Time evolution in quantum systems and stochastics

TIME EV OLUTION IN QUANTUM SYSTEMS AND STOCHASTICS ANAST ASIA DOIKOU 1 , SIMON J.A. MALHAM AND ANKE WIESE Sc hool of Mathematica l and Comput er Sciences, Department of Mathe matics, Heriot-W att Univ ersity , Edinburgh EH14 4AS, United K ingdom E-mail: a.doikou @hw.ac.uk, s.j.a.ma lham@hw.ac. uk a.wiese@ hw.ac.uk Abstract The time evolution problem for non-self adjoint second order differential oper- ators is stud ied b y means of the path integ ral form ulation. Explicit computa- tion of th e path integral via the u se of certain u nderlying sto chastic differential equations, which naturally emerge when computing the path integra l, leads to a universal expression for the associated measure regardless of t h e form of the differential op erators. The d iscrete non- linear h ierarc hy (D N LS) is then consid- ered and the corresp onding hierarch y of solv able, in prin cip le, SDEs is extracted. The first couple members of the hierarc hy corresp ond to the discrete sto chastic transp ort and heat equ ations. The discrete stochastic Burgers equ ation is also obtained through th e analogue of the Cole-Hopf transformation. The contin uum limit is also d iscussed. 1 In tro duction One of o ur main aims here is the solution o f the time evolution pro ble m a sso ciated to non self-adjoint operator s using the path in tegral formulation. W e consider the genera l second order differential op era tor ˆ L 0 , and the asso ciated time evolution problem: − ∂ t f (x , t ) = ˆ Lf (x , t ) =  ˆ L 0 + u (x)  f (x , t ) , (1.1) ˆ L 0 = 1 2 M X i,j =1 g ij (x) ∂ 2 ∂ x i ∂ x j + M X j =1 b j (x) ∂ ∂ x j , g (x) = σ (x) σ T (x) (1.2) where the diffusion ma trix g (x) and the ma tr ix σ (x) ar e in ge ne r al dynamical (de- pending on the fields x j ) M × M matr ices, while x and the drift b (x) are M vector fields with comp onents x j , b j resp ectively , and T denotes usua l transp o sition. The 1 This is based on a talk given by AD in ‘‘Quantum Theory and Symmetries XI’’, July 2019, Montreal, Canada. 1 op erator ˆ L 0 is not in genera l self-adjoint (Hermitian), therefore we also in tro duce the adjoint op er ator defined for an y suitable function f (x , t ) as ˆ L † 0 f (x , t ) = 1 2 M X i,j =1 ∂ 2 ∂ x i ∂ x j  g ij (x) f (x , t )  − M X j =1 ∂ ∂ x j  b j (x) f (x , t )  . (1.3) Then, tw o distinct time evolution equations emerge. 1. The F okker-Plank equation: ∂ t 1 f (x , t 1 ) = ˆ L † 0 f (x , t 1 ) (1.4) t 1 ≥ t 2 , with known initial co ndition f (x , t 2 ) = f 0 (x). 2. The Kolmogorov backw ard equation: − ∂ t 2 f (x , t 2 ) = ˆ L 0 f (x , t 2 ) (1.5) t 2 ≤ t 1 , with known final condition f ( x, t 1 ) = f f ( x ). T o simplify the complicated situation of non-cons ta nt diffusion coefficie nt s w e employ a lo cal change of frame, which r educes the ˆ L op erator to the simpler form with constant diffusion coe fficient s, and a n effective dr ift. W e then compute the generic path in tegral b y r equiring that the fields in volv ed satisfy discrete time a nalogues of sto chastic differential equations (SDEs). These equa tions natur ally emerge when computing the path integral via the time discretization scheme. This leads to the computation of the mea sure, which turns out to b e an infinite pro duct o f Gaussians. Our s econd ob jective is to explor e links b etw e en SDE s, and quantum in tegrable systems. T o illustrate these ass o ciations we