Intuitively the scaling of Planck constant scales up quantum lengths, in particular the size of causal diamond CD defined as intersection of future and past directd lightcones. This looks trivial but one one must describe precisely what is involved to check internal consistency and also to understand how to model the quantum phase transitions changing Planck constant. It turns out that the back of the Big Book along with CDs is analogous to Josephson junction which in presence of dissipation leads to the separation of charges to different pages. This might relate to the generation of matter antimatter asymmetry of the visible matter.
The first manner to understand the situation is to consider CD with a fixed range of M4 coordinates. The scaling up of the covariant Kähler metric of CD by r2=(hbar/hbar0)2 scales up the size of CD by r. Another manner to see the situation is by scaling up the linear M4 coordinates by r for the larger CD so that M4 metric becomes same for both CDs. The smaller CD is glued to the larger one isometrically together along (M2ÇCD) Ì CD anywhere in the interior of the larger CD. What happens is non-trivial for the following reasons.
- The singular coverings and factor spaces are different and M4 scaling is not a symmetry of the Kähler action so that the preferred extrema in the two cases do not relate by a simple scaling. The interpretation is in terms of the coding of the radiative corrections in powers of hbar to the shape of the preferred extremals. This becomes clear from the representation of Kähler action in which M4 coordinates have the same range for two CDs but M4 metric differs by r2 factor.
- In common M4 coordinates the M4 gauge part Aa of CP2 Kähler potential for the larger CD differs by a factor 1/r from that for the smaller CD. This guarantees the invariance of four-momentum assignable to Chern-Simons action in the phase transition changing hbar. The resulting discontinuity of Aa at M2 is analogous to a static voltage difference between the two CDs and M2 could be seen as an analog of Josephson junction. In absence of dissipation (expected in quantum criticality) the Kähler voltage could generate oscillatory fermion, em, and Z0 Josephson currents between the two CDs. Since Kähler gauge potential couples to quarks and leptons with opposite signs the current would be in opposite directions for quarks and leptons as well as for matter and antimatter. In presence of dissipation the currents would be ohmic and could force quarks and leptons and matter and antimatter to different pages of the Big Book and quarks inside hadrons would have nonstandard value of Planck constant.
- The discontinuities of Au and Af allow to assign electric and magnetic Kähler point charges QKe/m with M1 Ì M2 and having sign opposite to those assignable with dCD×CP2. It should be possible to identify physically M2, the line E1 representing quantization axis of angular momentum, and the position of QK.
For details and background see the updated chapter Quantum Hall effect and Hierarchy of Planck Constants of "Physics in Many-Sheeted Space-time".