Quantum Field Theory and Gravity: Conceptual and Mathematical ...

Shape Dynamics. An Introduction 259
argued in his Scholium in the Principia [8] that his invisible absolute motions
could be deduced from visible relative motions. This can be done but requires
more relative data than one would expect if only directly observable initial
data governed the dynamics. As we shall see, this fact, which is not widely
known, indicates how mechanics can be reformulated with less kinematic
structure than Newton assumed and simultaneously be made more predictive.
It is possible to create a framework that fully resolves the debate about the
nature of motion. In this framework, the fewest possible observable initial data
determine the observable evolution. 1
I show first that all candidate relational configurations 2 of the universe
have structures determined by a Lie group, which may be termed their structure
group. The existence of such a group is decisive. It leads directly to a
natural way to achieve the aim just formulated and to a characteristic universal
structure of dynamics applicable to a large class of systems. It is present
in modern gauge theories and, in its most perfect form, in general relativity.
However, the relational core of these theories is largely hidden because their
formulation retains redundant kinematic structure.
To identify the mismatch that shape dynamics aims to eliminate, the
first step is to establish the essential structure that Newtonian dynamics
employs. It will be sufficient to consider N, N ≥ 3, point particles interacting
through Newtonian gravity. In an assumed inertial frame of reference, each
particle a, a =1, ..., N, has coordinates x i a(t), i = x, y, z, that depend on t,
the Newtonian time. The x i a’s and t are all assumed to be observable. The
particles, assumed individually identifiable, also have constant masses m a .
For the purposes of our discussion, they can be assumed known.
Let us now eliminate potentially redundant structure. Newton granted
that only the inter-particle separations r ab , assumed to be ‘seen’ all at once,
are observable. In fact, this presupposes an external (absolute) ruler. Closer
to empirical reality are the dimensionless ratios
˜r ab :=
r √
ab
∑
, R rmh := rab 2 R , (1)
rmh
a