Path integral methods for the dynamics of stochastic and disordered systems

The idea of the Stratonovich convention is that, physically, ζ is a noise process with non-zero correlation time, so we should really write $\langle \zeta (t)\zeta (t^{\prime} )\rangle ={C}_{\zeta }(t-t^{\prime} )$ with C_ζ an even function, decaying quickly to zero for $| t-t^{\prime} |$ greater than some small correlation time ${\tau }_{0}$, and whose integral is $\int {\rm{d}}{t}\,{C}_{\zeta }(t)=2T$. Then, setting the field h to zero as it is no longer needed at this point:

$$\begin{eqnarray}&&\phi (t)=\phi (0){{\rm{e}}}^{-\mu t}+{\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \,{{\rm{e}}}^{-\mu (t-t^{\prime} )}\zeta (t^{\prime} )\end{eqnarray} \tag{ 6 }$$

so

$$\begin{eqnarray}&&\langle \phi (t)\zeta (t)\rangle ={\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \,{{\rm{e}}}^{-\mu (t-t^{\prime} )}{C}_{\zeta }(t-t^{\prime} )\simeq {\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \,{C}_{\zeta }(t-t^{\prime} )\simeq T,\end{eqnarray} \tag{ 7 }$$

where the second equality follows from the fast decay of C_ζ compared to the macroscopic timescales of ${ \mathcal O }(1/\mu )$ so that we can approximate ${{\rm{e}}}^{-\mu (t-t^{\prime} )}=1$. When considering $\langle \phi (t)\zeta (t^{\prime} )\rangle$ with $t\gt t^{\prime}$ (more precisely, $t-t^{\prime} \gg {\tau }_{0}$) on the other hand, all of the 'mass' of the correlation function is captured in the integration range; it therefore acts just like a δ-function and we get $\langle \phi (t)\zeta (t^{\prime} )\rangle =2T\exp [-\mu (t-t^{\prime} )]$ as expected from  (4) and (5).

To summarize, in the Stratonovich convention, the equal-time value R(t,t) = T/(2T) = 1/2 of the response function is half of that obtained in the limit $t^{\prime} \to t-0$. It is therefore also called the midpoint rule; see below.

The Ito convention effectively assumes that the noise $\zeta (t)$ acts 'after $\phi (t)$ has been updated', so it sets $\langle \phi (t)\zeta (t)\rangle ={\mathrm{lim}}_{t^{\prime} \to t+0}\langle \phi (t)\zeta (t^{\prime} )\rangle =0$, and hence also $R(t,t)=0$.

We will later look at path integral representations of the dynamics and so need a discretization of the stochastic process $\phi (t)$. We will now see that Ito and Stratonovich can be viewed as corresponding to different discretization methods.

Let us discretize time $t=n{\rm{\Delta }}$, with Δ a small time step eventually to be taken to zero, and write ${\phi }_{n}=\phi (t=n{\rm{\Delta }})$ and ${h}_{n}=h(t=n{\rm{\Delta }})$. The noise variables over the interval Δ are ${\zeta }_{n}={\int }_{n{\rm{\Delta }}}^{(n+1){\rm{\Delta }}}{\rm{d}}t\,\zeta (t)$, with $\langle {\zeta }_{m}{\zeta }_{n}\rangle =2T{\rm{\Delta }}{\delta }_{{mn}}$, where ${\delta }_{{mn}}$ is the Kronecker symbol. Then a suitable discrete version of  (1) is

$$\begin{eqnarray}&&{\phi }_{n+1}-{\phi }_{n}={\rm{\Delta }}[(1-\lambda )(-\mu {\phi }_{n}+{h}_{n})+\lambda (-\mu {\phi }_{n+1}+{h}_{n+1})]+{\zeta }_{n}\end{eqnarray} \tag{ 8 }$$

for any $\lambda \in [0,1];$ here we are evaluating (the non-noise part of) the right-hand side of  (1) as a weighted combination of the values at the two ends of the interval Δ. It is easy to solve this linear recursion exactly: from

$$\begin{eqnarray}&&{\phi }_{n+1}[1+{\rm{\Delta }}\lambda \mu ]={\phi }_{n}[1+{\rm{\Delta }}(\lambda -1)\mu ]+{\rm{\Delta }}[(1-\lambda ){h}_{n}+\lambda {h}_{n+1}]+{\zeta }_{n},\end{eqnarray} \tag{ 9 }$$

setting

$$\begin{eqnarray}&&c=\displaystyle \frac{1+{\rm{\Delta }}(\lambda -1)\mu }{1+{\rm{\Delta }}\lambda \mu }\end{eqnarray} \tag{ 10 }$$

and assuming the initial condition ${\phi }_{0}=0$, we have

$$\begin{eqnarray}&&{\phi }_{n}=\displaystyle \sum _{m=0}^{n-1}{c}^{n-m-1}\displaystyle \frac{{\rm{\Delta }}[(1-\lambda ){h}_{m}+\lambda {h}_{m+1}]+{\zeta }_{m}}{1+{\rm{\Delta }}\lambda \mu },\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\langle {\phi }_{n}\rangle =\displaystyle \frac{{\rm{\Delta }}}{1+{\rm{\Delta }}\lambda \mu }\left\{\lambda {h}_{n}+\displaystyle \sum _{m=1}^{n-1}\displaystyle \frac{{c}^{n-m-1}}{1+{\rm{\Delta }}\lambda \mu }{h}_{m}+{c}^{n-1}(1-\lambda ){h}_{0}\right\}.\end{eqnarray} \tag{ 12 }$$

From this we can read off the response function ${R}_{{nm}}=\partial \langle {\phi }_{n}\rangle /\partial ({\rm{\Delta }}{h}_{m});$ setting $n=t/{\rm{\Delta }}$, $m=t^{\prime} /{\rm{\Delta }}$ and taking ${\rm{\Delta }}\to 0$ we then get for the continuous time response

$$\begin{eqnarray}R(t,t^{\prime} )=\left\{\begin{array}{lll}0 & {\rm{for}} & t\lt t^{\prime} ,\\ \lambda & {\rm{for}} & t=t^{\prime} ,\\ \exp [-\mu (t-t^{\prime} )] & {\rm{for}} & t\gt t^{\prime} \mathrm{.}\end{array}\right.\end{eqnarray} \tag{ 13 }$$

The value of λ only affects the equal-time response; we see that $\lambda =1/2$ gives the Stratonovich convention, while $\lambda =0$ gives Ito. Note that in the discretization  (8), $\lambda =1/2$ corresponds to evaluating the change in ϕ at the midpoint of the interval between n and $n+1$ (hence 'midpoint rule'). Ito ($\lambda =0$) on the other hand just evaluates at the earlier time. Note also that for different times, $t\ne t^{\prime}$, the two discretization schemes are equivalent as expected.

The two conventions for multiplying equal-time fluctuations are, for the systems we will look at, simply different ways of describing the same time evolution $\phi (t)$. Cases where the noise strength is coupled to ϕ, such as in $\dot{\phi }=f(\phi )+g(\phi )\zeta$, are more serious: a convention for the equal-time product $g(\phi )\zeta$ has to be adopted, and the two conventions here actually give different stochastic processes $\phi (t);$ the corresponding Fokker–Planck equations differ by a non-trivial drift term [18].

For the simple cases we consider, where the noise power T does not depend on t or ϕ, Ito versus Stratonovich is basically a matter of taste—calculated expectation values of physical observables are the same in the two approaches. Stratonovich is the more 'physical' because it corresponds to a noise process ζ with small but non-zero correlation time; it also obeys all the usual rules for transformation of variables etc. Ito is more obvious from the discretized point of view—it is very much what one might naively program in a simulation. We will also see below that it can lead to technical simplifications in calculations.

Diagrams are just a pictorial way of keeping track of the various terms in the perturbation expansion (38), as evaluated using Wick's theorem. Having illustrated the equivalence between Ito and Stratonovich above, we stick to Ito ($\lambda =0$) for now, where the non-trivial part of the action is simply

$$\begin{eqnarray}&&{S}_{\mathrm{int}}=\displaystyle \frac{g}{6}{\rm{\Delta }}\displaystyle \sum _{n}{\eta }_{2n}{\eta }_{1n}^{3}.\end{eqnarray} \tag{ 44 }$$

To illustrate the diagrammatic notation, consider again the expansion for the normalization factor

$$\begin{eqnarray}&&\langle 1\rangle =\langle 1{\rangle }_{0}-\langle {S}_{\mathrm{int}}{\rangle }_{0}+\displaystyle \frac{1}{2}\langle {S}_{\mathrm{int}}^{2}{\rangle }_{0}+\cdots .\end{eqnarray} \tag{ 45 }$$

We have dealt with the zeroth and first order terms above. The second order term is

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\left(-\displaystyle \frac{g}{6}\right)}^{2}{{\rm{\Delta }}}^{2}\displaystyle \sum _{m,n}{\langle {\eta }_{2m}{\eta }_{1m}^{3}{\eta }_{2n}{\eta }_{1n}^{3}\rangle }_{0}.\end{eqnarray} \tag{ 46 }$$

Represent each of the summed over time indices m and n by a vertex with four 'legs' that symbolize the four η factors with the corresponding time index. Each vertex comes with a factor $-g/6$ from $-{S}_{\mathrm{int}}$. Both of the time indices are summed over and the result multiplied by Δ; in the limit ${\rm{\Delta }}\to 0$ these scaled sums of course become time integrals. The time indices are not fixed by our choice of observable to average, and are therefore called 'internal'—we will see external vertices in a moment. Now, having drawn the vertices, we can connect the legs in a number of ways, one for each pairing that Wick's theorem gives for the average in (46): a line between a vertex at m and one at n stands for a pair average $\langle {\eta }_{{\alpha }{m}}{\eta }_{{\beta }{n}}\rangle$ (${\alpha },{\beta }=1$ or 2). E.g. the diagram where the legs from both vertices are connected to legs from the same vertex only

represents all the pairings where ${\eta }_{n}$'s are connected to ${\eta }_{n}$'s only, and ${\eta }_{m}$'s to ${\eta }_{m}$'s only. At each vertex there are three choices for pairings of this form, so this diagram has the value

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{2}{\left(-\displaystyle \frac{g}{6}\right)}^{2}{{\rm{\Delta }}}^{2}\displaystyle \sum _{m,n}\,3\langle {\eta }_{2m}{\eta }_{1m}{\rangle }_{0}\langle {\eta }_{1m}{\eta }_{1m}{\rangle }_{0}\times 3\langle {\eta }_{2n}{\eta }_{1n}{\rangle }_{0}\langle {\eta }_{1n}{\eta }_{1n}{\rangle }_{0}\\ \quad =\displaystyle \frac{1}{2}{\left\{\left(-\displaystyle \frac{g}{6}\right){\rm{\Delta }}\displaystyle \sum _{n}3\langle {\eta }_{2n}{\eta }_{1n}{\rangle }_{0}\langle {\eta }_{1n}{\eta }_{1n}{\rangle }_{0}\right\}}^{2}\mathrm{.}\end{array}\end{eqnarray} \tag{ 48 }$$

This illustrates an important fact: if a diagram separates into subparts which are not connected, its value is just the product of the two diagrams separately (apart from the overall factor of 1/2 here, which comes from the expansion of $\exp (-{S}_{\mathrm{int}})$).

The diagram above certainly does not exhaust all the Wick pairings of  (46). So what are the other diagrams corresponding to  (46)? We can have either two mn-pairings, one nn and one mm; or four mn pairings. Altogether, one would therefore write  (46) in diagrams as:

We write down the value of the last of these: start with ${\eta }_{2m}$. From the diagram, this must be connected to a η on the other vertex, i.e. either ${\eta }_{2n}$ or one of the three ${\eta }_{1n}$. In the first case, each of the remaining ${\eta }_{1m}$ legs is connected to a ${\eta }_{1n}$ leg; there are $3\times 2\times 1=6$ ways of making those connections (pairings). In the second case, the ${\eta }_{2}$ factor from the second vertex, ${\eta }_{2n}$, must be connected to one of the three ${\eta }_{1m}$ vertices, and the remaining two ${\eta }_{1m}$ and ${\eta }_{1n}$ legs can be connected to each other in two ways. Thus the value of the diagram is:

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{(-g/6)}^{2}{{\rm{\Delta }}}^{2}\displaystyle \sum _{m,n}\{6\langle {\eta }_{2m}{\eta }_{2n}{\rangle }_{0}{(\langle {\eta }_{1m}{\eta }_{1n}{\rangle }_{0})}^{3}+18\langle {\eta }_{2m}{\eta }_{1n}{\rangle }_{0}\langle {\eta }_{1m}{\eta }_{2n}{\rangle }_{0}{(\langle {\eta }_{1m}{\eta }_{1n}{\rangle }_{0})}^{2}\}.\end{eqnarray} \tag{ 50 }$$

(Exercise: evaluate the remaining second order diagram and, as a check, count all pairings. The diagram just above dealt with $6+18=24$ pairings, while the first one corresponded to $3\times 3=9$ pairings. Since we have 8 η's in total, there are $7\times 5\times 3\times 1\,=8!/(4!2{!}^{4})=105$ pairings overall, hence the remaining diagram must correspond to $105-24-9=72$ pairings.)

In terms of diagrams, it is now quite easy to understand that $\langle 1{\rangle }_{S}=1$ as it should to all orders in g. Consider an arbitrary diagram in the expansion; let us assume it is connected, otherwise consider the subdiagrams separately. Now within the Ito convention the response function $\langle {\eta }_{1n}{\eta }_{2m}{\rangle }_{0}$ is non-zero only for $n\gt m$ (for Stratonovich, the value for n = m is also non-zero; for $n\lt m$ it is zero either way from causality). So to make the diagram non-zero, we have to connect the ${\eta }_{2{n}_{1}}$ from a given vertex, with time index n₁, say, to the ${\eta }_{1{n}_{2}}$ leg at another vertex, n₂. The ${\eta }_{2{n}_{2}}$ from that vertex must in turn be connected to ${\eta }_{1{n}_{3}}$ on another vertex and so on. Eventually, because we have a finite number of vertices, we must come back to our original vertex n₁. In the 'ring' sequence n₁, n₂, n₃, ..., n₁ there are at least two time indices that are in the 'wrong' order for the response function to be non-zero, so that the diagram contains at least one vanishing factor and thus vanishes itself.

