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The stationary phase approximation can be used to find an approximate value for the path integral \begin{equation}\int Dx e^{-S[x]} \approx e^{-S[\bar{x}]} \left(\det{\frac{\hat{A}}{2 \pi}}\right)^{-1/2}. \end{equation} Found by expanding around the stationarity of the functional $S$. \begin{equation} S[x] = S[\bar{x}+\lambda_x] \approx S[\bar{x}] + \frac{1}{2}\int dt \int dt' \lambda_x(t)A(t,t')\lambda_x(t'), \end{equation}

Where $A(t,t') = \frac{\delta^2 S[x]}{\delta x(t) \delta x(t')}\rvert_{x =\bar{x}}$, the first order term evaluates to zero and $\bar{x}$ extremizes $S$.

However, I am not sure how to deal with the case where the action depends on multiple coordinates; $S[x,y]$. I can also expand this action around its extreme point

\begin{equation} S[x] = S[\bar{x}+\lambda_x, \bar{y} + \lambda_y] \approx S[\bar{x},\bar{y}] + \frac{1}{2}\int dt \int dt' \lambda_x(t)A(t,t')\lambda_x(t') + \frac{1}{2}\int dt \int dt' \lambda_x(t)B(t,t')\lambda_y(t') + \frac{1}{2}\int dt \int dt' \lambda_y(t)C(t,t')\lambda_x(t') + \frac{1}{2}\int dt \int dt' \lambda_y(t)D(t,t')\lambda_y(t'), \end{equation}

Where $A,B,C,D$ are the corresponding functional derivatives.

But following this, I am unsure what the right next step is to actually get an approximate value for the path integral? If there were no cross-terms I would just reason; Ah we have some Gaussian integrals being added in the exponent so let's just multiply the corresponding determinants together.

Qmechanic
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incoherent_state
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1 Answers1

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FWIW, with multiple coordinates the functional determinant $\det H$ in the stationary phase approximation is over a Hessian $H_{jk}(t,t^{\prime})=\frac{\delta^2 S}{\delta x^j(t) \delta x^k(t^{\prime})}$ with both discrete and continuous indices. See also e.g. this related Phys.SE post.

Qmechanic
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