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I'm confused whether the Jacobian is needed in a path integral representation of a dynamical system, as I've seen multiple conventions in the existing literature.

For simplicity, let's just consider the following ODE: $$\dot{x} = F(x) + \eta(t),$$ where $F$ is some polynomial of $x$ of degree greater than $2$, and $\eta$ is some Gaussian white noise. Normally, the MSR (Martin-Siggia-Rose) path integral can be written as $$ \int DxD\eta \,\, P(\eta)\delta(\dot{x}-F(x)-\eta) \det\Big( \frac{\delta(\dot{x}-F(x))}{\delta x} \Big) \\ = \int DxDp \, \exp\big( -p^2 + ip(\dot{x}-F(x)) \big) \det\Big( \frac{\delta(\dot{x}-F(x))}{\delta x} \Big), $$ where $p$ is the "momentum" corresponding to the noise, and I'm interpreting the last term (the determinant) as some sort of Jacobian from the measure of $x$ to the measure of $F(x)$ (is this interpretation correct?), which in general does not vanish.

However, in the spin-glass notes by Castellani, the determinant seems to be excluded in his dynamical treatment of the PSM ($p$-spin spherical model) (see equation 93 on page 26): https://arxiv.org/abs/cond-mat/0505032: $$ \int DxD\eta P(\eta) \delta(\dot{x}-F(x)-\eta) ,$$ in which case the Jacobian would have result in the Hessian of the Ising energy function, which is clearly non-trivial if $p\geq 2$. My questions is, why is the Jacobian not considered in the dynamical treatment of the PSM?

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  • $\begingroup$ Apparently, the presence/absence of the determinant depends on whether one is doing the Stratonovich/Ito calculus. However, I do not understand mathematically why this is the case, so I'm leaving it as a comment for now. $\endgroup$
    – PeaBrane
    Dec 14 '20 at 23:41
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Discretisation

To compute the Jacobian we need to discretise time, $t=k\Delta$ with some small time-step $\Delta$, and integer $k\in [0, M]$, and work with a finite number of random variables $$x_k=x(k \Delta), \eta_k = \int_{k\Delta}^{(k+1)\Delta}\eta(t)\mathrm{d}t \tag{1}\label{discr}.$$ If $\eta(t)$ is Gaussian white noise at temperature T, the $\eta_n$'s are i.i.d. zero-mean Gaussian random variables with $\langle{\eta_k\eta_m\rangle}=2T\Delta\delta_{km}$.

In discretising the Langevin's equation $$ \dot{x} = F(x) + \eta(t) \tag{2}\label{langevin}$$ we have some freedom on when to evaluate $F$. For any $\alpha\in[0,1]$, the process $$ \frac{x_{k+1}-x_k}{\Delta} = (1-\alpha) F(x_{k})+\alpha F(x_{k+1})+\frac{1}{\Delta}\eta_k \tag{3}\label{discrLang}$$ converges to (\ref{langevin}) as $\Delta\to 0$. This is a generalisation of the Itô ($\alpha=0$) and Stratonovich ($\alpha=1/2$) conventions. This corresponds to specifying the same-time correlation $\langle x(t) \eta(t)\rangle$. In general, the same SDE can define different stochastic processes depending on $\alpha$. For example, that is the case in the presence of multiplicative noise. However, for the simpler case of (\ref{langevin}), the resulting process is the same for any $\alpha$. See this paper for a nice discussion of the difference between the two prescriptions, and this for further discussion and some recent developments.

Computing the Jacobian

The probability distribution of the discretised noise ($M$ iid Gaussians) is $$ P(\boldsymbol{\eta}) = \frac{1}{(4\pi T\Delta)^{M/2}} \mathrm{e}^{\frac{1}{4T\Delta}\sum_k \eta_k^2} .$$ We can now use (\ref{discrLang}) to change variables from $\eta$'s to $x$'s,

$$ P(\mathbf{x}) = P(\boldsymbol{\eta})\det \mathbf{J}(\mathbf{x}), \qquad \text{where}\ J_{km}(\mathbf{x})=\frac{\partial \eta_{k-1}}{\partial x_m} .$$ Note that because of how we defined (\ref{discr}) the index of $\eta_k$ runs from 0 to $M-1$, while that of $x_k$ from 1 to $M$ (the initial condition $x_0$ needs to be specified separately). From (\ref{discrLang}), $$ J_{km} = \delta_{k,m}\left[1-\alpha \Delta F'(x_k)\right]+\delta_{k,m+1} \left[-1-(1-\alpha) \Delta F'(x_k) \right].$$

That is, $\mathbf{J}$ is an $M\times M$ matrix with non-zero elements only on the diagonal, and just below it. Its determinant is the product of the elements on the diagonal,

$$ \det\mathbf{J} = \prod_{k=1}^m \left[1-\alpha \Delta F'(x_k)\right] = \mathrm{e}^{\sum_k\log[1-\alpha\Delta F'(x_k)]} = \mathrm{e}^{- \alpha\Delta\sum_k F'(x_k)+\mathcal{O}(\Delta)} .$$

Therefore in the $\Delta\to0$ limit, the MSR action gets, from the Jacobian, a term $$ -\alpha \int_0^t F(x(t'))\mathrm{d}t' .$$

Note that in the Itô convention ($\alpha=0$) $\mathbf{J}$ is the identity matrix, and therefore the term above vanishes, which is why it is usually chosen in this context (as mentioned in the paper by Castellani and Cavagna).

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