Switching quantum reference frames in the Nbody problem and the absence of global relational
perspectives

arXiv:1809.05093v3 [quant-ph] 16 Aug 2023

Augustin Vanrietvelde1,2,3,4 , Philipp A. Höhn5,2,6,7 , and Flaminia Giacomini8,2,7,9
1

Laboratoire Méthodes Formelles, Inria Saclay, France

2

Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Vienna

3

Quantum Group, Department of Computer Science, University of Oxford

4

Department of Physics, Imperial College London

5

Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa

6

Department of Physics and Astronomy, University College London, London

7

Faculty of Physics, University of Vienna

8

Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, Zürich, Switzerland

9

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

Given the importance of quantum reference systems to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of quantum reference
systems. Here, we expand on a unifying approach, begun in [1], which blends theoperational language of [2] with a gravity-inspired symmetry principle. The latter
enforces physics to be relational and leads, thanks to gauge related redundancies,
to a perspective-neutral structure which contains all frame choices at once and via
which frame perspectives can be consistently switched. Formulated in the language of
constrained systems, the perspective-neutral structure turns out to be the constraint
surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge
choice and involves a symmetry reduction procedure to an associated reduced phaseand Hilbert space. Quantum reference frame switches thereby amount to a particulargauge transformation.
Here, we show that they take the form of ‘quantum coordinate changes’. We
illustrate this method in a general mechanical particle model, namely the relational
N -body problem in three-dimensional space with rotational and translational symmetry. This model is particularly interesting because it features an analog of thegeneric Gribov problem so that globally valid gauge fixing conditions, and thereby
relational frame perspectives, are absent. We will show that the constraint surface
is topologically non-trivial and foliated by three-, five- and six-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. In consequence,
the N -body problem also does not admit globally valid canonically conjugate pairsof Dirac observables. These challenges notwithstanding, we exhibit how one can construct the quantum reference frame transformations for the three-body problem. Ourconstruction also sheds new light on the generic inequivalence of Dirac and reduced
quantization through its interplay with quantum frame perspectives.
Philipp A. Höhn: philipp.hoehn@oist.jp, corresponding author

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

1

1 Introduction
Reference frames appear ubiquitously in both quantum and gravitational physics. A
concrete description of some physical situation is usually given from the perspective ofsome appropriate choice of reference frame. In most cases, such a frame is considered
as an idealized external system that can be used as a vantage point but does not itself
back-react on the remaining physical systems.
Reference frames, however, are always physical systems and treating them fundamentally as being quantum in nature is inevitable in quantum gravity [3–18] and also inquantum information and foundations, for example once considering that measurements
are carried out with physical systems [19–36]. In fact, every physical measurement is relational: they relate some physical property of the measured systems to a physical propertyof the systems comprising the measurement device. For example, in a Stern-Gerlachmagnet, the spin of the measured particles is related to the (coherent) spin of the atoms
composing the magnet. Treating reference systems fundamentally as quantum objects is
ultimately a consequence of the universality of quantum theory, according to which all
physical systems are subject to its laws.
Both in quantum gravity and quantum foundations it therefore becomes crucial toclarify how to describe physics relative to quantum reference systems and how the descriptions relative to different such choices are related. Exploiting a fruitful interplay ofideas from both fields, we began developing a unifying method for transforming between
quantum reference systems in [1] that aims at ultimately encompassing both quantum andgravitational physics. The key ingredients of this method are a gravity inspired (gauge)
symmetry principle and the relational and operational approach to quantum reference
frames recently put forward in [2].
The symmetry principle is, in fact, inspired by Mach’s principle, which was a pivotal
motivation for the development of general relativity and is closely related to the diffeomorphism symmetry of the latter. Mach’s principle essentially states that all physics is
relational [4,37,38]. In particular, what inertial frames are is not determined with respect
to an absolute space (as in Newtonian physics), but by the remaining dynamical content
of the universe. We likewise employ the symmetry principle to enforce physical observables to be relational in the systems we consider and to generate an inherent redundancy
in their description. Exploiting this redundancy allows us to develop, as proposed in [39],a perspective-neutral meta-structure, which contains, so to speak, all frame perspectives
at once and via which they are changed.
This symmetry principle is implemented using the language of constrained Hamiltonian systems [40, 41], which also underlies the canonical formulation of general relativity
and quantum gravity [4, 13]. In our approach, the perspective-neutral structure corresponds to the constraint surface classically and to the gauge invariant physical Hilbert
space in the Dirac quantized theory. Taking the perspective of a specific frame is closely
related to imposing a gauge that fixes the redundancies in the description and changing
from one frame perspective to another amounts to a symmetry transformation via theperspective-neutral structure. A specific perspective relative to one choice of frame is classically encoded in a gauge-fixed reduced phase space and to a symmetry reduced Hilbert
space in the quantum theory. The latter will be the quantum analog of a gauge-fixed
reduced phase space, but not necessarily equivalent to a quantization of it. This gives rise
to a systematic method for changing between the perspectives relative to different choicesof quantum reference systems and can be applied to both temporal and spatial quantum
reference systems and in both quantum and gravitational physics. Compellingly, by al-

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

2

ways passing from one frame perspective through the perspective-neutral Hilbert space inorder to map to a new frame perspective, the quantum reference frame transformations
will take the form of ‘quantum coordinate transformations’.1
Our method extends the approach to changes of temporal quantum reference systems,
i.e. relational ‘clocks’, developed in [14–16] for models of quantum gravity and cosmology,
which precisely uses the above concepts and tools at a semiclassical level. Altogether, ourapproach can also be viewed as an expansion of the view, advocated in [44], that gauge
related redundancies are not just a mathematical artifact, but physically crucial for acompletely relational description of the world.
As already indicated, our operational language, ultimately originating in [2], also sheds
new light on the relation between Dirac and reduced quantization – i.e. on the relation
between quantizing first, then solving the constraints, and solving the constraints first,
then quantizing – which has been the topic of a lively debate in the literature [17,18,45–56].
While the agreement is that the two methods are generally not equivalent, the debate also
revolves around the question when one or the other would be the correct one to apply.
In our approach, the Dirac quantized theory provides the perspective-neutral structurefrom which internal perspectives are constructed via ‘quantum coordinate maps’. These
are the quantum analog of classical phase space reductions by gauge fixing and map tosymmetry reduced Hilbert spaces; they proceed as follows:
1. Trivialize the symmetry generating quantum constraints. That is, transform themin such a way that they only apply to (and fix) the subsystem one would like to
use as a reference and, accordingly, whose degrees of freedom one considers as theredundant ones in this specific description.
2. Condition on the corresponding classical gauge fixing conditions.
The symmetry reduced quantum theories so constructed – constituting the quantum frame
perspectives – will generally be unitarily inequivalent to the quantization of the classicallyreduced theories, i.e. the direct quantization of the classical frame perspectives. This
manifests the generic inequivalence between Dirac and reduced quantization; for example,
we will see that the Hamiltonians will differ. However, this is not a problem for ourframework of quantum frame covariance. As we shall argue, it is the Dirac quantizedtheory which should be given primacy as this is the one treating all degrees of freedom
on an equal footing and thereby translating the redundancy into the quantum theory
that enables one to define a notion of quantum covariance in the first place. In reduced
quantization one simply generally lacks the structure to relate different frame perspectives
and they will in general not be unitarily equivalent. In other words, both classically andin the quantum theory, we give primacy to the quantization of the perspective-neutral
structure over the quantization of the internal perspectives; it is the perspective-neutral
structure which links all the internal frame perspectives.
It should be emphasized that it is possible to render Dirac and reduced quantizationequivalent in the class of models considered here by invoking constructions exhibitedin [50, 53]. But this involves a non-standard factor ordering of the physical Hamiltonian
which essentially amounts to somewhat artificially adapting Dirac to reduced quantization
so that the two methods produce equivalent results. This leads to a serious drawback,1

