- Email: [email protected]
Hamiltonian approach to lagrangian gauge symmetries
30 September 1999
Physics Letters B 463 Ž1999. 248–251
Hamiltonian approach to lagrangian gauge symmetries R. Banerjee 1, H.J. Rothe 2 , K.D. Rothe
3
Institut fur ¨ Theoretische Physik - UniÕersitat ¨ Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 14 June 1999; received in revised form 13 August 1999; accepted 19 August 1999 Editor: P.V. Landshoff
Abstract We reconsider the problem of finding all local symmetries of a Lagrangian. Our approach is completely Hamiltonian without any reference to the associated action. We show that the restrictions on the gauge parameters entering in the definition of the generator of gauge transformations follow from the commutativity of a general gauge variation with the time derivative operation. q 1999 Elsevier Science B.V. All rights reserved.
The unravelling of gauge symmetries of a given action is an important problem which has received much attention in the past. Two main approaches have been followed in the literature: i. the Hamiltonian approach based on Dirac’s conjecture w1,2x, where a suitable combination of the first class constraints is shown to be a generator of local symmetries of the Lagrangian, and ii. a purely Lagrangian approach, based on techniques used for discussing differential equations which are unsolvable with respect to the highest derivatives w3–6x. With regard to the Hamiltonian approach essentially two different procedures have been followed recently: i. a hybrid approach where one departs from the requirement of the off-shell invariance of
1
E-mail: [email protected]. On leave of absence from S.N. Bose Natl. Ctr. for Basic Sc., Salt Lake, Calcutta 700091, India 2 E-mail: [email protected] 3 E-mail: [email protected]
the extended action 4 under the local symmetry transformations generated by the phase space constraints, and then imposes a gauge condition whereby all Lagrange multipliers associated with secondary first-class constraints vanish w7x; ii. a purely algebraic approach based on the Poisson algebra of gauge generators with the constraints and the canonical Hamiltonian w8,9x. In this case the restrictions on the gauge parameters have been obtained only for a special class of constrained systems. In this paper we present a simple algorithm, based entirely on the total Hamiltonian approach, for obtaining the generator of the most general symmetry transformation of a given action, without ever making an explicit reference to the action itself. The basic input is the commutativity of a general gauge variation with the time derivative operation. In order to simplify the discussion, and also for the sake of comparison, we restrict ourselves in the following to
4 By extended action we mean the action constructed in terms of the extended Hamiltonian, in Dirac’s terminology.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 9 7 7 - 6
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
Hamiltonian systems with only irreducible first class constraints. The extension to systems with mixed first and second class constraints then involves a trivial step which we shall comment on at the end of this paper. Consider a Hamiltonian system of 2 n degrees of freedom qi , pi ,i s 1 . . . ,n, described by a canonical Hamiltonian Hc and a complete and irreducible set of Žfirst class. primary constraints Fa1 f 0 Ž a1 s 1,PPP,r ., and secondary constraints Fa 2 f 0 Ž a 2 s r q 1,PPP, N ., where r is the rank of the Hessian associated with the Lagrangian in question. We collect these constraints into a single vector with components Fa ,a s 1,PPP, N. Following Dirac’s conjecture, we make the following ansatz for the generator of gauge transformation: N
Gs
Ý e a Ž t , p,q,Õ . Fa ,
Ž 1.
as1
where, as we shall see, it is in general necessary to allow the gauge parameters e a to depend not only explicitly on time, but also on all phase space variables, including the Lagrange multipliers Õ a1 4 Žand their time derivatives. entering in the total Hamiltonian, HT s Hc q Ý Õ a1Fa1 .
Ž 2.
a1
Here Hc is the canonical Hamiltonian, and Fa1 f 04 are the Žfirst class. primary constraints. An infinitesimal gauge transformation of a phase-space function F Ž q, p . is then given by
d F s w F ,F a x e a ,
Ž 3.
where a summation over repeated indices is henceforth implied. Note that the gauge parameters appear outside the Poisson bracket. In principle we could have included the gauge parameters inside the Poisson bracket. These different definitions are weakly equivalent. As a result of the algebra
w
HC ,Fa s VabF b
w
Fa ,F b s C ac bFc
x
x
,
249
whether the gauge parameters are kept inside or outside the Poisson bracket. The structure functions C ac b and Vab can in general depend on the phase space variables. We now notice that the action principle which leads to the Euler-Lagrange equations of motion, requires the commutativity of a general d variation with the time-differentiation. This commutativity need not hold for an arbitrary variation within the hamiltonian framework. Since our motivation is to abstract the symmetries of the action, we impose
d
d
d
d q ; i s 1, . . . ,n Ž 6. dt dt i as a fundamental requirement. This, as we now show, turns out to imply a non trivial condition on the gauge parameters and Lagrange multipliers. As shown by Dirac w1x, the Euler-Lagrange equations follow from the action principle d ST s 0, where ST is defined by qi s
ST s dt pi q˙i y HT .
