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Classical mechanics as a topological field theory
Volume 240, number 1,2
PHYSICS LETTERS B
19 April 1990
CLASSICAL M E C H A N I C S AS A T O P O L O G I C A L F I E L D T H E O R Y E. G O Z Z I Theory Division, CERN, CH-1211 Geneva 23, Switzerland and M. R E U T E R Institut fiir Theoretische Physik, Universitdt Hannover, Appelstrasse 2, D-3000 Hannover 1, FRG Received 19 December 1989
We show that our recent formulation of classical mechanics via path integrals has many features in common with Witten's topological field theories. The action of the theory is a pure BRS commutator, which arises when we "gauge-fix" the symmetry of arbitrary deformations in path space. This gauge fixing involves the hamiltonian H and the symplectic 2-form m on the phase space ./g2,. Imposing BRS-invariant boundary conditions, the path integral can be used to calculate "topological" invariants of the dynamical system (~//2,,co, H). Here we restrict ourselvesto the Euler number of the symplecticmanifold ,~2nand the Maslov indices of the dynamics. Most probably many other "topological" features of (~/¢2~,co, H) can be calculated using this path integral. Conventional (non-topological) classical mechanics is recovered by using boundary conditions which are not BRS invariant. They give rise to expectation values which are sensitive to the "local" details of the dynamics.
In two previous papers [ 1,2 ] we have given a pathintegral formulation o f classical m e c h a n i c s both in its lagrangian and in its h a m i l t o n i a n version. In the present paper we are going to discuss a rather surprising feature of the path integral for classical mechanics: it gives rise to a kind of topological field theory similar to the theories recently proposed by Witten [ 3 ]. In particular, its lagrangian ~ , turns out to be a pure BRS commutator. We shall see that the classical path integral can be used to compute "topological" ~ invariants of any d y n a m i c a l s y s t e m (og2,, co, H ) , where ~12, is a symplectic manifold [4] with symplectic 2-form ~o and H is the hamiltonian. The simplest example is provided by quantities which are insensitive to both "small" and "large" deformations ofo~ and H a n d therefore do not "feel" the dynamics. They are the ordinary topological invariants of Jg2~, ~ By "topological" we mean quantities insensitive to "small" deformations of (,g2n, oJ, H) but sensitive to "large" deformations.
the cohomology classes HP(J{2n, ~R) or, in particular, the Euler n u m b e r Z(dt2n)- There is also another class of invariants which are sensitive to the dynamics, but which do not change if we slightly perturb the dynamics. A typical example of this class is the Maslov index. We consider a h a m i l t o n i a n system with a time-independent H a m i l t o n i a n H(O a) defined on a 2N-dimensional phase space ~g2nwith local coordinates oa, a = 1.... , 2 n . T h e " s u p e r - h a m i l t o n i a n " reads
~{ =)tao)abObH + iffaO)acOc ObH b
( 1)
a n d the lagrangian is given by ~ = 2 a b a + igab a - ~ . (We use the notation of refs. [ 1,2], with which we assume the reader familiar in the following.) In the Schr6dinger picture ~( is essentially the same as the Lie derivative l h - i ~ acting on functions O(~ba, ca). Here h-= ( d H ) ~ is the h a m i l t o n i a n vector field associated to the hamiltonian, and the ghosts c a are a basis in the cotangent space T*J/¢2,. Now let us described what our approach to CM has in c o m m o n
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with topological field theory models [ 3 ]. The crucial observation is that the hamiltonian ~ and the lagrangian ~ are pure BRS variations. Using the results of refs. [ 1,2 ], one easily verifies that .~ = - i [ Q ,
[Q, h]]
(2)
and ~=-i[Q,
z],
(3)
where O=igaOS'2~, is the anti-BRS operator [1,2] and Z-g,[Oa-h~(O ) ]. Interpreting (3) in the language of the standard hamiltonian BRS approach to gauge theories, one would conclude that the lagrangian ~ is completely determined by the "gauge fixing function" Z, because the gauge-fixing condition enters the Batalin-Fradkin-Vilkovisky path integral precisely in the form o f a BRS c o m m u t a t o r [ Q, Z]. A natural question to ask is whether there exists a local symmetry which, upon gauge fixing, gives rise to the BRS invariance o f ~ . This is exactly what was shown to happen for the fermionic symmetry of Witten's original theory [ 5 ]. In the present case it is not difficult to find the underlying local symmetry. Note that in the expression for X, Hamilton's equation appears as a kind of gauge-fixing condition. In fact, let us consider a theory of paths O~(t) in phase space, which has an identically vanishing action S[O] = O. Clearly, for this trivial action each path is as good as any other path and so S is invariant under the transformation
O~(t)-~O~(t) + 8 0 ~ ( t ) ,
Z= f 90"(t) a[Oa-h(~) ]det[a~ot-O~,h"(O) ] , (5) Eq. (5) is exactly the starting point for the path-integral of classical mechanics which was proposed in refs. [ 1,2]. This shows that, in a sense, our path-integral can be considered a gauge-fixed version of the "topological action" S [ 0 ] = 0 , with Hamilton's equation playing the role of a gauge fixing condition. This turns the local symmetry (4) into the global BRS invariance. The next question to ask is whether the path-integral of classical mechanics can be used to compute topological invariants. Let us first clarify the meaning of "topological" in the present context. For the kind of topological field theory discussed by Witten [ 3 ], the expectation value ( [r ) of some operator (~ is a topological invariant if and only if it is invariant under infinitesimal changes in the metric g~,, of the riemannian manifold on which the theory is defined. Clearly, since we are working on a symplectic manifold (phase space), which in general does not have a riemannian structure, the topological invariance of some expectation value must have a different meaning. In our case a quantity is of topological nature if it does not depend on the dynamics given by the hamiltonian H, i.e., it does not change if we perform small deformations of H or of the associated vector field h. Let us change the hamiltonian vector field h" by an amount
(4)
where 30~(t) is a completely arbitrary deformation of the original path. Very formally, the partition function for this theory would be Z = f 2 0 ~ ( t ) , i.e., it is a functional integral over a constant. To give a meaning to Z, let us gauge fix the local ~2 ~'gauge" symmetry (4). We choose the following gauge condition: G ~- O~-ha( Ob) ~0 where h~- o)~b~bH is the vector field generated by some hamiltonian H. Obviously the paths obeying this condition are exactly the solutions of Hamilton's equation. In a heuristic manner we can now apply the F a d d e e v - P o p o v trick. We implement the gauge-fixing by a a-function a [ G ~ ] multiplied by the F a d d e e v - P o p o v determinant det
(8G~lSOb):
19 April 1990
a,,h"(0h) ---~l~(&).
(6)
Then eq. (3) shows that ~ changes by a BRS variation g,, cQ'=i[Q, Carla] .
