Supersymmetry in Stochastic Processes with Higher-Order Time

Home , Hagen Kleinert

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H x = Φ[ − i i F , p S − (1 + ,x, p, t f ( n replac- and , + ime x x i i i − R ,te the then ), ,c c, ¯ − 1 dtJx ) /ǫ t a .The ]. Q i ) On . F H = . i − (6) (7) iǫ, = 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 orthogonal 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 Itô- 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 associated 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 Itôchoice, 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 Itôstochastic integral [2]. The Itôintegral 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 . (12) paths, and can be treated by standard rules of path in- R tegration (e.g., perturbation expansion around Gaussian A parameter σ regulates the average size of deviations of measures). The price for this is the additional fermion x˙ from v in Eq. (11). If we regard the basic noise correla- interaction, which possesses as an attractive feature the tion functions as functional matrices (Dn)tt′ = hνntνnt′ i additional supersymmetry. for n = x, v, which act on functions of time as linear ′ The aim of our work is to extend this supersymmetric operators Dnft = dt (Dn)tt′ ft′ , the noise correlation functions associated with (12) are path integral representation to stochastic processes with R

2 −γt −1 2γt −1 −γt 2 Dv =1 − σ, Dx = σe −∂t e ∂t e . (13) H Q = dt(ic¯nδxn − pnδc¯n ) . (17) Substituting (11) into (10) we ﬁnd the two-derivative ver- n=1 XZ sion of (9),x ¨ + γx˙ + F (x)= η , driven by the combined σ It is readily veriﬁed that QH SH = 0, using the fact that noise 2 kmn c¯mck(δzk δzn Lm)cn ∼ n cn = 0 due to the Grass- ησ = νv +ν ˙x + γνx. (14) mann nature of cn. Explicitly, the Fermi part of the Paction SH reads P This noise is white for any choice of σ: ′ Sf = dt [¯cxc˙x +¯cvc˙v − c¯xcv +¯cvcxF (x)+γc¯vcv] . (18) hη i =0 , hη η ′ i = δ ′ . (15) σt σt σt tt Z

Let xt[η] be a solution of the original Langevin equation The Gaussian path integral over momenta in (16) has (9), and xt[ησ] a solution of the system (11), (10). The a meaning without time slicing, and can be performed property (15) implies that xt[ησ] has the same correlation to recover the Lagrangian version of the supersymmetric functions as xt[η], for any σ, thus describing the same action stochastic process. The freedom in choosing σ will later 2 be used to make the effective supersymmetric action local 1 −1 S = dt LnDn Ln + Sf . (19) in time. 2 n=1Z Once we have transformed Kramers’ process into a sys- X tem of coupled first-order Langevin equations (11) and The associated generator of supersymmetry is obtained H (10) which is a trivial extension of the first-order equa- from (17) by substituting into Q the solutions of the −1 tion (1) to a matrix form, there obviously exists a path Hamilton equations of motion pn = iDn Ln which ex- H H integral representation analogous to (6). It is for this tremize S (δpn S = 0), leading to reason that we have introduced a two noise variable and 2 a fluctuating relation betweenx ˙ and v in Eq. (11). There ˜ −1 Q = dt(icnδxn − iDn Lnδc¯n ) . (20) is a complication though in that the noise νx is no longer n=1Z white since Dx is nonlocal in time. However, as ob- X served above this does not affect the supersymmetry in The final step consists in integrating out the auxiliary the canonical form (6) of the path integral since the su- variable x2 = v, which only ap pears quadratically in the H persymmetry generator Q does not depend on Dx (in bosonic part of the action. Making use of the explicit contrast to Q). form of Dn given in (15), we obtain the Lagrangian form Thus, having established the supersymmetric path in- of the supersymmetric action tegral representation of the equivalent first-order stochastic system, our strategy is to integrate out all auxiliary 1 1 Sσ = dt Lt 1+ ∂tDx∂t Lt + Sf , (21) variables we have introduced and, thereby, derive the 2 1 − σ Z proper boundary conditions for the fermionic path in- where L is now the left-hand side of the initial equation tegral in the higher order stochastic processes as well as t (9), for the Kramers process at hand: L =x ¨ + γx˙ + to construct the supersymmetry generator in the initial t t t F (x ). At this stage, the effective action is nonlocal in configuration space. t time. Now we take advantage of the freedom in choosing To prepare the notation for the later generalization the parameter σ. We go to the limit σ → 0, in which to a stochastic differential equation with N derivatives, case D ∼ σ vanishes, reducing the action to the local we rename the variables x and v as x , with α = 1,N, x n form and for the moment N = 2. Only the equation for xN contains the force F = F (x1). The other equation just 1 2 S = S0 = dt Lt + Sf . (22) establishes a fluctuating equality betweenx ˙ 1 and x2, the 2 Z original process being described by x ≡ x1. Inserting the stochastic equations (10) and (11) into the exponent To find the generator of supersymmetry in this repre- of (12), we repeat the previous procedure and, choosing sentation, we omit δxN ≡ δv in (20), and replace xN ≡ v midpoint slicing with a = 1/2 àla Stratonovich, we ob- by the solution of the equation of motion tain the path integral representation of the generating −1 1 functional δvS˜ = −D Lx + (−∂t + γ)Lv =0 . (23) x 1 − σ H −S +i dtJx −1 −1 Z[J]= DpDxDc¯Dce , (16) In the limit σ → 0, Dx ∼ σ diverges, leading to an Z exact equality Lx = v − x˙ = 0, rather than the fluctu- 2 R 1 ating one (11). To take the limit σ → 0 in the operator SH = dt p D p − ip L +¯c δ L c . 2 n n n n n n xm n m (20), one must first substitute (23) into the would-be sin- n=1 Z −1 ˜ X gular term Dx Lx in Q, and then take the limit. The The generator of supersymmetry is supersymmetry generator assumes the final form

