In Wen XiaoGang's QFT of Many-Body Systems Sect 2.4.2, He studied the decay of a Meta-stable state via path integral method.

The real-time potential is

Real-time Potential

A state at $x=-a$ decays to $x=\infty$.

The imaginary-time potential is

Imaginary-time Potential

He claimed that in the integral of the quantum fluctuation of the bounce process, the determination of the operator $-\frac{d^2}{d\tau^2}+V''(x_{cl})$ has a negative eigenvalue $-\lambda$. So the quantum fluctuation contribution $K\sim det(-\frac{d^2}{d\tau^2}+V''(x_{cl}))<0$

However, if this is the case, then let $c$ be the amplitude of this negative mode. When integrating the quantum fluctuation, we would calculate $\int dc e^{\frac 12 \lambda c^2}$, which diverges. Does that make any sense?

P.S. After a few days' thought, I tend to believe that the reason is that the stationary phase approximation holds only in the vicinity of the classical path. So the integration of a quantum fluctuation mode should have an upper cut-off.

As to this system, the cliff of the real-time potential should have a bottom representing the true vacuum. So the high mountain of the imaginary-time potential should have a top. If we integrate the negative quantum fluctuation mode to infinity, we would climb over the mountain top and the approximation fails.

Is this idea correct? Is $K$ imaginary because of the magic of analytic continuation?

PS2. I've found the answer in The Uses of Instantons Sect 2.4 by Sidney Coleman. It turns that the considerations in my PS1 make sense.


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