  For a quantum field theory over four-dimensional Minkowski space at zero
temperature worldline holography states, that it can be expressed as a field
theory of its sources over five-dimensional AdS space to all orders in its
elementary fields, the fifth dimension being Schwinger's proper time of the
worldline formalism. For the finite temperatures studied here worldline
holography yields either a thermal AdS space or an AdS black hole as
five-dimensional manifolds. Comparing the values of the five-dimensional action
for the two alternatives does not predict a phase transition as a function of
the temperature. This absence is crucially linked to the used coordinates, and
worldline holography predicts the Fefferman-Graham form. For these coordinates
the only way to tilt the scales in favour of one or the other spacetime is to
switch between fermionic or bosonic elementary matter.
