#  Remarks About the Relationship Between Relational Physics and a Large
Kantian Component of the Laws of Nature

Sheldon Goldstein111Departments of Mathematics, Physics and Philosophy, Hill
Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road,
Piscataway, NJ 08854-8019, USA. E-mail<EMAIL_ADDRESS>and Nino
Zanghì222Dipartimento di Fisica dell’Università di Genova and INFN sezione di
Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

Relational mechanics is a reformulation of mechanics (classical or quantum)
for which space is relational. This means that the configuration of an
$N$-particle system is a shape, which is what remains when the effects of
rotations, translations and dilations are quotiented out. This reformulation
of mechanics naturally leads to a relational notion of time as well, in which
a history of the universe is just a curve in shape space without any reference
to a special parametrization of the curve given by an absolute Newtonian time.
When relational mechanics (classical or quantum) is regarded as fundamental,
the usual descriptions in terms of absolute space and absolute time emerge
merely as corresponding to the choice of a gauge. This gauge freedom forces us
to recognize that what we have traditionally regarded as fundamental in
physics might in fact be imposed by us through our choice of gauge. It thus
imparts a somewhat Kantian aspect to physical theory.

###### Contents

  1. 1 Relational Physics
    1. 1.1 Relational Space
    2. 1.2 Relational Time
    3. 1.3 Relational versus Relativistic
    4. 1.4 Relational Space, Relational Time, and Gauge Freedom
    5. 1.5 Why Go Relational?
  2. 2 Quantum Mechanics
    1. 2.1 Bohmian Mechanics
    2. 2.2 OOEOW
    3. 2.3 The Implications of Bohmian Mechanics
      1. 2.3.1 The Conditional Wave Function
      2. 2.3.2 The Conditional Probability Formula
    4. 2.4 Bohmian Motion on a Riemannian Manifold
  3. 3 Quantum Puzzles
    1. 3.1 The Wave Function of the Universe and Entropy
    2. 3.2 Nonnormalizability of the Wave Function of the Universe
    3. 3.3 Why vs. What
  4. 4 Wave Function as Law
    1. 4.1 $\psi$ vs. $\Psi$
    2. 4.2 A Kantian Component of Laws of Nature
  5. 5 Relational Quantum Mechanics
    1. 5.1 Shape Space as the Fundamental Configuration Space
      1. 5.1.1 Shape Space
      2. 5.1.2 Metrics on Shape Space and Best Matching
      3. 5.1.3 Invariant Metrics on Absolute Configuration Space
      4. 5.1.4 Conformal Factors
    2. 5.2 Free Motion on Shape Space
    3. 5.3 Fundamental Level vs. “Human” Level
    4. 5.4 Emergence of Interactions
  6. 6 Sources of Gauge Freedom
    1. 6.1 Classical Gauges
      1. 6.1.1 The Newton Gauge
      2. 6.1.2 The Expansion Gauge
      3. 6.1.3 A Note on Newtonian Gravitation
    2. 6.2 Quantum Gauges
      1. 6.2.1 Three Quantum Gauges
      2. 6.2.2 The Schrödinger gauge
  7. 7 Lessons from Relational Physics
    1. 7.1 Local Beables and Primitive Ontology
    2. 7.2 Projectivity
    3. 7.3 Non-Normalizability of the Wave function
    4. 7.4 Conditional Wave Function and Path Space
      1. 7.4.1 The Conditional Wave Function
      2. 7.4.2 The Conditional Probability Formula
      3. 7.4.3 Non-Normalizable Measures
      4. 7.4.4 The Physics of Sub-Systems
      5. 7.4.5 $|\psi|^{2}$ vs. $|\Psi|^{2}$
    5. 7.5 More Relational Points
      1. 7.5.1 Identical Particles
      2. 7.5.2 Different Ontologies
      3. 7.5.3 Geometry
      4. 7.5.4 Larger Groups
    6. 7.6 The Quantum Gauge?

##  1 Relational Physics

We shall address here three basic questions concerning relational physics:
What is it? What is its point? What can we learn from it? These issues have
been addressed in a long paper [1] published in 2020 with our friend and
colleague Detlef Dürr, who passed away in January 2021.

Relational physics has been revived in relatively recent times by Julian
Barbour and Bruno Bertotti, who, in a very inspiring and influential paper
published at the beginning of the eighties [2], transformed a long standing
philosophical controversy about the nature of space and time into a well-
defined physical problem. Our main focus here will be on gauge freedom and
objectivity versus subjectivity, probability, and the non-normalizability of
the quantum state.

###  1.1 Relational Space

The philosophical issue about the nature of space dates back to the dispute
between Isaac Newton, who favored and argued for the need of an absolute
theory of space and time, and Gottfried Wilhelm Leibniz, who insisted upon a
relational approach, also defended by Ernst Mach in the 19th century. To put
it briefly, the gist of the relational approach is that overall position,
orientation, and size are not relevant. This can be clarified by means of a
very elementary and simplified model of the universe.

Suppose we are given the configuration of a universe of $N$ particles. And
suppose we translate every particle of the configuration in the same direction
by the same amount. From a physical point of view it seems rather natural to
take the relational perspective that the two configurations of the universe so
obtained are physically equivalent or identical. Similarly for any rotation.
Going one step further, one regards two configurations of the universe
differing only by a dilation, i.e. by a uniform expansion or contraction, as
representing in fact the same physical state of the universe. The space of all
genuinely physically different possible configurations so obtained—taking into
account translations, rotations, and dilations—is usually called shape space.
The name shape space is indeed natural: only the shape of a configuration of
particles is relevant, not its position or orientation or overall size.

Shape space should be contrasted with absolute configuration space, the
totality of configurations of $N$ points in Euclidean three-dimensional space.

###  1.2 Relational Time

Once shape space is taken as fundamental, one should address how shapes
evolve. While positing an absolute Newtonian time as in classical mechanics is
compatable with relational space, it is natural to consider a relational
notion of time as well. According to relational time, global changes of speed
of the history of the universe give physically equivalent representations, so
that a history of the universe is just a curve in shape space without any
reference to a special parametrization of the curve given by any sort of
absolute Newtonian time.

Relational time should be contrasted with absolute or metrical time. By
metrical time we refer to any objective physical coordination of the
configurations along a geometrical path in a configuration space with the
points of a one-dimensional continuum: a (continuous) mapping from the
continuum onto the path. The continuum is usually represented by the real
numbers, but it need not be. However, it should be physically distinct from
the particular continuum that is the path itself.

Understood in this way, metrical time does not exist, from the relational
point of view, for the universe as a whole. However, for subsystems of the
universe metrical time naturally emerges: the continuum with which the
geometrical path corresponding to the evolution of the subsystem is
coordinated can be taken to be the path of its environment, with the obvious
mapping between the paths.

###  1.3 Relational versus Relativistic

Relational physics should be contrasted with relativistic physics. A simple
point: In relational physics the traditional separation of space and time is
retained. While configuration space is replaced by shape space, and time
becomes non-metrical, shape space retains an identity separate and distinct
from that of (non-metrical) time. This is in obvious contrast with
relativistic physics, in which space and time lose their separate identities
and are merged into a space-time.

Perhaps the most characteristic feature of relativity is the absence of
absolute simultaneity. Not so for relational physics. Since it retains the
separation of space and time, an absolute simultaneity is built into the very
structure of relational physics as described here. Nonetheless, there is a
sense in which simultaneity is lost. With relational time the notion of the
shape of the universe at “time $t$” is not physically meaningful. And with
what is meaningful—geometrical paths in the space of possible shapes—one can
no longer meaningfully compare or ask about the configurations for two
different possible histories at the same time. Given the actual configuration
of the universe, it is not meaningful to ask about the configuration of an
alternative history at that time without further specification of exactly what
that should mean.

One may inquire whether the relational point of view can be merged with or
extended to relativity. Can we achieve a relational understanding of space-
time? General relativity is certainly a step in that direction, but it does
not get us there. Space-time in general relativity is metrical—in a way that
neither space nor time are in relational physics. A complete extension, if at
all possible, is a real challenge.

Another possibility is that relativity is not fundamental, but is, instead, a
consequence of a suitable choice of gauge. This possibility, which is
suggested by the work of Bryce DeWitt [10] and Barbour and coworkers [5] would
be worth carefully exploring. For a bit of elaboration on this, see Section
7.5.3.

###  1.4 Relational Space, Relational Time, and Gauge Freedom

When physical theories are formulated in shape space, one should consider
first the simplest ones, namely the “free” theories based only on the
geometrical structures provided by the metric, without invoking any potential.
This is in contrast with theories formulated in absolute space, for which free
theories can’t begin to account for the experimental data. It is then natural
to ask: when we represent the theories in absolute space, what form do the
laws of motion take? Is the representation unique or are there various
representations yielding different-looking laws of motion, some unfamiliar and
some more or less familiar? Moreover, do interacting theories emerge with
nontrivial interactions, although in shape space the motion is free?

To answer these questions it is helpful to represent absolute configuration
space in geometrical terms as a fiber bundle, with shape space as base
manifold and the fibers generated by the similarity group, i.e, by
translations, rotations and dilations, which acting on configurations yield,
from a relational point of view, physically equivalent states. A
representation in absolute configuration space of the motion in shape space is
then given by a “lift” of the motion from the base into the fibers (more
details below).

Such lifts can rightly be called gauges. In the classical case it turns out
that in some gauges the law looks unfamiliar but there is (at least) one gauge
in which, after performing a time change (representing indeed another gauge
freedom when also time is seen as relational), the law of motion is Newtonian
with a potential appearing. So, it turns out that relational space and
relational time form a harmonious combination. More or less the same is true
for the quantum case. And similarly, as hinted above, the same might be true
for general relativity.

###  1.5 Why Go Relational?

One reason for turning to relational physics is this. Only relational
distinctions concerning space and time are observable. Thus, so the argument
might continue, since the positions of bodies are individuated only by
relative distances and angles, from an observational point of view, only
shapes or relational times are meaningful, while absolute space, absolute
time, and even absolute spacetime, do not have any clear observational
meaning. While having some force, this argument is too positivistic to be
taken as the main reason to go relational.

Here is another reason. Since the long-standing philosophical controversy
about the nature of space and time has now become a well-defined problem in
theoretical physics, we have a professional reason to go relational. It seems
worth exploring how far one can go in formulating physics in minimal
relational terms.

