a.k.aringazin et al- brst symmetry in cohomological hamiltonian mechanics

8/3/2019 A.K.Aringazin et al- BRST Symmetry in Cohomological Hamiltonian Mechanics

1/11

BRST SYMMETRY IN

COHOMOLOGICAL

HAMILTONIAN MECHANICS

A.K.Aringazin1,2, K.M.Aringazin1 and V.Arkhipov11Department of Theoretical Physics

Karaganda State UniversityKaraganda 470074, Kazakhstan

and2The Institute for Basic Research

P.O. Box 1577Palm Harbor, FL 34682, U.S.A.

April 28, 1994Hadronic J. 17 (1994) 429-439.

Abstract

We present BRST gauge fixing approach to cohomological Hamil-tonian mechanics. Considered as one-dimensional field theory, theHamiltonian mechanics appeared to be an example of topological fieldtheory, with the trivial underlying Lagrangian. Twisted (anti-)BRSTsymmetry is related to an exterior algebra. Correlation functions ofthe BRST observables are studied.

Permanent address.

0


2/11

1 The BRST approach

We consider one-dimensional field theory, in which the dynamical variable isthe map, ai(t) : M1 M2n, from one-dimensional space M1, t M1, to2n-dimensional symplectic manifold M2n.

The commuting fields ai = (p1, . . . , pn, q1, . . . , qn) are local coordinates on

the target space, phase space of Hamiltonian mechanics equipped with theclosed non-degenerate symplectic two-form, = 1

2ijda

i daj; ij = ji =const, ij

jk = ik.We start with the partition function

Z =

Da exp(iI0), (1)

where I0 =

dtL0, and take the Lagrangian to be trivial, L0 = 0. ThisLagrangian has symmetries more than the usual diffeomorphism invariance.

Trivial Lagrangians are known to be of much significance in the cohomo-logical quantum field theories[1]-[4], which can be derived by an appropriateBRST gauge fixing of a theory in which the underlying Lagrangian is zero.

We use the BRST gauge fixing scheme[5, 6] to fix the symmetry in (1)by introducing appropriate ghost and anti-ghost fields. The diffeomorphismsof M2n we are interested in are the symplectic diffeomorphisms, which leavethe symplectic tensor ij form invariant[7],

ai = hai, (2)

where h = hii is a Lie-derivative along the vector field h

i. To garantee theinvariance ofij, we take h

i to be Hamiltonian vector field[8], hi = ijjH(a),where we assume H to be Hamiltonian of classical mechanics.

By introducing the ghost field ci(t) and the anti-ghost field ci(t) , we writethe BRST version of the diffeomorphism (2),

sai = ici, sci = 0, sci = qi, sqi = 0, (3)

where the BRST operator is nilpotent, s2

= 0, and qi is a Lagrange multiplier.By an obvious mirror symmetry to the BRST transformations (3), we demandthe following anti-BRST transformations hold:

sai = ici, sci = 0, sci = qi, sqi = 0, (4)

1


3/11

Evidently, s2 = 0, and it can be easily checked that ss + ss = 0.The partition function (1) then becomes

Z =

DX exp(iI), (5)

where DX represents the path integral over the fields a,q,c, and c. Theaction I is trivial action I0 plus s-exact part,

I = I0 +

dtsB. (6)

Since s is nilpotent, I is BRST invariant for any choice of B, with sB having

ghost number zero. We face with the restriction implied by antisymmetricproperty of ij, and choose judiciously B to be linear in the fields,

B = ci(tai hi + ai + ijqj). (7)

where and are real parameters. The first two term in (7) give rise to theterms of the form (Lagrange multiplier)(gauge fixing condition) and theghost dependent part,

sB = qi(tai hi) + ici(tc

i shi) + (qiai + icic

i). (8)

As a feature of the theory under consideration, the -dependent term vanishesbecause ij is antisymmetric. To evaluate shi, we note that hi = (kh

i)ak

and, therefore, shi = ckkhi. Thus, the resulting Lagrangian becomes L =

L0 + Lgf,

L = L0 + qi(tai hi) + ici(t

ik kh

i)ck + (qiai + icic

i). (9)

L is BRST invariant by construction, and it can be readily checked that it isalso anti-BRST invariant, sL = 0.

