Institut für Theoretische Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Supersymmetry in Stochastic Processes with Higher-Order Time Derivatives

Hagen KLEINERT and Sergei V. SHABANOV

Abstract:

A supersymmetric path integral representation is developed for stochastic processes whose Langevin equation contains any number N of time derivatives, thus generalizing the Langevin equation with inertia studied by Kramers, where N=2. The supersymmetric action contains N fermion fields with first-order time derivatives whose path integral is evaluated for fermionless asymptotic states.

1. For a stochastic time-dependent variable obeying a first-order Langevin equation

 

driven by a white noise with , the correlation functions can be derived from a generating functional

 

with an action , and a Jacobian . We denote by the functional derivative of with respect to its argument. Explicitly: . The time variable is written as a subscript to have room for functional arguments after a symbol. It was pointed out by Parisi and Sourlas [1] that by expressing the Jacobian as a path integral over Grassmann variables

 

with a fermionic action

 

the combined action becomes invariant under supersymmetry transformations generated by the nilpotent ( ) operator

 

The supersymmetry implies QS=0.

The determinant (3) should not be confused with the partition function for fermions governed by the Hamiltonian associated with the action (4). Instead of a trace over external states it contains only the vacuum-to-vacuum transition amplitude for the imaginary-time interval under consideration. In the coherent state representation, and are set to zero at the initial and final times, respectively [2].

The path integral (2) can also be rewritten in a canonical Hamiltonian form by introducing an auxiliary Gaussian integral over momentum variables , and replacing by . The generator of supersymmetry for the canonical action is . This form has an important advantage to be used later that it does not depend explicitly on , so that the above analysis remains valid also for more general colored noises with an arbitrary correlation function .

Inserting (3) into (2), the generating functional becomes

 

This representation makes the stochastic process (1) equivalent to a supersymmetric quantum mechanical system in imaginary time. In the supersymmetric formulation of a stochastic process, there exists an infinity of Ward identities between the correlation functions which can be collected in the functional relation

 

valid for an arbitrary functional . The Ward identities simplify a perturbative computation of the correlation functions.

A proof of the equivalence of (1) to (2) requires a regularization of the path integral, most simply by time slicing. This is not unique, since there are many ways of discretizing the Langevin equation (1). If one sets for , , and , then the velocity may be approximated by . On the sliced time axis, the force may act at any time within the slice , which is accounted for by a parameter a and a discretization . Note that the discretized Langevin equation is assumed to be causal, meaning that given the initial value of the stochastic variable and the noise configurations , the Langevin equation uniquely determines the configurations of the stochastic variable at later time, . The simplest choice of the right-hand side of the Langevin equation compatible with the causality is to set it equal to . In general, one can replace by with A being an orthogonal matrix, . The latter is just an evidence of the symmetry of the stochastic process with the white noise with respect to orthogonal transformations .

Some specific values of the interpretation parameter a have been favored in the literature, with a=0 or 1/2 corresponding to the so-called Itô- or Stratonovich-related interpretation of the stochastic process (1), respectively [2, 3]. In the time-sliced path integral, these values correspond to a prepoint or midpoint sliced action [2, 4]. Emphasizing the a-dependence of the sliced action, we shall denote it by . This action is supersymmetric for any a: , and the sliced generator turns out to be independent of both the interpretation parameter a and the width of time slicing [2]. A shift of a changes the action by the Q-exact term,

 

where G is a function of a and a functional of . This makes the Ward identities independent of a, i.e. on the interpretation of the Langevin equation. Indeed, setting we find . Substituting this relation into (7) we observe that the a-dependence drops out from the Ward identities because of the supersymmetry and the nilpotency .

The simplest situation arises for the Itô choice, a=0. Then the sliced fermion determinant becomes a trivial constant independent of x. In the continuum limit of the path integral, however, this choice is inconvenient since then the limiting action cannot be treated as an ordinary time integral over the continuum Lagrangian. Instead, goes over into a so-called Itô stochastic integral [2]. The Itô integral calculus [3] differs in several respects form the ordinary one, most prominently by the property . This difficulty is avoided taking the Stratonovich value a=1/2, for which the continuum limit of is an ordinary integral [2, 5]. Splitting (8) as , the non-Stratonovich part vanishes in the continuum limit because Q does not depend on the slicing parameter , whereas G is proportional to [2]. For a=1/2, formula (6) has a conventional continuous interpretation as a sum over paths, and can be treated by standard rules of path integration (e.g., perturbation expansion around Gaussian measures). The price for this is the additional fermion interaction which possesses, as an attractive feature, the additional supersymmetry.

