space is a problem for further investigation. This work was supported in part by European Community Grant No INTAS-93-2058. The work of (A.V.G.) has been made possible by a fellowship of Tomalla Foundation (under the research program of ICFPM) and ISSEP Grant No A837-F. The work of (S.L.L) has been partially supported by RBRF Grant No 96-01-00482.
Phys. Lett. B 317 (1993) 354; Mod. Phys. Lett. A 8 ( 1993) 3757. 131 L. Brink and J.H. Schwarz, Phys. Lett. B 100 ( 1981) 310. [41 M.B. Green and J.H. Schwarz, Phys. Lett. B 136 ( 1984) 367. 151 L. Brink. M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505. [61 J.M. Evans, Phys. Len. B 233 (1989) 307; Nucl. Phys. B 331 (1990) 711. and S.L. Lyakhovich, [71 A.A. Deriglazov, A.V. Galajinsky preprint hep-th 9512036, 2lp.. to appear in NucI. Phys. B. [81 A. Dresse, J. Fisch, M. Henneaux and C. Schomblond, Phys. Len. B 210 (1988) 141. 191 P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, Belfer Graduate School of Science) (Academic, New York, 1964). [lOI D. Dalmazi, Phys. Lett. B 328 ( 1994) 43. 1111 1. Bengtsson and M. Cederwall, G&eborg preprint 84-21 (1984).
1996
PHYSICS
LETTERS B
Physics Letters B 386 (1996) 141-145
ELSEVIER
Consistent Dirac bracket for dynamical systems with infinitely reducible second class constraints A.A. Deriglazov ‘, A.V. Galajinsky, Department
S.L. Lyakhovich
ofTheoretical Physics, Tomsk State University, 634050 Tomsk, Russia Received 18 May 1996 Editor: PV. Landshoff
Abstract
The standard Dirac bracket construction is generalized to the case of infinitely reducible second class constraints. Reducibility of constraints is shown to manifest itself in weakening the Jacobi identity for the construction, that now holds on the second class constraint surface only. The proposed scheme is realized for N = 1 superparticle model in ten dimensions. P4CS: 04.60.D~; 11.3O.Pb Keywords: Covariant
quantization;
Mixed constraints;
Dirac bracket; Superparticle
1. A constructing of the Dirac bracket associated with reducible second class constraints is an important ingredient of covariant quantization scheme [ 1,2] for dynamical systems subject to mixed first and second class constraints. This construction to a great extent was designed to resolve covariant quantization problem of the superparticle [ 31 and superstring [4] theories (see Ref. [5] for a review). A complete constraint system of the models includes fermionic constraints which, being a mixture of half first class and half second class constraints, lie in the minimal spinor representation of the Lorentz group. Projectors that realize a covariant (reducible) split of the constraints into first and second class have been constructed in Refs. [ 6,7]. A recipe how to generalize the Dirac bracket to reducible case, i.e. to the case when second class constraints xa TZ0 satisfy the identity xaCaO = 0, where
’ E-mail: [email protected]. 0370-2693/%/$12.00 Copyright PII SO370-2693(96)00905-7
the C,” are (weak) null eigenvectors 2 of the Poisson brackets matrix AUp E {xn, xp}, has been formulated in Refs. [ 5,6,8]. The leading principle was to achieve a weak commutation of any quantity with the second class constraints. Presence of the property allows, by analogy with Dirac’s prescriptions of quantization, one to set the second class constraints strongly equal to zero and pass to the reduced phase space [ 91. It seems to be unnoticed, but the Jacobi identity was not automatically provided for the scheme. In the general case, however, the bracket of such a kind may possess the identity in a strong sense, weakly or has no the property at all. For instance the bracket of Ref. [6] possesses the Jacobi identity on the second
* In the general case the C,” themselves may possess null eigenvectors (the next stage of reducibility [ 1 ] ). The most interesting example is [5,6] Au0 = gu,.p,p, where prp is a projector p2 = p, pT = p and g is a reversible matrix. In this case C = 1 - p has evident null eigenvector pA with A being an arbitrary function.
