# 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).

## 1 Answer

The integral for the propagator in position space needs to be regulated and that is best looked up in a previous answer.

But as for the second part, canonical quantization and Fourier modes obeying Wick's theorem are mainly useful for QFTs that can be described in terms of finitely many fundamental fields which have a solvable weak coupling limit. Many interesting theories are not of this type, yet it turns out we can still learn a lot about them if they are conformal. This is the reason why CFT deserves to be its own subject. As you point out, the fundamental tool it uses is the OPE. This is a convergent expansion in CFTs but it is only asymptotic in general QFTs as I get into here.