Ž 7.
H
Moreover, the symmetries of the total action ST are also the symmetries of S s HdtLŽ q,q˙ ., once the Hamilton equation of motion for q˙i , defining the relation between q˙i , and the momenta pi as well as Lagrange multipliers, are used to eliminate the momenta and Lagrange multipliers in favour of all the velocities, including the undetermined ones. Since we are interested in the local symmetries of this action, we shall thus work with the total Hamiltonian. The equations of motion within the hamiltonian framework, are given by q˙i s w qi , HT x s w qi , Hc x q Õ a1 qi ,Fa1 ,
Ž 8.
and p˙i s w pi , HT x s w pi , Hc x q Õ a1 pi ,Fa1 , together with the constraint equations
Ž 4.
Fa1 s 0 .
Ž 5.
From Ž8. and Ž3. we obtain for the left hand side of Ž6.
this weak equivalence continues to be true for the Poisson brackets of d F with either the canonical Hamiltonian HC or the generator G. Since our analysis only involves this algebra, it is inconsequential
d q˙i s w qi , Hc x ,Fa e a q Õ a1 q d Õ a1 qi ,Fa1 ,
Ž 9.
qi ,Fa1 ,F b e b
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
250
and for the right hand side of the same equation, d d q s w qi ,Fa x , Hc e a q Õ a1e a w qi ,Fa x ,Fa1 dt i de a q w qi ,Fa x . Ž 10 . dt Equating both expressions and making use of the Jacobi identity, we obtain
w Hc ,Fa x ,qi e a q Õ a1 Fa1 ,Fa ,qi e a y d Õ a1 qi ,Fa1 q w qi ,Fa x
de a
s0 . Ž 11 . dt Using Ž4. and Ž5., we see that, on the constraint surface Fa s 04 , the above equation implies de b dt
y e a Vab q Õ a1 C ab1 a
EF b E pi
y d Õ a1
EFa1 E pi
s0 .
Now, the first class nature and linear independence Žirreducibility. of the constraints guarantees that each of these can be identified as a momentum conjugate to some coordinate, the precise mapping being effected by a canonical transformation. Since Ž12. holds for all i one is led to the conditions
d Õ b1 s de
de
b1
dt
y e a Vab 1 q Õ a1 C ab11a ,
b2
y e a Vab 2 q Õ a1 C ab12a . dt Note that in the above equations time derivative as given by 0s
de a
Ž 12 . Ž 13 .
de a dt
denotes the total
De a
q e a , HT , Ž 14 . dt Dt where, following the notation of Ref. w7x, D E E E s q Õ˙ a1 a1 q Õ¨ a1 a1 q . . . . Ž 15. Dt Et EÕ E Õ˙ The restrictions on the gauge parameters and the Lagrange multipliers found here are seen to agree with that of Ref. w7x, obtained by looking at the invariance of the total action considered as the gauge-fixed version of the extended action, defined in terms of the extended Hamiltonian s
HE s Hc q Ý j aFa ,
Ž 16 .
where the Lagrange multipliers j a 2 4 are required to vanish by imposing suitable gauge conditions. Our
analysis is also equally applicable to a dynamics determined by the extended Hamiltonian. The algebra now involves the full set of Žprimary and secondary. first class constraints, so that no condition emerges for the gauge parameters while the variation of the Lagrange multipliers j a is given by de a a dj a s y e b Vba q j c Ccb . Ž 17 . dt Hence one is free to choose the gauge parameters e a to be functions of time only. These equations again agree with those given in Ref. w7x, as obtained by requiring the invariance of the corresponding extended action. Let us conclude with some comments: Our requirement Ž6. only involved the relation between the ‘‘ velocities’’ and the canonical momenta and the arbitrary Lagrange multipliers. We have thus only used the ‘‘first’’ of Hamilton’s equations, i.e., Ž8.. Contrary to other procedures, our derivation was carried out on a purely Žtotal. Hamiltonian level. As such we could have equally well worked with gauge transformations in the form
d F s w F ,G x s F , e aFa ,
Ž 18 .
