We review highlights from string theory, black hole physics and doubly special relativity and some "thought" experiments which were suggested to probe the shortest distance and/or the maximum momentum at the Planck scale. The models which are designed to implement the minimal length scale and/or the maximum momentum in different physical systems are analysed entered the literature as the Generalized Uncertainty Principle (GUP). We compare between them. The existence of a minimal length and a maximum momentum accuracy is preferred by various physical observations. Furthermore, assuming modified dispersion relation allows for a wide range of applications in estimating, for example, the inflationary parameters, Lorentz invariance violation, black hole thermodynamics, Saleker-Wigner inequalities, entropic nature of the gravitational laws, Friedmann equations, minimal time measurement and thermodynamics of the high-energy collisions. One of the higher-order GUP approaches gives predictions for the minimal length uncertainty. Another one predicts a maximum momentum and a minimal length uncertainty, simultaneously. An extensive comparison between the different GUP approaches is summarized. We also discuss the GUP impacts on the equivalence principles including the universality of the gravitational redshift and the free fall and law of reciprocal action and on the kinetic energy of composite system. The concern about the compatibility with the equivalence principles, the universality of gravitational redshift and the free fall and law of reciprocal action should be addressed. We conclude that the value of the GUP parameters remain a puzzle to be verified.

Based on string theory, black hole physics, doubly special relativity and some 'thought' experiments, minimal distance and/or maximum momentum are proposed. As alternatives to the generalized uncertainty principle (GUP), the modified dispersion relation, the space noncommutativity, the Lorentz invariance violation, and the quantum-gravity-induced birefringence effects are summarized. The origin of minimal measurable quantities and the different GUP approaches are reviewed and the corresponding observations are analysed. Bounds on the GUP parameter are discussed and implemented in the understanding of recent PLANCK observations of cosmic inflation. The higher-order GUP approaches predict minimal length uncertainty with and without maximum momenta. Possible arguments against the GUP are discussed; for instance, the concern about its compatibility with the equivalence principles, the universality of gravitational redshift and the free fall and law of reciprocal action are addressed.

In order to characterize the higher order moments of the particle multiplicity, we implement the linear-sigma model with Polyakov-loop correction. We first studied the critical phenomena and estimated some thermodynamic quantities. Then, we compared all these results with the first-principle lattice QCD calculations. Then, the extensive study of non-normalized four moments is followed by investigating their thermal and density dependences. We repeat this for moments normalized to temperature and chemical potential. The fluctuations of the second order moment is used to estimate the chiral phase-transition. Then, we implement all these in mapping out the chiral phase transition, which shall be compared with the freeze-out parameters estimated from the lattice QCD simulations and the thermal models are compared with the chiral phase-diagram.

In the mean field approximation, the derivative of the grand potential, nonstrange and strange condensates, and the deconfinement phase transition in a thermal and dense hadronic medium are verified in the SU(3) Polyakov linear-cr model (PLSM). The chiral condensates ox and ay are analyzed with the goal of determining the chiral phase transition. The temperature and density dependences of the chiral mesonic phase structures are taken as free parameters and fitted experimentally. They are classified according to the scalar meson nonets: (pseudo)scalar and (axial) vector. For the deconfinement phase transition, the effective Polyakov-loop potentials < p and * are implemented. The in-medium effects on the masses of sixteen mesonic states are investigated. The results are presented for two different forms for the effective Polyakov-loop potential and compared with other models, which include and exclude the anomalous terms. It is found that the Polyakov-loop potential has considerable effects on the chiral phase transition so that the restoration of the chiral symmetry breaking becomes sharper and faster. Assuming that the Matsubara frequencies contribute to the meson masses, we have normalized all mesonic states with respect to the lowest frequency. By doing this, we characterize temperatures and chemical potentials at which the different meson states dissolve to free quarks. Different dissolving temperatures and chemical potentials are estimated. The different meson states survive the typically averaged QCD phase boundary, which is defined by the QCD critical temperatures at varying chemical potentials. The thermal behavior of all meson masses has been investigated in the large-V,. limit. It is found that, at high T, the scalar meson masses are T independent (except n and a). For the pseudoscalar meson masses, the large-A,, limit unifies the T dependences of the various states into a universal bundle. The same is also observed for axial and axial-vector meson masses.

In mean field approximation, the grand canonical potential of SU(3) Polyakov linear-σ model (PLSM) is analysed for chiral phase-transition, σ l and σ s and for deconfinement order-parameters, φ and φ * of light-and strange-quarks, respectively. Various PLSM parameters are determined from the assumption of global minimization of the real part of the potential. Then, we have calculated the subtracted condensates (∆ l,s ). All these results are compared with recent lattice QCD simulations. Accordingly, essential PLSM parameters are determined. The modelling of the relaxation time is utilized in estimating the conductivity properties of the QCD matter in thermal medium, namely electric [σ el (T )] and heat [κ(T )] conductivities. We found that the PLSM results on the electric conductivity and on the specific heat agree well with the available lattice QCD calculations. Also, we have calculated bulk and shear viscosities normalized to the thermal entropy, ξ/s and η/s, respectively, and compared them with recent lattice QCD. Predictions for (ξ/s)/(σ el /T ) and (η/s)/(σ el /T ) are introduced. We conclude that our results on various transport properties show some essential ingredients, that these properties likely come up with, in studying QCD matter in thermal and dense medium.