NUPHB114658114658S0550-3213(19)30144-010.1016/j.nuclphysb.2019.114658The Author(s)Quantum Field Theory and Statistical SystemsStochastic analysis & discrete quantum systemsAnastasiaDoikou⁎a.doikou@hw.ac.ukSimon J.A.Malhams.j.a.malham@hw.ac.ukAnkeWiesea.wiese@hw.ac.ukSchool of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United KingdomSchool of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghEH14 4ASUnited KingdomSchool of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom⁎Corresponding author.Editor: Hubert SaleurAbstractWe explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.1IntroductionLet Lˆ0 be a generic second order differential operator, and suppose f(x,t) satisfies the time evolution equation:(1.1)−∂tf(x,t)=Lˆf(x,t)=(Lˆ0+u(x))f(x,t),(1.2)Lˆ0=12∑i,j=1Mgij(x)∂2∂xi∂xj+∑j=1Mbj(x)∂∂xj,g(x)=σ(x)σT(x) where the diffusion matrix g(x) and the matrix σ(x) are in general dynamical (depending on the fields xj) M×M matrices, while x and the drift b(x) are M vector fields with components xj, bj respectively, and T denotes usual transposition. Within the quantum mechanics framework the Lˆ operator may be seen as the Hamiltonian of a system of M-interacting particles and the time evolution problem (1.2) can be thought of as a generalized imaginary time Schrödinger's equation. Feynman's path integral is a solution of (1.2) that can be explicitly calculated via a time discretization scheme, and can be physically interpreted as the probability of the system to progress from an initial state configuration x0 at time t0 to a final configuration xf at time tf. In the statistical physics language the imaginary time propagator provides the partition function of a given statistical system at temperature T∼1t (see for instance [1]).In the quantum mechanical set up the time evolution problem (1.2) is the commencing point, while from the stochastic analysis perspective the key object is a given stochastic differential equation (SDE) of the form(1.3)dxt=b(xt)dt+σ(xt)dwt, which yields the generator Lˆ0 of the stochastic process and the corresponding time evolution problem via Itô's formula. The SDE (1.3) a priori determines the probabilistic measure in the Feynman-Kac formula, which in turn provides the solution of the time evolution problem. w is also an M-vector with components wj, which are M-independent Wiener processes. Let us now recall the definition of the Wiener process (Brownian motion) (see e.g. [2–8]):•Wiener processThe one dimensional (or scalar) Wiener process wt is a stochastic process with the following properties:1.For s<t the increment wt−ws is a Gaussian with mean zero E(wt−ws)=0, and variance E((wt−ws)2)=t−s. Moreover, the increments associated to disjoint intervals are independent.2.wt is a continuous function.3.The process starts at t=0, i.e. w0=0. In the general case of M-dimensional Wiener processes wt is a vector field wt=(wt1,…wtj); the increments wtj−wsj are Gaussians with mean zero and E((wti−wsi)(wtj−wsj))=(t−s)δij, which loosely speaking (in the sense of averaging or as a scaling argument) can be read as: dwtidwtj=δijdt.Note that although wt is continuous, is nowhere differentiable. As pointed out the Lˆ0 operator and the associated time evolution equation can be obtained via Itô's formula, which is briefly reviewed below:•Itô's formulaConsider a function f(x,t), where x satisfies (1.3) then(1.4)df(x,t)=∂f∂tdt+∑j∂f∂xjdxj+12∑i,j∂2f∂xi∂xjdxidxj. Notice the appearance of the last term when formulating the differential above. Indeed, due to the fact that dxtj satisfy the SDE (1.3), and dwtidwtj=δijdt, the last term is not negligible. Using (1.3) and the scaling argument for the Wiener processes, expression (1.4) becomes(1.5)df(x,t)=(∂t+Lˆ0)f(x,t)dt+∑i,jσij(x)∂f(x,t)∂xidwj, providing the generator Lˆ0 (1.2), which can be thought of as a quantum mechanical Hamiltonian describing M interacting particles, or alternatively as an M-dimensional quantum mechanical Hamiltonian. The equation ∂t+Lˆ0=0 is obtained after taking the expectation value of (1.5). For a rigorous proof of (1.5) and detailed discussions on stochastic differential equations and in particular on Itô's lemma and Itô's calculus, Feynman-Kac formula, and Wiener processes we refer the interested reader to [2–8].After the brief review on the time evolution problem from the stochastic point of view we may now outline our perspective, and present the main objectives of the present analysis. The first main objective here is to solve the generic time dependent Schrödinger equation (time evolution equation) (1.1), by employing the path integral formulation. Then from the solution of the time evolution problem via the path integral (time discretization) formulation we will arrive at the SDE (1.3), and the Feynman-Kac formula (see also [9] and references therein). It is clear that the generic operator Lˆ0 is not in general self-adjoint (Hermitian), therefore we should also introduce the adjoint operator defined for any suitable function f(x,t) as(1.6)Lˆ0†f(x,t)=12∑i,j=1M∂2∂xi∂xj(gij(x)f(x,t))−∑j=1M∂∂xj(bj(x)f(x,t)). Hence, two distinct time evolution equations emerge:•Fokker-Plank equationFor t1≥t2 the Fokker-Plank or Kolmogorov forward equation is given by(1.7)∂t1f(x,t1)=Lˆ0†f(x,t1) with known initial condition f(x,t2)=f0(x).•Kolmogorov backward equationFor t2≤t1 the Kolmogorov backward equation is given by−∂t2f(x,t2)=Lˆ0f(x,t2) with known final condition f(x,t1)=ff(x).To solve the PDEs above we will follow as already mentioned the path integral formulation. We should note however, that formulating the path integral in the case of a generic Lˆ operator with non-constant (dynamical) diffusion coefficients represents a particular difficulty. More precisely, one has to deal with the situation of a non-Gaussian measure when computing the path integral in the standard way. To circumvent this complication in section 2 we employ the notion of quantum canonical transformation and reduce the operator Lˆ to a much simpler form with constant diffusion coefficients, but with an induced/effective drift. Moreover, by considering both Lˆ and a “complementary” operator Lˆs –it will be defined later in the text– we show that the drift plays the role of a super-potential, whereas Lˆ, Lˆs may be thought of as “super-partners” in the typical super-symmetric quantum mechanics sense (see also e.g. [10,11] and references therein). Indeed, (1.2) may be seen as a generalized imaginary time Schrödinger equation,Lˆ=∑j=1MDyj2+V(y) where Dyj is a covariant derivative and V(y) is some induced (effective) potential. At the level of the associated SDEs the existence of two equivalent SDEs corresponding to Lˆ in (1.2) and its simplified version with constant diffusion coefficient respectively is shown in subsection 2.1. Then Feynman's path integral is evaluated for the reduced Lˆ, Lˆs operators in subsection 3.1 and the findings obtained via the quantum canonical transform are confirmed. Moreover, direct connections with Girsanov's theorem [12] as well as super-symmetric quantum mechanics and gauge theories are made. The use of a precise time discretization scheme together with scaling arguments are essential for these connections.We also compute explicitly the generic path integral with dynamical diffusion coefficients in subsection 3.2 by requiring that the fields involved satisfy discrete time SDEs. This key assumption, which is natural in the context of the time discrete scheme for computing the propagator, leads to a straightforward computation of the corresponding probabilistic measure, that turns out to be an infinite product of Gaussians. Hence, the canonical transform implemented at the level of PDEs is also confirmed at the microscopic level of the path integral. It is worth emphasizing that in this framework solutions of the underlying SDEs are required in order to compute time expectation values (see also [13]). This formulation can be seen as an alternative to the customary Lagrangian formulation and perturbative methods used in quantum and statistical theories. Various relevant examples are considered in section 4, where the use of the standard semi classical approach and the stochastic approach are displayed. Typical examples such as the Ornstein-Uhlenbeck process, and the multidimensional Black-Scholes model are presented, and the use of the Feynman-Kac formula in the problem of quantum quenches is also discussed. The computation of the harmonic oscillator propagator (Mehler's formula) by means of stochastic analysis arguments is also presented.Our second main objective is to explore links between SDEs, quantum integrable systems and the Darboux-Bäcklund transformation [14–16], as discussed in section 5. To illustrate these associations between discrete quantum systems and SDEs we discuss typical exactly solvable discrete quantum systems, such as the Discrete non-linear Schrödinger hierarchy, in particular the first two non-trivial members i.e. the Discrete-self-trapping (DST) model and the discrete NLS (DNLS) model [17,18], as well as the Heisenberg (XXZ) quantum spin chain [19–21]. Interesting connections between the DST model and the Toda chain are shown, and the SDEs associated to the DST model are solved using integrator factor techniques. We also consider both DST and DNLS models at the continuum limit, and we explicitly show that the associated SDEs turn to well known solvable SPDEs i.e. the stochastic transport and the stochastic heat equation respectively. The stochastic heat equation in particular can be further mapped to the stochastic viscous Burgers equation (see for instance [22] and references therein). Moreover, the DST model in the presence of local defects is studied, the quantum Darboux-Bäcklund relations are derived for the defect and the corresponding SDEs are also identified. The algebraic picture is also engaged leading to a modified set of SDEs.2The quantum canonical transformationAs has been pointed out the formulation of the path integral solution in the case of a generic Lˆ operator with non-constant (dynamical) diffusion coefficients represents a particular difficulty, given that its explicit computation leads to a non-Gaussian measure. The main objective in this section is the transformation of the dynamical diffusion matrix in (1.2) into identity at the level of the PDEs. Then in the next subsection motivated by the result at the PDE level we shall be able to show the corresponding equivalence at the SDE level generalizing what is known as the Lamperti transform (see e.g. [3,4,23,24]). This result will be then utilized in section 3 for the explicit computation of the general path integral, and the derivation of the Feynman-Kac formula.Indeed, we will show in what follows that the general Lˆ operator can be brought into the less involved form:(2.1)Lˆ=12∑j=1M∂2∂yj2+∑j=1Mb˜j(y)∂∂yj+u(y) with an induced drift b˜(y). This can be achieved via a simple change of the parameters xj, which in the Riemannian geometry sense is nothing but a change of frame. Indeed, let us introduce a new set of parameters yj such that:(2.2)dyi=∑jσij−1(x)dxj,detσ≠0, then Lˆ can be expressed in the form (2.1), and the induced drift components are given as(2.3)b˜k(y)=∑jσkj−1(y)bj(y)+12∑j,lσjl(y)∂ylσkj−1(y). Bearing also in mind that ∑jσjlσkj−1=δkl, we can write in the compact vector/matrix notation:(2.4)b˜(y)=σ−1(y)(b(y)−12(∇yσT(y))T),∇y=(∂y1,…,∂yM) where one first solves for x=x(y) via (2.2).The latter statement can be seen in a more general algebraic (Hamiltonian) frame, which is more relevant for our purposes here, especially when considering the M-dimensional case, M→∞ i.e. in a 1+1 field theoretical setting. Let Xj,Pj be elements of the Heisenberg-Weyl algebra:(2.5)[Xi,Pj]=iδij, where i=−1. The contribution of the commutator above gives rise to the second term in the induced drift (2.4), and this is clearly a purely “quantum” effect. We shall be able in fact to reproduce this quantum effect via the computation of Feynman's path integral, and also we will be able to extract it at a fundamental SDE level. In the generic algebraic framework Lˆ is then expressed as:(2.6)Lˆ(X,P)=−12∑i,jgij(X)PiPj+i∑jbj(X)Pj+u(X). The elements X,P are M column vectors with components Xj,Pj, which in our setting are represented as:Xj↦xj,Pj↦−i∂xj.Now consider the following quantum canonical transformation Lˆ(X,P)↦Lˆ(Y,P):(2.7)Yi=∑j∫Xjσij−1(X′)dXj′,Pi=∑jσij(X)Pj where in the integral above and the derivative in (2.5) Xj are formally treated as parameters. After the canonical transformation (2.7) is implemented the operator Lˆ reads as(2.8)Lˆ(Y,P)=−12∑jPj2+i∑jb˜j(Y)Pj+u(Y) and naturally Yj,Pj are also elements of the Heisenberg-Weyl algebra (2.5) represented as:(2.9)Yj↦yj,Pj↦−i∂yj This procedure is analogous to the quantum Darboux-Bäcklund transformation studied in [16]; similar results are presented in [14,15], where the so-called Baxter's Q-operator [25] is used as the generating function of the quantum canonical transformation.We have established via the quantum canonical transform that the general diffusion matrix can turn to the identity. Let us now consider both L and(2.10)Lˆs=−12PTP+iPTb˜(Y)+u(Y). Lˆs not to be confused with the adjoint operator Lˆ†. The two operators coincide only when a purely imaginary drift b˜ is considered.•Remark 1: the drift as super-potentialWe introduce the following compact notation: ∀F,defineFD∈{F,Fs}. We can bring the LˆD operators to an even more familiar form by setting: P=P−ib˜(Y), where Yj,Pj are also elements of the Heisenberg-Weyl algebra (2.5). Then(2.11)LˆD=12PTP+VD(Y)VD(Y)=−12b˜T(Y)b˜(Y)−12∇Yb˜(Y)+uD(Y)us(Y)=u(Y)+∇Yb˜(Y), where VD are effective potentials, produced exclusively by the drift. In terms of differential operators then one obtains via:(2.12)LˆD=12∑jDyj2+VD(y) where Dyj=∂yj+b˜j(y) is the covariant derivative. Interestingly b˜ turns out to be the super-potential that produces V,Vs and satisfies the familiar Riccati equations (2.11). The statements at the algebraic level will be explicitly confirmed via the evaluation of the path integrals associated to both operators Lˆ,Lˆs.•Remark 2: Riccati equation & Girsanov's theoremNote that the Riccati equation (2.11) emerging from the identification of the effective potential due to the presence of the drift, -and hence the covariant derivative (2.12)- is essentially the origin of Girsanov's theorem on the change of probabilistic measure (we refer the interested reader to [2–8]) from the PDE point of view. This will be even more transparent in the next section from the explicit path integral computation.2.1Equivalence of SDEsThe generator Lˆ0 (1.2) as already discussed can be obtained via Itô's lemma from the SDE (1.3), while the operator in the simpler equivalent form established previously:(2.13)Lˆ0=12∑j=1M∂2∂yj2+∑j=1Mb˜j(y)∂∂yj is obviously obtained via the following simpler SDE(2.14)dyt=b˜(yt)dt+dwt, where recall b˜ is defined in (2.3), (2.4).The main aim now is to show that these two SDEs (1.3) and (2.14) are in fact equivalent. Indeed, let us start from (2.14), multiply both sides of the equation with σ(yt) and add Itô's correction i.e.(2.15)σ(yt)dyt+12dσ(yt)dyt=σ(yt)b˜(yt)dt+12dσ(yt)dyt+σ(yt)dwt. Recall the connection between Itô and Stratonovich calculus:(2.16)f(yt)∘dyt=f(yt)dyt+12df(yt)dytanddyidyj=δijdt where f(yt)∘dyt refers to Stratonovich calculus, and f(yt)dyt to Itô. More comments on this relation will be given in the next subsection, where the discrete time scheme is discussed via the path integral computation. Recall also the definition of the modified drift b˜ (2.3):σ(yt)b˜(yt)dt+12dσ(yt)dyt=b(yt)dt, then (2.15) becomesσ(yt)∘dyt=b(yt)dt+σ(yt)dwt. We now set(2.17)dxt=σ(yt)∘dyt and we immediately recover the original SDEdxt=b(xt)dt+σ(xt)dwt after solving y=y(x) through (2.17). Integration of (2.17) in the Stratonovich calculus frame – i.e. the usual calculus rules apply– yieldsxi=∑j∫yjσij(y′)∘dyj′, and the latter as expected is nothing but the inverse of (2.7).Naturally we could have started from (1.3) and ended up to (2.14) via the reverse process. Indeed, we start from (1.3) we multiply the equation by σ−1, add Itô's correction 12dσ−1(x)dx at both sides of the equation, and use (2.16). Then setting dy=σ−1(x)∘dx we end up to the desired equation (2.14). With this we conclude of our proof on the equivalence of the two SDEs (1.3) and (2.14). Relevant results on the change of the diffusion matrix from the SDE point of view via the so called Lamperti transform are presented in [23], (regarding the Lamperti transform we refer the interested reader to [3,4,24]).3The path integral3.1Identity diffusion matrixWe have established in the previous section that if the generic dynamical diffusion matrix is invertible then Lˆ can be brought into a simple form with identity diffusion matrix via the quantum canonical transformation. We basically confirm the results of section 2 via the explicit computation of the relevant path integral using the Trotter product formula (see for instance [9]). We review in this subsection the computation of the path integral in the case when the diffusion matrix is constant, but in the presence of a general drift. We explicitly compute in what follows the propagator associated to the operator Lˆ using the standard discrete time frame and make various interesting connections. More precisely:1.We make a direct connection with the discrete time version of Cameron-Martin-Girsanov theorem on the change of the probabilistic measure, which is also associated to the presence of a geometric phase in gauge theories.2.We show that the path integral can be alternatively computed by assuming the existence of underlying discrete time SDEs. This fundamental assumption leads to a straightforward computation of the associated measure, which turns out to be a Gaussian. This result is generalized in the next subsection where dynamical diffusion coefficients are also considered.3.We show that the drift, which is essentially a gauge field plays the role of the super-potential with Lˆ, Lˆs