  Certain aspects of the recently proposed antisymmetrised alpha particle
product state wave function, or THSR alpha cluster wave function, for the
description of the ground state in 8Be, the Hoyle state in 12C, and analogous
states in heavier nuclei, are elaborated in detail. For instance, the influence
of antisymmetrisation in the Hoyle state on the bosonic character of the alpha
particles is studied carefully. It is shown to be weak, so that bosonic aspects
are predominant. The de Broglie wave length of alpha particles in the Hoyle
state is shown to be much larger than the inter-alpha distance. It is pointed
out that the bosonic features of low density alpha gas states have measurable
consequences, one of which, that is enhanced multi-alpha decay properties,
likely already have been detected. Consistent with experiment, the width of the
proposed analogue to the Hoyle state in 16O at the excitation energy of
E_x=15.1 MeV is estimated to be very small (34 keV), lending credit to the
existence of heavier Hoyle-like states. The intrinsic single boson density
matrix of a self-bound Bose system can, under physically desirable boundary
conditions, be defined unambiguously. One eigenvalue then separates out, being
close to the number of alpha's in the system. Differences between Brink and
THSR alpha cluster wave functions are worked out. No cluster model of the Brink
type can describe the Hoyle state with a single configuration. On the contrary,
many superpositions of the Brink type are necessary, implying delocalisation
towards an alpha product state. It is shown that single alpha particle orbits
in condensates of different nuclei are almost the same. It is thus argued that
alpha particle antisymmetrised product states of the THSR type are a very
promising novel and useful concept in nuclear physics.