The moral of the story so far is: diagrams factorize over disconnected sub-diagrams, and the sub-diagrams with only internal (summed-over) vertices vanish. The latter are also called 'vacuum diagrams' in field theory.

Next we look at the diagrams for the propagator, which encapsulates correlation and response functions, $\langle {\eta }_{\alpha i}{\eta }_{\beta j}{\rangle }_{S}$ where $\alpha ,\beta \in \{1,2\}$. We now have two 'external' vertices with fixed time indices i and j; the propagator is represented as a double line between these two vertices. It is also called the 'full' or 'dressed' propagator, in contrast to the 'bare' propagator that is represented by the single lines in the diagrams and results from the pairings in Wick's theorem. To zeroth order in g, full and bare propagator are obviously equal. To first order, we have, from  (38),

$$\begin{eqnarray}&&{G}_{\alpha i,\beta j}=\langle {\eta }_{\alpha i}{\eta }_{\beta j}{\rangle }_{S}=\langle {\eta }_{\alpha i}{\eta }_{\beta j}{\rangle }_{0}-\langle {\eta }_{\alpha i}{\eta }_{\beta j}{S}_{\mathrm{int}}{\rangle }_{0}+\cdots \end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&=\,\langle {\eta }_{\alpha i}{\eta }_{\beta j}{\rangle }_{0}-(g/6){\rm{\Delta }}\displaystyle \sum _{n}\langle {\eta }_{\alpha i}{\eta }_{\beta j}{\eta }_{2n}{\eta }_{1n}^{3}{\rangle }_{0}+\cdots \end{eqnarray} \tag{ 52 }$$

In diagrams, we have two new contributions from the first order term, depending on whether we pair up ${\eta }_{i}$ with ${\eta }_{j}$ or with one of the ${\eta }_{n}$. We can therefore write the full propagator as

The second of these has a disconnected part with only internal vertices, so vanishes. The same argument applies to higher order diagrams as well: we only need to consider connected diagrams⁷ . Thus, evaluating the surviving diagram,

$$\begin{eqnarray}&&{G}_{\alpha i,\beta j}={G}_{\alpha i,\beta j}^{0}-(g/6){\rm{\Delta }}\displaystyle \sum _{n}\{3{G}_{\alpha i,1n}^{0}{G}_{1n,1n}^{0}{G}_{2n,\beta j}^{0}+3{G}_{\alpha i,2n}^{0}{G}_{1n,1n}^{0}{G}_{1n,\beta j}^{0}\}.\end{eqnarray} \tag{ 54 }$$

If we define a matrix ${{\rm{\Sigma }}}^{1}$ by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\gamma m,\delta n}^{1}=-(g/2){{\rm{\Delta }}}^{-1}{\delta }_{{mn}}{G}_{1n,1n}^{0}({\delta }_{\gamma ,1}{\delta }_{\delta ,2}+{\delta }_{\gamma ,2}{\delta }_{\delta ,1})\end{eqnarray} \tag{ 55 }$$

(where ${\delta }_{{ij}}$ is the Kronecker symbol as before) and agree to absorb a factor of Δ into matrix products, so that

$$\begin{eqnarray}&&{({AB})}_{\alpha i,\gamma k}={\rm{\Delta }}\displaystyle \sum _{\beta ,j}\,{A}_{\alpha i,\beta j}{B}_{\beta j,\gamma k}\end{eqnarray} \tag{ 56 }$$

then we can write  (54) in matrix form simply as

$$\begin{eqnarray}&&G={G}_{0}+{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}.\end{eqnarray} \tag{ 57 }$$

The way factors of Δ are absorbed here ensures that matrix multiplications become time integrals in the natural way for ${\rm{\Delta }}\to 0$, while the factor ${{\rm{\Delta }}}^{-1}{\delta }_{{mn}}$ in ${{\rm{\Sigma }}}^{1}$ becomes $\delta (t-t^{\prime} )$.

To first order in g, the above is the whole story for the propagator. Now look at the second order. Among the diagrams we have ones such as

These two only differ in how the internal vertices are labeled; since the latter are summed over, we can lump the two diagrams together into one unlabeled diagram. This just gives a factor of 2, which cancels exactly the prefactor $1/2!$ from the second order expansion of the exponential in  (38). Again, the same happens at higher orders: at ${ \mathcal O }({g}^{k})$ we have a prefactor of $1/k!$ but also k internal vertices which can be labeled in $k!$ different ways, so the unlabeled diagram has a prefactor of one. Thus, the unlabeled diagram

has the value (bearing in mind that the ${\eta }_{2}$ components at each of the internal vertices must be connected either to a ${\eta }_{1}$ leg at the other internal vertex, or to an external vertex, and that there are three choices at each vertex for which pair of ${\eta }_{1}$'s to connect to each other)

$$\begin{eqnarray}&&\begin{array}{l}3\times 3\times {(-g/6)}^{2}{{\rm{\Delta }}}^{2}\displaystyle \sum _{{mn}}({G}_{\alpha i,1m}^{0}{G}_{1m,1m}^{0}{G}_{2m,1n}^{0}{G}_{1n,1n}^{0}{G}_{2n,\beta j}^{0}\\ \quad +{G}_{\alpha i,1m}^{0}{G}_{1m,1m}^{0}{G}_{2m,2n}^{0}{G}_{1n,1n}^{0}{G}_{1n,\beta j}^{0}\\ \quad +{G}_{\alpha i,2m}^{0}{G}_{1m,1m}^{0}{G}_{1m,1n}^{0}{G}_{1n,1n}^{0}{G}_{2n,\beta j}^{0}\\ \quad +{G}_{\alpha i,2m}^{0}{G}_{1m,1m}^{0}{G}_{1m,2n}^{0}{G}_{1n,1n}^{0}{G}_{1n,\beta j}^{0})\mathrm{.}\end{array}\end{eqnarray} \tag{ 60 }$$

Using the definition  (55), one sees that in matrix form this is simply

$$\begin{eqnarray}&&{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}.\end{eqnarray} \tag{ 61 }$$

To make this simple form more obvious we have included in  (60) the term in the second line, which vanishes because it contains a zero ${G}_{22}^{0}$ factor. We have an example here of a 'one-particle reducible' (1PR) diagram, which can be cut in two by cutting just one bare propagator line, namely, the one in the middle. The result illustrates that the value of such diagrams factorizes into the pieces they can be cut into; e.g. the diagram

has the value ${G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}$. If we sum up all the diagrams of this form, we get

$$\begin{eqnarray}&&G={G}_{0}+{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}+{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}+{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}{{\rm{\Sigma }}}^{1}{G}_{0}\,+\,\ldots \,=\,{[{G}_{0}^{-1}-{{\rm{\Sigma }}}^{1}]}^{-1}\end{eqnarray} \tag{ 63 }$$

or

$$\begin{eqnarray}&&{G}^{-1}={G}_{0}^{-1}-{{\rm{\Sigma }}}^{1}.\end{eqnarray} \tag{ 64 }$$

The inverses are relative to the appropriate unit element ${\boldsymbol{I}}$ for our redefined matrix multiplication, which has elements ${I}_{\alpha i,\beta j}={{\rm{\Delta }}}^{-1}{\delta }_{\alpha \beta }{\delta }_{{ij}}$. (So ${A}^{-1}A={\boldsymbol{I}}$ means, because of the extra factor Δ in the matrix multiplication, that ${A}^{-1}$ is ${{\rm{\Delta }}}^{-2}$ times the conventional matrix inverse of A.)

The expression  (64) is an example of a resummation: we have managed to sum up an infinite subseries from among all the diagrams for the propagator. What if we want to sum up all the diagrams?

Again we can classify them into one-particle irreducible (1PI) diagrams, which cannot be cut in two by cutting a single line, and 1PR diagrams that factorize into their 1PI components. For example, to second order

More generally, if we denote the values of the different 1PI diagrams (defined analogously to ${{\rm{\Sigma }}}^{1}$, i.e. without the external G₀ legs) by ${{\rm{\Sigma }}}^{1}$, ${{\rm{\Sigma }}}^{2}$, ... then we can get all possible (1PI and 1PR) diagrams by 'stringing' together all possible combinations of 1PI diagrams:

$$\begin{eqnarray}&&G={G}_{0}\displaystyle \sum _{k=0}^{\infty }\,\displaystyle \sum _{{i}_{1}\ldots {i}_{k}}{{\rm{\Sigma }}}^{{i}_{1}}{G}_{0}\cdots {{\rm{\Sigma }}}^{{i}_{k}}{G}_{0}={G}_{0}\displaystyle \sum _{k=0}^{\infty }{({\rm{\Sigma }}{G}_{0})}^{k}={({G}_{0}^{-1}-{\rm{\Sigma }})}^{-1},\end{eqnarray} \tag{ 67 }$$

where ${\rm{\Sigma }}={\sum }_{i=1}^{\infty }{{\rm{\Sigma }}}^{i}$. Hence we see that the full propagator, with all diagrams summed up, can be written in the general form

$$\begin{eqnarray}&&{G}^{-1}={G}_{0}^{-1}-{\rm{\Sigma }},\end{eqnarray} \tag{ 68 }$$

where Σ, the so-called 'self-energy', is the sum of all 1PI diagrams. Equation  (68) is called the Dyson equation. An alternative form that is often useful is

$$\begin{eqnarray}&&{G}_{0}^{-1}G={\boldsymbol{I}}+{\rm{\Sigma }}G.\end{eqnarray} \tag{ 69 }$$

Let us write this out in terms of the separate blocks corresponding to correlation and response functions: we have

$$\begin{eqnarray}{G}_{0}=\left(\begin{array}{cc}{C}_{0} & {R}_{0}\\ {R}_{0}^{{\rm{T}}} & 0\end{array}\right)\qquad \Rightarrow \qquad {G}_{0}^{-1}=\left(\begin{array}{cc}0 & {({R}_{0}^{-1})}^{{\rm{T}}}\\ {R}_{0}^{-1} & -{R}_{0}^{-1}{C}_{0}{({R}_{0}^{-1})}^{{\rm{T}}}\end{array}\right)\end{eqnarray} \tag{ 70 }$$

and ${G}^{-1}$ has the same structure, so that also the 11 block of Σ must vanish,

$$\begin{eqnarray}{\rm{\Sigma }}=\left(\begin{array}{cc}0 & {{\rm{\Sigma }}}_{12}\\ {{\rm{\Sigma }}}_{12}^{{\rm{T}}} & {{\rm{\Sigma }}}_{22}\end{array}\right).\end{eqnarray} \tag{ 71 }$$

This can also be shown diagrammatically: a non-zero contribution to ${{\rm{\Sigma }}}_{11}$ would correspond to diagrams where the internal vertices that make connections to the two external vertices do so via ${\eta }_{1}$ legs. This leaves all the ${\eta }_{2}$ legs to be connected amongst the internal vertices, and then the same argument as for vacuum diagrams can be applied. Writing out  (69) we have thus

$$\begin{eqnarray}&&{({R}_{0}^{-1})}^{{\rm{T}}}{R}^{{\rm{T}}}={\boldsymbol{I}}+{{\rm{\Sigma }}}_{12}{R}^{{\rm{T}}},\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&0=0,\end{eqnarray} \tag{ 73 }$$

$$\begin{eqnarray}&&{R}_{0}^{-1}C-{R}_{0}^{-1}{C}_{0}{({R}_{0}^{-1})}^{{\rm{T}}}{R}^{{\rm{T}}}={{\rm{\Sigma }}}_{12}^{{\rm{T}}}C+{{\rm{\Sigma }}}_{22}{R}^{{\rm{T}}},\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&{R}_{0}^{-1}R={\boldsymbol{I}}+{{\rm{\Sigma }}}_{12}^{{\rm{T}}}R,\end{eqnarray} \tag{ 75 }$$

where we have abused the notation by writing ${\boldsymbol{I}}$ also for the non-zero M × M sub-blocks of the original $2M\times 2M$ matrix ${\boldsymbol{I}}$. The first and last of these equations are both equivalent to

$$\begin{eqnarray}&&{R}^{-1}={R}_{0}^{-1}-{{\rm{\Sigma }}}_{12}^{{\rm{T}}}\end{eqnarray} \tag{ 76 }$$

implying that ${{\rm{\Sigma }}}_{12}^{{\rm{T}}}$ acts like a self-energy for the reponse function. Rearranging, the components of the Dyson equation reduce to

$$\begin{eqnarray}&&{R}_{0}^{-1}R={{\rm{\Sigma }}}_{12}^{{\rm{T}}}R+{\boldsymbol{I}},\end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray}&&{R}_{0}^{-1}C={{\rm{\Sigma }}}_{12}^{{\rm{T}}}C+[{R}_{0}^{-1}{C}_{0}{({R}_{0}^{-1})}^{{\rm{T}}}+{{\rm{\Sigma }}}_{22}]{R}^{{\rm{T}}}.\end{eqnarray} \tag{ 78 }$$

Using  (76), the last equation can also be solved explicitly for C as

$$\begin{eqnarray}&&C=R[{R}_{0}^{-1}{C}_{0}{({R}_{0}^{-1})}^{{\rm{T}}}+{{\rm{\Sigma }}}_{22}]{R}^{{\rm{T}}}.\end{eqnarray} \tag{ 79 }$$

In the above we have defined Σ to be plus the sum of all 1PI diagrams; the opposite sign convention also appears in the literature.

Of course in general one cannot sum up all the diagrams for the self-energy. One must make some approximation. In  (64) we used just the lowest order diagram to approximate Σ. Can we easily improve the approximation? Yes, if we replace G₀ in the expression for Σ by the full propagator G; diagrammatically,

Which diagrams does this correspond to? This is easiest to find out order by order in g. To first order, Σ and G are

Re-inserting G into Σ we get its form to second order and therefore also G

and then we can iterate to get

So the simple operation of replacing G₀ by G effectively sums up an infinite series of 'tadpole' diagrams in the expansion for the self-energy. This is called 'mean field theory'. It gives exact results for many models with weak long-ranged interactions; for such models the diagrams that have not been included are negligible in the thermodynamic limit. The approximation is also called 'self-consistent one-loop' or 'Hartree–Fock' approximation.