being super-partners in the usual super-symmetric quantum mechanics sense, confirming the findings of the previous section.Our starting point is the time evolution equation (1.6), (1.7), (2.1):∂tf(y,t)=Lˆ†f(y,t), we then explicitly compute the propagator K(yf,yi|t,t′):(3.1)f(y,t)=∫∏j=1Mdyj′K(y,y′|t,t′)f(y′,t′)(3.2)=∫∏n=1N∏j=1Mdyjn∏n=1NK(yn+1,yn|tn+1,tn)f(y1,t1). In what follows we shall reproduce Itô's correction as the result of quantum/statistical fluctuations –i.e. use of suitable scaling arguments– and also derive the corresponding SDEs via the exact computation of the propagator K, assuming no a priori knowledge of stochastic analysis. Within this frame the discrete time analogue of Girsanov's theorem is a straightforward result.Employing the standard time discretization scheme or the semi-group property, as shown above, (see also for instance [9]) the path integral or time propagator can be expressed as(3.3)K(yf,yi|t,t′)=1(2π)NM∫dydpexp[−δ2∑n=1N∑j=1Mpjn2+i∑n=1N∑j=1Mpjn(Δyjn−δb˜jn(y))+δ∑n=1Nun(y)] where we definedy=∏n=2N∏j=1Mdyjn,dp=∏n=1N∏j=1Mdpjn, where δ=tn+1−tn and with boundary conditions: yf=yN+1,yi=y1,ti=t′=0 (t′ will be dropped henceforth for brevity), tf=t; pnj are also known as response variables. To obtain the latter formula we have inserted the unit N times, (12π∫dyjndpjneipjn(yjn−a)=1), associated to each component yj. After performing the Gaussian integrals with respect to the pjn parameters we conclude:(3.4)K(yf,yi|t)=∫dqexp[−∑j∑n(Δyjn−δb˜jn(y))22δ+δ∑nun(y)]dq=1(2πδ)NM2∏n=2N∏j=1Mdyjn where fn=fn(yn) and Δyjn=yjn+1−yjn. At this stage we are dealing with an M×N space-time lattice; n: time indices, j: space indices.We now expand the square in the exponential in (3.4), and consider the continuum time limit as follows (N→∞,δ→0):(3.5)∑n=1Nb˜nT(y)Δyn→∫b˜T(ys)dys(3.6)δ2∑n=1Nb˜nT(y)b˜n(y)→12∫0tb˜T(ys)b˜(ys)ds(3.7)δ∑n=1Nun(y)→∫0tu(ys)ds, where ∫0tb˜T(ys)dys corresponds to an Itô type integral, due to the chosen discretization scheme above. We shall discuss the Itô-Stratonovich integral correspondence later in this subsection, as these two types of integrals emerge due to two distinct time discretization schemes. The propagator then takes the form(3.8)K(yf,yi|t)=∫dPexp[−12∫0tb˜T(ys)b˜(ys)ds+∫b˜T(ys)dys+∫0tu(ys)ds] where we define(3.9)dP=limδ→0limN→∞1(2πδ)NM2exp[−∑n=1N∑j=1M(Δyjn)22δ]∏n=2N∏j=1Mdyjn.We can follow an alternative path to compute the propagator in a straightforward way by making explicit use of the corresponding SDEs. Recall expression (3.3) and assume that:(3.10)Δyn−δb˜n(y)=Δwn assuming also that wn are independent of the vector fields yn. After performing the standard Gaussian integrals in (3.3) subject to (3.10) we conclude that(3.11)K(yf,yi|t)=∫dMe∫0tu(ys)ds,dM=limδ→0limN→∞1(2πδ)NM2exp[−12δ∑n=1NΔwnTΔwn]∏n=2N∏j=1Mdwjn. Note that the presence of the Gaussians in the measure suggests that Δwnj are normal variables with zero mean and ΔwinΔwjn=δijδ (in the sense of expectation value). Moreover, we assume that w0=0, i.e. we are considering Wiener paths. This will become transparent in the next subsection when evaluating explicitly the measure for the general case of dynamical diffusion coefficients, using the Fourier representation of the Wiener paths. Expression (3.10) can be then seen as the discrete time analogue of the SDE (2.14). Explicit evaluation of the propagator via (3.8) naturally leads to the use of the Lagrangian formulation, which is usually implemented in quantum systems and quantum/statistical field theories. Evaluation of K(y,y′|t) on the other hand via (3.11) requires the solution of the corresponding SDEs.•Remark 3: Radon-Nikodym derivative & Girsanov's theoremNote that the ratio (3.8), (3.11)dMdP=exp[−12∫0tb˜T(ys)b˜(ys)ds+∫b˜T(ys)dys] turns out to be the Radon-Nikodym derivative in analogy to Girsanov's theorem [12] on the change of measure. Indeed, as already pointed out there is a one to one correspondence with the results of section 2 especially in relation to the notions of the effective potential V (2.11) and the covariant derivative (2.12).•Remark 4: Itô vs Stratonovich calculusWe can now reproduce the Itô-Stratonovich correspondence via distinct discretization schemes. The main reason we address this issue here is because Stratonovich type integrals follow the usual calculus rules, and are the ones that are usually employed in the path integral formulation in quantum/statistical physics. Let us focus again on the square of the exponential (3.4) and in particular on the following term:(3.12)∑j=1M∑n=1Nb˜jnΔyjn=∑j=1M∑n=1Nb˜jn+Δyjn−12∑j=1M∑n=1NΔb˜jnΔyjn where b˜jn+=12(b˜jn+1+b˜jn) and Δb˜jn=b˜jn+1−b˜jn. We also take into consideration the scaling rule, in the sense of averaging, ΔyinΔyjn∼δijδ, and consider the continuum time limit in (3.12), (δ→0,N→∞):(3.13)∑j=1M∑n=1Nb˜jnΔyjn→∑j=1M∫b˜j(ys)dysj,Itô integral(3.14)∑j=1M∑n=1Nb˜jn+Δyjn→∑j=1M∫b˜j(ys)∘dysj,Stratonovich integral(3.15)12∑j=1M∑n=1NΔb˜jnΔyjn→∑j=1M∫0t∂yjb˜j(ys)ds. The above lead to the Itô-Stratonovich correspondence (3.12)-(3.15):(3.16)∫b˜T(ys)dys=∫b˜T(ys)∘dys−12∫0t∇yb˜(ys)ds.•Remark 5: gauge theoriesIt is clear that the systems we are considering here (1.6), (1.7), (2.1) are typical gauge theories, with the vector field of the drift b˜ being the gauge field. After taking into consideration the Itô-Stratonovich correspondence discussed above (3.16) we may re-express the propagator (3.8) as:(3.17)K(yf,yi|t)=∫dPexp[−12∫0tb˜T(ys)b˜(ys)ds+∫b˜T(ys)∘dys−12∫0t∇yb˜(ys)ds+∫0tu(ys)ds]. Notice the second term in (3.8) (i.e. the Stratonovich type integral, i.e. standard calculus rules apply) corresponds to a standard geometric phase (see e.g. Berry phase [26]) arising in gauge theories, with typical example the Bohm-Aharonov effect [27]. It is clear that the first and third terms in the expression above can be combined with the potential u to yield an effective potential V, which is in agreement with the findings of section 2 and the associated Riccati equation (2.11) for the gauge field (super-potential).•Remark 6: super-symmetry & driftHaving introduced the Itô-Stratonovich correspondence at the microscopic level it is worth pointing out that the propagator Ks for the “super-partner” Ls (2.10) can also be evaluated and one concludes that the difference D=Ks−K (3.8) provides indeed the suitable expression in the context of super-symmetric quantum mechanics, related also to the index theorem (see for instance [10,11] and references therein),D=2∫dMˆexp[∫0tu(ys)ds]sinh(12∫0t∇ysb˜(ys)ds), where nowdMˆdP=exp[−12∫0tb˜T(ys)b˜(ys)ds+∫b˜T(ys)∘dys]. This result is also in line with the findings of section 2.3.2Dynamical diffusion matrixWe examine the general case with dynamical diffusion matrix (1.1), (1.2), although we showed its equivalence with a case of identity diffusion matrix and modified drift via the quantum canonical transform. The reason we view this case separately is because, as already mentioned, the underlying SDEs turn out to play a crucial role when computing the path integral. Indeed, instead of using the customary perturbative methods of computing the path integral one can alternatively exploit the solutions of the relevant SDEs (see also [13]). Specifically, our main aim in this subsection is the explicit computation of the propagator in the more general case of dynamical diffusion coefficients and drift. We first focus on the computation of the general path integral by employing the discrete analogue of the general SDEs (1.3) as a fundamental assumption naturally emerging after performing the relevant Gaussian integrals in the propagator.The connection with Girsanov's theorem and the appearance of the geometric term ∫0tb˜T(ys)∘dys requires the results of section 2 and the explicit computation of the first part of the previous subsection. A brief outline of the general path integral computation employing the approach of the previous subsection, which led to the discrete time version of Girsanov's theorem, is presented in the end of the subsection.Following the prescription of the preceding subsection we can compute the corresponding path integral for the general case (1.2) via (1.6), (1.7) and express it in the compact vector/matrix notation:(3.18)K(xf,xi|t)=1(2π)NM∫dxdpexp[−δ2∑n=1NpnTgn(x)pn+i∑n=1NpnT(Δxn−δbn(x))+δ∑n=1Nun(x)] where Δxn=xn+1−xn and recall the diffusion matrix is gn=σnσnT.We come now to our main assumption, letΔxn−δbn(x)=σn(x)Δwn and as in the previous subsection wn are independent of the vector fields xn, moreover wnj are independent normal variables and ΔwinΔwjn=δijδ. After performing the standard Gaussian integral:(3.19)∫dpnexp[−δ2pTngn(x)pn+ipnTσn(x)Δwn]=(detσn(x))−1(2πδ)M2e−12δΔwnTΔwn we conclude that(3.20)K(xf,xi|t)=∫dMe∫0tu(xs)ds,(3.21)dM=limδ→0limN→∞1(2πδ)NM2exp[−12δ∑n=1NΔwnTΔwn]∏n=1N(detσn(x))−1∏j=1M∏n=2Ndxjn, however: (detσn(x))−1∏j=1Mdxjn=∏j=1Mdwjn, which is a typical change of the volume element (see also relevant change of variables discussed in section 2 equ. (2.2)). We can now explicitly evaluate the measure from the explicit expression just