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Path integral in constrained classical mechanics
PhysicsLettersA 182 (1993) 179—183
PHYSICS LETTERS
North-Holland
A
Path integral in constrained classical mechanics A.A. Abrikosov Jr. Institutefor Theoretical andExperimentalPhysics, B. Cheremushkinskaya Street 25, 117259 Moscow, Russian Federation Received 10 August 1993; accepted for publication 8 September 1993 Communicated by V.M. Agranovich
The method ofpath integration in classical mechanics isextended to constrainedmotion andgeneralized Hamiltonian systems. It is shown that constraints lead to replacement of Poisson brackets by Dirac ones. In the presence of first class constraints additional “gauge fixing” conditions should be imposed. The resulting theory is BRST invariant.
1. Introduction Recently path integrals (P1) have become one of the basic concepts ofquantum mechanics [ 1,2 ] The classical limit is given by the contribution of the trajectory with minimal action which satisfies the classical equation of motion. It was found that classical mechanics (CM ) possesses an alternative exact P1 representation which reproduces trajectories without taking a limit h—~O [ 3 ] (In distinction to the quantum case the CM P1 are written for full probabilities and not for their amplitudes. ) The action of the CM P1 contains Grassmanian variables and exhibits BRST symmetry. There are indications that the CM P1 approach could be effective for the investigation of the influence of microscopic dynamics on large scale variables in macroscopic systems. Turbulence and stochasticity are tempting examples of such an interplay of large and small scales. It was shown that the unbroken supersymmetry of the CM P1 is the criterion of ergodicity [4] In kinetics P1 reproduce the hierarchy of the BBGKY equations in a simple and straightforward manner [5,6]. A promising field of application would be Euclidean quantum field theories which are known to be equivalent to classical statistical models. It turns out that slightly changing the action of the CM P1, one can treat a statistical model as a dynamical system [ 7 ] The introduction of time was extremely useful in quantum statistics and offers new possibilities for the classical cases as well. In quantum field theories difficulties often arise due to the complicated structure of the phase space and gauge invariance. Both these factors can be treated as a result of constraints imposed on phase coordinates. In order to incorporate these cases we extend the CM P1 to constrained systems and generalized Hamiltonian mechanics. That makes it possible to apply the P1 for classical statistics to Euclidean field theories. .
.
.
.
2. Constrained Hamiltonian mechanics First we recall the basic ideas of Dirac’s approach [8]. Let Hamiltonian H depend on 2n phase coordinates ~a
constrained by M conditions [9]
‘~m(ø)0,
m=l,...,M.
Elsevier SciencePublishers BY.
(1) 179
Volume 1 82, number 2,3
PHYSICS LETTERS A
1 5 November 1993
The constraints “am reduce the number of degrees of freedom and sometimes make physical quantities independent of certain variables. The latter case is referred to as the generalized Hamiltonian dynamics. An example is gauge invariance in field theories where Gauss’s law is the corresponding constraint. Let us analyze this phenomenon. Obviously the Hamiltonian of the constrained system is defined up to a linear combination of cb’s, ~2) H’=H+um~m. Suppose that the Hamiltonian equations (w’~is a symplectic two-form )
[91 (3)
~a.waba/1T1
together with constraints ( 1
) do not fix the functions ~ uniquely. Then generalized Hamiltonian behavior
arises.
The functions
m
should obey the consistency equations which follow from ( 1).
~~(Ø(t))=O.
(4)
The time derivatives should be expressed in terms of Poisson brackets with the Hamiltonian H’, ;~[H’, X1=w~l~~aaXabH1~m =0
(5)
.
Note that the constraints are to be imposed only after the calculation of the derivatives. This is denoted by the weak equality sign Substitution of the last formula into (4) gives ~.
