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i would like to evaluate $$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

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    $\begingroup$ Hello, have a look at this answer, I think it will be helpful: physics.stackexchange.com/a/488193/226902 $\endgroup$
    – Quillo
    Commented May 4, 2023 at 7:32
  • $\begingroup$ @Quillo but how do i deal with $\dot x$ ? do i treat it is a constant? $\endgroup$
    – Lendion
    Commented May 4, 2023 at 9:04
  • $\begingroup$ Hi @Lendion. Welcome to Phys.SE. Out of curiosity, where did the Lagrangian (2) come from? What are the boundary conditions? $\endgroup$
    – Qmechanic ♦
    Commented May 4, 2023 at 9:25
  • $\begingroup$ @Qmechanic the Lagrangian is in its current form because of a path integral on an auxiliary field introduced in an earlier part of the calculation, the problem itself is a statistical mechanics one $\endgroup$
    – Lendion
    Commented May 4, 2023 at 9:42
  • $\begingroup$ it was pointed out to me that this can also be solved exactly by means of fourier transform, however if someone could still give me some insight as to how to proceed with the saddle point approximation it would be appreciated as i suspect that it would also produce the same result in this specific case, while requiring a lighter computation $\endgroup$
    – Lendion
    Commented May 4, 2023 at 12:43

1 Answer 1

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The crossterms in OP's Lagrangian (2) are total derivative terms, so OP's partition function (1) with Dirichlet boundary conditions is essentially an ordinary harmonic oscillator (in Euclidean signature), which is e.g. done exactly in this Phys.SE post. The exact result coincides with the saddle point approximation.

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