discuss a t ypical exactly solv able discrete quantum s ystem, the dis crete non-linear Schr¨ odinger hier arch y . W e ex press the qua n- tum int egra ls of motion as seco nd o rder differential op erator s after a suitable r escaling of the fields and we then extract a hierarchy of asso ciated SDEs, which can b e in prin- ciple solved by mea ns of suitable integrator factors. The first tw o non-trivia l members of the hier arch y co rresp ond to the discrete sto chastic transp o r t and heat eq uations. The discr ete sto chastic Burger s e quation is also obtained from the discrete sto chastic heat equation through the analog ue of the Co le-Hopf transforma tion (see also relev ant [1]). Mo re details on the deriv ation o f the r ep o rted results ca n b e found in [2]. 2 Time ev olution and the F eynman-Kac form ula Before we compute the solution of the time e volution problem via the path integral formulation we s ha ll implement the quantum ca nonical tr ansform, that turns the dynamical diffusion matrix in (1.2) int o iden tit y a t the le vel of the PDEs. This result will b e then used for the e xplicit computation o f the genera l path int egra l, and the deriv atio n o f the F eynman-Ka c form ula [2]. 2 2.1 The quan tum c anonical transformation W e will show in what follows that the gener al ˆ L op era to r can b e brought into the less inv o lved form: ˆ L = 1 2 M X j =1 ∂ 2 ∂ y j 2 + M X j =1 ˜ b j (y) ∂ ∂ y j + u (y) (2.1) with a n induced drift ˜ b (y). This ca n b e achiev ed via a simple change of the para meters x j , which g eometrically is nothing but a change of frame. Indeed, let us in tro duce a new set of parameters y j such that [2]: d y i = X j σ − 1 ij (x) d x j , det σ 6 = 0 , (2.2) then ˆ L can b e expr essed in the form (2.1), a nd the induced drift comp onents are given as ˜ b k (y) = X j σ − 1 kj (y) b j (y) + 1 2 X j,l σ j l (y) ∂ y l σ − 1 kj (y) . (2.3) Bearing also in mind that P j σ j l σ − 1 kj = δ kl , we can write in the compact vector/matrix notation: ˜ b (y) = σ − 1 (y)  b (y) − 1 2 ( ∇ y σ T (y)) T  , ∇ y =  ∂ y 1 , . . . , ∂ y M  (2.4) where one first solves for x = x(y) via (2 .2). The transfor mation discussed ab ov e corres p o nds to a generaliza tion of the so called Lampe r ti tr ansform at the lev el of SDEs (we refer the interested reader to [2] and refere nc e s therein). 2.2 The path integral: F eynmann- Kac form ula W e are now in the p ositio n to solve the time ev olution pr oblem for the cons ider ably simpler op e rator (2.1). Our starting p oint is the time evolution equa tion (1.3), (1.4), (2.1): ∂ t f (y , t ) = ˆ L † f (y , t ) , we then explicitly compute the propagato r K (y f , y i | t, t ′ ): f (y , t ) = Z M Y j =1 dy ′ j K (y , y ′ | t, t ′ ) f (y ′ , t ′ ) (2.5) = Z N Y n =1 M Y j =1 dy j n N Y n =1 K (y n +1 , y n | t n +1 , t n ) f (y 1 , t 1 ) . (2.6) W e employ the standar d time discr etization scheme as shown ab ove, (se e a ls o for instance [3]), we insert the unit N times, ( 1 2 π R d y j n dp j n e i p jn (y jn − a ) = 1), for ea ch 3 comp onent y j , a nd we pe rform the Gaussia n integrals with respect to ea ch p j n pa- rameter. W e then conclude that the pa th in tegral can b e expressed as K (y f , y i | t, t ′ ) = Z d q exp h − X j X n  ∆y j n − δ ˜ b j n (y)  2 2 δ + δ X n u n (y) i (2.7) d q = 1 (2 π δ ) N M 2 N Y n =2 M Y j =1 d y j n (2.8) where f n = f n (y n ) a nd ∆y j n = y j n +1 − y j n . wher e δ = t