Generalized Uncertainty Principle: from the harmonic oscillator to a QFT toy model

Abstract

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg Uncertainty Principle into the Generalized Uncertainty Principle. In this work, we study the implications of a polynomial Generalized Uncertainty Principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the Generalized Uncertainty Principle.

Generalized uncertainty principle or curved momentum space?

“…where the coordinates transform according to the operator ordering imposed by geometric calculus [64] applied to momentum space. Note that other operator orderings would yield equivalent theories [65,66] which, however, would not manifestly unveil the nontriviality of momentum space. This transformation immediately implies that the Hamiltonian describing the dynamics of a nonrelativistic particle may be reexpressed as…”

Gravitational effects on the Heisenberg Uncertainty Principle: a geometric approach

“…In principle, the deformation parameter β is not fixed by the theory, although it is generally assumed to be of order unity in some models of string theory [15][16][17][18]. We also mention that many studies have recently attempted to constrain β by means of different quantum mechanical or field theoretical approaches [21][22][23][24][25][26][27][28][29][30][31][32][33][34] (see Ref. [35] for a recent review).…”