Covariant formulation of Generalised Uncertainty Principle

Abstract

We present a formulation of the generalised uncertainty principle based on commutator $\left[{\hat{x}}^i,{\hat{p}}_j\right]$ between position and momentum operators defined in a covariant manner using normal coordinates. We show how any such commutator can acquire corrections if the momentum space is curved. The correction is completely determined by the extrinsic curvature of the surface $p^2=$ constant in the momentum space, and results in non-commutativity of normal position coordinates $\left[{\hat{x}}^i,{\hat{x}}^j\right]\ne 0$. We then provide a construction for the momentum space geometry as a suitable four dimensional extension of a geometry conformal to the three dimensional relativistic velocity space - the Lobachevsky space - whose curvature is determined by the dispersion relation $F(p^2)=-m^2$, with $F(x)=x$ yielding the standard Heisenberg algebra.

arXiv comment6 pages, 3 figures, published as a Letter in Physical Review D

References in corpus (14)

• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• The motion of point particles in curved spacetime
• Universality of Quantum Gravity Corrections
• On Gravity and the Uncertainty Principle
• Minimal Length Uncertainty Relation and Ultraviolet Regularisation
• Self-consistency in Theories with a Minimal Length
• Self-Completeness and the Generalized Uncertainty Principle
• Multi-Particle States in Deformed Special Relativity
• Generalized uncertainty principles and quantum field theory
• Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry
• Generalized uncertainty principle or curved momentum space?
• Small scale structure of spacetime: van Vleck determinant and equi-geodesic surfaces
• A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations
• Normal coordinates based on curved tangent space

Cited by in corpus (6)

• Quantum-spacetime effects on nonrelativistic Schrödinger evolution
• Reinterpreting deformed Heisenberg algebras
• Momentum Gauge Fields and Non-Commutative Space-Time
• Momentum gauge fields from curved momentum space through Kaluza-Klein reduction
• Conditions for graviton emission in the recombination of a delocalized mass
• The Derivation of Phase-Space Metric in a Geometric Quantization Approach: General Relativity with Quantized Phase-Space Metric and Relative Spacetime