We have used a particular way of drawing diagrams above that does not distinguish between the physical field $\phi ={\eta }_{1}$ and the conjugate or 'response' field $\hat{\phi }={\eta }_{2}$. This approach has the virtue of making all diagrams look essentially like in a static (equilibrium) ${\phi }^{4}$ field theory. It does however mean that each diagram can group a number of terms, involving response or correlation functions depending where on each vertex the $\hat{\phi }$ variables are located in any given Wick pairing. If one wants to avoid this one can e.g. mark on each vertex the $\hat{\phi }$-leg by an outgoing arrow. Similarly in the response part of the propagator—the ${\eta }_{1}{\eta }_{2}$ sector in our previous convention—one would mark the $\hat{\phi }$ end by an arrow. The arrows then represent the flow of time because in all contractions of the type $\langle \hat{\phi }\phi \rangle$ the $\hat{\phi }$ must be at a time before the ϕ.

The rules for constructing diagrams are modified only slightly in this more detailed diagrammatic representation: an arrow from a vertex cannot connect back to the same vertex as this would give an equal-time response function, which vanishes in the Ito convention. At each vertex one has to consider the possible choices for how the leg with an arrow can be connected to other vertices or external nodes. Diagrams are non-vanishing only if the arrows give a consistent flow of time, i.e. arrows cannot form closed loops nor can two legs with outgoing arrows be connected.

Within the above convention, a propagator without an arrow on it is always a correlation function. In our example, the diagrammatic expansion up to second order of the correlation function then reads

while for the response function there are fewer diagrams that contribute:

Note that the expansions (85) and (86) are written as the direct analogs of (65) and differ from the latter only through the addition of the appropriate arrows. More compact expressions can be obtained in terms of the self-energy. For the correlation function specifically, the identity (79) may be more efficient for use in practical calculations. This is illustrated in the example in section 6 below, see (106) there.

So far we have discussed dynamics without applied fields h_n and with zero initial value ${\phi }_{0}$. These restrictions can be lifted, as we now explain. We continue to use the Ito convention ($\lambda =0$). It suffices to discuss non-zero fields, because a non-zero initial value ${\phi }_{0}=c$ can be produced by setting ${h}_{0}=c/{\rm{\Delta }}$. This gives ${\phi }_{1}=c+{ \mathcal O }({{\rm{\Delta }}}^{1/2})$, which in the continuous time limit ${\rm{\Delta }}\to 0$ approaches c. In the same limit the difference between ${\phi }_{0}$ and ${\phi }_{1}$ is immaterial, so the above choice of h₀ effectively fixes a non-zero initial condition.

For the perturbative approach to work, we need a quadratic action S₀ as our baseline while all non-quadratic terms are contained in the interacting part ${S}_{\mathrm{int}}$. If the field h_n is non-zero, this generates linear terms in the action. We then have two choices: either we include these linear terms in ${S}_{\mathrm{int}}$ and treat them perturbatively, or we transform variables by expanding the full action around a stationary or saddle point.

The first approach is relatively straightforward: one now has an extra vertex, with only one $\hat{\phi }$-leg, and the perturbative expansion is jointly in g and the field amplitude. For the propagator all contributing diagrams must have a total number of legs on the internal vertices that is even so the number of field vertices must also be even, hence the expansion is effectively in g and h². At ${ \mathcal O }({h}^{2})$ one has the extra diagram

where each dot with a single connection is an h-vertex⁸ , while at ${ \mathcal O }({h}^{2}g)$ one gets

We could represent the fact that all field vertices have a single $\hat{\phi }$-leg by an arrow pointing away from the vertex (see section 5.4). For the response function part of the propagator there would be an arrow pointing away from one of the external vertices as well so the ${ \mathcal O }({h}^{2})$ diagram in (87) vanishes as it contains an edge with opposing arrows. To a correlator $C(t,t^{\prime} )$ the diagram contributes in the continuous time limit

$$\begin{eqnarray}&&\displaystyle \int {{\rm{d}}{t}}_{1}{{\rm{d}}{t}}_{2}\,{R}_{0}(t,{t}_{1})h({t}_{1}){R}_{0}(t^{\prime} ,{t}_{2})h({t}_{2}).\end{eqnarray} \tag{ 89 }$$

The higher order diagrams can be evaluated similarly.

In the presence of a field even the mean $\langle {\phi }_{n}\rangle$ is in general non-zero. The perturbative expansion for this consists of diagrams with one external vertex and therefore requires an odd number of internal field vertices. The contributions to ${ \mathcal O }(h)$ and ${ \mathcal O }({hg})$ are

One sees that most of the new diagrams (87) and (88) in the propagator are products of diagrams like these. In fact one can convince oneself that if one considers the connected propagator, defined as $\langle {\eta }_{\alpha i}{\eta }_{\beta j}\rangle -\langle {\eta }_{\alpha i}\rangle \langle {\eta }_{\beta j}\rangle$, then as in the case without a field only the connected diagrams remain [19]; the lowest order one of these is the third diagram in (88).

Next we look at the saddle point approach, where one defines new variables relative to a stationary point of the action, thus eliminating any linear terms in the transformed action. Looking at (23) with ${\psi }_{n}=0$, the saddle point conditions with respect to ${\phi }_{n}$ and ${\hat{\phi }}_{n}$ give

$$\begin{eqnarray*}0 & = & {\hat{\phi }}_{n}-{\hat{\phi }}_{n-1}+{\hat{\phi }}_{n}{\rm{\Delta }}{f}_{n}^{\prime },\\ 0 & = & -2T{\hat{\phi }}_{n}+{\rm{i}}[-{\phi }_{n+1}+{\phi }_{n}+{\rm{\Delta }}({f}_{n}+{h}_{n})]\mathrm{.}\end{eqnarray*}$$

The 'initial' (at $n=M-1$) condition for the first of these equations, which results from stationarity with respect to ${\phi }_{M}$, is ${\hat{\phi }}_{M-1}=0$, and solving backwards in time then shows that ${\hat{\phi }}_{n}=0$ for all n at the saddle point. With this the second equation is just

$$\begin{eqnarray}&&{\phi }_{n+1}-{\phi }_{n}={\rm{\Delta }}({f}_{n}+{h}_{n})\end{eqnarray} \tag{ 91 }$$

which is the discrete time version of the deterministic (noise-free) time evolution $\dot{\phi }=f(\phi )+h$. If we call the solution of this ${\phi }^{* }$, then we need to use as new variables $\delta \phi =\phi -{\phi }^{* }$. In terms of these, by making use of the saddle point conditions, the (Ito) action reads

$$\begin{eqnarray}&&S=\displaystyle \sum _{n=0}^{M-1}\{T{\rm{\Delta }}{\hat{\phi }}_{n}^{2}-{\rm{i}}{\hat{\phi }}_{n}[-\delta {\phi }_{n+1}+\delta {\phi }_{n}+{\rm{\Delta }}\delta {f}_{n}]\}.\end{eqnarray} \tag{ 92 }$$

The difference $\delta {f}_{n}={f}_{n}-{f}_{n}^{* }\,=\,f({\phi }_{n}^{* }+\delta {\phi }_{n})-f({\phi }_{n}^{* })$ needs to be expanded in $\delta {\phi }_{n}$ to read off the resulting vertices in the interacting part of the action. For our example (32),

$$\begin{eqnarray}&&\delta f=-\mu \delta \phi -\displaystyle \frac{g}{3!}(3{\phi }^{* }{}^{2}\delta \phi +3{\phi }^{* }\delta {\phi }^{2}+\delta {\phi }^{3}).\end{eqnarray} \tag{ 93 }$$

If we include the $-\mu \delta \phi$ term in the unperturbed action S₀ then the latter has the same form as originally except for the change of variable from ϕ to $\delta \phi$, and the same diagrammatic expansion technique can be applied. The difference is that the interaction part ${S}_{\mathrm{int}}$ now contains three different kinds of vertices resulting from the terms proportional to g in (93), with two, three and four legs respectively and different time-dependent prefactors from the time dependence of ${\phi }^{* }$.

One point to bear in mind is that even though we have defined $\delta \phi$ relative to the saddle point solution, its average is not in general zero. To ${ \mathcal O }(g)$, for example, one has for $\langle \delta \phi (t)\rangle$ the diagram

coming from the $-(g/2){\phi }^{* }\delta {\phi }^{2}$ term in  (93), which evaluates to

$$\begin{eqnarray}&&\langle \delta \phi (t)\rangle =-\displaystyle \frac{g}{2}\int {\rm{d}}{t}^{\prime} \,{R}_{0}(t,t^{\prime} ){\phi }^{* }(t^{\prime} ){C}_{0}(t^{\prime} ,t^{\prime} ).\end{eqnarray} \tag{ 95 }$$

What if we want to improve the approximation to the self-energy further? The systematic approach is to include the lowest-order diagram not so far taken into account. We have the only first-order diagram already; the second-order 'tadpole' diagrams are also taken into account through self-consistency. The only missing second order diagram is therefore the 'watermelon' diagram

To work out what contribution to Σ this gives, let us revert temporarily to discrete time notation and label the left and right vertex m and n, respectively. The elements ${{\rm{\Sigma }}}_{1m,2n}$ of ${{\rm{\Sigma }}}_{12}$ correspond to those pairings where a ${\eta }_{1m}$ leg from vertex m is attached to an external vertex that would be on the left, and the ${\eta }_{2n}$ leg from vertex n attached to an external vertex on the right. Internally (among the remaining legs) we thus have two ${\eta }_{1m}{\eta }_{1n}$ pairings and one ${\eta }_{2m}{\eta }_{1n}$ pairing. To work out the prefactor of the diagram, note that there are three choices for the externally attached ${\eta }_{1m};$ there are three choices for which of the ${\eta }_{1n}$ to pair up with ${\eta }_{2m};$ and two more choices for how to make the two remaining ${\eta }_{1m}{\eta }_{1n}$ pairings. Thus, the diagram gives for ${{\rm{\Sigma }}}_{12}$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{1m,2n}=3\times 3\times 2{(-g/6)}^{2}{({C}_{{mn}}^{0})}^{2}{R}_{{nm}}^{0}=({g}^{2}/2){({C}_{{mn}}^{0})}^{2}{R}_{{nm}}^{0}.\end{eqnarray} \tag{ 114 }$$

For ${{\rm{\Sigma }}}_{22}$, we have both the ${\eta }_{2m}$ and ${\eta }_{2n}$ legs attached externally, and 6 choices for how to connect the three ${\eta }_{1m}$ and ${\eta }_{1n}$ legs internally, giving

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{2m,2n}=6{(-g/6)}^{2}{({C}_{{mn}}^{0})}^{3}=({g}^{2}/6){({C}_{{mn}}^{0})}^{3}.\end{eqnarray} \tag{ 115 }$$

We can again sum an infinite series of additional diagrams by replacing bare (${C}_{0},{R}_{0}$) by full quantities ($C,R$) here. Reverting to continuous time notation and including the first-order contribution in ${{\rm{\Sigma }}}_{12}$, we thus get for the self-energy in the self-consistent two-loop approximation

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{12}(t,t^{\prime} )=-(g/2)\delta (t-t^{\prime} )C(t,t)+({g}^{2}/2){C}^{2}(t,t^{\prime} )R(t^{\prime} ,t),\end{eqnarray} \tag{ 116 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{22}(t,t^{\prime} )=({g}^{2}/6){C}^{3}(t,t^{\prime} ).\end{eqnarray} \tag{ 117 }$$

A final comment: up to the order which we have considered, the self energy components ${{\rm{\Sigma }}}_{12}(t,t^{\prime} )$ and ${{\rm{\Sigma }}}_{22}(t,t^{\prime} )$ are simple functions of the correlation and response functions. This is not normally true once higher order diagrams are taken into account. For example, if we went to third order in g we would have to include the diagram

in the self energy; all other third order diagrams are automatically included by self-consistency. This diagram now has one internal vertex whose time index is not fixed by the two time indices that the self-energy carries. It therefore gives a contribution to ${{\rm{\Sigma }}}_{..}(t,t^{\prime} )$ which has an integral over this 'internal' time; e.g. for ${{\rm{\Sigma }}}_{22}$ we get a contribution

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{22}(t,t^{\prime} )\propto C(t,t^{\prime} )\int {\rm{d}}{t}^{\prime\prime} C(t,t^{\prime\prime} )C(t^{\prime} ,t^{\prime\prime} )[R(t,t^{\prime\prime} )C(t^{\prime} ,t^{\prime\prime} )+R(t^{\prime} ,t^{\prime\prime} )C(t,t^{\prime\prime} )].\end{eqnarray} \tag{ 119 }$$

Diagrams of even higher order contain additional time integrals; in general, then, the self-energy is a functional of the response and correlation functions.

An approximation, like that in (116) and (117), in which we keep only diagrams for ${\rm{\Sigma }}(t,t^{\prime} )$ that are functions of $G(t,t^{\prime} )$ and $C(t,t^{\prime} )$ has come to be called a mode coupling approximation. These have been studied extensively, often because they are exact for some mean-field (infinite-range) models [20].

In section 4, we discussed how the results of the perturbation theory are independent of the value of λ chosen in the discretization of the Langevin equation: the choice of λ changes the Jacobian but does not affect correlation functions and other observables. As alluded to already in section 3, there is another way of including the Jacobian in the path integral using Grassmann variables, which is both conceptually interesting and can often simplify notation. This observation was first made, and its consequences on dynamics explored, by Feigel'man and Tsvelik [21, 22], who followed earlier work by Parisi and Sourlas [23] on supersymmetric properties of an equilibrium system in a random external field. The current section is devoted to describing this approach.

