  The classical-field formalism has been widely applied in the calculation of
normal correlation functions, and the characterization of condensation, in
finite-temperature Bose gases. Here we discuss the extension of this method to
the calculation of more general correlations, including the so-called anomalous
correlations of the field, without recourse to symmetry-breaking assumptions.
Our method is based on the introduction of U(1)-symmetric classical-field
variables analogous to the modified quantum ladder operators of
number-conserving approaches to the degenerate Bose gas, and allows us to
rigorously quantify the anomalous and non-Gaussian character of the field
fluctuations. We compare our results for anomalous correlation functions with
the predictions of mean-field theories, and demonstrate that the nonlinear
classical-field dynamics incorporate a full description of many-body processes
which modify the effective mean-field potentials experienced by condensate and
noncondensate atoms. We discuss the role of these processes in shaping the
condensate mode, and thereby demonstrate the consistency of the Penrose-Onsager
definition of the condensate orbital in the classical-field equilibrium. We
consider the contribution of various noncondensate-field correlations to the
overall suppression of density fluctuations and interactions in the field, and
demonstrate the distinct roles of phase and density fluctuations in the
transition of the field to the normal phase.