The aim of our work is to extend this supersymmetric path integral representation to stochastic processes with higher time derivatives

 

where is a polynomial of any order N-1, thus producing N time derivatives on . This Langevin equation may account for inertia via a term in , and/or an arbitrary nonlocal friction where . The main problem is to find a proper representation of the more complicated determinant in terms of Grassmann variables. The standard formula (3), though formally applicable, does not provide a proper representation of the determinant of an operator with higher-order derivatives because of the boundary condition problem. This problem is usually resolved via an operator representation of the associated fermionic system. In the stochastic context it has so far been discussed only for the single time derivative [2]. In the first-order formalism, the fermion path integral can be defined in terms of coherent states [2] with the above discussed vacuum-to-vacuum boundary conditions. Higher-derivative theories, however, have many unphysical features, in particular states with negative norms [6, 7], and it is a priori unclear how to define the boundary conditions for the associated fermionic path integral. In gauge theories, the Faddeev-Popov ghosts give an example of a fermionic theory with higher-(second-)order derivatives. There, unphysical consequences of the negative norms of the ghost states are avoided by imposing the so called BRST invariant boundary conditions upon the path integral. For the above stochastic determinant with higher-order derivatives, the correct boundary condition are unknown.

2. The solution proposed by us in this work is best illustrated by first treating Kramers' process where one more time derivative is present, accounting for particle inertia, i.e. where for a unit mass . Omitting the time subscript of the stochastic variables, for brevity, we replace the stochastic differential equation (9) by two coupled first-order equations

  

There are now two independent noise variables, which fluctuate according to the path integral

 

A parameter regulates the average size of deviations of from v in Eq. (11). If we regard the basic noise correlation functions as functional matrices for n=x,v, which act on functions of time as linear operators , the noise correlation functions associated with (12) are

 

Substituting (11) into (10) we find the two-derivative version of (9), , driven by the combined noise

 

This noise is white for any choice of :

 

Let be a solution of the original Langevin equation (9), and a solution of the system (11), (10). The property (15) implies that has the same correlation functions as , for any , thus describing the same stochastic process. The freedom in choosing will later be used to make the effective supersymmetric action local in time.

Once we have transformed Kramers' process into a system of coupled first-order Langevin equations (11) and (10) which is a trivial extension of the first-order equation (1) to a matrix form, there obviously exists a path integral representation analogous to (6). It is for this reason that we have introduced a two noise variable and a fluctuating relation between and v in Eq. (11). There is a complication though in that the noise is no longer white since is nonlocal in time. However, as observed above this does not affect the supersymmetry in the canonical form (6) of the path integral since the supersymmetry generator does not depend on (in contrast to Q).

Thus, having established the supersymmetric path integral representation of the equivalent first-order stochastic system, our strategy is to integrate out all auxiliary variables we have introduced and, thereby, derive the proper boundary conditions for the fermionic path integral in the higher order stochastic processes as well as to construct the supersymmetry generator in the initial configuration space.

To prepare the notation for the later generalization to a stochastic differential equation with N derivatives, we rename the variables x and v as , with , and for the moment N=2. Only the equation for contains the force . The other equation just establishes a fluctuating equality between and , the original process being described by . Inserting the stochastic equations (10) and (11) into the exponent of (12), we repeat the previous procedure and, choosing midpoint slicing with a=1/2 à la Stratonovich, we obtain the path integral representation of the generating functional

 

The generator of supersymmetry is

 

It is readily verified that , using the fact that due to the Grassmann nature of . Explicitly, the Fermi part of the action reads

 

The Gaussian path integral over momenta in (16) has a meaning without time slicing, and can be performed to recover the Lagrangian version of the supersymmetric action

 

The associated generator of supersymmetry is obtained from (17) by substituting into the solutions of the Hamilton equations of motion which extremize ( ), leading to

 

The final step consists in integrating out the auxiliary variable , which only ap pears quadratically in the bosonic part of the action. Making use of the explicit form of given in (15), we obtain the Lagrangian form of the supersymmetric action

 

where is now the left-hand side of the initial equation (9), for the Kramers process at hand: . At this stage, the effective action is nonlocal in time. Now we take advantage of the freedom in choosing the parameter . We go to the limit , in which case vanishes, reducing the action to the local form

 

To find the generator of supersymmetry in this representation, we omit in (20), and replace by the solution of the equation of motion

 

In the limit , diverges, leading to an exact equality , rather than the fluctuating one (11). To take the limit in the operator (20), one must first substitute (23) into the would-be singular term in , and then take the limit. The supersymmetry generator assumes the final form

 

The action (22) provides us with the desired supersymmetric description of processes with second-order time derivatives. An important feature of the supersymmetry generated by Q is that the supermultiplet contains one boson field and two fermion fields. The reason for this is, of course, that a boson field with N time derivatives in the action carries N particles, each of which must have a supersymmetric fermionic partner. The fermion degrees of freedom have the conventional first order action, which permits us to impose the vacuum-to-vacuum boundary conditions within the coherent state representation of fermionic path integrals [2]. The boundary conditions for the bosonic path integral are the causal ones: and .

We have circumvented the problem of the boundary condition for the determinant of a higher-order operator by enlarging the number of Fermi fields, thereby reducing the problem to the known one for the determinant of the single-derivative operator. What happens if we integrate out the auxiliary Grassmann variables . In these variables, the action (18) is harmonic, driven by external forces and . After a quadratic completion the integration with the vacuum-to-vacuum boundary condition yields . The effective action for the other fermion pair becomes non-local

 

The total effective action is still supersymmetric. The supersymmetry is generated by the operator (24), if the last term in Q is dropped. The action (25) is the first-order action. So with the vacuum-to-vacuum boundary condition the integral over would also give a determinant. Thus we get the representation

 

Invoking the formula for the determinant of a block matrix, the non-locality in the second determinant can be removed, while maintaining the linearity in the time derivative

 

which is exactly the determinant arising from the two-noise process (10), (11). In this way we have represented the determinant of the second-order operator as a determinant of a first-order operator acting upon a higher-dimensional space for which the boundary conditions are known.

