From the path integral approach, we know that ## \displaystyle \langle x,t|x_i,0\rangle \propto \int_{\xi(0)=x_i}^{\xi(t_f)=x} D\xi(t) \ e^{iS[\xi]}##. Now, using ## |x,t\rangle=e^{-iHt}|x,0\rangle ##, ## |y\rangle\equiv |y,0\rangle ## and ## \sum_b |\phi_b\rangle\langle \phi_b|=1 ## where ## \{ \phi_b \} ## are the energy eigenstates we have:(adsbygoogle = window.adsbygoogle || []).push({});

## \langle x,t|x_i\rangle =\langle x| e^{iHt}|x_i\rangle=\sum_b \langle x|e^{iHt}|\phi_b\rangle \langle \phi_b|x_i\rangle=\sum_b \langle x|\phi_b\rangle e^{iE_bt}\langle \phi_b|x_i\rangle=\sum_b \phi_b(x) \phi_b^*(x_i) e^{iE_bt} ##

Now by doing a Wick rotation, ## t=it_E ## and taking the limit ## t_E\to \infty ##, we'll have:

##\displaystyle \langle x,i\infty|x_i\rangle \propto \phi_0(x) \Rightarrow \phi_0(x) \propto \int_{\xi(0)=x_i}^{\xi(t_E=\infty)=x} D\xi(t) \ e^{-S[\xi]} ##

Using a similar argument we can find:

##\displaystyle \phi_0^*(x) \propto \int_{\xi(t_E=-\infty)=x}^{\xi(0)=x_i} D\xi(t) \ e^{-S[\xi]} ##

The problem is, everywhere that I see this, the path integral for ## \phi_0(x) ## is from ## -\infty ## to 0 and the path integral for ## \phi_0^*(x) ## is from 0 to ## \infty ##, the opposite of what I got. What am I doing wrong?

Thanks

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# A Ground state wave function from Euclidean path integral

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