# Could a free conformal field theory be solved by second quantization?

I'm a beginner in CFT and I'm trying to understand whether or not standard tools in quantum field theory--more specifically second quantization--can be used to solve a conformal field theory. To begin, I'm trying to calculate the propagator of a free massless scalar field in 1+1D using second quantization, and compare it with CFT result.

The Action (Lagrangian) is $$S=\int\partial^\mu \phi\partial_\mu\phi~ dx dt.$$ The free field expansion of $$\phi$$ in e.g. Eq.(2.47) of Peskin's book is $$\phi(x)=\int\frac{dp^1}{2\pi}\frac{1}{\sqrt{2|p^1|}}(a_p e^{-i p\cdot x}+a^\dagger_p e^{i p\cdot x}).$$ Then we have $$\langle \phi(x)\phi(0)\rangle=\int_{-\infty}^\infty \frac{dp^1}{2\pi}\frac{1}{2|p^1|}e^{-ip\cdot x}=\int_0^\infty \frac{dp}{4\pi p} e^{-ip x^0}(e^{ip x^1}+e^{-ip x^1}).$$ But this doesn't even converge!

On the other side, notice that if we do a Wick rotation $$t\to i\tau$$, we get a free massless boson CFT in 2D Euclidean spacetime (up to some irrelevant boundary terms and an overall coefficient) $$S\sim\int\partial_{\bar{z}} \phi\partial_z\phi~ dz d\bar{z},$$ for which the standard CFT textbooks (e.g. DiFrancesco Chap.2.3.4) give $$\langle \phi(z)\phi(0)\rangle\propto \ln |z|.$$

What am I missing here? Is it at all possible to obtain the CFT results for free field using second quantization+ Wick rotation?

More generally, I'm wondering why standard tools we learned in QFT--second quantization (canonical commutation between $$\phi$$ and $$\pi$$, Fourier mode expansion in $$a_p,a_p^\dagger$$)--no longer appears in CFT (instead, I see a lot of OPE which I didn't encounter in QFT courses).