Self-quartic interaction for a scalar field in an extended DFR noncommutative space–time

Abstract

The framework Doplicher-Fredenhagen-Roberts (DFR) of a noncommutative (NC) space-time is considered as a alternative approach to study the NC space-time of the early Universe. In this formalism, the parameter of noncommutative θ µν is promoted to a coordinate of the space-time, and consequently, we are describing a field theory in a space-time with extra-dimension. Consequently, there is a canonical momentum associated to this new coordinate in which the effects of a new physics can emerge in the propagation of the fields along the extra-dimension. The Fourier space of this framework is automatically extended by the addition of new momenta components. The main concept that we would like to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. When the components of the new momentum are zero, the DFR approach is reduced to the usual NC case, in which θ µν is a antisymmetric constant matrix. We study a scalar field action with self-quartic interaction φ 4 ⋆ defined in the DFR NC spacetime, obtaining the Feynman rules in the Fourier space for the scalar propagator and vertex of the model. With these rules we are able to build out the radiative corrections to one loop order for the model propagator. The influence of the NC scale, as well as the propagation of the field in the extra-dimension, are analyzed in the ultraviolet divergences scenario. We investigate the actual possibility if this θ µν conjugate momentum has the property of healing the mixing IR/UV divergences that emerges in this recently new NC spacetime quantum field theory.

The standard electroweak model in the noncommutative DFR space–time

“…An an extended Hilbert space with ten dimensions (four Minkowskian and six θ coordinates) was constructed. Recently, it was demonstrated [15] precisely that the formalism has a conjugate momentum for the θ coordinate and several elements of a DFR QFT were constructed [22][23][24]. However, to avoid unitarity we will keep the time components of θ µν equal to zero, namely, θ 0i = 0.…”

“…In the last few years, we have been concentrated in the construction of a NC QFT concerning the DFR formalism [22][23][24][25]. In this work we, following this DFR QFT quest, have constructed the DFR basis for a standard model after the construction of a DFR metric tensor.…”

Classical Mechanics and Quantum Fields in Noncommutative Phase Spaces

“…Namely, it does not require any quantum gravity scenario. The approach here is very different from the ones investigated in [18][19][20]. Here, we will show a direct approach to construct the scalar field.…”

“…In papers published by two of us [18][19][20], one can see that the construction of the commutation relations between the bosonic/fermionic fields with themselves and with its associated momenta are missing. It is our intention in this section to fill this gap.…”

Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequence

“…Here a general solution for the DFRA complex scalar fields has also been constructed using the Green's function technique. It has been shown that, in DFRA scalar model with φ 4 interaction at the one-loop level, the UV/IR mixing does not exist [19]. The covariant Dirac equation has been derived in DFR space-time from the generalised DFRA Klein-Gordon equation [20].…”