0 1996 Elsevier Science B.V. All rights reserved.
AA. Derigluzou et al./Physics Letters B 386 (1996) 141-145
142
class constraint surface only [ 71, while certain cycles break the identity for the bracket of Ref. [ lo]. Since, in passing to quantum description, correct correspondence rules imply the Jacobi identity for the classical counterpart of the quantum commutator, the question of how to ensure the property for the scheme acquires a great importance. In this letter we propose the general scheme for constructing the consistent covariant Dirac bracket associated with infinitely reducible second class constraints. It will be shown that building such a bracket is equivalent to solving certain system of algebraic equations in appropriately extended phase space. As an application we realize the scheme for the Brink-Schwarz superparticle model in D = 10.
[ 81. Note as well that rank A 1= n/2 by virtue of Eq. (3), which correctly reflects reducibility of the constraints involved. To construct a bracket under which any quantity weakly 3 commutes with the second class constraints, let us find a (symmetric) matrix zap being inverse to the AaP in the following sense: Ai =p-
(4)
Taking into account consequences p+Kp-
= 0,
+2_ + P -P 1 p+p- = 0, satisfying
-2
P
=p-9
(1)
p++p-=1
the conditions
LY,‘, X,‘> M 0, L&Y;1 {x,3 xi} = Asp = (P-A’P-
= 0, lap,
(5)
+p+&p+,
(6)
where A1 is some matrix consistent with Eq. (4) and & is an arbitrary matrix. As will be shown, only the first term from Eq. (6) is essential in building the Dirac bracket. In what follows we will also suppose that the following conditions {x,ttclA~)
=o,
{,&tcIA2)
(7)
x 0
hold 4 . With the & at hand, generalization of the Dirac bracket to reducible case present no special problem. The suitable ansatz is {A, B}D = {A, B} - {A,x,}~~~{x& - {A,
(CIA~}Q~~~Z{$C~,
B},
B} (8)
where 6 is the inverse matrix to VA2B2= {@AZ,I,~B?}, 00 = 1. For specific models the brackets like Eq. (8) were also considered in Refs. [ 5,6,10]. Several comments on the structure of the bracket are relevant here. First, any quantity A weakly commutes with the second class constraints under the bracket (8)
(2)
{A,$B~}D =O,
where x * = p*x. From Eqs. (2) it follows the determining equation for the projector operators A*p+ M 0.
=0
one comes to the decomposition a = p-alp-
2. Consider a model constraint system of which includes (among other) fermionic constraints xn z o,Ly= l,... , n, and suppose half of them to be first class and another half to be second, i.e {xa, xp} -_ A& and rank A*1 = n/2. Since in the original phase space r, covariant separation of the constraints may present a nontrivial task [ 5, 111, we extend this space to r* = (r, radd ) , where r&d is some set of additional variables. It is implied that nondynamical character of the variables is provided by some new constraints. Denote first and second class constraints of this type as ‘PA,(r*)MOand~IA2(r*)M0,Al=l ,..., nl,Al= 1,. . , n2, respectively and we suppose the latter to be linearly independent. To split the constraints xa z 0 in reducible and equivalent manner, we suppose the existence in the space r* of a pair of (strict) projectors
p-hp+
of this equality
{A, X,}D = {A, P,~}x~ In the geometrical brackets matrix
= 0.
(9)
terms it means that the Poisson
(3)
Since the second class constraints have been extracted by means of a strict projector, they are infinitely reducible and automatically satisfy the criterium of Ref.
’ Here and up to second 4 Equations represent the
later by the term “weak equality” we mean equality class constraints. (7) are only technical restrictions that allow one to bracket (8) in the simplest form.
