Time Evolution in Quantum Systems
and Stochastics
Anastasia Doikou, Simon J. A. Malham, and Anke Wiese
Abstract The time evolution problem for non-self-adjoint second order differential
operators is studied by means of the path integral formulation. Explicit computation
of the path integral via the use of certain underlying stochastic differential equations,
which naturally emerge when computing the path integral, leads to a universal
expression for the associated measure regardless of the form of the differential
operators. The discrete non-linear hierarchy (DNLS) is then considered and the
corresponding hierarchy of solvable, in principle, SDEs is extracted. The first couple
members of the hierarchy correspond to the discrete stochastic transport and heat
equations. The discrete stochastic Burgers equation is also obtained through the
analogue of the Cole–Hopf transformation. The continuum limit is also discussed.
Keywords Time evolution · Non-self-adjoint operators · Quantum integrability ·
Hierarchy of SDEs
One of our main aims here is the solution of the time evolution problem associated
with non-self-adjoint operators using the path integral formulation. We consider
the general second order differential operator ˆ
L 0 , and the associated time evolution
problem:
−∂ t f (x, t) = ˆ
Lf (x, t) =
ˆ
L 0 + u(x)
f (x, t),
(1)
ˆ
L 0 =
1
2
M
i,j =1
g ij (x)
∂ 2
∂x i ∂x j
+
M
j =1
b j (x)
∂
∂x j
,
g(x) = σ (x)σ
T (x), (2)
where the diffusion matrix g(x) and the matrix σ (x) are in general dynamical
(depending on the fields x j ) M ×M matrices, while x and the drift b(x) are M vector
A. Doikou () · S. J. A. Malham · A. Wiese
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK
e-mail: a.doikou@hw.ac.uk; s.j.a.malham@hw.ac.uk; a.wiese@hw.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_49
523
and Stochastics
Anastasia Doikou, Simon J. A. Malham, and Anke Wiese
Abstract The time evolution problem for non-self-adjoint second order differential
operators is studied by means of the path integral formulation. Explicit computation
of the path integral via the use of certain underlying stochastic differential equations,
which naturally emerge when computing the path integral, leads to a universal
expression for the associated measure regardless of the form of the differential
operators. The discrete non-linear hierarchy (DNLS) is then considered and the
corresponding hierarchy of solvable, in principle, SDEs is extracted. The first couple
members of the hierarchy correspond to the discrete stochastic transport and heat
equations. The discrete stochastic Burgers equation is also obtained through the
analogue of the Cole–Hopf transformation. The continuum limit is also discussed.
Keywords Time evolution · Non-self-adjoint operators · Quantum integrability ·
Hierarchy of SDEs
One of our main aims here is the solution of the time evolution problem associated
with non-self-adjoint operators using the path integral formulation. We consider
the general second order differential operator ˆ
L 0 , and the associated time evolution
problem:
−∂ t f (x, t) = ˆ
Lf (x, t) =
ˆ
L 0 + u(x)
f (x, t),
(1)
ˆ
L 0 =
1
2
M
i,j =1
g ij (x)
∂ 2
∂x i ∂x j
+
M
j =1
b j (x)
∂
∂x j
,
g(x) = σ (x)σ
T (x), (2)
where the diffusion matrix g(x) and the matrix σ (x) are in general dynamical
(depending on the fields x j ) M ×M matrices, while x and the drift b(x) are M vector
A. Doikou () · S. J. A. Malham · A. Wiese
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK
e-mail: a.doikou@hw.ac.uk; s.j.a.malham@hw.ac.uk; a.wiese@hw.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_49
523