Abstract
Using a quantization of the nonassociative and noncommutative Snyder phi^4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter beta. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative phi^4 products can generate different momentum-conserving integrals. Finally most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore infrared divergences should exist, in general, in the Snyder phi^4 theory. We consider infrared divergences at the limit p -> 0 as UV-IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative phi^4 products. We also discuss the extrapolation of the Snyder deformation parameter beta to negative values as well as certain general properties of one-loop quantum corrections in Snyder phi^4 theory at the zero-momentum limit.
Highlights
Several well-known arguments indicate that at very short spacetime distances the very concept of point and localizability may no longer be adequate
Using a quantization of the nonassociative and noncommutative Snyder φ4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D, 4, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter β
We consider infrared divergences at the limit p → 0 as UV/IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative φ4 products
Summary
Several well-known arguments indicate that at very short spacetime distances the very concept of point and localizability may no longer be adequate. Quantum properties of these sibling theories are not as easy to characterize as the Moyal theories [29,31] In his seminal paper, Snyder [32], assuming a noncommutative structure of spacetime and a deformation of the Heisenberg algebra, observed that it is possible to define a discrete spacetime without breaking the Lorentz invariance. From the underlying mathematics, like L∞ algebras (see [45] and references within [46]), new structures arise, for example the star-product algebra of functions, which were studied through nongeometric strings, probing noncommutative and nonassociative deformations of closed string background geometries [47,48,49], see the celebrated paper by Kontsevich [50].
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