We examine in greater detail the proposal that time is the conjugate of the
constants of nature. Fundamentally distinct times are associated with different
constants, a situation often found in “relational time” settings. We show in
detail how in regions dominated by a single constant the Hamiltonian constraint
can be reframed as a Schrodinger equation in the corresponding time, solved in
the connection representation by outgoing-only monochromatic plane waves moving
in a “space” that generalizes the Chern-Simons functional. We pay special
attention to the issues of unitarity and the measure employed for the inner
product. Normalizable superpositions can be built, including solitons,
“light-rays” and coherent/squeezed states saturating a Heisenberg uncertainty
relation between constants and their times. A healthy classical limit is
obtained for factorizable coherent states, both in mono-fluid and multi-fluid
situations. For the latter, we show how to deal with transition regions, where
one is passing on the baton from one time to to another, and investigate the
fate of the subdominant clock. For this purpose minisuperspace is best seen as
a dispersive medium, with packets moving with a group speed distinct from the
phase speed. We show that the motion of the packets’ peaks reproduces the
classical limit even during the transition periods, and for subdominant clocks
once the transition is over. Deviations from the coherent/semi-classical limit
are expected in these cases, however. The fact that we have recently
transitioned from a decelerating to an accelerating Universe renders this
proposal potentially testable, as explored elsewhere.
