Helium Atom Spectrum in Non-Commutative Space

In the context of the field theory we show that for particles with opposite non-commutativity, in QED, there is no correction for the two-body bound state at the tree level. Consequently, the Helium atom is considered in the non-commutative space. It is shown that the effects of spatial non-commutativity appear at the tree level and they are of the order α. ∗ email: [email protected] † email: [email protected] Non-commutative spaces and their phenomenological aspects have been recently considered by many authors [1-6]. Among the others P. M. Ho and H. C. Kao in their paper [1] has shown that there is not any non-commutative correction to the Hydrogen atom spectrum at the tree level owing to the opposite non-commutativity of the particles with opposite charges. Although such corrections have been done in the non-commutative quantum mechanics (NCQM) framework but one can easily see that in the tree level each bound state diagram, in QED, contains only two vertices with opposite charges. Therefore taking θ with opposite sign in each diagram, results in no correction at this level. These arguments become more clear if one notices the Bethe-Salpeter (BS) equation [7]. The BS equation provides a covariant description for bound states of two body systems. This equation for the bound state of a fermion-(anti)fermion system has the form (Fig. 1): Ψ(p, q) = S(p)S(q) ∫ dkI(k; p, q)Ψ(p + k, q − k), (1) where Ψ(p, q) is the BS amplitude for the bound state, S(p) and S(q) are the fermion field propagators and I(k; p, q) is the kernel of the interaction which is the sum of all possible irreducible graphs. The effects of non-commutativity are introduced in QED by modifying the vertices and including 3-photon and 4-photon interactions. Therefore, in non-commutative QED (NCQED) the kernel of Eq.(1) can be easily constructed by using the Feynman rules of the theory which are completely given in references [5, 8]. For instance, for the Hydrogen atom the tree level calculations correspond to the ladder approximation for the kernel with free particle propagators (Fig. 2). At this level in the center of mass frame one has I l θ(k; p,−p) = e i 2 +−I (k), (2) where the superscript l stands for ladder and I (k) is the interaction kernel in the commutative space in the ladder approximation. The parameter of non-commutativity θ, is an antisymmetric real valued tensor defined as follows: θ = −i [x, x ] . (3) Since θ0i 6= 0 leads to some problems with the concept of causality and the unitarity of the field theories [9], it is assumed that space and time commute, i.e. θ0i = 0. The subscripts ± in Eq.(2) refers to the assumption that one should associate different parameters of non-commutativity to the coordinates of particles with different charges ±e [1]. Obviously for θ+ = −θ−, Eq.(2) gives I l θ(k; p,−p) = I (k). (4) In the case of positronium, there is also an annihilation diagram in the lowest order. In such a diagram, each vertex contains a factor e i 2 p.θ.(−p) = 1, thus there is not any non-commutativity correction at this level due to the annihilation graph. Consequently for two-body bound states in QED there is not any corrections at the tree level and for such systems the lowest order contribution of θ comes from the one-loop diagrams and is negligible. Figure 1: Bethe-Salpeter equation for two particles bound state. Figure 2: Bethe-Salpeter equation in the ladder approximation. The coupling constant g± = −ieγμe − i 2 ±, where θ± are the non-commutativity parameter corresponding to the particles with charges ±e. In contrast, in the Helium atom there are two electrons with the same parameter of noncommutativity and the corrections at the tree level is expected. Therefore, we explore the effects of non-commutativity of space on the spectrum of the Helium atom, in NCQM, by considering θ− = θ and θ+ = −θ for the electron and the proton, respectively. To this end we consider the Schrödinger equation for the Helium atom as follows: i ∂ ∂t ψ (x1,x2,x; t) = H (x1,x2,x) ⋆ ψ (x1,x2,x; t) , (5) in which xi is the position of the i-th electron, x is the coordinate of the nucleus and H = − ∇x1 2m − ∇x2 2m − ∇x 2M + V (x1,x2,x) , (6) where m and M are the masses of the electron and the nucleus, respectively, and V (x1,x2,x) = − Ze2 |x1 − x| − Ze2 |x2 − x| + e2 |x1 − x2| . (7) Now we define the center of mass coordinates as R = m (x1 + x2) +Mx 2m+M , rj = xj − x j = 1, 2. (8) The non-commutativity algebra, [x1,x2] = 0, [xj ,x] = 0, j = 1, 2 [xj,xj ] = +iθ, j = 1, 2 [x,x] = −iθ. (9) leads to the following algebra for the center of mass coordinates: [r1, r2] = −iθ, [rj ,R] = i m+M 2m+M θ = iηθ, j = 1, 2, [rj , rj ] = 0, j = 1, 2, [R,R] = i 2m2 −M2 (2m+M)2 θ = iβθ. (10) Considering Eq.(10) and the translation rj → rj + 1 2 ηθ.K, (11) in which K is the wave vector of the center of mass [1], one can show that