Nonlocal Gauge Theories Including Quantum Gravity

In this chapter we review nonlocal field theory (infinite derivative field theory). We start with the discussion of the main peculiarities of nonlocal field theory on the example of d = 4 scalar ϕ4 model. The nonlocal ϕ4 model is ultraviolet finite, unitary, and macrocausal. We consider the model with an infinite number of local scalar fields ϕn(x) and local interactions among the local fields ϕn(x). An account of infinite number of local fields ϕn(x) leads to nonlocal and ultraviolet finite theory. We discuss the generalization of nonlocal ϕ4 model to the case of abelian and nonabelian gauge field theories. In contrast to the case of ϕ4 model, nonlocal gauge theories are superrenormalizable but not ultraviolet finite. The introduction of nonlocality allows to make Feynman integrals ultraviolet finite except some finite number of one-loop integrals. Also we review the main results obtained in nonlocal quantum gravity, and we point out that the nonlocal generalization of Einstein gravity can make the theory ultraviolet finite except one-loop level. Especially interesting is nonlocal generalization of renormalizable Stelle gravity which allows to get rid of the problems with negative norm states at least for free graviton propagator. On the example of axial electrodynamics, we discuss gauge theories with γ5-anomalies and show how nonlocal gauge field theories can help to make such theories meaningful. Also we consider nonlocal generalization of standard nonsupersymmetric SU(5) Georgi-Glashow GUT and show that it is possible to solve the problems with the proton lifetime and the Weinberg angle without introduction of additional particles in the spectrum. Nonlocal scale Λ responsible for ultraviolet cutoff coincides (up to some factor) with GUT scale MGUT ≈ 3 ⋅ 1016 GeV.