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In calculating the false vacuum decay, the main contribution to the imaginary energy part of the Euclidean path integral comes from the bounce solution. And we somehow apply saddle point approximation to it. Among all the deformations, most make it a minimum. But, since there is one negative eigenvalue, deformation along that eigenfunction makes the Euclidean action not a strict local minimum. Then, how is the approximation still valid and the result still physical?

What adds to the apparent mathematical scam is that, the original quantity $\langle 0\vert e^{-HT}\vert0\rangle$ should be a real quantity without the imaginary part, which obviously contradicts an complex-valued result.

I think people who are familiar with this topic and able to answer the question don't need further details.

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  • $\begingroup$ Possible duplicate: How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue? $\endgroup$
    – Qmechanic ♦
    Commented Oct 14, 2023 at 10:30
  • $\begingroup$ @Qmechanic Yes, after reading part of Marino's book, I kind of get what's happening mathematically. It's obscure how the analytic continuation should be explicitly done even for a one dimensional particle. Is analytic continuation carried out for the negative eigenvalue after it is solved? How should complex eigenvalues be present during the continuation? Is it equivalent to some kind of analytic deformation of the potential? I can't answer these questions, let alone why the imaginary part above and below the branch cut can be identified with the decay rate. $\endgroup$
    – Bababeluma
    Commented Oct 14, 2023 at 19:10

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You deform the path of integration for the coeffeicient of that mode so that it leaves the real axis at 90 degrees at the point where the coefficient reaches the stationary point. In this way the integration for the imaginary part of the action is the usual steepest descent path.

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  • $\begingroup$ Yes, after reading Marino's book, I kind of get what's happening mathematically and what you're talking about. It's obscure how the analytic continuation should be explicitly done even for a one dimensional particle. Is analytic continuation carried out for the negative eigenvalue after it is solved? How should complex eigenvalues be present during the continuation? Is it equivalent to some kind of analytic deformation of the potential? I can't answer these questions, let alone why the imaginary part above and below the branch cut can be identified with the decay rate. $\endgroup$
    – Bababeluma
    Commented Oct 14, 2023 at 19:04
  • $\begingroup$ You are computing the ground-state energy density ${\mathcal E}$. Twice the imaginary part of the energy density gives the decay rate per unit volume: $P_{\rm survive}=e^{-\Gamma t}$, $ \Gamma = 2{\rm Im}({\mathcal E})(Vol)$ $\endgroup$
    – mike stone
    Commented Oct 15, 2023 at 11:42
  • $\begingroup$ my problem can be well summarized by quoting Schwartz's words in his 2016 paper "Direct Approach to Quantum Tunneling": What seems clear is that there is a mathematical consistent way to define the imaginary part for the real quantity Z(T) in the $T\rightarrow \infty$"limit. However, there is a surprising lack of commentary on the connnection between this imaginary quantity and the physical decay rate. So I don't think there is a good answer to what I'm asking, for the moment, even in the science community. $\endgroup$
    – Bababeluma
    Commented Oct 15, 2023 at 13:29

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