Un-equivalency Theorem of Deformed Heisenberg-Weyl’s Algebra in Noncommutative Space

An extensively tacit understandings of equivalency between the deformed HeisenbergWeyl algebra in noncommutative space and the undeformed Heisenberg-Weyl algebra in commutative space is elucidated. Equivalency conditions between two algebras are clarified. It is explored that the deformed algebra related to the undeformed one by a non-orthogonal similarity transformation. Furthermore, non-existence of a unitary similarity transformation which transforms the deformed algebra to the undeformed one is demonstrated. The un-equivalency theorem between the deformed and the undeformed algebras is fully proved. Elucidation of this un-equivalency theorem has basic meaning both in theory and practice. Spatial noncommutativity is an attractive basic idea for a long time. Recent interest on this subject is motivated by studies of the low energy effective theory of D-brane with a nonzero Ns NS B field background [1–3]. It shows that such low energy effective theory lives on noncommutative space. For understanding low energy phenomenological events quantum mechanics in noncommutative space (NCQM) is an appropriate framework. NCQM have been extensively studied and applied to broad fields [4–16]. But up to now it is not fully understood. In literature there is an extensively tacit understandings about equivalency between the deformed Heisenberg-Weyl algebra in noncommutative space and the undeformed Heisenberg-Weyl algebra in commutative space. As is well known that the deformed phase space variables are related to the undeformed ones by a linear transformation. This leads to such tacit understandings of equivalency between two algebras. In this paper we elucidate this subtle point. First we clarify equivalency conditions between two algebras. Then we demonstrate that the deformed algebra is related to the undeformed one by a similarity transformation with a non-orthogonal real matrix. Furthermore, we prove that a unitary similarity transformation which transforms two algebras to each other does not exist. The results are summarized in the un-equivalency theorem between two algebras. Because the deformed and undeformed Heisenberg-Weyl algebras are, respectively, the foundations of noncommutative and commutative quantum theories, elucidation of the un-equivalency theorem has significant meaning both in theory and practice. One expects that essentially new effects of spatial noncommutativity may emerge from noncommutative quantum theories. This depends on the fact that the deformed Heisenberg-Weyl algebra is not equivalent to the undeformed one. The un-equivalency theorem shows that explorations of essentially new effects of spatial noncommutativity emerged from noncommutative quantum theories can be expected. In order to develop the NCQM formulation we need to specify the phase space and the Hilbert space on which operators act. The Hilbert space is consistently taken to be exactly the same as the Hilbert space of the corresponding commutative system [4]. As for the phase space we consider both position-position noncommutativity (positiontime noncommutativity is not considered) and momentum-momentum noncommutativity