n +1 − t n and with b oundar y conditions: y f = y N +1 , y i = y 1 , t i = t ′ = 0 ( t ′ will b e dropp ed henceforth for brevity), t f = t . W e recall expression (2.7) a nd we make the fundamental assumption [2]: ∆y n − δ ˜ b n (y) = ∆w n (2.9) assuming a lso that w nj are Brownian paths (see for instance [4] on Wiener pro cesses), i.e. (2 .9) is the discr ete time analo g ue of an SDE. After a change of the volume elemen t in (2.7), sub ject to (2.9), we c onclude (see [2 ] for the deta iled computatio n): K (y f , y i | t ) = Z d M e R t 0 u (y s ) ds , (2.10) d M = lim δ → 0 lim N →∞ 1 (2 π δ ) N M 2 exp h − 1 2 δ N X n =1 ∆w T n ∆w n i N Y n =2 M Y j =1 d w j n . (2.11) W e may no w ev a lua te the measure: in the co ntin uum time limit (2.11), we consider the F ourie r r e presentation on [0 , t ] for w s , i.e. Wiener’s repr esentation of the Brownian path [4]: w s = f 0 √ t s + r 2 t X k> 0 f k ω k sin ( ω k s ) , ω k = 2 π k t . (2.12) f 0 = w t √ t and f k , k ∈ { 0 , 1 , . . . } are M vectors with co mp onents f kj , j ∈ { 1 , 2 , . . . , M } being s tandard normal v a r iables. W e are interested in the computation of the meas ure in the co ntin uum limit N → ∞ , δ → 0, and we als o r ecall the following b oundary conditions: w( s = 0) = 0 , w( s = t ) = w t , then d M = e − 1 2 t w T t w t (2 π t ) M 2 d M 0 d M 0 = Y k ≥ 1 M Y j =1 d f kj √ 2 π exp[ − 1 2 X k ≥ 1 X j f 2 kj ] . (2.13) The measure natur a lly is expres sed as an infinite pro duct o f Gaussians re gardless of the sp ecific forms of the diffusion coefficie nt s and the drift. Having co mputed the propaga tor explicitly (2.10) we conclude that equation (2.6) can b e then expressed as f (x f , t f ) = Z d M e R t 0 u (x s ) ds f 0 (x 0 ) , f 0 (x 0 ) = f (x 0 , t 0 ) 4 which is precisely the F eynman-Ka c formula, a nd describ es the time evolution o f a given initial profile f 0 (x 0 ) to f (x f , t f ) a solution of the F ok ker-Planck equatio n. One could have star ted from the Kolmog orov backward equatio n and computed the path int egra l bac kwards in time: f (x 0 , t 0 ) = Z d M e R t 0 u (x s ) ds f f (x f ) , f f (x f ) = f (x f , t f ) . In this ca se the F eynman-Kac for mula describ es the reversed time evolution o f a given final s ta te f f (x f ), to a pr e vious state f (x 0 , t 0 ) a solution of the Kolmo gorov backw ard equation. One of the main aims is the computation of exp ectation v alues: hO (x s ) i = E t  O (x s ) e R t 0 u (x s ) ds  E t  e R t 0 u (x s ) ds  , 0 ≤ s ≤ t (2.14) where we define via (2.10), (2 .13) E t  O (x s )  = Z d w t d M O (x s ) 0 ≤ s ≤ t. (2.15) (2.14) can be us e d provided that solutions of the a sso ciated SDEs are av ailable, so that the fields x tj are expre ssed in terms of the v aria bles w tj . 3 The quan tum (D)NLS and a hierarc h y of S(P)DEs W e s tart our ana lysis with the DNLS mo del, with the cor resp onding quantum La x op erator given by [5], [6 ], L j ( λ ) =  λ + Θ j + z j Z j z j Z j 1  z j , Z j are canonical [ z i , Z j ] = − δ ij , and we consider the ma p: z j 7→ x j , Z j 7→ ∂ x j . (3.1) Let us now define the g enerating function of the integrals of motion of the system: t ( λ ) = tr  L M ( λ ) . . . L 2 ( λ ) L 1 ( λ )  . (3.2) Indeed, the ex pa nsion of ln ( t ( λ )) = P M k =0 I k λ k provides the lo cal integrals o f motion (see e.g. [8]). W e keep here terms up to thir d o rder in the expansion o f ln ( t ) and b y suitably scaling the