08.10.15 · stat-mech / qft

Stochastic perturbation theory and the tree expansion

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Anchor (Master): Parisi & Wu, *Sci. Sin.* 24, 483 (1981) (stochastic quantisation and the perturbative equivalence); Damgaard & Hüffel, *Phys. Rep.* 152, 227 (1987) (the canonical review, §§4-6 on the diagrammatic expansion and stochastic regularisation); Floratos & Iliopoulos, *Nucl. Phys. B* 214, 392 (1983) (proof of equivalence to ordinary perturbation theory); Zwanziger, *Nucl. Phys. B* 192, 259 (1981) (stochastic gauge fixing); Martin, Siggia & Rose, *Phys. Rev. A* 8, 423 (1973) (the response-field functional)

Intuition Beginner

Stochastic quantisation says you can compute the predictions of a quantum field theory by simulating a noisy ball that rolls downhill on the action landscape and watching where it settles. The previous units treated that picture numerically: run the noisy equation on a computer and average. This unit does the same calculation with pencil and paper, by solving the noisy equation step by step instead of running it.

The trick is to treat the interaction strength as small. Switch the interaction off and the noisy ball obeys a simple linear law: each random kick decays away at a fixed rate, and the resting wobble of the ball is exactly the free propagator of the field theory. That solved free motion is the starting point.

Now switch the interaction back on a little at a time. The first correction takes the free wobble, feeds it through the interaction, and feeds the result back into the motion. The second correction does this twice, and so on. Each correction is a small recipe built out of free wobbles joined together by the interaction.

When you draw these recipes, they look like little branching trees: a root where you read off the answer, branches where the interaction splits one line into several, and leaves where the random kicks enter. Adding up all the trees, and pairing their random-kick leaves, rebuilds the ordinary Feynman diagrams of the theory once you wait long enough for the ball to settle.

Visual Beginner

A side-by-side schematic. On the left, the noisy ball relaxing on the action landscape with a labelled fictitious-time axis. On the right, a rooted tree: a single root node at the bottom (the point where the answer is read off), interaction vertices branching upward, and open leaves at the top marked with wavy lines for the random kicks. A dashed arc pairs two leaves to show a noise contraction collapsing two trees into one closed diagram.

The picture captures the whole construction: a tree is one term of the step-by-step solution of the noisy equation, the leaves are where the noise enters, and pairing the leaves of two trees produces one Feynman diagram of the field theory in the long-time limit.

Worked example Beginner

Take a single mode (one number $ϕ$ , no lattice) with free rate $a$ and a small cubic interaction, and find the leading correction to its resting wobble.

Step 1. Without interaction, the noisy equation reads "the rate of change of $ϕ$ equals minus $a$ times $ϕ$ , plus a random kick." The solution at fictitious time $τ$ is the sum of all past kicks, each faded by the factor $e^{- a (τ - s)}$ for a kick at earlier time $s$ . This faded-memory factor is the free retarded propagator. The resting variance works out to $1/ a$ , which is the free propagator value.

Step 2. Turn on a small interaction of strength $g$ , contributing a term $- g ϕ^{2}$ to the rate of change. Write $ϕ = ϕ_{0} + g ϕ_{1} + \dots$ , where $ϕ_{0}$ is the free solution from Step 1.

Step 3. Match the terms of size $g$ . The first correction $ϕ_{1}$ obeys the same free equation but driven by $- ϕ_{0}^{2}$ instead of a kick. So $ϕ_{1}$ is the faded-memory factor applied to $- ϕ_{0}^{2}$ : take the free solution, square it, feed it back through the same memory kernel.

Step 4. Read off the diagram. The correction $ϕ_{1}$ has one vertex (where the square happens) and two free lines feeding in, joined to the root by one more free line. Drawn out, it is a small tree with one branch point.

Step 5. Average to get the resting wobble. Squaring $ϕ_{0}$ and averaging pairs up its kicks, leaving one closed free line at the vertex. The result is a number proportional to $g$ times the free propagator, evaluated in the long-time limit. This is the tadpole correction, the same first correction the ordinary perturbation theory of this model gives.

What this tells us: solving the noisy equation order by order produces the perturbative corrections directly, drawn as trees, and the long-time average reproduces the standard answer.

Check your understanding Beginner

Formal definition Intermediate+

Work on a finite lattice $Λ$ (or in momentum space; the construction is identical) with a real scalar field and Euclidean action $$ S[\phi] = \tfrac{1}{2}\sum_{x,y}\phi(x),K(x,y),\phi(y) + \tfrac{g}{3!}\sum_x \phi(x)^3 , $$ where $K = - Δ_{Λ} + m^{2}$ is the free kinetic operator. (We use a cubic interaction for notational economy; the $ϕ^{4}$ case is identical with $S^{'''} = g δ$ .) The Langevin equation in fictitious time $τ \geq 0$ is $$ \partial_\tau \phi(x,\tau) = -\frac{\delta S}{\delta\phi(x)}\big[\phi(\cdot,\tau)\big] + \eta(x,\tau) = -\big(K\phi\big)(x,\tau) - \tfrac{g}{2}\phi(x,\tau)^2 + \eta(x,\tau), $$ with Gaussian white noise $⟨ η (x, τ) η (y, τ^{'})⟩ = 2 δ_{x y} δ (τ - τ^{'})$ and $⟨ η ⟩ = 0$ .

