A Contextual Planck Parameter and the Classical Limit in Quantum Cosmology

We propose that whatever quantity controls the Heisenberg uncertainty relations (for a given complementary pair of observables) it should be identified with an effective Planck parameter. With this definition it is not difficult to find examples where the Planck parameter depends on the region under study, varies in time, and even depends on which pair of observables one focuses on. In quantum cosmology the effective Planck parameter depends on the size of the comoving region under study, and so depends on that chosen region and on time. With this criterion, the classical limit is expected, not for regions larger than the Planck length, l(P), but for those larger than l(Q) =(l(P)(2)H(-1))(1/3), where H is the Hubble parameter. In theories where the cosmological constant is dynamical, it is possible for the latter to remain quantum even in contexts where everything else is deemed classical. These results are derived from standard quantization methods, but we also include more speculative cases where ad hoc Planck parameters scale differently with the length scale under observation. Even more speculatively, we examine the possibility that similar complementary concepts affect thermodynamical variables, such as the temperature and the entropy of a black hole.

Automated summary

The text introduces the concept of effective Planck parameter and discusses its dependence on the region under study, time variations, and the specific pair of observables. It also explores how the classical limit may be expected for regions beyond a certain length scale, and mentions the possibility of quantum effects in contexts where the cosmological constant remains dynamic. Additionally, it touches upon speculative cases where Planck parameters scale differently with length scales, and the potential impact of complementary concepts on thermodynamic variables like temperature and black hole entropy.