H’=[H’,
JP]+Um[1m,
H’]~O
(6)
,
where W= ~b 1. If the rank of the matrix Xi,, ~ [~ “1 iS less than M the following happens. First of all there are compatibility conditions for the system of inhomogeneous equations. They can lead to secondary constraints I~ ~ of the following form, I-~~c~[H,eb1]
(7)
.
Certainly the consistency conditions (6 ) should be valid for H= 1~“ as well. Fortunately the corresponding coefficients u11 ~ 0 and there is no need to consider secondary constraints on equal grounds with the primary ones. Thus the number of unknowns does not increase. It can happen thattmafter taking into and system, higher constraints k”, v~’ ~“, beingstill are defined up toaccount solutionsallofthe thesecondary homogeneous vm= V7,’v~with arRgE
(8)
The functions ‘I-~weakly commuting with each other and with the Hamiltonian H’. [H’,1~]~,J’~]~0,
~t,v=l
k,
(9)
are called first class constraints. The presence of the arbitrary functions v(t) testifies that the phase manifold includes nonphysical (gauge) degrees of freedom. The constraints I~act as generators of gauge transformations [8]. One can eliminate the ambiguity imposing k additional constraints x’ noncommuting with the ‘~, 180
Volume 1 82, number 2,3
1 5 November 1993
PHYSICS LETTERS A
det[I~,~~]~O.
(10)
The consistency conditions for ~ will fix Vi’. Their values will depend on the particular choice oft’s but that will not affect physical (gauge invariant) quantities. Note that this “gauge fixing” diminishes the dimensionality of the phase manifold by the doubled number of first class constraints, dim ~+ dimX= 2k.
3. The path integral Let us now introduce constraints into the CM path integral. Suppose first that all of them are second class. Then, after taking into account the secondary constraints RgI=M, and eqs. (6 ) have a single solution. The classical trajectory Ø~1(t),Ø~~(0) = 0, is the solution of Hamilton’s equations (3 ) supplemented by the consistency conditions (6), a=l,...,2n,
øaWabôb(H+Um~m),
(lla) 1=1, ...,M.
0lwab0aJjl3b(H+Um~Jm)0
(llb)
The starting point for defining the CM P1 [3
< pf ~S( t, 0 )I ø~ > = ô(Øf
_
Øc1
] is the product of integrals called an evolution operator, 2~ØkdTMuk fT Øcl(kt/N, 0,) ) N.—! 111 o( Uk U (0k) ) rN—i N o2n(0k
J
(t, 0, )) = N—~oo lim
fl
k=O
—
d
k=O
.
k=O
(12)
In the limit N—boo one obtains the path integral ofthe functional ô’~-functionwith support on the classical trajectory. Functions of the phase coordinates Ø~ 1(t)change according to the formula g(Ø(t))= $<ØIs(t,o)IØi>g(Øi)d2nØi.
(13)
Among others this equation describes the evolution ofthe distribution functionfwhich is of fundamental importance [10]. One can transform expression ( 12 ) substituting the set of equations ( 1 1 ) for th~arguments of the ~-functions. Of course that will generate the functional determinant, <øfIS(t, 0) IØ~> =<ØIdetôW~~ôôb~J
[d2nØ][dMU]ô2n(~_WabôbHF)öM(Wabôa~ôbH~)IØj).
(14)
It is easy to put this equation into the following form,
0)101>
=
The Lagrangian functions of fields ~
$ [~-~-—~][~~] [d~dc]
=
~
+~
[d5db] exp(i
$
~‘
dt).
(15)
contains two parts. The first comes from the integral representation of the ô-
0 and u, (16)
~
The second depends on Grassmanian “ghosts” which represent the functional Jacobian, ~‘
Note that it differs from the usual Lagrange function.
