Supersymmetry in Stochastic Processes with Higher-Order Time Derivatives Hagen KLEINERT and Sergei V. SHABANOV ∗ † Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany The supersymmetry implies QS = 0. A supersymmetric path integral representation is devel- The determinant (3) should not be confused with the oped for stochastic processes whose Langevin equation con- partition function for fermions governed by the Hamil- tains any number N of time derivatives, thus generalizing the tonian associated with the action (4). Instead of a Langevin equation with inertia studied by Kramers, where trace over external states it contains only the vacuum- N = 2. The supersymmetric action contains N fermion fields to-vacuum transition amplitude for the imaginary-time with first-order time derivatives whose path integral is evalu- ated for fermionless asymptotic states. interval under consideration. In the coherent state rep- resentation,c ¯t and ct are set to zero at the initial and 1. For a stochastic time-dependent variable xt obeying a first-order Langevin equation final times, respectively [2]. The path integral (2) can also be rewritten in a canon- ical Hamiltonian form by introducing an auxiliary Gaus- Lt[x] ≡ x˙ t + F (xt)= ηt, (1) sian integral over momentum variables pt, and replac- H 2 driven by a white noise ηt with hηti = 0, hηtηt′ i = δtt′ , ing Sb by Sb = dt (pt /2 − iptLt). The generator the correlation functions hx ··· x i can be derived from of supersymmetry for the canonical action is QH = t1 tn R a generating functional dt (ictδxt + ptδc¯t ). This form has an important advan- tage to be used later that it does not depend explicitly R i dtJx −Sb+i dtJx on D , so that the above analysis remains valid also for Z[J]= he i = Dx ∆ e , (2) t more general colored noises with an arbitrary correlation R Z R function hηatηbt′ i = (Dab)tt′ 6= δtt′ . 1 2 with an action Sb = 2 dt Lt , and a Jacobian ∆ = Inserting (3) into (2), the generating functional be- det δx ′ Lt. We denote by δx ′ Lt the functional deriva- comes t R t tive of Lt[x] with respect to its argument. Explicitly: H ′ ′ i dtJx −S −Sf +i dtJx δxt′ Lt = [∂t + F (xt)]δtt . The time variable is written as Z[J]=he i= DpDxDc¯Dc e . (6) a subscript to have room for functional arguments after R Z R a symbol. It was pointed out by Parisi and Sourlas [1] that by expressing the Jacobian ∆ as a path integral over This representation makes the stochastic process (1) Grassmann variables equivalent to by supersymmetric quantum mechanical system in imaginary time. In the supersymmetric formu- lation of a stochastic process, there exists an infinity of ∆= Dc¯Dce−Sf (3) Ward identities between the correlation functions which Z can be collected in the functional relation with a fermionic action −SH H arXiv:quant-ph/9705042v1 24 May 1997 DpDxDc¯Dce Q Φ=0, (7) ′ ′ S = dtdt c¯ δ ′ L c ′ = dt c¯ (∂ + F )c , (4) f t xt t t t t t Z Z Z valid for an arbitrary functional Φ ≡ Φ[p, x, c,c¯ ]. The the combined action S ≡ Sb+Sf becomes invariant under Ward identities simplify a perturbative computation of supersymmetry transformations generated by the nilpo- the correlation functions. 2 tent (Q = 0) operator A proof of the equivalence of (1) to (2) requires a regu- larization of the path integral, most simply by time slic- Q = dt (icδx − iLtδc¯) (5) ing. This is not unique, since there are many ways of dis- Z cretizing the Langevin equation (1). If one sets ti = iǫ, for i =0, 1, 2,...,M, xi = xti , and Fi = F (xi), then the velocityx ˙ may be approximated by (xi − xi−1)/ǫ. On the sliced time axis, the force F (xt) may act at any time ∗Humboldt fellow; on leave from Laboratory of Theoretical within the slice (ti,ti−1), which is accounted for by a pa- Physics, JINR, Dubna, Russia. † rameter a and a discretization F → aFi + (1 − a)Fi−1. Email: [email protected]; [email protected] Note that the discretized Langevin equation is assumed berlin.de; URL: http://www.physik.fu-berlin.de/˜kleinert. to be causal, meaning that given the initial value of Phone/Fax: 0049/30/8383034 the stochastic variable x0 and the noise configurations 1 η0, η1, ..., ηM−1, the Langevin equation uniquely deter- higher time derivatives mines the configurations of the stochastic variable at later time, x1, x2, ..., xM . The simplest choice of the right- Lt = γ(∂t)x ˙ t + F (xt)= ηt , (9) hand side of the Langevin equation compatible with the where γ is a polynomial of any order N − 1, thus pro- causality is to set it equal to ηi−1. In general, one