1. Generating Functional Approach
(Martin-Siggia-Rose Formalism)
Arkajyoti Manna
Ramakrishna Mission Vivekananda University
arkajyoti1@live.com
November 19, 2015
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 1 / 14

2. Overview
1 Parisi-Wu approach for Generating Functional
In this approach we will show that the correlation function (averaged
over the noise distribution) corresponds to usual correlation function
in Quantum Field Theory in some limit.This is due to Parisi and Wu.
2 Construction of Generating Functional
Here we will set up the Martin-Siggia-Rose generating functional for
a stochastic system which is governed by Langevin equation.And we
take a gaussian white noise as a random source.
3 Example:Ornstein-Uhlenbeck process
We will demonstrate how to set up a generating functional for this
process and how to calculate the corrlation function.
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 2 / 14

3. The Parisi-Wu approach for generating
functional
Quantum(in Euclidean field theory) correlation functions(for euclidean
action S[φ])
0|Tφ(x1)φ(x2)|0 =
Dφ[φ(x1)φ(x2)]exp(−S[φ])
Dφexp(−S[φ])
Parisi Wu approach:
1.φ(x) → φ(x, τ)(fields are coupled to a random noise source).Here the
system is supposed to be in equilibrium at τ → ∞ and the Euclidean
action becomes: SE = dτd4xL(φ(x, τ), ∂φ(x, τ))
2.
∂φ(x, τ)
∂τ
= −
δS[φ]
δφ
+ η(x, τ) ::Langevin Eqn.
η(x1, τ1)η(x2, τ2) = 2δ(x1 − x2)δ(τ1 − τ2) ::Gaussian noise
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 3 / 14

4. The Parisi-Wu approach for generating
functional
3.
φ(x1, τ1)φ(x2, τ2) η =
DηP(η)φ(x1, τ1)φ(x2, τ2)
DηP(η)
::correlation function
where P(η) is a normalized distribution.
4.Introduce Probability distribution P(φ, τ) satisfying Fokker-Planck
eqn.
∂P(φ, τ)
∂τ
=
δ2P
δφ2
+
δ
δφ
P
δS
δφ
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 4 / 14

5. The Parisi-Wu approach for generating
functional
Then the correlation function can be given by
φ(x1, τ1)φ(x2, τ2) η =
DφP(φ, τ)φ(x1)φ(x2)
DφP(φ, τ)
5.Then we recast the Fokker-Planck eqn.
(−)HFPψ[φ, τ] =
∂ψ[φ, τ]
∂τ
(−)HFP =
δ2
δφ2
−
1
4
δS
δφ
+
1
2
δ2S
δφ2
ψ(φ, τ) = P(φ, τ)eS[φ]/2
here HFP is positive semi-definite.
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 5 / 14

6. The Parisi-Wu approach for generating
functional
6.Now we observe that if ψn[φ] ∝ e−S[φ]/2
HFPψn(φ) = Enψn(φ)
ψ[φ, τ] =
n
cnψn(φ)e−2Enτ
here τ is Euclidean 5-th time.
7.Then we take the limit τ → ∞ and find that
lim
τ→∞
φ(x1, τ)φ(x2, τ) η =
Dφ[φ(x1)φ(x2)]exp(−S[φ])
Dφexp(−S[φ])
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 6 / 14

7. MSR generating functional
Now we can write the MSR generating functional
Z[J, ˜J]η =
j
Dφj (x, τ)D ˜φj (x, τ)exp d4
xLeff (φj , ˜φj )
×exp


j
d4
xdτi(Jφj + ˜J ˜φj )


Leff =
j
dτ i ˜φj (τ)(∂τ φj + ∂j H) − T|˜φj (τ)|2
+trln(Tjk(τ, τ ))
Here ˜φ(x, τ) is the conjugate(unphysical) field. And we use τ → iτ
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 7 / 14

8. MSR generating functional
So the correlation function becomes
φ(x1, τ1)φ(x2, τ2) η = lim
τ→∞
(−)
1
Z[0]
δ2Z[J, ˜J]
δJ(x1)δJ(x2) J=0
(1)
where Z[0] = 1.
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 8 / 14

9. Example:Ornstein-Uhlenbeck process
The Ornstein-Uhlenbeck process is described by the SDE
∂τ x(τ) + αx(τ) − η(τ) = 0, α > 0
The corresponding generating functional is given by
Z[J, ˜J = 0] = D[x(τ)]D[˜x(τ)]
×exp i dτ˜x(∂τ x(τ) + αx(τ)) + J(τ)x(τ) × exp −σ dτ|˜x(τ)|2
Here as the conjugate variable ˜x(τ) is unphysical one we set ˜J = 0.
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 9 / 14

10. Example:Ornstein-Uhlenbeck process
Then we perform the integral over x(τ) which gives delta function
restricting the auxiliary variable ˜x(τ).And finally we write the generating
functional
Z[J, ˜J = 0] = Aexp −σ dk
|J(k)|2
(k2 + α2)
(2)
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 10 / 14

11. Some Remarks
1.Here we only deal with scalar fields.
2.In MSR formalism we do not have to normalize the Z[0] as it is unity by
construction.
3.In this approach we assume the SDE is well defined and initial conditions
are given.Hence the solutions are unique.There is no spontaneous
symmetry breaking.
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 11 / 14

12. References
E.Gozzi (1983)
Functional-integral approach to Parisi-Wu stochastic quantization:Scalar theory
PHYSICAL REVIEW D vol-28,number-8,page-1922-1930.
Cirano Dominicis and Irene Giadina
Random Fields And Spin Glasses
Cambridge University Press
Giorgio Parisi
Statistical Field Theory
Addison-Wesley Publishing House
Inordinatum
A quick introduction to the Martin-Siggia-Rose formalism
P. Damgaard and H. Huffel
Stochastic Quantization
North-Holland, Amsterdam
D. Gangopadhyay,A. Chatterjee and P. Majumdar
Functional stochastic quantization and perturbation theory
IOP SCIENCE
Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 12 / 14

13. Further developement
For other functional approach(other than MSR formalism) which does not
invoke extra unphysical fields see::
Effective Action for Stochastic Partial Differential Eqns.::David
Hochberg,Carmen Molina-Paris,Juan Perez-Mercader and Matt
Visser::arXiv:cond-mat
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14. Arkajyoti Manna (RKMVU) MSR formalism November 19, 2015 14 / 14