181
Volume 1 82, number 2,3
~
.1
9~g1~Ca~
a
~
PHYSICS LETTERS A
-a\(
a
The explicit formula for ~ _~7~
j~a(_0)a/~9b8ch1~)
1 5 November 1993
a\~
(17a)
is 1bW~’ôaTI’8h’I~ih” ~
~
( I 7b)
Formulae ( 1 5 )— ( 1 7 ) constitute the P1 representation of classical mechanics with second class constraints. They are valid for generalized Hamiltonian systems with constraints of the first class ( 8 ). ( 9 ) after eliminating gauge freedom by means of the supplementary conditions ( 1 0) It is easy to integrate ( 1 4 ) with respect to the auxiliary variables u, ~ and b, b. Then only the first terms in and 2~will remain and the symplectic form w’~”will change into .
_~7~
(18)
~
This corresponds to switching from Poisson to Dirac brackets, (19)
[fg]D=—w~aafabg.
The new symplectic form w~is a projection of the old one ~ onto the manifold of smaller dimension set off by constraints. The evolution operator is given now by the CM PT with the Lagrangian ~
(20)
.
In conclusion let us note the two facts. Firstly, the constrained theory is BRST and anti-BRST symmetric [ 3, 1 1 Namely the action generated by the Lagrangian (20 ) is invariant with respect to transformations, Øa~Oa+jWab~cb~
Øa~Oa+~-ca ~,
Ca~~Ca_fWa~b,
Ca~Ca+’~,a~,
1.
(21a)
Aa~Ca~a,Ca,
~a,C’~~ta,~’
(2lb)
~ are parameters. The symmetries are generated by the charges
QBRST
.
a~
1C ~
~ BRST
• ab— ~ . 1WD Caf~b
~
The presence of the w~’forms in the definition of the charge Secondly the change of variables ca=(expJw~ôbôCHdt)Cb(0),
Q makes it compatible with the constraints.
~=~(0)(exPJw~abaCHdT)
(23)
makes the ghost part of £f’D trivial: ~ The determinant in eq. ( 1 4 ) appears to be a multiplicative constant det 8/0t independent of the dynamics. This simplifies applications if zero boundary conditions on ghosts are acceptable ~2
4. Summary We have extended the CM path integral to constrained systems and generalized Hamiltonian dynamics. We began by deriving the representation with Lagrangian fields for constraints (15 )—( 17). Eliminating auxiliary S2
The non-zero boundary conditions on C, c give a possibility to study the dynamics of differential forms on the phase space.
182
Volume 182, number 2,3
PHYSICS LETTERS A
1 5 November 1993
fields led to the standard form of the CM P1 but to the replacement of Poisson brackets by Dirac ones, see eq. ( 19). The action proved to be BRST symmetric.
Acknowledgement I would like to thank E. Gozzi for warm hospitality and friendly discussions. It is a pleasure to acknowledge L. Alvarez-Gaumé and A.A. Migdal for elucidating remarks.
References [1 ] R.P. Feynman and AR. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [2] L.D. Faddeev andA.A. Slavnov, Gauge fields: introduction to quantum theory (Benjamin/Cummings, Mento Park, 1991). [3] E. Gozzi, M. Reuterand W.D. Thacker, Phys. Rev. D 40 ( 1989) 3633.
[4]E. Gozzi and M. Reuter, Phys. Lett. B 233 ( 1989) 383; 238 ( 1990) 451 (E). [5 ] E. Gozzi and M. Reuter, A generalized BBGKY hierarchy from the classical path-integral, in: Quarks, symmetries and strings, eds. M. Kaku et al. (World Scientific, Singapore, 1991). [6] T. Jolicoeur and J.C. Le Guillou, Singapore, Phys. Rev. A 40 (1989) 5815. [7]A.A. Abrikosov, Nucl. Phys. B 382 (1992) 581. [8] P.A.M. Dirac, Lectures on quantum mechanisms (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [9] V.1. Arnold, Mathematical methods ofclassical mechanisms (Springer, Berlin, 1978). [10] E.M. Lifshitz and L.P. Pitayevski, Physical kinetics (Pergamon, Oxford, 1981). [11] C. Becci, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287; IV. Tyutin and P.N. Lebedev, Physical Institute preprint 39 (1975).