Before getting into the details of the supersymmetric formalism, we need to familiarize ourselves with Grassmann variables. Grassmann variables are charaterized by the fact that they anticommute with each other. Multiplication of them is also associative; i.e. for Grassmann variables ${\xi }_{i}$, we have

$$\begin{eqnarray}&&{\xi }_{i}({\xi }_{j}{\xi }_{k})=({\xi }_{i}{\xi }_{j}){\xi }_{k}\ \ {\rm{(associative)}},\end{eqnarray} \tag{ 120 }$$

$$\begin{eqnarray}&&{\xi }_{i}{\xi }_{j}=-{\xi }_{j}{\xi }_{i}\ \ ({\rm{anti}}\text{-}{\rm{commutative}}).\end{eqnarray} \tag{ 121 }$$

A consequence of anti-commutation is that

$$\begin{eqnarray}&&{\xi }_{i}^{n}=0\ \ \forall n\gt 1.\end{eqnarray} \tag{ 122 }$$

Grassmann variables can be added to each other and also multiplied by complex numbers: one says formally that they form an algebra, i.e. a vector field over the complex numbers endowed with a multiplication. This means that, given (122), the most general functions of one and two Grassmann variables can be written respectively as

$$\begin{eqnarray}&&f({\xi }_{1})={c}_{0}+{c}_{1}{\xi }_{1},\end{eqnarray} \tag{ 123 }$$

$$\begin{eqnarray}&&f({\xi }_{1},{\xi }_{2})={c}_{0}+{c}_{1}{\xi }_{1}+{c}_{2}{\xi }_{2}+{c}_{12}{\xi }_{1}{\xi }_{2},\end{eqnarray} \tag{ 124 }$$

where ${c}_{0},{c}_{1},{c}_{2}$ and c₁₂ are arbitrary complex numbers.

Integration and differentiation for Grassmann variables are defined by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{\xi }_{i}}{\xi }_{j}={\delta }_{{ij}},\ \ \ \int {\rm{d}}\xi =0,\ \ \ \int {\rm{d}}\xi \,\xi =1,\end{eqnarray} \tag{ 125 }$$

and these lead to the following formulae that we will use later:

$$\begin{eqnarray}&&\int {\rm{d}}\xi (a+b\xi )=b,\ \ \ \int {\rm{d}}{\xi }_{1}{\rm{d}}{\xi }_{2}\,{\xi }_{2}{\xi }_{1}=1,\end{eqnarray} \tag{ 126 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}\xi }f(\xi )=\displaystyle \frac{{\rm{d}}}{{\rm{d}}\xi }(a+b\xi )=b,\ \ \ \displaystyle \frac{{\rm{d}}}{{\rm{d}}{\xi }_{1}}{\xi }_{2}{\xi }_{1}=-{\xi }_{2}.\end{eqnarray} \tag{ 127 }$$

As a consequence of the above, and using two independent sets of Grassmann variables ${\xi }_{n}$ and ${\overline{\xi }}_{n}$, one has the following important representation of the determinant of a matrix ${\boldsymbol{A}}$:

$$\begin{eqnarray}&&| {\boldsymbol{A}}| =\int D[\xi \overline{\xi }]\exp \left\{\displaystyle \sum _{{mn}}{\overline{\xi }}_{m}{A}_{{mn}}{\xi }_{n}\right\},\end{eqnarray} \tag{ 128 }$$

where $D[\xi \overline{\xi }]={\prod }_{n}{\rm{d}}{\xi }_{n}{\rm{d}}{\overline{\xi }}_{n}$. The integrand on the right-hand side of (128) defines a Gaussian measure for Grassmann variables under which we have $\langle {\xi }_{m}{\overline{\xi }}_{n}\rangle =-{({{\boldsymbol{A}}}^{-1})}_{{mn}}$.

Employing the representation (128) for the determinant of a matrix, we can write the Jacobian that appears in the transformation (18) in the following way. First, we recall that the non-zero elements of the Jacobian are

$$\begin{eqnarray}&&\partial {\eta }_{n}/\partial {\phi }_{n+1}=1-{\rm{\Delta }}\lambda {f}_{n+1}^{\prime },\end{eqnarray} \tag{ 129 }$$

$$\begin{eqnarray}&&\partial {\eta }_{n}/\partial {\phi }_{n}=-1-{\rm{\Delta }}(1-\lambda ){f}_{n}^{\prime }\end{eqnarray} \tag{ 130 }$$

and therefore

$$\begin{eqnarray}&&J[\phi ]=\int D[\xi \overline{\xi }]\exp \left\{\displaystyle \sum _{n}{\overline{\xi }}_{n}(1-{\rm{\Delta }}\lambda {f}_{n+1}^{\prime }){\xi }_{n+1}+\displaystyle \sum _{n}{\overline{\xi }}_{n}[-1-{\rm{\Delta }}(1-\lambda ){f}_{n}^{\prime }]{\xi }_{n}\right\},\end{eqnarray} \tag{ 131 }$$

where ${\xi }_{n}$ (with n = 1,..., M) and ${\overline{\xi }}_{n}$ (with $n=0,\ldots ,\,M-1$) are Grassmann numbers.

The dynamical partition function is defined as before in (26) but with the average now also involving integration over the Grassmann variables,

$$\begin{eqnarray}&&\langle \ldots {\rangle }_{S}=\int \displaystyle \frac{{\rm{d}}\phi \,{\rm{d}}\hat{\phi }}{{(2\pi )}^{M}}\displaystyle \prod _{n}{\rm{d}}{\xi }_{n}{\rm{d}}{\overline{\xi }}_{n}\,\ldots \,\exp (-S)\equiv \int D[\phi \hat{\phi }]D[\xi \overline{\xi }]\,\ldots \,\exp (-S).\end{eqnarray} \tag{ 132 }$$

The action reads

$$\begin{eqnarray}S & = & \displaystyle \sum _{n}(T{\rm{\Delta }}{\hat{\phi }}_{n}^{2}-{\rm{i}}{\hat{\phi }}_{n}\{-{\phi }_{n+1}+{\phi }_{n}+{\rm{\Delta }}[(1-\lambda ){f}_{n}+\lambda {f}_{n+1}]\})\\ & & -\displaystyle \sum _{n}{\overline{\xi }}_{n}(1-{\rm{\Delta }}\lambda {f}_{n+1}^{\prime }){\xi }_{n+1}-\displaystyle \sum _{n}{\overline{\xi }}_{n}(-1-{\rm{\Delta }}(1-\lambda ){f}_{n}^{\prime }){\xi }_{n},\end{eqnarray} \tag{ 133 }$$

where the second line replaces the last term in  (25). In the continuous time limit ${\rm{\Delta }}\to 0$ this can be written as

$$\begin{eqnarray}&&S=\int {\rm{d}}{t}\{T{\hat{\phi }}^{2}+{\rm{i}}\hat{\phi }[{\partial }_{t}\phi -f(\phi )]\}-\int {\rm{d}}{t}\,\overline{\xi }[{\partial }_{t}-f^{\prime} (\phi )]\xi .\end{eqnarray} \tag{ 134 }$$

Let us see how the inclusion of the Grassmann 'ghosts' works out for our example case of $f(\phi )=-\mu \phi -(g/3!){\phi }^{3}$. Going back to discretized time temporarily will help us understand how to treat equal-time correlations. With $f(\phi )$ as given, the action can be written as

$$\begin{eqnarray}&&S={S}_{0}+{S}_{\mathrm{int}},\end{eqnarray} \tag{ 135 }$$

$$\begin{eqnarray}{S}_{0} & = & \displaystyle \sum _{n}(T{\rm{\Delta }}{\hat{\phi }}_{n}^{2}-{\rm{i}}{\hat{\phi }}_{n}\{-{\phi }_{n+1}+{\phi }_{n}-\mu {\rm{\Delta }}[(1-\lambda ){\phi }_{n}+\lambda {\phi }_{n+1}]\})\\ & & -\displaystyle \sum _{n}\{{\overline{\xi }}_{n}(1+{\rm{\Delta }}\lambda \mu ){\xi }_{n+1}+{\overline{\xi }}_{n}[-1+{\rm{\Delta }}(1-\lambda )\mu ]{\xi }_{n}\},\end{eqnarray} \tag{ 136 }$$

$$\begin{eqnarray}{S}_{\mathrm{int}} & = & {\rm{i}}\displaystyle \frac{g}{3!}{\rm{\Delta }}\displaystyle \sum _{n}{\hat{\phi }}_{n}[(1-\lambda ){\phi }_{n}^{3}+\lambda {\phi }_{n+1}^{3}]\\ & & -{\rm{\Delta }}\displaystyle \frac{g}{2}\displaystyle \sum _{n}[{\overline{\xi }}_{n}\lambda {\phi }_{n+1}^{2}{\xi }_{n}+{\overline{\xi }}_{n}(1-\lambda ){\phi }_{n}^{2}{\xi }_{n}]\mathrm{.}\end{eqnarray} \tag{ 137 }$$

The coefficient matrix ${\boldsymbol{A}}$ appearing in the ghost term of the bare action S₀ has entries $1+{\rm{\Delta }}\lambda \mu$ on the main diagonal and $-1+{\rm{\Delta }}(1-\lambda )\mu$ on the diagonal below. This matrix is easily inverted to show that the ghost covariance is

$$\begin{eqnarray}&&\langle {\xi }_{m}{\overline{\xi }}_{n}{\rangle }_{0}=-\displaystyle \frac{1}{1+{\rm{\Delta }}\lambda \mu }{\left(\displaystyle \frac{1-{\rm{\Delta }}(1-\lambda )\mu }{1+{\rm{\Delta }}\lambda \mu }\right)}^{m-n-1}=-\exp [-\mu (m-n-1){\rm{\Delta }}]\end{eqnarray} \tag{ 138 }$$

for $m\gt n$ and 0 otherwise; the last expression applies for ${\rm{\Delta }}\to 0$. The ghost correlator is therefore causal and reads in the continuum limit:

$$\begin{eqnarray}&&\langle \xi (t)\overline{\xi }(t^{\prime} ){\rangle }_{0}=-{\rm{\Theta }}(t-t^{\prime} )\,\exp [-\mu (t-t^{\prime} )].\end{eqnarray} \tag{ 139 }$$

While this is λ-independent, the dependence on λ reappears in how equal-time Wick contractions are treated in the perturbative expansion in powers of $-{S}_{\mathrm{int}}$. To see this, note that up to vanishing corrections the interacting action is

$$\begin{eqnarray}\begin{array}{rcl}{S}_{\mathrm{int}} & = & {\rm{i}}\displaystyle \frac{g}{3!}{\rm{\Delta }}\displaystyle \sum _{n}[\lambda {\hat{\phi }}_{n-1}+(1-\lambda ){\hat{\phi }}_{n}]{\phi }_{n}^{3}\\ & & -{\rm{\Delta }}\displaystyle \frac{g}{2}\displaystyle \sum _{n}[\lambda {\overline{\xi }}_{n-1}+(1-\lambda ){\overline{\xi }}_{n}]{\phi }_{n}^{2}\ {\xi }_{n}\mathrm{.}\end{array}\end{eqnarray} \tag{ 140 }$$

The square brackets in the first line, including the factor i, define what we previously called the response field ${\eta }_{2n}$, which has equal-time correlator with ${\phi }_{n}$ of ${R}_{{nn}}^{0}=\lambda$. Similarly we could now define the combination in square brackets in the second line as a new Grassmann response field ${\tilde{\xi }}_{n}$, which has equal-time correlator with ξ of $\langle {\xi }_{n}{\tilde{\xi }}_{n}{\rangle }_{0}=\lambda \langle {\xi }_{n}{\overline{\xi }}_{n-1}{\rangle }_{0}=-\lambda$ using  (138). If for simplicity of notation we do not distinguish between $\tilde{\xi }$ and $\overline{\xi }$ (and similarly ${\eta }_{2}$ and $\hat{\phi }$) then the upshot of this discussion is that for ${\rm{\Delta }}\to 0$ one can work directly with the continuous-time version

$$\begin{eqnarray}&&{S}_{\mathrm{int}}=\displaystyle \frac{g}{3!}\int {\rm{d}}{t}({\rm{i}}\hat{\phi }{\phi }^{3}-3\overline{\xi }{\phi }^{2}\xi )\end{eqnarray} \tag{ 141 }$$

of the interacting action, provided that one remembers that the equal-time ghost correlator has to be set to $\langle \xi (t)\overline{\xi }(t){\rangle }_{0}=-\lambda$ in any Wick contractions, and similarly $\langle \phi (t)i\hat{\phi }(t){\rangle }_{0}=\lambda$. Note that there are never any contractions between ordinary and Grassmann variables because the average of any single Grassmann variable over a (Grassmann) Gaussian vanishes.