In the delta function gauge, i.e. at = 0, it reproduces exactly, up toL0, the Lagrangian, which has been derived in the path integral approach to

Hamiltonian mechanics by Gozzi and Reuter[9]-[11], via the Faddeev-Popovmethod. Indeed, by integrating out the fields q, c and c we obtain from(5) the partition function Z =

Da (a acl)exp(iI0), where a

icl denotes

solutions of the Hamiltons equation tai = hi, which had been used as a

starting point of the approach, with I0 = 0. The delta function constraint

2


4/11

plays, evidently, the role of the Faddeev-Popov gauge fixing condition, whichis in effect the Hamiltons equation.

Thus, the theory (5) represents an example of one-dimensional cohomo-logical field theory, in the sense that the Lagrangian (9) is BRST exact,with trivial underlying Lagrangian L0. The theory (5) can be thought as atopological phase of the Hamiltonian mechanics.

The (anti-)BRST symmetry is an inhomogeneous part of larger symme-try of the theory, inhomogeneous symplectic ISp(2) group symmetry, gen-erated by the charges, Q = iciqi, Q = ici

ijqj, C = cici, K =

1

2ijc

icj,and K = 1

2ij cicj[9]. Here, Q and Q are the BRST and anti-BRST op-

erators, respectively. ISp(2) algebra reflects the Cartan calculus on sym-

plectic manifold M2n

, with the correspondences, ci

dai

T M2n

andqi ii T

M2n[11].

2 Physical states

If we consider deformations ai along the solution of the Hamiltons equation,then in order for ai+ai to still be a solution it has to satisfy the deformationequation ta

i = hi. This equation is the equation for Jacobi field, ai T M2n, which can be thought of as the bosonic zero mode. This mode is

just compensated by anti-commuting zero-mode through the ghost dependent

term, in the Lagrangian (9), = 0.The Hamilton function H associated with (9),

H = qihi + icic

kkhi aiqi iC, (10)

covers, at = 0, in the ghost-free part the usual Liouvillian L = hii ofordinary classical mechanics derived by von Neumann[12], in the operatorapproach. H is the generalization of the Liouvillian to describe an evolutionof the p-form (p-ghost) probability distributions, = (a,c,t), instead of theusual distribution function (zero-form), governed by the Liouville equation,t(a, t) = L(a, t).

In the following, we use the delta function gauge omitting the -dependentterms in (10), one of which is the ghost number operator C.To study the physical states, that is the states which are BRST and anti-

BRST invariant, Q = Q = 0, we exploit the identification of twisted(anti-)BRST operator algebra with an exterior algebra[13]. Conventionally,

3


5/11

twist is used to obtain BRST theory from a supersymmetric one[1]. Below,we use a kind of twist to obtain, conversely, supersymmetry from the BRSTinvariance. Defining the twisted operators[14]

Q = eHQeH, Q = e

HQeH, (11)

where 0 is a real parameter, one can easily find that they are conservednilpotent supercharges, and {Q, Q} = 2iH. Consequently, these super-charges, together with H, build up N = 2 supersymmetry, which has beenfound[9] to be a fundamental property of the Hamiltonian mechanics. Par-ticularly, this supersymmetry is related[16] to the regular-unregular motiontransitions in Hamiltonian systems. Also, it has been proven[14] that the

cohomologies ofQ and Q are both isomorphic to the de Rham cohomologyso one can associate, in a standard way, some elliptic complex to them.

Thus, the following identifications can be made: d Q, d

Q, =dd

+ d

d {Q, Q} = 2iH, and (1)p (1)C, where exterior

derivative d acts on p-forms, p, and C is the ghost number. The coho-

mology groups Hp(M2n) = {ker d/imd p} are finite when M2n is com-

pact. According to the Hodge-de Rham theorem, canonical representativesof the cohomology classes Hp(M2n) are harmonic p-forms, = 0. Theyare closed p-forms minimizing the functional E() =

M2n ||

2, i.e. d = 0and d = 0. One can then define Bp as the number of independent harmonicforms, i.e. Bp() = dim{ker

p}. Formally, Bp continuosly varies with

but, being a discrete function, it is independent on so that one can findBp by studying the vacua of the Hamilton function, H = 0.