183
PHYSICS LETTERS
North-Holland
A
Path integral in constrained classical mechanics A.A. Abrikosov Jr. Institutefor Theoretical andExperimentalPhysics, B. Cheremushkinskaya Street 25, 117259 Moscow, Russian Federation Received 10 August 1993; accepted for publication 8 September 1993 Communicated by V.M. Agranovich
The method ofpath integration in classical mechanics isextended to constrainedmotion andgeneralized Hamiltonian systems. It is shown that constraints lead to replacement of Poisson brackets by Dirac ones. In the presence of first class constraints additional “gauge fixing” conditions should be imposed. The resulting theory is BRST invariant.
1. Introduction Recently path integrals (P1) have become one of the basic concepts ofquantum mechanics [ 1,2 ] The classical limit is given by the contribution of the trajectory with minimal action which satisfies the classical equation of motion. It was found that classical mechanics (CM ) possesses an alternative exact P1 representation which reproduces trajectories without taking a limit h—~O [ 3 ] (In distinction to the quantum case the CM P1 are written for full probabilities and not for their amplitudes. ) The action of the CM P1 contains Grassmanian variables and exhibits BRST symmetry. There are indications that the CM P1 approach could be effective for the investigation of the influence of microscopic dynamics on large scale variables in macroscopic systems. Turbulence and stochasticity are tempting examples of such an interplay of large and small scales. It was shown that the unbroken supersymmetry of the CM P1 is the criterion of ergodicity [4] In kinetics P1 reproduce the hierarchy of the BBGKY equations in a simple and straightforward manner [5,6]. A promising field of application would be Euclidean quantum field theories which are known to be equivalent to classical statistical models. It turns out that slightly changing the action of the CM P1, one can treat a statistical model as a dynamical system [ 7 ] The introduction of time was extremely useful in quantum statistics and offers new possibilities for the classical cases as well. In quantum field theories difficulties often arise due to the complicated structure of the phase space and gauge invariance. Both these factors can be treated as a result of constraints imposed on phase coordinates. In order to incorporate these cases we extend the CM P1 to constrained systems and generalized Hamiltonian mechanics. That makes it possible to apply the P1 for classical statistics to Euclidean field theories. .
.
.
.
2. Constrained Hamiltonian mechanics First we recall the basic ideas of Dirac’s approach [8]. Let Hamiltonian H depend on 2n phase coordinates ~a
constrained by M conditions [9]
‘~m(ø)0,
m=l,...,M.
Elsevier SciencePublishers BY.
(1) 179
Volume 1 82, number 2,3
PHYSICS LETTERS A
1 5 November 1993
The constraints “am reduce the number of degrees of freedom and sometimes make physical quantities independent of certain variables. The latter case is referred to as the generalized Hamiltonian dynamics. An example is gauge invariance in field theories where Gauss’s law is the corresponding constraint. Let us analyze this phenomenon. Obviously the Hamiltonian of the constrained system is defined up to a linear combination of cb’s, ~2) H’=H+um~m. Suppose that the Hamiltonian equations (w’~is a symplectic two-form )
[91 (3)
~a.waba/1T1
together with constraints ( 1
) do not fix the functions ~ uniquely. Then generalized Hamiltonian behavior
arises.
The functions
m
should obey the consistency equations which follow from ( 1).
~~(Ø(t))=O.
(4)
The time derivatives should be expressed in terms of Poisson brackets with the Hamiltonian H’, ;~[H’, X1=w~l~~aaXabH1~m =0
(5)
.
Note that the constraints are to be imposed only after the calculation of the derivatives. This is denoted by the weak equality sign Substitution of the last formula into (4) gives ~.
H’=[H’,
JP]+Um[1m,
H’]~O
(6)
,
where W= ~b 1. If the rank of the matrix Xi,, ~ [~ “1 iS less than M the following happens. First of all there are compatibility conditions for the system of inhomogeneous equations. They can lead to secondary constraints I~ ~ of the following form, I-~~c~[H,eb1]
(7)
.