Notice that some authors distinguish between quantum reference frames, i.e., where the reference frame
is treated as a physical system and has a quantum state associated to it, and quantum coordinates [36,42, 43], where the reference frame transformation is generalised but reference frames are not treated asphysical systems. Here, we do not make this distinction, and use both interchangeably.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

3however: the so modified Dirac quantized theory is then equivalent to the quantizationof only one of the classical frame perspectives and explicitly depends on this choice.
Specifically, the modification of the physical Hamiltonian then depends on the choiceof perspective, with different choices leading to inequivalent results, manifesting that thedirect quantizations of the different classical frame perspectives are unitarily inequivalent.
In particular, constructing the quantum frame perspectives either by directly quantizing theclassical frame perspectives or by quantum reduction from the perspective-neutral Hilbert
space are in general two physically inequivalent procedures. We comment further on thisbelow and in Appendix D.
In [1], we have introduced the general conceptual underpinning of this approach andillustrated it in a simple N -particle model in one-dimensional space with a single linear translation generator constraint. Accordingly, globally valid gauge fixing conditions
and thus globally valid relational perspectives on the physics were possible in this simple
model. Using our new method, we recovered in [1] some of the quantum frame transformations of [2], which were constructed using a different approach, thereby embedding themin a perspective-neutral framework. These transformations were also employed to showin [1] how entanglement and classicality of an observed system depend on the quantum
frame perspective.
Here, our aim is to generalize our approach, begun in [1], by applying it to a significantly more complicated model, namely the relational N -body problem in threedimensional space which now features not only translational, but also rotational invariance. This is a very general mechanical particle model and it will serve to substantiate
both our conceptual and technical line of argumentation, proving the capability of ourapproach.
The most crucial difference to the model in [1] is the absence of globally valid internal
perspectives, i.e. any internal perspective will fail to fully describe all physical situations.
This is analogous to the absence of global coordinates on generic spacetime manifolds
and links with the conceptual discussion in [1]. The origin of this property in the present
model is the additional rotational symmetry. The gauge orbits of the rotation group in
the constraint surface are compact (and thus closed) and this results in the impossibilityof finding globally valid gauge fixing conditions. This feature is a mechanical analog of the
Gribov problem in gauge field theories, and it is also the reason why globally valid pairs ofcanonically conjugate gauge invariant (i.e. Dirac) observables are absent. In consequence,
it is only possible to develop non-global descriptions of the physics relative to a particularreference frame in both the classical and quantum theory. This is closely related to theglobal problem of time in general relativistic systems [7, 8, 14–18]. Concretely, this meansthat generic relational clocks (temporal quantum reference systems) in these systems
feature turning points – and, accordingly, can start running ‘backwards’ – and thus cannot
be used as globally valid temporal references. Therefore, non-global relational descriptions
appear, and should be expected, in most interesting physical scenarios, as argued alsoin [1]. The below is still a fairly benign illustration of this Gribov problem.
It must be emphasized, however, that this is not a problem for the physics per se, but
only of descriptions of it. In fact, the perspective-neutral structures in both the classicaland quantum theory can be formulated globally without any issues. Challenges only arise
when defining specific perspectives and so much of our attention here will be devoted to
exhibiting how to describe physics relative to non-global relational perspectives and how
to nevertheless switch between them via a perspective-neutral structure that itself can beformulated globally. Moreover, one should keep in mind that in this work, the internal
perspectives only fail on very specific, highly degenerate configurations (typically, when

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

4all particles are aligned); and moreover that they all fail on the same configurations for
the N = 3 case which we focus on.2 The first of these features shows that the situation isless severe than, for example, in the case of Schwarzschild coordinates, which leave out aconsequent chunk of spacetime; the second puts it in contrast with the case of sphericalcoordinates, in which the failure happens over an arbitrary axis that bears no relationshipwith the physical configuration.
The rest of the article is arranged as follows. In sec. 2, we introduce the classical N body problem in three-dimensional Newtonian space, but subject to a global rotational
and translational invariance, which renders the spatial physics fully relational and leads
to a redundancy in the description. Here, we discuss all the technical challenges thatarise due to the appearance of the (mechanical analog of the) Gribov problem and how
one can nevertheless choose a frame perspective. In sec. 3, we quantize the three-bodyproblem in both the Dirac and reduced method. Our main ambition here is to construct explicitly the quantum symmetry reduction, i.e. quantum coordinate maps, fromthe Dirac to the reduced quantum theories in the particular frame perspectives and tocompare the result to the quantization of the classical frame perspectives. We finally thenuse this quantum reduction method to also construct the quantum reference frame transformation that takes one from a quantum frame perspective to another. Importantly,
this transformation proceeds by inverting the quantum reduction, mapping back to theperspective-neutral Hilbert space and then performing the forward quantum reduction to
the desired perspective, hence a ‘quantum coordinate transformation’. Despite the absence of global classical relational perspectives, these quantum frame changes are unitary
for regular Hilbert space states because the pathological configurations comprise a set ofmeasure zero; for distributional states, the situation differs somewhat. We illustrate thetransformation on example states which show the quantum frame dependence of entanglement, before concluding in sec. 4. For better readability, we have moved most technicaldetails to various appendices and for better orientation, we also provide here a table ofcontents.

Contents
1

Introduction

22Internal perspectives in the classical N -body problem
2.1 A toy model for Mach’s principle in 3D space . . . . . . . . . . . . . . . .
2.2 Choosing an internal perspective = choosing a gauge . . . . . . . . . . .
2.3 Switching internal perspectives . . . . . . . . . . . . . . . . . . . . . . . .

6
6
10
15

3

Quantization and relative states
3.1 Reduced quantization – quantizing classical internal perspectives . . . . .
3.2 Dirac quantization – the perspective-neutral quantum theory . . . . . . .
3.2.1 Constructing the translation invariant Hilbert space . . . . . . . .
3.2.2 Translational reduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Rotational invariance . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Rotational reduction to A’s perspective . . . . . . . . . . . . . . . . . . .
3.3.1 Rotational reduction . . . . . . . . . . . . . . . . . . . . . . . . . .

16
16
18
19
21
23
23
24

2

However, the second point will not be true anymore for N > 3; see the discussion at the very end of
Section 3.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

5

3.4
3.5
4

3.3.2 Relation with reduced quantization of A’s classical perspective . .
Transformations between relative states . . . . . . . . . . . . . . . . . . .
Remarks on the unitarity of the quantum frame transformations . . . . .

Conclusions

28
30
32
33

A The rotation and translation invariant N -body problem
A.1 The rotation and translation invariant Lagrangian . . . . . . . . . . . . .
A.2 Dependence of constraints on total collisions and collinearity . . . . . . .
A.3 Independent Dirac observables on generic N particle configurations . . . .
A.4 Absence of global canonically conjugate Dirac observables . . . . . . . . .
A.5 Proof of the three-body gauge-fixing in the 3D case . . . . . . . . . . . . .
A.6 Solving constraints for the redundant momenta . . . . . . . . . . . . . . .
A.7 Switching internal perspectives classically . . . . . . . . . . . . . . . . . .

34
34
36
37
38
39
41
41

B Using spherical coordinates
B.1 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Spherical harmonics states . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Spherical coordinates operators . . . . . . . . . . . . . . . . . . . . . . . .