A stronger reason, which is indeed the one we favor, has to do with the status
and the nature of the wave function. Since at this stage, this sounds totally
incomprehensible, we shift gears and turn to quantum mechanics.

##  2 Quantum Mechanics

Standard quantum mechanics is problematic. It is often said that the
fundamental problem with quantum mechanics is the measurement problem, or,
more or less equivalently, Schrödinger’s cat paradox. However, the measurement
problem, as important as it is, is nonetheless but a manifestation of a more
basic difficulty with standard quantum mechanics: it is not at all clear what
quantum theory is about. Indeed, it is not at all clear what quantum theory
actually says. Is quantum mechanics fundamentally about measurement and
observation? Or is it about the behavior of suitable fundamental microscopic
entities, elementary particles and/or fields? To use the words of Bell,
standard formulation of quantum mechanics are “unprofessionally vague and
ambiguous.” A formulation free of these shortcomings is Bohmian mechanics [7,
6, 11, 12, 8].

###  2.1 Bohmian Mechanics

Bohmian mechanics is a theory providing a description of reality, compatible
with all of the quantum formalism, but free of any reference to observables or
observers in its formulation. In Bohmian mechanics a system of particles is
described in part by its wave function, evolving according to Schrödinger’s
equation, the central equation of quantum theory. However, the wave function
provides only a partial description of the system. This description is
completed by the specification of the actual positions of the particles. The
latter evolve according to the “guiding equation,” which expresses the
velocities of the particles in terms of the wave function. Thus in Bohmian
mechanics the configuration of a system of particles evolves via a
deterministic motion choreographed by the wave function.

Bohmian mechanics might be regarded as the minimal completion of Schrödinger’s
equation, for a non-relativistic system of particles, to a theory describing a
genuine motion of particles. For Bohmian mechanics the state of a system of
$N$ particles is described by its wave function
$\Psi{}=\Psi{}(\vec{\mathsf{q}}_{1},\dots,\vec{\mathsf{q}}_{N})=\Psi{}({\bm{q}})$,
a complex- (or spinor-) valued function on the space of possible
configurations ${\bm{q}}$ of the system, together with its actual
configuration $\bm{Q}$ defined by the actual positions
$\vec{\mathsf{Q}}_{1},\dots,\vec{\mathsf{Q}}_{N}$ of its particles. The theory
is then defined by two evolution laws. One is Schrödinger’s equation

$i\hbar\frac{\partial\Psi{}}{\partial t}=H\Psi{}\,,$ (1)

for $\Psi{}=\Psi{}_{t}$, the wave function at time $t$, where $H$ is the non-
relativistic (Schrödinger) Hamiltonian, containing the masses $m_{\alpha}$,
$\alpha=1,\ldots,N$, of the particles and a potential energy term $V$. For
spinless particles, it is of the form

$H=-\sum_{\alpha=1}^{N}\frac{{\hbar}^{2}}{2m_{\alpha}}\vec{\mathsf{\nabla}}^{2}_{\alpha}+V\,,$
(2)

where
$\vec{\mathsf{\nabla}}_{\alpha}=\frac{\partial\,\;}{\partial\vec{\mathsf{q}}_{\alpha}}$
is the gradient with respect to the position of the $\alpha$-th particle. The
other law is the the guiding law, given by the equation

$\
\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=\frac{\hbar}{m_{\alpha}}\mathrm{Im}\frac{\Psi{}^{*}\vec{\mathsf{\nabla}}_{\alpha}\Psi{}}{\Psi{}^{*}\Psi{}}(\vec{\mathsf{Q}}_{1},\dots,\vec{\mathsf{Q}}_{N})\,.$
(3)

for
$\bm{Q}=\bm{Q}(t)=(\vec{\mathsf{Q}}_{1}(t),\dots,\vec{\mathsf{Q}}_{N}(t))$,
the configuration at time $t$ (with $\mathrm{Im}$ denoting the imaginary part
of the complex quantity that follows it). If $\Psi{}$ is spinor-valued, the
products in numerator and denominator in (3) should be understood as scalar
products. If external magnetic fields are present, the gradient should be
understood as the covariant derivative, involving the vector potential. For
spinless particles, setting $\Psi{}=Re^{iS/\hbar}$ (the polar representation
of $\Psi$), the guiding equation (3) becomes

$\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=\frac{\hbar}{m_{\alpha}}\mathrm{Im}\frac{\vec{\mathsf{\nabla}}_{\alpha}\Psi{}}{\Psi{}}(\vec{\mathsf{Q}}_{1},\dots,\vec{\mathsf{Q}}_{N})=\frac{1}{m_{\alpha}}\vec{\mathsf{\nabla}}_{\alpha}S(\vec{\mathsf{Q}}_{1},\dots,\vec{\mathsf{Q}}_{N})$
(4)

For an $N$-particle system Schrödinger’s equation and the guiding equation,
together with the detailed specification of the Hamiltonian $H$, completely
define the Bohmian motion of the system.

###  2.2 OOEOW

Bohmian mechanics is for us the simplest version of quantum mechanics: Given
Schrödinger’s equation, it has the Obvious Ontology Evolving in the Obvious
Way.

For a theory of particles, in addition to the wave function you’ve got the
actual positions of the particles. This is a rather obvious ontology. We
usually emphasize this point by saying that the positions of the particles
provide the primitive ontology of the theory. In so saying we wish to convey
that the whole point of the theory—and the whole point of the wave function—is
to define a motion for the particles, and it’s in terms of this motion that
pointers end up pointing and experiments end up having results, the kinds of
results that it was the whole point of quantum mechanics to explain. So the
connection to physical reality in the theory is via what we’re calling the
primitive ontology of the theory, in Bohmian mechanics the positions of the
particles.

The only thing Bohmian mechanics adds is a first-order equation of motion for
how the positions of particles evolve. This equation, the new equation in
Bohmian mechanics, is kind of an obvious equation. It is more or less the
first thing you would guess if you asked yourself, What is the simplest motion
of the particles that could reasonably be defined in terms of the wave
function?

There are indeed many (more or less) obvious ways of guessing the velocity
field function ${\bm{v}}={\bm{v}}^{\Psi}$ in the right hand side of (3). The
simplest way, for particles without spin, is the following: Begin with the de
Broglie relation ${{\bm{p}}}=\hbar{{\bm{k}}}$, a remarkable and mysterious
distillation of the experimental facts associated with the beginnings of
quantum theory—and itself a relativistic reflection of the first quantum
equation, namely the Planck relation $E=h{\it\nu}$. The de Broglie relation
connects a particle property, the momentum ${{\bm{p}}}=m{{\bm{v}}}$, with a
wave property, the wave vector ${{\bm{k}}}$. Understood most simply, it says
that the velocity of a particle should be the ratio of $\hbar{{\bm{k}}}$ to
the mass of the particle. But the wave vector ${{\bm{k}}}$ is defined only for
a plane wave. For a general wave $\psi$, the obvious generalization of
${{\bm{k}}}$ is the local wave vector $\mathbf{\nabla}S({{\bm{q}}})/\hbar$,
where $S$ is the phase of the wave function (defined by its polar
representation, see above). With this choice the de Broglie relation becomes
${\bm{v}}=\nabla S/m$, the right hand side of which is our first guess for
${\bm{v}}^{\psi}.$

We note that the de Broglie relation also immediately yields Schrödinger’s
equation, giving the time evolution for $\psi$, as the simplest wave equation
that reflects this relationship. This is completely standard. In this simple
way, the defining equations of Bohmian mechanics can be regarded as flowing in
a natural manner from the first quantum equation $E=h\nu$.

The quantum continuity equation is the key for a route which is meaningful
also for particles with spin. This equation, an immediate consequence of
Schrödinger’s equation, involves a quantum probability density $\rho$ and a
quantum probability current $J$. Since densities and currents are classically
related by $J=\rho\mbox{\sl v}$, it requires little imagination to set
${\bm{v}}^{\psi}=J/\rho$.

Another way to arrive at a formula for ${\bm{v}}^{\psi}$ is to invoke
symmetry. Since the space-time symmetry of the non-relativistic Schrödinger
equation is that of rotations, translations, time-reversal, and invariance
under Galilean boosts, it is natural to demand that this Galilean symmetry be
retained when Schrödinger’s equation is combined with the guiding equation. As
described in [11], this leads to a specific formula for ${\bm{v}}^{\psi}$ as
the simplest possibility.

There are several other natural ways to arrive at ${\bm{v}}^{\psi}$, but we
shall give no more. All these routes, those we’ve explicitly mentioned and
those to which we’ve alluded, yield in fact exactly the same formula for
${\bm{v}}^{\psi}$, though the explicit form may appear different in some
cases.

###  2.3 The Implications of Bohmian Mechanics

While the formulation of Bohmian mechanics does not involve the notion of
quantum observables, as given by self-adjoint operators—so that its
relationship to the quantum formalism (the familiar axioms of quantum theory)
may at first appear somewhat obscure—it can in fact be shown that Bohmian
mechanics embodies quantum randomness, as expressed by Born’s rule, and the
entire quantum formalism based on self-adjoint operators as observables as the
very expression of its empirical import [12, Ch.2 and 3]. The most notably
implication of Bohmian mechanics is the fact that in a universe governed by
Bohmian mechanics there are sharp, precise, and irreducible limitations on the
possibility of obtaining knowledge, limitations that can in no way be
diminished through technological progress leading to better means of
measurement. This absolute uncertainty is in precise agreement with
Heisenberg’s uncertainty principle.

We shall review below the key points that lead to these implications.

####  2.3.1 The Conditional Wave Function

In physics we are usually concerned not with the entire universe but with
subsystems of the universe, for example with a hydrogen atom or a pair of
entangled photons. The quantum mechanical treatment of such systems involves
the quantum state of that system, often given by its wave function—not the
wave function of the universe. Yet, the notion of the wave function of a
subsystem is rather elusive. In standard quantum mechanics the state of a
subsystem is usually described in terms of its reduced density matrix.

However, Bohmian mechanics provides a precise formulation and understanding of
this notion in terms of the conditional wave function [11]

$\psi(x)=\Psi(x,Y)\,,$ (5)

where $\Psi=\Psi(q)=\Psi(x,y)$ is the wave function of the universe, with $x$
and $y$ the generic variables for the configurations of the system and its
environment (everything else), respectively, and where $Y$ is the actual
configuration of the environment. The conditional wave function of a Bohmian
system behaves exactly as one would expect the wave function of a system to
behave, with respect to both dynamics and statistics.