3 or any combination of these)—as long as the slicing is Q = dt [icxδx − i(−∂t + γ)Ltδc¯ − iLtδc¯ ] . (24) x v done equally in the bosonic and the fermionic actions. In Z Section 4, the procedure will be generalized to a friction The action (22) provides us with the desired supersym- coefficient γ which is a function of x. metric description of processes with second-order time derivatives. An important feature of the supersymme- 3. We now generalize our construction to stochastic try generated by Q is that the supermultiplet contains processes of an arbitrary order N. As a result we shall one boson field and two fermion fields. The reason for arrive at a supersymmetric extension of general higher this is, of course, that a boson field with N time deriva- order Lagrangian systems with a supermultiplet of N tives in the action carries N particles, each of which must fermion fields which all possess a good quantum theory have a supersymmetric fermionic partner. The fermion due to their first-order dynamics. degrees of freedom have the conventional first order ac- Consider a system of coupled stochastic processes tion, which permits us to impose the vacuum-to-vacuum boundary conditions within the coherent state represen- N tation of fermionic path integrals [2]. The boundary con- LN =x ˙ N + γn−1xn + F (x1)= νN ; (28) n=1 ditions for the bosonic path integral are the causal ones: X L =x ˙ − x = ν , n = N − 1,N − 2, ..., 1 , (29) xt=0 = x0 andx ˙ t=0 = v0 =x ˙ 0. n n n+1 n We have circumvented the problem of the boundary condition for the determinant of a higher-order operator where x1 ≡ x. This stochastic process is equivalent to by enlarging the number of Fermi fields, thereby reducing the original one if we assume the noise average as being −Sν the problem to the known one for the determinant of the taken with the weight e , generalizing that in (12) to single-derivative operator. What happens if we integrate N−1 out the auxiliary Grassmann variablesc ¯v, cv. In these 1 1 2 1 2 Sν = dt νN + (ΛN−nνn) , (30) variables, the action (18) is harmonic, driven by external 2 " 1 − σ σn # ′ Z n=1 forcesc ¯x and cxF (x). After a quadratic completion the X integration with the vacuum-to-vacuum boundary condi- N−1 n n−m where σ = n=1 σn and Λn = m=0 γN−m∂t , γN ≡ tion yields det(∂t + γ). The effective action for the other 1. As for N = 2, equations (28) and (29) can b e com- fermion pair becomes non-local P P N−1 bined into a single equation Lt = νNt+ n=1 ΛN−nνnt ≡ η . From (30) follows that hη i = 0 and hη η ′ i = δ ′ . −1 ′ σt σt σt σt tt Sf = dt c¯xc˙x +¯cx(∂t + γ) (F (x)cx) . (25) Thus the correlation functions of the systemP (28) are the Z same as of the original one. Note also that the combined The total effective action S = Sb + Sf is still supersym- noise correlation functions do not depend on the param- metric. The supersymmetry is generated by the operator eters σn. We shall assign some specific values to the σ’s (24), if the last term in Q is dropped. The action (25) to simplify the sequel formalism. is the first-order action. So with the vacuum-to-vacuum The Hamiltonian path integral for the stochastic sys- boundary condition the integral overc ¯x,cx would also tem (28) and (29) has the form (16), where the label n give a determinant. Thus we get the representation runs now from 1 to N. With the same extension of the index sum, the operator QH in (17) generates supersym- −1 ′ ∆ = det[∂t + γ] det[∂t + (∂ + γ) F (x)] . (26) metry. The noise correlation functions (13) are general- N−1 † −1 Invoking the formula for the determinant of a block ma- ized to DN =1− n=1 σn and Dn = σn(ΛN−nΛN−n) . After integrating out the momenta pn we arrive at the trix, the non-locality in the second determinant can be re- P moved, while maintaining the linearity in the time deriva- action (19) with the extended sum, and the generator of tive supersymmetry assumes the form (20) with the extended sum. ∂ + γ F ′ Integrating out the auxiliary variables x is now tech- ∆ = det t , (27) n −1 ∂t nically more involved, but the integral is still Gaussian. A successive integration is possible by observing that the which is exactly the determinant arising from the two- fermion action does not depend on the variables xn for noise process (10), (11). In this way we have represented n > 1, the stochastic process being nonlinear only in the determinant of the second-order operator as a deter- the physical variable x1 ≡ x. The classical equations of minant of a first-order operator acting upon a higher- ˜ motion δxn S = 0 can be written in the form dimensional space for which the boundary conditions are −1 −1 −1 known. − Dn−1Ln−1 − ∂tDn Ln + γn−1DN LN =0 , (31) Thus, with the help of two coupled equations driven by auxiliary noises we have succeeded in giving a unique for n = 2, 3, ..., N. Combining the equations for n = N meaning to the path integral representation of the and n = N − 1, and the result with the equation for Kramers process. The final path integral can be time- n = N − 2, and so on, we derive the relation sliced in any desired way (prepoint, postpoint, midpoint,

