Generalized Wick Theorems in Conformal Field Theory and the Borcherds Identity

Abstract

As a counterpart of the well-known generalized Wick theorem by Bais et. al. in 1988 for interacting fields in two dimensional conformal field theory, we present a new contour integral formula for the operator product expansion of a normally ordered operator and a single operator on its right hand. Quite similar to the original Wick theorem for the opposite order operator product, it expresses the contraction i. e. the singular part of the operator product expansion as a contour integral of only two terms, each of which is a product of a contraction and a single operator. We discuss the relation between these formulas and the Borcherds identity satisfied by the quantum fields associated with the theory of vertex algebras. 5/28 local. The following proposition is known as the Dong's lemma. 5) Proposition 3 (Ref. 6, Proposition 2.1.5.) If A(z), B(z) and C(z) are mutually local fields, then A(z) (m) B(z) and C(z) are local. 7/28

A new generalized Wick theorem in conformal field theory

“…He also pointed out its equivalence to the generalized Wick theorem (1). A proof of this equivalence is available in [8]…”

“…As far as the author knows, no one had correctly answered this question yet. So we tried to answer it and found the following expression [8].…”

“…In section 3 we present examples of the calculations of the operator product expansions by using the generalized Wick theorems including fermionic fields. For reasons of simplicity, I will omit almost all mathematically rigorous arguments which are available in [8] for interested readers.…”

Новая Обобщенная Теорема Вика В Конформной Теории Поля

“…Насколько известно автору, пока никто не дал корректного ответа на этот вопрос. Мы попытались найти этот ответ и получили следующее выражение [3]:…”

“…В разделе 3 мы приводим примеры вычисления операторных разложений путем использования обощенной теоремы Вика, содержащей фермионные поля. В целях упрощения изложения почти все строгие математические рассуждения опущены, интересующийся читатель может найти их в работе [3].…”