inv olved fields, we obta in the first three lo cal integrals o f motion 5 of the q uantum DNLS hierarch y (keeping the suitably scaled terms) [5]: H 1 = M X j x j ∂ x j H 2 = 1 2 M X j =1 x 2 j ∂ 2 x j − M X j =1 ∆ (1) (x j ) ∂ x j H 3 = 1 2 M X j =1 x 2 j ∂ 2 x j − ν M X j =1 ∆ (2) (x j ) ∂ x j + (hig he r order terms) ... (3.3) where we hav e chosen Θ j = 1, and H 1 = I 1 , H 2 = − I 2 + 1 2 I 1 , H 3 = − 1 3  I 3 + I 2 − 1 2 I 1 ), ν = 1 3 . W e also define: ∆ (1) z j = z j +1 − z j , ∆ (2) z j = z j +2 − 2 z j +1 + z j . The next or der in the expansion provides H 4 , which is the Hamiltonian of the quantum version of complex mKdV system and so on. The eq uations of motion (classica l and quantum) asso ciated e.g . to H (2) can be derived via the zero curv a ture condition or Heisenberg’s equa tion (recall a ls o (3.1)): dz j dt = − ∆ (1) z j + z 2 j Z j . (3.4) Similar equations can b e obtained fo r H 3 , but are omitted here for brevit y . The Hamiltonians H 2 , 3 are of the form (1.2), and the corresp onding set of SDEs are [2] d x tj = − ν k ∆ ( k − 1) x t dt + x tj d w tj . (3.5) where k ∈ { 2 , 3 } and ν 2 = 1 , ν 3 = 1 3 . ν k can b e set equa l to one henceforth, after suitably re scaling time. B y comparing (3 .4) and (3.5 ) ( k = 2) we obs erve that the non-linearity a pp e aring in (3.4 ) is repla ced by the m ultiplicative noise in (3.5). Let us now derive the solution of the set of SDEs (3.5) intro ducing s uitable inte- grator factor s (see e.g. [10]). Let us consider the ge ner al set of SDES d x tj = b j (x t ) dt + x tj d w tj . W e in tro duce the following set of integrator factor s: F j ( t ) = ex p  − Z t 0 d w sj + 1 2 Z t 0 ds  (3.6) and define the new fields: y tj = F j ( t )x tj , then one obtains a different ial equatio n for the vector field y: d y t dt = A ( t )y t ⇒ y t = P ex p  Z t 0 A ( s ) ds  y 0 . (3.7) F or instance in the ca se of (3.5), for k = 2 , the M × M matrix A is given a s A ( t ) = M X j =1  e j j − B j ( t ) e j j +1  , B j ( t ) = exp  ∆ (1) (w tj )  , 6 where e ij are M × M matrices with entries ( e ij ) kl = δ ik δ j l . F or k = 3 , the A matrix inv o lves also terms e j j +2 , and so on. The solutio n (3.7) ca n be expre ssed as a formal series expa nsion P exp  Z t 0 A ( s ) ds  = ∞ X n =0 Z t 0 Z t n 0 . . . Z t 2 0 dt n dt n − 1 . . . dt 1 A ( t n ) A ( t n − 1 ) . . . A ( t 1 ) , t ≥ t n ≥ t n − 1 . . . ≥ t 2 . Remark 1. The discr ete version of the sto chastic Burger s e quation c an b e obtained fro m the dis- crete sto chastic hea t equation through the ana lo gue of the Cole - Hopf transforma tion. Indeed, by setting x j = e y j , in (3 .5) ( k = 3): d y j = −  e ∆y j  e ∆y j +1 − 1  −  e ∆y j + 1   dt + d w j , (3.8) where for simplicity we hav e set ∆ (1) = ∆. By a lso s etting u j = ∆y j , we o btain a discrete version of the sto c hastic Burger s equa tion du j = −  e u n +1  e u j +2 − e u j  − 2  e u j +1 − e u j   dt + ∆ d w j . (3.9) Assuming the s caling ∆ y j ∼ δ , w e ex pa nd the ex p o nentials and keep up to second order terms in (3.8), (3.9): d y j = −  ∆ (2) y j +  ∆y j  2 + O ( δ 3 )  + d w j (3.10) du j = −  ∆ (2) u j + ∆ u 2 j + O ( δ 3 )  + ∆ d w j . (3.11) The second of the equa tio ns above provides a go od approximation for the discrete viscous Burg ers equation, as will be also clear in the next subsection. 