####  2.3.2 The Conditional Probability Formula

In Bohmian mechanics, for a non-relativistic system of particles, the
configuration of a system is regarded as random, with randomness corresponding
to the quantum equilibrium distribution $\mu^{\psi}$ given by $|\psi|^{2}dq$.
What this actually means, in a deterministic theory such as Bohmian mechanics,
is a delicate matter, involving a long story [11] with details and
distinctions that we shall ignore here. However, a crucial ingredient for that
analysis—for an understanding of the origin of quantum randomness in a
universe governed by Bohmian mechanics—is the fundamental conditional
probability formula for the conditional distribution of the configuration
$X_{t}$ of a system at time $t$ given that of its environment $Y_{t}$ at that
time:

$P^{\Psi_{0}}(X_{t}\in dx\,|Y_{t})=|\psi_{t}(x)|^{2}dx,$ (6)

where $\Psi_{0}$ is the initial wave function of the universe and
$P^{\Psi_{0}}$ is the probability distribution on trajectories arising from
the Bohmian dynamics with an initial quantum equilibrium distribution, and
$\psi_{t}$ is the (normalized) conditional wave function (5) of the system at
time $t$.

A crucial ingredient in proving (6) is equivariance [11]: if at any time the
configuration of a Bohmian system is randomly distributed according to
$|\Psi|^{2}$ at that time then at any other time $t$ it will be distributed
according to $|\Psi_{t}|^{2}$. Equivariance is an immediate consequence of the
continuity equation arising from the Schrödinger equation:

$\frac{\partial\rho^{\Psi}}{\partial t}+\operatorname{div}J^{\Psi}=0\,,$ (7)

with $\rho^{\Psi}=|\Psi|^{2}$, the quantum equilibrium distribution, and

$J^{\Psi}=\rho^{\Psi}{\bm{v}}^{\Psi}\,,$ (8)

the quantum probability current, where ${\bm{v}}^{\Psi}$ is the Bohmian
velocity in the right hand side of (4).

###  2.4 Bohmian Motion on a Riemannian Manifold

We conclude this section with a mathematical observation, whose importance
will become clear later on when we discuss the quantum mechanics of shapes.
Note that, given $V$, the Bohmian mechanics defined by equations (1), (2), and
(4) depends only upon the Riemannian structure $g=g_{e}$ given by the standard
Euclidean metric on configuration space
$[g_{e}]_{ij}=m_{\alpha_{i}}\delta_{ij}$, where the $i$-th component refers to
the $\alpha_{i}$-th particle. In terms of this Riemannian structure, the
evolution equations (4) and (2) become

$\displaystyle\frac{dQ}{dt}$
$\displaystyle=\hbar\,\mathrm{Im}\frac{\nabla_{g}\Psi{}}{\Psi{}}$ (9)
$\displaystyle i\hbar\frac{\partial\Psi{}}{\partial t}$
$\displaystyle=-\frac{\hbar^{2}}{2}\Delta_{g}\Psi{}+V\Psi{}\,,$ (10)

where $\Delta_{g}$ and $\nabla_{g}$ are, respectively, the Laplace-Beltrami
operator and the gradient on the configuration space equipped with this
Riemannian structure. But there is nothing special about this particular
Riemannian structure. Indeed, equations (9) and (10) as such hold very
generally on any Riemannian manifold. Thus, the formulation of a Bohmian
dynamics on a Riemannian manifold requires only as basic ingredients the
differentiable and metric structure of the manifold.

##  3 Quantum Puzzles

The Bohmian perspective allows us to illuminate some quantum puzzles, somewhat
hidden in other formulations of quantum mechanics. These puzzles find their
natural resolution in a proper relational reformulation of quantum mechanics.
We shall elaborate in later sections.

###  3.1 The Wave Function of the Universe and Entropy

The fundamental equation for the wave function of the universe in canonical
quantum cosmology is the Wheeler-DeWitt equation [9]

$\mathscr{H}\varPsi=0\,,$ (11)

for a wave function $\Psi(q)$ of the universe, where $q$ refers to
3-geometries and to whatever other stuff is involved, all of which correspond
to structures on a 3-dimensional space. In this equation $\mathscr{H}$ is a
sort of generalized Laplacian, a cosmological version of a Schrödinger
Hamiltonian $H$. And like a typical $H$, it involves nothing like an explicit
time-dependence. But unlike Schrödinger’s equation, the Wheeler-DeWitt
equation has on one side, instead of a time derivative of $\Psi$, simply 0.
Its natural solutions are thus time-independent.

That this is so is in fact the problem of time in quantum cosmology. We live
in a world where things change. But if the basic object in the world is a
timeless wave function, how does change come about? Much has been written
about this problem of time. A great many answers have been proposed. But what
we want to emphasize here is that from a Bohmian perspective the timelessness
of $\Psi$ is not a problem. Rather it is just what the doctor ordered.

The fundamental role of the wave function in Bohmian mechanics is to govern
the motion of something else. Change fundamentally occurs in Bohmian mechanics
not so much because the wave function changes but because the thing $Q$ that
it governs changes, according to a law

$dQ/dt=v^{\varPsi}(Q)$ (12)

determined by the wave function. The problem of time vanishes entirely from a
Bohmian point of view.

But there is a difficulty that arises in connection with the stationarity of
the universal wave function. According to standard formulations of quantum
statistical mechanics, thermodynamics seems to depend only upon the wave
function. But if the initial wave function is stationary, it can’t be
responsible for the irreversible behavior of our universe, and in particular
of entropy increase. So what is the origin of the arrow of time? And why does
entropy increase?

###  3.2 Nonnormalizability of the Wave Function of the Universe

The solutions of Wheeler-DeWitt equation typically fail to be normalizable. So
what could the associated non-normalizable “probabilities” given by
$|\Psi|^{2}$ physically mean? How can the quantum equilibrium analysis we
hinted at above, that allows us to recover the entire quantum formalism, get
off the ground for such a non-normalizable measure?

###  3.3 Why vs. What

From a Bohmian perspective there is another puzzle. One may ask, Why should
the motion of configurations be governed by a wave function?

This problem is rather subtle and elusive, hardly understandable at all in the
standard formulation of quantum mechanics, as well as in other more precise
versions of quantum mechanics, such as Many Worlds or GRW, in which the wave
function is all there is. We believe that relational physics sheds light on
this problem and shall elaborate on this in Section 5. We also wish to note,
concerning quantum mechanics, the contrast between asking why and asking what.
One often finds derivations of quantum mechanics from some principle or other,
which thus seem to answer the why question. But in most, if not all, such
cases, the answer leaves us with little or no understaning of what quantum
mechanics actually says about physical reality. This is not good. The
appropriate time to ask why is after the what question has been satisfactorily
addressed. With Bohmian mechanics we are at such a time.

##  4 Wave Function as Law

The most puzzling issue in the foundations of quantum mechanics is perhaps
that of the status of the wave function of a system in a quantum universe. We
have suggested elsewhere that the wave function should be regarded as
nomological, nomic—that it’s really more in the nature of a law than a
concrete physical reality.

Thoughts in this direction might arise when you consider the unusual way
Bohmian mechanics is formulated, and the unusual way the wave function behaves
in Bohmian mechanics. The wave function of course affects the behavior of the
configuration, i.e., of the particles. This is expressed by the guiding
equation (12). But in Bohmian mechanics there’s no back action, no effect in
the other direction, of the configuration upon the wave function, which
evolves autonomously via Schrödinger’s equation, in which the actual
configuration $Q$ does not appear.

A second point is that for a multi-particle system the wave function
$\Psi{}({\bm{q}})=\Psi{}(\vec{\mathsf{q}}_{1},\dots,\vec{\mathsf{q}}_{N})$ is
not a field on physical space, but on configuration space, the set of all
hypothetical configurations of the system. For a system of more than one
particle that space is not physical space. What this suggests to us is that
you should think of the wave function as describing a law and not as some sort
of concrete physical reality. After all (12) is an equation of motion, a law
of motion, and the whole point of the wave function here is to provide us with
the law, i.e., with the right hand side of this equation.

###  4.1 $\psi$ vs. $\Psi$

There are, however, problems with regarding the wave function as nomological.
Laws aren’t supposed to be dynamical objects, they aren’t supposed to change
with time, but the wave function of a system typically does. And laws are not
supposed to be things that we can control. But the wave function is often an
initial condition for a quantum system. We often, in fact, prepare a system in
a certain quantum state, that is, with a certain wave function. We can in this
sense control the wave function of a system. But we don’t control a law of
nature. This makes it a bit difficult to regard the wave function as
nomological.

But with regard to this difficulty it’s important to recognize, as already
suggested, that there’s only one wave function we should be worrying about,
the fundamental one, the wave function $\Psi$ of the universe. In Bohmian
mechanics, the wave function $\psi$ of a subsystem of the universe is defined
in terms of the universal wave function $\Psi$ according to (5). Thus, to the
extent that we can grasp the nature of the universal wave function, we should
understand as well, by direct analysis, also the nature of the objects that
are defined in terms of it, and in particular we should have no further
fundamental question about the nature of the wave function of a subsystem of
the universe.

Note that from this perspective, the most important equation in quantum
mechanics, the time-dependent Schrödinger’s equation should be regarded as a
phenomenological equation describing how the conditional wave function $\psi$
of a subsystem evolves—under suitable conditions of decoupling from its
environment—in a universe having a suitable stationary universal wave function
$\Psi$.

###  4.2 A Kantian Component of Laws of Nature

Laws are supposed to be simple. But here a distinction should be made.
Consider the velocity field in the right hand side of (12),

$v^{\Psi}\\!(Q)=v(\Psi,Q)\,.$

If we regard $\Psi$ as a law, $v^{\Psi}$ should be a simple function of $Q$.
On the other hand, if we regard $\Psi$ as part  of a law, $v(\Psi,Q)$ should
be a simple function of $\Psi$ and $Q$.

So we would like to find a compelling law of motion, or a compelling principle
yielding such a law, that can be expressed in terms of a $v^{\Psi}$ for a
suitable $\Psi$. We don’t have that. But, as we shall explain, the flexibility
and gauge freedom afforded by relational physics might help us reach our goal.