(7)
This statement is analogous to the property of the theories in ref. [2] that the metric variation of the lagrangian (the e n e r g y - m o m e n t u m tensor) is a BRS commutator. Now we investigate under which conditions the expectation value 7
( ( ' ( t ) ) = f g O " ( t ) . . . C ' ( O ( t ) .... ) e x p ( i f d t f f ~) 7"
(8) ~2 Here and in the following "local" means local in t. 138
of some local operator C = ( ( ¢ a ( t ) , 2a(t), c"(t),
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#a(t) ) is invariant under (6): 8 , ( ( 5 ( t ) ) =0. The integral (8) is over fields defined in [ - T , T]. It includes the integration over all fields at t = - T and over ~ and c a at t = + T. Note that these boundary conditions are BRS invariant, because of the BRS invariance of both the action and the measure, we can prove in the usual way [ 3 ] that the expectation value of any BRS variation vanishes: ( [ Q, ~ ] ) = 0. Exploiting this fact, it is easy to see that the r/variation of ( C~(t) ) is given by
8.< (5'(t)F=+I[Q,C;'(t)] f dte.rl~> , (9) and that ( 6' (t) > is a (non-zero) topological invariant, if we require that [Q, (! ( t ) ] = 0 ,
T
Zpbc: f ~ ( t ) . . . e x p ( i f d t ' ~ )
=
f
d2nOo d 2ncoK(~oCo, a a
0
TI¢g, cg, O) ,
c~, ti)=o(Zn)(f~'~--oal(tf, 0i))
X~(2n)(Cf--cal(tf,
Ci, [q~])) •
(12)
Periodic boundary conditions are indeed preserved under BRS transformations. We first compute Zpbc in the limit T ~ 0 and show later on that Zpbc actually does not depend on T. For closed paths and short times Tthe kernel K receives contributions only from the points were the vector field h a = 09ab0Jt vanishes, i.e., for the critical points of the hamiltonian H. Let us assume for simplicity that H has only isolated, nondegenerate ~s critical points at O~p). We then have K(q~, c~, Tl~g, cg, 0 ) = 6(en) (cg)
det[~bh"(O(p))] ~(2,)(~)~_O~p) ) X (p)~ idet[Obh~(O(p))] I
(I0)
These conditions are very similar to the analogous requirements for topological Yang-Mills theory or the topological a-model, say. The observable (5 must be BRS closed, but not exact, and the q variation (or the g ~ variation) o f C has to be a BRS commutator. If these requirements are met, ( ~' ) does not depend on the details of the dynamics prescribed by H. As a first example, let us consider the simplest operator satisfying (10), namely the unit operator (5:= 1. Computing its expectation value with free boundary conditions as in (8) we would obtain ( ( 5 ) = 0 because of the ghosts' zero-modes ~3. However this is not so for periodic boundary conditions (pbc)~4. Requiring oa (0) = ~ (T) and c ~(0) = c ~(T) for some fixed T > 0, we are led to
pbc
K ( 0 f , cf, tf I~f,
(13)
where the sum extends over all critical points in dg2~. Using (13) in ( 11 ) one finds
(5(t)#[Q,(.)],
6, d ' ( t ) = [Q, ( ' ) ] .
19 April 1990
(ii)
where [ 1 ]
Zpbc = (p) 2
det[OaObH(¢(p))] = ~ ( - - 1 ) ip. [det[OaObH(~(p))]l (p)
Here ip denotes the index of the critical point (p), i.e., the number of negative eigenvalues of the hessian of H a t (p) (which in local coordinates has components 0,0y(O~p)) ). The above result for Zpbc has a well-known interpretation [ 7 ] : it is the Morse theory representation of the Euler n u m b e r z o f the manifold on which the "Morse function" H is defined. Thus we have found that Zob~ = Z ( J[2n ) .
This answer is indeed independent of the dynamics of the system under consideration. The hamiltonian acts only as a Morse function and hence Zpbc contains only information about the topology of phase space, but not about the dynamics generated by H. Next we show that eq. ( 14) holds for any T > 0 and not only for T ~ 0 . The argument we use is quite standard in the discussion of the Witten index ofsupersymmetric quantum mechanics [ 8,9 ]. Recalling the operatorial formulation of the path integral [ 1,2], we may write eq. (11) as Zpbc = T r [ ( - ) F e x p ( - i T , # ) ] .
~3 A way out is to insert a "'gauge"-fixing function N(2 (0), c(0 ), ( ( 0 ) ) for these zero modes, as we did in the appendix of ref. [2]. ~4 This corresponds to a choice of a particular N function.
(14)
(15)
,5 This restriction is not essential. Along the lines of ref. [6 ] one could generalize the discussion to include degenerate Morse theory. 139
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Here the trace " T r " is performed over a complete set of functions 0,,(0 a, c a) or, equivalently, a set of antisymmetric tensor fields Q!P~. .~( O), p=O,1 ..... 2n. The "fermion number" operator F counts the degree of the respective differential form, hence ( - ) F = + 1 ( -- 1 ) f o r p even (odd). To give a meaning to eq. ( 15 ), we have to introduce a scalar product for forms. This is done by choosing an arbitrary riemannian metric g,~, o n ~,~/¢2n- Then, by a well known argument [ 8,9 ], the trace reduces to a sum over harmonic forms or, equivalently, to a trace in the Rham cohomology group H v (,,,//2,,, ~1):
,
(r'}))(t)=pFa ...... (O(t))~m(t)ca~(t)...C"~(t).
The RHS ofeq. (16) has a well-known interpretation in terms of the Lefschetz coincidence theorem [7], namely it is the Lefschetz number Lef[exp ( - Tlh)] of the mapping induced by e x p ( - T l h ) on phase space. The general Lefschetz theorem, for an arbitrary mapping K of some manifold into itself, expresses alternating sums like (16) in terms of local data of the fixed point set of the respective mapping. It can be shown that Lef[K] is always integer, and that it does not depend on the riemannian metric chosen. Furthermore, Lef[K] is a homotopic invariant of K. This implies that, if K is homotopic to the identity map, K ~ id, the traces o f H p are simply given by the Betti numbers bP: (17)
In our case K - exp ( - Tlh) is generated by a continuous time evolution and Kis in fact homotopic to the identity (with T playing the role of the hornotopy parameter). Therefore eq. (16) becomes Zpbc= ~ 2 np = 0 tI, - - 1 )PbP=z(d/[2n ) which coincides with the result (14) of the explicit evaluation of the classical path integral. Since K ~ id for any value of T, the relation above shows that Zpb~ does not depend on T. The reader will have realized, by now, that our classical path-integral has many features in c o m m o n with the partition function (with pbc) of supersymmetric quantum mechanics on a curved manifold [ 8,10 ]. Now let us try to find more general topological invariants ( ( c ) . Assuming that ~' does not depend on H, eq. (10) requires that (~ has a vanishing commutator with Q, but cannot be written as the BRS vari140
~)9~(t) = [Q, (~ ~ ( t ) ]
(18)
(16)
p~0
Trn~[K] = T r n ~ [ i d ] =dimHP(./ff2n, 91)=6 p .
ation of something. Operators with this property are in one-to-one correspondence with representatives F(P)=F(p) cttha I / k ...A0a~ofthecohomologyclasses -al,..ap ~ 'f" HP(.J/2n, ~). If we define (f(v°)(t)=F~f!,p(O(t)) Xcal(1)ca2(t).,.cap(t), it easily follows, from the operatorial rules derived in refs. [ 1,2 ], that (~fg) is BRS closed, but not exact. If ((~FO~(t)) is to be topological invariant, it must not depend on t. This would be guaranteed if~'; ~9) were a BRS-commutator. It is easy to verify that this is indeed the case:
with
2n
Zpb,.= ~ ( - 1 ) P T r n ~ [ e x p ( - T / ~ ) ] .