To see how the perturbation theory with ghosts works in practice, consider the response function to first order in the perturbation:

$$\begin{eqnarray}&&\langle \phi (t)\hat{\phi }(t^{\prime} ){\rangle }_{S}=\langle \phi (t)\phi (t^{\prime} )\exp [-{S}_{\mathrm{int}}]{\rangle }_{0}=\langle \phi (t)\hat{\phi }(t^{\prime} )[1-{S}_{\mathrm{int}}]{\rangle }_{0}+{ \mathcal O }({g}^{2})\end{eqnarray} \tag{ 142 }$$

$$\begin{eqnarray}&&=\langle \phi (t)\hat{\phi }(t^{\prime} ){\rangle }_{0}-\displaystyle \frac{{\rm{i}}{g}}{3!}\displaystyle \int {\rm{d}}{t}^{\prime\prime} \langle \phi (t)\hat{\phi }(t^{\prime} )\hat{\phi }(t^{\prime\prime} ){\phi }^{3}(t^{\prime\prime} ){\rangle }_{0}\\ &&+\,\displaystyle \frac{g}{2}\int {\rm{d}}{t}^{\prime\prime} \langle \phi (t)\hat{\phi }(t^{\prime} ){\phi }^{2}(t^{\prime\prime} )\bar{\xi }(t^{\prime\prime} )\xi (t^{\prime\prime} ){\rangle }_{0}+{ \mathcal O }({g}^{2}).\end{eqnarray} \tag{ 143 }$$

The physical piece of this is the first term and the contraction without equal-time response factors in the first integral:

$$\begin{eqnarray}&&\langle \phi (t)\hat{\phi }(t^{\prime} )\rangle =\langle \phi (t)\hat{\phi }(t^{\prime} ){\rangle }_{0}-\displaystyle \frac{{\rm{i}}{g}}{2}\int {\rm{d}}{t}^{\prime\prime} \langle \phi (t)\hat{\phi }(t^{\prime\prime} ){\rangle }_{0}\langle \phi (t^{\prime\prime} )\hat{\phi }(t^{\prime} ){\rangle }_{0}\langle {\phi }^{2}(t^{\prime\prime} ){\rangle }_{0}.\end{eqnarray} \tag{ 144 }$$

When $\lambda \ne 0$, i.e. for any convention other than Ito, there are two other non-zero contractions of the first integral:

$$\begin{eqnarray}\begin{array}{rcl} & & -\displaystyle \frac{g}{2}\langle \phi (t)\hat{\phi }(t^{\prime} ){\rangle }_{0}\displaystyle \int {\rm{d}}{t}^{\prime\prime} \langle {\phi }^{2}(t^{\prime\prime} ){\rangle }_{0}\langle \phi (t^{\prime\prime} ){\rm{i}}\hat{\phi }(t^{\prime\prime} ){\rangle }_{0}\\ & & \ \ \ \ -g\displaystyle \int {\rm{d}}{t}^{\prime\prime} \langle \phi (t)\phi (t^{\prime\prime} ){\rangle }_{0}\langle \phi (t^{\prime\prime} )\hat{\phi }(t^{\prime} ){\rangle }_{0}\langle \phi (t^{\prime\prime} ){\rm{i}}\hat{\phi }(t^{\prime\prime} ){\rangle }_{0}\mathrm{.}\end{array}\end{eqnarray} \tag{ 145 }$$

In addition the two possible Wick contractions of the ghost term in the second line of  (143) give, bearing in mind that $\langle \overline{\xi }\xi {\rangle }_{0}=-\langle \xi \overline{\xi }{\rangle }_{0}$,

$$\begin{eqnarray}\begin{array}{rcl} & & -\displaystyle \frac{g}{2}\langle \phi (t)\hat{\phi }(t^{\prime} ){\rangle }_{0}\displaystyle \int {\rm{d}}{t}^{\prime\prime} \langle {\phi }^{2}(t^{\prime\prime} ){\rangle }_{0}\langle \xi (t^{\prime\prime} )\overline{\xi }(t^{\prime\prime} ){\rangle }_{0}\\ & & \ \ \ \ -g\displaystyle \int {\rm{d}}{t}^{\prime\prime} \langle \phi (t)\phi (t^{\prime\prime} ){\rangle }_{0}\langle \phi (t^{\prime\prime} )\hat{\phi }(t^{\prime} ){\rangle }_{0}\langle \xi (t^{\prime\prime} )\overline{\xi }(t^{\prime\prime} ){\rangle }_{0}\mathrm{.}\end{array}\end{eqnarray} \tag{ 146 }$$

Because $\langle \phi \,{\rm{i}}\hat{\phi }{\rangle }_{0}=\lambda$ at equal-time while the ghost correlator has the opposite sign $\langle \xi \overline{\xi }{\rangle }_{0}=-\lambda$, the terms in  (145) and  (146) exactly cancel each other as they should.

In other words, whether or not we include the Jacobian, the extra terms  (145) do not appear in the perturbation theory, either because they are simply equal to zero, or because they cancel with the additional terms  (146) that arise from the ghost correlation functions from the Grassmann representation of the Jacobian. As will be elaborated in the next section, this is formally a consequence of a symmetry of the theory that represents the fact that the path probabilities are normalized to one.

Let us now define the superfield Φ as

$$\begin{eqnarray}&&{\rm{\Phi }}=\phi +\overline{\theta }\xi +\overline{\xi }\theta +{\rm{i}}\overline{\theta }\theta \hat{\phi },\end{eqnarray} \tag{ 147 }$$

where θ and $\overline{\theta }$ are themselves Grassmann variables and are sometimes referred to as Grassmann time. The action  (134) can then be written in terms of Φ in a compact way that reveals unexpected symmetries of the problem.

To see this, it is convenient to define

$$\begin{eqnarray}&&V(\phi )=-{\int }^{\phi }{\rm{d}}\phi ^{\prime} f(\phi ^{\prime} ),\end{eqnarray} \tag{ 148 }$$

so $f(\phi )=-{\rm{d}}{V}(\phi )/{\rm{d}}\phi$. The existence of the potential V is crucial for the supersymmetric description; it cannot be used for systems of more than one variable when these are non-equilibrium in the sense that the drift f cannot be written as a gradient.

In our current one-dimensional example where a potential can always be defined we have, by Taylor expansion around $V(\phi )$ and throwing away terms that vanish because of (122),

$$\begin{eqnarray}&&V({\rm{\Phi }})=V(\phi )-(\overline{\theta }\xi +\overline{\xi }\theta +{\rm{i}}\overline{\theta }\theta \hat{\phi })f(\phi )-\displaystyle \frac{1}{2}{(\overline{\theta }\xi +\overline{\xi }\theta +{\rm{i}}\overline{\theta }\theta \hat{\phi })}^{2}f^{\prime} (\phi )\end{eqnarray} \tag{ 149 }$$

$$\begin{eqnarray}&&=\,V(\phi )-(\overline{\theta }\xi +\overline{\xi }\theta +{\rm{i}}\overline{\theta }\theta \hat{\phi })f(\phi )+\overline{\theta }\theta \overline{\xi }\xi f^{\prime} (\phi ),\end{eqnarray} \tag{ 150 }$$

which leads to

$$\begin{eqnarray}&&\int {\rm{d}}\theta {\rm{d}}\overline{\theta }\,V({\rm{\Phi }})=-{\rm{i}}\hat{\phi }f(\phi )+\overline{\xi }f^{\prime} (\phi )\xi ,\end{eqnarray} \tag{ 151 }$$

giving us two of the terms in  (134).

The remaining terms can be written in terms of derivatives of Φ. We have

$$\begin{eqnarray}&&{\partial }_{\theta }{\rm{\Phi }}=-\overline{\xi }-{\rm{i}}\overline{\theta }\hat{\phi },\end{eqnarray} \tag{ 152 }$$

$$\begin{eqnarray}&&{\partial }_{\overline{\theta }}{\rm{\Phi }}=\xi +{\rm{i}}\theta \hat{\phi },\end{eqnarray} \tag{ 153 }$$

$$\begin{eqnarray}&&\theta {\partial }_{t}{\rm{\Phi }}=\theta {\partial }_{t}\phi +\theta \overline{\theta }{\partial }_{t}\xi .\end{eqnarray} \tag{ 154 }$$

If we then evaluate the quantity

$$\begin{eqnarray}&&T\int {\rm{d}}\theta {\rm{d}}\overline{\theta }({\partial }_{\theta }{\rm{\Phi }})({\partial }_{\overline{\theta }}{\rm{\Phi }}),\end{eqnarray} \tag{ 155 }$$

we find that we get the $T{\hat{\phi }}^{2}$ term in the action (134). Similarly, we find

$$\begin{eqnarray}&&\int {\rm{d}}\theta {\rm{d}}\overline{\theta }({\partial }_{\theta }{\rm{\Phi }})(\theta {\partial }_{t}{\rm{\Phi }})=-{\rm{i}}\hat{\phi }{\partial }_{t}\phi +\overline{\xi }{\partial }_{t}\xi ,\end{eqnarray} \tag{ 156 }$$

which are the negatives of the terms in (134) involving time derivatives. Putting all these results together and defining ${\rm{d}}\tau \equiv {\rm{d}}{t}\,{\rm{d}}\theta \,{\rm{d}}\overline{\theta }$, we can write the action in the form

$$\begin{eqnarray}&&S=\int {\rm{d}}\tau \{{\partial }_{\theta }{\rm{\Phi }}[T{\partial }_{\overline{\theta }}{\rm{\Phi }}-\theta {\partial }_{t}{\rm{\Phi }}]+V({\rm{\Phi }})\}.\end{eqnarray} \tag{ 157 }$$

It will be handy to introduce the notation

$$\begin{eqnarray}&&{D}=T{\partial }_{\overline{\theta }}-\theta {\partial }_{t},\end{eqnarray} \tag{ 158 }$$

$$\begin{eqnarray}&&\overline{{D}}={\partial }_{\theta }\end{eqnarray} \tag{ 159 }$$

so that

$$\begin{eqnarray}&&S=\int {\rm{d}}\tau [\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}+V({\rm{\Phi }})].\end{eqnarray} \tag{ 160 }$$

Up to here the formalism is general enough—subject to the existence of the potential V—that it can describe also non-stationary dynamics, e.g. relaxation to equilibrium. From now on we restrict ourselves further by assuming we have a stationary state. The supersymmetric action then has several symmetries; the obvious one is time translation invariance. But it is also invariant under other 'translations' that involve shifts in the Grassmann times θ and $\overline{\theta }$. In what follows, we identify these invariances and investigate their physical meanings [15, 24].

Consider a 'translation' generated by the operator

$$\begin{eqnarray}&&{D}^{\prime} \equiv {\partial }_{\overline{\theta }}.\end{eqnarray} \tag{ 161 }$$

This produces a shift $\overline{\theta }\to \overline{\theta }+\epsilon$ and a change in Φ:

$$\begin{eqnarray}&&{\rm{\Phi }}\to {\rm{\Phi }}+\epsilon {\partial }_{\overline{\theta }}{\rm{\Phi }},\end{eqnarray} \tag{ 162 }$$

where is a Grassmann 'infinitesimal' that acts like a separate Grassmann variable. Now, by using (153) (or simply by substituting $\overline{\theta }\to \overline{\theta }+\epsilon$ in the definition (147) of the superfield), we find

$$\begin{eqnarray}&&{\rm{\Phi }}\to {\rm{\Phi }}+\epsilon \xi +{\rm{i}}\epsilon \theta \hat{\phi }=(\phi +\epsilon \xi )+\overline{\theta }\xi +(\overline{\xi }+{\rm{i}}\epsilon \hat{\phi })\theta +{\rm{i}}\overline{\theta }\theta \hat{\phi }.\end{eqnarray} \tag{ 163 }$$

But according to the definition of the superfield, whatever appears in the expression (163) not mutiplied by θ, $\overline{\theta }$ or $\overline{\theta }\theta$ should be identified as the new ϕ, and whatever appears multiplied (from the right) by θ should be identified as the new $\overline{\xi }$. The component fields therefore transform as

$$\begin{eqnarray}&&\phi \to \phi +\epsilon \xi ,\end{eqnarray} \tag{ 164 }$$

$$\begin{eqnarray}&&\xi \to \xi ,\end{eqnarray} \tag{ 165 }$$

$$\begin{eqnarray}&&\overline{\xi }\to \overline{\xi }+{\rm{i}}\epsilon \hat{\phi },\end{eqnarray} \tag{ 166 }$$

$$\begin{eqnarray}&&\hat{\phi }\to \hat{\phi }.\end{eqnarray} \tag{ 167 }$$

This transformation is called supersymmetric because it mixes 'bosonic' degrees of freedom (i.e. ϕ and $\hat{\phi }$) with 'fermionic' (i.e. Grassmann) ones.

To verify that such a transformation is indeed a symmetry of the model, we proceed analogously to what we would do to test, say, rotational invariance in an ordinary field theory with vector fields. We start by performing the 'rotation' (164)–(167) on the superfield components in each term of the integrand in (160), i.e. we substitute the transformed Φ and carry out the required derivatives with respect to time and Grassmann time. In general, this leads to a change in the integrand, which we then integrate over τ (i.e. over θ, $\overline{\theta }$ and t). If the result is zero we have a symmetry. As an example, let us see how the shift generated by ${D}^{\prime}$ affects the 'kinetic' term $\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}$ in the action. From (152) we see, using (166) and (167), that

$$\begin{eqnarray}&&\overline{{D}}{\rm{\Phi }}\to \overline{{D}}{\rm{\Phi }}-{\rm{i}}\epsilon \hat{\phi }.\end{eqnarray} \tag{ 168 }$$

For the change in ${\rm{D}}{\rm{\Phi }}$, we see from (153) that the first term ${\partial }_{\overline{\theta }}{\rm{\Phi }}$ does not change as it involves only ξ and $\hat{\phi }$, and these do not change under (165) and (167). From (154), (164) and (165), the change in $\theta {\partial }_{t}{\rm{\Phi }}$ is

$$\begin{eqnarray}&&\theta {\partial }_{t}{\rm{\Phi }}\to \theta {\partial }_{t}{\rm{\Phi }}-\epsilon \theta \dot{\xi }.\end{eqnarray} \tag{ 169 }$$

Putting these contributions together, the change in $\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}$ is

$$\begin{eqnarray}&&(-\overline{\xi }-{\rm{i}}\overline{\theta }\hat{\phi })\epsilon \theta \dot{\xi }-{\rm{i}}\epsilon \hat{\phi }[T(\xi +{\rm{i}}\theta \hat{\phi })-\theta \dot{\phi }-\theta \overline{\theta }\dot{\xi }].\end{eqnarray} \tag{ 170 }$$

Only terms proportional to $\theta \overline{\theta }$ will survive the integration over θ and $\overline{\theta }$, but these cancel:

$$\begin{eqnarray}&&-{\rm{i}}\epsilon \theta \overline{\theta }\hat{\phi }\dot{\xi }+{\rm{i}}\epsilon \theta \overline{\theta }\hat{\phi }\dot{\xi }=0.\end{eqnarray} \tag{ 171 }$$

A similar calculation shows that the 'potential' term $\int {\rm{d}}\tau \,V({\rm{\Phi }})$ is also unchanged. Thus, the action S is invariant under ${D}^{\prime} ={\partial }_{\overline{\theta }}$.