4 N−n −1 1 k k The functions λx,v are subject to the condition Dn Ln = (−1) γn+k∂t LN , (32) 1 − σ ′ " k=0 # λ = γ − 1 , λ = F − λ . (37) X x v x having inserted DN = 1 − σ and with n = 2, 3, ..., N. With the noise average defined by (12) and the condition These expressions may be substituted into the action (37), the stochastic system (35), (36) is equivalent to the (19), and the generator (20). As in the case N = 2, original system Lt =x ¨ + γ(x)x ˙ + F (x)= η. the supersymmetric Lagrangian action and the opera- The difference between (36) and (11) is just the ex- tor Q turn out to have a smooth limit σn → 0. Since tra force λx, which does not affect the derivation of the Dn ∼ 1/σn, we see from (32) that Ln → 0, and we associated supersymmetric action. Repeating calcula- recover the physical relationsx ˙ n = xn+1 and, hence, n tions of section 2, we arrive at the supersymmetric action xn = ∂t x. The action assumes the form (22), with Lt of S = S + S where Eq. (9). The generator of supersymmetry becomes b f

N N−n 1 2 Sb= dt (¨x + γ(x)x ˙ + F (x)) , (38) Q = i dt c δ − (−1)kγ ∂kL δ . 2 1 x n+k t t c¯n Z Z ( n=1 " k=0 # ) X X Sf= dt [¯cxc˙x +¯cvc˙v +¯cxcx(γ(x) − 1)+¯cvcv (33) Z ′ −c¯vcx(F (x) − γ(x)+1) − c¯xcv] . (39) For convenience, we give the fermion action explicitly:

