Dynamics on Noncommutative Orlicz Spaces
- PDF / 385,077 Bytes
- 22 Pages / 612 x 792 pts (letter) Page_size
- 57 Downloads / 188 Views
Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES∗ L.E. LABUSCHAGNE† DSI-NRF CoE in Mathematics and Statistics Science, Focus Area for PAA, Internal Box 209, School of Mathematics and Statistics Science NWU, PVT. BAG X6001, 2520 Potchefstroom, South Africa E-mail : [email protected]
W.A. MAJEWSKI Focus Area for PAA, North-West-University, Potchefstroom, South Africa E-mail : [email protected] Abstract Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces hLcosh −1 , L log(L + 1)i, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we “complete” the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair hL∞ , L1 i. As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26]. Key words
detailed balance; Orlicz space; Markov semigroup; completely positive
2010 MR Subject Classification
1
46L55; 47L90; 46L51; 46L52; 46E30; 81S99; 82C10
Introduction
This paper completes a sequence of ideas detailing the utility of Orlicz spaces for quantum physics, with the primary thesis being that the pair of Orlicz spaces hLcosh −1 , L log(L + 1)i offers a more natural home for regular observables and states with “good” entropy, than the more common pair of hL∞ , L1 i. We pause to trace this evolution of ideas, before highlighting the significance of the present work. ∗ Received July, 16, 2019; revised January 14, 2020. The contribution of L. E. Labuschagne is based on research partially supported by the National Research Foundation (IPRR Grant 96128). Any opinion, findings and conclusions or recommendations expressed in this material, are those of the author, and therefore the NRF do not accept any liability in regard thereto. † Corresponding author: L.E. LABUSCHAGNE.
1250
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
The origins of a quantity representing something like entropy may be found in the work of Ludwig Boltzmann. In his study of the dynamics of rarefied gases, Boltzmann formulated the so-called spatially homogeneous Boltzmann equation as far back as 1872, namely Z Z ∂f1 = dΩ d3 v2 I(g, θ)|v2 − v1 |(f1′ f2′ − f1 f2 ), ∂t where f1 ≡ f (v1 , t), f2′ ≡ f (v2′
Data Loading...
