12.6. Quantum Pythagorean theorem A regularization of the normal order : A(x)B(x) : of two operators can be obtained by the limit limη→0 : A(x−η/2)B(x+η/2) : and an average on all directions of η, so that the final expression is invariant under the rotations. The average is equivalent to the substitution ημην/|η|2→ 12

δμν.

Use this regularization and the bosonization formulae of the text to prove the quantum version of the Pythagorean theorem

(: cosϕ :)2 +(: sinϕ :)2 = −1 4

(∂ϕ)2.

(Observe that (: cosϕ :)2 =: cos2 ϕ :).