The free Ornstein-Uhlenbeck solution is obtained by dropping the $g$ term. Its causal (retarded) Green function is the free retarded propagator $$ G_R(x,y;\tau,\tau') = \big(e^{-K(\tau-\tau')}\big)(x,y),\theta(\tau-\tau'), $$ the operator exponential of $- K$ in fictitious time, supported only on $τ > τ^{'}$ . In momentum space $G_{R} (k; τ, τ^{'}) = e^{- (k^{2} + m^{2}) (τ - τ^{'})} θ (τ - τ^{'})$ . The free field is the noise convolved with $G_{R}$ : $$ \phi_0(x,\tau) = \int \mathrm{d}\tau' \sum_y G_R(x,y;\tau,\tau'),\eta(y,\tau'). $$

The stochastic propagator is the free two-point function, $$ D(x,y;\tau,\tau') = \langle\phi_0(x,\tau),\phi_0(y,\tau')\rangle = \int \mathrm{d}s,\sum_z 2,G_R(x,z;\tau,s),G_R(y,z;\tau',s), $$ where the noise contraction has been used. In momentum space, $$ D(k;\tau,\tau') = \frac{e^{-(k^2+m^2)|\tau-\tau'|} - e^{-(k^2+m^2)(\tau+\tau')}}{k^2+m^2}. $$

The stochastic (tree) perturbation expansion is the formal series $ϕ = \sum_{n \geq 0} g^{n} ϕ_{n}$ obtained by Picard iteration of the Langevin equation: substituting the series and matching powers of $g$ gives the recursion $$ \phi_n(x,\tau) = -\tfrac{1}{2}\int\mathrm{d}\tau'\sum_y G_R(x,y;\tau,\tau')\sum_{p+q=n-1}\phi_p(y,\tau'),\phi_q(y,\tau'), \qquad n\ge 1, $$ with $ϕ_{0}$ the free solution above. Each $ϕ_{n}$ is a sum over rooted trees: the root is the external $(x, τ)$ , each internal node is a cubic vertex contributing the factor from $S^{'''}$ , each edge is a factor $G_{R}$ , and the leaves are insertions of the noise $η$ . The recursion's convolution structure (a root edge feeding into a product of subtrees) is exactly the rooted-tree composition law.

A correlation function is then computed by taking expectations: $⟨ ϕ (x_{1}, τ_{1}) \dots ⟩$ expands as a sum over trees whose leaves are paired by $⟨ η η ⟩ = 2 δ$ . Each pairing glues two leaves and produces an internal line carrying the noise factor; trees thereby close into graphs. The equilibrium correlator is the equal-fictitious-time limit $$ G_2(x,y) = \lim_{\tau\to\infty}\langle\phi(x,\tau),\phi(y,\tau)\rangle, $$ which the next section identifies with the ordinary Euclidean two-point function.

Counterexamples to common slips

The internal lines are retarded, not symmetric. An internal edge of a tree is $G_{R}$ (causal, $θ (τ - τ^{'})$ ), not the symmetric $D$ . The symmetric stochastic propagator $D$ appears only after two trees are glued at a noise pair. Using $D$ for internal tree edges double-counts the noise.
The expansion has trees, not loops, at the level of $ϕ_{n}$ . Loops appear only when noise leaves are contracted in expectation values. Each $ϕ_{n}$ itself is a genuine tree (no internal loops), which is why the construction is called a tree expansion. Mistaking $ϕ_{2}$ for a one-loop object conflates the field expansion with the correlator expansion.
The equilibrium limit is required. At finite $τ$ the propagator $D (k; τ, τ)$ carries the transient term $- e^{- 2 (k^{2} + m^{2}) τ} / (k^{2} + m^{2})$ . Dropping the $τ \to \infty$ limit leaves an initial-condition transient that is not part of the Euclidean amplitude.

Key theorem with proof Intermediate+

Theorem (equilibrium tree expansion reproduces the free propagator; Parisi-Wu 1981 ^[Parisi-Wu], Damgaard-Hüffel 1987 ^{[Damgaard-Huffel]}). For the free theory ( $g = 0$ ), the equal-fictitious-time stochastic propagator converges, as $τ \to \infty$ , to the static Euclidean propagator: $$ \lim_{\tau\to\infty} D(k;\tau,\tau) = \frac{1}{k^2+m^2} = \big\langle\phi(k),\phi(-k)\big\rangle_{\text{Euclidean}} . $$ Moreover, the order- $g$ correction to the two-point function generated by the tree recursion equals, in the same limit, the standard first-order Euclidean Feynman correction.

Proof. Set $ω_{k} = k^{2} + m^{2}$ . From the momentum-space stochastic propagator, $$ D(k;\tau,\tau) = \frac{1 - e^{-2\omega_k\tau}}{\omega_k}. $$ Because $ω_{k} > 0$ (with $m > 0$ ), the transient $e^{- 2 ω_{k} τ} \to 0$ as $τ \to \infty$ , leaving $lim_{τ \to \infty} D (k; τ, τ) = 1/ ω_{k}$ . This is the free Euclidean propagator, establishing the zeroth-order statement.

For the order- $g$ correction, expand the two-point function $⟨ ϕϕ ⟩ = ⟨ ϕ_{0} ϕ_{0} ⟩ + g (⟨ ϕ_{0} ϕ_{1} ⟩ + ⟨ ϕ_{1} ϕ_{0} ⟩) + O (g^{2})$ . The first correction $ϕ_{1}$ from the recursion is $$ \phi_1(x,\tau) = -\tfrac{1}{2}\int\mathrm{d}\tau'\sum_y G_R(x,y;\tau,\tau'),\phi_0(y,\tau')^2 . $$ Compute $⟨ ϕ_{0} (x, τ) ϕ_{1} (z, τ)⟩$ . Inserting $ϕ_{1}$ and using Wick's rule for the Gaussian $ϕ_{0}$ , the three $ϕ_{0}$ factors pair as one contraction between the external $ϕ_{0} (x, τ)$ and one leg of the vertex, and one self-contraction of the remaining two legs of $ϕ_{0} (y, τ^{'})^{2}$ : $$ \langle\phi_0(x,\tau),\phi_1(z,\tau)\rangle = -\tfrac{1}{2}\cdot 2\int\mathrm{d}\tau'\sum_y G_R(z,y;\tau,\tau'),D(x,y;\tau,\tau'),D(y,y;\tau',\tau'), $$ the combinatorial factor $2$ counting the two legs that can connect to the external line. The tadpole factor $D (y, y; τ^{'}, τ^{'})$ is the equal-point free propagator; the retarded line $G_{R}$ and the cross propagator $D$ join the vertex to the two external points.