AA. Deriglazov et al./Physics
-
‘dik;kt+bA2f?A2B2 a ~~~B2dj
is degenerate (corankp’i null eigenvectors I)ija,j$A? = 0,
pijajx;
= 122+ n/2)
M0
(10) and (weak)
- 0. Jxp -
143
M ,+, is reversible Map@* = aay and consequently &A = -fi(&M)A?.For the case concerned, however, we deal with Eq. (4), from which it immediately follows the another relation
(11)
are normals to the second class constraint surface. It is worth mentioning that the latter condition can be strengthened to a strict equality #jp-Pa. a
Letters B 386 (1996) 141-145
(12)
The first term in r.h.s. of Eq. (16) reproduces the situation of the standard Dirac bracket. The remaining terms are manifestation of reducibility of the constraints involved and it is their contribution into the cycle ( 15) which breaks the identity. To suppress the contribution let us impose the following restrictions
Second, there is a natural arbitrariness in definition of the bracket being related to the possibility to add arbitrary combination of second class constraints to the ansatz (8). By this reason, the contribution of the second term from Eq. (6) into the bracket (8) can always be neglected, if one takes into account the identity
Considering now Eq. ( 15) with p’j being presented in the form ( 10) and making use of Eq. ( 16) and ( 17) one can get, after tedious calculations, the equality
p;“&x,
( -l)‘j”pk’Ylpij
= -(Z”p;“)x,.
(13)
{PZ
XY)
(‘kA2493
= 0,
+ cycle(ijk)
-(-1)
I’+& = 0.
+ cycle( ijk) .
Third, by construction the bracket (8) possesses the graded symmetry, linearity and acts in a functional space as differentiation. Thus, the only conventional property to be discussed is the Jacobi identity. It is straightforward to check that for the bracket ( 8) the identity does not hold. We will show, however, that imposing further restrictions on the structure of the projectors p,fp and the constraints @AZM 0 one can provide the property on the second class constraint surface. For this purpose consider the graded cycle
+ (-_l)~~~~k~a~,ii
(15)
which identically vanishes for the ordinary Dirac bracket and ensures the Jacobi identity for the construction. The proof of the identity in the irreducible case is, actually, based on the fact that the matrix of Poisson brackets of second class constraints, say
=
l,Q+fitE,+%,) [WkL-Wk”;;,~Xp~PyapX;““I]
The latter circumstance means that one can search for the A in the form h = p-&lp- and consider that (14)
(17)
=o,
x wiGnx,
[ ps’” &P
+ ;,iPp;q
&/y$LPj ( 18)
Taking into account now the identity ( 13) one concludes that, under the restrictions ( 17)) the Jacobi identity for the bracket (8) holds on the second class constraint surface. Thus, Eqs. (l), (3), (4), (7) and (17) represent the sufficient conditions for the Dirac bracket associated with infinitely reducible second class constraints to exist. Several remarks seem to be relevant. First, although some properties of the bracket (8) (a commutation of any quantity with second class constraints and the Jacobi identity) hold on second class constraint surface only, this bracket is consistent with setting all second class constraints strongly equal to zero. Consequently, the proposed construction do generalize the Dirac bracket to reducible case. Second, in passing to quantum description (when second class constraints operators are treated as strong equations) the standard correspondence rules (A, B}D + [A, sl may be imposed. Third, for some models the suggested scheme
AA. Deriglazov et d/Physics
144
can be realized without extension of the original phase space. The example is D = 9 massive superparticle with the Wess-Zumino term [ 6,7]. Forth, making use of the natural arbitrariness one can always continue the bracket (8) up to the one with the strong Jacobi identity. It should be noted, however, that for certain models such a continuation may conflict with manifest Poincare covariance [ 71. 3. As an example of application of the scheme proposed we consider D = 10 superparticle model with the action [ 31 (we use the notations from Refs. [ 6,7] )
Letters B 386 (1996) 141-145
to the Hamiltonian formalism one can find the following constraints for the model (22) Pe M 0,
7r2 = 0;
(24a)
ps - if!F7r,,
Xn : (PA,
:
$A?