because of the first-class nature of Hc and G. However, on the Lagrangian level, the two ways, Ž3. and Ž18. of writing the gauge transformation matters, since it is to be a symmetry also away from the constrained surface. As one easily checks, by explicitly looking at the off-shell invariance of the action Ž7., it is the definition, as given by Ž3., which leads to the transformation law Ž12. for the Lagrange multipliers, and conditions Ž13. on the gauge parameters. Finally we emphasize that the same discussion applies to the case where also second class constraints are present, with the simple replacement of Hc by the first class operator H Ž1., defined in the standard way by adding the contribution of the second class constraints whose Lagrange multipliers are now completely fixed. Acknowledgements One of the authors ŽR.B.. would like to thank the Alexander von Humboldt Foundation for providing financial support making this collaboration possible.
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
References w1x P.A.M. Dirac, Can. J. Math. 2 Ž1950. 129; Lectures on Quantum Mechanics, Yeshiva University, 1964. w2x M.E.V. Costa, H.O. Girotti, T.J.M. Simoes, Phys. Rev. D 32 Ž1985. 405. w3x E.C.G. Sudarshan, N. Mukunda, Classical Dynamics: a Modern Perspective, Wiley, New York, 1974. w4x D.M. Gitman, I.V. Tyutin, Quantization of fields with constraints, Springer-Verlag, Heidelberg, 1990.
251
w5x M. Chaichian, D. Louis Martinez, J. Math. Phys. 35 Ž1994. 6536. w6x A. Shirzad, J. Phys. A 31 Ž1998. 2747. w7x M. Henneaux, C. Teitelboim, J. Zanelli, Nucl. Phys. B 332 Ž1990. 169. w8x J. Gomis, M. Henneaux, J.M. Pons, Class. Quantum Grav. 7 Ž1990. 1089. w9x A. Cabo, M. Chaichian, D. Louis Martinez, J. Math. Phys 34 Ž1993. 5646.
Physics Letters B 463 Ž1999. 248–251
Hamiltonian approach to lagrangian gauge symmetries R. Banerjee 1, H.J. Rothe 2 , K.D. Rothe
3
Institut fur ¨ Theoretische Physik - UniÕersitat ¨ Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 14 June 1999; received in revised form 13 August 1999; accepted 19 August 1999 Editor: P.V. Landshoff
Abstract We reconsider the problem of finding all local symmetries of a Lagrangian. Our approach is completely Hamiltonian without any reference to the associated action. We show that the restrictions on the gauge parameters entering in the definition of the generator of gauge transformations follow from the commutativity of a general gauge variation with the time derivative operation. q 1999 Elsevier Science B.V. All rights reserved.
The unravelling of gauge symmetries of a given action is an important problem which has received much attention in the past. Two main approaches have been followed in the literature: i. the Hamiltonian approach based on Dirac’s conjecture w1,2x, where a suitable combination of the first class constraints is shown to be a generator of local symmetries of the Lagrangian, and ii. a purely Lagrangian approach, based on techniques used for discussing differential equations which are unsolvable with respect to the highest derivatives w3–6x. With regard to the Hamiltonian approach essentially two different procedures have been followed recently: i. a hybrid approach where one departs from the requirement of the off-shell invariance of
1
E-mail: [email protected]. On leave of absence from S.N. Bose Natl. Ctr. for Basic Sc., Salt Lake, Calcutta 700091, India 2 E-mail: [email protected] 3 E-mail: [email protected]
the extended action 4 under the local symmetry transformations generated by the phase space constraints, and then imposes a gauge condition whereby all Lagrange multipliers associated with secondary first-class constraints vanish w7x; ii. a purely algebraic approach based on the Poisson algebra of gauge generators with the constraints and the canonical Hamiltonian w8,9x. In this case the restrictions on the gauge parameters have been obtained only for a special class of constrained systems. In this paper we present a simple algorithm, based entirely on the total Hamiltonian approach, for obtaining the generator of the most general symmetry transformation of a given action, without ever making an explicit reference to the action itself. The basic input is the commutativity of a general gauge variation with the time derivative operation. In order to simplify the discussion, and also for the sake of comparison, we restrict ourselves in the following to
4 By extended action we mean the action constructed in terms of the extended Hamiltonian, in Dirac’s terminology.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 9 7 7 - 6
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
Hamiltonian systems with only irreducible first class constraints. The extension to systems with mixed first and second class constraints then involves a trivial step which we shall comment on at the end of this paper. Consider a Hamiltonian system of 2 n degrees of freedom qi , pi ,i s 1 . . . ,n, described by a canonical Hamiltonian Hc and a complete and irreducible set of Žfirst class. primary constraints Fa1 f 0 Ž a1 s 1,PPP,r ., and secondary constraints Fa 2 f 0 Ž a 2 s r q 1,PPP, N ., where r is the rank of the Hessian associated with the Lagrangian in question. We collect these constraints into a single vector with components Fa ,a s 1,PPP, N. Following Dirac’s conjecture, we make the following ansatz for the generator of gauge transformation: N
Gs
Ý e a Ž t , p,q,Õ . Fa ,
Ž 1.
