The MSR Formalism

The Martin–Siggia–Rose (MSR) formalism (also known as the Janssen–de Dominicis formalism) provides a field-theoretic framework for studying stochastic differential equations (SDEs) using path-integral methods.

From Langevin Equations to Path Integrals

Consider a stochastic process described by a Langevin equation:

\[\frac{\partial \phi_i(x,t)}{\partial t} = -\frac{\delta \mathcal{H}[\phi]}{\delta \phi_i(x,t)} + \eta_i(x,t)\]

where \(\phi_i\) is the physical field of interest, \(\mathcal{H}[\phi]\) is an energy functional, and \(\eta_i\) is Gaussian white noise with correlator

\[\langle \eta_i(x,t)\,\eta_j(x',t') \rangle = 2 D_{ij}\,\delta(x-x')\,\delta(t-t').\]

The MSR procedure recasts averages over noise realisations as a field-theoretic path integral by introducing an auxiliary response field \(\psi_i\) (sometimes written \(\tilde\phi_i\) or \(\hat\phi_i\)):

\[\langle \mathcal{O}[\phi] \rangle = \int \mathcal{D}\phi\,\mathcal{D}\psi\; \mathcal{O}[\phi]\; e^{-S[\phi,\psi]}\]

The two field types in sft-wick correspond to:

• \(\phi\) — the physical field (FieldType.PHYSICAL)

• \(\psi\) — the response field (FieldType.RESPONSE)

Free and Interaction Actions

The MSR action splits into a free (Gaussian) part and an interaction part:

\[S[\phi,\psi] = S_0[\phi,\psi] + S_{\mathrm{int}}[\phi,\psi]\]

The free action \(S_0\) is quadratic in the fields and determines the two-point functions (propagators). The interaction action \(S_{\mathrm{int}}\) contains all nonlinear terms — these are the terms represented by Vertex objects in sft-wick.

Propagators

The free two-point functions derived from \(S_0\) are:

Contraction	Propagator	Physical meaning
\(\langle \phi_i(x)\,\phi_j(x') \rangle_{S_0}\)	\(C_{ij}(x,x')\)	Correlation (equal-time or two-time)
\(\langle \phi_i(x)\,\psi_j(x') \rangle_{S_0}\)	\(R_{ij}(x,x')\)	Response (retarded Green’s function)
\(\langle \psi_i(x)\,\psi_j(x') \rangle_{S_0}\)	\(0\)	Vanishes by construction

The vanishing of the \(\psi\)–\(\psi\) contraction is a fundamental consequence of the MSR construction (the noise is integrated out exactly) and is the key constraint exploited by sft-wick for efficient enumeration of Wick contractions.

Note

Convention in sft-wick: for the response propagator R, the physical field is always placed on the left:

\[R_{ij}(x,x') = \langle \phi_i(x)\,\psi_j(x') \rangle_{S_0}\]

Itô Convention and Causality

The response propagator is retarded: \(R(t,t') \propto \Theta(t - t')\). The Itô discretisation convention sets \(\Theta(0) = 0\), which implies:

\[R(x,x) = 0\]

This eliminates equal-point response contractions (self-response tadpoles). More generally, any closed loop of response propagators vanishes by causality:

\[R(a_1,a_2)\,R(a_2,a_3)\,\cdots\,R(a_n,a_1) = 0\]

because the retarded nature of \(R\) would require \(t_1 > t_2 > \cdots > t_n > t_1\), which is impossible.

In sft-wick, both rules are applied by default when ito=True.

Response Phase Convention

In some formulations of the MSR action, the contraction of a physical field with a response field picks up a phase:

\[\langle \phi_i(a)\,\psi_j(b) \rangle_{S_0} = -\mathrm{i}\,R_{ij}(a,b)\]

When response_phase=True (the default in compute_moment()), each term in the final result is multiplied by \((-\mathrm{i})^n\), where n is the number of response propagators in that term.

The Partition Function \(Z=1\)

A remarkable property of the MSR formalism is that the partition function equals unity:

\[Z = \int \mathcal{D}\phi\,\mathcal{D}\psi\; e^{-S[\phi,\psi]} = 1.\]

This means there is no denominator in the perturbative expansion of moments, greatly simplifying the diagrammatic analysis (no vacuum diagram subtraction is needed).

Perturbative Expansion

Since \(Z=1\), the expectation value of an observable \(\mathcal{O}[\phi]\) can be expanded directly:

\[\langle \mathcal{O} \rangle_S = \sum_{n=0}^{N} \frac{(-1)^n}{n!}\, \langle \mathcal{O}\, S_{\mathrm{int}}^{\,n} \rangle_{S_0}\]

Each term in the sum involves a Gaussian expectation \(\langle\cdots\rangle_{S_0}\), which is evaluated via Wick’s theorem (see Wick’s Theorem). The function compute_moment() implements this expansion up to any finite order N.