An analogous calculation shows that the kinetic term is not invariant under shifts generated by $\overline{{D}}={\partial }_{\theta }$. However, we can try combining a shift in θ with one in time, using the generator

$$\begin{eqnarray}&&\overline{{D}}^{\prime} ={\partial }_{\theta }+\alpha \overline{\theta }{\partial }_{t}\end{eqnarray} \tag{ 172 }$$

and see whether there is a value of α for which $\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}$ is invariant. Now the transformation of ξ acquires a new term proportional to ${\partial }_{t}\phi$,

$$\begin{eqnarray}&&\xi \to \xi +\epsilon ({\rm{i}}\hat{\phi }-\alpha {\partial }_{t}\phi ),\end{eqnarray} \tag{ 173 }$$

and $\hat{\phi }$ is also changed, proportional to ${\partial }_{t}\overline{\xi }$:

$$\begin{eqnarray}&&\hat{\phi }\to \hat{\phi }-{\rm{i}}(\alpha {\partial }_{t}\overline{\xi })\epsilon .\end{eqnarray} \tag{ 174 }$$

The remaining variables ϕ and $\overline{\xi }$ transform according to

$$\begin{eqnarray}&&\phi \to \phi +\overline{\xi }\epsilon ,\end{eqnarray} \tag{ 175 }$$

$$\begin{eqnarray}&&\overline{\xi }\to \overline{\xi },\end{eqnarray} \tag{ 176 }$$

Then, after some algebra, we find that, under our trial $\overline{{D}}^{\prime}$, $\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}$ is changed by

$$\begin{eqnarray} & & -\overline{\xi }[(1-\beta \theta \overline{\theta }{\partial }_{t})({\rm{i}}\hat{\phi }-\alpha \dot{\phi })+(\alpha -\beta )\theta {\partial }_{t}\overline{\xi }]\epsilon -{\rm{i}}\overline{\theta }\hat{\phi }({\rm{i}}\hat{\phi }-\alpha \dot{\phi })\epsilon \\ & & -{\rm{i}}(\alpha -\beta )\hat{\phi }\overline{\theta }\theta {\partial }_{t}\overline{\xi }\epsilon -\alpha \overline{\theta }({\partial }_{t}\overline{\xi })\epsilon [\xi +\theta ({\rm{i}}\hat{\phi }-\beta \dot{\phi })]\end{eqnarray} \tag{ 177 }$$

(here $\beta =1/T$). Integrating over θ and $\overline{\theta }$ leaves

$$\begin{eqnarray}&&-\beta \overline{\xi }{\partial }_{t}({\rm{i}}\hat{\phi }-\alpha \dot{\phi })\epsilon -\alpha ({\partial }_{t}\overline{\xi })\epsilon ({\rm{i}}\hat{\phi }-\beta \dot{\phi })-{\rm{i}}(\alpha -\beta )\hat{\phi }{\partial }_{t}\overline{\xi }\epsilon ,\end{eqnarray} \tag{ 178 }$$

so we see that with the choice $\alpha =\beta$ this just reduces to

$$\begin{eqnarray}&&-\beta {\partial }_{t}[\overline{\xi }({\rm{i}}\hat{\phi }-\dot{\phi })]\epsilon .\end{eqnarray} \tag{ 179 }$$

This vanishes on integration over t (for stationary initial and final states), proving the invariance of this part of the action. Again the proof of invariance for the $V({\rm{\Phi }})$ term is similar to that for ${D}^{\prime}$, so the total action is invariant under $\overline{{D}}^{\prime}$.

The reader who is uneasy with the formal manipulations using superfields here can check these results by applying the transformations (164)–(167) for ${D}^{\prime}$ and (173)–(176) for $\overline{{D}}^{\prime}$ directly to ϕ, $\hat{\phi }$, ξ and $\overline{\xi }$, in the form (134) of the action not using superfields.

To see the meaning of these supersymmetries, we consider the supercorrelation function

$$\begin{eqnarray}&&Q(1,2)=\langle {\rm{\Phi }}(1){\rm{\Phi }}(2)\rangle ,\end{eqnarray} \tag{ 180 }$$

where 1 stands for $({t}_{1},{\theta }_{1},\overline{{\theta }_{1}})$ and analogously for 2. When we use (147) and expand the product, many terms vanish, either because the averages are of products of Grassmann variables with ordinary ones or of a pair of $\hat{\phi }$'s. The remaining terms are

$$\begin{eqnarray}&&Q(1,2)=\langle {\phi }_{1}{\phi }_{2}\rangle +{\overline{\theta }}_{1}{\theta }_{2}\langle {\xi }_{1}{\overline{\xi }}_{2}\rangle +{\theta }_{1}{\overline{\theta }}_{2}\langle {\overline{\xi }}_{1}{\xi }_{2}\rangle +{\rm{i}}{\overline{\theta }}_{1}{\theta }_{1}\langle {\hat{\phi }}_{1}{\phi }_{2}\rangle +{\rm{i}}{\overline{\theta }}_{2}{\theta }_{2}\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle ,\end{eqnarray} \tag{ 181 }$$

where ${\phi }_{1}$ means $\phi ({t}_{1})$, etc.

From the invariance of the action under ${D}^{\prime}$ (translations in $\overline{\theta }$), we have

$$\begin{eqnarray}{D}^{\prime} Q(1,2) & = & ({{D}}_{1}^{\prime }+{{D}}_{2}^{\prime })Q(1,2)=({\partial }_{{\overline{\theta }}_{1}}+{\partial }_{{\overline{\theta }}_{2}})Q(1,2)\\ & = & {\theta }_{2}(\langle {\xi }_{1}{\overline{\xi }}_{2}\rangle +{\rm{i}}\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle )+{\theta }_{1}(-\langle {\overline{\xi }}_{1}{\xi }_{2}\rangle +{\rm{i}}\langle {\hat{\phi }}_{1}{\phi }_{2}\rangle )=0.\end{eqnarray} \tag{ 182 }$$

The vanishing of the term proportional to ${\theta }_{2}$ says that the ghost correlation function $\langle {\xi }_{1}{\overline{\xi }}_{2}\rangle$ has to be the negative of the response function ${\rm{i}}\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle$ (as in (139)). The vanishing of the term proportional to ${\theta }_{1}$ says the same thing if we notice that the ghost correlation function here is $\langle \overline{\xi }\xi \rangle$, not $\langle \xi \overline{\xi }\rangle$. Thus, invariance under ${D}^{\prime}$, through its enforcement of the cancellation of disconnected diagrams, is just conservation of probability.

Analogously, for the $\overline{{D}}^{\prime}$ symmetry (172), we have

$$\begin{eqnarray}&&\overline{{D}}^{\prime} Q(1,2)=({\overline{{D}}}_{1}^{\prime }+{\overline{{D}}}_{2}^{\prime })Q(1,2)=({\partial }_{{\theta }_{1}}+{\partial }_{{\theta }_{2}}+\beta {\overline{\theta }}_{1}{\partial }_{{t}_{1}}+\beta {\overline{\theta }}_{2}{\partial }_{{t}_{2}})Q(1,2)=0.\end{eqnarray} \tag{ 183 }$$

This gives

$$\begin{eqnarray}&&({\overline{\theta }}_{1}-{\overline{\theta }}_{2})(\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle -\langle {\hat{\phi }}_{1}{\phi }_{2}\rangle )+\beta {\overline{\theta }}_{1}{\partial }_{{t}_{1}}\langle {\phi }_{1}{\phi }_{2}\rangle +\beta {\overline{\theta }}_{2}{\partial }_{{t}_{2}}\langle {\phi }_{1}{\phi }_{2}\rangle =0.\end{eqnarray} \tag{ 184 }$$

For ${t}_{1}\gt {t}_{2}$, $\langle {\hat{\phi }}_{1}{\phi }_{2}\rangle$ vanishes, so we have, also using ${\partial }_{{t}_{2}}\langle {\phi }_{1}{\phi }_{2}\rangle =-{\partial }_{{t}_{1}}\langle {\phi }_{1}{\phi }_{2}\rangle$ from time translation invariance,

$$\begin{eqnarray}&&({\overline{\theta }}_{1}-{\overline{\theta }}_{2})(\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle +\beta {\partial }_{{t}_{1}}\langle {\phi }_{1}{\phi }_{2}\rangle )=0.\end{eqnarray} \tag{ 185 }$$

Thus,

$$\begin{eqnarray}&&\langle {\phi }_{1}{\hat{\phi }}_{2}\rangle =-\beta {\partial }_{{t}_{1}}\langle {\phi }_{1}{\phi }_{2}\rangle ,\end{eqnarray} \tag{ 186 }$$

which is the FDT.

To summarize, the theory has three invariances: time translation, ${D}^{\prime}$ and $\overline{{D}}^{\prime}$. ${D}^{\prime}$ expresses conservation of probability, and $\overline{{D}}^{\prime}$ expresses the FDT, i.e. the fact that the system is in equilibrium.

So far, we have treated a single-site problem. It is straightforward to extend the formalism to multiple degrees of freedom ${\phi }_{i}$. However, as emphasized before, the supersymmetric construction is permitted only when the drift is the negative gradient of a potential, ${f}_{i}=-\partial V/\partial {\phi }_{i}$. Otherwise the $\overline{{D}}^{\prime}$ supersymmetry fails. This means that the FDT is not obeyed; the system, even though it may possess a steady state, is not in equilibrium. (Of course, it is well-known from simple arguments making no reference to supersymmetry that models with non-gradient drifts, as arising e.g. from asymmetric coupling matrices, do not satisfy the FDT.)

In addition to the insight it provides into the symmetries of the problem, the supersymmetric formulation can also be of practical advantage in calculations. For example, in diagrammatic perturbation theory, one needs only draw the diagrams for the static problem. Another example can be found in Biroli's analysis [11] of dynamical TAP equations for the p-spin glass, where the entire structure of the argument and the equations could be carried over from the static treatment of Plefka [25]. Here we sketch how to do diagrammatic perturbation theory in the superfield language and show, in a simple example, how it reduces to the diagrams we had in the MSR formalism with the $\langle \phi (t)\phi (t^{\prime} )\rangle$ and $\langle \phi (t)\hat{\phi }(t^{\prime} )\rangle$ correlation functions [20].

We write our standard model  (14) in the form

$$\begin{eqnarray}&&S={S}_{0}+{S}_{\mathrm{int}},\end{eqnarray} \tag{ 187 }$$

with

$$\begin{eqnarray}&&{S}_{0}=\int {\rm{d}}\tau \left(\overline{{D}}{\rm{\Phi }}{D}{\rm{\Phi }}+\tfrac{1}{2}\mu {{\rm{\Phi }}}^{2}\right),\end{eqnarray} \tag{ 188 }$$

and

$$\begin{eqnarray}&&{S}_{\mathrm{int}}=\displaystyle \frac{g}{4!}\int {\rm{d}}\tau \,{{\rm{\Phi }}}^{4}.\end{eqnarray} \tag{ 189 }$$

To do perturbation theory, we can expand $\exp (-{S}_{\mathrm{int}})$ in g and apply Wick's theorem to evaluate the resulting averages, just as we did in section 5. Wick's theorem holds also for Grassmann variables although in principle one has to be careful with sign changes that arise from changing the order of the variables when performing contractions. For example, $\langle {\xi }_{1}{\overline{\xi }}_{1}{\xi }_{2}{\overline{\xi }}_{3}\rangle =\langle {\xi }_{1}{\overline{\xi }}_{1}\rangle \langle {\xi }_{2}{\overline{\xi }}_{3}\rangle -\langle {\xi }_{1}{\overline{\xi }}_{3}\rangle \langle {\xi }_{2}{\overline{\xi }}_{1}\rangle$. Fortunately this is not an issue in our context as we only need averages of powers of the superfield Φ, and from (147) Φ only contains products of pairs of Grassmann variables: commuting such pairs through each other never gives any minus signs.

The first thing we need for the perturbation theory is the correlation function or propagator of the non-interacting system. Integrating (188) by parts, we get

$$\begin{eqnarray}&&{S}_{0}=\int {\rm{d}}\tau \left(-{\rm{\Phi }}\overline{{D}}{D}{\rm{\Phi }}+\tfrac{1}{2}\mu {{\rm{\Phi }}}^{2}\right).\end{eqnarray} \tag{ 190 }$$

It is convenient to write this as

$$\begin{eqnarray}&&{S}_{0}=\tfrac{1}{2}\int {\rm{d}}\tau [(-{\rm{\Phi }}({[\overline{{D}},{D}]}_{-}+{[\overline{{D}},{D}]}_{+}){\rm{\Phi }}+\mu {{\rm{\Phi }}}^{2}),\end{eqnarray} \tag{ 191 }$$

where ${[\ldots ]}_{-}$ and ${[\ldots ]}_{+}$ denote the commutator and anti-commutator respectively. It is then simple to show that

$$\begin{eqnarray}&&{[\overline{{D}},{D}]}_{+}=-{\partial }_{t}\end{eqnarray} \tag{ 192 }$$

and

$$\begin{eqnarray}&&{[\overline{{D}},{D}]}_{-}=2T{\partial }_{\theta }{\partial }_{\overline{\theta }}+2\theta {\partial }_{\theta }{\partial }_{t}-{\partial }_{t}.\end{eqnarray} \tag{ 193 }$$

The term in (191) involving the anticommutator can be neglected because it vanishes on time integration, so we can write S₀ in the appealing form

$$\begin{eqnarray}&&{S}_{0}=\tfrac{1}{2}\int {\rm{d}}\tau {\rm{\Phi }}(-{{D}}^{(2)}+\mu ){\rm{\Phi }},\end{eqnarray} \tag{ 194 }$$

with⁹

$$\begin{eqnarray}&&{{D}}^{(2)}\equiv {[\overline{{D}},{D}]}_{-}.\end{eqnarray} \tag{ 195 }$$

From this, we identify

$$\begin{eqnarray}&&{Q}_{0}^{-1}=-{{D}}^{(2)}+\mu \end{eqnarray} \tag{ 196 }$$

as the inverse of the free propagator:

$$\begin{eqnarray}&&(-{{D}}^{(2)}+\mu ){Q}_{0}(1,2)=\delta (1,2)\equiv \delta ({t}_{1}-{t}_{2})({\overline{\theta }}_{1}-{\overline{\theta }}_{2})({\theta }_{1}-{\theta }_{2}),\end{eqnarray} \tag{ 197 }$$

where we have used the fact that $({\overline{\theta }}_{1}-{\overline{\theta }}_{2})({\theta }_{1}-{\theta }_{2})$ acts as a delta-function in the Grassmann times. Now, multiplying by $({{D}}^{(2)}+\mu )$ from the left, using the fact that ${({{D}}^{(2)})}^{2}={\partial }_{t}^{2}$, and Fourier transforming in time, we arrive at

$$\begin{eqnarray}&&{Q}_{0}({\theta }_{1},{\overline{\theta }}_{1},{\theta }_{2},{\overline{\theta }}_{2};\omega )=\displaystyle \frac{1}{{\omega }^{2}+{\mu }^{2}}[2T+({\overline{\theta }}_{1}-{\overline{\theta }}_{2})({\theta }_{1}-{\theta }_{2})\mu +({\overline{\theta }}_{1}+{\overline{\theta }}_{2})({\theta }_{1}+{\theta }_{2}){\rm{i}}\omega ].\end{eqnarray} \tag{ 198 }$$

Back in the time domain, this is

$$\begin{eqnarray}&&{Q}_{0}(1,2)=\displaystyle \frac{T}{\mu }\exp (-\mu | t| )+({\overline{\theta }}_{2}-{\overline{\theta }}_{1}){\theta }_{2}\exp (-\mu t){\rm{\Theta }}(t)+({\overline{\theta }}_{1}-{\overline{\theta }}_{2}){\theta }_{1}\exp (\mu t){\rm{\Theta }}(-t),\end{eqnarray} \tag{ 199 }$$

where $t={t}_{1}-{t}_{2}$. It is comforting to confirm that we can get this result in another, simpler way, using the representation (181) with the equalities implied by ${D}^{\prime}$ invariance (182):

$$\begin{eqnarray}&&Q(1,2)=\langle {\phi }_{1}{\phi }_{2}\rangle +({\overline{\theta }}_{2}-{\overline{\theta }}_{1})({\theta }_{2}\langle {\phi }_{1}{\rm{i}}{\hat{\phi }}_{2}\rangle -{\theta }_{1}\langle {\rm{i}}\hat{{\phi }_{1}}{\phi }_{2}\rangle ).\end{eqnarray} \tag{ 200 }$$

Using the free correlation and response functions (3) and (4) here then leads to (199).