Ideally, the suitable $\Psi$ referred to above should be simple. Failing that,
it should at least obey a simple law, such as the Wheeler-DeWitt equation.
From a fundamental point of view, it might be a complete accident that $\Psi$
obeys such an equation. It might just happen to do so. The fact that the
equation is satisfied might have nothing to do with why the fundamental
dynamics is of the form (12). But as long as $\Psi$ does satisfy the equation,
by accident or not, all the consequences of satisfying it would follow.

The point here is to distinguish what is really fundamental from what we have
traditionally regarded as fundamental. The latter might, in fact, be imposed
by us. We express this by saying that the laws of nature might have a Kantian
component. And that Kantian component might be much larger than suggested so
far, as will become clearer in the next section devoted to relational quantum
mechanics.

##  5 Relational Quantum Mechanics

At the end of Section 1 we said that the stronger reason for us to go
relational has to do with understanding the status and the nature of the wave
function. At the beginning of Section 3, we claimed that the quantum puzzles
presented there find their natural resolution in a proper relational
reformulation of quantum mechanics.

It’s time to intertwine all these threads. And the key question to start with
is the one we asked in the last paragraph of the previous section, What is
really fundamental?

###  5.1 Shape Space as the Fundamental Configuration Space

Let us go back to the toy model of an $N$-particle universe introduced in
Section 1.1. The space of all its genuinely physically different possible
configurations is shape space. Mathematically, this is described as follows.

####  5.1.1 Shape Space

The totality of configurations
${\bm{q}}=(\vec{\mathsf{q}}_{1},\ldots,\vec{\mathsf{q}}_{N})$ of $N$ points in
Euclidean three-dimensional space forms the configuration space
$\boldsymbol{\mathcal{Q}}=\\{{\bm{q}}\\}=\mathbb{R}^{3N}$ of an $N$-particle
system. We shall call $\boldsymbol{\mathcal{Q}}$ the absolute configuration
space. On $\boldsymbol{\mathcal{Q}}$ there is a natural action of the
similarity transformations of Euclidean space, namely rotations, translations
and dilations, since each of them acts naturally on each component of the
configuration vector in physical Euclidean space. The totality of such
transformations form the group $G$ of similarity transformations of Euclidean
space. Since the shape of a configuration is what remains when the effects
associated with rotations, translations and dilations are filtered away, the
totality of shapes, i.e., the shape space, is the quotient space
$\mathcal{Q}\equiv\boldsymbol{\mathcal{Q}}/G$, the set of equivalence classes
with respect to the equivalence relations provided by the similarity
transformations of Euclidean space.

As such, shape space is not in general a manifold. To transform it into a
manifold some massaging is needed (e.g., by excluding from
$\boldsymbol{\mathcal{Q}}$ coincidence points and collinear configurations),
but we shall not enter into this.333For more details on this issue, see, e.g.,
[14] and the references therein. Here we shall assume that the appropriate
massaging of $\boldsymbol{\mathcal{Q}}$ has been performed and that
$\mathcal{Q}$ is a manifold. Since the group of similarity transformations has
dimension $7$ (3 for rotations + 3 for translations + 1 for dilations), the
dimension of $\mathcal{Q}\equiv\boldsymbol{\mathcal{Q}}/G$ is $3N-7$ for
$N\geq 3$. For $N=1$ and $N=2$ shape space is trivial (it contains just a
single point). $N=3$ corresponds to the simplest non-trivial shape space; it
has dimension $2$.

Accordingly, it is helpful to represent absolute configuration space
$\boldsymbol{\mathcal{Q}}$ in geometrical terms as a fiber bundle, with shape
space $\mathcal{Q}$ as the base manifold and with the fibers generated by the
the similarity group $G$ group. The points on each fiber represent physically
equivalent states. See Fig. 1.

Figure 1: Absolute configuration space $\boldsymbol{\mathcal{Q}}$ and shape
space $\mathcal{Q}$ (for a system of three particles). The fiber above shape
$q$ consists of absolute configurations ${\bm{q}}$ differing by a similarity
transformation of Euclidean space and thus representing the same shape $q$.
Real change of shape occurs only by a displacement to a neighboring fiber
$q+dq$. Only the orthogonal component $d{{\bm{q}}}_{\perp}$ of $d{{\bm{q}}}$
represents real change, while the vertical displacement $d{{\bm{q}}}_{\|}$
does not contribute; ${{\bm{q}}}+d{{\bm{q}}}_{\perp}$ is the absolute
configuration in the fiber above $q+dq$ closest to $q$ in the sense of the
$g_{B}$-distance (best matching).

####  5.1.2 Metrics on Shape Space and Best Matching

The topology of shape space is well defined by its construction as a quotient
space, but topology, of course, does not fix a metric. A metric should provide
more, namely a natural notion of distance on $\mathcal{Q}$. And since each
point in $\mathcal{Q}$ represents a class of configurations of $N$ particles
related by a similarity transformation, the distance between two elements of
$\mathcal{Q}$ induced by the metric should not recognize any absolute
configurational difference due to an overall translation, or rotation, or
dilation. In other words, it should provide a measure of the _intrinsic_
difference between two absolute configurations (that is, not involving any
consideration regarding how such configurations are embedded in Euclidean
space).

It turns out [1] that a metric on absolute configuration space
$\boldsymbol{\mathcal{Q}}$ that is invariant under the group $G$ of similarity
transformations of Euclidean space, given by a suitable “conformal factor” (to
be explained below), defines canonically a metric on shape space
$\mathcal{Q}$.

To understand why this is so, consider the representation of absolute
configuration space $\boldsymbol{\mathcal{Q}}$ as a fiber bundle with each
fiber being homeomorphic to $G$ and $\mathcal{Q}$ being its base space (see
Fig. 1). So, if $g$ is a metric invariant under any element of $G$, the
tangent vectors at each point ${\bm{q}}\in\boldsymbol{\mathcal{Q}}$ are
naturally split into “vertical” and “horizontal,” where by “naturally” we mean
that the splitting itself is invariant under the action of $G$. The vertical
ones correspond to (infinitesimal) displacements along the fiber through
${\bm{q}}$ and the horizontal ones are those that are orthogonal to the fiber,
i.e., to the vertical ones, according to the relation of orthogonality defined
by $g$. More precisely, if $d{\bm{q}}$ is an infinitesimal displacement at
${\bm{q}}$, we have

$d{\bm{q}}=d{\bm{q}}_{\|}+d{\bm{q}}_{\perp}\quad\text{with}\quad
g(d{\bm{q}}_{\|},d{\bm{q}}_{\perp})=0$

(see Fig. 1), with $d{\bm{q}}_{\|}$ vertical and $d{\bm{q}}_{\perp}$
horizontal.

The corresponding Riemannian metric on $\mathcal{Q}$ is defined as follows.
Let $q$ be a shape, ${\bm{q}}$ any absolute configuration in the fiber above
$q$, and $d{\bm{q}}$ any displacement at ${\bm{q}}$. Since $g$ is invariant
under the group $G$, the length of $d{\bm{q}}_{\perp}$ has the same value for
all absolute configurations ${\bm{q}}$ above $q$. Thus we may set the length
of $dq$ equal to that of $d{\bm{q}}_{\perp}$ and hence obtain the Riemannian
metric $g_{B}$ on $\mathcal{Q}$

$g_{B}(dq,dq)=g(d{\bm{q}}_{\perp},d{\bm{q}}_{\perp})\,.$ (13)

The subscript $B$ stands for Barbour and Bertotti, as well as base and best
matching. In fact, The distance on $\mathcal{Q}$ induced by $g_{B}$ is exactly
the one resulting from applying Barbour’s best matching procedure. Consider
two infinitesimally close shapes, $q$ and $q+dq$, and let ${\bm{q}}$ be any
absolute representative of $q$, i.e., any point in the fiber above $q$. The
$g_{B}$-distance between these shapes is then given by the $g$-length of the
vector $d{\bm{q}}$ such that (i) $d{\bm{q}}$ is orthogonal to the fiber above
$q$ and (ii) ${\bm{q}}+d{\bm{q}}$ is an absolute representative of $q+dq$. It
follows that ${\bm{q}}+d{\bm{q}}$ is the absolute configuration closest to
${\bm{q}}$ in the fiber above $q+dq$. Thus the $g_{B}$-distance is the “best
matching” distance.

####  5.1.3 Invariant Metrics on Absolute Configuration Space

So we need to show how to construct an invariant metric $g$ on
$\boldsymbol{\mathcal{Q}}$. Let $g_{e}$ be the mass-weighted Euclidean metric
on $\boldsymbol{\mathcal{Q}}$ with positive weights $m_{\alpha}$,
$\alpha=1,\ldots,N$, (the masses of the particles), in particle coordinates
${\bm{q}}=(\vec{\mathsf{q}}_{1},\ldots,\vec{\mathsf{q}}_{\alpha},\ldots,\vec{\mathsf{q}}_{N})$
given by

$ds^{2}=\sum_{\alpha=1}^{N}m_{\alpha}d\vec{\mathsf{q}}_{\alpha}\cdot
d\vec{\mathsf{q}}_{\alpha}\,,$ (14)

i.e., with $[g_{e}]_{ij}=m_{\alpha_{i}}\delta_{ij}$, where the $i$-th
component refers to the $\alpha_{i}$-th particle. The corresponding line
element is

$|d{\bm{q}}|=\sqrt{\sum_{\alpha=1}^{N}m_{\alpha}d\vec{\mathsf{q}}_{\alpha}\cdot
d\vec{\mathsf{q}}_{\alpha}}\,.$ (15)

The metric defined by (14) is invariant under rotations and translations, but
not under a dilation ${\bm{q}}\to\lambda{\bm{q}}$, where $\lambda$ is a
positive constant. Invariance under dilations is achieved by multiplying
$|d{\bm{q}}|^{2}$ by a scalar function $f({\bm{q}})$ that is invariant under
rotations and translations and is homogeneous of degree $-2$. We call $f$ a
conformal factor. So, for any choice of $f$,

$g=fg_{e}\,,\quad\text{i.e,}\quad
g(d{\bm{q}},d{\bm{q}})=f({\bm{q}})|d{\bm{q}}|^{2}\,,$ (16)

is an invariant metric on $\boldsymbol{\mathcal{Q}}$, yielding the metric on
shape space

$g_{B}(dq,dq)=f({\bm{q}})|d{\bm{q}}_{\perp}|^{2}\,.$ (17)

For the associated line element we shall write

$\displaystyle
ds=|dq|=\sqrt{g_{B}(dq,dq)}=\sqrt{f({\bm{q}})}\,|d{\bm{q}}_{\perp}|\,.$ (18)

####  5.1.4 Conformal Factors

Many choices of conformal factors are possible. One that was originally
suggested by Barbour and Bertotti is444Here and in the following examples the
conformal factors are modulo dimensional factors.