43
43
43
44

C Rotational reduction in the quantum theory
C.1 The rotation generator trivialization map and its action on states . . . . .
C.2 Trivializing the rotation constraints . . . . . . . . . . . . . . . . . . . . . .
C.3 Dirac observables and rotational reduction . . . . . . . . . . . . . . . . . .

44
44
46
47

D Illustration of the Hamiltonian discrepancy between Dirac and reduced quantization48

2 Internal perspectives in the classical N -body problem
2.1 A toy model for Mach’s principle in 3D space
We employ a toy model3 for Mach’s principle in a system of N interacting particles inthree-dimensional space with translation and rotation invariance. This gauge symmetry
implies that the position, orientation and motion of the particles with respect to the
Newtonian background space has no physical significance and that the gauge invariant
information is purely relational. As such, our system bears some resemblance to the
Barbour-Bertotti and related models [37, 38, 57, 58], except that here we will not havedilation or Hamiltonian constraints.
For simplicity and in analogy to [1], we restrict to unit mass particles and a Euclidean
phase space Q = R3N so that our phase space, coordinatized by particle positions ⃗qi andmomenta p⃗i , is T ∗ Q = R6N . We take an N particle system whose Lagrangian, described inAppendix A.1, features the desired gauge symmetries and upon Legendre transformation
leads to the following six (primary) constraints on phase space:4Pa =X

pai ≈ 0 ,

(1a)

i
3

We thank T. Koslowski for suggesting this model.

4

Henceforth, the Einstein convention holds and repeated spatial indices are implicitly summed over.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

6

Ra =

X

ϵabc qib pci ≈ 0 ,

(1b)

i

where a, b, c ∈ {x, y, z} denote spatial components. These constraints are independent5(almost everywhere, see below) and first-class, clearly constituting the generators of the
Euclidean group:{P a , P b } = 0 ,{Ra , Rb } = ϵabc Rc ≈ 0 ,{P a , Rb } = −ϵabc P c ≈ 0 .

(2)They are conserved so that no secondary constraints arise. The total Hamiltonian following from the Lagrangian of Appendix A.1 is
N
1 X
a a
a a
p⃗i 2 + V ({|⃗qi − ⃗qj |}N
i,j=1 ) + λ P + µ R ,2 i=1

Htot =(3)

where the λa and µa are six a priori arbitrary Lagrange multipliers, accounting for gauge
freedom, and the potential V depends only on the relative absolute distances between theparticles.
Consequently, the six constraints generate gauge transformations on the constraint
surface; clearly, the P a induce global translations, namely infinitesimally(

qia → qia + {qia , P b } εb = qia + εa ,pai → pai + {pai , P b } εb = pai ,(4)while the Ra are generators of global rotations, infinitesimally given by(

qia → qia + {qia , Rb } εb = qia + ϵabc εb qic ,pai → pai + {pai , Rb } εb = pai + ϵabc εb pci .

(5)

A subtlety arises: while the six constraints (1) are independent on generic pointsof the constraint surface C in phase space (so generically, the gauge orbits will be sixdimensional), there exist special pathological points on C (thus a priori allowed), where
the constraints become partially dependent. These correspond to N particle collisions
and situations when all particles are collinear. As shown in Appendix A.2, in thesecases, only three and five constraints are independent, respectively, and the corresponding
configurations reside in lower dimensional gauge orbits, so that the constraint surface is, infact, foliated by six-, five- and three-dimensional gauge orbits. This is a direct consequenceof the compactness of the pure rotation orbits and the (complete or partial) rotational
invariance of these special configurations. Given that total collisions and collinearity are
a set of measure zero, almost all gauge orbits are six-dimensional. However, the lower
dimensional ones are topologically relevant and have physical consequences: globally valid
gauge conditions will not be possible because on the lower dimensional orbits one cannot
fully fix the rotational gauge freedom as some rotations will act trivially (see Appendix
A.2 and the discussion in sec. 2.2).6 Since we will, again, interpret choices of internal
perspectives as gauge fixings, this is the origin of why there will be no globally validinternal perspectives in this model.
5
In the N = 2 case, the system (1) is not independent and there are only five independent constraints,
which induces some peculiarities. In the rest of the paper, unless stated otherwise, we will assume N ≥ 3.
6
Also, as a manifestation of the Gribov problem, other gauge fixing surfaces might miss the lower
dimensional orbits entirely.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

7This implies significant repercussions for the reduced phase space, which is the set of
all gauge orbits (every gauge orbit is the equivalence class corresponding to one physicalstate). Mathematically it is the quotient Pred = C/∼, where ∼ identifies points if theylie in the same orbit, and so will be topologically non-trivial because of the differing
dimensions of the gauge orbits.
Clearly, this also has consequences for gauge invariance and, specifically, gauge invariant functions, i.e. Dirac observables. Before we discuss these consequences, we recall thatany function can be expressed in terms of our basic phase space variables ⃗qi , p⃗j and inquire
what the most general form of a Dirac observable is. Given that the full set of rotations
only leaves inner products invariant, any Dirac observable must be a function of innerproducts of our basic variable vectors. These inner products must also be translation
invariant and so any Dirac observable must be a function of(⃗qi − ⃗qj ) · (⃗qk − ⃗ql ),(⃗qi − ⃗qj ) · p⃗k ,p⃗i · p⃗j ,i, j, k, l = 1, . . . , N , (6)which comprise all rotation and translation invariant inner products quadratic in our basic
variables.7 In particular, the unconstrained part of Htot is obviously also gauge invariant.
Again, there is a redundancy among the elementary Dirac observables (6). Indeed, theN particles have 3N coordinates and the symmetry consists of three global translations
and three global rotations so there can only be (at most) 3N − 6 independent gauge invariant configuration degrees of freedom. Similarly, there are 3N momentum coordinatesand six constraints (1) that can be solved for momenta so there can likewise only be (atmost) 3N − 6 independent gauge invariant momentum degrees of freedom. For intuitionof these statements, it is helpful to visualize the N particle motion and its gauge invariant
information geometrically: a generic N particle configuration (i.e., no total collision or
collinearity) corresponds to a triangulation with N vertices in 3D Euclidean space (seefig. 1) whose 3N − 6 edges are labeled by relative distances ⃗qi − ⃗qj from among the(square roots of the) configuration observables in (6). In this manner, we argue geometrically in Appendix A.3 that for generic N particle configurations there are indeed 3N − 6
independent configuration and 3N − 6 independent momentum Dirac observables.
But what about N particle collisions and total collinearity where we saw that thesymmetry generators become dependent? We prove in Appendix A.4 that (i) none of theelementary Dirac observables (6) are independent on the N particle collisions residing inthree-dimensional gauge orbits, and (ii) only 2(N − 1) of them are independent on totally
collinear N particle configurations residing in five-dimensional gauge orbits.
As we also show in Appendix A.4, this has a severe ramification: canonically conjugate
or affine 8 pairs of Dirac observables do not exist globally on C. Accordingly, there areno canonically (or affinely) conjugate coordinates covering all of the reduced phase space,manifesting its topologically non-trivial nature alluded to above. A priori this will complicate also the reduced quantization of the physics in any of the internal perspectives, as we
can not simply apply canonical or affine quantization methods [59] to the reduced phasespace as a whole without restricting the dynamics and introducing boundary conditions.
In this sense, also the internal perspectives in the quantum theory will be non-global.
Let us now construct canonically conjugate Dirac observables that are ‘as global as
possible’. Constructing sets of independent (non-global) canonically conjugate Dirac observables for the N -body problem depends on N and so here we shall only provide such a
7
Note that also all Dirac observables which involve cross products will be functions of these combinations.
8
Two phase space variables q, π form an affine (rather than canonical) pair if {q, π} = q. That is, π
is the generator of dilatations (rather than translations) of q.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