above in the continuum limit.11When keeping N finite we have from (3.18)dM=(detσ(x1))−11(2πδ)M2exp[−12δ∑n=1NΔw1TΔw1]dM−. The factor in front of dM− is nothing but the infinitesimal kernel K(x2,x1|δ→0)=(detσ(x1))−1δ(Δw1) compatible with the initial conditions of K. The pre-factor (detσ(x1))−1 is essentially absorbed in the measure via the change of the volume element (detσ(xn))−1∏j=1Mdxjn=∏j=1Mdwjn.To compute the corresponding measure (3.21) (see also footnote 1)(3.22)dM=limδ→0limN→∞1(2πδ)NM2exp[−12δ∑n=1NΔwnTΔwn]∏j=1M∏n=2Ndwjn in the continuum limit we consider the Fourier representation on [0,t] for ws, i.e. Wiener's representation of the Brownian path:(3.23)ws=f0ts+2t∑k>0fkωksin(ωks),ωk=2πkt. f0=wtt and fk,k∈{0,1,…} are M vectors with components fkj,j∈{1,2,…,M} being standard normal variables. We are interested in the computation of the measure in the continuum limit N→∞,δ→0, and we also recall the following boundary conditions: w(s=0)=0,w(s=t)=wt. Taking all the above considerations into account we conclude:(3.24)dM=e−12twtTwt(2πt)M2dM0dM0=∏k≥1∏j=1Mdfkj2πexp[−12∑k≥1∑jfkj2]. The measure naturally is expressed as an infinite product of Gaussians regardless of the specific forms of the diffusion coefficients and the drift, which is indeed an elegant result. However, the price one pays is that solutions of the SDEs, available in general via iteration techniques, are now required. In any case this scheme can be seen as an alternative to the established perturbative approach employed in quantum mechanical systems and quantum field theories (see also [13]), therefore algebraic and numerical techniques developed in solving SDEs will be of great relevance (see e.g. [6,28] and references therein). The significant observation is the existence of the heat kernel in front of dM0. This boundary term plays a crucial role when computing explicitly the propagator or in general when computing time expectation values as will be transparent later in the text. In the more standard case where Lˆ is expressed as 12∑i∂xi2+u(x) the underlying SDE is(3.25)dxt=dwt⇒xs=x0+ws. It is clear that in more complicated cases, i.e. in the presence of drift the relation between wt and xt may become significantly involved depending on the specific form of the associated SDEs. A few simple illustrative examples are discussed in section 4.Alternatively, the problem described above can be thought of as a typical optimal control problem (feedback problem). More precisely, the minimization of the action ∫0tds(12ηsTηs−u(xs)) should be required subject to the equation x˙t=b(xt)+σ(xt)ηt. Then linearization methods. i.e. suitable perturbations around the optimal solution, can be applied leading to certain Riccati type equations. This point of view is naturally closer to the more established semi-classical Lagrangian approach. We shall report on these approaches in more detail in a separate publication [30].•Remark 7: Feynman-Kac formulaHaving computed the propagator explicitly (3.20) we conclude that equation (3.2) can be then expressed asf(xf,tf)=∫dMe∫0tu(x)dsf0(x0),f0(x0)=f(x0,t0) which is precisely the Feynman-Kac formula, and describes the time evolution of a given initial profile f0(x0) to f(xf,tf) a solution of the Fokker-Planck equation.One could have started from the Kolmogorov backward equation and computed the path integral backwards in time, which would have led to the time-reversed version of the Feynman-Kac formula:f(x0,t0)=∫dMe∫0tu(x)dsff(xf),ff(xf)=f(xf,tf). In this case the Feynman-Kac formula describes the reversed time evolution of a given final state ff(xf), backwards in time, to a previous state f(x0,t0) a solution of the Kolmogorov backward equation.Let us also briefly outline the re derivation of the path integral in the case of dynamical diffusion matrices generalizing the computations of the previous subsection, which led to the discrete time version of Girsanov's theorem. We start from the general expression (3.18) and perform the Gauss integrals (set Bn(x)=Δxn−δbn(x))∫dpnexp[−δ2pTngn(x)pn+ipnTBn(x)]=(detσn(x))−1(2πδ)M2e−12δ(σn−1(x)Bn(x))Tσn−1(x)Bn(x). Let us now focus on the term σn−1(x)Bn(x), which can be expressed asσn−1(x)Bn(x)=(σn+)−1(x)Δxn−12Δσn−1(x)σn+(x)(σn+)−1(x)Δxn−δσn−1(x)bn(x), in accordance to our usual notation: (σn+)−1=12(σn+1−1+σn−1),Δσn−1=σn+1−1−σn−1. We set(3.26)Δyn=(σn+)−1(x)Δxn, which is the discrete time analogue of definition (2.17). Using (3.26) we conclude:σn−1(x)Bn(x)=Δyn−12Δσn−1(y)σn(y)Δyn−δσn−1(y)bn(y)=Δyn−δb˜n(y) after solving xn=xn(yn) via (3.26) as in the continuous time case discussed in section 2. The definition of b˜ in the expression above is apparently the discrete time analogue of (2.3). Bearing also in mind the volume element change (detσn(x))−1∏j=1Mdxjn=∏j=1Mdyjn, we conclude that the propagator (3.18) coincides with expression (3.4), thus all computations of subsection 3.1 apply and the findings of section 2 are once more confirmed.3.2.1The case detσ=0We shall briefly discuss in what follows the special situation where detσ=0, in this case where the σ-matrix has at least one zero eigenvalue. Let us consider the quite general scenario, where σ is given asσ(x)=(σ¯(x)m×m0(M−m)×m0m×(M−m)0(M−m)×(M−m)). Then the corresponding SDEs are expressed as follows(3.27)dxti=bi(xt)dt+∑j=1mσ¯iji(xt)dwtj,i≤mdxti=bi(xt)dt,i>m. The associated generator Lˆ0 is then given as(3.28)Lˆ0=∑i,j=1mgij(x)∂2∂xi∂xj+∑i=1Mbi(x)∂∂xi, g=σ¯σ¯T.Following the findings of section 4 and assuming that detσ≠0, we consider the following simple change of variables:dyti=∑i,j=1mσ¯ij−1(xt)dxtj,i≤mdyti=dxti,i>m, then the generator takes the simpler formLˆ0=∑i=1m∂2∂yi2+∑i=1mb˜i(y)∂∂yi+∑i=m+1Mbi−(y)∂∂yi and the SDE (3.27) becomesdyti=b˜i(yt)dt+dwtj,i≤mdyti=bi−(yt)dt,i>m, b˜ is an m-vector defined asb˜(ys)=σ¯−1(ys)(b(ys)−12(∂ys−σ¯T(ys))T),∂y−=(∂y1,…,∂ym), and b− is an M−m vector with components bi(y), i∈{m+1,…,M}. Typical systems with such a behavior are systems in the presence of discontinuities/defects or interfaces. The DNLS/DST model in the presence of local defects is a such a system and will be examined in the subsequent section.4Lagrangian versus stochastic formulationIn this section we compute the propagators for certain prototype models via the Lagrangian and stochastic approaches. More precisely, in the next subsection we evaluate Feynman's path integral (3.8), for a system of M-damped harmonic oscillators with linear drift i.e. the case of M-dimensional Ornstein-Uhlenbeck process using the standard Lagrangian point of view. In subsection 4.2 we review the computation of the propagator for the 1-dimensional harmonic oscillator via the stochastic approach (see also [7]).We also comment on generic processes associated to more complicated SDEs as well as on possible future directions combining the two approaches i.e. Lagrangian approach and perturbation theory versus stochastic methods and use of solutions of SDEs. Here we basically consider simple prototypical models such as the harmonic oscillator and the Ornstein-Uhlenbeck process, in order to review how the two approaches work. However the goal is the study of more involved examples, where both perturbative/semi-classical methods and the stochastic approach can be tested and compared.4.1Lagrangian approach: multidimensional Ornstein-Uhlenbeck processIn this subsection we compute the propagator (3.8) for a system of M-damped oscillators using the standard for physicists formulation i.e. the Lagrangian description. The quantum Hamiltonian in this case is given by (2.1) with:b˜(y)=−Θy,u(y)=−12yTΘˆTΘˆy, where Θ,Θˆ are constant M×M matrices. Notice in this case we deal with a non-self-adjoint operator. From the stochastic point of view this is a multidimensional Ornstein-Uhlenbeck process (see also expression (3.8)) with relevant SDEs given by:(4.1)dyt=−Θytdt+dwt.We shall express the path integral (3.8) in a form familiar for physicists (we have suppressed the time subscript s in this subsection for simplicity)(4.2)K(yf,yi|t)=∫dqexp[−∫0tL(y,y˙)ds], and the Lagrangian is given (3.8), (2.11):L(y,y˙)=12y˙Ty˙+12b˜T(y)b˜(y)−b˜T(y)y˙+12∇yb˜(y)−u(y) the third term in the expression above comes from the term ∫0tb˜T∘dy in (3.17), and y˙=dyds, (recall Remarks 4, 5). Indeed, notice that here all the involved integrals are Stratonovich type integrals, hence the extra term 12∇yb˜(y) (Ito's contribution (3.17)). The astute reader might be perplexed by the appearance of y˙ in the expression above, however this is nothing but a convenient choice of notation.The Lagrangian can be then expressed in components as:(4.3)L(y,y˙)=12∑iy˙i2+12∑iΩij2yiyj+∑i,jΘijy˙iyj−trΘ2 where Ω2=ΘTΘ+ΘˆTΘˆ and is by construction symmetric. Notice that in the special case where Θ is a symmetric matrix the third term of the expression above can be expressed as a total derivative giving rise to a purely boundary term after integration.Let us now follow the typical Lagrangian approach in computing the path integral (4.2) and focus on the case where Θ is symmetric. Let y=z+w, where z is the