Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time

Abstract

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows one to straightforwardly define a quantum motion, a Bohmian mechanics, on shape space. We show how this motion gives rise to the more or less familiar theory in absolute space and time. We find that free motion on shape space, when lifted to configuration space, becomes an interacting theory. Many different lifts are possible corresponding in fact to different choices of gauges. Taking the laws of Bohmian mechanics on shape space as physically fundamental, we show how the theory can be statistically analyzed by using conditional wave functions, for subsystems of the universe, represented in terms of absolute space and time.

Appendix: Some Facts About Second-Order Partial Differential Operators

In local coordinates, any second order partial differential operator (PDO) with real coefficients, self-adjoint with respect to some volume element $\mu (dq)$ is of the form

$$\begin{aligned} {\mathsf {L}}=\sum _{ij} A^{ij}\partial _i \partial _j + \sum _i B^i\partial _i + C\end{aligned}$$

(154)

with symmetric matrix function $A= (A^{ij} )$, vector $B =(B^i)$, and scalar $C$. So it can be compactly written as

$$\begin{aligned} {\mathsf {L}}= A\nabla \nabla + B \cdot \nabla +C\,. \end{aligned}$$

(155)

Note that, while the explicit functions $A= A(q)$, $B= B(q)$ and $C=C(q)$ depend on the coordinate system chosen, A is in fact a tensor, so that the fact that two second order PDOs L and $L'$ have equal A-parts (pure second-derivative parts) is invariant.

Note that for a Laplace–Beltrami operator $C=0$. Moreover, we shall need the A-part of the Laplace–Beltrami operator with respect to the invariant metric $g= f g_e$, where $g_e$ is the mass-weighted Euclidean metric. According to (68), we have

$$\begin{aligned} A= { f ^{-1}} I \,, \end{aligned}$$

where I is the identity matrix.

Here are some relevant facts: Suppose ${\mathsf {L}}$ and ${\mathsf {L}}'$ are second order PDOs on a manifold M. If

• 1. they have the same A-part, and

• 2. are self-adjoint with respect the same measure $\mu$,

then they differ by at most a multiplication operator $D=D(q)$, i.e.,

$$\begin{aligned} {\mathsf {L}}' = {\mathsf {L}}+ D\,. \end{aligned}$$

This is so because their difference, which must be of the form $B\cdot \nabla + D$, must also be self adjoint. Since D is as well, $B\cdot \nabla$ must also be. But for no measure $\mu$ can $B\cdot \nabla$ be self-adjoint on $L^2(d\mu )$, unless $B=0$. Moreover, if ${\mathsf {L}}$ has no $C$-part, then

$$\begin{aligned} D = {\mathsf {L}}' 1 \,,\end{aligned}$$

where 1 is the constant function equal to 1.