Thus, with the help of two coupled equations driven by auxiliary noises we have succeeded in giving a unique meaning to the path integral representation of the Kramers process. The final path integral can be time-sliced in any desired way (prepoint, postpoint, midpoint, or any combination of these)--as long as the slicing is done equally in the bosonic and the fermionic actions. In Section 4, the procedure will be generalized to a friction coefficient which is a function of x.

3. We now generalize our construction to stochastic processes of an arbitrary order N. As a result we shall arrive at a supersymmetric extension of general higher order Lagrangian systems with a supermultiplet of N fermion fields which all possess a good quantum theory due to their first-order dynamics.

Consider a system of coupled stochastic processes

  

where . This stochastic process is equivalent to the original one if we assume the noise average as being taken with the weight , generalizing that in (12) to

 

where and . As for N=2, equations (28) and (29) can b e combined into a single equation . From (30) follows that and . Thus the correlation functions of the system (28) are the same as of the original one. Note also that the combined noise correlation functions do not depend on the parameters . We shall assign some specific values to the 's to simplify the sequel formalism.

The Hamiltonian path integral for the stochastic system (28) and (29) has the form (16), where the label n runs now from 1 to N. With the same extension of the index sum, the operator in (17) generates supersymmetry. The noise correlation functions (13) are generalized to and . After integrating out the momenta we arrive at the action (19) with the extended sum, and the generator of supersymmetry assumes the form (20) with the extended sum.

Integrating out the auxiliary variables is now technically more involved, but the integral is still Gaussian. A successive integration is possible by observing that the fermion action does not depend on the variables for , the stochastic process being nonlinear only in the physical variable . The classical equations of motion can be written in the form

for n=2,3,...,N. Combining the equations for n=N and n=N-1, and the result with the equation for n=N-2, and so on, we derive the relation

 

having inserted and with n=2,3,...,N. These expressions may be substituted into the action (19), and the generator (20). As in the case N=2, the supersymmetric Lagrangian action and the operator Q turn out to have a smooth limit . Since , we see from (32) that , and we recover the physical relations and, hence, . The action assumes the form (22), with of Eq. (9). The generator of supersymmetry becomes

 

For convenience, we give the fermion action explicitly:

 

The operator (33) transforms the original stochastic variable into the Grassmann variable , , whereas all the fermionic variables are transformed into some functions of the only bosonic variable x . The fermionic action (34) is constructed in such a way that depends only on . The terms containing the other Grassmann variables are cancelled amongst each other. The -term is cancelled against the term resulting from , i.e. . It is important to realize that the fermions are coupled with each other, and thus belong to an irreducible supermultiplet. The number of fermion is equal to the highest order of the time derivative entering the bosonic action, as observed before for N=2.

4. The idea of splitting the higher order Langevin equation into a system of coupled first-order stochastic processes with a combined noise can also be applied to construct a supersymmetric quantum theory associated with the higher order stochastic process where the coefficients are functions of . We illustrate this with the example of Kramers' process with the friction coefficient being a function of the stochastic variable .

A straightforward replacement of by in (10) would yield a problem because the combined noise appears to be a function of , making the system (10), (11) inequivalent to the original stochastic process (if the Gaussian distributions for the auxiliary noises are assumed). To resolve this problem, we take two coupled non-linear first-order processes

  

The functions are subject to the condition

 

With the noise average defined by (12) and the condition (37), the stochastic system (35), (36) is equivalent to the original system .

The difference between (36) and (11) is just the extra force , which does not affect the derivation of the associated supersymmetric action. Repeating calculations of section 2, we arrive at the supersymmetric action where

  

The supersymmetry generator has the form

 

It is not hard to verify that QS=0.

If we set to be independent of x in (39), the fermionic action does not turn into (18), in contrast to what one might expect. The reason is that the fermionic path integral exhibits a large symmetry associated with general canonical transformations on the Grassmann phase space spanned by and c. Recall that under canonical transformations the canonical one-form is invariant up to a total differential . Also the measure remains unchanged. Thus there exists infinitely many equivalent supersymmetric representations of the same stochastic process. The situation is similar to the BRST symmetry [7] in gauge theories where the BRST charge is defined up to a general canonical transformation. This freedom can be used to simplify the fermionic action or the Fermi-part of the supersymmetry generator.

This formal invariance of the continuum phase-space path integral measure with respect to canonical transformations has been studied thoroughly [8] for bosonic phase spaces. A regularization of the continuum phase-space path integral measure with respect to canonical transformations on a phase space which is a Grassmann manifold is still an open problem.

Acknowledgment:
The authors are grateful to Drs. Glenn Barnich and Axel Pelster for many useful discussions, and to Prof. John Klauder for comments.

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