can M ducing N time derivatives on xt. This Langevin equa- replace ηi−1 by 1 Aj−1,i−1ηi−1 with A being an or- thogonal matrix, AT A = 1. The latter is just an evi- tion may account for inertia via a term m ∂t in γ(∂t), dence of the symmetryP of the stochastic process with the and/or an arbitrary nonlocal friction dτγτ x˙ t−τ = N−1 γ ∂+1x where γ = dτγ (−τ)n/n!. The main white noise with respect to orthogonal transformations n=0 n t t n τ R η → (Aη) . problem is to find a proper representation of the more t t P R Some specific values of the interpretation parameter complicated determinant ∆ = det δxt′ Lt in terms of a have been favored in the literature, with a = 0 or Grassmann variables. The standard formula (3), though 1/2 corresponding to the so-called Itˆo- or Stratonovich- formally applicable, does not provide a proper represen- related interpretation of the stochastic process (1), re- tation of the determinant of an operator with higher- spectively [2,3]. In the time-sliced path integral, these order derivatives because of the boundary condition values correspond to a prepoint or midpoint sliced ac- problem. This problem is usually resolved via an op- tion [2,4]. Emphasizing the a-dependence of the sliced erator representation of the associated fermionic system. H In the stochastic context it has so far been discussed only action, we shall denote it by Sa . This action is super- H H for the single time derivative [2]. In the first-order for- symmetric for any a: Q Sa = 0, and the sliced genera- H malism, the fermion path integral can be defined in terms tor Q = i(ici∂xi +pi∂c¯i ) turns out to be independent of both the interpretation parameter a and the width ǫ of coherent states [2] with the above discussed vacuum- of time slicingP [2]. A shift of a changes the action by the to-vacuum boundary conditions. Higher-derivative theo- Q-exact term, ries, however, have many unphysical features, in partic- ular states with negative norms [6,7], and it is a priori H H H Sa+δa = Sa + δaQ G, (8) unclear how to define the boundary conditions for the as- sociated fermionic path integral. In gauge theories, the where G is a function of a and a functional of p, x, c,¯ c. Faddeev-Popov ghosts give an example of a fermionic the- This makes the Ward identities independent of a, i.e. ory with higher-(second-)order derivatives. There, un- on the interpretation of the Langevin equation. In- physical consequences of the negative norms of the ghost − H − H H deed, setting δa = −a we find e Sa = e S0 eaQ Φ ≡ states are avoided by imposing the so called BRST invari- H −S0 H ′ e (1 + Q Φa). Substituting this relation into (7) we ant boundary conditions upon the path integral. For the observe that the a-dependence drops out from the Ward above stochastic determinant with higher-order deriva- H H identities because of the supersymmetry Q S0 = 0 and tives, the correct boundary condition are unknown. the nilpotency (QH )2 = 0. 2. The solution proposed by us in this work is best The simplest situation arises for the Itˆochoice, a = 0. illustrated by first treating Kramers’ process where one Then the sliced fermion determinant ∆ becomes a triv- more time derivative is present, accounting for particle ial constant independent of x. In the continuum limit inertia, i.e. where γ(∂ )= ∂ + γ for a unit mass m ≡ 1. of the path integral, however, this choice is inconvenient t t Omitting the time subscript of the stochastic variables, since then the limiting action S cannot be treated as an 0 for brevity, we replace the stochastic differential equation ordinary time integral over the continuum Lagrangian. (9) by two coupled first-order equations Instead, Sa=0 goes over into a so-called Itˆostochastic in- tegral [2]. The Itˆointegral calculus [3] differs in several Lv =v ˙ + γv + F (x)= νv, (10) respects form the ordinary one, most prominently by the property dx 6= dtx˙. This difficulty is avoided taking Lx =x ˙ − v = νx, (11) the Stratonovich value a = 1/2, for which the contin- There are now two independent noise variables, which uum limitR of S Ris an ordinary integral [2,5]. Splitting 1/2 fluctuate according to the path integral (8) as Sa = S1/2 + (a − 1/2)Q G, the non-Stratonovich part vanishes in the continuum limit because Q does not depend on the slicing parameter ǫ, whereas G is pro- hF [x, v]i = DνxDνvF [x, v] portional to ǫ → 0 [2]. For a = 1/2, formula (6) has Z 2 2 −1/2 dt[ν /2(1−σ)+(ν ˙ +γν ) /2σ] a conventional continuous interpretation as a sum over ×e v x x .