Non-relativistic scalar field on the quantum plane

We apply the coherent state approach to the non-commutative plane to check the one-loop finiteness of the twofour-point functions of a non-relativistic scalar field theory in 2 + 1 dimensions. We show that the two-point and four-po functions of the model are finite at one-loop level and one recovers the divergent behavior of the model in the limit θ → 0+ by appropriate redefinition of the non-commutativity parameter.  2005 Elsevier B.V.Open access under CC BY license.


Introduction
The origin of the recent interests in non-commutative field theories refers to the fact that they may appear naturally in string/M-theories [1,2]. Historically it was a hope that the idea of non-commutative space-time may provide a mechanism, which can cure the UV divergent behavior of the quantum field theory [3]. However, Filk pointed out that the noncommutative field theories exhibit the same divergent behavior of the usual commutative field theories [4]. In non-commutative field theories one replaces the space-time coordinates x µ with operatorsx µ , which E-mail address: st_a.jahan@mail.urmia.ac.ir (A. Jahan).
do not commute with each other, i.e., (1) x µ ,x ν = iθ µν , with θ µν as a real antisymmetric matrix. Thus because of the uncertainty relation (2) x µ x ν |θ µν | 2 induced by the relation (1), short distance scales in the x µ direction correspond to the large distance scales in the x ν direction and vice versa. So there will be a mixing between the ultraviolet and infrared behaviors of the field theories in non-commutative space-times. Such a problem, which is called the "UV/IR mixing problem", is characteristic of the models based on the -product approach [5].
In a non-commutative space-time the usual product between the fields must be replaced by the Weyl- Moyal bracket or -product defined as The -product is the usual starting point in the most studies about the non-commutative field theories since it encodes the non-commutativity of space-time for the product of the several fields defined in the same point in the following sense Recently, authors in [6,7] have developed a new approach to study the non-commutative space-time.
Their main idea is to use the expectation values of the operators between the coherent states of the non-commutative space-time instead of using the -product (3). In this approach the free particle propagator acquires a Gaussian damping factor, which may act as a probable mechanism to make the whole formalism of the perturbation theory finite. So in the coherent state approach the non-commutativity of spacetime manifests itself by modification of the propagators rather than the interaction vertices and this is to say that there is no UV/IR mixing problem in coherent state approach to the non-commutative space-time.
Non-commutative non-relativistic scalar field theory is studied by the authors in [8] at one-loop level. Their argument, which is based on the -product approach, implies that the UV/IR mixing is not only characteristic feature of the relativistic fields but also it takes place in non-relativistic fields. In this Letter we apply the formalism based on the coherent state approach to the non-commutative plane in order to investigate the UV finiteness of the two-point and four-point functions of a non-relativistic scalar field on a quantum plane. To this aim we give a brief review of the coherent state approach to the quantum plane in the next section. In the subsequent section we apply the formalism based on the coherent state approach to the two-point and four-point functions of the model at one-loop level. In particular, we demonstrate that one obtains the divergent behavior of the model on the usual commutative plane from the finite non-commutative version of the model when the non-commutativity between coordinates disappears.

Coherent state approach to the non-commutative plane
We consider a non-commutative plane described by the coordinates y 1 and y 2 satisfying with corresponding raising/lowering operators as Coherent states corresponding to the above operators are defined as the eigenstates |α in the following sense then the mean position of the particle on the quantum plane is defined as More generally any function F (ŷ) on the non-commutative plane will be replaced by its mean value as So in coherent state formalism the fuzziness of space manifests itself by smearing the fields over space by modification of the kernel of integral via a Gaussian damping factor as in (9). Thus by means of the general recipe for any auxiliary kernel f (k), one immediately finds the non-relativistic free particle momentum space propagator as (settingh = m = 1)

Self-interacting scalar field on the quantum plane
We consider a self-interacting model of the nonrelativistic boson field characterized by the Lagrangian One-loop correction to the two-point function comes from the term (Fig. 1) where the UV cut-off parameter Λ is introduced. This divergent term can be removed by adding the appropriate counter term to the Lagrangian (13). However, as we shall see later, this expression is no longer divergent on the quantum plane. The one-loop quantum correction associated with the four-point correction is given by (Fig. 2) [9] where the unprimed and primed quantities refer respectively to the incoming and outgoing particles. So once again we are left with a divergent term (in this case logarithmically divergent) but as the case of two-point function, the expression (15) is no longer UV divergent on the quantum plane and one obtains a finite result for the integral (15), when the noncommutativity of plane is taken into account.  First let us focus on the two-point function. From (11) we have −iΣ (1) which, in contrast to (14), is obviously a finite result. By redefinition of the non-commutativity parameter as θ = 4/Λ 2 the expression (17) in the limit θ → 0 + coincides with Eq. (14) in the limit Λ → ∞, So one recovers the one-loop correction to the twopoint function on the usual commutative plane from the corresponding term on the quantum plane in the limit θ → 0 + .
Next we have the one-loop correction to the fourpoint function on the quantum plane as with k 0 = |k 0 |. After integration over ω and shifting k → k + k 0 /2, we arrive at where we have invoked the well-known formula Integral (21) is singular at 0 (ω 0 − k 2 0 4 )/2 (see Appendix A). Hence the Cauchy principal value of the integral in (21) must be evaluated carefully. We proceed further by noting that For the first integral of right-hand side of Eq. (23) we have where the exponential function E i (αx) is defined as The symbol γ stands for the Euler-Mascheroni constant. By substituting the results obtained above in (21) we are finally left with Now let us explore the θ → 0 + limit of the oneloop contribution to the four-point function on the quantum plane. To this end we expand the factor exp(−θω 0 /2) in a power series as where we have set θ = 4/Λ 2 once again and ignored the constant −γ − ln 2 since Λ 2 is a very large quantity.
Hence as the case of two-point function the UV divergent one-loop four-point function on the usual commutative plane can be recovered from the corresponding (finite) term on the quantum plane in the limit θ → 0 + . So as the results (17) and (27) imply, we get finite result for the two-point and four-point functions on the quantum plane. In the limit θ → 0 + , i.e., when the non-commutativity between the coordinates disappears, both of the two-point and four-point functions diverge and by suitable redefinition of the non-commutativity parameter one can recover the divergent behavior of the two-point and four-point functions on the usual commutative plane.

Conclusion
We studied the UV finiteness of the non-relativistic scalar field theory on the quantum plane by employing the formalism based on the coherent state approach. We were able to show that in contrast to the arguments based on the -product approach, the coherent state approach provides a formalism in which the oneloop two-point and four-point functions of the model are finite on the quantum plane. The coherent state approach to the non-commutative spaces (space-times) aims to cure the short distance behavior of the free particle propagator along the reasoning based on the existence of a minimum length and the relatively simple physical set up studied through the Letter provides an example in which the coherent state approach to the non-commutative plane works well at least in one-loop approximation. It is interesting to note that in the limit θ → 0 + , i.e., when the non-commutativity between the coordinates disappears, both of the two-point and four-point functions diverge and one recovers the divergent behavior of the two-point and four-point functions on the usual commutative plane.