deterministic contribution i.e. the solution of the classical equations of motion and w is the random contribution, where we assume the following boundary conditions: z(0)=z0,z(t)=zf and w(0)=w(t)=0. Then we obtain(4.4)exp[−∫0tL(y,y˙)ds]=exp[−∫0t(L˜(z,z˙)ds+L˜(w,w˙)+F(z,z˙;w,w˙))ds]×e−12zfTΘzf+12z0TΘz0+trΘt2 where we define(4.5)L˜(z,z˙)=12∑iz˙i2+12∑iΩij2zizj. The classical equations of motion are obtained via the Euler-Lagrange equations:(4.6)∂L˜∂zj=∂∂s(∂L˜∂z˙j)⇒z¨−Ω2z=0. The general solution of (4.6) is given by(4.7)z(s)=sinh(sΩ)(sinh(tΩ)−1zf−sinh(tΩ)−1cosh(tΩ)z0)+cosh(sΩ)z0 where recall we have considered z(0)=z0,z(t)=zf.The last term inside the integral in (4.4) is linear in w, and disappears via the classical equations of motion, whereas the classical Lagrangian will produce only boundary terms due to (4.6). The path integral is then expressed as(4.8)K(yf,0|t)=e−12zfTz˙f+12z0Tz˙0−12zfTΘzf+12z0TΘz0+Θt2∫dwe−∫0tL˜(w,w˙)dsdw=limδ→0limN→∞1(2πδ)NM2∏n=2N∏j=1Mdwjn. Let as first compute the classical contribution of the path integral (4.8) using the solution (4.7) of the classical equations of motion. Indeed, substituting zf,z˙f via (4.7) we immediately obtain(4.9)exp[−12zfTz˙f+12z0Tz˙0−12zfTΘzf+12z0TΘz0+trΘt2]=exp[−12tr((zfzfT+z0z0T)cosh(tΩ)Ωsinh(tΩ)−1)+tr(z0zfTΩsinh(tΩ)−1)−12(zfTΘzf−zT0Θz0)]etrΘt2. To compute the quantum contribution of the path integral (4.8) we consider a Fourier transform expansion for the random part of the fields (see also subsection 3.2)(4.10)ws=2t∑k≥1sin(ωks)ωkfk,ωk=kπt fk are M-vector fields with components fkj. After some straightforward substitutions via (4.10) we obtain(4.11)∫dwe−∫0tL˜(w,w˙)ds=1(2πt)M2∫dfexp[−12∑k≥1fkT(IM+t2k2π2Ω2)fk],=1(2πt)M2E(exp[−12∑k≥1fkT(t2k2π2Ω2)fk])df=∏k≥1∏j=1Mdfkj2π. As already mentioned earlier in the text the appearance of y˙ in the expression above although perhaps perplexing is nothing but a convenient choice of notation. In any case, we are able to obtain the correct result, having suitably regularized the coefficients of the Fourier transform (4.10).After performing the Gaussian integrals in (4.11) we express the quantum contribution as(4.12)∫dwe−∫0tL˜(w,w˙)ds=1(2πt)M2∏k≥1det(Im+t2k2π2Ω2)−12. Recalling the infinite product identity∏k≥1(1+a2k2)−1=πasinh(πa) we eventually obtain(4.13)∫dwe−∫0tL˜(w,w˙)ds=1(2πt)M2det((tΩ)−1sinh(tΩ))−12.Putting together the classical and quantum contributions (4.9), (4.13) and recalling that y0,f=z0,f we conclude (4.8):(4.14)K(yf,y0|t)=1(2πt)M2det((tΩ)−1sinh(tΩ))−12×exp[−12tr((yfyfT−y0y0T)Θ)+trΘt2]×exp[−12tr((yfyfT+y0y0T)cosh(tΩ)Ωsinh(tΩ)−1)+tr(y0yfTΩsinh(tΩ)−1)]. In the case where the potential is zero Θˆ=0, we obtain the multidimensional Ornstein-Uhlenbeck propagator. In the one dimensional case in particular the generic expression above reduces to the familiar propagator:(4.15)K(y,x|t)=Θπ(1−e−2Θt)exp[−Θ(y−xe−Θt)21−e−2Θt]. It is clear that the Lagrangian description provides a straightforward way of computing the propagator of the Ornstein-Uhlenbeck process.Let us comment on the case where Θ contains an anti symmetric part as well, i.e. Θ=Σ+A where Σ is the symmetric part and A is the anti symmetric one. From a physical viewpoint this case corresponds to the presence of an angular momentum term in the Lagrangian, whereas from the computational point of view there are two non-trivial contributions when computing the propagator. First the classical equations of motion are modified due to the existence of the anti-symmetric part. More precisely, from the Euler-Lagrange equations one obtains, the following classical equations of motion:z¨+2Az˙−Ω2z=0, thus the classical contribution (4.9) is also modified accordingly. More importantly there is a highly non-trivial contribution when computing the quantum part (4.10), indeed when Θ is symmetric as described above the term e−∫0tds∑i,jΘij(w˙iwj−wiw˙j) gives a purely boundary contribution. However, when there is an anti-symmetric part then an extra non-trivial term in the integral (4.10) appears, which should be taken into consideration. The computation of the quantum contribution when Θ is anti-symmetric and for Ω=0 is essentially equivalent to the problem of computing the characteristic function for the Levy area:(4.16)A=∫dm0exp[∑i>jΘij∫(wsidwsj−wsjdwsi)] where dm0 is given in (3.24), and ws is given in (3.23) (see also e.g. [29] and references therein). This is a significant issue, which however will be addressed in a separate publication [30].4.2Stochastic approach: one dimensional harmonic oscillatorOur goal in this subsection is to rederive the propagator for the one dimensional quantum harmonic oscillator using probabilistic techniques, although the analytical form of the Feynman's path integral via the Lagrangian (semi-classical) formulation is widely known in the quantum physics community, given by the celebrated Mehler formula. Consider the single particle quantum harmonic oscillator, with potential u(x)=ω22x2 for x∈R, where ω>0 is a fixed parameter. We derive the formula here from the probabilistic perspective. Using (3.11), (3.24), we evaluate:K(y,x|t)=∫dMexp(−12ω2∫0t|xs|2ds)=e−(x−y)22t2πtE(exp(−12ω2∫0t|xs|2ds):xt=y,x0=x). For x=0 this is a standard quadratic functional of Brownian motion studied extensively by Pitman and Yor, see for example [7].We now derive this result for x≠0. Let ws be a standard Wiener process (w0=0). Then the required Brownian bridge satisfying x0=x and xt=y is given by (3.25) wt=y−x and as Wiener proved a standard Wiener process has the Fourier representation on [0,t] given in (3.23), where fn are normal random variables, (wt=tf0). Substituting this Fourier series into the prescription for xs above(4.17)xs=x+st(y−x)+2tπ∑n≥11nfnsin(nπst).Let us directly substitute the Fourier series for xs into the integral in the exponent. After performing standard trigonometric integrals we conclude−12ω2∫0t|xs|2ds=−12∑n⩾1(anfn2+bnfn)−12c, where we definean:=ω2t2π2n2,bn:=ω2(2t)3/2π2n2(x−(−1)ny)andc:=ω2t3(x2+xy+y2).The expectation value for a generic term in the sum, namely E(exp(−12anξ2−12bnξ)), with ξ a standard normal random variable. Directly computing we findE(exp(−12anξ2−12bnξ))=12πexp(12ω4bn24(1+ω2an))∫Rexp(−12(1+ω2an)(ξ+C)2)dξ=exp(12ω4bn24(1+ω2an))⋅1(1+ω2an)1/2, where C=ω2bn/(2(1+ω2an)).Substituting these last expressions into the expectation of interest, we findE(exp(−12ω2∫0t|xs|2ds))=exp(−ω2t6(x2+xy+y2))(∏n=1∞n2n2+ω2t2/π2)1/2×exp((2t)38ω4π4∑n⩾1n−4(x−(−1)ny)21+ω2t2/(n2π2))=exp(−ω2t6(x2+xy+y2))(ωtsinh(ωt))1/2exp(t3ω4π4∑n⩾1x2+y2−2(−1)nxyn2(n2+ω2t2/π2)), where we have substituted for an and bn from the expressions above and also used Euler's formula for the infinite product shown.We now focus on the terms in the exponent of the final factor. By direct computation we observe thatt3ω4π4∑n⩾1x2+y2−2(−1)nxyn2(n2+ω2t2/π2)=tω2π2((x2+y2)(π26+π2ω2t212(1−ωtcoth(ωt))+2xy(π26+π2ω2t212(1−ωtcoth(ωt)))−xy(π26+4π2ω2t212(1−12ωtcoth(12ωt)))).Suitably combining the above contributions we conclude that the quadratic functional of the Brownian bridge associated with the scalar single particle quantum harmonic oscillator can be then expressed as followsE(exp(−12ω2∫0t|xs|2ds):xt=y,x0=x)=(ωtsinh(ωt))1/2exp(−ω2(x2+y2)coth(ωt)+ωxysinh(ωt)+(x−y)22t), which givesK(y,x|t)=(ω2πsinh(ωt))1/2exp(−ω2(x2+y2)coth(ωt)+ωxysinh(ωt)), and this is precisely Mehler's formula. We observe that the proof above reveals that the physicist's proof of Mehler's formula via the action S[x] is much more succinct. Note that the extension of Mehler's formula for the M-particle harmonic oscillator by means of probabilistic techniques is given in [30], but also the results of the previous section from the semi-classical point of view are relevant when considering the case of zero drift.4.2.1Examples of stochastic processes & applicationsIn general, one of the main aims is the computation of expectation values using the universal expression (3.20), (3.24) and solutions of the underlying SDEs. In particular, in statistical and quantum physics the quantities of significance are (we use here the familiar in physics community notation for expectation values 〈O〉):(4.18)〈O(xs)〉=Et(O(xs)e∫0tu(xs)ds)Et(e∫0tu(xs)ds),0≤s≤t where we define via (3.20), (3.24)(4.19)Et(O(xs))=∫dwtdMO(xs)0≤s≤t. Formula (4.18) can be practically used provided that solutions of the associated SDEs (SPDEs in the continuum space limit, i.e. in quantum/statistical field theories) are available so that the fields xtj are expressed in terms of the normal variables wtj. Alternatively, in the cases of generic drift and/or potential in order to obtain the propagator and expectation values one can in principle start from (3.8) and apply semi-classical methods, perturbation theory, and Green's function techniques as is customary in quantum and statistical field theory.Let us briefly discuss below a few illustrative examples of simple processes, in the absence of potential u, which will also be used later in the text when discussing two fundamental integrable quantum systems, i.e. the DST and XXZ models. In the examples