Pass to momentum space and take $τ \to \infty$ . The retarded line gives $\int_{0}^{\infty} d u e^{- ω_{k} u} = 1/ ω_{k}$ once the fictitious-time integral over the vertex position is performed, and the two cross propagators contribute $1/ ω_{k}$ each in the equilibrium limit, while the tadpole loop $\int_{q} D (q; \infty) = \int_{q} 1/ ω_{q}$ is the standard one-loop integral. Assembling the factors, $$ \lim_{\tau\to\infty} g,\langle\phi\phi\rangle^{(1)}(k) = -\frac{g}{2},\frac{1}{\omega_k}\Big(\int_q \frac{1}{\omega_q}\Big)\frac{1}{\omega_k}, $$ which is precisely the first-order self-energy insertion $- \frac{g}{2} G_{0} (k) Σ_{1} G_{0} (k)$ of ordinary $ϕ^{3} / ϕ^{4}$ Euclidean perturbation theory, with $G_{0} = 1/ ω_{k}$ and the tadpole $Σ_{1} = \int_{q} 1/ ω_{q}$ . $□$

Bridge. This identification builds toward the all-orders equivalence proved in the next tier and appears again in every later use of the response-field formalism, because the foundational reason the stochastic and ordinary expansions agree is structural: the equal-time equilibrium limit of the retarded fictitious-time dynamics is exactly the static propagator, so each tree, once its noise leaves are contracted, carries the same momentum-space weight as the Feynman graph it draws. The central insight is that the causal line $G_{R}$ and the symmetric line $D$ are not independent inputs — $D$ is built from two $G_{R}$ factors glued by the noise — which is exactly the fluctuation-dissipation relation in fictitious time, and this is exactly the structure the Martin-Siggia-Rose doubled-field formalism makes manifest. Putting these together, the tree expansion generalises ordinary perturbation theory by adding a fictitious-time bookkeeping that the equilibrium limit then erases, leaving the standard diagrams; the bridge is the observation that the transient terms $e^{- ω_{k} τ}$ are precisely the memory of the initial condition, and their decay is what equilibration means.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the stochastic propagator $D (k; τ, τ^{'})$ by contracting two free fields $ϕ_{0} = G_{R} * η$ using $⟨ η η ⟩ = 2 δ$ , and confirm the equal-time form $D (k; τ, τ) = (1 - e^{- 2 ω_{k} τ}) / ω_{k}$ .

Hint

$D (k; τ, τ^{'}) = \int_{0}^{\infty} d s 2 G_{R} (k; τ, s) G_{R} (k; τ^{'}, s)$ with the causal $θ$ functions limiting the $s$ range.

Answer

With $ϕ_{0} (k, τ) = \int d s G_{R} (k; τ, s) η (k, s)$ and $⟨ η (k, s) η (- k, s^{'})⟩ = 2 δ (s - s^{'})$ , $$ D(k;\tau,\tau') = 2\int_0^{\min(\tau,\tau')}!!\mathrm{d}s, e^{-\omega_k(\tau-s)}e^{-\omega_k(\tau'-s)} = 2,e^{-\omega_k(\tau+\tau')}\int_0^{\min(\tau,\tau')}!!e^{2\omega_k s},\mathrm{d}s. $$ The integral is $(e^{2 ω_{k} m i n (τ, τ^{'})} - 1) / (2 ω_{k})$ , giving $$ D(k;\tau,\tau') = \frac{e^{-\omega_k|\tau-\tau'|} - e^{-\omega_k(\tau+\tau')}}{\omega_k}. $$ Setting $τ^{'} = τ$ : $D (k; τ, τ) = (1 - e^{- 2 ω_{k} τ}) / ω_{k}$ . Rubric: full credit for the $s$ -integral, the general form, and the equal-time limit.

Exercise 4 (medium, short-answer).

Explain why each term $ϕ_{n}$ of the Langevin Picard iteration is a sum over rooted trees and contains no loops, while loops appear only in expectation values.

Hint

Look at the recursion: $ϕ_{n}$ is one root edge $G_{R}$ feeding into a product of lower-order $ϕ_{p} ϕ_{q}$ . A product of subtrees joined at a single new vertex cannot create a cycle.

Answer

The recursion $ϕ_{n} = - \frac{1}{2} G_{R} * \sum_{p + q = n - 1} ϕ_{p} ϕ_{q}$ builds $ϕ_{n}$ from a single new vertex (the squaring) whose output runs along one new $G_{R}$ edge to the root, with two independent lower-order subtrees $ϕ_{p}, ϕ_{q}$ as inputs. Attaching disjoint subtrees to a fresh vertex by a single edge is precisely the inductive construction of a rooted tree; no edge ever connects two nodes already linked through the root, so no cycle forms. Each $ϕ_{n}$ is therefore a forest of rooted trees with noise leaves. Loops arise only when one takes an expectation value and the noise leaves of the trees are paired by $⟨ η η ⟩ = 2 δ$ : a pairing that joins two leaves within the same correlator closes a cycle, producing the loop of the corresponding Feynman graph. Rubric: full credit for the tree-from-recursion argument and the leaves-pairing-makes-loops distinction.

Exercise 5 (medium, short-answer).

State the fluctuation-dissipation relation between $G_{R}$ and $D$ in fictitious time and explain why it makes the noise-strength normalisation $2$ in $⟨ η η ⟩ = 2 δ$ the unique choice that yields the equilibrium propagator $1/ ω_{k}$ .

Hint

$D$ is two retarded lines glued by the noise. Track how the noise prefactor $2 c$ propagates into the equilibrium value $D (k; \infty) = c / ω_{k}$ .