8

B

rAB

D

C

γ

F

E

rAC

A
(a)

B
A

C
(b)

Figure 1: (a) Illustration of the three-body problem and gauge invariant configuration degrees of
freedom. Our three configuration Dirac observables from the (almost everywhere) conjugate pairs in (7)are functions of these: ρBA , ρCA are simply the logs of the relative distances rAB = |⃗qA − ⃗qB |, rAC =|⃗qA − ⃗qC | between A and B, C, respectively, while u is (minus) the cotangent of the relative angle
between B and C as seen from A. (b) Illustration of a generic situation in the six-body problem as atriangulation. In order to fully localize particle F relative to particles A to E, three relative distances
between F and the rest are sufficient and not all relative distances are independent.

set for the three-body problem. Labeling the three particles as before by A, B, C, we assembled a convenient set of 3N − 6 = 3 (almost everywhere) independent and canonicallyconjugate Dirac observable pairs for our purposes as follows
ρBA := ln rBA = ln |⃗qB − ⃗qA | ,pρBA := p⃗B · (⃗qB − ⃗qA ) ,ρCA := ln rCA = ln |⃗qC − ⃗qA | ,pρCA := p⃗C · (⃗qC − ⃗qA ) ,(⃗qB − ⃗qA ) · (⃗qC − ⃗qA ),u := − cot γ = −(⃗qB − ⃗qA ) × (⃗qC − ⃗qA )rCA
pu := sin γ p⃗C · cos γ (⃗qC − ⃗qA ) −(⃗qB − ⃗qA ) .
rBA

(7)

We choose this set as it has a direct geometrical interpretation: the configuration Diracobservables are the logarithms of the relative distances rBA , rCA ∈ [0, ∞) of B and C from
A and (minus) the cotangent of the angle γ ∈ [0, π] in the triangle between B and C at
A (see fig. 1a). However, in contrast to rBA , rCA , γ, our geometric variables ρBA , ρCA , ueach take value in all of R which will be more convenient for quantization. (We chooseu = − cot γ so that u goes from −∞ to +∞ as γ runs from 0 to π.) The momentum
Dirac observables are projections of the particle momenta in the appropriate directions.
The same construction can be carried out, of course, for permutations of A, B, C with theobvious meaning.
Clearly, this set will not define global coordinates on the reduced phase space asρBA , ρCA , pu become singular when A collides with either B or C and u is undefinedin those cases and when the three particles are collinear. (These configurations lie inthe boundary of the gauge invariant configuration space and are a set of measure zero[60].) For other non-global constructions of canonically conjugate Dirac observables forthe three-body problem, e.g., in terms of Jacobi (mass-weighted difference) vectors, see[58, 60, 61].
Our treatment for these illnesses is to dynamically restrict the model: we shall henceforth assume (but not explicitly write) that the potential V ({|⃗qi − ⃗qj |}i,j=A,B,C ) becomes
infinitely repulsive when any of the relative distances vanishes. For example, we might
imagine three electrons and a Coulomb potential. This rules out any particle collisions(for finite energies) and so the three-dimensional gauge orbits will no longer become dyAccepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

9

namically accessible. Note that the pairs (ρBA , pρBA ) and (ρCA , pρCA ) will then indeed
be canonically conjugate on the dynamically accessible configurations and also pu willremain finite. Only u will, strictly speaking, remain undefined when A, B, C are collinear,although γ is well-defined then. Since these configurations also remain a set of measurezero in the boundary of the gauge invariant configuration space [60], we shall simply interpret the divergence of u for collinearity as ‘infinite distance’ in the variable u. In thissense, after dynamical restriction, our variables (7) are ‘as good as it gets’ and canonicallyconjugate on the dynamically accessible regions of C.
In summary, owing to the gauge symmetry, the N particle system has no physicallymeaningful absolute position, orientation and motion in space, yet relative position, orientation and motion among the particles is gauge invariant. Building up on our discussionin [1], we interpret the structures described here and, in particular, all the structure on theconstraint surface C as a perspective-neutral meta-structure which, thanks to its inherent
redundancy, contains all perspectives at once. Choosing a specific internal perspective ofa particle frame will again amount to a gauge fixing, as we shall discuss shortly.
We noted that there cannot be a globally valid internal perspective on account of the
topological non-trivialities of the gauge orbits. However, it must be emphasized that all
ensuing challenges only arise when collecting the structures necessary for moving into a
specific internal perspective. The perspective-neutral structure has no problems per se as
C can be consistently and globally described in terms of the original phase space variables⃗qi , p⃗j and, while Dirac observables are necessary for gauge invariance, global canonicallyconjugate pairs of them are not required.

2.2 Choosing an internal perspective = choosing a gauge
In order to completely fix the gauge, we have to impose six gauge fixing conditions as wehave six independent first class constraints on generic points of C. Altogether, as alreadynoted, we then haveF = 6N − 12(8)independent gauge invariant phase space degrees of freedom away from pathological configurations – the dimension of the reduced phase space.9
Suppose we want to describe the physics from the internal perspective of particle A.
In complete analogy to the purely translation invariant case of [1], we define A to be theorigin from which all distances are measured:!

aχa = qA= 0,

a = x, y, z ,(9)!

where the symbol = means that we demand the equality to hold. However, we alsoneed to fix the rotational gauge symmetry. Before we move on, we already note that
there can be no globally valid gauge fixing conditions that will pierce every gauge orbit.
This follows from pure dimension counting. Indeed, a set of gauge fixing conditions that
also completely fixes the rotational symmetry for generic configurations must define a(6N − 12)-dimensional gauge fixing surface G within the (6N − 6)-dimensional constraint
surface C. Locally, within C, G will be described by six independent conditions. We
noted that the gauge orbits in which the total collisions reside are three-dimensional (seeAppendix A.2) and so these will locally, within C, be described by 6N − 9 independentconditions. Clearly, it is impossible to satisfy the 6 + 6N − 9 = 6N − 3 > 6N − 6
9Except for N = 2, where F = 2.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

10

z

B

qBz
A

y

C

qCz

θC

x

qCx

⌃

Figure 2: Illustration of the gauge-fixing procedure (9–12) for the three-body problem. We choose A
as the origin from which to measure all distances, B to define the z-axis and C to define the x-z-plane.
This procedure works in the same way for N > 3 particles by choosing a three-body subsystem.

independent conditions simultaneously within C. The analogous state of affairs holds forthe five-dimensional orbits in which totally collinear configurations reside. Hence, any
gauge fixing surface that fixes the rotational symmetry for generic configurations will
necessarily miss the lower-dimensional gauge orbits. This is an incarnation of the Gribov
problem. Without a global gauge fixing surface, we will also not have a globally validinternal perspective.
We proceed by focusing on generic configurations and accept that we will have to missthe lower-dimensional orbits. In order to complete the choice of reference frame, we now
also need to define the three axes of space as seen from particle A, thereby fixing the threerotational gauge degrees of freedom. This has to be done in terms of the other particles
as these provide the only physically meaningful reference for A. We have illustrated ourgauge-fixing procedure for better visualization in fig. 2. First, picking another particle
B, we can get two gauge conditions by fixing the direction in which A sees B, i.e., thedirection (but not the norm) of ⃗qBA = ⃗qB − ⃗qA ≈ ⃗qB . Choosing it as A’s z-axis imposes!

yϕ1 = qBA= 0,!

xϕ2 = qBA
= 0.