To see how the diagrammatics work in this formalism, consider the second-order watermelon graph  (113) for the self-energy,

$$\begin{eqnarray}&&{\rm{\Sigma }}(1,2)={\left(-\displaystyle \frac{g}{4!}\right)}^{2}\times 4\times 4\times 3!\times Q{(1,2)}^{3}=\displaystyle \frac{{g}^{2}}{3!}Q{(1,2)}^{3}.\end{eqnarray} \tag{ 201 }$$

(Here we are doing the resummed expansion in which Σ is a functional of the full correlation function Q, not just Q₀, as in section 6.)

From the representation  (200), the part of $Q(1,2)$ with no Grassmann times multiplying it is the correlation function, and the parts mutiplied by $\pm ({\overline{\theta }}_{2}-{\overline{\theta }}_{1}){\theta }_{\mathrm{1,2}}$ are the retarded and advanced response functions, respectively. Expanding $Q{(1,2)}^{3}$, we get

$$\begin{eqnarray}&&{\rm{\Sigma }}(1,2)=\displaystyle \frac{{g}^{2}}{3!}[\langle {\phi }_{1}{\phi }_{2}{\rangle }^{3}+3\langle {\phi }_{1}{\phi }_{2}{\rangle }^{2}({\overline{\theta }}_{2}-{\overline{\theta }}_{1})({\theta }_{2}\langle {\phi }_{1}{\rm{i}}{\hat{\phi }}_{2}\rangle -{\theta }_{1}\langle {\rm{i}}{\hat{\phi }}_{1}{\phi }_{2}\rangle )].\end{eqnarray} \tag{ 202 }$$

This has the same form as (200). We note that the components of ${\rm{\Sigma }}(1,2)$ here are just the ${ \mathcal O }({g}^{2})$ contributions to the self-energies ${{\rm{\Sigma }}}_{12}$, ${{\rm{\Sigma }}}_{21}$ and ${{\rm{\Sigma }}}_{22}$ that we found in the conventional MSR theory (116) and (117), including the factor of 3 in ${{\rm{\Sigma }}}_{12}$ and ${{\rm{\Sigma }}}_{21}$.

It is useful to discuss in more detail how perturbation-theoretic corrections to $Q(1,2)$ in the supersymmetric formulation correspond to results obtained in the conventional MSR theory. The two theories differ superficially in two ways: (1) In the conventional theory we have to keep track of two kinds of correlations functions ($\langle \phi \phi \rangle$ and ${\rm{i}}\langle \phi \hat{\phi }\rangle$) while in the supersymmetric formulation we have only one (super)field and need only draw the diagrams we would have in statics. (2) In the supersymmetric theory both real and Grassmann times of intermediate vertices in the graphs are integrated over, while in the conventional theory only ordinary times are integrated over. To compare the two ways of doing the calculation, we consider the first term obtained in expanding the Dyson equation in ${\rm{\Sigma }}(1,2)$:

$$\begin{eqnarray}&&{\rm{\Delta }}Q(1,4)=\int {\rm{d}}{\tau }_{2}{\rm{d}}{\tau }_{3}\,Q(1,2){\rm{\Sigma }}(2,3)Q(3,4).\end{eqnarray} \tag{ 203 }$$

To resolve ${\rm{\Delta }}Q(1,4)$ into components, we use first the fact that Grassmann factors of the form $({\overline{\theta }}_{2}-{\overline{\theta }}_{1}){\theta }_{2}$ are idempotent under convolution, e.g.

$$\begin{eqnarray}&&\int {\rm{d}}{\theta }_{2}{\rm{d}}{\overline{\theta }}_{2}\,({\overline{\theta }}_{2}-{\overline{\theta }}_{1}){\theta }_{2}({\overline{\theta }}_{3}-{\overline{\theta }}_{2}){\theta }_{3}=({\overline{\theta }}_{3}-{\overline{\theta }}_{1}){\theta }_{3}.\end{eqnarray} \tag{ 204 }$$

That means that the retarded part of ${\rm{\Delta }}Q(1,4)$ (the part proportional to $({\overline{\theta }}_{4}-{\overline{\theta }}_{1}){\theta }_{4}$) is, in the notation we have used earlier ($\langle \phi ({t}_{1})\phi ({t}_{2})\rangle =C({t}_{1},{t}_{2})$, ${\rm{i}}\langle \phi ({t}_{1})\hat{\phi }({t}_{2})\rangle =R({t}_{1},{t}_{2})$)

$$\begin{eqnarray}&&\int {{\rm{d}}{t}}_{2}{{\rm{d}}{t}}_{3}\,R({t}_{1},{t}_{2})\left[\displaystyle \frac{{g}^{2}}{3!}\,3{(C({t}_{2},{t}_{3}))}^{2}R({t}_{2},{t}_{3})\right]R({t}_{3},{t}_{4}),\end{eqnarray} \tag{ 205 }$$

and analogously for the advanced part.

We also get a contribution to ${\rm{\Delta }}Q(1,4)$ from the first term in ${\rm{\Sigma }}(2,3)$. Since that term contains no Grassmann times, this contribution will contain a factor

$$\begin{eqnarray}&&\int {\rm{d}}{\theta }_{2}{\rm{d}}{\overline{\theta }}_{2}\,Q(1,2)\cdot \int {\rm{d}}{\theta }_{3}{\rm{d}}{\overline{\theta }}_{3}\,Q(3,4).\end{eqnarray} \tag{ 206 }$$

These integrations pick out factors of the retarded function $R({t}_{1},{t}_{2})$ and the advanced function $R({t}_{4},{t}_{3})$, respectively, so we find a contribution, involving no Grassmann times, of

$$\begin{eqnarray}&&\int {{\rm{d}}{t}}_{2}{{\rm{d}}{t}}_{3}\,R({t}_{1},{t}_{2})\left[\displaystyle \frac{{g}^{2}}{3!}{(C({t}_{2},{t}_{3}))}^{3}\right]R({t}_{4},{t}_{3}).\end{eqnarray} \tag{ 207 }$$

These are exactly the second-order contributions to $R({t}_{1},{t}_{4})$ and $C({t}_{1},{t}_{4})$ that we would find in the conventional formulation from expanding the Dyson equation to first order in the self-energies ${{\rm{\Sigma }}}_{12}$ and ${{\rm{\Sigma }}}_{22}$ in (116) and (117), again up to ${ \mathcal O }(g)$ terms not written explicitly here. Thus, because of the algebra of the Grassmann times in the supersymmetric formulation, the results of the multiplication and convolution of the supercorrelation functions, when reduced to components, reproduce the terms found in the conventional MSR theory. This result extends to all graphs in perturbation theory, because it depends only on (1) the idempotency of factors like $({\overline{\theta }}_{2}-{\overline{\theta }}_{1}){\theta }_{2}$ and (2) the fact that multiple correlator lines between a pair of interaction vertices, like those in (201), combine as in (202).

The generating functional for the dynamics is defined as

$$\begin{eqnarray}&&Z[\psi ]=\left\langle \exp \left(\displaystyle \sum _{i}{\psi }_{i}(t){\sigma }_{i}(t)\right)\right\rangle ,\end{eqnarray} \tag{ 240 }$$

where the average denoted by $\langle \cdots \rangle$ is over the distribution of trajectories generated according to the probability distribution  (235) and the ${\psi }_{i}$ are fields that allow one to obtain the statistics of the ${\sigma }_{i}$ by taking derivatives. The key to writing a path integral representation of the generating functional in this case is to work with the local fields h_i(t), which are continuous for $N\to \infty$, and not the original spin variables:

$$\begin{eqnarray}\begin{array}{rcl}Z[\psi ] & = & \mathrm{Tr}\displaystyle \int D[h]\displaystyle \prod _{t}\delta ({\boldsymbol{h}}(t)-{{\boldsymbol{h}}}^{\mathrm{ext}}-{\boldsymbol{J}}{\boldsymbol{\sigma }}(t))\\ & & \times \exp \left(\displaystyle \sum _{t}\,\,[{\boldsymbol{\psi }}(t+1)+{\boldsymbol{h}}(t)]\cdot {\boldsymbol{\sigma }}(t+1)\right.-\displaystyle \sum _{i,t}\,\,\mathrm{lc}({h}_{i}(t))),\end{array}\end{eqnarray} \tag{ 241 }$$

where ${\boldsymbol{J}}$ is the interaction matrix, $\mathrm{Tr}$ indicates a sum over all spin trajectories $\{{\boldsymbol{\sigma }}(t)\}$ and similarly $D[h]$ an integral over all field trajectories $\{{\boldsymbol{h}}(t)\}$. The discrete time variable range is $t=0,1,\ldots ,T-1$ where T is the final time.

The delta function in (241) is introduced to enforce the definition (236) and can be written as a Fourier transform, leading to

$$\begin{eqnarray}&&Z[\psi ]=\mathrm{Tr}\int D[h\hat{h}]\exp (-{S}_{\sigma }),\end{eqnarray} \tag{ 242 }$$

$$\begin{eqnarray}&&{S}_{\sigma }=-\displaystyle \sum _{t}\,{\rm{i}}\hat{{\boldsymbol{h}}}(t)\cdot [{\boldsymbol{h}}(t)-{{\boldsymbol{h}}}^{\mathrm{ext}}-{\boldsymbol{J}}{\boldsymbol{\sigma }}(t)]-\displaystyle \sum _{t}\,[{\boldsymbol{\psi }}(t+1)+{\boldsymbol{h}}(t)]\cdot {\boldsymbol{\sigma }}(t+1)+\displaystyle \sum _{i,t}\,\mathrm{lc}({h}_{i}(t)).\end{eqnarray} \tag{ 243 }$$

This expression can be used in several ways. One is to average over the distribution of ${\boldsymbol{J}};$ the second is to derive mean-field equations for the system. We do not go into the first route here, i.e. the quenched averaged dynamics of the system, as this is both very involved and also discussed in detail elsewhere [39]. However, as an example of how one can use the path integeral formulation (242) of the generating functional, we consider in the following subsections the simple saddle point approximation to the path integral as well as possible improvements to this.

To derive the saddle point equations, we first perform the trace in (242), which for a single timestep involves

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \sum _{{\boldsymbol{\sigma }}(t)}\,\exp [-{\rm{i}}\hat{{\boldsymbol{h}}}(t)\cdot {\boldsymbol{J}}{\boldsymbol{\sigma }}(t)+({\boldsymbol{\psi }}(t)+{\boldsymbol{h}}(t-1))\cdot {\boldsymbol{\sigma }}(t)]\\ \quad =\displaystyle \prod _{i}\displaystyle \sum _{{\sigma }_{i}=\pm 1}\exp \left(\left[{\psi }_{i}(t)+{h}_{i}(t-1)-{\rm{i}}\displaystyle \sum _{j}{\hat{h}}_{j}(t){J}_{{ji}}\right]{\sigma }_{i}(t)\right)\end{array}\end{eqnarray} \tag{ 244 }$$

$$\begin{eqnarray}&&=\,\displaystyle \prod _{i}2\cosh ({\psi }_{i}(t)+{h}_{i}(t-1)-{\rm{i}}{({{\boldsymbol{J}}}^{{\rm{T}}}\hat{{\boldsymbol{h}}}(t))}_{i}).\end{eqnarray} \tag{ 245 }$$

Using this we can write

$$\begin{eqnarray}&&Z[\psi ]=\int D[h\hat{h}]\exp (-S),\end{eqnarray} \tag{ 246 }$$

$$\begin{eqnarray}\begin{array}{rcl}S & = & -\displaystyle \sum _{t}\,{\rm{i}}\hat{{\boldsymbol{h}}}(t)\cdot [{\boldsymbol{h}}(t)-{{\boldsymbol{h}}}^{\mathrm{ext}}]-\displaystyle \sum _{i,t}\,[\mathrm{lc}({\psi }_{i}(t+1)\\ & & +{h}_{i}(t)-i{({{\boldsymbol{J}}}^{{\rm{T}}}\hat{{\boldsymbol{h}}}(t+1))}_{i})-\mathrm{lc}({h}_{i}(t))].\end{array}\end{eqnarray} \tag{ 247 }$$

The saddle point equations for stationarity of S with respect to the h_i(t) and ${\hat{h}}_{i}(t)$ are then

$$\begin{eqnarray}&&\displaystyle \frac{\partial S}{\partial {h}_{i}(t)}=-{\rm{i}}{\hat{h}}_{i}(t)-\tanh ({\psi }_{i}(t+1)+{h}_{i}(t)-{\rm{i}}{({{\boldsymbol{J}}}^{{\rm{T}}}\hat{{\boldsymbol{h}}}(t+1))}_{i})+\tanh ({h}_{i}(t))=0,\end{eqnarray} \tag{ 248 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial S}{\partial {\hat{h}}_{i}(t)}=-{\rm{i}}[{h}_{i}(t)-{h}_{i}^{\mathrm{ext}}]+{\rm{i}}\displaystyle \sum _{j}{J}_{{ij}}\tanh ({\psi }_{j}(t)+{h}_{j}(t-1)-{\rm{i}}{({{\boldsymbol{J}}}^{{\rm{T}}}\hat{{\boldsymbol{h}}}(t))}_{j})=0.\end{eqnarray} \tag{ 249 }$$