Answer

The relation is $D = G_{R} * (21) * G_{R}^{T}$ : the symmetric correlation propagator is two retarded responses joined by the noise kernel. With a general noise strength $⟨ η η ⟩ = 2 c δ$ , repeating Exercise 3 gives $D (k; \infty) = c / ω_{k}$ . The equilibrium Euclidean propagator must be $1/ ω_{k}$ , so $c = 1$ , i.e. the prefactor $2$ is fixed. This is the fictitious-time fluctuation-dissipation theorem: the noise strength (fluctuation) and the relaxation rate $ω_{k}$ in $G_{R}$ (dissipation) are locked together so that the stationary variance reproduces the temperature-one Gibbs propagator. Any other normalisation rescales the equilibrium measure to $e^{- S / c} / Z$ . Rubric: full credit for the $G_{R}$ -noise- $G_{R}$ structure and the $c = 1$ fixing argument.

Exercise 6 (hard, short-answer).

Show that the order- $g$ tadpole correction to the equilibrium two-point function computed by the tree expansion equals the ordinary first-order $ϕ^{3}$ self-energy insertion $- \frac{g}{2} G_{0} Σ_{1} G_{0}$ with $Σ_{1} = \int_{q} 1/ ω_{q}$ .

Hint

Use the result of the Key Theorem: one vertex, one tadpole loop $D (y, y)$ , and two external propagators, all taken to the $τ \to \infty$ limit.

Answer

The first-order correlator correction is $g ⟨ ϕ_{0} ϕ_{1} ⟩ + (x \leftrightarrow z)$ with $$ \langle\phi_0(x,\tau)\phi_1(z,\tau)\rangle = -\int\mathrm{d}\tau'\sum_y G_R(z,y;\tau,\tau'),D(x,y;\tau,\tau'),D(y,y;\tau',\tau'), $$ the factor $\frac{1}{2} \cdot 2 = 1$ from the two attachable legs cancelling the symmetry factor. In the equilibrium limit each cross propagator $D \to 1/ ω_{k}$ , the closed line $D (y, y) \to \int_{q} 1/ ω_{q} = Σ_{1}$ , and the retarded line integrated over the vertex time gives $\int_{0}^{\infty} d u e^{- ω_{k} u} = 1/ ω_{k}$ . Collecting, the symmetrised correction is $$ \lim_{\tau\to\infty} g,\langle\phi\phi\rangle^{(1)}(k) = -\frac{g}{2},\frac{1}{\omega_k},\Sigma_1,\frac{1}{\omega_k} = -\frac{g}{2},G_0(k),\Sigma_1,G_0(k), $$ the standard first-order self-energy insertion. The fictitious-time integrals have collapsed to the static loop integral, with no acausal or transient remainder. Rubric: full credit for the vertex/loop/external assembly and the matching to $- \frac{g}{2} G_{0} Σ_{1} G_{0}$ .

Exercise 7 (hard, short-answer).

Explain how the same tree expansion arises from the Martin-Siggia-Rose response-field generating functional, identifying $G_{R}$ and $D$ with the propagators of the doubled field $(ϕ, \hat{ϕ})$ .

Hint

Enforce the Langevin equation with a delta functional written via a conjugate field $\hat{ϕ}$ , then average over the Gaussian noise to produce a quadratic $\hat{ϕ}$ term.

Answer

Write the noise average of an observable as a path integral over $ϕ$ with the Langevin equation imposed by a functional delta: $δ [\partial_{τ} ϕ + K ϕ + \frac{g}{2} ϕ^{2} - η] = \int D \hat{ϕ} exp (i \int \hat{ϕ} (\partial_{τ} ϕ + K ϕ + \frac{g}{2} ϕ^{2} - η))$ . Averaging over the Gaussian noise $η$ with $⟨ η η ⟩ = 2 δ$ removes $η$ and produces the term $- \int \hat{ϕ} \hat{ϕ}$ (strength $2$ ). The result is the MSR action $S_{MSR} [ϕ, \hat{ϕ}] = \int \hat{ϕ} (\partial_{τ} ϕ + K ϕ) - \int \hat{ϕ}^{2} + \frac{g}{2} \int \hat{ϕ} ϕ^{2}$ . The free part is quadratic in $(ϕ, \hat{ϕ})$ ; inverting it gives the response propagator $⟨ ϕ \hat{ϕ} ⟩ = G_{R}$ (the retarded Green function of $\partial_{τ} + K$ ) and the correlation propagator $⟨ ϕϕ ⟩ = D$ (two $G_{R}$ joined by the $\hat{ϕ}^{2}$ vertex), while $⟨ \hat{ϕ} \hat{ϕ} ⟩ = 0$ . The cubic vertex $\hat{ϕ} ϕ^{2}$ has one $\hat{ϕ}$ leg, forcing exactly one $G_{R}$ out of each vertex toward the root — this is the rooted-tree structure. So the Feynman rules of the doubled theory reproduce the stochastic tree expansion term by term. Rubric: full credit for the delta-functional/ $\hat{ϕ}$ construction, the noise average giving $\hat{ϕ}^{2}$ , and the propagator identification.

Advanced results Master

Theorem (equivalence of stochastic and ordinary perturbation theory; Parisi-Wu 1981 ^[Parisi-Wu], Floratos-Iliopoulos 1983 ^{[Floratos-Iliopoulos]}). Let $S [ϕ]$ be a polynomial Euclidean action with a stable free part $K = - Δ + m^{2} > 0$ . The equal-fictitious-time, $τ \to \infty$ limit of the connected correlation functions generated by the stochastic (tree) expansion of the Langevin equation $\partial_{τ} ϕ = - δ S / δ ϕ + η$ coincides, order by order in the coupling and graph by graph, with the connected Euclidean Feynman expansion generated by the measure $e^{- S} / Z$ .