(10a)

(10b)

There is then only one continuous gauge freedom left, corresponding to the rotation of
direction z around the origin. To fix it, we consider a third particle C, and fix it to lie inthe (x, z) plane:!
yϕ3 = qCA= 0.
(11)
In fact, we have only blocked all continuous gauge transformations, but there remainsome allowed discrete gauge transformations, namely, the rotations of angle π around oneof the axes, whose effect is to invert the two other axes. Indeed, those transformations
leave invariant the gauge-fixing conditions (9–11). This is a consequence of the fact thatthe rotational orbits are compact so that the hyperplane defined by (9–11) pierces them
multiple times. To get rid of this residual gauge symmetry, it is necessary and sufficient
to fix the orientation of two of the axes. This can be done with conditions, which fix thepositive part of the z and x axes through:
z
qBA
> 0,

(12a)

x
qCA
> 0.

(12b)

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

11

With those conditions, B is now constrained to lie on the semi -axis [Az), and C to lie
on the half -plane spanned by the axis (Az) and the semi-axis [Ax), see fig. 2. Indeed,
it is clear from the previous discussion and that in Appendix A.2 that we could not fix
the gauge completely, using B and C as material references, if at least one of them werecoincident with A or the three particles were collinear. Altogether, we then have a systemof 12 second-class constraints (given by (1), (9–12)), that is, a completely gauge-fixedsystem, as proven in Appendix A.5.
In summary, a complete gauge fixation is given by the choice of three particles A,B,C 10
out of the N particles, provided they are not collinear. We can denote such a gauge choice
as [A,B,C]. Physically, such a gauge choice corresponds to an operational definition of
the axes with which A assesses positions of other particles in space: B provides with thefirst axis of reference z, and C defines the plane (Axz). This information is then sufficientfor A to build a non-ambiguous set of orthogonal axes (x, y, z). One can also see thisprocedure as an operational construction of spherical coordinates: the origin is defined as
being particle A; the zenith direction z, to which the polar angle θ is relative, is defined as
the direction of particle B; and the plane (Axz), to which the azimuth angle ϕ is relative,
is defined as the one in which C is lying.
We need to check that the gauge choice is consistent with the equations of motionq̇ia =∂Htot= pai + λa + ϵabc µb qic ,
∂pai

(13a)

∂Htot
∂V
= ϵabc µb pci − a .
a
∂qi
∂qi

(13b)

ṗai = −The conservation of our gauge conditions fixes the vectors ⃗λ and µ⃗ as defined in (3),⃗λ = −⃗
pA ,

(14a)

px
µy = − BA
z ,
qB

(14b)

pyBA
z ,
qB

(14c)

µx =

1 
qz y 
µz = − x pyCA − C
z pBA ,
qC
qB

(14d)

where we write pij = pi − pj . Putting (14) into (13) then gives us the equations of motion
for all remaining particles as seen from A’s frame. Note that the validity of the latterz > 0 is implied by our dynamical restriction of infinite repulsion
requires (12) to hold; qB
x = 0 is not dynamically ruled out and happens on the measure
on collisions. However, qC
zero set on the boundary of the gauge invariant configuration space when u → ±∞.
Let us now limit ourselves to N = 3 particles for clarity. The canonically conjugate
10
The most general case of gauge choice would be given by introducing gauge parameters, which allow
for example to fix A to lie not at the origin, but at any position in space - the same goes for the directionsof B and C. We will not consider in detail this formal possibility which does not add to the physicalmeaning of this analysis.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

12

Dirac observables (7) take the following form in our gauge (see fig. 2 for illustration):zρBA ≡ ρB := ln rB = ln qB,

ρCA ≡ ρC := ln rC = lnqx )2 + (q z )2 ,
(qC
C

z z
pρBA ≡ pρB := qBpB ,xzpρCA ≡ pρC := pxC qC+ pzC qC
, (15)qzu ≡ uC := − cot θC = − C
x ,
qC
y
z x
x z
pu ≡ puC := sin2 θC (qC
p C − qC
pC ) = sin2 θC RC.
z , r as the distances of B and C as seen from A, while θHence, we may interpret qB
CC
becomes the angle which A sees between B and C. Note also that C’s angular momentum
around y is essentially conjugate to the angle between B and C.
After having gauge fixed our model, we need to replace the Poisson by Dirac brackets[40, 41]{F, G}D := {F, G} − {F, Λα }(C −1 )αβ {Λβ , G} ,(16)

where Λα , α = 1, . . . , 12, runs over the 12 second class constraints (1) and (9–12) and C −1
is given in (88) in Appendix A.5, to construct the bracket structure for our gauge-fixed
x , q y , q z , q x , q y , q y and their
reduced phase space. The Dirac brackets of the gauge-fixed qA
A A B B C
conjugated momenta vanish and we can now drop these variables (the latter being solved
z , q z , q x ) ∈ R ×R×R (duefor by the constraints, see Appendix A.6). By contrast, the (qB++
C C
to (12)11 ) and their pzB , pxC , pzC remain canonically conjugate also with respect to {., .}D ,so we could use them as canonical coordinates on the dynamically accessible region of
the reduced phase space. However, for quantization it will be more convenient to use
the radial and angle coordinates (ρBA , ρCA , u) ∈ R3 and their respective momenta in (15)which remain canonically conjugate in {., .}D there too.
Before we continue, let us make an important remark. Denote the reduced phase space,which we have obtained after gauge fixing to A’s perspective, by PBC|A as it encodes thephysics of B and C relative to A. Strictly speaking, PBC|A is not actually equivalent
to the abstract – and, in fact, perspective-neutral – reduced phase space Pred = C/∼,
alluded to in sec. 2.1. Indeed, PBC|A is equivalent to the intersection C ∩ GBC|A (and canbe canonically embedded as such into C), where GBC|A is the gauge fixing surface definedby the gauge conditions (9–12)). As argued above, GBC|A entirely misses the lowerdimensional gauge orbits for dimensional reasons and so does not intersect each gauge
orbit once and only once. Hence, total collisions and total colinearity are contained in Pred ,but not in PBC|A . The non-equivalence, of course, only concerns a set of measure zero,from which we have also dynamically ruled out the total collisions. The configurations
left out in PBC|A for N = 3 do not depend on our choice of perspective; indeed, gaugefixing to B’s perspective (i.e. considering PAC|B ) or to C’s perspective (PAB|C ) would
leave out exactly the same configurations. This is why the internal perspectives will
still be equivalent with one another, even though they are not fully equivalent with theperspective-neutral Pred . However, the situation changes for N > 3, where gauge choices
relative to distinct particles can fail on different configurations. We have also argued
above that there are no global gauge fixing conditions, due to the Gribov problem, andso one cannot obtain a gauge fixed phase space that is fully equivalent to Pred . We canthus take PBC|A (or any similarly gauge fixed phase space) as a best possible descriptionof the perspective-neutral Pred relative to a choice of reference frame. We shall work with
PBC|A and refer to it as the reduced phase space in A perspective.
11
The gauge invariant configuration space of the three-body problem is homeomorphic to half of R3
with all collisional and collinear configurations residing in the boundary [60].