$\displaystyle
f({\bm{q}})=f_{a}({\bm{q}})\equiv\left(\sum_{\alpha<\beta}\frac{m_{\alpha}m_{\beta}}{|\vec{\mathsf{q}}_{\alpha}-\vec{\mathsf{q}}_{\beta}|}\right)^{2}\,.$
(19)

Another example is

$\displaystyle f({\bm{q}})=f_{b}({\bm{q}})\equiv{\mathrm{L}^{-2}}\,,$ (20)

where

$\displaystyle\mathrm{L}^{2}$
$\displaystyle=\sum_{\alpha}m_{\alpha}{\vec{\mathsf{\mathcal{q}}}_{\alpha}}^{2}=\frac{1}{\sum_{\alpha}m_{\alpha}}\sum_{\alpha<\beta}m_{\alpha}m_{\beta}|\vec{\mathsf{q}}_{\alpha}-\vec{\mathsf{q}}_{\beta}|^{2}$
(21)

with
$\vec{\mathsf{\mathcal{q}}}_{\alpha}=\vec{\mathsf{q}}_{\alpha}-\vec{\mathsf{q}}_{\text{cm}}\,,$
the coordinates relative to the center of mass

$\displaystyle\vec{\mathsf{q}}_{\text{cm}}=\frac{\sum_{\alpha}m_{\alpha}\vec{\mathsf{q}}_{\alpha}}{\sum_{\alpha}m_{\alpha}}\,.$
(22)

$I\equiv\mathrm{L}^{2}$ is sometimes called (but the terminology is not
universal) the moment of inertia of the configuration ${\bm{q}}$ about its
center of mass. This quantity is half the trace of the moment of inertia
tensor $\mathsf{M}$,

$\displaystyle\mathrm{L}^{2}=\frac{1}{2}\operatorname{Tr}\mathsf{M}\,.$ (23)

We recall that $\mathsf{M}=\mathsf{M}({\bm{q}})$, the tensor of inertia of the
configuration ${\bm{q}}$ about any orthogonal cartesian system $x$,$y$,$z$
with origin at the center of mass of the configuration ${\bm{q}}$, has matrix
elements given by the standard formula

$\displaystyle
M_{ij}=\sum_{\alpha=1}^{N}m_{\alpha}(\rho_{\alpha}^{2}\delta_{ij}-\rho_{\alpha
i}\rho_{\alpha j})\,,$ (24)

where $i,j=x,y,z$, $\rho_{\alpha x}\equiv x_{\alpha}$, $\rho_{\alpha y}\equiv
y_{\alpha}$, $\rho_{\alpha z}\equiv z_{\alpha}$, and
$\rho_{\alpha}^{2}=x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}$.

A choice of conformal factor that has not been considered in the literature is

$\displaystyle
f({\bm{q}})=f_{c}({\bm{q}})\equiv\mathrm{L}^{-\frac{8}{7}}(\det\mathsf{M})^{-\frac{1}{7}}\,.$
(25)

Since $\det\mathsf{M}$ scales as $\mathrm{L}^{6}$, $f({\bm{q}})$ given by (25)
scales as it should, namely, as $\mathrm{L}^{-2}$. Though at first glance this
choice seems unnatural, it is in fact very natural once the motion of shapes
is analyzed from a quantum perspective [1].

Finally, we give other two examples:

$\displaystyle f({\bm{q}})$
$\displaystyle=f_{d}({\bm{q}})\equiv\sum_{\alpha<\beta}\frac{m_{\alpha}m_{\beta}}{|\vec{\mathsf{q}}_{\alpha}-\vec{\mathsf{q}}_{\beta}|^{2}}$
(26) $\displaystyle f({\bm{q}})$
$\displaystyle=f_{g}({\bm{q}})\equiv\mathrm{L}^{-1}\sum_{\alpha<\beta}\frac{m_{\alpha}m_{\beta}}{|\vec{\mathsf{q}}_{\alpha}-\vec{\mathsf{q}}_{\beta}|}\,.$
(27)

The first one corresponds to a natural modification of the Newtonian
gravitational potential and the second, discussed in [3], corresponds to a
dynamics very close to that of Newtonian gravity.

###  5.2 Free Motion on Shape Space

All one needs to define the simplest motion on shape space, be it classical or
quantum (Bohmian), is the metric $g_{B}$ on shape space. In the classical
case, such a metric $g_{B}$ directly yields a law of free motion on shape
space, that is, geodesic motion with constant speed.

In the quantum (Bohmian) case, the situation is similar. Equations (9) and
(10) define immediately free quantum motion on shape space with Riemannian
metric $g=g_{B}$ as the motion on shape space given by the evolution equations

$\displaystyle\frac{dQ}{dt}$
$\displaystyle=\hbar\,\mathrm{Im}\frac{\nabla_{B}\Psi{}}{\Psi{}}$ (28)
$\displaystyle i\hbar\frac{\partial\Psi{}}{\partial t}$
$\displaystyle=-\frac{\hbar^{2}}{2}\Delta_{B}\Psi{}$ (29)

where $\Delta_{B}$ and $\nabla_{B}$ are, respectively, the Laplace-Beltrami
operator and the gradient on the configuration space equipped with the
Riemannian metric (17).

We have insisted on free motion, because it is the simplest motion on shape
space, which has, surprisingly, a very rich structure. Of course, one may
introduce a potential term both in the classical and the quantum case in a
straightforward way: (minus) the gradient of the potential in the right hand
side of the geodesic equation for the classical case and the usual potential
term in the right hand side of Schrödinger’s equation.

This is all there is to say about the formulations of classical and quantum
theories on shape space, modulo a caveat: so defined, classical and quantum
motion rely on an external absolute time, which, from a relational
perspective, is problematical, see below.

###  5.3 Fundamental Level vs. “Human” Level

Let us draw some morals from the foregoing. First of all, it is rather clear
that to define motion on shape space is very easy. The hard work, basically in
the quantum case [1], involves the transition from the fundamental level of
shape space to what could be called the “human” level of absolute
configuration space. We say “human” because it is the description we usually
use. Even if shape space is fundamental, we humans don’t typically formulate
things in terms of shape space. That doesn’t come quite so naturally to us,
even if shape space is in a deeper sense more fundamental and natural.
Traditionally we do things in this way for good human reasons and probably for
many others.

The level of absolute configuration space is a sort of emergent level because
it is not fundamental. How things are described at that level is imposed by
us. It is “human” in the sense that it is based on our choice of description.
This is why we consider it a Kantian component of physical theory. As we shall
elaborate below, much of what we’ve taken as fundamental is not fundamental,
but imposed by us by a proper choice of gauge. Interactions provide an
important example.

###  5.4 Emergence of Interactions

Given a motion in shape space, there is a huge host of motions in absolute
space that are compatible with it, the only constraint being that they should
project down to the motion in shape space. In other words, recalling the fiber
bundle structure of absolute configuration space described above, we have that
a representation in absolute configuration space of the motion in shape space
is given by a “lift” of the motion from the base into the fibers.

Such lifts can rightly be called gauges. In the classical case it turns out
that in some gauges the law looks unfamiliar but there is (at least) one gauge
in which, after performing a time change—representing indeed another gauge
freedom when also time is seen as relational—the law of motion is Newtonian
with a potential appearing. We call it the Newton gauge. The potential depends
on the choice of the invariant metric (16) in absolute configuration space,
with the various possibilities for the conformal factor, some of them listed
at the end of Section 5.1.

More or less the same is true for the quantum case, where however the gauge
yielding ordinary Bohmian mechanics in absolute configuration space—which we
call the Schrödinger gauge—emerges only for a stationary, i.e. time-
independent, wave function (such as with the Wheeler-DeWitt equation) on shape
space. This again involves regarding time as being relational, with an
external absolute time playing no physical role.

Also here, while the fundamental physics is given by a free Bohmian dynamics
in shape space, in the Schrödinger gauge potential terms appear. One potential
term is determined by the scalar curvature induced by the invariant metric on
absolute configuration space. Another potential term arises from the gauge
freedom we have to lift the Laplace-Beltrami operator from shape space to
absolute configuration space, where more gauge freedom arises from allowing
transformations of the lifted wave function.

##  6 Sources of Gauge Freedom

The main sources of gauge freedom are a lift from the base into the fibers and
a time change, both for the classical and quantum case. For the quantum case
there is in addition a gauge transformation

$\Psi\mapsto\Psi^{\prime}=F\Psi\,,\quad\text{ with $F$ real.}$ (30)

The implications of this will be discussed below in detail.

###  6.1 Classical Gauges

In the classical case, consider the free motion $Q=Q(t)$ on shape space, that
is, the geodesic motion with constant speed. A very natural lift of this
motion in absolute configuration space is its horizontal lift, that is, a
motion $\bm{Q}=\bm{Q}(t)$ in absolute configuration space that starts at some
point ${\bm{q}}_{1}$ on the fiber above $q_{1}$ and is horizontal, i.e., the
infinitesimal displacements $d\bm{Q}$ are all horizontal. (Note that the final
point ${\bm{q}}_{2}$ in the fiber above $q_{2}$ is then uniquely determined.)
We call this the invariant (or horizontal) gauge.

It can be shown [1] that in this gauge, the familiar total momentum
$\vec{\mathsf{P}}$, total angular momentum $\vec{\mathsf{J}}$ and (maybe less
familiar) dilational momentum $D$ are all zero, i.e.,

$\displaystyle\vec{\mathsf{P}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}m_{\alpha}\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=0$
(31) $\displaystyle\vec{\mathsf{J}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}m_{\alpha}\vec{\mathsf{Q}}_{\alpha}\times\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=0$
(32) $\displaystyle{D}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}m_{\alpha}\vec{\mathsf{Q}}_{\alpha}\cdot\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=0\,.$
(33)

####  6.1.1 The Newton Gauge

By a suitable change of speed, that is by a suitable time change
$t^{\prime}=t^{\prime}(t)$, namely,

$\displaystyle\frac{v}{\sqrt{f}}\frac{dt}{dt^{\prime}}=\sqrt{2f}\,,\quad\text{i.e.,}\quad\frac{dt^{\prime}}{dt}=\frac{v}{\sqrt{2}f}\,,$
(34)

we obtain another gauge that we call the Newton gauge, a gauge in which the
motion is Newtonian, i.e., it satisfies Newton’s equation $F=ma$ for suitable
$F$. More precisely, the particle positions $\vec{\mathsf{Q}}_{\alpha}$,
$\alpha=1,\ldots,N$, forming the configuration $\bm{Q}$ then satisfy Newton’s
equations

$\displaystyle
m_{\alpha}\frac{d^{2}\vec{\mathsf{Q}}_{\alpha}}{d{t^{\prime}}^{2}}=-\vec{\mathsf{\nabla}}_{\alpha}V(\vec{\mathsf{Q}}_{1},\ldots\vec{\mathsf{Q}}_{N})\,.$
(35)

with $V({\bm{q}})=-f({\bm{q}})$, the conformal factor.