below we use solutions of the underlying SDEs in order to compute expectation values.1.Multidimensional Brownian motion with constant diffusion matrixThe corresponding SDEs and solutions are given by(4.20)dyt=σdwt⇒ysj=y0j+∑jσijwsk where σ is a non-dynamical (i.e. does not depend on the fields yj) M×M matrix. Let us compute the first couple of expectation values (first couple of moments): Et(ysj) and Et(ysiys′j). Indeed, one readily finds via (4.19)Et(ysj)=y0j,Et(ysiys′j)=Et(y0iy0j)+Et(∑k,lσikσjlwskws′l)=y0iy0j+gijs′,s′≤s and we have used the fact that(4.21)Et(wsj)=0,(4.22)Et(wsiws′j)=δijs′.s′≤s. Equation (4.21) is obvious via (4.19), an explicit proof of (4.22) is presented below.We focus on the one dimensional case, but the multidimensional generalization is straightforward. This expectation value can be readily computed using the fact that ws−ws′ and ws′ are independent increments, which leads to:(4.23)Et(wsws′)=Et(ws′2)=s′,s′≤s. However, let us show explicitly (4.23), via the use of (4.19) and Wiener's representation (3.23). Let 0≤s′≤s≤t:Et(wsws′)=∫dwt2πt∏k≥1dfk2πexp[−wt22t−∑k≥1fk22](wt2t2ss′+2t∑k≥1sin(ωks)sin(ωks′)ωk2fk2)=ss′t−t∑k≥1cos(2kπs+s′2t)−cos(2kπs−s′2t)k2π2. Taking into consideration the identity∑k≥1cos(2kπx)k2π2=B2(x),0≤x≤1, where B2 is the second Bernoulli polynomial B2(x)=x2−x+16, we concludeEt(wsws′)=ss′t−t(B2(s+s′2t)−B2(s−s′2t))=s′, which immediately leads to Et((ws−ws′)2)=s−s′.2.Multi-dimensional geometric Brownian motion (Black-Scholes model)In this case the SDEs read as(4.24)dxtj=bjxtjdt+xtj∑kSjkdwtk both bj and Sij are non-dynamical quantities. The obvious change of variable xi=eyj, also in the spirit of the quantum canonical transform, leads to the simplified version of the equations above(4.25)dytj=b˜jdt+∑kSjkdwtk⇒ysj=y0j+b˜js+∑kSikwsk where b˜j=bj−gjj2. It is then straightforward to compute expectation values via the solutions (4.25) and (4.21), (4.22)Et(xsj)=Et(eysj)=x0jebjs, and similarly, using the definition (4.19) and the change of variables we obtain:Et(xtixtj)=x0ix0je(bi+bj)tegijt.As in the previous example we find the propagator via the universal expressions (3.20), (3.24) and (4.25). From the solution of (4.25)wt=S−1(Δy−b˜t) here Δy=yt−y0. Then substituting the above in (3.24) we conclude that the propagator is (3.20)K(yt,y0|t)=e−12t(Δy−b˜t)Tg−1(Δy−b˜t)detS(2πt)M2. The propagator can also be expressed in terms of the vector field xt, recall yj=logxj.3.Feynman-Kac & quantum quenchesWe illustrate below how the Feynman-Kac formula (3.1), (3.11) can be utilized for the study of quantum quenches. We recall that a quantum quench is a process where a quantum system is arranged in an initial state, which is an eigenstate of a Hamiltonian H0, and then the system evolves in time under a different Hamiltonian H=H0+H1.The Feynman-Kac formula provides indeed the time evolution of an initial state f0(x)=f(x,0) at t=0 to a state f(y,t) under the Hamiltonian H=Lˆ as discussed in section 3. We consider here as an example the simple Hamiltonian H of M-harmonic oscillators, and as an initial profile function i.e. initial quantum state the ground state of the Hamiltonian H0=12∑jm∂xj2−∑j=1mΩj22xj2 that describes m<M harmonic oscillators:(4.26)f0(x)=exp(−12∑j=1mΩjxj2) with corresponding eigenvalue E0(m)=∑j=1mΩj2.The time propagator K associated to the Hamiltonian of M-harmonic oscillators is known and is given by the generalized Mehler formula. In fact, this formula can be easily obtained from expression (4.14), after setting yf=y,y0=x, Θ=0 and considering for simplicity Ω to be diagonal, we conclude that the propagator for the system of M-harmonic oscillators is given as:(4.27)K(y,x|t)=1(2π)M2∏j=1M(Ωjsinh(tΩj))12×exp(−12∑j=1M(xj2+yj2)Ωjcosh(tΩj)sinh(tΩj)+∑j=1MxjyjΩjsinh(tΩj)).Substituting K (4.27) and f0 (4.26) in (3.1) and performing the Gaussian integrals involved we conclude(4.28)f(y,t)=∏j=1me−tΩj2∏j=m+1M(1cosh(tΩj))12×exp(−12∑j=1mΩjyj2−12∑j=m+1MΩjcoth(tΩj)yj2). Consider now the behavior of the system after long enough time t≫1, i.e. when the system reaches equilibrium, then the state f(y,t) is given by the ground state of the Hamiltonian of M-harmonic oscillators(4.29)f(y,t)∼e−tE0(M)exp(−12∑j=1MΩjyj2).Of course the system we examine here is simple and the underlying SDE is easy to solve. When considering more complicated diffusion reaction systems, as will be discussed in the next section, the Feynman-Kac can be utilized provided that solutions of the SDE are available. The time evolution of more involved systems will be discussed in future works (see e.g. [30]).5Discrete integrable quantum systems & SDEsIn this section we explore some interesting links between discrete quantum systems and SDEs. The correspondence between the quantum equations of motion and the SDEs is discussed and typical examples of exactly solvable quantum systems and the associated SDEs are presented.To illustrate the main ideas of the quantum canonical transformation and the quantum equations of motion in connection with SDEs we present below typical examples of integrable quantum systems: the discrete NLS model and the Heisenberg (XXZ) quantum spin chain. As in the classical case integrability at the quantum level can also be described via the Lax pair (Lm,Am) which depends on quantum fields and a spectral parameter in general. The Lax pair satisfies the quantum auxiliary problem and hence the zero curvature conditiondLm(λ)dt=Am+1(λ)Lm(λ)−Lm(λ)Am(λ) providing the quantum equations of motion in analogy to Heisenberg's picture [16,21]. The Lax pair formulation ensures the existence of many conserved charges. Indeed, definet(λ)=tr0T0(λ) where T is the monodromy -a discrete path integral–is given as(5.1)T0(λ)=L0M(λ)L0M−1(λ)…L01(λ). Using the quantum zero curvature condition and assuming periodic boundary conditions it is shown in straightforward manner that: dt(λ)dt=0, i.e. t is the generator of the conserved quantities: t(λ)=∑kIkλkLet us clarify the index notation in the expressions above, the indices 0 (auxiliary) and m∈{1,…,M} (quantum) correspond to the position in the M+1 tensor productL0m(λ)=∑a,beab(0)⊗I(1)…⊗I(m−1)⊗lab(m)(λ)⊗…⊗I(M) where eab are in general N×N matrices with elements: (eab)cd=δacδbd. Notice that the quantum indices are suppressed in the expression for the monodromy (5.1) for brevity.The algebraic formulation of quantum integrability on the other hand is based on the existence of a quantum R-matrix that satisfies the Yang-Baxter equation [21,25]. This provides a rather stronger frame in the sense that guarantees the existence of many charges in involution, i.e. ensures the existence of a family of mutually commuting quantum charges: [t(λ),t(λ′)]=0. In this setting the L operator satisfies the fundamental algebraic relation [20,21],(5.2)R(λ1−λ2)(L(λ1)⊗I)(I⊗L(λ2))=(I⊗L(λ2))(L(λ1)⊗I)R(λ1−λ2) R(λ)∈End(V⊗V) and L(λ)∈End(V)⊗A; A is the underlying deformed algebra defined by (5.2). More specifically, equation (expressed in the index free notation) (5.2) acts on V⊗V⊗AR(λ)=∑a,b,c,dRab,cd(λ)eab⊗ecd⊗II⊗L(λ)=∑i,jeab⊗I⊗lab(λ)L(λ)⊗I=∑a,bI⊗eab⊗lab(λ), lij(λ)∈A and T(λ)∈End(V)⊗A⊗M, i.e. the monodromy is a tensor representation of (5.2). In both examples we are considering here the associated R-matrix is the Yangian or the trigonometric XXZ R matrix (see [20,21] and references therein).5.1The quantum discrete NLS hierarchyWe start our analysis with the DNLS model, with the corresponding Lax operator given by [14,18]Lj(λ)=(λ+NjzjZj1). Nj=Θj+zjZj. Using the fact that L satisfies (5.2) together with the quantum zero curvature condition one can also construct the A operators for the whole hierarchy [16]. In particular, the A-operators associated to the second and third conserved charges of the hierarchy is(5.3)Aj(1)(λ)=(λzjZj−10),Aj(2)(λ)=(λ2−zjZj−1λzj−zjNj+zj+1λZj−1−Zj−1Nj−1+Zj−2zjZj−1). Notice that the first charge gives the numbers of particles of the DNLS model (corresponds to a system of M harmonic oscillators), the second charge corresponds to the momentum of the model also known as the DST model [17], whereas the third charge is the DNLS Hamiltonian [18]. We shall examine below the second and third charges, identify the corresponding SDEs, and study their continuum limits, which produce the stochastic transport and stochastic heat equation correspondingly.5.1.1The DST modelLet us first focus on the second charge (momentum) is given by [14,17](5.4)H(1)=12∑j=1Mzj2Zj2+∑j=1M(cjzj−zj+1)Zj and via (5.2) one obtains: [zi,Zj]=−δij. The equations of motion can now be derived via the zero curvature condition or Heisenberg's equation and they read as(5.5)dzjdt=(cjzj−zj+1)+zj2Zj.Consider now the following map(5.6)zj↦xj,Zj↦∂xj then the Hamiltonian is expressed as:(5.7)H(1)=12∑j=1Mxj2∂xj2+∑j=1M(cjxj−xj+1)∂xj and we have considered periodic boundary conditions xj=xj+M. The Hamiltonian is apparently of the form (1.2), and the corresponding set of SDEs are given by(5.8)dxtj=(cjxtj−xtj+1)dt+xtjdwtj. If we compare the SDEs (5.8) with the quantum equations of motion (5.5) we observe that the