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

13

Using the constraint solutions for the redundant momenta (see Appendix A.6), we cannow express our Hamiltonian (3) in terms of the surviving degrees of freedom on PBC|A :HBC|A = (pzB )2 + (pzC )2 + (pxC )2 + pzB pzC +

pxC y
y 2
z
z
x
(R
)
−
C
z )2
z RC + V (qB , qC , qC ) . (17)
(qB
qB
1

This is the Hamiltonian for the dynamics of particles B and C, as seen by A. The termpzB pzC and the lack of 1/2 factors in front of the squared momenta correctly take into
account the relative forces originating in the potential in (3) (e.g., an interaction between
A and C will also affect the position of B relative to A even if B and C do not interact,px yy 2see [1] for further details). The term (qz1 )2 (RCconstitutes an effective potential) − qzC RC
B
B
that generally becomes infinitely repulsive on approach of particle A (where our condition(12a) no longer holds) because B is used to relationally define the non-negative z-direction
from A’s perspective. This singularity in the effective potential can be viewed in someanalogy to the coordinate singularity in the Schwarzschild spacetime: it is not a physicalsingularity, but merely signifies that the chosen phase space coordinates become invalid,
here due to the global gauge-fixing issues elaborated on above. Indeed, the physicalHamiltonian Htot does not feature any such singular behaviour that is independent of theactual potential V .
This Hamiltonian can also be expressed in terms of the (gauge-fixed) canonicallyconjugate Dirac observables (15) as
1
HBC|A = hµν (ρB , ρC , uC )pµ pν + V (ρB , ρC , uC ) ,2(18)

where the index µ is such that pµ runs over pρB , pρC , puC in (15) and we have the physicalconfiguration space metrichµν = 2 ×


e

(19)
− 12 e−ρB −ρC √ uC 2
1+u

−2ρB

 1
 − e−ρB −ρC √ uC2
 2
p 1+uC
1 −ρB −ρC−2e
1 + u2CC

e−2ρCp− 21 e−ρB −ρC 1 + u2Cp− 21 e−ρB −ρC 1 + u2Cp− 21 e−ρB −ρC 1 + u2C3(e−2ρB + e−2ρC )(1 + u2C )2 + e−ρB −ρC uC (1 + u2C ) 2with determinant3

h = 6e−2ρB −2ρC (1 + u2C ) 2q(e−2ρB + e−2ρC ) 1 + u2C + e−ρB −ρC uC



.

(20)

It can be checked that h > 0 for all finite values of ρB , ρC , uC so that h = |h|. However,clearly for ρB , ρC → −∞ and uC → ±∞, i.e. the configurational pathologies, the metriccomponents diverge. This expression will be useful for directly quantizing A’s classicalperspective, i.e. for reduced quantization, and subsequent comparison with Dirac quantization of the perspective-neutral structure.
It is instructive to look at what (17) becomes, had we permitted the particles to havedifferent masses mi :HBC|A =

1
1 1
1
1 1
1
1 z z
1 1
z 2
z 2
x 2
2 ( mA + mB )(pB ) + 2 ( mA + mC )(pC ) + 2 ( mA + mC )(pC ) + mA pB pC
y 2
yz , qz , qx ) .
+ m1B (qz1 )2 (RC) − m1A q1z pxC RC+ V (qB
C C
B
B

Note that in the limit mA → ∞ this Hamiltonian becomes of standard form (with effective
potential), in agreement with the fact that an infinite mass reference system constitutes
an inertial frame.
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

14

2.3 Switching internal perspectives
Going from the internal perspective of reference frame A to that of, e.g., particle C
amounts to a gauge transformation plus a swap of what one considers the redundant andthe relevant Dirac observables (e.g., one would have to exchange the A and C labels in(7, 15)). This requires one to firstly embed the reduced phase space in A perspective into
the perspective-neutral constraint surface, subsequently performing the pertinent gaugetransformation and, finally, projecting again to the reduced phase space in C perspective.
We shall only be schematic here as the situation is geometrically transparent. The detailsof the following discussion can be found in Appendix A.7 (see also [1]).
There exists a canonical embedding map12
ιBC|A : PBC|A ,→ C ,(21)

with image C ∩ GBC|A , that can be reversed by a ‘projection’πBC|A : C ∩ GBC|A → PBC|A ,(22)which drops all redundant embedding information, so that πBC|A ◦ ιBC|A = IdPBC|A . By
exchanging A and C labels, the same construction holds for the reduced phase space
PAB|C in C perspective.
Switching from A to C perspective requires the gauge transformation αA→C , generated
by the constraints P a , Ra , that maps one embedding C ∩GBC|A to the other C ∩GAB|C . We
emphasize that here we take GAB|C as being the analogous gauge fixing condition to (9–12), except that A and C are everywhere exchanged. That means, we use the analogous
gauge fixing procedure as in fig. 2, but seen from the perspective of particle C. It isclear that the sought-after transformation amounts to a few translations and rotationsof the particles. In Appendix A.7, we construct it explicitly and show how this yields a
map SA→C : PBC|A → PAB|C , that we also spell out explicitly in coordinates, and which
satisfies the following commutative diagram:
Pred = C/∼−1
ζAC ∩ GBC|A

ζC

αA→C

C ∩ GAB|C

ιBC|A

PBC|A

πAB|C
SA→C

PAB|C

For completeness, we have here also included the perspective-neutral reduced phase space
Pred = C/ ∼; ζA is the map that associates with each element of C/∼, i.e. gauge orbit inC, the point on it defined by the intersection with the gauge-fixing surface GBC|A . Owing
to the global gauge fixing issues, this map is not globally defined on Pred and its inverse
is to be understood accordingly. The same applies to ζC .
Note that, in the N = 3 case to which this paper restricts itself, α is an isomorphism,
and all perspectives are therefore fully equivalent. This is due to the fact that all perspectives leave out the same set of points (those for which the three particles are aligned).
12
The physical interpretation of PBC|A as the physics seen by A singles out this embedding, which
otherwise would be ambiguous.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

15This will however not be the case anymore for N > 3, where there can be inequivalent
perspectives, due to the different possible choices of three-points subsets in this case; see
the discussion at the end of Section 3.
Altogether, our change of internal perspective first maps the old perspective backinto the perspective-neutral structure, carries out a gauge transformation, and finallyprojects into a new perspective. The diagram highlights the similarity to coordinatetransformations: defining φA := πBC|A ◦ζA , and similarly for C, we have SA→C = φC ◦φ−1A .

3 Quantization and relative states
We shall now quantize the perspective-neutral structure using the Dirac method. This
will endow the Dirac quantized theory with the interpretation of the perspective-neutral
quantum theory from which we will then derive the quantum descriptions relative to internal frame perspectives through a quantum symmetry reduction procedure that will
constitute the ‘quantum coordinate maps’. We shall support this interpretation by exploiting these quantum coordinate maps to derive the transformations that take one fromthe internal perspective of one quantum reference frame to that of another. On accountof the exhibited absence of globally valid internal perspectives, the reduced quantum
theories and the transformations between them will also fail to be globally valid, when
we take distributional states (classical configurations) into account. For general Hilbert
space elements, however, the pathological configurations, being a set of measure zero [60]in the integration, will essentially be irrelevant. Our exposition below will thereby serve
to substantiate our discussion in [1] by extending it to a rather general particle model.
In order to compare the symmetry reduced quantum theories, obtained upon applying
the quantum coordinate maps to the perspective-neutral quantum theory, with the direct
quantization of the classical frame perspectives, we shall begin with the reduced quantization of the latter. As we shall see, the two quantizations will not agree in general,
exemplifying the generic difference between Dirac and reduced quantization. We will conclude from this that in general we have to give primacy to Dirac quantization because it
treats all degrees of freedom on an equal footing and translates the classical redundancy
in the description into the quantum theory which ultimately is a prerequisite for defining
a notion of quantum frame covariance.
Given that we will encounter a considerable number of Hilbert spaces and transformations between them, we organize the main steps in fig. 3 to support the orientation
within our construction.