19 April 1990
(19)
The observables ~,}9~ and (!'~)) are analogous to Witten's [3] operators Wo and W~ out o f which the Donaldson invariants are constructed. Since in our case the "base manifold" is one-dimensional, there are only two classes of such operators. Now let us look at the expectation values (6~9) ) and ((!'}))). Here we have to distinguish carefully between the path integral defined on the interval [ - T , + T] where the fields 0" and c" are integrated independently at t = _+ T, and the path integral for a compactified base manifold S ~ where the points ( 0 a ( - T), c a ( - T)) and (oa( + T), C~( + T) ) are identified so that there is only one integration over O ( - T ) = O ( T ) and c " ( T ) =c~( - T). For the first type of boundary condition and the operators ( ~ 9 ) ( t = 0 ) , we find for the path integral (8)
< e',~.°'(o)) = J dx(t) d x ( O ) d x ( - T ) K ( x ( T ) , T I x ( O ) ,
O)
× C'~9)(x(O))K(x(O),OIx(-T),-t)
= f dr(O) (,'gg~(x(O)),
(20)
where we have used the notation x = (0% ca). Obviously ((~l;o)) does not give rise to interesting topological invariants since for p ~ 2 n this expectation value vanishes because of the ghost integration. For p=2n one obtains fd2n 0 Fo ....... e ........ which is proportional to the volume of phase space (if b o = b 2n= 1 ). We stress, however, that on [ - T, + T] it is pos-
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sible in principle to have a non-zero expectation value o f an o p e r a t o r like 6~ °) with non-vanishing ghost charge. The reason is that, as in any phase-space path integral over some interval [ - T, T] [ 1 1 ] there is a m i s m a t c h between the position c a and the m o m e n tum Ca integration: including the position integrations at t = _+ T, there is one more position integration than m o m e n t u m integration. This means that the measure ~ c ~ ¢ is not neutral u n d e r the ghost charge transformation. The situation is different if we compactify [ - T , T] to the circle S ~, i.e., if we integrate over closed paths only. Then the c and &integrations match and only operators o f vanishing ghost n u m b e r can have a non-zero expectation value ~6. Since both 6'~-°) and 6 ' ~ contain only c's, but no ¢'s, no further invariants b e y o n d Zvb¢ can be c o m p u t e d in this way. Before discussing a further class o f "topological" invariants, we briefly m e n t i o n a slight generalization o f our path integral. We consider N distinguishable species o f particles, all o f which evolve according to H a m i l t o n ' s equation with the same h a m i l t o n i a n vector field h ~ - O)abOaH. The time-evolution kernel for their generalized densities 0(0~, c~, 0~, c~, ..., 0~, C~v) ~7 is given by a p r o d u c t of the 0-functions as in eq. (12)
Ku({Ok,d, {Ck,r}, tfl {Ok,,}, {Ok,i}, /i' N
= l-[ K(0k,f, Ck,f, tfl0~,i, Ck,i, ti k=l
:f20~.~2k,a~C~-~(k,,~exp(iidt~u ) .
(21,
ti
The path-integral representation o f KN has the same form as that o f K b u t with ~ 0 a ( t ) ~ 2 , ( t ) . . . replaced by ~ 0 ~ ( t ) ~ 2 a k .... and the lagrangian S~ replaced by ~N = E k V = I ~ ( ~ , /]-a,k, C~, ga,k)- Since the various particle species do not " i n t e r a c t " , the resulting Nparticle theory is simply the N-fold tensor product o f the original one. The action ~N = fdt ~ is invariant under a set o f N i n d e p e n d e n t BRS transformations o f the form given in refs. [ 1,2 ]. ~6 This follows from the vanishing index of the kinetic operator of the ghosts. The situation is similar as to that of topological quantum mechanics, see ref. [ 12]. ,7 This function gives the "super-probability" to find a particle of species 1 at (0~, c~), of species 2 at (0~, c~ ), etc.
19 April 1990
We are now going to discuss another kind o f " t o p ological" observables which are sensitive to m and H us. This discussion is specific to the path-integral o f classical mechanics and we do not know if it has a counterpart in quantum field theory. The point is that up to now, in order to integrate over closed paths, we had to specify this by an explicit b o u n d a r y condition 0 a ( 0 ) = 0 a ( T ) . This condition selects those initial points 0~ for which the vector field h a, or the hamiltonian H respectively, admits closed trajectories of length T. Typically there will be only very few such trajectories; in fact, it is known that they form a set o f measure zero. F o r a generic h a m i l t o n i a n H, almost all initial conditions 0 a ( 0 ) will give rise to trajectories which never close. There exists a special class o f hamiltonians, however, which has the property that they generate closed orbits in phase space for any initial point 0~. This kind o f h a m i l t o n i a n naturally appears in integrable systems, for instance. Let us assume we are given an integrable h a m i l t o n i a n system (J/2n, ~o, H ) so that we can transform from the 0 a coordinates to action-angle variables (0,, J,), i = 1, ..., n. Considering the actions J i = J ~ ( 0 a) as the generators o f a family o f canonical transformations, their integral curves generate the basic homology cycles on the tori. Starting from a given point 0 a ( 0 ) and following the trajectory induced by Ji according to ~a=o)at'ot, J , ( O ) , we generate the ith cycle Fi. A given initial point is equivalent to a set o f numerical values {Ji, i = 1.... , n} for the action variables. This set defines a certain n-torus in ,~#2n. The curves F~, i = 1, ..., n generated by Js(O~) starting from 0 a ( 0 ) lie on this torus for all t. F r o m the path-integral point o f view it is attractive to use Js(O), rather than H, as a hamiltonian, because for this class of hamiltonians the pathintegral is saturated by closed paths even in the case when we work on { - T, T] with i n d e p e n d e n t 0 a and c a integration at t = _+ T ~9. We begin with a very simple example. We take the path integral (8) over [ - T, T] with i n d e p e n d e n t integrations at the end points and replace the h a m i l t o n i a n H by the actions 3",.. Furthermore we pick a certain representative F = F , d0 ~ o f H I ('.#2n, ~]~ ) and define the observable ~8 Again we stress the meaning of "topological" (or sensitive) here: it means that they do not "feel" small deformations of o), ,;¢, but only "large" deformations. ~9 Recall that in this situation we can have (¢~)¢0 even if Cf contains ghosts. 141
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~ ( 0 g ) = 6<2~)(0"(0) -Og)a~")(c~(O) )
× ~dt' fa(ob(l'))oa(t'),
(22)
which is similar to eq. (19) for p = 1..N is a functional of the (closed) path O~(t ' ) and the t'-integration is over a full period. Clearly .~- is BRS invariant, since a BRS transformation leads to a homotopic deformation of the path which does not change the integral or the RHS of (22) since d F = 0 . Moreover, 0~(0) would change by an amount ec~(0), which is zero due to the second 6- function. If we now compute the expectation value ( J ) it is easy to see that the path integral calculates fFad6 ~ along the classical trajectories ~ generated by J, starting from 0g:
19 April 1990
which describes the change of the position of the particle at time t due to a small change of the initial conditions. Note that this matrix is symplectic, S~nbE Sp (2n), since it is the jacobian matrix of a canonical transformation [ 4 ]. Now divide the phase-space coordinates (0 ~, ..., 02n) into a set of n "positions" q~ and n "momenta" p', i= 1.... , n, and then form the ( n × n ) matrices SUo(t)-S~(t) and Sqp(t)~-S~. As was shown by Littlejohn and Robbins [14], these matrices can be used to compute the Maslov index/~, associated to F~: g~ = 2 W[ det{Sg + iSg } ] d,
~i~ilndet[Sg(t)+iSqp(t)].
=2
(26)
F,
(.~7(0g)) = ~ d t ' f,(O~l(t',O~))O~,(t',Obo).
(23)
Even this somewhat trivial example nicely illustrates the interplay between dynamics (or topological) invariance, BRS invariance and homotopic invariance. l f F is closed, .Y is BRS invariant and therefore ( . ~ ) is insensitive to the details of the "gauge-fixing", i.e., of the hamiltonian. On the other hand, Fbeing closed means that }r,F is invariant under homotopic deformations of F,, and the BRS transformation ~b~(t) 0 a (t) + Ec~(t) provides precisely such an infinitesimal transformation. Nevertheless, contrary to )~(~t6~), say, ( , ~ ) depends on the hamiltonian, since replacing Js by Jk, k ¢ i, for instance, F is integrated over the cycle Fk which yields a different result in general. Now let us apply the above method to the Maslov index [12-14] #~ entering the semiclassical quantization condition Jl = (rti + l / / i ) h, r / , ¢ Z
(24)
Traditionally the Maslov index was calculated by investigating the caustics of lagrangian manifolds, but here we shall follow the approach of Littlejohn et al. [ 14 ], who interpreted #~ as a winding number in the Sp(2n) group manifold. To each cycle F~ generated by Ji we can associate an integer/t~ by the following construction, let F~ be parameterized by O~(t), t~ [0,2~z]. We define the 2~ periodic matrix function
O0~(t) Sg(t) = 00b(0),
First of all this formula expresses #i as a winding number W in the complex plane: the integral on the RHS of eq. (26) counts the number of times the value of the function det [S q + iS q ] encircles the origin of the complex plane, in a period of oscillation of 0. Eq. (26) can be used to assign an integer W[S] to any periodic, symplectic matrix function S. Clearly W[S] can depend only on the homotopy class of S. Since ~(Sp (2n) ) = Z, these classes are labelled by a single integer (the winding number), which turns out ~1oto be given by W[S]. We now proceed to give a pathintegral representation for #~. The crucial observation is that the matrix Sg(t) ofeq. (25) is essentially the same as our ghost fields. In fact, if ~ ( t ) evolves with the "hamiltonian" J,, this matrix function is a solution of
[O,6g - Obf/(O(t) ) ]S)(t) = 0 ,
with the initial condition S g ( 0 ) = 6 g (here j ~ = o)ab0bJi).Comparing eqs. (27) with the equations of motion of the ghosts [ 1,2 ], we see that, for each fixed value of the lower index c, S~ evolves in the same way as the ghost c b. This means that ab object like Sg appears naturally in the N-particle path integral of eq. (21) if we put N=2n. Using the initial condition c~(0) = ~f, for the 2n "species" of ghosts, we can recover S~, at a later time by Sg(t)=(O/d()c~b~(t). Here ~ is an anticommuting constant introduced because Sg is a commuting object, whereas the ghosts
(25) t~10 For a proof see the appendix of ref. [ 15 ].