The proof is organisational. At order $n$ the stochastic expansion is a sum over rooted trees with $n$ vertices and noise leaves; taking the expectation pairs the leaves via $⟨ η η ⟩ = 2 δ$ . Each pairing of two trees produces a graph whose internal lines are either retarded ( $G_{R}$ , from a tree edge) or symmetric ( $D$ , from a glued leaf pair). The fictitious-time integrals over vertex positions are then performed. The retarded structure makes every such integral convergent and causal, and in the $τ \to \infty$ limit the standard manipulation $\int_{0}^{\infty} d u e^{- ω u} = 1/ ω$ collapses each fictitious-time line to a static propagator factor $1/ ω_{k}$ . Floratos and Iliopoulos show that the sum over the relative orderings of the vertices, weighted by the retarded $θ$ -functions, reconstructs exactly the symmetric momentum-space integrand of the corresponding Feynman graph, with the correct symmetry factor; no graph is missing and none is overcounted. The crucial cancellation is that acausal contributions, which would spoil the match, are forbidden by the $θ$ -functions in $G_{R}$ .

Theorem (stochastic regularisation; Damgaard-Hüffel 1987 ^{[Damgaard-Huffel]}). Smearing the noise correlator in fictitious time, $⟨ η (k, τ) η (- k, τ^{'})⟩ = 2 R_{Λ} (k) δ (τ - τ^{'})$ with a regulator $R_{Λ} (k) \to 1$ as $Λ \to \infty$ , deforms the stochastic propagator to $D_{Λ} (k; τ, τ) = R_{Λ} (k) (1 - e^{- 2 ω_{k} τ}) / ω_{k}$ and provides a gauge-invariant ultraviolet regularisation of every loop integral, since the regulator enters only through the noise and never through the drift.

Because the regulator sits in the noise correlator rather than in the action, the equilibrium measure remains $e^{- S} / Z$ up to the regulator, and in gauge theories the gauge invariance of the drift is untouched. This is the mechanism behind the claim that stochastic quantisation furnishes a perturbation theory that respects gauge symmetry without a symmetry-breaking momentum cutoff — the regulator multiplies the noise, and the Ward identities, which follow from the gauge invariance of the drift $- δ S / δ ϕ$ , survive.

Theorem (ghost-free gauge perturbation theory; Zwanziger 1981 ^[Zwanziger]). In Yang-Mills theory the stochastic tree expansion, augmented by Zwanziger's gauge-orbit drift, produces a Lorentz-covariant perturbation theory whose propagators are the transverse and (gauge-parameter-dependent) longitudinal parts of the response propagator $G_{R}$ , with no Faddeev-Popov ghost fields and no Gribov ambiguity. The gauge-orbit drift selects a representative on each orbit without altering gauge-invariant correlators; in the small-fluctuation expansion it reproduces the conventional covariant-gauge propagator, but the ghost determinant of the Faddeev-Popov construction is replaced by the manifestly positive noise measure.

Synthesis. The stochastic tree expansion puts together the foundational reason stochastic quantisation is more than a Monte Carlo device: solving the Langevin equation perturbatively gives a self-contained, diagrammatic perturbation theory whose equilibrium limit is exactly ordinary Euclidean field theory, and this is exactly the analytic counterpart of the numerical equilibrium identification of 08.10.08. The central insight is that the two propagators of the theory, the retarded $G_{R}$ and the symmetric $D$ , are not independent: $D$ is two $G_{R}$ lines glued by the noise, which is exactly the fictitious-time fluctuation-dissipation relation, and this relation is what the Martin-Siggia-Rose doubled-field action of 08.10.14 makes manifest by inverting a quadratic form in $(ϕ, \hat{ϕ})$ . Putting these together, the tree expansion generalises the Feynman-diagram expansion by adding a causal fictitious-time bookkeeping that the equilibrium limit erases — the transients $e^{- ω_{k} τ}$ are the initial-condition memory, and their decay is equilibration — so the bridge is the recognition that ordinary perturbation theory is the $τ \to \infty$ shadow of a manifestly causal stochastic process.

The construction is dual to the Fokker-Planck operator route of 08.10.02: where the Fokker-Planck equation describes the relaxation of the distribution, the tree expansion describes the relaxation of the trajectory, and both converge to the same Gibbs measure. The same insight generalises to gauge theories, where the gauge invariance of the drift makes stochastic regularisation respect the Ward identities and Zwanziger's gauge-orbit drift removes the Faddeev-Popov ghosts; and it generalises laterally to all of stochastic field theory, since the rooted trees with retarded internal lines are precisely the diagrams of the response-field formalism that governs critical dynamics, driven diffusion, and surface growth. This pattern recurs wherever a Euclidean theory is reconstructed as the equilibrium of a noisy process.

Full proof set Master

Proposition (causality of the free propagator forces the rooted-tree structure). Every term $ϕ_{n}$ of the Picard iteration of $\partial_{τ} ϕ = - K ϕ - \frac{g}{2} ϕ^{2} + η$ is a finite sum of rooted trees whose internal edges are retarded propagators $G_{R}$ , whose internal nodes are cubic vertices, and whose leaves are noise insertions; no $ϕ_{n}$ contains an internal loop.

Proof. Induct on $n$ . For $n = 0$ , $ϕ_{0} = G_{R} * η$ is a single edge from a noise leaf to the root — a tree. Assume the claim for all $m < n$ . The recursion gives $$ \phi_n(x,\tau) = -\tfrac{1}{2}\int\mathrm{d}\tau'\sum_y G_R(x,y;\tau,\tau')!!\sum_{p+q=n-1}!!\phi_p(y,\tau'),\phi_q(y,\tau'). $$ By hypothesis each $ϕ_{p}, ϕ_{q}$ is a sum of rooted trees. The operation forms, for each pair, a new vertex at $(y, τ^{'})$ whose two upper edges are the roots of a $ϕ_{p}$ -tree and a $ϕ_{q}$ -tree, and whose single lower edge $G_{R} (x, y; τ, τ^{'})$ runs to the root $(x, τ)$ . Distributing the sums, $ϕ_{n}$ is a sum of such objects, each a rooted tree with one more vertex than its subtrees. The new vertex connects two disjoint subtrees by a single edge to a fresh root, so the graph remains acyclic: adding a vertex of degree three that joins two previously disconnected components and the root creates no cycle. Hence every $ϕ_{n}$ is a sum of rooted trees with $G_{R}$ edges and noise leaves, with no internal loop. $□$

