  A generical formalism for the discussion of Brownian processes with
non-constant particle number is developed, based on the observation that the
phase space of heat possesses a product structure that can be encoded in a
commutative unit ring. A single Brownian particle is discussed in a Hilbert
module theory, with the underlying ring structure seen to be intimately linked
to the non-differentiability of Brownian paths. Multi-particle systems with
interactions are explicitly constructed using a Fock space approach. The
resulting ring-valued quantum field theory is applied to binary branching
Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The
presented formalism permits the application of the full machinery of quantum
field theory to Brownian processes.