In this regard, one should observe that if time is relational, changes of
speed, such as that given by (34), provide equivalent representations of the
same motion. Accordingly, the use of one time variable instead of another is a
matter of convenience, analogous to the choice of a gauge. The choice of time
variable that leads from the invariant gauge to the Newton gauge is the gauge
fixing condition

$\displaystyle
E=\frac{1}{2}\left|\frac{d\bm{Q}}{dt^{\prime}}\right|^{2}+V=0\,,$ (36)

that is, that the total Newtonian energy be zero. Of course, other gauges
could be useful, as discussed next.

####  6.1.2 The Expansion Gauge

Another gauge that could be useful is what could be called the expansion
gauge, a gauge in which you do not use a horizontal lift, but instead have a
uniform expansion of configurations taking place—in the horizontal gauge the
total overall size does not change: the universe would not expand since the
total dilational momentum is zero in that gauge[3]. In contrast, in the
expansion gauge configurations and thus the universe would expand. Why one
would consider such a gauge? One reason is that in this way one might simplify
the form of the gravitational forces (see below), so that they would become
more familiar in this gauge. In this regard, note that since the motion is not
horizontal, after suitable time change the Newtonian form of the equation of
motion would involve other terms, in addition to the forces generated by the
conformal factor. Another reason is that one would obtain in this way a
striking explanation of why we think that the universe is expanding.

####  6.1.3 A Note on Newtonian Gravitation

We have seen that in the Newton gauge, when the physical law on shape space is
free motion (or even non-free motion), the potential $V=-f$ appears, where $f$
is the conformal factor. We mentioned some choices for $f$ in Section 5.1.4.
No such choices, which are necessarily functions homogenous of degree $-2$,
seem to yield exactly the Newtonian gravitational potential $U_{g}$. While we
believe the detailed exploration of the implications of the models discussed
here is worthwhile, we nonetheless regard the models explored in this paper,
both classical and quantum, as toy models, so that such an analysis of them,
with the expectation of recovering well established physics, might be somewhat
inappropriate or premature.

However, it should be observed that some of the conformal factors given in
Sect. 5.1.4, e.g., $f_{a}$ and $f_{g}$, indeed give rise to a force law in the
Newton gauge that is very close to that of the Newtonian gravitational force.
Note for example that for the conformal factor $f_{g}$ the corresponding
potential is of the form $V_{g}=\mathrm{L}^{-1}U_{g}$, where $U_{g}$ is the
Newton gravitational potential and $\mathrm{L}$ in the Newton gauge is a
constant of the motion. The force arising from this potential adds to the
Newtonian force a very small centripetal correction that allows
$I=\mathrm{L}^{2}$, the moment of inertia about the center of mass, to remain
constant [3]. It is worth exploring these possibilities in the expansion
gauge.

###  6.2 Quantum Gauges

Let $Q=Q(t)$ be a Bohmian motion in shape space, that is, a solution of (28)
with the wave function $\Psi{}$ being a solution of Schrödinger’s equation
(29) on shape space. As in the classical case, we wish to characterize motions
in absolute space that are compatible with motions in shape space, that is,
motions $\bm{Q}=\bm{Q}(t)$ in $\boldsymbol{\mathcal{Q}}$ that project down to
$Q=Q(t)$ in $\mathcal{Q}$, i.e., such that

$\displaystyle\pi(\bm{Q}(t))=Q(t)\,,$ (37)

where $\pi$ is the canonical projection from $\boldsymbol{\mathcal{Q}}$ to
$\mathcal{Q}$. Clearly, there are a great many possibilities for compatible
motions in absolute configuration space.

As in the classical case, one may restrict the possibilities by considering
natural gauges. And as in the classical case, where one looks for gauges such
that the absolute motions satisfy Newton’s equations, in the quantum case we
now look for gauges such that the compatible motions on
$\boldsymbol{\mathcal{Q}}$ are themselves Bohmian motions, i.e. motions
generated by a wave function in the usual sort of way.

####  6.2.1 Three Quantum Gauges

Suppose that we proceed as in the classical case and take a horizontal lift of
a motion $Q=Q(t)$ in shape space, that is, an absolute motion for which the
infinitesimal displacements $d\bm{Q}$ are all horizontal. Let us now consider
the lift to $\boldsymbol{\mathcal{Q}}$ of a wave function $\Psi{}$ on
$\mathcal{Q}$, namely, the wave function $\widehat{\Psi}_{1}$ on absolute
configuration space such that

$\widehat{\Psi}_{1}({\bm{q}})=\Psi{}(q)$ (38)

for any point ${\bm{q}}$ on the fiber above $q$. Let $\nabla_{g}$ be the
gradient with respect to the invariant measure (16). Then the vector
$\nabla_{g}\widehat{\Psi}_{1}({\bm{q}})$ in $\boldsymbol{\mathcal{Q}}$ is
horizontal and the motions on $\boldsymbol{\mathcal{Q}}$ defined by

$\displaystyle\frac{d\bm{Q}}{dt}=\hbar\,\mathrm{Im}\frac{\nabla_{g}\widehat{\Psi}_{1}}{\widehat{\Psi}_{1}}$
(39)

are horizontal lifts of motions on $\mathcal{Q}$. So, in the quantum case,
horizontality is immediate.

Let us now consider the time evolution of the lifted wave function
$\widehat{\Psi}_{1}$ on $\boldsymbol{\mathcal{Q}}$. Let $\widehat{\Delta}_{B}$
be a lift to absolute configuration space of the Laplace-Beltrami operator
$\Delta_{B}$ on shape space, namely an operator on $\boldsymbol{\mathcal{Q}}$
such that

$\displaystyle\widehat{\Delta}_{B}\widehat{\Psi}_{1}=\Delta_{B}\Psi{}\,.$ (40)

Then

$\displaystyle i\hbar\frac{\partial\widehat{\Psi}_{1}}{\partial
t}=\widehat{\mathsf{H}}_{1}\widehat{\Psi}_{1}\,,\quad\text{with}\quad\widehat{\mathsf{H}}_{1}=-\frac{\hbar^{2}}{2}\widehat{\Delta}_{B}\,.$
(41)

It might seem natural to guess that $\widehat{\Delta}_{B}$ coincides with
$\Delta_{g}$, the Laplace-Beltrami operator with respect to $g$, but this is
wrong; nor is $\widehat{\mathsf{H}}_{1}$ a familiar sort of Schrödinger
Hamiltonian, with or without a potential term.

While $\widehat{\Psi}_{1}$ need not obey any familiar Schrödinger-type
equation, one may ask whether there exists a gauge equivalent wave function
that does. By gauge equivalent wave function we mean this: If one writes
$\widehat{\Psi}_{1}$ as $Re^{(i/\hbar)S}$ one sees that the velocity field
given by (39) is just $\nabla_{g}S$, so transformations of the wave function

$\displaystyle\widehat{\Psi}_{1}\to\widehat{\Psi}_{1}^{\prime}=F\widehat{\Psi}_{1}\,,$
(42)

where $F$ is a positive function, do not change its phase and thus the
velocity.

It turns out that there exists a positive function $F$ such that
$\widehat{\Psi}_{3}=F\widehat{\Psi}_{1}$ satisfies a Schrödinger-type equation
on absolute configuration space for a suitable potential $V$, namely,

$\displaystyle i\hbar\frac{\partial\widehat{\Psi}_{3}}{\partial
t}=\widehat{\mathsf{H}}_{3}\widehat{\Psi}_{3}$ (43)

with

$\displaystyle\widehat{\mathsf{H}}_{3}$
$\displaystyle=-\frac{\hbar^{2}}{2}\sum_{\alpha=1}^{N}\vec{\mathsf{\nabla}}_{\alpha}\cdot\frac{1}{fm_{\alpha}}\vec{\mathsf{\nabla}}_{\alpha}+V$
(44)
$\displaystyle=-\frac{\hbar^{2}}{2}\bm{\nabla}\bm{\cdot}\frac{1}{f}\bm{\nabla}+V\,,$
(45)

where $\bm{\nabla}$ and $\bm{\nabla}\bm{\cdot}$ are the gradient and
divergence with respect to the mass-weighted Eucidean metric (14), i.e.,

$\displaystyle\bm{\nabla}=\left(\frac{1}{m_{1}}\vec{\mathsf{\nabla}}_{1},\ldots,\frac{1}{m_{N}}\vec{\mathsf{\nabla}}_{N}\right)$
(46)

and

$\displaystyle\bm{\nabla}\bm{\cdot}=\left(\vec{\mathsf{\nabla}}_{1},\ldots,\vec{\mathsf{\nabla}}_{N}\right)\bm{\cdot}\,.$
(47)

Here $f$ is the conformal factor, and

$\displaystyle V$ $\displaystyle=V_{1}+V_{2}$ (48) $\displaystyle V_{1}$
$\displaystyle=-\frac{\hbar^{2}}{2}\frac{\widehat{\Delta}_{B}J^{1/2}}{J^{1/2}}$
(49) $\displaystyle V_{2}$
$\displaystyle=-\frac{\hbar^{2}}{2}f^{\frac{n}{4}}\Delta_{g}\big{(}f^{-\frac{n}{4}}\big{)}\,,$
(50)

with

$\displaystyle J=\mathrm{L}f^{7/2}\sqrt{\det\mathsf{M}}\,,$ (51)

where $\mathrm{L}=\mathrm{L}({\bm{q}})$ is given by equation (21) and
$\mathsf{M}=\mathsf{M}({\bm{q}})$ is the tensor of inertia of the
configuration ${\bm{q}}$ about any orthogonal cartesian system $x$,$y$,$z$
with origin in its center of mass and with matrix elements given by (24), and
where $\widehat{\Delta}_{B}$ is the canonical lift of $\Delta_{B}$ (see [1]
for details on this and a proof of the aforementioned statements).