Proposition (noise contraction closes trees into Feynman graphs). For a connected correlation function $⟨ ϕ (x_{1}, τ) \dots ϕ (x_{r}, τ)⟩$ , expanding each field in trees and contracting all noise leaves by $⟨ η η ⟩ = 2 δ$ produces a sum over connected graphs in which each contracted leaf pair becomes an internal symmetric line $D$ and each surviving tree edge a retarded line $G_{R}$ ; the loop number of each graph equals the number of independent noise contractions minus the number of trees plus one.

Proof. A connected correlator of $r$ external fields expands as a product of $r$ tree sums, each rooted at an external point. Each tree carries noise leaves; the Gaussian average pairs the leaves two at a time (odd products vanish), each pairing contributing $⟨ η η ⟩ = 2 δ$ and gluing two leaves into a single line. By the stochastic-propagator identity, a glued pair of retarded edges sharing the noise integration variable is exactly $D$ . The retarded edges internal to a tree are unaffected and remain $G_{R}$ . The resulting graph $G$ has vertices = the cubic interaction nodes, lines = the $G_{R}$ tree edges and the $D$ contraction lines, and external legs at the roots. Connectedness of the correlator is preserved because we restrict to pairings that link all $r$ trees into one component. The first Betti number (loop number) of a connected graph is $L = E - V + 1$ in the appropriate counting; each independent noise contraction that is not needed to connect a new tree adds one independent cycle, giving $L = (# contractions) - (# trees) + 1$ . $□$

Proposition (equilibrium limit collapses fictitious-time integrals to static propagators). Let a connected graph from the previous proposition have vertices at fictitious times $τ_{1}, \dots, τ_{V}$ , all external points at common time $τ$ , internal retarded lines $G_{R} (k; τ_{i}, τ_{j}) = e^{- ω_{k} (τ_{i} - τ_{j})} θ (τ_{i} - τ_{j})$ , and contraction lines $D$ . Then $$ \lim_{\tau\to\infty}\int!\Big(\prod_i\mathrm{d}\tau_i\Big),(\text{integrand}) = \Big(\text{the static Euclidean amplitude with each line replaced by }1/\omega_k\Big). $$

Proof. Fix the spatial momenta and perform the $V$ fictitious-time integrals. Order the vertices by the retarded $θ$ -functions; for each fixed ordering the integrand is a product of decaying exponentials $e^{- ω (τ_{i} - τ_{j})}$ over the tree edges and the symmetric kernels $D$ over contraction lines. Each $D (k; τ_{i}, τ_{j}) = (e^{- ω_{k} ∣ τ_{i} - τ_{j} ∣} - e^{- ω_{k} (τ_{i} + τ_{j})}) / ω_{k}$ ; as every $τ_{i} \to \infty$ with differences held in the integration, the second (transient) term vanishes and $D \to e^{- ω_{k} ∣ τ_{i} - τ_{j} ∣} / ω_{k}$ . Integrating the innermost vertex time over its allowed half-line gives a factor $\int_{0}^{\infty} d u e^{- ω u} = 1/ ω$ ; iterating outward, each fictitious-time integral contributes one inverse-frequency factor and the sum over orderings reassembles the symmetric static integrand. The product of $1/ ω$ factors over all lines is the static Euclidean amplitude. The transient boundary terms $\propto e^{- ω τ}$ all vanish in the limit, leaving no initial-condition dependence. $□$

Proposition (one-loop $ϕ^{4}$ self-energy from the tree expansion). For the $ϕ^{4}$ action $S = \frac{1}{2} ϕ K ϕ + \frac{λ}{4 !} ϕ^{4}$ , the order- $λ$ equilibrium self-energy computed by the stochastic expansion is the momentum-independent mass shift $Σ_{1} = \frac{λ}{2} \int_{q} 1/ (q^{2} + m^{2})$ , identical to the standard one-loop tadpole.

Proof. For $ϕ^{4}$ the drift is $- K ϕ - \frac{λ}{3 !} ϕ^{3}$ , so the first correction is $ϕ_{1} = - \frac{1}{3 !} G_{R} * ϕ_{0}^{3}$ . The order- $λ$ piece of $⟨ ϕϕ ⟩$ is $⟨ ϕ_{0} ϕ_{1} ⟩ + (sym)$ . Wick-contracting the four $ϕ_{0}$ fields (external $ϕ_{0}$ with one vertex leg, and the remaining two vertex legs self-contracted), the combinatorial factor is $3$ from choosing the leg that connects to the external line, cancelling part of the $1/3!$ . In the equilibrium limit the external lines give $1/ ω_{k}$ each, the self-contraction gives the tadpole loop $\int_{q} D (q; \infty) = \int_{q} 1/ (q^{2} + m^{2})$ , and the retarded vertex line integrates to $1/ ω_{k}$ . Assembling, the amputated self-energy is $Σ_{1} = \frac{λ}{2} \int_{q} 1/ (q^{2} + m^{2})$ , momentum-independent, matching the conventional $ϕ^{4}$ one-loop tadpole exactly. $□$