3.1 The con tin uum mo dels and SPDEs It will b e instructiv e to co nsider the con tinuum limits o f the Hamiltonia ns H 2 , H 3 (3.3) and the resp ective SDEs. Aft er cons idering the thermo dynamic limit M → ∞ , δ → 0 ( δ ∼ 1 M ) we obtain x tj → ϕ ( x, t ) , x tj +1 − x tj δ → ∂ x ϕ ( x, t ) , δ X j f j → Z dx f ( x ) , w tj → W ( x, t ) , (3.12) where the Wiener field or Br ownian sheet W ( x, t ) is p er io dic and squar e integrable in [ − L, L ], a nd is repr esented as [4 ] W ( x, t ) = √ L π X n ≥ 1 1 n X ( n ) t cos  nπ x L  + Y ( n ) t sin  nπ x L  ! , (3.13) 7 X ( n ) t , Y ( n ) t are indep e ndent Brownian motio ns. In the co nt inuum limit the Hamilto- nians (3.3) become the Hamiltonians of q ua ntum NLS hier arch y: H ( k ) c = Z dx  1 2 ϕ 2 ( x ) ˆ ϕ 2 ( x ) − ∂ ( k − 1) x ϕ ( x ) ˆ ϕ ( x )  , k = 2 , 3 (3.14) where  ϕ ( x ) , ˆ ϕ ( y )  = δ ( x − y ), ( ˆ ϕ ( x ) ∼ ∂ ∂ ϕ ( x ) ) a nd the SDEs (3.5) become the sto chastic transp o rt ( k = 2) a nd hea t equatio n ( k = 3) with m ultiplicative noise: ∂ t ϕ ( x, t ) = − ∂ k − 1 x ϕ ( x, t ) + ϕ ( x, t ) ˙ W ( x, t ) . The sto chastic heat equation ca n b e ma ppe d to the sto chastic Hamilton-J a cobi a nd viscous Burg ers equations [1]. Indeed, we set: ϕ = e h , u = ∂ x h then (3 .15): ∂ t h ( x, t ) = − ∂ 2 x h ( x, t ) − ( ∂ x h ( x, t )) 2 + ˙ W ( x, t ) ∂ t u ( x, t ) = − ∂ 2 x u ( x, t ) − 2 u ( x, t ) ∂ x u ( x, t ) + ∂ x ˙ W ( x, t ) . (3.15) Connections b etw een the SDEs and the quantum Darb o ux trans forms [11], [1 2] can be also studied. The classical Dar bo ux-B¨ acklund tr ansformatio n [13], [14] provides an efficient w ay to find so lutio ns of integrable PDEs.The key question is how this transformatio n can facilitate the solution of SDEs [1 ], [1 5], [16]. Ac knowledgmen ts AD ackno wledges supp ort from the EPSRC r esearch gr a nt: E P/R00 9465/ 1. References [1] I. Corwin, Exactly solving the KPZ e quation , arXiv.1804.057 21 [math.PR] . [2] A. Doi k ou, S.J.A. Malham and A. Wiese, Nucl. Phys. B945 (2019) 114658. [3] B. Sim on, F unctional Inte gr ation and Quantum Physics , AMS Publis hi ng, (2005). [4] C. Prevˆ ot and M. R¨ ockne r, A c oncise co urse on sto chastic p artial differ e ntial e quations , Springer, (2007). [5] A. K undu and O. Ragnisco, J. Phys. A27 (1994) 6335. [6] E. Sklya nin, In Inte gr able systems: fr om c lassic al to quantum , (1999), Montreal, CRM Pro c. Lecture Notes (V ol. 26, pp. 227-250), nlin/0009009 . [7] A.C. Scott and J.C. Eilb eck, Phys. Lett. A119 (1986) 60. [8] V.E. Korepin, N .M. Bogoliub ov and A.G. Izergin, Quantum inverse sc attering metho d and c or- r elation functions , Cambridge Universit y Pr ess, (1983). [9] A. Doi k ou, I. Findlay and S. Sklav eniti, Nucl. Phys. B941 (2019) 376. [10] B. Oksendal, Sto c hastic differ ential e quations: intr o duction and applic ations , Springer, (2003). [11] C. Korff, J. Phys. A49 (2016) 104001. [12] A. Doikou and I. Fi ndla y , The quantum auxiliary line ar pr oblem & quantum D arb oux-B¨ acklund tr ansformations , arXiv:1706.06052 [math-p h] . [13] V.E. Zakharo v and A.B. Shabat, F unct. Anal. Appl. 13 (1979) 166. [14] V.B. Matv eev and M.A. Sall e, Darb oux tr ansformations and solitons , Springer-V erlag, (1991). [15] N. O’Connell, In Memoriam Mar c Y or - S´ eminair e de Pr ob abilit´ es XL VII . Lecture Notes in Mathematics, vol 2137. Spri nger, Cham, (201 5). [16] A. Doiko u, S.J. A. Mal ham, I. Stylianidis and A. Wiese, Applic ations of Gr assmannian and gr aph flows to nonline ar systems , arXiv:1905.053 5 [ma th.AP] . 8