last term in (5.5) is replaced by the multiplicative noise in (5.8). It is worth noting that in the special case that cj≫1 the second term in the drift is neglected and the SDEs (5.8) reduce to the ones of the multidimensional Black-Scholes model (4.24).Let us apply the quantum canonical transformation as described in section 2. Indeed, we define the new variables yj via (2.2)dyi=xi−1dxi⇒xi=Aieyi where Ai are integration constants. Then the Hamiltonian of the DST model can be re-expressed in terms of the new variables as a Hamiltonian with identity diffusion matrix(5.9)H(1)=12∑j=1M∂yj2+∑j=1M(Cj+Bjeyj+1−yj)∂yj, Cj=cj−12,Bj=−Aj+1Aj−1, and the corresponding set of SDEs aredytj=(Cj+Bjeytj+1−ytj)dt+dwtj. Connection with the quantum Darboux transforms introduced in [16] would also be a very interesting direction to pursue. Note that in [15,16] an alternative version of the quantum discrete NLS model is studied i.e. the so called q-Boson or quantum Ablowitz-Ladik model. Recall that in the classical case the Darboux-Bäcklund transformation [31,32] provides an efficient way to find solutions of integrable non-linear PDEs. In particular, the transformation connects solutions of the same or different integrable PDEs. At the quantum level the transformation connects different realizations of the underlying quantum algebra. The pertinent question is how this transformation affects the associated SDEs. For instance does the quantum Darboux transformation connect solutions of different SDEs? This is a significant open question, which needs to be systematically addressed. In fact, a novel generalization of the “dressing” method has been introduced for classical PDEs [33], whereas a similar formulation is suitably extended in the context of SPDEs [34]. For a relevant discussion on stochastic Bäcklund transformation see also [35].An interesting observation can be made on the connection of the DST model and the Toda chain. Let us also consider the adjoint operator of (5.9), which reads asH(1)†=12∑j=1M∂yj2−∑j=1Mb˜j(y)∂yj+∑j=1MB˜jeyj+1−yj B˜j=Bj−Bj−1. Then the self-adjoint operator H=12(H(1)+H(1)†) is expressed asH=12∑j=1M∂yj2+∑j=1MB˜jeyj+1−yj, which is nothing but the Hamiltonian of the Toda chain.Let us now derive the solution of the set of SDEs (5.8) introducing suitable integrator factors (we refer the interested reader to [4] on integrator factors in SDEs). Let us consider the general set of SDESdxtj=bj(xt)dt+xtjdwtj, for any drift b. We introduce the following set of integrator factors:(5.10)Fj(t)=exp(−∫0tdwsj+12∫0tds) and define the new fields: ytj=Fj(t)xtj, then one obtains a differential equation for the vector field y. Indeed, from (5.8), (5.10)(5.11)d(Fj(t)−1ytj)=dxtj⇒dytj=Fj(t)bj(F−1k(t)ytk)dt and we bear in mind that the usual calculus rules apply for the LHS of the first equation above.Let us focus now on the SDEs of the DST model (5.8); in this case (5.11) leads to the ODE:dytdt=A(t)yt, where the M×M matrix A is given as (we have set cj=1 in (5.8))A(t)=I−B=∑j=1M(ejj−Bj(t)ejj+1), and we define(5.12)Bj(t)=Fj(t)Fj+1−1(t)=exp(wtj+1−wtj). The formal solution of the latter linear problem is the path ordered exponential (monodromy):yt=Pexp(∫0tA(s)ds)y0, which can be expressed as a formal series expansion:Pexp(∫0tA(s)ds)=∑n=0∞∫0t∫0tn…∫0t2dtndtn−1…dt1A(tn)A(tn−1)…A(t1),t≥tn≥tn−1…≥t2. Note that A is an upper triangular local matrix, thus products of the matrices preserve the triangular structure, but not the locality. In fact, this formal series expansion provides time-like non-local charges of the theory in analogy to the non-local charges associated to representations of deformed algebras in quantum and classical integrable models.It will be instructive to consider the DST model (5.4) and the respective SDEs (5.8) in the continuum limit. Let us set cj=1 in (5.7), then after suitably re-scaling the fields and considering the thermodynamic limit M→∞,δ→0 (δ∼1M) we obtain(5.13)xtj→φ(x,t)xtj+1−xtjδ→∂xφ(x,t)δ∑jfj→∫dxf(x)wtj→W(x,t), a brief discussion on the representation of the Brownian sheet W(x,t) (space-time white noise) is presented in the last section. In the continuum limit the Hamiltonian of the DST model (5.7) becomes the Hamiltonian of an 1+1 dimensional quantum field theoryHc(1)=∫dx(12φ2(x)φˆ2(x)−∂xφ(x)φˆ(x)), where the issue of the ultra locality of underlying algebra has been taken into consideration: [φ(x),φˆ(y)]=δ(x−y), (φˆ(x)=∂∂φ(x)) and the SDEs (5.8) become the stochastic transport equation with multiplicative noise:∂tφ(x,t)=−∂xφ(x,t)+φ(x,t)W˙(x,t) where W˙(x,t)=dWdt. It is worth noting that in general, the continuum limit of SDEs will lead to non-local SPDEs, given that the diffusion matrix is generically a full matrix, and the drift describes in principle non local interactions. However, in the case of diagonal (or slightly off diagonal) diffusion matrices and local drift one obtains local SPDEs as the example described above.5.1.2The DNLS modelWe come now to the next member of the DNLS Hierarchy i.e. the quantum DNLS model (see e.g. [18] and [36] and references therein). We directly express the Hamiltonian in terms of differential operators after taking into consideration the map (5.6):(5.14)H(2)=12∑j=1M(xj(xj+1−xj)∂xj2+xj+12∂xj∂xj+1−(xj−2xj+1+xj+2)∂xj) and we read the diffusion matrix with entries(5.15)gjj=xj(xj+1−xj),gjj+1=gj+1j=xj+12,gij=0,|i−j|>1, as well as drift components bj(x)=−12(xj−2xj+1+xj+2), which in the continuum space limit discussed below provides a second derivative of the field.Using the dictionary described above (5.13) we can readily write down the continuum limit of the DNLS Hamiltonian(5.16)Hc(2)=12∫dx(φ2(x)φˆ2(x)−∂x2φ(x)φˆ(x)) as well as the continuum limit of the DNLS SDEs, which is nothing but the stochastic heat equation with multiplicative noise:(5.17)∂tφ(x,t)=−12∂x2φ(x,t)+φ(x,t)W˙(x,t). The latter equation is solvable, and can also be mapped to the stochastic viscous Burgers equation (see for instance [22]). Indeed, we set: φ=eh,u=∂xh then (5.17) becomes,(5.18)∂th(x,t)=−12∂x2h(x,t)−12(∂xh(x,t))2+W˙(x,t)∂tu(x,t)=−12∂x2u(x,t)−u(x,t)∂xu(x,t)+∂xW˙(x,t). The latter is precisely the viscous Burgers equation with additive noise.5.2The XXZ quantum spin chainWe briefly discuss another prototype integrable model, the XXZ spin 12 quantum spin chain and the associated SDEs. The XXZ Hamiltonian has the familiar form(5.19)H=−14∑j=1M(σjxσj+1x+σjyσj+1y+Δσjzσj+1z)+ξ8I σx,σy,σz are the familiar 2×2 Pauli matrices, which form a two dimensional representation of the sl2 algebra, Δ=cosh(μ), μ the anisotropy parameter, ξ is an arbitrary constant. The sl2 algebra has generators Sj, j∈{1,2,3} that satisfy:(5.20)[Si,Sj]=2iϵijkSk. The spin S∈R representation of the algebra in terms of differential operators is given as(5.21)S1↦−(x2−1)∂x+S(x+x−1)S2↦−i((x2+1)∂x+S(x−1−x))S3↦2x∂x. In the case where S is an integer or half integer one obtains the n=2S+1 dimensional representation of sl2 with a corresponding orthonormal basis given as: eα=xS−α+1,α∈{1,2,…,n}. This n dimensional space is equipped with an inner product in the unit circle (see also relevant discussion in [37]).Motivated by the form of the XXZ Hamiltonian we replace the Pauli matrices in (5.19) with the spin 12 two dimensional representation of sl2 expressed in terms of differential operators (5.21). Then (5.19) becomes a typical diffusion reaction Hamiltonian(5.22)H=∑j(12(xj+12+xj2−2Δxjxj+1)∂xj∂xj+1+14(xj2(xj+1−1+xj−1−1)−(xj+1+xj−1))∂xj−18(xjxj+1−1+xj−1xj+1)+ξ2xj2∂xj2+ξ2xj∂xj) from which one immediately reads the diffusion matrix g as well as the drift b, and thus the associated set of SDEs (1.3). Recall also g=σσT one then can identify σ and hence σ−1 in order to apply the results of section 2. Notice that last two terms in (5.22) correspond to the last term of (5.19) after substituting the identity with ∑j(σjz)2, given that (σjz)2=I2×2.The Hamiltonian above can also be expressed in terms of the variables yj,∂yj (xj=eyj) as(5.23)H=∑j(cosh(yj−yj+1)−Δ)∂yj∂yj+1+ξ2∂yj2+ξ2∂yj+12∑j(sinh(yj−yj+1)+sinh(yj−yj−1))∂yj−14∑jcosh(yj−yj+1). Both expressions (5.22), (5.23) may be useful especially when considering the continuum limit of the model and making connections with certain quantum field theories, this is an important aspect, which however will be studied separately. Relevant recent results on stochastic XXZ spin chain are presented in [38].