142
(27)
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are anticommuting. Now it is easy to express/~i as an expectation value of the form T
f ~O~.(t) ~2a,k... (('exp(i f dt~'~),
((()=
--T
(28) with t~ [ - 7", + T], T>~2g, and independent integrations at the end points. We define the observables as ((0~) 2n
= 2 [1 ~ 2 " ) ( ¢ ~ ( 0 ) - 0 g ) k=l
2n
l-[ ~ ( 2 " ) ( c ~ ( 0 ) - ( ~ ) k=l
2zt
Xf dt lndet(~([cq(t)+icq(t)])
(29)
0
The first 0-function fixes a c o m m o n starting point for all 2n paths we follow, and the second assigns the initial values to the ghosts. Obviously when computing the expectation value ( 6 i ( ¢ ~ ) ), the path integral is saturated by a single classical trajectory on which the observable ~ci(0~) is evaluated. This means that ( ~ ( 0 ~ ) ) =~t~ for any initial point 0~. The observable (!:( ~ ) is BRS invariant expect for the 6 function fixing the starting point. U n d e r the BRS transformation it is changed by an amount ekc~ = ( e a ~ 1. To obtain a fully BRS invariant observable we can omit the 0-function, which fixes the initial point 0o, from (29) or, more precisely, use the operator 6 = £2- ~f d2nOoX ( ( 0 0 ) where ,Q is the volume of the phase space ~t2. Obviously, since all trajectories generated by Jj lead to the same Maslov index, we have again ( ( ! ' ) =/z~ but this time with a manifestly BRS-invariant operator (~.. Applying the above arguments, the BRS invariance of C! implies the invariance o f / 4 under "small" deformations of J~, which in turn translates into small deformations of the original hamiltonian H which gave rise to the actions {Ji}. Note again that "topological" invariance has a different meaning for Zpb¢ and for the Maslov index. In the first case "global" properties of only ,/g2, mattered, whereas in the second case we picked up "global" or ("large") properties of the whole triplet that characterize the dynamical systems. In our analysis the
(dg2,,~, H)
~J Here ekis the transformation parameter for the "species" k. ~2 If.I/2, is non-compact, one might need a sort of"infra-red regularization" to define C.
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cohomological meaning of the Maslov index is not obvious. We will come back in the future [ 16 ] to this issue by giving a path-integral representation o f the Maslov index in the Arnol'd version [ 13 ] where the cohomological meaning is transparent. We close with a few remarks on the meaning of BRS-invariant boundary conditions. O f course, a priori, classical mechanics is not a topological field theory and the time evolution of some phase-space density 0 ( 0 ) or its generalization 0(0, c) has little to do with topology. The time evolution of these functions is described by a path integral whose boundary conditions are BRS invariant. Hence the dynamics of 0 indeed "feels" the way in which we "gauge fix" the symmetry (4), i.e., it does depend on the local form of the hamiltonian which enters the gauge fixing condition. It is only for the very limited class of BRS-invariant observables together with BRS-invariant boundary conditions that we may obtain a topological invariant. We can rephrase this by looking at the theory in its Schr~dinger picture (see refs. [1,2]). In general we are dealing with functions which are not BRS invariant. Requiring them to be annihilated by Ca) = 0 means that, for 0-forms (for example), the only allowed or "physical" state is 0 ( 0 ) = c o n s t . Clearly, from the usual dynamical point of view, this is a rather uninteresting situation. Similarly, for a general p-form one selects the closed forms only. If we make the additional requirement that 0 should be of the form 04= Q ( ' ), we see that the "physical-state space" of the classical particle is given by the de R h a m cohomology classes HP (.~//2n, ~ ) . What is meant by "physical" in the above context, is that in these states the gaugefixing, and hence the dynamics, is not felt. In the future [ 16 ] we will come back to the issue of the physical-state condition but using a different BRS generator. This generator is the one we get once we study the symmetry of Zcm with the N-function #13 inserted. Let us remember that the N-function acts as a "gauge-fixing" for the ghost zero modes and so the expectation values of every observable have to be Nindependent: this will modify our original BRS charge. Using this different BRS charge, the new physical-state condition acquires a meaning once we work in the GNS representation of classical mechanics, and there we that it ensures the of the classical probability.
not
Q, QO- Cb3bO(O",
feel
positivity
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W e w i s h to w a r m l y t h a n k o u r f r i e n d Silvio Sorella for m a n y useful d i s c u s s i o n s . ~13This is the function that we introduced in the appendix of ref. [2] (and previously in this paper) to cure the problem of the ghost zero-modes.
References [ 1 ] E. Gozzi, Phys. Lett. B 201 (1988) 525 [2] E. Gozzi, M. Reuter and W.D. Thacker, Phys. Rev. D 40 (10) (1989) 3363. [3] E. Witten, Phys. Lett. B 206 (1988) 601; Commun. Math. Phys. 117 (1988) 353; 118 (t988) 411; L. Baulieu and I.M. Singer, NucL Phys. (Proc. Suppl. ) B 5 (1988) 12; S. Ouvry et al., Phys. Lett. B 220 (1989) 159; S. Sorella, Phys. Lett. B 228 (1989) 69; A. Blasi and R. Collina, Phys. Lett. B 222 (1989) 419. [4] V.I. Arnol'd Mathematical methods of classical mechanics (Springer, Berlin, 1978 ).
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[5] J.M. Labastida and M. Pernici, Phys. Lett. B 212 (1988) 56. [ 6 ] J.M. Labastida, Commun. Math. Phys. 123 ( 1989 ) 641. [7]W. Greub, S. Halperin and R. Vanstone, Connections, curvature and cohomology (Academic Press, New York, 1972); J. Milnor, Morse theory (Princeton U.P., Princeton, NJ, 1963). [8] E. Witten, Nucl. Phys. B 202 (1982) 253. [9] E. Witten, J. Diff. Geom. 17 (1982) 661. [ 10] L. Baulieu and B. Grossman, Phys. Lett. B 212 (1988) 351; L. Baulieu, CERN preprint CERN-TH 5537/89; D. Birmingham et al., Phys. Lett. B 214 (1988) 381; B. Grossman, Rockefeller University preprints. [ 11 ] R.J. Rivers, Path-integral methods in quantum field theory (Cambridge U.P., Cambridge, 1987). [ 12] V.I. Arnol'd, Funkt. Anal. Appl. 1 (1967) 1. [13]V. P. Maslov and M.V. Fredoriuk, Semi-classical approximation in quantum mechanics (Reidel, Dordrecht, 1981 ). [ 14] R.G. Littlejohn and J.M. Robbins, Phys. Rev. A 36 (1987) 2953. [15] R.G. Littlejohn, Phys. Rep. 138 (1986) 193. [ 16 ] E. Gozzi and M. Reuter, work in progress.
