A unifying perspective on the Moyal and Voros products and their physical meanings

Abstract

The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative nonrelativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.

Aspects of finite electrodynamics inD= 3 dimensions

“…In this connection, it may be recalled that the essential idea of this development is to define the fields as mean value over coherent states of the noncommutative plane, such that a star product needs not be introduced. We further note that recently it has been shown that the coherent state approach can be summarized through the introduction of a new multiplication rule which is known as Voros star-product [10,11], [12,13,14]. Nevertheless, and most importantly, physics turns out be independent from the choice of the type of product [15].…”

Remarks on the interquark potential in the presence of a minimal length

“…Also, from (24) it is easy see that the introduction of a minimal length preserves the asymptotically free behavior at short distance characterizing the non-Abelian character of the strong interactions.…”

“…Interestingly, it is observed that the term proportional to θ is ultraviolet finite. Another important point to be mentioned in our discussion comes from the explicit appearance of the cutoff Λ in (24). Evidently, at distances higher than 1 / Λ the interaction potential (24) is convergent.…”

Finite axionic electrodynamics from a new non-commutative approach

“…To do this, the source δ (3) (x − y) is replaced by the smeared source e θ 2 ∇ 2 δ (3) (x − y). Hence expression (18) reduces to…”

“…The key ingredient of this development is to define the fields as mean value over coherent states of the noncommutative plane, such that a star product needs not be introduced. More recently, it has been shown that the coherent state approach can be summarized through the introduction of a new multiplication rule which is known as Voros star-product [14,15], [16,17,18]. Anyway, physics turns out be independent from the choice of the type of product [19].…”