In the Hamiltonian mechanics, such an equation plays the role of theergodicity condition[15, 17]. Thus, the extended ergodicity condition canbe thought of as the condition for to be harmonic form, or, equivalently,supersymmetric (Ramond) ground state.

To ensure the solution in the p-ghost sector to be non-degenerate onetherefore should have Bp = 1. For the standard de Rham complex, Bpsare simply Betti numbers, with

(1)pBp being Euler characteristic ofM

2n.Recent studies of the physical states[17, 18] showed that the only physi-

cally relevant solution comes from the 2n-ghost sector, and has specificallythe Gibbs state form, = Kn exp(H). The other ghost sectors yieldsolutions of the form = Kp exp(+H), for the even-ghost sectors. Intwo-dimensional case, analysis shows that does not depend on , for theodd-ghost sectors (see Appendix).

4


6/11

Note that in the limit 0, we recover the classical Poincare integralinvariants Kp, p = 1, . . . n, as the solutions ofH = 0. They are fundamentalBRST invariant observables of the theory, {Q, Kp} = 0, since in the untwist-ing limit, 0, the supersymmetry generators Q and Q become theBRST and anti-BRST generators, respectively. Geometrically, this followsfrom dKp = 0 since Kp = p and d = 0. Similarly, Kps are anti-BRSTinvariant observables.

Also, it should be noted that there seems to be no relation of the super-symmetric Hamilton function H to the Morse theory[13] due to the genericabsence of terms quadratic in hi. The reason is that the symplectic two-form is closed so that this does not allow one to construct, or obtain by

the BRST gauge fixing procedure, non-vanishing terms quadratic in fields,except for the ghost-antighost. On the other hand, such terms, which arenatural in (cohomological) quantum field theories, would produce stochas-tic contribution (Gaussian noise) to the equations of motion[19] that would,clearly, spoil the deterministic character of the Hamiltonian mechanics.

3 Correlation functions

Let us now turn to consideration of the correlation functions of the BRSTobservables. The BRST invariant observables of interest are of the form

OA = Ai1ip(a)ci1

cip

, which are p-forms on M2n

, A p

. One can easilyfind that {Q, OA} = 0 if and only if A is closed since {Q, OA} = OdA.Therefore, the BRST observables correspond to the de Rham cohomology.

The basic field ai(t) is characterized by homotopy classes of the mapM1 M2n. Clearly, they are classes of conjugated elements of the funda-mental homotopy group 1(M

2n), in our one-dimensional field theory, sinceclosed paths make M1 = S1 (M1 = R1 is homotopically trivial). Hence, weshould study (quasi-)periodical trajectories in M2n characterized by a period, t S1. This case is of much importance in a general framework since itallows one to relate the correlation functions to the Lyapunov exponents[20],positive values of which are well known to be a strong indication of chaos in

Hamiltonian systems, and to the Kolmogorov-Sinai entropy.IfN is a closed submanifold of (compact) M2n representing some homol-

ogy class of codimension m (2n-m cycle), then, by Poincare duality, we havem-dimensional cohomology class A (m-cocycle), which can be taken to have

5


7/11

delta function support on N[21]. Thus, any closed form A is cohomologiousto a linear combination of the Poincare duals of appropriate Ns. The generalcorrelation function is then of the form,

OA1(t1) OAm(tm), (12)

where Ak are the Poincare duals of the Ns. Our aim is to find the con-tribution to this correlation function on S1 coming from a given homotopyclass of the maps S1 M2n. The conventional techniques with the modulispace M[1] consisting of the fields ai(t) of the above topological type canbe used here owing to the BRST symmetry. The non-vanishing contributionto (12) can only come from the intersection of the submanifolds Lk M

consisting of as such that ai

(t) Nk, and we obtain familiar formula[22],OA1(t1) OAm(tm)S1 = #m Lk

, relating the correlation function to

the number of intersections. As it was expected, the correlation function doesnot depend on time but only on the indeces of the BRST observables. ThePoincare invariants, Kps, as the BRST observables, correspond to homotopi-cally trivial sector since ij are constant coefficients, for which case as arehomotopically constant maps, ai(t) = ai(t0), and, therefore, the correlationfunction (12) can be presented as

M2n A1 Am.

