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arXiv:2503.09111

Feynman Propagator of the Arthurs-Kelly system at the Planck Scale

Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park

Published Mar 12, 2025• 0 citations
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Abstract

The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in the Arthurs-Kelly system. The Feynman propagator for this system is exactly derived within the first order of the GUP parameter $β$. The application of it in the early universe stage is briefly discussed.

DOI
10.48550/arxiv.2503.09111

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The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in the Arthurs-Kelly system. The Feynman propagator for this system is exactly derived within the first order of the GUP parameter βββ. The application of it in the early universe stage is briefly discussed.
Nonrelativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in ddd-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.
Several theories of quantum gravity propose the existence of a minimal measurable length and maximum measurable momentum near the Planck scale. When integrated into the framework of quantum mechanics, such restrictions lead to the generalization of the Heisenberg uncertainty principle, which is commonly referred to as the generalized uncertainty principle (GUP). The GUP has been applied to a plethora of physical models in order to provide further insights on our understanding of the Universe, both in the macroscopic and the microscopic regime. Nevertheless, the corrections related to GUP are expected to be extremely small, which thus renders experimental verification a difficult task. Therefore, phenomenological approaches on a qualitative level are typically addressed with the aim of enhancing our possibility of detecting these tiny effects. Motivated by a recent work on a universal decoherence mechanism stemming from the standard formulation of GUP, we extend the ideas contained therein by considering a more general form of GUP which includes linear and quadratic momentum terms to examine its implications. To achieve such a picture, we equip the deformation parameter appearing in front of all momentum terms with a stochastic nature, so that the effective decoherence mechanism emerges by taking the average over fluctuations. We find that, despite the apparently small differences among the two generalizations of the Heisenberg uncertainty principle, the consequences at the level of the universal decoherence mechanism they entail is significant, as they predict decoherence times that are completely uncorrelated and distinct.
We propose a simple analytical model for Harmonic oscillator potential that is associated to the generalized uncertainty principle (GUP) in the presence of energy-dependent mass. A formal condition connected with these systems is based on the mathematical structure of modified quantum mechanics. Consequently, thanks to Feynman’s path integrals formalism, we are able to treat the reorganization of wave functions. Furthermore, in order to convert this path formalism into a local representation, we will use a coordinate transformation method at different usual point interval, which is what these systems assume. In each application, the propagator is calculated, the wave functions and the energy eigenvalues are obtained with the corresponding curves. The special cases are then deduced.
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 (GUP), which manifests a minimal Planck length in quantum spacetime, is central in various quantum gravity theories and has been widely used to describe the Planck-scale phenomenon. Here, we propose a thought experiment based on GUP – as a quantum version of Galileo's falling bodies experiment – to show that the experimental results cannot be consistently described in quantum mechanics. This paradox arises from the interaction of two quantum systems in an interferometer, a photon and a mirror, with different effective Planck constants. Our thought experiment rules out the widely used GUP, and establishes a Quantum Coupling Principle that two physical systems of different effective Planck constants cannot be consistently coupled in quantum mechanics. Our results point new directions to quantum gravity.