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CLASSICAL M E C H A N I C S AS A T O P O L O G I C A L F I E L D T H E O R Y E. G O Z Z I Theory Division, CERN, CH-1211 Geneva 23, Switzerland and M. R E U T E R Institut fiir Theoretische Physik, Universitdt Hannover, Appelstrasse 2, D-3000 Hannover 1, FRG Received 19 December 1989
We show that our recent formulation of classical mechanics via path integrals has many features in common with Witten's topological field theories. The action of the theory is a pure BRS commutator, which arises when we "gauge-fix" the symmetry of arbitrary deformations in path space. This gauge fixing involves the hamiltonian H and the symplectic 2-form m on the phase space ./g2,. Imposing BRS-invariant boundary conditions, the path integral can be used to calculate "topological" invariants of the dynamical system (~//2,,co, H). Here we restrict ourselvesto the Euler number of the symplecticmanifold ,~2nand the Maslov indices of the dynamics. Most probably many other "topological" features of (~/¢2~,co, H) can be calculated using this path integral. Conventional (non-topological) classical mechanics is recovered by using boundary conditions which are not BRS invariant. They give rise to expectation values which are sensitive to the "local" details of the dynamics.
In two previous papers [ 1,2 ] we have given a pathintegral formulation o f classical m e c h a n i c s both in its lagrangian and in its h a m i l t o n i a n version. In the present paper we are going to discuss a rather surprising feature of the path integral for classical mechanics: it gives rise to a kind of topological field theory similar to the theories recently proposed by Witten [ 3 ]. In particular, its lagrangian ~ , turns out to be a pure BRS commutator. We shall see that the classical path integral can be used to compute "topological" ~ invariants of any d y n a m i c a l s y s t e m (og2,, co, H ) , where ~12, is a symplectic manifold [4] with symplectic 2-form ~o and H is the hamiltonian. The simplest example is provided by quantities which are insensitive to both "small" and "large" deformations ofo~ and H a n d therefore do not "feel" the dynamics. They are the ordinary topological invariants of Jg2~, ~ By "topological" we mean quantities insensitive to "small" deformations of (,g2n, oJ, H) but sensitive to "large" deformations.
the cohomology classes HP(J{2n, ~R) or, in particular, the Euler n u m b e r Z(dt2n)- There is also another class of invariants which are sensitive to the dynamics, but which do not change if we slightly perturb the dynamics. A typical example of this class is the Maslov index. We consider a h a m i l t o n i a n system with a time-independent H a m i l t o n i a n H(O a) defined on a 2N-dimensional phase space ~g2nwith local coordinates oa, a = 1.... , 2 n . T h e " s u p e r - h a m i l t o n i a n " reads
~{ =)tao)abObH + iffaO)acOc ObH b
( 1)
a n d the lagrangian is given by ~ = 2 a b a + igab a - ~ . (We use the notation of refs. [ 1,2], with which we assume the reader familiar in the following.) In the Schr6dinger picture ~( is essentially the same as the Lie derivative l h - i ~ acting on functions O(~ba, ca). Here h-= ( d H ) ~ is the h a m i l t o n i a n vector field associated to the hamiltonian, and the ghosts c a are a basis in the cotangent space T*J/¢2,. Now let us described what our approach to CM has in c o m m o n
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with topological field theory models [ 3 ]. The crucial observation is that the hamiltonian ~ and the lagrangian ~ are pure BRS variations. Using the results of refs. [ 1,2 ], one easily verifies that .~ = - i [ Q ,
[Q, h]]
(2)
and ~=-i[Q,
z],
(3)
where O=igaOS'2~, is the anti-BRS operator [1,2] and Z-g,[Oa-h~(O ) ]. Interpreting (3) in the language of the standard hamiltonian BRS approach to gauge theories, one would conclude that the lagrangian ~ is completely determined by the "gauge fixing function" Z, because the gauge-fixing condition enters the Batalin-Fradkin-Vilkovisky path integral precisely in the form o f a BRS c o m m u t a t o r [ Q, Z]. A natural question to ask is whether there exists a local symmetry which, upon gauge fixing, gives rise to the BRS invariance o f ~ . This is exactly what was shown to happen for the fermionic symmetry of Witten's original theory [ 5 ]. In the present case it is not difficult to find the underlying local symmetry. Note that in the expression for X, Hamilton's equation appears as a kind of gauge-fixing condition. In fact, let us consider a theory of paths O~(t) in phase space, which has an identically vanishing action S[O] = O. Clearly, for this trivial action each path is as good as any other path and so S is invariant under the transformation
O~(t)-~O~(t) + 8 0 ~ ( t ) ,
Z= f 90"(t) a[Oa-h(~) ]det[a~ot-O~,h"(O) ] , (5) Eq. (5) is exactly the starting point for the path-integral of classical mechanics which was proposed in refs. [ 1,2]. This shows that, in a sense, our path-integral can be considered a gauge-fixed version of the "topological action" S [ 0 ] = 0 , with Hamilton's equation playing the role of a gauge fixing condition. This turns the local symmetry (4) into the global BRS invariance. The next question to ask is whether the path-integral of classical mechanics can be used to compute topological invariants. Let us first clarify the meaning of "topological" in the present context. For the kind of topological field theory discussed by Witten [ 3 ], the expectation value ( [r ) of some operator (~ is a topological invariant if and only if it is invariant under infinitesimal changes in the metric g~,, of the riemannian manifold on which the theory is defined. Clearly, since we are working on a symplectic manifold (phase space), which in general does not have a riemannian structure, the topological invariance of some expectation value must have a different meaning. In our case a quantity is of topological nature if it does not depend on the dynamics given by the hamiltonian H, i.e., it does not change if we perform small deformations of H or of the associated vector field h. Let us change the hamiltonian vector field h" by an amount
(4)
where 30~(t) is a completely arbitrary deformation of the original path. Very formally, the partition function for this theory would be Z = f 2 0 ~ ( t ) , i.e., it is a functional integral over a constant. To give a meaning to Z, let us gauge fix the local ~2 ~'gauge" symmetry (4). We choose the following gauge condition: G ~- O~-ha( Ob) ~0 where h~- o)~b~bH is the vector field generated by some hamiltonian H. Obviously the paths obeying this condition are exactly the solutions of Hamilton's equation. In a heuristic manner we can now apply the F a d d e e v - P o p o v trick. We implement the gauge-fixing by a a-function a [ G ~ ] multiplied by the F a d d e e v - P o p o v determinant det
(8G~lSOb):
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a,,h"(0h) ---~l~(&).
(6)
Then eq. (3) shows that ~ changes by a BRS variation g,, cQ'=i[Q, Carla] .
