The generalized uncertainty principle (GUP) has brought the idea of the existence of a minimum measurable length in quantum physics. Depending on this GUP, the nonrelativistic Hamiltonian at the Planck scale is modified. In this paper, we construct the kernel for this GUP-corrected Hamiltonian for a free particle by applying the Hamiltonian path integral approach and checking the validity conditions for this kernel thoroughly. Interestingly, the probabilistic interpretation of this kernel induces a momentum upper bound in the theory which is comparable with GUP-induced maximum momentum uncertainty.

A particular framework for Quantum Gravity is the Doubly Special Relativity (DSR) formalism that introduces a new observer independent scale, the Planck energy. Our aim in this paper is to study the effects of this energy upper bound in relativistic thermodynamics. We have explicitly computed the modified equation of state for an ideal fluid in the DSR framework. In deriving our result we exploited the scheme of treating DSR as a non-linear representation of the Lorentz group in Special Relativity.

Quantum Gravity framework motivates us to find new theories in which an observer independent finite energy upper bound (preferably Planck Energy) exists. We have studied the modifications in the thermodynamical properties of a photon gas in such a scenario where we have an invariant energy scale. We show that the density of states and the entropy in such a framework are less than the corresponding quantities in Einstein's Special Relativity (SR) theory. This result can be interpreted as a consequence of the deformed Lorentz symmetry present in the particular model we have considered.

Very Special Relativity (VSR) framework, proposed by Cohen and Glashow [1], demonstrated that a proper subgroup of the Poincaré group, (in particular ISIM(2)), is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. Subsequently a deformation of the latter, DISIM b (2), was suggested by Gibbons, Gomis and Pope [2].In the present work, we introduce a novel Non-Commutative (NC) spacetime structure, underlying the DISIM b (2). This allows us to construct explicitly the DISIM b (2) generators, consisting of a sector of Lorentz rotation generators and the translation generators. Exploiting the Darboux map technique, we construct a point particle Lagrangian that lives in the NC phase space proposed by us and satisfies the modified dispersion relation proposed by Gibbons et. al. [2]. It is interesting to note that in our formulation the momentum algebra becomes non-commutative.

In this Letter, we consider a Non-Commutative (NC) extension of AdS-CFT correspondence and its effects on holographic superconductors. NC corrections are incorporated via the NC generalization of Schwarzschild black hole metric in AdS with the probe limit. We study NC effects on the relations connecting the charge density and the critical temperature of the Holographic Superconductors. Furthermore, condensation operator of the superconductor has been analyzed. Our results suggest that generically, NC effects increase the critical temperature of the holographic superconductor.