These equations can be written in a simpler form in terms of the magnetizations in the system biased by ψ, which are generally given by ${m}_{i}(t)={\partial }_{{\psi }_{i}(t)}\mathrm{ln}Z$. In the saddle point approximation $Z\approx {Z}_{s}\equiv \exp (-S)$, and because S is stationary with respect to h and $\hat{h}$,

$$\begin{eqnarray}&&{m}_{i}^{s}(t)=-{\partial }_{{\psi }_{i}(t)}S=\tanh ({\psi }_{i}(t)+{h}_{i}^{s}(t-1)-{\rm{i}}{({{\boldsymbol{J}}}^{{\rm{T}}}\cdot {\hat{{\boldsymbol{h}}}}^{s}(t))}_{i})\equiv {\mu }_{i}(t).\end{eqnarray} \tag{ 250 }$$

Here we have used 's' superscripts to denote saddle point values and introduced ${\mu }_{i}(t)$ as a convenient abbreviation for later use. The saddle point equations then simplify to

$$\begin{eqnarray}&&{\rm{i}}{\hat{h}}_{i}^{s}(t)+{m}_{i}^{s}(t+1)-\tanh ({h}_{i}^{s}(t))=0,\end{eqnarray} \tag{ 251 }$$

$$\begin{eqnarray}&&{h}_{i}^{s}(t)={h}_{i}^{\mathrm{ext}}+\displaystyle \sum _{j\,}{J}_{{ij}}{m}_{j}^{s}(t).\end{eqnarray} \tag{ 252 }$$

For the physical dynamics we want the solution at $\psi =0$, which we denote with a '0' superscript. One can show that this has ${\hat{h}}_{i}^{0}(t)=0$. Indeed, if one considers the dynamics over a finite number of timesteps $t=1,2,\ldots ,T$, then both ${\boldsymbol{h}}(t)$ and $\hat{{\boldsymbol{h}}}(t)$ are defined over the range $t=0,1,\ldots ,T-1$ and accordingly one finds that in the first saddle point equation  (248) taken at $t=T-1$, the $\hat{{\boldsymbol{h}}}(t+1)$-term inside the tanh is absent. For $\psi =0$ this equation then dictates ${\hat{h}}_{i}^{0}(T-1)=0$, and working backwards in time from there one finds recursively ${\hat{h}}_{i}^{0}(t)=0$ for all t. Accordingly (251) and (252) simplify to the standard mean field equations for the magnetizations,

$$\begin{eqnarray}&&{m}_{i}^{0}(t+1)=\tanh ({h}_{i}^{0}(t)),\end{eqnarray} \tag{ 253 }$$

$$\begin{eqnarray}&&{h}_{i}^{0}(t)={h}_{i}^{\mathrm{ext}}+\displaystyle \sum _{j\,}{J}_{{ij}}{m}_{j}^{0}(t).\end{eqnarray} \tag{ 254 }$$

Note that the saddle point value of the action (247) is then zero for $\psi =0$, as expected from the normalization condition $Z[0]=1$.

One can go beyond the saddle point approximation and take into account the Gaussian fluctuations around the saddle point to obtain a better estimate of the generating functional and hence of the equations of motion for the m_i(t). The Gaussian corrections to the log generating functional are

$$\begin{eqnarray}&&A[\psi ]=\mathrm{ln}\int D[h\hat{h}]\exp \left(-\displaystyle \frac{1}{2}{[\delta h\hat{h}]}^{{\rm{T}}}\cdot {\partial }^{2}S[\delta h\ \hat{h}]\right)=-\displaystyle \frac{1}{2}\mathrm{ln}| {\partial }^{2}S| ,\end{eqnarray} \tag{ 255 }$$

where $\delta h$ indicates the deviation of h from its saddle point value. We denote by ${\partial }^{2}S$ the matrix of the second derivatives of S with respect to h and $\hat{h}$ calculated at the saddle point; this has entries

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}S}{\partial {h}_{i}(t)\partial {h}_{j}(t^{\prime} )}={\delta }_{{ij}}{\delta }_{{tt}^{\prime} }[{\mu }_{i}^{2}(t+1)-\,{\tanh }^{2}({h}_{i}^{s}(t))],\end{eqnarray} \tag{ 256 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}S}{\partial {h}_{i}(t)\partial {\hat{h}}_{j}(t^{\prime} )}=-{\rm{i}}{\delta }_{{ij}}{\delta }_{{tt}^{\prime} }+i{\delta }_{t+1,t^{\prime} }{J}_{{ji}}[1-{\mu }_{i}^{2}(t+1)],\end{eqnarray} \tag{ 257 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}S}{\partial {\hat{h}}_{i}(t)\partial {\hat{h}}_{j}(t^{\prime} )}={\delta }_{{tt}^{\prime} }\displaystyle \sum _{k}{J}_{{ik}}{J}_{{jk}}[1-{\mu }_{k}^{2}(t)].\end{eqnarray} \tag{ 258 }$$

From the corrected log generating functional $\mathrm{ln}Z[\psi ]={\mathrm{ln}Z}_{s}[\psi ]+A[\psi ]$ we obtain a corrected expression for the magnetization

$$\begin{eqnarray}&&{m}_{i}(t)={\left.\displaystyle \frac{\partial \mathrm{ln}Z[\psi ]}{\partial {\psi }_{i}(t)}\right|}_{\psi =0}={m}_{i}^{0}(t)+{\left.\displaystyle \frac{\partial A[\psi ]}{\partial {\psi }_{i}(t)}\right|}_{\psi =0}.\end{eqnarray} \tag{ 259 }$$

To calculate the ψ-derivative of A at $\psi =0$ we can either appeal to numerical methods or evaluate A approximately. One relatively simple approximation, which will give us an intuitive feeling for the corrections and turns out to capture the lowest order corrections in J, is to only keep the equal-time ($t=t^{\prime} )$ elements of the matrix ${\partial }^{2}S$, i.e. to discard the second term in  (257). The matrix then separates into blocks corresponding to the different timesteps t. Each of those blocks is of size $2N\times 2N$. Let us order the elements so that the top left N × N sub-block contains the entries from  (256). This block, which we will denote by ${{\boldsymbol{\alpha }}}_{t}$, is diagonal and vanishes at the physical saddle point, where ${\psi }_{i}(t)=0$ and ${\hat{h}}_{i}^{s}(t)={\hat{h}}_{i}^{0}(t)=0$. The two off-diagonal blocks  (257) are $-{\rm{i}}{\boldsymbol{I}}$ within our approximation. If we call the bottom right N × N block ${{\boldsymbol{\beta }}}_{t}$, then we have by a standard determinant block identity

$$\begin{eqnarray}&&A[\psi ]=-\displaystyle \frac{1}{2}\displaystyle \sum _{t}\mathrm{ln}\,| {\boldsymbol{I}}+{{\boldsymbol{\alpha }}}_{t}{{\boldsymbol{\beta }}}_{t}| =-\displaystyle \frac{1}{2}\displaystyle \sum _{t}\mathrm{Tr}\,\mathrm{ln}({\boldsymbol{I}}+{{\boldsymbol{\alpha }}}_{t}{{\boldsymbol{\beta }}}_{t}).\end{eqnarray} \tag{ 260 }$$

Near the physical saddle point ${{\boldsymbol{\alpha }}}_{t}$ is small so we can linearize the logarithm to get

$$\begin{eqnarray}&&A[\psi ]\approx -\displaystyle \frac{1}{2}\displaystyle \sum _{t}\mathrm{Tr}({{\boldsymbol{\alpha }}}_{t}{{\boldsymbol{\beta }}}_{t})\end{eqnarray} \tag{ 261 }$$

$$\begin{eqnarray}&&\,=-\displaystyle \frac{1}{2}\displaystyle \sum _{{it}}[{\mu }_{i}^{2}(t+1)-\,{\tanh }^{2}({h}_{i}^{s}(t))]\displaystyle \sum _{k}{J}_{{ik}}^{2}[1-{\mu }_{k}^{2}(t)].\end{eqnarray} \tag{ 262 }$$

Linearizing then also the diagonal entries of ${{\boldsymbol{\alpha }}}_{t}$ that appear in the first square bracket, and accordingly setting the last factor to its value at the physical saddle point gives

$$\begin{eqnarray}&&\begin{array}{l}A[\psi ]\approx -\displaystyle \sum _{{it}}({\psi }_{i}(t+1)-{\rm{i}}{({{\boldsymbol{J}}}^{{\rm{T}}}\cdot {\hat{{\boldsymbol{h}}}}^{s}(t+1))}_{i})[1-{m}_{i}^{0}{(t+1)}^{2}]\\ \,\times {m}_{i}^{0}(t+1)\displaystyle \sum _{k}{J}_{{ik}}^{2}[1-{m}_{k}^{0}{(t)}^{2}].\end{array}\end{eqnarray} \tag{ 263 }$$

When we take the derivative of this with respect to ${\psi }_{i}(t)$, we do in principle get a term from the dependence of ${\hat{{\boldsymbol{h}}}}^{s}$ on ψ. But as ${\hat{{\boldsymbol{h}}}}^{s}$ is multiplied by a factor of ${\boldsymbol{J}}$, this will give us a contribution to the derivative that is of higher order in J than the main term from the explicit dependence on ${\psi }_{i}(t)$. Discarding this higher order contribution—as our second approximation in addition to neglecting correlations between different time steps in ${\partial }^{2}S$—yields for the derivative

$$\begin{eqnarray}&&{A}_{i}(t)\equiv \displaystyle \frac{\partial A[\psi ]}{\partial {\psi }_{i}(t)}\approx -[1-{m}_{i}^{0}{(t)}^{2}]{m}_{i}^{0}(t)\displaystyle \sum _{k}\,{J}_{{ik}}^{2}[1-{m}_{k}^{0}{(t-1)}^{2}].\end{eqnarray} \tag{ 264 }$$

Here we have implicitly already set $\psi =0$ because we linearized around the physical saddle point throughout. Using this result in (259), we obtain the corrected magnetization as

$$\begin{eqnarray}&&{m}_{i}(t)={m}_{i}^{0}(t)-[1-{m}_{i}^{0}{(t)}^{2}]\left\{{m}_{i}^{0}(t)\displaystyle \sum _{k}{J}_{{ik}}^{2}[1-{m}_{k}^{0}{(t-1)}^{2}]\right\}.\end{eqnarray} \tag{ 265 }$$

This is the result of our naive approach to including Gaussian corrections in the saddle point integral. Note that because of the normalization $Z[0]=1$, we do not expect Gaussian corrections to $\mathrm{ln}Z[0]$ at the physical saddle point. The approximation we made in evaluating $A[\psi ]$ as a sum of local-in-time terms preserves this requirement: $A[0]=0$ as is clear from  (260) given that ${{\boldsymbol{\alpha }}}_{t}=0$ when $\psi =0$.

It is instructive to compare  (265) with the so-called dynamical TAP equations for our system [12]

$$\begin{eqnarray}&&{m}_{i}^{\mathrm{TAP}}(t)=\tanh \left({h}_{i}^{\mathrm{ext}}+\displaystyle \sum _{j}{J}_{{ij}}{m}_{j}^{\mathrm{TAP}}(t)-{m}_{i}^{\mathrm{TAP}}(t)\displaystyle \sum _{k}{J}_{{ik}}^{2}[1-{m}_{k}^{\mathrm{TAP}}{(t-1)}^{2}]\right),\end{eqnarray} \tag{ 266 }$$

which for stationary magnetizations reduce to the better-known equilibrium TAP equations [40, 41]. The negative term inside the hyperbolic tangent is known as the Onsager correction, which improves on the naive estimate of the mean effective field acting on a spin.

One now observes that the term inside the curly brackets on the right-hand side of our simpler corrected equation  (265) is exactly of the form of this Onsager correction, so there is a close relation between the two approaches. However, there are two significant differences. First, while the Onsager term in (266) appears inside the $\tanh ,$ in (265) it is outside, as if we had linearized in the correction. This is important: it means there is nothing on the right-hand side of (265) to stop the magnetizations from going outside the physical range $[-1,1]$. The origin of this difference is the fact that while the dynamical TAP method corrects the naive estimate of the field acting on a spin, the Gaussian fluctuation approach described in the previous subsection directly estimates corrections for the magnetizations themselves.

The second difference between (265) and (266) is that in the latter it is the corrected magnetizations ${m}_{i}^{\mathrm{TAP}}$ itself that appear in the Onsager term, making the approximation self-consistent. In our naive method, on the other hand, the Onsager term is evaluated at the uncorrected saddle-point magnetizations.

The dynamical TAP equations (266) can be derived by the Plefka approach [25]. This is an elegant method that captures corrections to the effective fields as explained above, ensuring in contrast to (265) that the magnetizations remain in the physical range even when corrections to the naive mean field estimate are large. This is achieved by working not with the generating functional directly (or the Helmholtz free energy in the equilibrium case), but with its Legendre transform with respect to the magnetizations (the analogue of the equilibrium Gibbs free energy). In this way, one essentially approximates the system with an independent spin model in which each spin feels an effective field ${h}_{i}^{\mathrm{eff}}$. This leads to expressions of the form ${m}_{i}(t)=\tanh ({h}_{i}^{\mathrm{eff}}(t-1))$ for the magnetizations, which therefore always lie between −1 and 1. The effective fields are determined so as to best match the Gibbs free energy of the original model, by a perturbation expansion to second order in J.

Interestingly, the philosophy of working with the Legendre transform can also be applied to the approach we have described above, which initially has a rather different starting point, namely a saddle point approximation with Gaussian corrections. We show this in the Appendix, where we demonstrate that by switching to the Gibbs free energy and keeping only first order terms in the corrections A_i one can retrieve exactly the dynamical TAP equations. We refer the reader to [42, 43] for applications of the same idea in the context of the equilibrium Ising and Potts models.