Particular kind of observables, we turn to consider, is of the form[20]

OA = ci1 cip(a(t0) a0)ci1 cip

. After normal ordering, this observable can be thought of as the operatorcreating p ghosts (p-volume form in T M2n) at some time t0 and point a0 M2n, and then annihilating them at some later time t. Certainly, we shouldintroduce also time-ordering to define this operator correctly, but this willnot be a matter in the correlation function since we dealing with t S1.Indeed, the correlation function OA(t)S1 p(, a0) does not depend onspecific time, and we indicate this fact analytically by labeling p with theperiod . An important result[20] is that the higher order largest Lyapunovexponents are found to be related to this correlation function by

lp(a0) =p

m=1

lim

sup1

ln m(, a0). (13)

The partition function (5), for the p-form sector, Zp() = T rS1 exp(iHpt),takes the normalized form[20]

Zp() = T rS1p(t, a)/T rS11, (14)

6


8/11

where T rS1 denotes integral over all the solutions ai characterized by period

.The problem in computing Zp() arises due to the fact that the integral

over all the as is obviously abandoned. The finite value can be obtained byrealizing that the integral can be replaced with a sum over all the homotopyclasses of as mentioned above. Explicit computation can be performed, forexample, by realizing M2n as a covering space, f : M2n Y2n, of a suitablelineary connected manifold Y2n having the same fundamental group as S1,1(Y

2n, t0) 1(S1, t0). Since Y

2n has 0(Y2n, t0) = 0, the set of preimages

is discrete, f1(t0) = {a0, a1, . . .}. The number of elements of f1(t0) and

of the monodromy group G = 1(Y2n, t0)/f1(M

2n, a0) coincides due to

canonical one-to-one correspondence between f1

(t0) and G, and does notdepend on t0 due to 0(Y

2n, t0) = 0. Hence,

Zp() =G

p(, a). (15)

which is finite if G is a finite group. Clearly, Zps are topological entities,which can be used to define topological entropy[20] of the Hamiltonian sys-tem, with

2n(1)pZp being Euler characteristic of the symplectic manifold

M2n[11].

4 Concluding remarks

After having analyzed the main ingredients of the construction, we make fewcomments.

Note that the -dependent terms in the Lagrangian can be absorbed byslightly generalizing of the time derivative, t Dt = t + , which lookslike a covariant derivative, in one-dimensional case.

As we have already mentioned, the symmetries of the resulting Hamil-tonian (10) appeared to be even more than the (anti-)BRST symmetry wehave demanded upon. So, the reason of the appearence of these additionalsymmetries, K, K, and C, should be clarified, in the context of the BRST

gauge fixing approach. Surely, these symmetries are natural, and establishthe Poincare integral invariants, its conjugates, and the ghost-number con-servation.

We note that there is a tempting possibility to start with a non-trivialtopologically invariant action I0, if exist, instead of the trivial one. The

7


9/11

problem is to construct an appropriate topological Lagrangian L0, for whichI0 will not be dependent on the metric on M1, a positive definite functionof time, g = g(t), that is, g = arbitrary, I0 = 0. In this way, by the useof the BRST gauge fixing scheme, which is presently known to be the onlymethod to deal with topologically invariant theories, one would construct akind of topological classical mechanics.

Also, it is interesting to make BRST gauge fixing for the theory, withan explicit accounting for the energy conservation. Due to the fact thatthe (2n 1)-dimensional submanifold, M2n1 M2n, of constant energy,H(a) = E, is invariant under the Hamiltonian flow, and the p-forms evolveto p-forms on M2n1[20], one deals in effect with a reducible action of the

symplectic diffeomorphisms, for which case more refined Batalin-Vilkoviskygauge fixing method[23] should be applied, instead of the usual BRST oneused in this paper.

