Scalar-Scalar Bound State in Noncommutative Space and Normal Zeeman Effect

Bethe-Salpeter equation in the non-commutative space for a scalarscalar bound state in the ladder approximation with instantaneous interaction is considered. We show that spatial non-commutativity leads to the normal Zeeman effects on the spectra of two body bound state. e-mail: [email protected] e-mail: [email protected] Non-commutativity of space-time has been recently a subject of intense interest both in quantum mechanics and quantum field theory [1]-[6]. In this paper, we would like to study the effects of such a non-commutativity on spectra of bound state of two scalar particles. Bethe-Salpeter (BS) equation [7, 8] is the usual tool for computing, for instance, the electromagnetic form factors and relativistc spectra of two body bound states. In the following analysis we examine scalar-scalar bound state spectra. To this end, the BS equation for two scalar particle is Γ(p1, p2) = ∫ dkI(k, p1, p2)D(p1 + k, p2 − k)Γ(p1 + k, p2 − k). (1) Γ(p1, p2) is the bound state vertex function and D(p1, p2) is given by D(p1, p2) = D(p1)D(p2), (2) where D(p) is the scalar field propagator, which is usually approximated by its free form as D(p) = 1 p −m + iǫ . (3) I(k; p1, p2, θ) is the interaction kernel in the non-commutative space-time and depends on the parameter of non-commutativity [1] θ = −i[x, x ]. (4) In general it is not possible to find the exact solutions of the BS equation Eq.(1). Therefore we consider the ladder approximation and assume the instantaneous interactions [9]. Consequently one can rewrite the interaction kernel in the well known form I(k; p1, p2, θ) = exp [ik ∧ (p1 − p2)] I(k ̄ ), (5) where p ∧ q = 1 2 θpμqν . It is shown that θ 0i 6= 0 lead to some problems with unitarity of field theories and the concept of causality [2, 3]. Therefore, we consider θ = 0. We define E to be the mass of the bound state in the center of momentum (CM) frame and t and T , the particles individual CM bound state energy. Thus the CM energy-momenta of the particles would be p1 = (p ̄ , t+ w) and p2 = (−p ̄ , T − w) and we have E = t+ T [10]. 1 Defining φ(p ̄ ) = 1 2πi ∫ dwD(p1, p2)Γ(p1, p2), one can show that Eq.(1) leads to the Eq.(6)