There is a 2-gauge, intermediate between the 1-gauge and the 3-gauge, that we
shall not describe further here.

####  6.2.2 The Schrödinger gauge

We shall now describe what we think is appropriate to be called the
Schrödinger gauge, the true quantum analogue of the Newton gauge. If we take
into account that time is relational, as we should, the fundamental equation
for the wave function on shape space is presumably the stationary equation

$\displaystyle-\frac{\hbar^{2}}{2}\Delta_{B}\Psi{}=\mathscr{E}\Psi{}\,,$ (52)

where $\mathscr{E}$ is any given fixed constant (for example $\mathscr{E}=0$).

As before, let $\widehat{\Psi}_{1}$ be the lift of $\Psi{}$ to
$\boldsymbol{\mathcal{Q}}$, so that $\widehat{\Psi}_{1}$ satisfies the
equation

$\displaystyle(\widehat{\mathsf{H}}_{1}-\mathscr{E})\widehat{\Psi}_{1}=0\,,$
(53)

with $\widehat{\mathsf{H}}_{1}$ a lift of $\mathsf{H}$ as in (41), and the
evolution on the absolute configuration space $\boldsymbol{\mathcal{Q}}$ is
still given by (39). But now, for relational time, motions following the same
path with different speeds are the same motion. So, in the formula for the
gradient on the right hand side of (39), $\nabla_{g}=f^{-1}\bm{\nabla}$, we
may regard $f$ as a change of speed defining a new time variable that for the
sake of simplicity we shall still call $t$ (a “time change”). Then in absolute
space the guiding equation (39) becomes

$\displaystyle\frac{d\vec{\mathsf{Q}}_{\alpha}}{dt}=\frac{\hbar}{m_{\alpha}}\,\mathrm{Im}\frac{\vec{\mathsf{\nabla}}_{\alpha}\widehat{\Psi}_{1}}{\widehat{\Psi}_{1}}\,.$
(54)

Again, $\widehat{\Psi}_{1}$ need not obey any familiar stationary Schrödinger-
type equation. However, as before, we may exploit gauge freedom to transform
(53) into a stationary Schrödinger-type equation. Indeed, we now have an even
greater gauge freedom in changing the wave function and the Hamiltonian, see
[1]. In particular, there is a gauge, the Schrödinger gauge, in which (52)
becomes

$\displaystyle\widehat{\mathsf{H}}_{S}\Phi=0$ (55)

with

$\displaystyle\widehat{\mathsf{H}}_{S}=-\frac{\hbar^{2}}{2}\bm{\nabla}^{2}+U\,,$
(56)

where $\bm{\nabla}^{2}=\bm{\nabla}\bm{\cdot}\bm{\nabla}$ is the mass-weighed
Euclidean Laplacian, and

$\displaystyle
U=f(V_{1}-\mathscr{E})-\frac{\hbar^{2}}{8}\frac{n-2}{n-1}fR_{g}\,,$ (57)

where $R_{g}$ is the scalar curvature of the invariant metric $g$ and $n=3N$.

##  7 Lessons from Relational Physics

In this closing section we shall further consider what we can learn from
relational physics.

###  7.1 Local Beables and Primitive Ontology

We have stressed elsewhere the importance of certain local beables, what we
call the primitive ontology of a theory—stuff in space evolving in time—which
is what the theory is fundamentally about. Relational physics requires a more
flexible understanding of this notion.

To appreciate this point, note that for relational space the state of the
universe at a particular location is not, in and of itself, meaningful. In
that sense, for relational physics, there are no local beables in the usual
sense, and locality itself can’t easily be meaningfully formulated.

Similarly one can’t meaningfully consider the behavior of individual particles
without reference to other particles, since there is no absolute space in
which an individual particle could be regarded as moving. And even for a pair
of particles, to speak meaningfully of the distance between them, a third
particle would be required, to establish a scale of distance. And similarly
for galaxies.

Relational physics is, in fact, genuinely holistic, and suggests the holistic
character of quantum physics associated with entanglement and quantum
nonlocality.

###  7.2 Projectivity

Compare the guiding equation (3) with the same equation but with the
denominator $\Psi^{*}\Psi$ omitted. The latter simpler equation might be
regarded as in some sense more fundamental, with (3) itself regarded as
arising from a convenient choice of gauge corresponding to the freedom of
time-change associated with relational time.

The time-parameter corresponding to the use of the denominator in (3) has the
nice feature that the dynamics using that time-parameter depends on fewer
details of the wave function than would be the case if the denominator were
deleted: with the denominator the dynamics depends only on the ray of $\Psi$,
with $\Psi$ and $c\Psi$ yielding the same dynamics for any constant $c\neq 0$.
This has a particularly nice implication for the behavior of subsystems.

With this choice of time-parameter the dynamics for a subsystem will often not
depend upon the configuration of its environment, with the subsystem evolving
according to an autonomous evolution involving only the configuration and the
(conditional) wave function of the subsystem itself [11]. This would happen
when the subsystem is suitably decoupled from its environment, for example for
a product wave function when there is no interaction between system and
environment. Without the denominator this would not be true, and there would
appear to be an additional nonlocal dependence of the behavior of a subsystem
on that of its environment that would not be present with a time-parameter
associated with the use of the usual denominator.

###  7.3 Non-Normalizability of the Wave function

Note that since it is translation and scaling invariant, the wave function in
the Schrödinger gauge or in any of the gauges discussed in Sect.6.2) are not
normalizable. In other words, the corresponding $|\widehat{\Psi}|^{2}$
measures are non-normalizable. However, since the non-normalizability arises
from unobservable (and, from a shape space point of view, unphysical)
differences and dimensions it should somehow not be a problem.

Nonetheless, the real question is how the empirical distributions arising from
the fundamental shape space level are related to those coming from the physics
in a gauge. While the different gauges, such as the Schrödinger gauge,
correspond to theories that, we argued, are empirically equivalent to the
fundamental shape space theory, that was only in purely dynamical terms. We
have not yet addressed the possible differences in empirical distributions
that may arise. We would like to see that they don’t, i.e., that probabilities
naturally associated with relational physics on shape space yield familiar
Born-rule probabilities when the physics is lifted to absolute configuration
space.

There are several considerations that suggest that the non-normalizability
should not be a genuine problem. First of all, as just mentioned, the non-
normalizability arises only from non-observable dimensions, suggesting that it
should be physically irrelevant. Moreover, it is the universal wave function
$\widehat{\Psi}$ (in any of the gauges) that is not normalizable. But the
universal wave function is rarely used in practice. In quantum mechanics we
usually deal, not with the entire universe, but with small subsystems of the
universe. The wave functions with which we usually deal are thus conditional
wave functions, and there seems to be no reason why these should fail to be
normalizable.

As already said, non-normalizable wave functions tend to occur in quantum
cosmology. Such wave functions would normally be regarded as problematical and
unphysical (since the formal structures of orthodox quantum mechanics, with
their associated probabilities, are crucially based on the notion of a Hilbert
space of square-integrable, i.e. normalizable, wave functions). However, for
our analysis in [1] the nonnormalizability of the measure on configuration
space turned out to be crucial for its success (see below). Hence what from an
orthodox perspective is a vice is transformed into a virtue in relational
Bohmian mechanics.

###  7.4 Conditional Wave Function and Path Space

To substantiate the last statement we mention a few ingredients of our
analysis in [1], without any pretension of completeness.

####  7.4.1 The Conditional Wave Function

The very notion of conditional wave function (5) presupposes the possibility
of expressing the configuration $Q$ of the universe as a pair $(X,Y)$
consisting of the configuration $X$ of a system, the $x$-system, and the
configuration $Y$ of its environment (the $y$-system). The holistic character
of shape space physics emphazized above does not allow for this, that is,
there is no natural product structure

$\mathcal{Q}=\mathcal{Q}_{sys}\times\mathcal{Q}_{env}$

for shape space: Here the system is a collection of (labelled) particles with
its own shape space $\mathcal{Q}_{sys}=\mathcal{X}$, the set of possible
shapes $X$ of the system, and the environment consists of the rest of the
particles of the universe, with shape space
$\mathcal{Q}_{env}=\mathcal{Y}=\\{Q_{env}=Y\\}$, with $Y$ the shape associated
with the particles (labels) of the environment. The crucial fact is that it is
not true that

$\mathcal{Q}=\mathcal{X}\times\mathcal{Y}.$

$X$ and $Y$ don’t involve sufficient information to determine the complete
shape $Q$. What is missing is the spatial relationship between these shapes.

Nonetheless we have that $\mathcal{Q}$ can be identified with
$\mathcal{X}_{Y}\times_{y}\mathcal{Y}=\\{(X,Y)|Y\in\mathcal{Q}_{env},X\in\mathcal{Q}_{Y}\\}\,,$
where $\mathcal{Q}_{Y}=\\{Q\in\mathcal{Q}\,|\,Q_{env}=Y\\}.$ We may then
define the conditional wave function for the subsystem, for
$Y\in\mathcal{Q}_{env}$ and universal wave function $\Psi$, by

$\psi(x)=\Psi(x,Y),\quad x\in\mathcal{Q}_{Y}.$ (58)

This looks like the usual conditional wave function, but it is important to
bear in mind that, unlike with the usual conditional wave function, here $x$
represents the shape of the universe for a fixed $Y$ and there are no obvious
natural coordinates to efficiently describe it.

The conditional wave function (58) can be lifted to absolute configuration
space and one has the formula

$\widehat{\psi}(\bm{x})=\widehat{\Psi}(\bm{x},\bm{Y}),$ (59)

Here the lifts $\bm{x}$ of $x$ and $\bm{Y}$ of $Y$ involve something akin to
the choice of a frame of reference in absolute space, and the “hat” on $\Psi$
refers to any of the gauge equivalent representations of the universal wave
function that we have described in Section 6.2.

####  7.4.2 The Conditional Probability Formula

One might wonder whether something analogous to (6) holds true. The answer is
yes, but with several provisos.

It turn out that in the gauges 1 and 3 described in Section 6.2, indeed
something similar holds. More precisely, in the 3-gauge the conditional
distribution is exactly given by the modulus square of the conditional wave
function (59) as in (6). The proof of this is rather intricate and lengthy[1].