•The Ising modelTwo special cases of interest arise as suitable limits of the XXZ spin chain: the XX model when Δ=0 which describes the free fermion point of the sine-Gordon model, and the Ising model when Δ→∞, allow also ξ=−2ξˆΔ. Let us focus here on the simple case of the Ising model in the presence of an external longitudinal magnetic field, emerging directly from (5.23) (we have considered the open spin chain for convenience):H=ξˆ2∑j=1M−1∂yj2+12∑j=1M−1∂yj∂yj+1+ξˆ2∑j=1M∂yj+a2∂yM2, where a2+1a=ξˆ. The diffusion matrix is now constant and it reads as:g=∑j=1M−1(ξˆejj+ejj+1+ej+1j)+aeMM⇒σ=1a(a∑j=1Mejj+∑j=1M−1ejj+1) hence the SDEs are given asdytj=ξˆ2dt+1a(adwtj+dwtj+1),1≤j≤M−1,dytM=ξˆ2dt+adwtM. The solutions of the associated SDEs are then easily determined:(5.24)ysj−y0j=ξˆ2s+1a(awsj+wsj+1)1≤j≤M−1ysM−y0M=ξˆ2s+awsM. The variables xj=eyj are also immediately identified via the solution above. Expectation values can readily be computed using the solutions above as described in section 3 (see also the example in section 4.1).The addition of a transverse magnetic field, e.g. h=ξ2∑j(S1−iS2), to the free Hamiltonian leads to a set of SDEs with the same constant diffusion matrix as above, but with drift components proportional to eytj. The solution of the associated SDEs in this case is a more involved problem, but it is the first task in order to be able to identify expectation values.As a general remark let us note that in the case of exactly solvable models, such as for instance the system of M-harmonic oscillators, the DNLS and XXZ models, the spectrum and eigenstates of Hamiltonians are available via e.g. the Bethe ansatz formulation [20,21] (see also [39–41]). Having a complete basis of eigenstates available one can then express the solution of the time evolution problem in terms of this basis. This is rather the standard way for studying the time evolution problem when dealing with quantum mechanical systems with a complete basis of the Hamiltonian eigenstates available. However, in the case of non-Hermitian Hamiltonians, as are the general operators (1.2), the challenging issue of the existence of a complete basis needs to be addressed. From this point of view the study of the time evolution problem via the path integral approach as described in section 3 offers a more suitable frame.5.3Quantum Darboux transformation & defectsIt will be useful for our purposes here to consider the DST model in the presence of integrable local defects [36], [42], [43]. We will derive the associated quantum Darboux-Bäcklund transform for the DNLS model as a defect matrix and shall also identify the corresponding SDEs. The monodromy in this case is modified as (the auxiliary index is suppressed below for brevity)T(λ)=LM(λ)…Dm(λ)…L1(λ) where we consider the defect matrix D to be of the generic formDm(λ)=(λ+αmβmγmλ−αm). The matrix D will be explicitly derived via the zero curvature condition on the defect point i.e. the t part of the quantum Darboux-Bäcklund transformation [16].In the algebraic scheme preserving integrability requires that D also satisfies (5.2), hence it turns out that α,β,γ are the generators of sl2. Integrability in the weaker sense via the Lax pair description on the other hand leads to the zero curvature condition for D (the t-part of the Bäcklund transformation (5.25))(5.25)dDm(λ)dt=A˜m(λ)Dm(λ)−Dm(λ)Am(λ) where A˜m in general isA˜m(λ)=(λz˜mZ˜m−10), but in our case A˜m=Am+1. We first focus on the Lax pair description and we assume that Aj are given in (5.3) for all the sites of the chain, that is we assume the existence of the fields zm,Zm. Then solving (5.25) we obtain the quantum Darboux-Bäcklund relations:(5.26)βm=zm−z˜mγm=Z˜m−1−Zm−1αm2=ζ−(zm−z˜m)(Z˜m−1−Zm−1), where the expression for α follows from requiring the quantum determinant of the matrix D is a constant. In our case here we treat D as a defect matrix: z˜m=zm+1,Z˜m−1=Zm. The corresponding conserved charge in the presence of the local defects then reads as [43](5.27)H=12∑j≠mzj2Zj2+(∑j≠mcjzj−∑j≠m,m−1zj+1)Zj−zm+1Zm−1−βmZm−1−γmzm+1+αm22, then via relations (5.26) and recalling the map (5.6) the Hamiltonian is rewrittenH=12∑j≠mxj2∂xj2+(∑j≠mcjxj−∑j≠m,m−1xj+1)∂xj+12(xm+1∂xm−1−xm∂xm−1−xm+1∂xm−xm∂xm+ζ). One can immediately read the corresponding SDEs, indeed for j≠m,m−1 they are given by (5.8) anddxtm−1=(cm−1xtm−1+12(xtm+1−xtm))dt+xtm−1dwtm−1dxtm=−12(xtm+1+xtm)dt. We conclude from the equations above that the σ matrix in this case is not invertible as it has one zero eigenvalue, in particular σ=diag(x1,x2,…xm−1,0,xm+1,…xM), therefore the setting described in subsection 3.2.1 can be implemented.Let us also employ the algebraic setting, in this case D is a representation of the algebra (5.2), therefore as mentioned α,β,γ are elements of sl2 expressed in the Chevalley-Serre basis:β=12(S1−iS2),γ=12(S1+iS2),α=12S3. In the spin S representation (5.21) they are given by the map:βm↦−xn2∂xm+Sxm,γm↦∂xm+Sxm−1αm↦xm∂xm, and the Hamiltonian (5.27) becomesH=12∑j≠mxj2∂xj2+(∑j≠mcjxj−∑j≠m,m−1xj+1)∂xj−xm+1∂xm−1+(xm2∂xm−Sxm)∂xm−1−(∂xm+Sxm−1)xm+1+12xm2∂xm2+12xm∂xm. The corresponding SDEs are given as before via (5.8) for j≠m,m−1, anddxtm−1=(cm−1xtm−1−xtm+1−Sxtm)dt+xtm−1dwtm−1+2xtmdwtmdxtm=(12xtm−xtm+1)dt+xtmdwtm+2xtmdwtm−1. Notice that the diffusion matrix is not diagonal anymore and interestingly the σ matrix is invertible, therefore the quantum canonical transformation can be implemented subject to certain modifications associated to the presence of the defect. This model is integrable in the strong sense, and the Bethe ansatz techniques can be applied so the spectrum and eigenstates are available. It is also worth noting that in the algebraic setting one obtains deformed A operators around the defect point:Am(λ)=(λβm+zm+1Zm−10),Am−1=(λzm+1γm+Zm−10). The operators above become the usual bulk ones (5.3) only in the continuum limit, via analyticity conditions implemented around the defect point. These analyticity conditions provide exactly Bäcklund type relations for the fields [43]. The classical equations of motion, which structurally coincide with the quantum ones are available for the model in the presence of local defects.6Generalizations & commentsWe may now discuss certain generalizations associated to discrete quantum systems as well as quantum field theories. Let us first consider the obvious extension from SDEs associated to vector fields x to SDEs for generic matrix or tensor fields. Indeed, let us consider the tensor field Y with components Yi1i2...id,d∈N, then the generator Lˆ0 is expressedLˆ0=gi1...id;j1...jd(Y)∂2∂Yi1...id∂Yj1...jd+bi1...id(Y)∂∂Yi1...id,ik∈{1,…,M}. gi1...id;j1...jd and bi1...id are the generalized tensor diffusion coefficients and drift components respectively, and we use the standard convention where repeated indices are summed. In the d=2 case for instance the generator above is associated to the partition function of a matrix model (see e.g. review articles [44,45] and references therein, and [46] in relation to integrable models).The SDEs associated to the generator -via the generalized Itô formula- are then given asdYti1...id=bi1...id(Yt)dt+σi1...id;j1...jd(Yt)dwtj1...jd provided that the tensor Wiener processes satisfy:dwti1...iddwtj1...jd=δi1j1…δidjddt, and thus gi1...id;j1...jd=σi1...id;k1...kdσj1...jd;k1...kd. In the continuum limit (M→∞), which is of particular interest when studying quantum/statistical field theories at finite temperature and SPDEs, the tensor fields become continuous space random fields depending on t and the continuum space parameters x∈Rd, i.e.Yti1...id→φ(x,t),wti1...id→W(x,t) and the SDEs become SPDEs; W(x,t) are the multi-dimensional Wiener fields or Brownian sheets (white space-time noise). This description is also in line with the notion of stochastic quantization in quantum field theory (see for example [47,48]).We come now to the issue of representing the Brownian sheet (we refer the interested reader to [5]). Let ek,k∈N be an orthonormal basis of a separable Hilbert space H consisting of eigen-vectors of an operator Q with corresponding eigenvalues λk. Then the H valued stochastic process Wt,t∈[0,t] (a Q-Wiener process) is represented asW(t)=∑k∈Nλkβk(t)ek where βk(t) are independent real valued Brownian motions. Let us present as an example the one spatial dimensional case, then the Wiener field, which is periodic and square integrable in [−L,L] is(6.1)W(x,t)=Lπ∑n≥11n(Xt(n)cos(nπxL)+Yt(n)sin(nπxL)), Xt(n),Yt(n) are independent Brownian motions. In this case we have used as the operator Q the inverse Laplacian: Q=(∂x2)−1. The representation of higher dimensional Brownian sheets becomes a more complicated issue involving multidimensional Fourier transforms. We have already considered two fundamental examples i.e. the DST and DNLS models that in the continuum limit led to the stochastic transport and the stochastic heat equations respectively.The computation of expectation values via the solution of the associated SDEs for the DST (DNLS) and XXZ models is one of our main future goals. We have already at our disposal a formal series solution for the DST model, whereas solutions for the XXZ SDEs (or special cases such as the XX model) need to be derived using for instance the change of variables introduced in section 2. Moreover, we have discussed the issue of space like defects as a way to tackle the generic case where detσ=0, and we have provided a particular example i.e. the DST model in the presence of local defects. A relevant significant issue is the effect of non-trivial space and time like boundary conditions on the form of the generator of the Itô process as well as the form of the SDEs. 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