3.1 Reduced quantization – quantizing classical internal perspectives
We begin by quantizing the reduced phase space PBC|A in A perspective. For convenience, we choose the canonically conjugate set (15), as the configuration Dirac observables ρB , ρC , u take value in all of R.13 We promote these variables directly to operators
satisfying the canonical commutation relations14ρρu[ρ̂B , π̂B] = [ρ̂C , π̂C] = [ûC , π̂C] = i,(23)

on the Hilbert space HBC|A = L2 (R3 , dρB dρC duC ), see fig. 3. We have called our momentum operators π̂ in order to distinguish them from the momentum operators that we
13

Otherwise, we would have to deal with global non-trivialities [59] in the quantization.

14

Henceforth, we work in units where h̄ = 1.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

16⃗
⃗ =R=⃗⃗ χ=ϕ=0P

original phase space T ∗ Q ≃ R18

PBC|A

Dirac quantization

Hkinδ(P̂ a )reduced quantization
TA,BC

HTI
δ(R̂a )

χ=0|
A ⟨⃗
TI
TI
HA,BC
HBC|A
a +R̂a )
δ(R̂B
C

TA,BC

Hphys

phys
HA,BC

χ=0|
A ⟨⃗

a +R̂a )
δ(R̂B
C

phys
HBC|A

RB,C

phys
HB,C|A

⃗
⟨ϕ=0|

HBC|A

Figure 3: Diagram of the two quantization methods of sec. 3 for the three-body problem and theirrelation (see also Table 1 for a summary of the notations used). Each column represents a step fromthe Dirac to the reduced quantum theory, as explained in the main text. Vertical arrows on the left sidecorrespond to constraint imposition. The horizontal arrows between Hilbert spaces are all isometries andcorrespond to the quantum reduction steps. They are a sequence of: (1) trivialization of the translation
generators, (2) conditioning on the classical conditions χ⃗ = 0 that fix the translational gauge freedom,⃗ = 0(3) trivialization of the rotation generators, and (4) conditioning on the classical conditions ϕfixing the rotational gauge freedom. The two squared diagrams among Hilbert spaces on the lower left
are commutative. For better visualization, we have summarized the relevant phase and Hilbert spaces⃗ = 0| is not globally defined owing toappearing in this diagram in a table. The last reduction map ⟨ϕ
the global gauge fixing issues. It does, however, act as an isometry on its image.

will introduce for the Dirac quantization. In position representation, we will representρ= −i∂ρB , etc. An arbitrary reduced quantum state can now be writtenthem as usual π̂Bas|ψ⟩BC|A =ZdρB dρC duC ψBC|A (ρB , ρC , uC ) |ρB ⟩ |ρC ⟩ |uC ⟩(24)

and we choose the standard inner product R3 dρB dρC uC ϕ∗BC|A (ρB , ρC , uC )ψBC|A (ρB , ρC , uC ).
To quantize the reduced Hamiltonian (18), we choose the standard operator ordering inwhich the kinetic term in (18) is promoted to an operator which is equivalent to (minus)
the covariant Laplace-Beltrami operator associated with the metric hµν in (19),15 whichyields
1
1
1
1
red
(25)
ĤBC|A
= |h|− 4 π̂µ |h| 2 hµν π̂ν |h|− 4 + V (ρ̂B , ρ̂C , ûC ) .
2
Being the direct quantization of A’s classical frame perspective on the dynamics ofparticles B and C, we can interpret the result for the time being as one a priori possible
quantum theory describing B and C relative to A. We will later argue, however, that
this is not the appropriate quantum theory describing the perspective of the quantum
reference frame associated with A. Notice that this quantum theory does not include
total collisions (which, in any case, we had ruled out dynamically) or total collinearity ofthe three particles because neither did PBC|A . Indeed, collisions happen for ρB , ρC → −∞R

15

More precisely, the covariant Laplace-Beltrami operator is usually (e.g., see [50]) given in the form
−1/4−∆h = |h|−1/4 p̂µ |h|1/2 hµν p̂ν |h|−1/4 , where p̂p∂µ |h|−1/4 , and both ∆h and p̂µ are selfR n µ = −i|h|∗ µadjoint with respect to the inner product dq|h| ϕ̃ (q )ψ̃(q µ ), i.e. one which differs from ours throughpp
the density |h|. However, it can be easily checked that −∆h in the measure dq n |h| is equivalent to11

1|h|− 4 π̂µ |h| 2 hµν π̂ν |h|− 4 in the measure dq n if the wave functions in the two representations are relatedby ψ̃ = |h|1/4 ψ.
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

17and total collinearity as uC → ±∞. Given the normalization conditions in HBC|A , any
ψBC|A (ρB , ρC , uC ) will have to have vanishing support there. In this sense, it thus is aquantum theory that is not globally valid. Also, the Hamiltonian (25), while precluding
collisions, can still evolve to states with collinear situations in finite time even for initialstates that do not have support on it. However, since the pathological configurations
comprise a measure zero set, this is not a serious problem and the evolution should be welldefined for a dense set of states. It is only an issue when considering distributional stateswith support on these pathologies. Given the absence of global relational perspectives,
we have to accept this peculiarity for the direct quantization of the classical perspectiveof A.
Later, we will compare this reduced quantized theory to the reduced theory obtainedby symmetry reducing the Dirac quantized model. While their structure will look very
similar, they will be inequivalent, unless certain somewhat artificial choices are madein Dirac quantization. On the one hand, this will demonstrate that a quantum frame
perspective will generally not be equivalent to the direct quantization of a classical frame
perspective. On the other, it will nevertheless provide intuition for why we will identify
the quantum symmetry reduced theories with quantum frame perspectives.

3.2 Dirac quantization – the perspective-neutral quantum theory
Next, we quantize the perspective-neutral structure the Dirac way. That is, we quantize
the original phase space T ∗ Q ≃ R18 , by promoting all canonical pairs (qia , pai ) to operators
on a kinematical Hilbert space Hkin := L2 (R9 ), satisfying the canonical commutation
relations [q̂ia , p̂bj ] = i δ ab δij . This permits us to quantize the constraints (1) and our aim
would be to find physical states |ψ⟩phys that solve them:P̂ a |ψ⟩phys =p̂ai |ψ⟩phys = 0 ,(26a)ϵabc q̂ib p̂ci |ψ⟩phys = 0 .

(26b)

X
i=A,B,CR̂a |ψ⟩phys=

X
i=A,B,CIn addition, we quantize the Hamiltonian (3) in a standard way, as
1 ˆ 2 ˆ 2 ˆ 2(p⃗A + p⃗B + p⃗C ) + V (|⃗qd
qd
qd
(27)
AB |, |⃗
AC |, |⃗BC |) .
2
The classical constraint algebra (2) of the Euclidean group directly attains a quantum
representation:Ĥtot =

[P̂ a , P̂ b ] = 0 ,[R̂a , R̂b ] = iϵabc R̂c ,[P̂ a , R̂b ] = −iϵabc P̂ c .

(28)

Notice that the three conditions (26b) are equivalent to a single (quadratic) constraint:⃗R̂ 2 |ψ⟩phys = 0 .

