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.

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

References in corpus (5)

• Universality of Quantum Gravity Corrections
• Multi-Particle States in Deformed Special Relativity
• Generalized uncertainty principles and quantum field theory
• Generalized uncertainty principle or curved momentum space?
• A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations

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• The Derivation of Phase-Space Metric in a Geometric Quantization Approach: General Relativity with Quantized Phase-Space Metric and Relative Spacetime