Connections Master

Langevin updates and lattice numerics 08.10.08. That unit proves the numerical equilibrium identification — the Langevin trajectory samples $e^{- S} / Z$ — and runs it on a computer. The present unit is its analytic counterpart: instead of integrating the noisy equation numerically, one solves it perturbatively, and the same equilibrium limit that the simulation reaches by waiting is the $τ \to \infty$ limit that collapses the tree diagrams to Feynman amplitudes. The Zwanziger stochastic gauge fixing established there is what makes the present perturbation theory ghost-free.
Fokker-Planck equation and equilibrium distribution 08.10.02. The tree expansion is the trajectory-side dual of the Fokker-Planck operator route. Where 08.10.02 evolves the probability density toward $e^{- S} / Z$ via the operator $L^{*}$ and reads off equilibrium from its kernel, the present unit evolves the field trajectory via the retarded propagator and reads off correlators from the equal-time limit. Both converge to the same Gibbs measure; the retarded propagator $G_{R}$ here is the Green function of the same generator whose adjoint is the Fokker-Planck operator there.
The Martin-Siggia-Rose response-field formalism 08.10.14. The tree expansion is the MSR diagrammatics. Introducing the conjugate response field $\hat{ϕ}$ turns the noise-averaged Langevin dynamics into a path integral whose propagators are exactly the retarded response $G_{R} = ⟨ ϕ \hat{ϕ} ⟩$ and the correlation $D = ⟨ ϕϕ ⟩$ of this unit, and the cubic vertex $\hat{ϕ} ϕ^{2}$ — one response leg per vertex — enforces the rooted-tree structure. The present unit derives the same diagrams directly from iterating the equation; 08.10.14 packages them as a generating functional.
$ϕ^{4}$ theory and the Dyson series 08.10.03. The ordinary perturbative expansion of $ϕ^{4}$ correlators — the Dyson series and its Feynman diagrams — is precisely what the stochastic tree expansion reproduces in the equilibrium limit. The present unit shows that the same diagrams, with the same symmetry factors, emerge from a manifestly causal fictitious-time process, giving an independent derivation of the Feynman rules of 08.10.03.
One-loop renormalisation in $ϕ^{4}$ 08.10.06. The one-loop tadpole and self-energy that 08.10.06 renormalises are recovered here from the order- $λ$ tree expansion, with the loop integral $\int_{q} 1/ (q^{2} + m^{2})$ arising from a noise contraction rather than a Feynman loop. Stochastic regularisation, by smearing the noise, supplies an ultraviolet regulator for exactly these integrals that respects gauge symmetry, complementing the dimensional and momentum-cutoff schemes of 08.10.06.

Historical & philosophical context Master

Stochastic quantisation was proposed by Giorgio Parisi and Yong-Shi Wu in their 1981 paper Perturbation theory without gauge fixing (Sci. Sin. 24, 483) ^[Parisi-Wu]. Their central claim had two faces. The first, developed numerically in 08.10.08, is that the Euclidean measure $e^{- S} / Z$ is the equilibrium of a Langevin process in a fictitious fifth time. The second, the subject of this unit, is perturbative: the very same Langevin equation, solved order by order in the coupling, generates a diagrammatic perturbation theory that reproduces the ordinary Feynman expansion in the equilibrium limit. Parisi and Wu emphasised the title's promise — without gauge fixing — because the equilibrium measure is gauge-invariant, so the perturbation theory inherits gauge invariance without the Faddeev-Popov ghost machinery. The free building block they iterate around is the Ornstein-Uhlenbeck process, whose retarded Green function $e^{- ω (τ - τ^{'})} θ (τ - τ^{'})$ goes back to Uhlenbeck and Ornstein's 1930 analysis of Brownian motion ^{[Uhlenbeck-Ornstein]}.

The equivalence to ordinary perturbation theory, asserted by Parisi and Wu, was proved in detail by Emmanuel Floratos and John Iliopoulos in 1983 (Nucl. Phys. B 214, 392) ^{[Floratos-Iliopoulos]}, who showed graph by graph that the sum over vertex orderings weighted by the retarded $θ$ -functions reconstructs the symmetric Feynman integrand. Daniel Zwanziger's 1981 stochastic gauge fixing (Nucl. Phys. B 192, 259) ^[Zwanziger] supplied the gauge-theory completion, replacing the Faddeev-Popov determinant by a gauge-orbit drift and evading the Gribov ambiguity. The diagrammatic and regularisation machinery — the rooted-tree organisation, the stochastic propagator $D$ , and stochastic regularisation via smeared noise — was consolidated in Poul Damgaard and Helmuth Hüffel's 1987 Physics Reports review (152, 227) ^{[Damgaard-Huffel]}, still the canonical reference. The tree expansion is also the perturbative face of the response-field formalism introduced by Paul Martin, Eric Siggia, and Harvey Rose in 1973 (Phys. Rev. A 8, 423) ^{[Martin-Siggia-Rose]} for classical statistical dynamics; the recognition that stochastic quantisation and the MSR formalism share the same $(G_{R}, D)$ propagator structure unifies particle-physics stochastic quantisation with condensed-matter critical dynamics. Philosophically, the construction realises a striking idea: the timeless, Euclidean correlation functions of a quantum field theory are the long-time shadow of a manifestly causal classical stochastic process living in one extra dimension — quantum fluctuations re-expressed as the equilibrium fluctuations of a noisy relaxation.

Bibliography Master

@article{ParisiWu1981,
  author  = {Parisi, Giorgio and Wu, Yong-Shi},
  title   = {Perturbation theory without gauge fixing},
  journal = {Scientia Sinica},
  volume  = {24},
  year    = {1981},
  pages   = {483--496}
}

@article{DamgaardHuffel1987,
  author  = {Damgaard, Poul H. and H{\"u}ffel, Helmuth},
  title   = {Stochastic quantization},
  journal = {Physics Reports},
  volume  = {152},
  number  = {5-6},
  year    = {1987},
  pages   = {227--398}
}

@article{FloratosIliopoulos1983,
  author  = {Floratos, Emmanuel G. and Iliopoulos, John},
  title   = {Equivalence of stochastic and canonical quantization in perturbation theory},
  journal = {Nuclear Physics B},
  volume  = {214},
  year    = {1983},
  pages   = {392--404}
}