In the Schrödinger gauge, however, a conceptual issue arises as to exactly
what of physical significance this conditional distribution represents. After
all, the transition to the Schrödingergauge required relational time, but if
we take relational time seriously, what is physical is not the configuration
$Q_{t}$ of the universe at some time $t$, but the geometrical path of the full
history of the configuration, with no special association of the
configurations along a path with times. In this (more physical) framework, the
conditional distribution of the configuration $X_{t}$ of a subsystem given the
configuration $Y_{t}$ of its environment is not meaningful.

What is meaningful is (i) a probability distribution $\mathbb{P}^{\Psi}$ on
the space $\mathscr{P}$ of (geometrical, i.e. unparametrized) paths and (ii)
the conditional distribution relative to $\mathbb{P}$ of the configuration
$X_{Y}$ of the subsystem when the path $\gamma\in\mathscr{P}$ has
environmental configuration $Y,$ given that the path passes through a
configuration with environment $Y$, $Y\in\gamma$. In this more general
framework, one arrives at a conditional probability formula analogous to to
(6), namely,

$\mathbb{P}^{\Psi}(X_{Y}\in
dx|Y\in\gamma)=|\widehat{\psi}_{S}(\bm{x})|^{2}d\bm{x}\,,$ (60)

where the subscript $S$ refers to the Schrödinger gauge. Also the proof of
this theorem is rather intricate and lengthy. An assumption of this theorem is
what we have called in [1] the “existence of clock variables” for the Bohmian
dynamics, which a mild assumption for a sufficiently structured universe.

####  7.4.3 Non-Normalizable Measures

A crucial ingredient of our analysis is the connection between a measure on
path space and a (stationary) non-normalizable measure on configuration space.
By the very nature of this connection the measure on configuration space and
its associated wave function in fact has to be non-normalizable—just to
rehearse the point made at the end of Section 7.3 that the nonnormalizability
of the universal wave function is good!

####  7.4.4 The Physics of Sub-Systems

On the absolute configuration space level the dynamics and the probabilities
for subsystems should be of the usual form. While it is true that on the
universal level the connection between $|\Psi|^{2}$ and probability, or, more
precisely, typicality, would be broken, this would not be visible in any of
the familiar every-day applications of quantum mechanics, which are concerned
only with subsystems and not with the entire universe. This is the key to
approach the problem of entropy increase raised in Section 3.1.

####  7.4.5 $|\psi|^{2}$ vs. $|\Psi|^{2}$

The patterns described by the quantum equilibrium hypothesis will be typical
with respect to a measure, not on absolute configuration space, but on shape
space, on the fundamental level, which is fine. There is a widespread
misconception with respect to Bohmian mechanics that $|\Psi|^{2}$ for the
universe and $|\psi|^{2}$ for subsystems play, physically and conceptually,
similar roles. They do not, since the role of $|\Psi|^{2}$ is typicality while
that of $|\psi|^{2}$ is probability. If this distinction is too subtle, the
fact that, from a relational perspective, these objects live on entirely
different levels of description, $|\Psi|^{2}$ on the fundamental level, i.e.,
on shape space, and $|\psi|^{2}$ on absolute configuration space, might make
it easier to appreciate how very different they are.

###  7.5 More Relational Points

####  7.5.1 Identical Particles

There is one rather conspicuous relational aspect that we’ve ignored. For
indistinguishable particles we should have taken one further quotient and
enlarged the similarity group $G$ to include the relevant permutations of
particle labels. Presumably, this would not have any relevant implications in
the classical case, but in the quantum case would lead to the correct quantum
description of the dynamics of bosonic and fermionic shapes [16].

####  7.5.2 Different Ontologies

We do not regard particles as the inevitable ultimate building blocks of the
universe. Far from it. Our insistence on particles is mostly methodological,
not metaphysical. The point is that the particle ontology already suffices to
highlight the relevant conceptual issues of relational physics. Our
analysis—in particular that concerning the wave function of the universe,
subsystems, and probability—really depends only on rather general qualitative
features of the structure of quantum relational physics, not on the details of
any specific ontology.

####  7.5.3 Geometry

We have assumed that the geometrical relations among the particles are those
of Euclidean geometry. But we know that the geometry of space need not be
Euclidean and may not be fixed at all, but be dynamically determined by the
matter content of the universe.

One is thus naturally lead to geometrodynamics, the dynamics of 3-geometries.
According to John Archibald Wheeler, superspace, the totality $\mathcal{Q}$ of
3-geometries, should be regarded as the true arena of general relativity (see,
e.g., [17]).

In many respects, superspace is analogous to shape space. The totality
$\boldsymbol{\mathcal{Q}}$ of Riemannian metrics on 3-space is analogous to
absolute configuration space, and the group $G$ of diffeomorphisms of 3-space
is analogous to the similarity group of Euclidean space. The fiber bundle
structure $\mathcal{Q}=\boldsymbol{\mathcal{Q}}/G$ (modulo singularities) has
been highlighted by Bryce DeWitt [10].

Now DeWitt in [10] and many others after him have regarded four-dimensional
spacetime as fundamental and the dynamics of 3-geometries as gauge, arising
from the gauge freedom given by the so-called lapse and shift functions.
However, one could take the opposite view and consider the 3-geometry as
fundamental (analogously to a shape of particles) and four-dimensional
spacetime as arising in a suitable gauge from a timeless history of the
universe, i.e., a path, in superspace. This view, that has been explored by
Barbour and collaborators [5], leads to the conclusion that relativity
(general and special) need not be regarded as fundamental, but can be regarded
as arising from a suitable choice of gauge. Again we encounter here a striking
manifestation of a Kantian component of physical laws.

####  7.5.4 Larger Groups

For the case of an $N$-particle universe, we have already indicated that a
natural extension is to include in $G$ the relevant permutations of particle
labels. Are there other natural extensions? How far can the gauge group $G$ be
extended?

In the case of 3-geometries, Barbour and collaborators have explored the
possibility of extending the group of diffeomorphism to include the conformal
transformations of the 3-geometries. One arrives in this way at a conformal
superspace [4, 13, 15]. They perform some massaging to arrive at general
relativity (in a suitable gauge). But maybe there is no need of any massaging,
nor any need of arriving exactly at general relativity—to recover it as an
approximation of a more fundamental theory is sufficient.

###  7.6 The Quantum Gauge?

Is there a gauge that could be called the quantum gauge? We do not know the
answer, but we find that the question itself is very interesting. It is
related to the “why”-question we asked at the end of Section 3.3.

Let us explain what we mean. We started with the Bohmian motion on shape space
governed by the universal wave function $\Psi$. Everything we discussed in the
quantum case had as a basic ingredient the universal wave function $\Psi$. But
one should contemplate the possibility of starting with something that is not
really quantum, not a motion guided by a suitable $\Psi$.

Suppose we have a motion in shape space that is not of the usual Bohmian form,
with no wave function $\Psi$ involved in defining the law of motion of the
configuration. In other words, suppose we discover a natural motion on shape
space that is not quantum at all. But suppose also that when one takes into
account the freedom of time change and the freedom of how the motion is lifted
to absolute space, that is, that when one exploits all the gauge freedom we
have discussed above, it turns out that the law of motion can indeed be cast
into the usual Bohmian form, with a wave function guiding it in the usual way.
If that were the case, quantum mechanics, while entirely absent from the
fundamental (shape space) level, would arise as a choice of gauge in absolute
configuration space. If this were the case, we would have another instance,
maybe the most surprising, of what we have called a Kantian component of
physical laws—that much of what we take as fundamental is not fundamental, but
imposed by us by a proper choice of gauge.

In view of this—which at the moment is very speculative—and in view of what
has been said before about relativity—much less speculative—it seems to us not
unreasonable to entertain the following possibility: that quantum mechanics
and relativity, the two great revolutions of twentieth century physics, may
not be fundamental at all but imposed on nature by us.

## References

  * [1] Dürr D., Goldstein S., and Zanghi N.:  Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time.  Journal of Statistical Physics 180, 92–134 (2020).
  * [2] Barbour, J. B., Bertotti, B.:  Mach’s principle and the structure of dynamical theories.  Proc. Roy. Soc. Lond. A, 382, 295–306 (1982).
  * [3] Barbour, J.:  Scale-invariant gravity: particle dynamics.  Classical and Quantum Gravity, 20, 1543 (2003).
  * [4] Barbour, J.:  Shape dynamics. An introduction.  In: Quantum Field Theory and Gravity, 257–297. Springer (2012).
  * [5] Barbour, J., Foster, B. Z., Ó Murchadha, N.:  Relativity without relativity.  Classical and Quantum Gravity, 19, 3217 (2002).
  * [6] Bell, J.S.:  Speakable and unspeakable in quantum mechanics.  Cambridge University Press (1987).
  * [7] Bohm, D.:  A suggested interpretation of the quantum theory in terms of” hidden” variables. I.  Physical Review, 85, 166 (1952).
  * [8] Bricmont, J.:  Making sense of quantum mechanics.  Springer Science & Business Media (2016).
  * [9] DeWitt, B. S.: Quantum Theory of Gravity  Quantum Theory of Gravity.  Physical Review, 160, 1113 (1967).
  * [10] DeWitt, B. S.:  Spacetime as a sheaf of geodesics in superspace.  In: Relativity, 359–374, Springer, (1970).
  * [11] Dürr, D., Goldstein, S., Zanghì, N.:  Quantum equilibrium and the origin of absolute uncertainty.  Journal of Statistical Physics, 67, 843–907 (1992).
  * [12] Dürr, D., Goldstein, S., Zanghì, N.:  Quantum physics without quantum philosophy.  Springer Science & Business Media (2012).
  * [13] Gomes, H., Gryb S., Koslowski, T.:  Einstein gravity as a 3d conformally invariant theory.  Classical and Quantum Gravity, 28, 045005 (2011).
  * [14] Le, H., Kendall, D.G.:  The Riemannian structure of Euclidean shape spaces: a novel environment for statistics.  The Annals of Statistics, 21, 1225–1271 (1993).
  * [15] Mercati, F.:  Shape dynamics: relativity and relationalism.  Oxford University Press (2018).
  * [16] Mellini, A. :  Ph.D. Thesis.  In preparation.
  * [17] Misner, C.W., Thorne, K.S., and Wheeler, J.A.  Gravitation.  W.H. Freeman (1973).