Generalized uncertainty principle or curved momentum space?

Abstract

The concept of minimum length, widely accepted as a low-energy effect of quantum gravity, manifests itself in quantum mechanics through generalized uncertainty principles. Curved momentum space, on the other hand, is at the heart of similar applications such as doubly special relativity. We introduce a duality between theories yielding generalized uncertainty principles and quantum mechanics on nontrivial momentum space. In particular, we find canonically conjugate variables which map the former into the latter. In that vein, we explicitly derive the vielbein corresponding to a generic generalized uncertainty principle in $d$ dimensions. Assuming the predominantly used quadratic form of the modification, the curvature tensor in momentum space is proportional to the noncommutativity of the coordinates in the modified Heisenberg algebra. Yet, the metric is non-Euclidean even in the flat case corresponding to commutative space, because the resulting momentum basis is noncanonical. These insides are used to constrain the curvature and the deviation from the canonical basis.

basically the published version, result unchanged

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Cited by in corpus (21)

• 30 years in: Quo vadis generalized uncertainty principle?
• The minimal length: a cut-off in disguise?
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Quantum-spacetime effects on nonrelativistic Schrödinger evolution
• Gravitational effects on the Heisenberg Uncertainty Principle: a geometric approach
• Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: new route to non-extensive thermostatistics
• Spin operator, Bell nonlocality and Tsirelson bound in quantum-gravity induced minimal-length quantum mechanics
• Relativistic Extended Uncertainty Principle from Spacetime Curvature
• Black hole interior quantization: a minimal uncertainty approach
• Covariant formulation of Generalised Uncertainty Principle
• Reinterpreting deformed Heisenberg algebras
• Constraining GUP Models Using Limits on SME Coefficients
• A framework for nonrelativistic isotropic models based on generalized uncertainty principles
• Space and time transformations with a minimal length
• Penetration of Arbitrary Double Potential Barriers with Probability Unity: Implications for Testing the Existence of a Minimum Length
• Momentum gauge fields from curved momentum space through Kaluza-Klein reduction
• Covariant dynamics on the energy-momentum space: scalar field theory
• Squeezing of light from Planck-scale physics
• Non-local quantum field theory from doubly special relativity
• Towards quantum Mechanics on the curved cotangent bundle
• Formal developments in curved momentum space: the quantum field theory roadmap