Relativistic generalized uncertainty principle for a test particle in four-dimensional spacetime
Abstract
The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, 〈xμ〉⟨xμ⟩, becomes uncertain. There exists a specific range of coordinates where the accessibility to 〈(xμ)2〉⟨(xμ)2⟩ is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value 〈xμ〉⟨xμ⟩. This expectation value is dependent on 〈(xμ)2〉⟨(xμ)2⟩, which is itself limited by a minimum measurable length Δx2min. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between Δxmin and √−〈g〉μν is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between Δxmin and the Planck length ℓp is established with a proportionality factor of √−〈g〉μν√β2, where β2 denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction ψ(x,t), leading to an unrealistic Δp. It is also noted that the lower limit of 〈(pμ)2〉 is directly proportional to the fundamental tensor.