N The supersymmetry generator has the form Sf = dt c¯nc˙n − c¯ncn+1 "n=1 Q = dt (icxδx − iLtδc¯v − i(−∂t + 1)Ltδc¯x ) . (40) Z X N Z ′ + cN γn−1cn + F (x)c1 . (34) It is not hard to verify that QS = 0. n=0 !# If we set γ to be independent of x in (39), the fermionic X action does not turn into (18), in contrast to what one The operator (33) transforms the original stochastic vari- might expect. The reason is that the fermionic path able x = x into the Grassmann variable c , Qx = ic , 1 1 1 integral exhibits a large symmetry associated with gen- whereas all the fermionic variables are transformed into eral canonical transformations on the Grassmann phase some functions of the only bosonic variable x. The space spanned byc ¯ and c. Recall that under canoni- fermionic action (34) is constructed in such a way that cal transformations the canonical one-form c¯ dc is QS depends only on c . The terms containing the other n n n f 1 invariant up to a total differential dF (¯c,c). Also the mea- Grassmann variables are cancelled amongst each other. sure dc¯ dc remains unchanged. Thus thereP exists in- The c -term is cancelled against the term resulting from n n n 1 finitely many equivalent supersymmetric representations QS , i.e. Q(S + S ) = 0. It is important to realize b b f of theQ same stochastic process. The situation is similar that the fermions are coupled with each other, and thus to the BRST symmetry [7] in gauge theories where the belong to an irreducible supermultiplet. The number of BRST charge is defined up to a general canonical trans- fermion is equal to the highest order of the time deriva- formation. This freedom can be used to simplify the tive entering the bosonic action, as observed before for fermionic action or the Fermi-part of the supersymmetry N = 2. generator. 4. The idea of splitting the higher order Langevin This formal invariance of the continuum phase-space equation into a system of coupled first-order stochastic path integral measure with respect to canonical trans- processes with a combined noise can also be applied to formations has been studied thoroughly [8] for bosonic construct a supersymmetric quantum theory associated phase spaces. A regularization of the continuum phase- with the higher order stochastic process where the coeffi- space path integral measure with respect to canonical cients γn are functions of xt. We illustrate this with the transformations on a phase space which is a Grassmann example of Kramers’ process with the friction coefficient manifold is still an open problem. being a function of the stochastic variable xt. A straightforward replacement of γ by γ(x) in (10) Acknowledgment: would yield a problem because the combined noise ησ The authors are grateful to Drs. Glenn Barnich and Axel appears to be a function of xt, making the system (10), Pelster for many useful discussions, and to Prof. John (11) inequivalent to the original stochastic process (if the Klauder for comments. Gaussian distributions for the auxiliary noises are assumed). To resolve this problem, we take two coupled non-linear first-order processes

Lv =v ˙ + v + λv(x)= νσ , (35)

Lx =x ˙ − v + λx(x)= νσ . (36)

5 [1] G. Parisi and N. Sourlas, Phys.Rev.Lett. 43, 744 (1979); Nucl.Phys. B206, 321 (1982); M.V. Feigel’man and A.M. Tsvelik, Sov.Phys. JETP, 56, 823 (1982); Phys.Lett. 95A, 469 (1983); For a comprehensive review see J. Zinn-Justin, Quan- tum Field Theory and Critical Phenomena (2nd Edition, Clarendon Press, Oxford, 1993). [2] H. Ezawa and J. R. Klauder, Prog.Thor.Phys. 74, 104 (1985); L.P. Singh and F. Steiner, Phys.Lett. 166B, 155 (1986); H. Nakazato, K. Okano, L. Schülke and Y. Yamahaka, Nucl.Phys. B346, 611 (1990). [3] C.W. Gardiner, Handbook of Stochastic Methods (Springer Series in Synergetics, Vol. 13, Springer, Berlin, 1983). [4] H. Kleinert, Path Integrals in Quantum Mechanics, Statis- tics and Polymer Physics, World Scientific, Second Edi- tion, 1995. [5] In Section 10.5 of Ref. [4] it is shown that the correct time slicing of an interaction dt qF˙ (q) in a path integral is of the midpoint type, corresponding to a = 1/2. Sometimes this is referred to as the midpointR prescription for defining the sliced action, but it can actually be derived from the short-time action along a classical orbit. [6] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79, 145 (1950); M.V. Ostrogradsky, Mem. Acad. Sci. St-Petersburg, 6, 385 (1850); See also: E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cam- bridge University Press, Cambridge, 1959); and Section 17.3 in Vol. II of H. Kleinert, Gauge Fields in Condensed Matter, (World Scientific, Singapore, 1989). [7] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992). [8] J.R. Klauder, Ann. Phys. (NY), 188, 120 (1988).