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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 945 (2019) 114658

www.elsevier.com/locate/nuclphysb

Stochastic analysis & discrete quantum systems

Anastasia Doikou

∗

, Simon J.A. Malham, Anke Wiese

School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

Received 23

October 2018; received in revised form 11 April 2019; accepted 24 May 2019

Available

online 31 May 2019

Editor: Hubert

Saleur

Abstract

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 quan-

tum

Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are

derived.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

addresses: a.doikou@hw.ac.uk (A. Doikou), s.j.a.malham@hw.ac.uk (S.J.A. Malham), a.wiese@hw.ac.uk

(A. Wiese).

https://doi.org/10.1016/j.nuclphysb.2019.114658

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

2 A. Doikou et al. / Nuclear Physics B 945 (2019) 114658

1. Introduction

Let

be a generic second order differential operator, and suppose f(x, t) satisﬁes the time

evolution equation:

−∂

f(x,t)=

Lf (x,t)=



+u(x)



f(x,t), (1.1)



i,j=1

(x)

∂

∂x



j=1

(x)

∂

∂x

,g(x) = σ(x)σ

(x) (1.2)

where the diffusion matrix g(x) and the matrix σ(x) are in general dynamical (depending on the

ﬁelds x

) M ×M matrices, while x and the drift b(x) are M vector ﬁelds with components x

respectively, and

denotes usual transposition. Within the quantum mechanics framework

the

L 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 e

xplicitly calculated via a

time discretization scheme, and can be physically interpreted as the probability of the system to

progress from an initial state conﬁguration x

at time t

to a ﬁnal conﬁguration x

at time t

. In

the statistical physics language the imaginary time propagator provides the partition function of

a given statistical system at temperature T ∼

(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

=b(x

)dt +σ(x

)dw

, (1.3)

which yields the generator

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. wis

also an M-vector with components w

, which are M-independent Wiener processes. Let us now

recall the deﬁnition of the Wiener process (Brownian motion) (see e.g. [2–8]):

• Wiener process

The one dimensional (or scalar) W

iener process w

is a stochastic process with the following

properties:

1. For s<

t the increment w

− w

is a Gaussian with mean zero E



− w



= 0, and

variance E



−w

)



=t −s. Moreover , the increments associated to disjoint intervals

are independent.

2. w

is a continuous function.

3. The process starts at t =0, i.e. w

=0.

In the general case of M -dimensional W

iener processes w

is a vector ﬁeld w



, ...w



; the increments w

− w

are Gaussians with mean zero and E



−

)(w

− w

)



= (t − s)δ

, which loosely speaking (in the sense of averaging or as a

scaling argument) can be read as: dw

=δ

dt.

Note that although w

is continuous, is nowhere differentiable.

As pointed out the

operator and the associated time evolution equation can be obtained via

Itô’s formula, which is brieﬂy reviewed below:

A. Doikou et al. / Nuclear Physics B 945 (2019) 114658 3

• Itô’s formula

Consider a function f(x, t), where x

satisﬁes (1.3) then

df (x,t)=

∂f

∂t

dt +



∂f

∂x



i,j

∂

∂x

. (1.4)

Notice the appearance of the last term when formulating the differential above. Indeed, due to

the fact that dx

satisfy the SDE (1.3), and dw

=δ

dt, the last term is not negligible.

Using (1.3) and the scaling argument for the Wiener processes, expression (1.4) becomes

df (x,t)=



∂



f(x,t)dt +



i,j

(x)

∂f (x,t)

∂x

, (1.5)

providing the generator

(1.2), which can be thought of as a quantum mechanical Hamil-

tonian describing M interacting particles, or alternatively as an M -dimensional quantum

mechanical Hamiltonian. The equation ∂

= 0is obtained after taking the expectation

value of (1.5). For a rigorous proof of (1.5) and detailed discussions on stochastic differen-

tial 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 re

view 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 inte

gral 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

is not in general self-adjoint (Hermitian), therefore we should also introduce

the adjoint operator deﬁned for any suitable function f(x, t) as

†

f(x,t)=



i,j=1

∂

∂x



(x)f (x,t)



−



j=1

∂

∂x



(x)f (x,t)



. (1.6)

Hence, two distinct time evolution equations emerge:

• Fokker-Plank equation

For t

≥t

the Fokker-Plank or Kolmogorov forward equation is given by

∂

f(x,t

) =

†

f(x,t

) (1.7)

with known initial condition f(x, t

) =f

(x).

• Kolmogorov backward equation

For t

≤t

the Kolmogorov backward equation is given by

−∂

f(x,t

) =

f(x,t

)

with known ﬁnal condition f(x, t

) =f

(x).

To solv

e 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

L operator

4 A. Doikou et al. / Nuclear Physics B 945 (2019) 114658

with non-constant (dynamical) diffusion coefﬁcients represents a particular difﬁculty. More pre-

cisely, 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 no-

tion of quantum canonical transformation and reduce the operator

L to a much simpler form

with constant dif

fusion coefﬁcients, but with an induced/effective drift. Moreover, by consider-

ing both

L and a “complementary” operator

–it will be deﬁned later in the text– we show that

the drift plays the role of a super-potential, whereas

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,

L =



j=1

+V(y)

where D

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

L 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

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 e

xplicitly the generic path integral with dynamical diffusion coefﬁcients in

subsection 3.2 by requiring that the ﬁelds involved satisfy discrete time SDEs. This key assump-

tion, 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 le

vel

of PDEs is also conﬁrmed at the microscopic level of the path integral. It is worth emphasiz-

ing that in this framework solutions of the underlying SDEs are required in order to compute

time e

xpectation 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. T

ypical 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 objecti

ve 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 tw

o 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 solv

ed 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

A. Doikou et al. / Nuclear Physics B 945 (2019) 114658 5

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.

2. The quantum canonical transformation

As has been pointed out the formulation of the path integral solution in the case of a generic

L 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 dif

fusion 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

xplicit computation of the general path integral, and the derivation of the Feynman-Kac formula.

Indeed, we will sho

w in what follows that the general

L operator can be brought into the less

involved form:

L =



j=1

∂

∂y



j=1

(y)

∂

∂y

+u(y) (2.1)

with an induced drift

b(y). This can be achieved via a simple change of the parameters x

, which

in the Riemannian geometry sense is nothing but a change of frame. Indeed, let us introduce a

new set of parameters y

such that:



−1

(x)dx

, det σ = 0, (2.2)

then

L can be expressed in the form (2.1), and the induced drift components are given as

(y) =



−1

(y)b

(y) +



j,l

(y)∂

−1

(y). (2.3)

Bearing also in mind that



−1

=δ

, we can write in the compact vector/matrix notation:

b(y) =σ

−1

(y)



b(y) −

(∇

(y))



, ∇



∂

,...,∂



(2.4)

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 rele

vant for our purposes here, especially when considering the M-dimensional case,

M →∞i.e. in a 1 +1 ﬁeld theoretical setting. Let X

, P

be elements of the Heisenberg-Weyl

algebra:





=iδ

, (2.5)

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

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