(7)
This statement is analogous to the property of the theories in ref. [2] that the metric variation of the lagrangian (the e n e r g y - m o m e n t u m tensor) is a BRS commutator. Now we investigate under which conditions the expectation value 7
( ( ' ( t ) ) = f g O " ( t ) . . . C ' ( O ( t ) .... ) e x p ( i f d t f f ~) 7"
(8) ~2 Here and in the following "local" means local in t. 138
of some local operator C = ( ( ¢ a ( t ) , 2a(t), c"(t),
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#a(t) ) is invariant under (6): 8 , ( ( 5 ( t ) ) =0. The integral (8) is over fields defined in [ - T , T]. It includes the integration over all fields at t = - T and over ~ and c a at t = + T. Note that these boundary conditions are BRS invariant, because of the BRS invariance of both the action and the measure, we can prove in the usual way [ 3 ] that the expectation value of any BRS variation vanishes: ( [ Q, ~ ] ) = 0. Exploiting this fact, it is easy to see that the r/variation of ( C~(t) ) is given by
8.< (5'(t)F=
T
Zpbc: f ~ ( t ) . . . e x p ( i f d t ' ~ )
=
f
d2nOo d 2ncoK(~oCo, a a
0
TI¢g, cg, O) ,
c~, ti)=o(Zn)(f~'~--oal(tf, 0i))
X~(2n)(Cf--cal(tf,
Ci, [q~])) •
(12)
Periodic boundary conditions are indeed preserved under BRS transformations. We first compute Zpbc in the limit T ~ 0 and show later on that Zpbc actually does not depend on T. For closed paths and short times Tthe kernel K receives contributions only from the points were the vector field h a = 09ab0Jt vanishes, i.e., for the critical points of the hamiltonian H. Let us assume for simplicity that H has only isolated, nondegenerate ~s critical points at O~p). We then have K(q~, c~, Tl~g, cg, 0 ) = 6(en) (cg)
det[~bh"(O(p))] ~(2,)(~)~_O~p) ) X (p)~ idet[Obh~(O(p))] I
(I0)
These conditions are very similar to the analogous requirements for topological Yang-Mills theory or the topological a-model, say. The observable (5 must be BRS closed, but not exact, and the q variation (or the g ~ variation) o f C has to be a BRS commutator. If these requirements are met, ( ~' ) does not depend on the details of the dynamics prescribed by H. As a first example, let us consider the simplest operator satisfying (10), namely the unit operator (5:= 1. Computing its expectation value with free boundary conditions as in (8) we would obtain ( ( 5 ) = 0 because of the ghosts' zero-modes ~3. However this is not so for periodic boundary conditions (pbc)~4. Requiring oa (0) = ~ (T) and c ~(0) = c ~(T) for some fixed T > 0, we are led to
pbc
K ( 0 f , cf, tf I~f,
(13)
where the sum extends over all critical points in dg2~. Using (13) in ( 11 ) one finds
(5(t)#[Q,(.)],
6, d ' ( t ) = [Q, ( ' ) ] .
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(ii)
where [ 1 ]
Zpbc = (p) 2
det[OaObH(¢(p))] = ~ ( - - 1 ) ip. [det[OaObH(~(p))]l (p)
Here ip denotes the index of the critical point (p), i.e., the number of negative eigenvalues of the hessian of H a t (p) (which in local coordinates has components 0,0y(O~p)) ). The above result for Zpbc has a well-known interpretation [ 7 ] : it is the Morse theory representation of the Euler n u m b e r z o f the manifold on which the "Morse function" H is defined. Thus we have found that Zob~ = Z ( J[2n ) .
This answer is indeed independent of the dynamics of the system under consideration. The hamiltonian acts only as a Morse function and hence Zpbc contains only information about the topology of phase space, but not about the dynamics generated by H. Next we show that eq. ( 14) holds for any T > 0 and not only for T ~ 0 . The argument we use is quite standard in the discussion of the Witten index ofsupersymmetric quantum mechanics [ 8,9 ]. Recalling the operatorial formulation of the path integral [ 1,2], we may write eq. (11) as Zpbc = T r [ ( - ) F e x p ( - i T , # ) ] .
~3 A way out is to insert a "'gauge"-fixing function N(2 (0), c(0 ), ( ( 0 ) ) for these zero modes, as we did in the appendix of ref. [2]. ~4 This corresponds to a choice of a particular N function.
(14)
(15)
,5 This restriction is not essential. Along the lines of ref. [6 ] one could generalize the discussion to include degenerate Morse theory. 139
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Here the trace " T r " is performed over a complete set of functions 0,,(0 a, c a) or, equivalently, a set of antisymmetric tensor fields Q!P~. .~( O), p=O,1 ..... 2n. The "fermion number" operator F counts the degree of the respective differential form, hence ( - ) F = + 1 ( -- 1 ) f o r p even (odd). To give a meaning to eq. ( 15 ), we have to introduce a scalar product for forms. This is done by choosing an arbitrary riemannian metric g,~, o n ~,~/¢2n- Then, by a well known argument [ 8,9 ], the trace reduces to a sum over harmonic forms or, equivalently, to a trace in the Rham cohomology group H v (,,,//2,,, ~1):
,
(r'}))(t)=pFa ...... (O(t))~m(t)ca~(t)...C"~(t).
The RHS ofeq. (16) has a well-known interpretation in terms of the Lefschetz coincidence theorem [7], namely it is the Lefschetz number Lef[exp ( - Tlh)] of the mapping induced by e x p ( - T l h ) on phase space. The general Lefschetz theorem, for an arbitrary mapping K of some manifold into itself, expresses alternating sums like (16) in terms of local data of the fixed point set of the respective mapping. It can be shown that Lef[K] is always integer, and that it does not depend on the riemannian metric chosen. Furthermore, Lef[K] is a homotopic invariant of K. This implies that, if K is homotopic to the identity map, K ~ id, the traces o f H p are simply given by the Betti numbers bP: (17)
In our case K - exp ( - Tlh) is generated by a continuous time evolution and Kis in fact homotopic to the identity (with T playing the role of the hornotopy parameter). Therefore eq. (16) becomes Zpbc= ~ 2 np = 0 tI, - - 1 )PbP=z(d/[2n ) which coincides with the result (14) of the explicit evaluation of the classical path integral. Since K ~ id for any value of T, the relation above shows that Zpb~ does not depend on T. The reader will have realized, by now, that our classical path-integral has many features in c o m m o n with the partition function (with pbc) of supersymmetric quantum mechanics on a curved manifold [ 8,10 ]. Now let us try to find more general topological invariants ( ( c ) . Assuming that ~' does not depend on H, eq. (10) requires that (~ has a vanishing commutator with Q, but cannot be written as the BRS vari140
~)9~(t) = [Q, (~ ~ ( t ) ]
(18)
(16)
p~0
Trn~[K] = T r n ~ [ i d ] =dimHP(./ff2n, 91)=6 p .
ation of something. Operators with this property are in one-to-one correspondence with representatives F(P)=F(p) cttha I / k ...A0a~ofthecohomologyclasses -al,..ap ~ 'f" HP(.J/2n, ~). If we define (f(v°)(t)=F~f!,p(O(t)) Xcal(1)ca2(t).,.cap(t), it easily follows, from the operatorial rules derived in refs. [ 1,2 ], that (~fg) is BRS closed, but not exact. If ((~FO~(t)) is to be topological invariant, it must not depend on t. This would be guaranteed if~'; ~9) were a BRS-commutator. It is easy to verify that this is indeed the case:
with
2n
Zpb,.= ~ ( - 1 ) P T r n ~ [ e x p ( - T / ~ ) ] .
