  We present a formalization of the spine change of measure approach for
branching diffusions that improves on the scheme laid out for branching
Brownian motion in Kyprianou (2004) ["Travelling wave solutions to the KPP
equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] which
itself made use of earlier works of Lyons et al (1997) ["A conceptual proof of
the Kesten-Stigum theorem for multi-type branching processes", Classical and
modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York,
pp181-185]. We use our new formulation to interpret certain `Gibbs-Boltzmann'
weightings of particles and use this to give a new, intuitive and proof of a
more general `Many-to-One' result which enables expectations of sums over
particles in the branching diffusion to be calculated purely in terms of an
expectation of one particle. Significantly, our formalization has provided the
foundations that facilitate a variety of new, greatly simplified and more
intuitive proofs in branching diffusions: see, for example, the L^p convergence
of additive martingales in Hardy and Harris (2006) ["Spine proofs for
L^p-convergence of branching-diffusion martingales", arXiv:math.PR/0611056],
the path large deviation results for branching Brownian motion in Hardy and
Harris (2006) ["A conceptual approach to a path result for branching Brownian
motion", Stochastic Processes and their Applications,
doi:10.1016/j.spa.2006.05.010] and the large deviations for a continuous-typed
branching diffusion in Git et al (2006) ["Exponential growth rates in a typed
branching diffusion", Annals Applied Prob., (under revision)] and Hardy and
Harris (2004) ["A spine proof of a lower-bound for a typed branching
diffusion", no.0408, Mathematics Preprint, University of Bath].
