Noncommutative Quantization in 2D Conformal Field Theory

The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time and change the compactification radius appropriately. The four point function is deformed, preserving, nevertheless, the sl(2,C) invariance. Finally the first Ward identity of the deformed theory is derived. Email: [email protected] Email: [email protected] 1 In this letter we explore a possible way of quantizing a string based on a noncommutative harmonic oscillator as opposed to the ordinary commutative one [1]. This noncommutativity is neither the one in the D brane worldvolume that arises in the quantization of open strings ending on D branes with background B field [2], nor the one introduced in M-theory compactified on T 2 [3]. To study the modifications that occur in bosonic string theory, the simplest possible noncommutative harmonic oscillator in two dimensions is employed. We assume that the position and momentum operators satisfy the following commutation relations [4]: [q̂ n, p̂ j n] = ih̄δij ; [q̂ 1 n, q̂ 2 n] = iθ/n; [p̂ 1 n, p̂ 2 n] = −inθ. (1) An operator representation of the q̂ n(p̂ i n) that realizes the commutation relations (1) is: q̂ n = √ α 2n (an + a 1 n † ) (2) p̂n = −i √ n 2α [(h̄an − iθan)− (h̄an † + iθan † )] (3) q̂ n = √ 1 2nα [(h̄an − iθan) + (h̄an † + iθan † )] ≡ √ α 2n (An + A † n) (4) p̂n = −i √ nα 2 (an − an † ), (5)