# Jacobian in dynamic path integral

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?

• 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. Dec 14 '20 at 23:41

### 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).