Appendix

Two-dimensional phase space, n = 1, is characterized by the coefficients ofthe symplectic tensor 12 = 21 = 1; a1 = p, a2 = q. The general expansionof the distribution reads

(a, c) = 0 + 1c

1

+ 2c

2

+ 12c

1

c

2

.

The general system of the ground state equations, Q = Q = 0, reduces,in this case, to (see Ref.[18] for details)

c1(1 h1)0 = 0, c2(2 h2)0 = 0,

c1(1 + h1)12 = 0, c2(2 + h2)12 = 0,

c1c2[(1 h1)2 (2 h2)1)] = 0,

(1 + h1)2 (2 + h2)1 = 0.

Here, i /ai

and hi H(a1

, a2

)/ai

(i = 1, 2). This system of equa-tions immediately implies, for the ghost-free sector 0 and the two-ghostsector 12,

0 = 0e+H, 12 = e

H

8


10/11

These ghost sectors define scalar distributions, with commuting coefficients0 and 12.

The equations for the vector distribution, (1, 2), can be rewrittenin the following form:

12 21 = 0, 2(h12 h21) = 0,

or, taking > 0,

h2h1

11 21 = 11h2h1

, 2 =h2h1

1.

The characteristic equations for the non-homogeneous first-order partial dif-ferential equation above are

da1

ds=

h2h1

,da2

ds= 1,

d1ds

= 11h2h1

,

from which one can easily find the first and the second integrals

U1 =

(h1da1 + h2da

2), U2 =h2h1

1.

The general solution then is of the form (U1, U2) = 0, that is, we can write

1 =h1h2

f(U1),

and, accordingly, 2 = f(U1), where f is an arbitrary function. We see thatthe solutions for the one-ghost (odd-ghost) sector, 1 and 2, characterizingthe vector distribution , do not depend on the parameter > 0. Thisremarkable property might go beyond the two-dimensional case.

References

[1] E.Witten, Comm. Math. Phys. 117,353 (1988) 353.

[2] E.Witten, Comm. Math. Phys. 118,411 (1988) 411.

[3] D.Birmingham, M.Blau, M.Rakowski, Phys. Rep. 209,129 (1991) 129.

9


11/11

[4] B.A.Dubrovin, Comm. Math. Phys. 141,195 (1992) 195.

[5] L.Baulieu, I.M.Singer, Nucl. Phys. Proc. Sup. 15B,12 (1988) 12.

[6] E.Bergshoeff, A.Sevrin, X.Shen, Phys. Lett. B 296,95 (1992) 95.

[7] E.Gozzi, M.Reuter, INFN preprint INFN/AE-93/10 (1993).

[8] R.Abraham and J.Marsden, Foundations of mechanics (Benjamin, NewYork, 1978).

[9] E.Gozzi, M.Reuter, Phys. Lett. B 233,383 (1989).



[12] J.von Neumann, Ann. Math. 33,587; 789 (1932).

[13] E.Witten, J. Diff. Geom. 17,661 (1982).

[14] E.Gozzi, M.Reuter, W.D.Thacker, Phys. Rev. D 46,757 (1992).

[15] V.I.Arnold, A.Avez, Ergodic problems of classical mechanics (Benjamin,New York, 1968).

[16] E.Gozzi, INFN preprint INFN/AE-91/20 (1991).

[17] E.Gozzi, M.Reuter, W.D.Thacker, Chaos, Soliton and Fractals 2,411(1992).

[18] A.K.Aringazin, Phys. Lett B 314,333 (1993); Hadronic J. 16,385(1993).

[19] G.Parisi, N.Sourlas, Nucl. Phys. B 206,321 (1982).

[20] E.Gozzi, M.Reuter, INFN preprint UTS-DFT-92-32 (1992).

[21] R.Bott, L.Tu, Differential forms in algebraic topology (Springer, Berlin,1982).

[22] E.Witten, Int. J. Mod. Phys. A6,2775 (1991).

[23] M.Henneaux, Phys. Rep. 126,1 (1985).

10

a.k.aringazin et al- brst symmetry in cohomological hamiltonian mechanics

Documents