Abstract
In this paper we will study Lorentz-invariant, infinite derivative quantum field theories, where infinite derivatives give rise to non-local interactions at the energy scale Ms, beyond the Standard Model. We will study a specific class, where there are no new dynamical degrees of freedom other than the original ones of the corresponding local theory. We will show that the Green functions are modified by a non-local extra term that is responsible for acausal effects, which are confined in the region of non-locality, i.e. Ms−1. The standard time-ordered structure of the causal Feynman propagator is not preserved and the non-local analog of the retarded Green function turns out to be non-vanishing for space-like separations. As a consequence the local commutativity is violated. Formulating such theories in the non-local region with Minkowski signature is not sensible, but they have Euclidean interpretation. We will show how such non-local construction ameliorates ultraviolet/short-distance singularities suffered typically in the local quantum field theory. We will show that non-locality and acausality are inherently off-shell in nature, and only quantum amplitudes are physically meaningful, so that all the perturbative quantum corrections have to be consistently taken into account.
Highlights
Introduction and beyond2 derivativesIn nature, a simple 2 derivative field theory is able to capture aspects of local interactions - both in a classical and in a quantum sense
We will show that the retarded Green function becomes acausal due to non-locality and as a consequence we show that local commutativity is violated
In the case of non-local interactions, f (2) = 0, the Green function general relativity (GR) shows an acausal behavior, i.e. it is non-vanishing for space-like separations, we can not use the first inequality x0 − x 0 ≥ |x − x |, x 0 ≥ |x − y|, as done above for the local case
Summary
A simple 2 derivative field theory is able to capture aspects of local interactions - both in a classical and in a quantum sense. Higher derivative kinetic terms may harbor certain kind of classical and quantum instabilities depending on the nature of the sign of the kinetic terms; for example, Ostrógradsky instability [1] can arise, due to the fact that the Hamiltonian density is unbounded from below This classical instability can be seen at a quantum level, in the Lagrangian formalism, especially when there are extra propagating degrees of freedom, which comes with a negative residue in the propagator- an indication of a ghost-like degree of freedom. There is one particular avenue, where higher derivatives play a very significant role which is massless gravitational interaction It has been known for a while that the quadratic curvature theory of gravity is renormalizable in 4 dimensions [2],1 but contains a massive spin-2 Weyl ghost as a dynamical degree of freedom.
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