1. Generating Functional Approach for Stochastic Differential Eqn.
(Martin Siggia Rose Formalism)
Arkajyoti Manna∗
Ramakrishna Mission Vivekananda University
Belur,India
November 11, 2015
Abstract
In this paper we have discussed the generating functional approach due to Martin-Siggia-Rose to construct
the corrrelation function of the field variables of a system which is governed by the Langevin Eqn. with Gaussian
white noise.First,we discuss how can one correspond the stochastic correlation function to the usual correlation
function in QFT.Then we set up the Martin-Siggia-Rose generating functional from which we can calculate the
correlation function.Finally we find the correlation function for the Ornstein-Uhlenbeck process.
1 Introduction
Stochastic differential equations are basic tool for describ-
ing a system with noise.These euations can also be recasted
as field theoretic generating functional from Functional
formalism.Martin-Siggia-Rose provided an approach how
to construct a generating functional from the dynam-
ics(with noise) given.The Parisi-Wu stochastic quantiza-
tion enables us to calculate the correlation directly from
the generating functional.
2 Parisi-Wu approach
Parisi and Wu proposed the following method to show
that the correlation function of a stochastic process can be
given by the quantum mechanical generating functional(in
some limit).
1.They assume that the system is governed by the
Langevin eqn.
∂φ(x, τ)
∂τ
= −
δS[φ]
δφ
+ η(x, τ) (1)
where η(x, τ) is the gaussian white noise and τ is the
stochastic 5-th time. 2.Now they take the limit τ → ∞ in
the stochastic correlation function.That is
lim
τ→∞
φ(x1, τ)φ(x2, τ) η = (2)
Dφ[φ(x1)φ(x2)]exp(−S[φ])
Dφexp(−S[φ])
To show this Parisi and Wu introduce the probability
P(φ, τ) of having field configuration φ(x, τ) at τ and ex-
press the time evolution of it as a Schrodinger eqn. The
evolution can be given by Fokker-Planck eqn.
∂P(φ, τ)
∂τ
=
δ2
P
δφ2
+
δ
δφ
P
δS
δφ
(3)
which is also given by
HF P ψ =
∂ψ
∂τ
(4)
HF P =
δ2
δφ2
−
1
4
δS
δφ
+
1
2
δ2
S
δφ2
ψ(φ, τ) = P(φ, τ)eS[φ]/2
Parisi and Wu showed that
lim
τ→∞
P(φ, τ) = exp(−S[φ]) (5)
Once this has been established we can safely construct the
generating functional in the usual way and then take the
limit.
3 Martin Siggia Rose generating
functional
First we observe the identity of delta functional
Z[0] = 1 =
j
Dφj(τ)δ
∂φj
∂τ
+
∂H[φ, π]
∂φj
− η(x, τ) (6)
×det
∂
∂τ
δ(τ − τ )δjk +
∂2
H
∂φj∂φk
δ(τ − τ )
∗arkajyoti1@live.com

2. And now we expand the δ functional in terms of another
auxiliary field ˜φ(x, τ) and write the generating functional
in terms of two fields.The MSR generating functinal for
two sources is then given by
Z[J, ˜J]η =
j
Dφj(x, τ)D ˜φj(x, τ) (7)
×exp d4
xLeff (φj, ˜φj) exp


j
d4
xdτ(Jφj + ˜J ˜φj)


where the effective Lagrangian can be given by
Leff =
j
dτ i˜φj(τ)(∂τ φj + ∂jH) − T|˜φj(τ)|2
(8)
+trln(Tjk(τ, τ ))
Here the generating functional is already averaged over
the noise distribution. Now all the correlation function be-
tween the solution of the Langevin eqn. can be constructed
from the above functional by usual method we do and take
the limit τ → ∞.And as Z[0] is already normalised to unity
so we don’t have to normalize Z[J, ˜J].Here the auxiliary
field ˜φ(x, τ) is unphysical.One can extend this approach to
systems that is governed by other than Langevin eqn.
4 The Ornstein-Uhlenbeck Pro-
cess
Now we can write the generating functional for single par-
ticle mechanics.So
Z[J, ˜J]η = Dx(τ)D˜x(τ)
× exp dτ(Jx(τ) + ˜J ˜x(τ)) + L(x, ˜x)
where the Lagrangian is given by-
L(x, ˜x) = dτ i˜x(τ)(∂τ x(τ) + ∂xH) − |˜x(τ)|2
(9)
Here we neglect the Jacobian for now.
The Ornstein-Uhlenbeck process is one of the stochastic
processes that can be solved exactly.The SDE of Ornstein-
Uhlenbeck process is given by
∂τ x(τ) + αx(τ) − η(τ) = 0, α > 0 (10)
Where η is a gaussian noise.Now as the auxiliary variable
˜x(τ) is unphysical, we set ˜J = 0.So the generating func-
tional for this process will be
Z[J, ˜J = 0] = D[x(τ)]D[˜x(τ)] (11)
×exp dτ(i˜x(∂τ x(τ) + αx(τ))) + J(τ)x(τ)
×exp −σ dτ|˜x(τ)|2
Here σ is the varience of the gaussian noise distribution.
Then the integral over the x(τ) variable can be done and
finally we get the generating functional
Z[J, ˜J = 0] = Aexp −σ dk
| ˜J(k)|2
(k2 + α2)
(12)
So as Z[J, ˜J = 0] can be calculated exactly ,we don’t need
perturbation expansion of Z[J, ˜J = 0].
5 Conclusion
The MSR formalism introduce an extra conju-
gate(unphysical) field which will be extremely helpful
to deal system with diffusion and provides one to extend
some formulation for conservative system to these sys-
tems.The role of jacobian is more involved in perturbation
expansion which goes beyond our present discussion.
6 Reference
1.Functional-integral approach to Parisi-Wu stochastic
quantization:Scalar theory::E.Gozzi(1983),
PHYSICAL REVIEW D
2.Random Fields And Spin Glasses::Cirano Dominicis and
Irene Giadina::Cambridge University Press
3.Statistical Field Theory::Giorgio Parisi::Addison-Wesley
Publishing House
4.A quick introduction to the Martin-Siggia-Rose formal-
ism::Inordinatum-a website
5.Functional stochastic quantization and perturbation
theory::Debashis Gangopadhyay,Ashok Chatterjee and
Parthasarathi Majumdar::IOP SCIENCE
7 Acknowledgement
1.Prof. Parthasarathi Majumdar,for providing resources
on stochastic quantization
2.Prof. Somendra Mohan Bhattacharjee ,for useful discus-
sion on MSR formalism
3.My friends Sourav Laha and Jagannath Santara for use-
ful help on latex typeset.