path integral - Solving Quantum Tunnelling Without Wick Rotation

Edit

It seems that I haven't written my question clearly enough, so I will try to develop more using the example of quantum tunnelling. As a disclaimer, I want to state that my question is not about how to perform a Wick rotation in the path integral formulation!

Let's look a the probability of quantum tunnelling in the path integral formulation. The potential is given by $V[x(t)]=(x(t)^2-1)^2$, which has two minima at $x=\pm x_m=\pm1$. Given that the particle starts at $t=-\infty$ at $x=-x_m$, what is the probability that it is at $x=x_m$ at $t=\infty$. The probability amplitude is given by

$$K(x_m,-x_m,t)=\langle x_m|e^{-i \hat H t}|-x_m \rangle$$

The usual trick is to Wick rotate $t\to-i\tau$, compute everything in imaginary time using a saddle point approximation and at the end of the calculation rotate back to real time. I understand how it works. No problem with that.

What I want to understand is

• how can I do the calculation without using the Wick rotation?



• how does this solution connect to the Euclidean formulation?

In principle, we should be able to do the calculation with the path integral formulation in real time

$$\int Dx(t) e^{i S[x(t)]/\hbar}$$

In the stationary phase approximation we look for a complex path $x(t)$ which minimizes the action, and expand about this point.

Choose $m =1$ for simplicity. The equation of motion is

$$\ddot x-2 x+2x^3=0$$

which has no real solution, i.e. no Newtonian (classical) solution. But there is a complex function that solves it: $x_s(t)=i \,\tan(t)$. One problem is that it behaves pretty badly. If anyway I accept this a correct solution, I should be able to compute the gaussian fluctations, add up all the kinks/antikinks, etc. and recover the correct result (usually obtained with the euclidean action and $\tau\to -it$). Am I right?

So my question is: is it possible to do the calculation that way, and if so, how is it related to the trick of going back and forth in imaginary time?

Original

I have a question on the mathematical meaning of the Wick rotation in path integrals, as it is use to compute, for instance, the probability of tunneling through a barrier (using instantons).

I am aware that when computing an ordinary integral using the Stationary Phase Approximation

$\int dx e^{i S(x)/\hbar}$

with $x$ and $S$ real, one should look at the minimum of $S(z)$ in the whole complex plane, which can be for instance on the Imaginary axis.

In the case of a path integral, one wants to compute

$\int Dx(t) e^{i S[x(t)]/\hbar}$

and there is a priori no reason that the "classical path" from $x_a(t_a)$ to $x_b(t_b)$ (i.e. that minimizes $S[x(t)]$) should lie on the real axis. I have no problem with that. What I don't really get is the meaning of the Wick rotation $t\to -i\tau$ from a (layman) mathematical point of view, because it is not as if the function $x(t)$ is taken to be imaginary (say, $x(t)\to i x(t)$), but it is its variable that we change !

In particular, if I discretize the path-integral (which is what one should do to make sense of it), I obtain

$\int \prod_n d x_n e^{i S(\{x_n\})/\hbar}$.

where $S(\{x_n\})=\Delta t\sum_n\Big\{ (\frac{x_{n+1}-x_n}{\Delta t})^2-V(x_n)\Big\}$

At this level, the Wick rotation applies on the time slice $\Delta t\to -i\Delta \tau$ and does not seem to be a meaningful change of variable in the integral

I understand that if I start with an evolution operator $e^{-\tau \hat H/\hbar}$ I will get the path integral after Wick rotation, but it seems to be a convoluted argument.

The question is : Is it mathematically meaningful to do the Wick rotation directly at the level of the path-integral, and especially when it is discretized?