Two-dimensional bosonic field theory

I'm struggeling with the following question:

Consider a two-dimensional bosonic field theory defined by the following action $$S =\frac{k}{2} \int dx_{1}dx_2 [(∂x_1 φ(x_1, x_2))^2 + (∂x_2 φ(x_1, x_2))^2 ]$$ with a Green function $$G(x) ≡ <φ(x_1, x_2)φ(0, 0)> = −\frac{1}{4\pi K} \ln( |x|^2),$$ $$x = (x_1, x_2).$$ Derive a quantum analogue of the Pythagoras theorem $$: \cos(√4πKφ(x)) :^2 +: \sin(√4πKφ(x)) :^2 =?$$ What happens when the arguments of cos and sin would have a different prefactor in front of φ(x) (e.g $$\cos(\sqrt{\alpha\phi}$$) for some $$\alpha\phi \neq \sqrt{4\pi K}$$?

I defined the following vertex operator:

$$V_{\alpha} = :e^{i\alpha \phi}:$$

expanded the functions and I got:

$$: \cos(√4πKφ(x)) :^2 +: \sin(√4πKφ(x)) :^2 = \frac{1}{2}(:e^{i\alpha \phi}::e^{-i\alpha \phi}:+:e^{-i\alpha \phi}::e^{i\alpha \phi}:)$$

But can I add these two normal ordered operators?

$$\frac{1}{2}(:e^{i\alpha \phi}::e^{-i\alpha \phi}:+:e^{-i\alpha \phi}::e^{i\alpha \phi}:) = :e^{i\alpha \phi}::e^{-i\alpha \phi}:$$

• Could you provide a reference? – LucashWindowWasher Apr 19 at 3:11
• Yes, you can. See (6.63)-(6.66) in di Francesco. – mavzolej Apr 19 at 4:17