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(19)
The observables ~,}9~ and (!'~)) are analogous to Witten's [3] operators Wo and W~ out o f which the Donaldson invariants are constructed. Since in our case the "base manifold" is one-dimensional, there are only two classes of such operators. Now let us look at the expectation values (6~9) ) and ((!'}))). Here we have to distinguish carefully between the path integral defined on the interval [ - T , + T] where the fields 0" and c" are integrated independently at t = _+ T, and the path integral for a compactified base manifold S ~ where the points ( 0 a ( - T), c a ( - T)) and (oa( + T), C~( + T) ) are identified so that there is only one integration over O ( - T ) = O ( T ) and c " ( T ) =c~( - T). For the first type of boundary condition and the operators ( ~ 9 ) ( t = 0 ) , we find for the path integral (8)
< e',~.°'(o)) = J dx(t) d x ( O ) d x ( - T ) K ( x ( T ) , T I x ( O ) ,
O)
× C'~9)(x(O))K(x(O),OIx(-T),-t)
= f dr(O) (,'gg~(x(O)),
(20)
where we have used the notation x = (0% ca). Obviously ((~l;o)) does not give rise to interesting topological invariants since for p ~ 2 n this expectation value vanishes because of the ghost integration. For p=2n one obtains fd2n 0 Fo ....... e ........ which is proportional to the volume of phase space (if b o = b 2n= 1 ). We stress, however, that on [ - T, + T] it is pos-
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sible in principle to have a non-zero expectation value o f an o p e r a t o r like 6~ °) with non-vanishing ghost charge. The reason is that, as in any phase-space path integral over some interval [ - T, T] [ 1 1 ] there is a m i s m a t c h between the position c a and the m o m e n tum Ca integration: including the position integrations at t = _+ T, there is one more position integration than m o m e n t u m integration. This means that the measure ~ c ~ ¢ is not neutral u n d e r the ghost charge transformation. The situation is different if we compactify [ - T , T] to the circle S ~, i.e., if we integrate over closed paths only. Then the c and &integrations match and only operators o f vanishing ghost n u m b e r can have a non-zero expectation value ~6. Since both 6'~-°) and 6 ' ~ contain only c's, but no ¢'s, no further invariants b e y o n d Zvb¢ can be c o m p u t e d in this way. Before discussing a further class o f "topological" invariants, we briefly m e n t i o n a slight generalization o f our path integral. We consider N distinguishable species o f particles, all o f which evolve according to H a m i l t o n ' s equation with the same h a m i l t o n i a n vector field h ~ - O)abOaH. The time-evolution kernel for their generalized densities 0(0~, c~, 0~, c~, ..., 0~, C~v) ~7 is given by a p r o d u c t of the 0-functions as in eq. (12)
Ku({Ok,d, {Ck,r}, tfl {Ok,,}, {Ok,i}, /i' N
= l-[ K(0k,f, Ck,f, tfl0~,i, Ck,i, ti k=l
:f20~.~2k,a~C~-~(k,,~exp(iidt~u ) .
(21,
ti
The path-integral representation o f KN has the same form as that o f K b u t with ~ 0 a ( t ) ~ 2 , ( t ) . . . replaced by ~ 0 ~ ( t ) ~ 2 a k .... and the lagrangian S~ replaced by ~N = E k V = I ~ ( ~ , /]-a,k, C~, ga,k)- Since the various particle species do not " i n t e r a c t " , the resulting Nparticle theory is simply the N-fold tensor product o f the original one. The action ~N = fdt ~ is invariant under a set o f N i n d e p e n d e n t BRS transformations o f the form given in refs. [ 1,2 ]. ~6 This follows from the vanishing index of the kinetic operator of the ghosts. The situation is similar as to that of topological quantum mechanics, see ref. [ 12]. ,7 This function gives the "super-probability" to find a particle of species 1 at (0~, c~), of species 2 at (0~, c~ ), etc.
19 April 1990
We are now going to discuss another kind o f " t o p ological" observables which are sensitive to m and H us. This discussion is specific to the path-integral o f classical mechanics and we do not know if it has a counterpart in quantum field theory. The point is that up to now, in order to integrate over closed paths, we had to specify this by an explicit b o u n d a r y condition 0 a ( 0 ) = 0 a ( T ) . This condition selects those initial points 0~ for which the vector field h a, or the hamiltonian H respectively, admits closed trajectories of length T. Typically there will be only very few such trajectories; in fact, it is known that they form a set o f measure zero. F o r a generic h a m i l t o n i a n H, almost all initial conditions 0 a ( 0 ) will give rise to trajectories which never close. There exists a special class o f hamiltonians, however, which has the property that they generate closed orbits in phase space for any initial point 0~. This kind o f h a m i l t o n i a n naturally appears in integrable systems, for instance. Let us assume we are given an integrable h a m i l t o n i a n system (J/2n, ~o, H ) so that we can transform from the 0 a coordinates to action-angle variables (0,, J,), i = 1, ..., n. Considering the actions J i = J ~ ( 0 a) as the generators o f a family o f canonical transformations, their integral curves generate the basic homology cycles on the tori. Starting from a given point 0 a ( 0 ) and following the trajectory induced by Ji according to ~a=o)at'ot, J , ( O ) , we generate the ith cycle Fi. A given initial point is equivalent to a set o f numerical values {Ji, i = 1.... , n} for the action variables. This set defines a certain n-torus in ,~#2n. The curves F~, i = 1, ..., n generated by Js(O~) starting from 0 a ( 0 ) lie on this torus for all t. F r o m the path-integral point o f view it is attractive to use Js(O), rather than H, as a hamiltonian, because for this class of hamiltonians the pathintegral is saturated by closed paths even in the case when we work on { - T, T] with i n d e p e n d e n t 0 a and c a integration at t = _+ T ~9. We begin with a very simple example. We take the path integral (8) over [ - T, T] with i n d e p e n d e n t integrations at the end points and replace the h a m i l t o n i a n H by the actions 3",.. Furthermore we pick a certain representative F = F , d0 ~ o f H I ('.#2n, ~]~ ) and define the observable ~8 Again we stress the meaning of "topological" (or sensitive) here: it means that they do not "feel" small deformations of o), ,;¢, but only "large" deformations. ~9 Recall that in this situation we can have (¢~)¢0 even if Cf contains ghosts. 141
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~ ( 0 g ) = 6<2~)(0"(0) -Og)a~")(c~(O) )
× ~dt' fa(ob(l'))oa(t'),
(22)
which is similar to eq. (19) for p = 1..N is a functional of the (closed) path O~(t ' ) and the t'-integration is over a full period. Clearly .~- is BRS invariant, since a BRS transformation leads to a homotopic deformation of the path which does not change the integral or the RHS of (22) since d F = 0 . Moreover, 0~(0) would change by an amount ec~(0), which is zero due to the second 6- function. If we now compute the expectation value ( J ) it is easy to see that the path integral calculates fFad6 ~ along the classical trajectories ~ generated by J, starting from 0g:
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which describes the change of the position of the particle at time t due to a small change of the initial conditions. Note that this matrix is symplectic, S~nbE Sp (2n), since it is the jacobian matrix of a canonical transformation [ 4 ]. Now divide the phase-space coordinates (0 ~, ..., 02n) into a set of n "positions" q~ and n "momenta" p', i= 1.... , n, and then form the ( n × n ) matrices SUo(t)-S~(t) and Sqp(t)~-S~. As was shown by Littlejohn and Robbins [14], these matrices can be used to compute the Maslov index/~, associated to F~: g~ = 2 W[ det{Sg + iSg } ] d,
~i~ilndet[Sg(t)+iSqp(t)].
=2
(26)
F,
(.~7(0g)) = ~ d t ' f,(O~l(t',O~))O~,(t',Obo).
(23)
Even this somewhat trivial example nicely illustrates the interplay between dynamics (or topological) invariance, BRS invariance and homotopic invariance. l f F is closed, .Y is BRS invariant and therefore ( . ~ ) is insensitive to the details of the "gauge-fixing", i.e., of the hamiltonian. On the other hand, Fbeing closed means that }r,F is invariant under homotopic deformations of F,, and the BRS transformation ~b~(t) 0 a (t) + Ec~(t) provides precisely such an infinitesimal transformation. Nevertheless, contrary to )~(~t6~), say, ( , ~ ) depends on the hamiltonian, since replacing Js by Jk, k ¢ i, for instance, F is integrated over the cycle Fk which yields a different result in general. Now let us apply the above method to the Maslov index [12-14] #~ entering the semiclassical quantization condition Jl = (rti + l / / i ) h, r / , ¢ Z
(24)
Traditionally the Maslov index was calculated by investigating the caustics of lagrangian manifolds, but here we shall follow the approach of Littlejohn et al. [ 14 ], who interpreted #~ as a winding number in the Sp(2n) group manifold. To each cycle F~ generated by Ji we can associate an integer/t~ by the following construction, let F~ be parameterized by O~(t), t~ [0,2~z]. We define the 2~ periodic matrix function
O0~(t) Sg(t) = 00b(0),
First of all this formula expresses #i as a winding number W in the complex plane: the integral on the RHS of eq. (26) counts the number of times the value of the function det [S q + iS q ] encircles the origin of the complex plane, in a period of oscillation of 0. Eq. (26) can be used to assign an integer W[S] to any periodic, symplectic matrix function S. Clearly W[S] can depend only on the homotopy class of S. Since ~(Sp (2n) ) = Z, these classes are labelled by a single integer (the winding number), which turns out ~1oto be given by W[S]. We now proceed to give a pathintegral representation for #~. The crucial observation is that the matrix Sg(t) ofeq. (25) is essentially the same as our ghost fields. In fact, if ~ ( t ) evolves with the "hamiltonian" J,, this matrix function is a solution of
[O,6g - Obf/(O(t) ) ]S)(t) = 0 ,
with the initial condition S g ( 0 ) = 6 g (here j ~ = o)ab0bJi).Comparing eqs. (27) with the equations of motion of the ghosts [ 1,2 ], we see that, for each fixed value of the lower index c, S~ evolves in the same way as the ghost c b. This means that ab object like Sg appears naturally in the N-particle path integral of eq. (21) if we put N=2n. Using the initial condition c~(0) = ~f, for the 2n "species" of ghosts, we can recover S~, at a later time by Sg(t)=(O/d()c~b~(t). Here ~ is an anticommuting constant introduced because Sg is a commuting object, whereas the ghosts
(25) t~10 For a proof see the appendix of ref. [ 15 ].
