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Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems…

Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be…

Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics so to introduce a minimal uncertainty in position in quantum mechanics. Such a minimal uncertainty…

Fring and al in their paper entitled "Strings from position-dependent noncommutativity" have introduced a new set of noncommutative space commutation relations in two space dimensions. It had been shown that any fundamental objects…

In this paper we consider two kinds of noncommutative space-time commutation relations in two-dimensional configuration space and feature the absolute value of the minimal length from the generalized uncertainty relations associated to the…

The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory lead to this result. The key ingredients…

The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance…

More recently in [J. Phys. A: Math. Theor. 53, 115303 (2020)], we have introduced a set of noncommutative algebra that describes the space-time at the Planck scale. The interesting significant result we found is that the generalized…

Non-Hermitian dynamics in quantum systems have unveiled novel phenomena, yet the implementation of valid non-Hermitian quantum measurement remains a challenge, because a universal quantum projective mechanism on the complete but skewed…

The minimal-length paradigm, a possible implication of quantum gravity at low energies, is commonly understood as a phenomenological modification of Heisenberg's uncertainty relation. We show that this modification is equivalent to a…

More recently, we have proposed a set of noncommutative space that describes the quantum gravity at the Planck scale [J. Phys. A: Math. Theor. 53, 115303 (2020)]. The interesting significant result we found is that, the generalized…

The existence of a minimal observable length has long been suggested, in quantum gravity, as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal…

In an attempt to generalize general relativity, we propose a new Hermitian theory of gravity. Space-time is generalized to space-time-momentum-energy and both the principles of general covariance and equivalence are extended. The theory is…

We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in…

Many modern theories which try to unify gravity with the Standard Model of particle physics, as e.g. string theory, propose two key modifications to the commonly known physical theories: i) the existence of additional space dimensions; ii)…

The geometry of Hamiltonian's eigenstates is encoded in the quantum geometric tensor (QGT). It contains both the Berry curvature, central to the description of topological matter and the quantum metric. So far the full QGT has been measured…

The possibility of a minimal physical length in quantum gravity is discussed within the asymptotic safety approach. Using a specific mathematical model for length measurements ("COM microscope") it is shown that the spacetimes of Quantum…

Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations…

The existence of a minimal measurable length is a common feature of various approaches to quantum gravity such as string theory, loop quantum gravity and black-hole physics. In this scenario, all commutation relations are modified and the…

Broad arguments indicate that quantum gravity should have a minimal length scale. In this essay we construct a minimum length model by generalizing the time-position and energy-momentum operators while keeping much of the structure of…