  The fluctuations of scalar fields, that are invariant under rotations of the
worldvolume, in Euclidian signature, can be described by a system of Langevin
equations. These equations can be understood as defining a change of variables
in the functional integral for the noise, with which the physical degrees of
freedom are in equilibrium. The absolute value of the Jacobian of this change
of variables therefore repackages the fluctuations. This provides a new way of
relating the number and properties of scalar fields with the consistent and
complete description of their fluctuations and is another way of understanding
the relevance of supersymmetry, which, in this way, determines the minimal
number of real scalar fields (e.g. two in two dimensions, four in three
dimensions and eight in four dimensions), in order for the system to be
consistently closed. The classical action of the scalar fields, obtained in
this way, contains a surface term and a remainder, in addition to the canonical
kinetic and potential terms. The surface term describes possible flux
contributions in the presence of boundaries, while the remainder describes
additional interactions, that can't be absorbed in a redefinition of the
canonical terms. It is, however, through its combination with the surface term
that the noise fields can be recovered, in all cases. However their identities
can be subject to anomalies. What is of particular, practical, interest is the
identification of the noise fields, as functions of the scalars, whose
correlation functions are Gaussian. This implies new identities, between the
scalars, that can be probed in real, or computer, experiments.
