随机分析与离散量子系统的相互作用

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"随机分析和离散量子系统" 本文深入探讨了随机分析理论与离散量子力学系统之间的相互联系。研究的核心内容包括了费曼-卡克公式和卡梅隆-马丁-吉尔萨诺夫定理这两个重要的数学工具。费曼-卡克公式是量子力学中的一个基本概念，它允许通过路径积分的方法来计算量子系统的演化。而卡梅隆-马丁-吉尔萨诺夫定理则是概率论中的一个关键定理，它在随机过程的分析中扮演着重要角色。文章指出，当涉及到一般动态扩散系数时，可以利用量子规范变换来计算时间传播器。量子规范变换是一种在量子力学中改变系统哈密顿量而不改变物理结果的方法，这对于理解和处理复杂系统至关重要。通过对路径积分的显式计算，作者们得到了与扩散系数和漂移形式无关的相关度量的通用表达式。这个发现揭示了一个有趣的现象：漂移在超对称量子力学的框架下实际上起到了超电势的作用。超对称量子力学是量子力学的一个扩展，其中包含了一对相互对偶的粒子，这一理论在理论物理和数学物理中都有广泛的应用。此外，文中还讨论了一些具体的示例，如Ornstein-Uhlenbeck过程和多维Black-Scholes模型。Ornstein-Uhlenbeck过程常用于描述金融市场的随机波动，而Black-Scholes模型则是期权定价的经典理论。这些例子有助于读者更好地理解随机分析和量子力学概念在实际问题中的应用。进一步，作者提出了量子可积系统的实例，比如量子离散非线性层次结构（DNLS）和XXZ自旋链。DNLS和XXZ自旋链是量子力学中具有特殊性质的系统，它们允许精确解。通过这些系统，作者展示了量子（可积）系统与随机微分方程（SDE）之间的具体联系。例如，NLS层次的前两个成员的SDE在连续极限下分别对应随机传递和随机热方程。这显示了量子力学系统在连续极限下的行为可以被经典随机过程所描述。在离散NLS模型中，量子Darboux矩阵被计算出来并作为缺陷矩阵出现，同时导出了相关的SDE。Darboux矩阵在量子力学中用于构造可积系统的解析解，而缺陷矩阵则描述了系统的不连续性或边界条件。这种计算方法为理解和模拟量子系统在随机环境中的行为提供了新的视角。这篇论文通过深入研究随机分析与离散量子系统的相互作用，不仅丰富了我们对这两个领域的理解，也为未来跨学科的研究开辟了新的途径。无论是对于理论物理学家、数学家还是金融工程师，这篇文章都提供了宝贵的洞察力和实用工具。

A. Doikou et al. / Nuclear Physics B 945 (2019) 114658 7



(y) (2.12)

where D

= ∂

(y) is the covariant derivative. Interestingly

b turns out to be the

super-potential that produces V, V

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

• Remark 2: Riccati equation & Girsanov’s theorem

Note that the Riccati equation (2.11)e

merging from the identiﬁcation of the effective poten-

tial 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 inter-

ested reader to [2–8]) from the PDE point of vie

w. This will be even more transparent in the

next section from the explicit path integral computation.

2.1. Equivalence of SDEs

The generator

(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:



j=1

∂

∂y



j=1

(y)

∂

∂y

(2.13)

is obviously obtained via the following simpler SDE

b(y

)dt +dw

, (2.14)

where recall

b is deﬁned in (2.3), (2.4).

The main aim no

w 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(y

) and add Itô’s

correction i.e.

σ(y

)dy

dσ(y

)dy

=σ(y

)

b(y

)dt +

dσ(y

)dy

+σ(y

)dw

. (2.15)

Recall the connection between Itô and Stratonovich calculus:

f(y

) ◦ dy

=f(y

)dy

df (y

)dy

and dy

=δ

dt (2.16)

where f(y

) ◦dy

refers to Stratonovich calculus, and f(y

)dy

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(y

)

b(y

)dt +

dσ(y

)dy

=b(y

)dt,

then (2.15) becomes

σ(y

) ◦ dy

=b(y

)dt +σ(y

)dw

We now set

=σ(y

) ◦ dy

(2.17)

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

and we immediately recover the original SDE

=b(x

)dt +σ(x

)dw

after solving y = y(x) through (2.17). Integration of (2.17)in the Stratonovich calculus frame –

i.e. the usual calculus rules apply– yields







) ◦ dy



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

dσ

−1

(x)dxat

both sides of the equation, and use (2.16). Then setting dy =σ

−1

(x) ◦dxwe 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], (re

garding the Lamperti transform we

refer the interested reader to [3,4,24]).

3. The path integral

3.1. Identity diffusion matrix

We have established in the previous section that if the generic dynamical diffusion matrix

is invertible then

L 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 T

rotter 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

L using the standard discrete time frame and mak

various interesting connections. More precisely:

1. We mak

e 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 sho

w 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 gen-

eralized in the next subsection where dynamical diffusion coefﬁcients are also considered.

3. We sho

w that the drift, which is essentially a gauge ﬁeld plays the role of the super-potential

with

being super-partners in the usual super-symmetric quantum mechanics sense,

conﬁrming the ﬁndings of the previous section.

Our starting point is the time e

volution equation (1.6), (1.7), (2.1):

∂

f(y,t)=

†

f(y,t),

we then explicitly compute the propagator K(y

, y

|t, t



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