@article{Zwanziger1981,
  author  = {Zwanziger, Daniel},
  title   = {Covariant quantization of gauge fields without {G}ribov ambiguity},
  journal = {Nuclear Physics B},
  volume  = {192},
  year    = {1981},
  pages   = {259--269}
}

@article{MartinSiggiaRose1973,
  author  = {Martin, P. C. and Siggia, E. D. and Rose, H. A.},
  title   = {Statistical dynamics of classical systems},
  journal = {Physical Review A},
  volume  = {8},
  year    = {1973},
  pages   = {423--437}
}

@article{UhlenbeckOrnstein1930,
  author  = {Uhlenbeck, G. E. and Ornstein, L. S.},
  title   = {On the theory of the {B}rownian motion},
  journal = {Physical Review},
  volume  = {36},
  year    = {1930},
  pages   = {823--841}
}

@book{ZinnJustin2002,
  author    = {Zinn-Justin, Jean},
  title     = {Quantum Field Theory and Critical Phenomena},
  publisher = {Oxford University Press},
  series    = {International Series of Monographs on Physics},
  volume    = {113},
  edition   = {4th},
  year      = {2002}
}

@book{Parisi1988,
  author    = {Parisi, Giorgio},
  title     = {Statistical Field Theory},
  publisher = {Addison-Wesley},
  year      = {1988}
}

Prerequisites

08.10.08
08.10.02

Tier anchors

beginner: Parisi, *Statistical Field Theory* (Addison-Wesley, 1988), Ch. 19 (Langevin equation and the diagram-expansion narrative); Damgaard & Hüffel, *Phys. Rep.* 152 (1987) §§1-2 (motivation for solving the noisy equation order by order); Zee, *Quantum Field Theory in a Nutshell*, 2e (Princeton, 2010), Ch. I.7 (Feynman diagrams as bookkeeping for a perturbative expansion)
intermediate: Damgaard & Hüffel, *Stochastic Quantization*, *Phys. Rep.* 152 (1987) §4 (perturbative solution of the Langevin equation, the retarded propagator, and the equilibrium limit); Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, 4e (Oxford, 2002), Ch. 17 (stochastic quantisation, the response field, and the doubled-field action); Floratos & Iliopoulos, *Nucl. Phys. B* 214 (1983) 392 (equivalence of stochastic and conventional perturbation theory)
master: Parisi & Wu, *Sci. Sin.* 24, 483 (1981) (stochastic quantisation and the perturbative equivalence); Damgaard & Hüffel, *Phys. Rep.* 152, 227 (1987) (the canonical review, §§4-6 on the diagrammatic expansion and stochastic regularisation); Floratos & Iliopoulos, *Nucl. Phys. B* 214, 392 (1983) (proof of equivalence to ordinary perturbation theory); Zwanziger, *Nucl. Phys. B* 192, 259 (1981) (stochastic gauge fixing); Martin, Siggia & Rose, *Phys. Rev. A* 8, 423 (1973) (the response-field functional)

References

Parisi, G. & Wu, Y.-S. — Perturbation theory without gauge fixing · Sci. Sin. 24, 483 (1981) — the founding paper of stochastic quantisation. The Euclidean correlation functions are obtained as equal-fictitious-time limits of the solution of a Langevin equation $\partial_\tau\phi=-\delta S/\delta\phi+\eta$; expanding that solution iteratively in the coupling and contracting the noise reproduces, in the $\tau\to\infty$ limit, the ordinary Euclidean Feynman amplitudes order by order
Damgaard, P. H. & Hüffel, H. — Stochastic quantization · Phys. Rep. 152, 227 (1987), §§4-6 — the canonical review: the perturbative solution of the field Langevin equation, the retarded free propagator $G_R$ and the stochastic propagator $D$, the rooted-tree organisation of the iterative solution, the proof that the equal-time equilibrium limit recovers conventional perturbation theory, and stochastic regularisation
Floratos, E. G. & Iliopoulos, J. — Equivalence of stochastic and canonical quantization in perturbation theory · Nucl. Phys. B 214, 392 (1983) — a direct order-by-order proof that the perturbative expansion generated by the Langevin equation, after the noise contractions and the equal-time equilibrium limit, coincides graph by graph with the standard Euclidean Feynman expansion; clarifies the role of the retarded propagator and the absence of acausal contributions
Zwanziger, D. — Covariant quantization of gauge fields without Gribov ambiguity · Nucl. Phys. B 192, 259 (1981) — stochastic gauge fixing: a drift along gauge orbits added to the Langevin equation provides a Lorentz-covariant gauge fixing that requires no Faddeev-Popov ghosts and evades the Gribov ambiguity; the stochastic perturbation expansion inherits this ghost-free structure
Martin, P. C., Siggia, E. D. & Rose, H. A. — Statistical dynamics of classical systems · Phys. Rev. A 8, 423 (1973) — the response-field (MSR) generating functional for a classical stochastic dynamics: a conjugate field $\hat\phi$ doubles the degrees of freedom and turns the Langevin average into a field-theory path integral whose propagators are exactly the retarded response $G_R=\langle\phi\hat\phi\rangle$ and the correlation $D=\langle\phi\phi\rangle$ of the tree expansion
Uhlenbeck, G. E. & Ornstein, L. S. — On the theory of the Brownian motion · Phys. Rev. 36, 823 (1930) — the Ornstein-Uhlenbeck process: the linear Langevin equation $\dot x=-\gamma x+\eta$ whose retarded Green function $e^{-\gamma(\tau-\tau')}\theta(\tau-\tau')$ and stationary variance are the free building blocks the stochastic perturbation expansion iterates around
Zinn-Justin, J. — Quantum Field Theory and Critical Phenomena · International Series of Monographs on Physics 113 (Oxford, 4th ed., 2002), Ch. 17 — stochastic quantisation, the Langevin equation as a field theory in one extra dimension, the response-field formulation, supersymmetry of the doubled action, and the perturbative equivalence with ordinary field theory

Estimated time

beginner: 20m
intermediate: 50m
master: 90m