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are anticommuting. Now it is easy to express/~i as an expectation value of the form T
f ~O~.(t) ~2a,k... (('exp(i f dt~'~),
((()=
--T
(28) with t~ [ - 7", + T], T>~2g, and independent integrations at the end points. We define the observables as ((0~) 2n
= 2 [1 ~ 2 " ) ( ¢ ~ ( 0 ) - 0 g ) k=l
2n
l-[ ~ ( 2 " ) ( c ~ ( 0 ) - ( ~ ) k=l
2zt
Xf dt lndet(~([cq(t)+icq(t)])
(29)
0
The first 0-function fixes a c o m m o n starting point for all 2n paths we follow, and the second assigns the initial values to the ghosts. Obviously when computing the expectation value ( 6 i ( ¢ ~ ) ), the path integral is saturated by a single classical trajectory on which the observable ~ci(0~) is evaluated. This means that ( ~ ( 0 ~ ) ) =~t~ for any initial point 0~. The observable (!:( ~ ) is BRS invariant expect for the 6 function fixing the starting point. U n d e r the BRS transformation it is changed by an amount ekc~ = ( e a ~ 1. To obtain a fully BRS invariant observable we can omit the 0-function, which fixes the initial point 0o, from (29) or, more precisely, use the operator 6 = £2- ~f d2nOoX ( ( 0 0 ) where ,Q is the volume of the phase space ~t2. Obviously, since all trajectories generated by Jj lead to the same Maslov index, we have again ( ( ! ' ) =/z~ but this time with a manifestly BRS-invariant operator (~.. Applying the above arguments, the BRS invariance of C! implies the invariance o f / 4 under "small" deformations of J~, which in turn translates into small deformations of the original hamiltonian H which gave rise to the actions {Ji}. Note again that "topological" invariance has a different meaning for Zpb¢ and for the Maslov index. In the first case "global" properties of only ,/g2, mattered, whereas in the second case we picked up "global" or ("large") properties of the whole triplet that characterize the dynamical systems. In our analysis the
(dg2,,~, H)
~J Here ekis the transformation parameter for the "species" k. ~2 If.I/2, is non-compact, one might need a sort of"infra-red regularization" to define C.
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cohomological meaning of the Maslov index is not obvious. We will come back in the future [ 16 ] to this issue by giving a path-integral representation o f the Maslov index in the Arnol'd version [ 13 ] where the cohomological meaning is transparent. We close with a few remarks on the meaning of BRS-invariant boundary conditions. O f course, a priori, classical mechanics is not a topological field theory and the time evolution of some phase-space density 0 ( 0 ) or its generalization 0(0, c) has little to do with topology. The time evolution of these functions is described by a path integral whose boundary conditions are BRS invariant. Hence the dynamics of 0 indeed "feels" the way in which we "gauge fix" the symmetry (4), i.e., it does depend on the local form of the hamiltonian which enters the gauge fixing condition. It is only for the very limited class of BRS-invariant observables together with BRS-invariant boundary conditions that we may obtain a topological invariant. We can rephrase this by looking at the theory in its Schr~dinger picture (see refs. [1,2]). In general we are dealing with functions which are not BRS invariant. Requiring them to be annihilated by Ca) = 0 means that, for 0-forms (for example), the only allowed or "physical" state is 0 ( 0 ) = c o n s t . Clearly, from the usual dynamical point of view, this is a rather uninteresting situation. Similarly, for a general p-form one selects the closed forms only. If we make the additional requirement that 0 should be of the form 04= Q ( ' ), we see that the "physical-state space" of the classical particle is given by the de R h a m cohomology classes HP (.~//2n, ~ ) . What is meant by "physical" in the above context, is that in these states the gaugefixing, and hence the dynamics, is not felt. In the future [ 16 ] we will come back to the issue of the physical-state condition but using a different BRS generator. This generator is the one we get once we study the symmetry of Zcm with the N-function #13 inserted. Let us remember that the N-function acts as a "gauge-fixing" for the ghost zero modes and so the expectation values of every observable have to be Nindependent: this will modify our original BRS charge. Using this different BRS charge, the new physical-state condition acquires a meaning once we work in the GNS representation of classical mechanics, and there we that it ensures the of the classical probability.
not
Q, QO- Cb3bO(O",
feel
positivity
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W e w i s h to w a r m l y t h a n k o u r f r i e n d Silvio Sorella for m a n y useful d i s c u s s i o n s . ~13This is the function that we introduced in the appendix of ref. [2] (and previously in this paper) to cure the problem of the ghost zero-modes.
References [ 1 ] E. Gozzi, Phys. Lett. B 201 (1988) 525 [2] E. Gozzi, M. Reuter and W.D. Thacker, Phys. Rev. D 40 (10) (1989) 3363. [3] E. Witten, Phys. Lett. B 206 (1988) 601; Commun. Math. Phys. 117 (1988) 353; 118 (t988) 411; L. Baulieu and I.M. Singer, NucL Phys. (Proc. Suppl. ) B 5 (1988) 12; S. Ouvry et al., Phys. Lett. B 220 (1989) 159; S. Sorella, Phys. Lett. B 228 (1989) 69; A. Blasi and R. Collina, Phys. Lett. B 222 (1989) 419. [4] V.I. Arnol'd Mathematical methods of classical mechanics (Springer, Berlin, 1978 ).
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[5] J.M. Labastida and M. Pernici, Phys. Lett. B 212 (1988) 56. [ 6 ] J.M. Labastida, Commun. Math. Phys. 123 ( 1989 ) 641. [7]W. Greub, S. Halperin and R. Vanstone, Connections, curvature and cohomology (Academic Press, New York, 1972); J. Milnor, Morse theory (Princeton U.P., Princeton, NJ, 1963). [8] E. Witten, Nucl. Phys. B 202 (1982) 253. [9] E. Witten, J. Diff. Geom. 17 (1982) 661. [ 10] L. Baulieu and B. Grossman, Phys. Lett. B 212 (1988) 351; L. Baulieu, CERN preprint CERN-TH 5537/89; D. Birmingham et al., Phys. Lett. B 214 (1988) 381; B. Grossman, Rockefeller University preprints. [ 11 ] R.J. Rivers, Path-integral methods in quantum field theory (Cambridge U.P., Cambridge, 1987). [ 12] V.I. Arnol'd, Funkt. Anal. Appl. 1 (1967) 1. [13]V. P. Maslov and M.V. Fredoriuk, Semi-classical approximation in quantum mechanics (Reidel, Dordrecht, 1981 ). [ 14] R.G. Littlejohn and J.M. Robbins, Phys. Rev. A 36 (1987) 2953. [15] R.G. Littlejohn, Phys. Rep. 138 (1986) 193. [ 16 ] E. Gozzi and M. Reuter, work in progress.