as1
where, as we shall see, it is in general necessary to allow the gauge parameters e a to depend not only explicitly on time, but also on all phase space variables, including the Lagrange multipliers Õ a1 4 Žand their time derivatives. entering in the total Hamiltonian, HT s Hc q Ý Õ a1Fa1 .
Ž 2.
a1
Here Hc is the canonical Hamiltonian, and Fa1 f 04 are the Žfirst class. primary constraints. An infinitesimal gauge transformation of a phase-space function F Ž q, p . is then given by
d F s w F ,F a x e a ,
Ž 3.
where a summation over repeated indices is henceforth implied. Note that the gauge parameters appear outside the Poisson bracket. In principle we could have included the gauge parameters inside the Poisson bracket. These different definitions are weakly equivalent. As a result of the algebra
w
HC ,Fa s VabF b
w
Fa ,F b s C ac bFc
x
x
,
249
whether the gauge parameters are kept inside or outside the Poisson bracket. The structure functions C ac b and Vab can in general depend on the phase space variables. We now notice that the action principle which leads to the Euler-Lagrange equations of motion, requires the commutativity of a general d variation with the time-differentiation. This commutativity need not hold for an arbitrary variation within the hamiltonian framework. Since our motivation is to abstract the symmetries of the action, we impose
d
d
d
d q ; i s 1, . . . ,n Ž 6. dt dt i as a fundamental requirement. This, as we now show, turns out to imply a non trivial condition on the gauge parameters and Lagrange multipliers. As shown by Dirac w1x, the Euler-Lagrange equations follow from the action principle d ST s 0, where ST is defined by qi s
ST s dt pi q˙i y HT .
Ž 7.
H
Moreover, the symmetries of the total action ST are also the symmetries of S s HdtLŽ q,q˙ ., once the Hamilton equation of motion for q˙i , defining the relation between q˙i , and the momenta pi as well as Lagrange multipliers, are used to eliminate the momenta and Lagrange multipliers in favour of all the velocities, including the undetermined ones. Since we are interested in the local symmetries of this action, we shall thus work with the total Hamiltonian. The equations of motion within the hamiltonian framework, are given by q˙i s w qi , HT x s w qi , Hc x q Õ a1 qi ,Fa1 ,
Ž 8.
and p˙i s w pi , HT x s w pi , Hc x q Õ a1 pi ,Fa1 , together with the constraint equations
Ž 4.
Fa1 s 0 .
Ž 5.
From Ž8. and Ž3. we obtain for the left hand side of Ž6.
this weak equivalence continues to be true for the Poisson brackets of d F with either the canonical Hamiltonian HC or the generator G. Since our analysis only involves this algebra, it is inconsequential
d q˙i s w qi , Hc x ,Fa e a q Õ a1 q d Õ a1 qi ,Fa1 ,
Ž 9.
qi ,Fa1 ,F b e b
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
250
and for the right hand side of the same equation, d d q s w qi ,Fa x , Hc e a q Õ a1e a w qi ,Fa x ,Fa1 dt i de a q w qi ,Fa x . Ž 10 . dt Equating both expressions and making use of the Jacobi identity, we obtain
w Hc ,Fa x ,qi e a q Õ a1 Fa1 ,Fa ,qi e a y d Õ a1 qi ,Fa1 q w qi ,Fa x
de a
s0 . Ž 11 . dt Using Ž4. and Ž5., we see that, on the constraint surface Fa s 04 , the above equation implies de b dt
y e a Vab q Õ a1 C ab1 a
EF b E pi
y d Õ a1
EFa1 E pi
s0 .
