Antimatter as dark matter?

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.

1. 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.

2. In common M⁴ coordinates the M⁴ gauge part A_a 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 A_a 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.

3. The discontinuities of A_u and A_f allow to assign electric and magnetic Kähler point charges Q_K^e/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 Q_K.

For details and background see the updated chapter Quantum Hall effect and Hierarchy of Planck Constants of "Physics in Many-Sheeted Space-time".