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Quantum Mechanics on SO(3) via Non-commutative Dual Variables
We formulate quantum mechanics on SO(3) using a non-commutative dual space
representation for the quantum states, inspired by recent work in quantum
gravity. The new non-commutative variables have a clear connection to the
corresponding classical variables, and our analysis confirms them as the
natural phase space variables, both mathematically and physically. In
particular, we derive the first order (Hamiltonian) path integral in terms of
the non-commutative variables, as a formulation of the transition amplitudes
alternative to that based on harmonic analysis. We find that the non-trivial
phase space structure gives naturally rise to quantum corrections to the action
for which we find a closed expression. We then study both the semi-classical
approximation of the first order path integral and the example of a free
particle on SO(3). On the basis of these results, we comment on the relevance
of similar structures and methods for more complicated theories with
group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam
models.Comment: 29 pages; matches the published version plus footnote 7, a journal
reference include
A q-analogue of convolution on the line
In this paper we study a q-analogue of the convolution product on the line in
detail. A convolution product on the braided line was defined algebraically by
Kempf and Majid. We adapt their definition in order to give an analytic
definition for the q-convolution and we study convergence extensively. Since
the braided line is commutative as an algebra, all results can be viewed both
as results in classical q-analysis and in braided algebra. We define various
classes of functions on which the convolution is well-defined and we show that
they are algebras under the defined product. One particularly nice family of
algebras, a decreasing chain depending on a parameter running through (0,1],
turns out to have 1/2 as the critical parameter value above which the algebras
are commutative. Morerover, the commutative algebras in this family are
precisely the algebras in which each function is determined by its q-moments.
We also treat the relationship between q-convolution and q-Fourier transform.
Finally, in the Appendix, we show an equivalence between the existence of an
analytic continuation of a function defined on a q-lattice, and the behaviour
of its q-derivatives.Comment: 31 pages; many small corrections; accepted by Methods and
Applications of Analysi
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis
The novelty of the Jean Pierre Badiali last scientific works stems to a
quantum approach based on both (i) a return to the notion of trajectories
(Feynman paths) and (ii) an irreversibility of the quantum transitions. These
iconoclastic choices find again the Hilbertian and the von Neumann algebraic
point of view by dealing statistics over loops. This approach confers an
external thermodynamic origin to the notion of a quantum unit of time (Rovelli
Connes' thermal time). This notion, basis for quantization, appears herein as a
mere criterion of parting between the quantum regime and the thermodynamic
regime. The purpose of this note is to unfold the content of the last five
years of scientific exchanges aiming to link in a coherent scheme the Jean
Pierre's choices and works, and the works of the authors of this note based on
hyperbolic geodesics and the associated role of Riemann zeta functions. While
these options do not unveil any contradictions, nevertheless they give birth to
an intrinsic arrow of time different from the thermal time. The question of the
physical meaning of Riemann hypothesis as the basis of quantum mechanics, which
was at the heart of our last exchanges, is the backbone of this note.Comment: 13 pages, 2 figure
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