Path integral for nonrelativistic generalized uncertainty principle corrected Hamiltonian

Abstract

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.

Generalized uncertainty principle and point interaction

“…This expression exactly coincides with the result of Ref. [8,9], where the Kernel is derived by direct path-integral. Of course, when α = 0, Eq.…”

“…The non-relativistic quantum mechanics with GUP-corrected Hamiltonian was examined by Schrödinger approach [7] and Feynman's path-integral approach [8,9]. In the usual quantum mechanics the transition amplitude K[q f , t f : q 0 , t 0 ] from (t 0 , q 0 ) to (t f , q f ), which is usually called Kernel [10], is calculated by a path-integral…”

Differentiable-path integrals in quantum mechanics

“…It is interesting to note that the coefficient accompanying p 2 , act's as a coupling: it is the system's effective low energy "memory" of its high-energy (short-time) properties. As we shall mention in VII A, this may have some relation with the concept of Generalized Uncertainty Principle [22][23][24][25] which proposes a modified commutator of the form (39).…”

“…Generalized Uncertainty Principle and Minimal Lenght. Another connection may be established with the approach of Generalized Uncertainty Principle (GUP) and Minimal Length [22][23][24][25], as we already pointed out at (39). In recent years, there has been increasing interest in studying the possible existence of a minimal length scale motivated by string theory, loop quantum gravity, and non-commutative geometry.…”

“…Now we turn to compute the sum (25), keeping in mind the case α > 2 so that all series converge. We are not interested in exact numerical values, but rather only on the dependence of the series upon the parameters A and α, and therefore we will only seek for upper bounds as means of estimating the various functions involved.…”