Time-Space Noncommutativity: Quantised Evolutions

Abstract

In previous work, we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al.. Here we extend it to certain noncommutative versions of the cylinder, ${\mathbb{R}}^3$ and $\mathbb{R}\times S^3$. In all these models, only discrete time translations are possible, a result known before in the first two cases. One striking consequence of quantised time translations is that even though a time independent Hamiltonian is an observable, in scattering processes, it is conserved only modulo $\frac{2\pi }{\theta }$, where $\theta$ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing $a$ and length $L=Na$, only momentum mod $\frac{2\pi }{L}$ is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.

25 pages, LaTex; minor corrections, references corrected

References in corpus (3)

• Space-Time Noncommutativity, Discreteness of Time and Unitarity
• Discrete Time from Quantum Physics
• One Particle Hilbertspace of 2+1 dimensional Gravity using Non commuting Coordinates

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• Noncommutative BTZ Black Hole and Discrete Time
• Noncommutative Differential Forms on the kappa-deformed Space
• Matrix Model Approach to Cosmology
• Energy nonconservation as a link between $f(R,T)$ gravity and noncommutative quantum theory
• Signature change in matrix model solutions
• Quantum statistics and noncommutative black holes
• Matrix Model Cosmology in Two Space-time Dimensions
• Noncommutative spaces, the quantum of time and the Lorentz symmetry
• Non-Pauli Effects from Noncommutative Spacetimes
• Light Propagation in a Background Field for Time-Space Noncommutativity and Axionic Noncommutative QED
• Free-fall in a uniform gravitational field in non-commutative quantum mechanics
• On space-time noncommutative theories at finite temperature
• Discrete Time Evolution and Energy Nonconservation in Noncommutative Physics
• Noncommutative Static Strings from Matrix Models
• Quantum Aspects of the Noncommutative Sine-Gordon Model
• Relativistic Particles on Quantum Space-time
• Phase structure of fuzzy black holes
• Noncommutative Coordinates Invariant under Rotations and Lorentz Transformations
• Gauge Theories and Non-Commutative Geometry
• Perturbed nonlinear models from Noncommutativity
• Waves on Noncommutative Spacetimes