Quantum mechanics and the generalized uncertainty principle

Abstract

The generalized uncertainty principle has been described as a general consequence of incorporating a minimal length from a theory of quantum gravity. We consider a simple quantum mechanical model where the operator corresponding to position has discrete eigenvalues and show how the generalized uncertainty principle results for minimum uncertainty wave packets.

Effects of the Generalized Uncertainty Principle on the Inflation Parameters

“…18 As a result, the standard uncertainty relation of quantum mechanics is replaced by a gravitational uncertainty relation which contains a minimal observable length of the order of the Planck length. [19][20][21][22][23] The very notion of space-time in this quantum gravity era cannot be probed more precisely than this minimal observable length. [1][2][3][4][5][6]16,17 These fundamental properties of space-time in the quantum gravity era result in a variety of phenomenologically interesting outcomes for the rest of physics (see for instance Refs.…”

An Exact Solution of the Dirac Oscillator Problem in the Context of Generalized Uncertainty Principle

“…Non-commutative geometries [11], black hole physics [12] and the holographic principle [13] also furnish the more strong evidence for the occurrence of the minimal length. This modified commutation relation leads to the standard Heisenberg uncertainty relation as .This modification of uncertainty relation is usually termed the generalized uncertainty principle (GUP) [14] or the minimal length uncertainty principle. In the context of the GUP the momentum operator acts as At the present time, one of the common aspects of the notion of the GUP is to unify quantum mechanics with general relativity in the realm of quantum gravity [15].…”

Properties of a discrete quantum field theory

“…While the interpretation of the theory as the quantization of a classical system may be absent, the theory is nevertheless a welldefined quantum system. One straightforward technique for producing a well-defined quantum mechanics on a discrete configuration space is to consider momentum space as being compact, in which case it is the position space that becomes discrete [2]. If one properly defines a Hamiltonian on this discrete space, one can represent various dynamics and one can produce in the continuum limit (where the discretization becomes small) various continuum Hamiltonians (for example the free particle) [3].…”