Abstract
We study the $\phi_{\star}^4$ model for a scalar field in a linearization of the Snyder model, using the methods of the Worldline Formalism. Our main result is a master equation for the 1-loop n-point function. From this we derive the renormalization of the coupling parameters of the theory and observe the appearance of a $\phi^6$ divergent contribution that opens the question of whether this theory is renormalizable or not. Additionally, we observe that some terms in the renormalized action can be interpreted as coming from an effective metric proportional to the square of the field.
Highlights
We study the φ4⋆ model for a scalar field in a linearization of the Snyder model, using the methods of the worldline formalism
Many theoretical arguments point at the conclusion that the present understanding of the structure of spacetime must be modified at short distances if one wants to reconcile quantum mechanics and general relativity
One of the oldest attempts in this sense is the idea of a noncommutative geometry
Summary
Many theoretical arguments point at the conclusion that the present understanding of the structure of spacetime must be modified at short distances if one wants to reconcile quantum mechanics and general relativity. Since Feynman’s original idea to express some QFT quantities in terms of path integrals in a first quantization language [17], the formalism has been applied to several computations, among them in the calculation of gravitational anomalies [18], in quantum gravity [19], Abelian and non-Abelian gauge theories [20] and on manifolds with boundaries [21]. The strength of this formalism lies in its possibility to handle symmetries and the way they are automatically displayed in the simplified results. Some intermediate results regarding the computation of the 4-point function are written in Appendix C
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