Certainly the consistency conditions (6 ) should be valid for H= 1~“ as well. Fortunately the corresponding coefficients u11 ~ 0 and there is no need to consider secondary constraints on equal grounds with the primary ones. Thus the number of unknowns does not increase. It can happen thattmafter taking into and system, higher constraints k”, v~’ ~“, beingstill are defined up toaccount solutionsallofthe thesecondary homogeneous vm= V7,’v~with arRgE
(8)
The functions ‘I-~weakly commuting with each other and with the Hamiltonian H’. [H’,1~]~,J’~]~0,
~t,v=l
k,
(9)
are called first class constraints. The presence of the arbitrary functions v(t) testifies that the phase manifold includes nonphysical (gauge) degrees of freedom. The constraints I~act as generators of gauge transformations [8]. One can eliminate the ambiguity imposing k additional constraints x’ noncommuting with the ‘~, 180
Volume 1 82, number 2,3
1 5 November 1993
PHYSICS LETTERS A
det[I~,~~]~O.
(10)
The consistency conditions for ~ will fix Vi’. Their values will depend on the particular choice oft’s but that will not affect physical (gauge invariant) quantities. Note that this “gauge fixing” diminishes the dimensionality of the phase manifold by the doubled number of first class constraints, dim ~+ dimX= 2k.
3. The path integral Let us now introduce constraints into the CM path integral. Suppose first that all of them are second class. Then, after taking into account the secondary constraints RgI=M, and eqs. (6 ) have a single solution. The classical trajectory Ø~1(t),Ø~~(0) = 0, is the solution of Hamilton’s equations (3 ) supplemented by the consistency conditions (6), a=l,...,2n,
øaWabôb(H+Um~m),
(lla) 1=1, ...,M.
0lwab0aJjl3b(H+Um~Jm)0
(llb)
The starting point for defining the CM P1 [3
< pf ~S( t, 0 )I ø~ > = ô(Øf
_
Øc1
] is the product of integrals called an evolution operator, 2~ØkdTMuk fT Øcl(kt/N, 0,) ) N.—! 111 o( Uk U (0k) ) rN—i N o2n(0k
J
(t, 0, )) = N—~oo lim
fl
k=O
—
d
k=O
.
k=O
(12)
In the limit N—boo one obtains the path integral ofthe functional ô’~-functionwith support on the classical trajectory. Functions of the phase coordinates Ø~ 1(t)change according to the formula g(Ø(t))= $<ØIs(t,o)IØi>g(Øi)d2nØi.
(13)
Among others this equation describes the evolution ofthe distribution functionfwhich is of fundamental importance [10]. One can transform expression ( 12 ) substituting the set of equations ( 1 1 ) for th~arguments of the ~-functions. Of course that will generate the functional determinant, <øfIS(t, 0) IØ~> =<ØIdetôW~~ôôb~J
[d2nØ][dMU]ô2n(~_WabôbHF)öM(Wabôa~ôbH~)IØj).
(14)
It is easy to put this equation into the following form,
0)101>
=
The Lagrangian functions of fields ~
$ [~-~-—~][~~] [d~dc]
=
~
+~
[d5db] exp(i
$
~‘
dt).
(15)
contains two parts. The first comes from the integral representation of the ô-
0 and u, (16)
~
The second depends on Grassmanian “ghosts” which represent the functional Jacobian, ~‘
Note that it differs from the usual Lagrange function.
