Covariant GUP Deformed Hamilton-Jacobi Method

Abstract

We first briefly revisit the original Hamilton-Jacobi method and show that the Hamilton-Jacobi equation for the action I of tunneling of a fermionic particle from a charged black hole can be written in the same form of that for a scalar particle. On the other hand, various theories of quantum gravity suggest the existence of a minimal length scale, incorporating of which into quantum mechanics implies a modification of the uncertainty principle. In the scenario incorporating the generalized uncertainty principle (GUP) into a quantum field theory (QFT) in a covariant way, we derive the deformed model-independent KG/Dirac and Hamilton-Jacobi equations using the methods of effective field theory. For this Lorentz invariant GUP modified QFT, we find that the effect of GUP on the Hamilton-Jacobi equations is simply to “renormalize” the mass of the emitted particles, from m to meff. Therefore, in this scenario, the Hawking temperature of a black hole does not receive any corrections from the GUP effect.