# Wick rotation and Exponential mapping of an "imaginary" differential operator acting on a real-valued wavefunction

The position shift operator $$T^{a}$$ (where $$a \in \mathbb{R}$$ ) takes a real valued wavefunction $$\psi$$ on $$\mathbb{R}$$ to its translation $$\psi_{a}$$, $$T^{a} \psi(x)=\psi_{a}(x)=\psi(x+a)$$. A practical operational calculus representation of the linear operator $$T^{a}$$ in terms of the plain derivative $$\frac{d}{d x}$$ was introduced by Lagrange, $$T^{a}=e^{a\frac{d}{d x}}.$$

Of course this is the well known unitary translation operator in quantum mechanics and the exponent can be written to express it in terms of the momentum operator $$T^{a}=e^{a\frac{i}{\hbar}(\frac{\hbar}{i}\frac{d}{d x})} =e^{a\frac{i}{\hbar}\hat{P_x}}.$$

But what about $$T^{ia}=e^{ia\frac{d}{d x}}=e^{\frac{-a}{\hbar}(\frac{\hbar}{i}\frac{d}{d x})}=e^{\frac{-a}{\hbar}\hat{P_x}}$$ where $$\psi$$ is a real valued wavefunction of a real variable, $$i$$ the imaginary unit and $$a\in \mathbb{R}$$. It's meaningful to consider its action on a real-valued function? $$T^{ia}\psi(x)=e^{ia\frac{d}{d x}}\psi(x)=???$$ What would be the expected result?

There's any connection to the Wick rotation? It's possible to perform a Wick rotation in the variable to get a meaningful result in some way?

I doubt there is a deep connection with Wick's rotation. The main purpose behind the imaginary prefactor is mathematical convenience. Indeed, the standard shift operator $$e^{a\partial_x}$$ is anti-unitary (for a real $$a$$) because the derivative $$\partial_x$$ itself is skew-hermitian, while $$i\partial_x$$ is a hermitian operator, which makes $$e^{ia\partial_x}$$ unitary.
Of course, the initially real-valued wavefunction becomes complex-valued, because $$e^{ia\partial}\psi(x) = \psi(x+ia)$$, but it is not a problem in quantum mechanics, since the only meaningful quantity $$-$$ because it embodies the measured physical observables of the system $$-$$ turns out to be the probability distribution represented by $$|\psi(x+ia)|^2$$ after translation, which is indeed real thanks to the modulus.