Time-Space Noncommutativity: Quantised Evolutions

Abstract

In previous work [7], we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al. [1]. Here we extend it to certain noncommutative versions of the cylinder, R 3 and R × S 3 . In all these models, only discrete time translations are possible, a result known before in the first two cases [2]-[6]. 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 2π θ , where θ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing a and length L = N a, only momentum mod 2π 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.

Noncommutative BTZ black hole and discrete time

“…Quantum theories on the noncommutative space-time cylinder have been previously studied. [21], [22] Other novel results were shown in addition to the discrete time spectrum, which may also apply here. Among them is the result that time-independent Hamiltonians are conserved only up to modulo 2π/θ.…”

Waves on noncommutative spacetimes

“…The formulation of unitary qf t ′ s when θ 0i = 0 is nontrivial and was carefully done already by Doplicher et al [1]. In recent papers [4,5], Balachandran et al formulated unitary quantum mechanics when θ 0i = 0 basing themselves on the ideas of Doplicher et al Consequences of spacetime noncommutativity in the quantum mechanics of atoms and molecules have been explored by Balachandran and Pinzul [6].…”

Discrete time evolution and energy nonconservation in noncommutative physics

“…This fact leads to the possibility of energy nonconservation in scattering processes [2]: it is conserved only modulo 2π θ , as we shall see in this paper. (See also [7,8,9] in this connection.…”

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AbstractTime-space noncommutativity leads to quantisation of time [1] and energy nonconservation [2] when time is conjugate to a compact spatial direction like a circle. Such a possibility arises for example in theories with a compact extra dimension with which time does not commute.

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