181
Volume 1 82, number 2,3
~
.1
9~g1~Ca~
a
~
PHYSICS LETTERS A
-a\(
a
The explicit formula for ~ _~7~
j~a(_0)a/~9b8ch1~)
1 5 November 1993
a\~
(17a)
is 1bW~’ôaTI’8h’I~ih” ~
~
( I 7b)
Formulae ( 1 5 )— ( 1 7 ) constitute the P1 representation of classical mechanics with second class constraints. They are valid for generalized Hamiltonian systems with constraints of the first class ( 8 ). ( 9 ) after eliminating gauge freedom by means of the supplementary conditions ( 1 0) It is easy to integrate ( 1 4 ) with respect to the auxiliary variables u, ~ and b, b. Then only the first terms in and 2~will remain and the symplectic form w’~”will change into .
_~7~
(18)
~
This corresponds to switching from Poisson to Dirac brackets, (19)
[fg]D=—w~aafabg.
The new symplectic form w~is a projection of the old one ~ onto the manifold of smaller dimension set off by constraints. The evolution operator is given now by the CM PT with the Lagrangian ~
(20)
.
In conclusion let us note the two facts. Firstly, the constrained theory is BRST and anti-BRST symmetric [ 3, 1 1 Namely the action generated by the Lagrangian (20 ) is invariant with respect to transformations, Øa~Oa+jWab~cb~
Øa~Oa+~-ca ~,
Ca~~Ca_fWa~b,
Ca~Ca+’~,a~,
1.
(21a)
Aa~Ca~a,Ca,
~a,C’~~ta,~’
(2lb)
~ are parameters. The symmetries are generated by the charges
QBRST
.
a~
1C ~
~ BRST
• ab— ~ . 1WD Caf~b
~
The presence of the w~’forms in the definition of the charge Secondly the change of variables ca=(expJw~ôbôCHdt)Cb(0),
Q makes it compatible with the constraints.
~=~(0)(exPJw~abaCHdT)
(23)
makes the ghost part of £f’D trivial: ~ The determinant in eq. ( 1 4 ) appears to be a multiplicative constant det 8/0t independent of the dynamics. This simplifies applications if zero boundary conditions on ghosts are acceptable ~2
4. Summary We have extended the CM path integral to constrained systems and generalized Hamiltonian dynamics. We began by deriving the representation with Lagrangian fields for constraints (15 )—( 17). Eliminating auxiliary S2
The non-zero boundary conditions on C, c give a possibility to study the dynamics of differential forms on the phase space.
182
Volume 182, number 2,3
PHYSICS LETTERS A
1 5 November 1993
fields led to the standard form of the CM P1 but to the replacement of Poisson brackets by Dirac ones, see eq. ( 19). The action proved to be BRST symmetric.
Acknowledgement I would like to thank E. Gozzi for warm hospitality and friendly discussions. It is a pleasure to acknowledge L. Alvarez-Gaumé and A.A. Migdal for elucidating remarks.
References [1 ] R.P. Feynman and AR. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [2] L.D. Faddeev andA.A. Slavnov, Gauge fields: introduction to quantum theory (Benjamin/Cummings, Mento Park, 1991). [3] E. Gozzi, M. Reuterand W.D. Thacker, Phys. Rev. D 40 ( 1989) 3633.
[4]E. Gozzi and M. Reuter, Phys. Lett. B 233 ( 1989) 383; 238 ( 1990) 451 (E). [5 ] E. Gozzi and M. Reuter, A generalized BBGKY hierarchy from the classical path-integral, in: Quarks, symmetries and strings, eds. M. Kaku et al. (World Scientific, Singapore, 1991). [6] T. Jolicoeur and J.C. Le Guillou, Singapore, Phys. Rev. A 40 (1989) 5815. [7]A.A. Abrikosov, Nucl. Phys. B 382 (1992) 581. [8] P.A.M. Dirac, Lectures on quantum mechanisms (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [9] V.1. Arnold, Mathematical methods ofclassical mechanisms (Springer, Berlin, 1978). [10] E.M. Lifshitz and L.P. Pitayevski, Physical kinetics (Pergamon, Oxford, 1981). [11] C. Becci, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287; IV. Tyutin and P.N. Lebedev, Physical Institute preprint 39 (1975).
183