:
=
TA,l
Pnr
-
M 0;
(24b)
0;
(24~)
OTT,, =
0,
Pnt + &I =O,
mpn
=
0,
(24d)
%-B,, %O
where we denoted momenta conjugate to the variables (e,x”*,p”‘,8”,A”*,Bm)
s=
I d+12,
Constraint
(19)
system of the theory is of the form: 7rX) 2
Pe = 0,
0
x z pH - iW”l77,, M 0,
(20)
where (pe, T,,,, porn) are momenta conjugate to the variables (e, xrn, 0(l) respectively. Evaluating the Poisson bracket of the spinor constraints ( {x”, rm} = a”,,,
as %-B,) respectively. The constraints (24a), (24~) are first class. Among the fermionic constraints (24b) half are first class and another half are second class. Constraint system (24d) is second class and implies that the pairs (p”‘, rpm ) , ( B” , TTB,,) are trivial. To split the constraints xa M 0 into first and second class in covariant manner, one has to find a projector pf being a (weak) null eigenvector of the matrix A* = 2i( F’r,,, ) . The result proves to be of the form (Pr 9~lll, 7TI,l?I 7PBn 7Th,
pf = ;(l+K),
p-+(1-K),
{O”* Psp) = -Fp) 1
K= {xa,
xp}
= 2i(r”*~,,),p,
one concludes that there are eight first class and eight second class constraints among xu M 0. To construct the Dirac bracket for the case concerned, let us rewrite the theory (19) in the equivalent form S=
2
(21)
dT{p,(An’-i/T”&)
+B,(A”‘-k”‘)
- $1,
flnrnllT
A n I?*,
(TTA)~ - IT~A~
(25)
K2=l.
and the constraint
algebra is
{x,‘, x,‘,
= 0,
{XZ, xp}
{x;,/@
= 20-I”*r,,p-),p
{,&VA,}
=
= 0, - Asp,
./ (22)
where the new variables p”, A”, B”’ have been introduced. Taking into account the equations of motion for the variables 6S
~=Pm+B,,=O,
-
6S
Sp”l
gq= - iW,b)
{,&+Az}
=
{,&A}
=o
Inverting Fq. (4)
further the matrix A,,
0,
(26)
in accordance
with
A,, - &,, = 0, h”P =
= ep,,, - (A,
0,
= 0,
2i(J(n-A)2
(23)
one can easily show that the theory (22) is on-shell equivalent to the original superparticle ( 19). Passing
p-m’s(pA,)““p-Pn - m2A2 + (TA))
’
iA=pand inserting the result into Eq. (8) one gets
(27)
AA. Deriglazov et ok/Physics
{A,C}D = {AX}
-{A x,1
Letters B 386 (1996) 141-145
145
References (p+Anp-)@
2i(-/(n-A)*
- m2A2 + (?rA))
txp1 Cl
III LA. Batalin and ES. Fradkin, Lebedev Inst. Preprint No. 259 (1982);
Lebedev Inst. Preprint No. 165 ( 1983).
t21 I.A. Batalin and I.V. Tyutin, Nucl. Phys. B 381 (1992) 619;
Since the conditions (7), (17) hold, one concludes that Eqs. (25), (27) and (28) represent the consistent covariant Dirac bracket for the superparticle model in ten dimensions. To quantize the theory it is suffice to realize the algebra of quantum observables with respect to the bracket (28) and set the first class constraint operators to vanish on the physical states. The explicit representation for the algebra in a certain Hilbert space is a problem for further investigation. This work was supported in part by European Community Grant No INTAS-93-2058. The work of (A.V.G.) has been made possible by a fellowship of Tomalla Foundation (under the research program of ICFPM) and ISSEP Grant No A837-F. The work of (S.L.L) has been partially supported by RBRF Grant No 96-01-00482.
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