Noncommutative Model with Spontaneous Time Generation and Planckian bound

Abstract

We illustrate the thesis that if time did not exist, we would have to create it if space is noncommutative, and extend functions by something like Schroedinger's equation. We propose that the phenomenon is a somewhat general mechanism within noncommutative geometry for `spontaneous time generation'. We show in detail how this works for the $s{u}_2$ algebra $[x_i,x_j]=2ı\lambda {\epsilon }_{ij}{}^k{x}_k$ as noncommutative space, by explicitly adjoining the forced time variable. We find the natural induced noncommutative Schroedingers equation and show that it has the correct classical limit for a particle of some mass $m\ne 0$, which is generated as a second free parameter by the theory. We show that plane waves exist provided $|\vec{p}|<\pi /2\lambda$, i.e. we find a Planckian bound on spatial momentum. We also propose dispersion relations $|{\del p^0\over\del \vec p}|=|\tan(λ|\vec p|)|/mλ$ for the model and explore some elements of the noncommutative geometry. The model is complementary to our previous bicrossproduct one.

Some corrections to quantum polar coordinates formulae in sec 6. 19 pages ams-latex and 2 .eps figures

References in corpus (13)

• Lorentz invariance with an invariant energy scale
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Doubly Special Relativity theories as different bases of $\kappa$--Poincaré algebra
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories
• Coherent State Induced Star-Product on $R_{\lambda }^3$ and the Fuzzy Sphere
• Noncommutative geometry of angular momentum space U(su(2))
• Topological field theory and the quantum double of SU(2)
• Fuzzy Actions and their Continuum Limits
• Quantization by cochain twists and nonassociative differentials
• Noncommutative cohomology and electromagnetism on $C_q[S{L}_2]$ at roots of unity
• Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
• Duality Principle and Braided Geometry

Cited by in corpus (20)

• New realizations of Lie algebra kappa-deformed Euclidean space
• Covariant realizations of kappa-deformed space
• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Quantization by cochain twists and nonassociative differentials
• Almost commutative Riemannian geometry: wave operators
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
• Algebraic approach to quantum gravity II: noncommutative spacetime
• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• q-Deformation and Semidualisation in 3d Quantum Gravity
• Fuzzy and discrete black hole models
• The Poincaré group as a Drinfel'd double
• Quantum mechanics without spacetime: a case for noncommutative geometry
• Some comments on the universal constant in DSR
• Relativistic Planck-scale polymer
• Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere
• Drinfel'd double structures for Poincaré and Euclidean groups
• q-Fuzzy spheres and quantum differentials on B_q[SU_2] and U_q(su_2)
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Poisson Principal Bundles
• Algebraic approach to quantum gravity III: noncommmutative Riemannian geometry