Now, the first class nature and linear independence Žirreducibility. of the constraints guarantees that each of these can be identified as a momentum conjugate to some coordinate, the precise mapping being effected by a canonical transformation. Since Ž12. holds for all i one is led to the conditions
d Õ b1 s de
de
b1
dt
y e a Vab 1 q Õ a1 C ab11a ,
b2
y e a Vab 2 q Õ a1 C ab12a . dt Note that in the above equations time derivative as given by 0s
de a
Ž 12 . Ž 13 .
de a dt
denotes the total
De a
q e a , HT , Ž 14 . dt Dt where, following the notation of Ref. w7x, D E E E s q Õ˙ a1 a1 q Õ¨ a1 a1 q . . . . Ž 15. Dt Et EÕ E Õ˙ The restrictions on the gauge parameters and the Lagrange multipliers found here are seen to agree with that of Ref. w7x, obtained by looking at the invariance of the total action considered as the gauge-fixed version of the extended action, defined in terms of the extended Hamiltonian s
HE s Hc q Ý j aFa ,
Ž 16 .
where the Lagrange multipliers j a 2 4 are required to vanish by imposing suitable gauge conditions. Our
analysis is also equally applicable to a dynamics determined by the extended Hamiltonian. The algebra now involves the full set of Žprimary and secondary. first class constraints, so that no condition emerges for the gauge parameters while the variation of the Lagrange multipliers j a is given by de a a dj a s y e b Vba q j c Ccb . Ž 17 . dt Hence one is free to choose the gauge parameters e a to be functions of time only. These equations again agree with those given in Ref. w7x, as obtained by requiring the invariance of the corresponding extended action. Let us conclude with some comments: Our requirement Ž6. only involved the relation between the ‘‘ velocities’’ and the canonical momenta and the arbitrary Lagrange multipliers. We have thus only used the ‘‘first’’ of Hamilton’s equations, i.e., Ž8.. Contrary to other procedures, our derivation was carried out on a purely Žtotal. Hamiltonian level. As such we could have equally well worked with gauge transformations in the form
d F s w F ,G x s F , e aFa ,
Ž 18 .
because of the first-class nature of Hc and G. However, on the Lagrangian level, the two ways, Ž3. and Ž18. of writing the gauge transformation matters, since it is to be a symmetry also away from the constrained surface. As one easily checks, by explicitly looking at the off-shell invariance of the action Ž7., it is the definition, as given by Ž3., which leads to the transformation law Ž12. for the Lagrange multipliers, and conditions Ž13. on the gauge parameters. Finally we emphasize that the same discussion applies to the case where also second class constraints are present, with the simple replacement of Hc by the first class operator H Ž1., defined in the standard way by adding the contribution of the second class constraints whose Lagrange multipliers are now completely fixed. Acknowledgements One of the authors ŽR.B.. would like to thank the Alexander von Humboldt Foundation for providing financial support making this collaboration possible.
R. Banerjee et al.r Physics Letters B 463 (1999) 248–251
References w1x P.A.M. Dirac, Can. J. Math. 2 Ž1950. 129; Lectures on Quantum Mechanics, Yeshiva University, 1964. w2x M.E.V. Costa, H.O. Girotti, T.J.M. Simoes, Phys. Rev. D 32 Ž1985. 405. w3x E.C.G. Sudarshan, N. Mukunda, Classical Dynamics: a Modern Perspective, Wiley, New York, 1974. w4x D.M. Gitman, I.V. Tyutin, Quantization of fields with constraints, Springer-Verlag, Heidelberg, 1990.
251
w5x M. Chaichian, D. Louis Martinez, J. Math. Phys. 35 Ž1994. 6536. w6x A. Shirzad, J. Phys. A 31 Ž1998. 2747. w7x M. Henneaux, C. Teitelboim, J. Zanelli, Nucl. Phys. B 332 Ž1990. 169. w8x J. Gomis, M. Henneaux, J.M. Pons, Class. Quantum Grav. 7 Ž1990. 1089. w9x A. Cabo, M. Chaichian, D. Louis Martinez, J. Math. Phys 34 Ž1993. 5646.