(29)

The shape of this algebra permits us to decompose the constraint imposition into convenient steps:
1. We construct the translation invariant Hilbert space HTI (see fig. 3) by firstly solving
the translation constraints P̂ a . Given that the latter have a continuous spectrum
around zero, their solutions will not be normalizable in Hkin so that HTI will not
actually be a proper subspace of it. A new translation invariant inner product,
normalizing translation invariant states will be required. This will be the topic ofsec. 3.2.1.
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

18

2. Since [P̂ a , R̂b ] = −iϵabc P̂ c , the angular momentum constraints R̂a will commute
with the translation generators P̂ a on HTI . Hence, the R̂a will leave HTI invariant
and we can consistently treat them as translation invariant observables on it. We
could therefore then simply impose (26b) on HTI . Given that the spectrum of theR̂a is discrete, solutions to them will be normalizable in HTI and so the translation
invariant inner product will also constitute the physical inner product. That is, thephysical Hilbert space Hphys (see fig. 3) of translation and rotation invariant states(26) will be a proper subspace of HTI .
Notice that the reverse ordering of these steps would be more cumbersome as thetranslation generators would a priori not leave a rotation invariant Hilbert space invariant.
Nevertheless, for our purposes it will be even more convenient to proceed slightly
differently. Our ultimate goal is to construct the transformation that switches the internal
perspective from HBC|A to HAB|C , i.e. from A to C perspective. To that end, it will
not actually be necessary to explicitly construct Hphys , although the transformation willswitch via Hphys from one perspective to another. While constructing Hphys is certainly
feasible, rotation invariant states of three particles will be unnecessarily convoluted andwe shall therefore abstain from explicitly exhibiting them.
Instead, we will sidestep the construction of Hphys by including an additional step
in-between steps 1 and 2 above. Indeed, we will insert a partial reduction step through amap TA,BC that ‘trivializes’ the translation generator constraints, pushing all redundancy
TI
onto particle A and yielding the Hilbert space HA,BCin fig. 3. This step, followed by asubsequent conditioning A ⟨⃗q = 0|, is the quantum analog of imposing the classical gauge
TI
condition ⃗qA = 0 and will yield the translation invariant two-body Hilbert space HBC|A,
see fig. 3. At this stage, we have a two-body problem of B and C, and imposing rotational
invariance will become substantially simpler. In this manner, we explicitly constructphys
in fig. 3, replacing step 2 above that would yield Hphys . We show, however,
HBC|Athat the imposition of the rotation constraints δ(R̂a ) commutes with the trivializationphyswill thereby be equivalent to Hphys . This procedure will be
map TA,BC and that HBC|A
detailed in secs. 3.2.2 and 3.2.3.
The remaining reduction steps to HBC|A in fig. 3 will be discussed in sec. 3.3.
3.2.1

Constructing the translation invariant Hilbert space

Given the linearity of the translation generators P̂ a , it is not difficult to construct thetranslation invariant Hilbert space HTI . As this step is already carried out in [1],16 weshall be brief here and refer to this reference for further details.
We invoke group averaging [13,62,63] to define an (improper) projector onto solutionsof (26a):17⃗
δ(P̂ ) : Hkin → HTI|ψ⟩kin 7→ |ψ⟩TI :=



1
(2π)3

Z +∞

a

dsx dsy dsz eis P̂a|ψ⟩kin .

(30)

−∞

16
In [1], this step is carried out for the one-dimensional three-body problem, however, extension to three
dimensions is trivial and amounts to simply ‘attaching a vector symbol ⃗ to the basic variables’.
17

See [64] for an alternative method that adapts the underlying Hilbert space topology.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

19

PBC|AClassical phase space after reduction to the perspective of A

Hkin

Kinematical (‘total’) Hilbert space

HTI

Translation-invariant Hilbert space

TI
HA,BC

Image of the prior under translational trivialization TA,BC

TI
HBC|A

Image of the prior under reduction by A ⟨χ = 0|

Hphys

Physical (i.e. translation and rotation-invariant) Hilbert space

phys
HA,BC

Image of the prior under transformation TA,BC

phys
HBC|A

Image of the prior under reduction by A ⟨χ = 0|

phys
HB,C|A

Image of the prior under the rotational trivialization RB,C

HBC|A

Can be defined as both:(i) Image of the prior under conditioning by ⟨θB , φB , φC = 0|(ii) Result of the quantization of PBC|A
Table 1: A summary of the notations used.

A general translation invariant state then takes any of the following forms|ψ⟩TI =

Z
Z

=
Z

=

TI
d3 p⃗B d3 p⃗C ψBC|A(⃗
pB , p⃗C ) |−⃗pB − p⃗C ⟩A |⃗pB ⟩B |⃗pC ⟩C
TI
d3 p⃗A d3 p⃗C ψAC|B(⃗
pA , p⃗C ) |⃗pA ⟩A |−⃗
pA − p⃗C ⟩B |⃗
pC ⟩C

(31)

TI
d3 p⃗A d3 p⃗B ψAB|C(⃗
pA , p⃗B ) |⃗pA ⟩A |⃗
pB ⟩B |−⃗
pA − p⃗B ⟩C ,depending on which particle’s momentum is solved for, where for later use we have defined
TI
ψBC|A
(⃗
pB , p⃗C ) := ψ kin (−⃗pB − p⃗C , p⃗B , p⃗C ) ,
TI
ψAC|B
(⃗
pA , p⃗C ) := ψ kin (⃗pA , −⃗
pA − p⃗C , p⃗C ) ,

(32)

TI
ψAB|C
(⃗
pA , p⃗B ) := ψ kin (⃗pA , p⃗B , −⃗pA − p⃗B ) .

This yields three different descriptions of the same translation invariant state |ψ⟩TI andwill be of use later, corresponding to different internal frame perspectives. The translation
invariant inner product, normalizing these states, reads⃗(ψ TI , ϕTI )TI := kin ⟨ψ| δ(P̂ ) |ϕ⟩kin ,(33)

where ⟨·|·⟩ denotes the inner product of Hkin . Based on these structures, one can Cauchy
complete the space of solutions of (26a) to a Hilbert space HTI .
The conjugate Dirac observables (in perspective-neutral form) (7) are, in particular,translation invariant. We can thus represent them already on HTI . We choose a symmetricAccepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

20

factor ordering:qb⃗q 2 ,

ρ̂BA = lnBAqb⃗q 2 ,

ρ̂CA = lnCA1p̂ρBA = (p̂⃗B · ⃗q̂BA + ⃗q̂BA · p̂⃗B ) ,2
1p̂ρCA = (p̂⃗C · ⃗q̂CA + ⃗q̂CA · p̂⃗C ) ,
2

dγ ,
b
(34)
û = −|⃗
qBA × ⃗qCA |−1 · ⃗q̂BA · ⃗q̂CA = −cot

1
b
b
p̂u =
+ u2 )−1/2 eρ̂CA −ρ̂BA ⃗qˆBA ) · p̂⃗C(−û (1+ u2 )−1 ⃗q̂CA − (12bb+ p̂⃗C · (−û (1+ u2 )−1 ⃗q̂CA − (1+ u2 )−1/2 eρ̂CA −ρ̂BA ⃗q̂BA )=1 ddγ eρ̂CA −ρ̂BA ⃗dγ ⃗(sin γ cosq̂ CA − sinq̂ BA ) · p̂⃗C2dγ cosdγ eρ̂CA −ρ̂BA ⃗dγ ⃗+p̂⃗C · (sinq̂ CA − sinq̂ BA ) .

Some of these operators have to be understood in terms of spectral decomposition. As
noted above, Hphys is a proper subspace of HTI so that these operators here already define a representation of the Dirac observables (7) on Hphys too. It is clear that some ofthese Dirac observable operators will be unbounded on states with support on total collisions. Since these pathological configurations are a set of measure zero [60], there is still achance that these operators may be densely defined and ultimately essentially self-adjoint.
However, we shall not enter into such subtleties here, as our focus is on symmetry reduction to a quantum frame perspective; our construction will thus be somewhat informal.
We also recall that we have dynamically ruled out total collisions through a potentialthat becomes infinitely repulsive there. The quantized translation and rotation invariant
Hamiltonian (27) is an observable on both HTI and Hphys . In particular, on Hphys (which
we do not construct explicitly), this will mean that, starting with finite-energy states that
do not have support on total collisions, will not dynamically produce states that do if
evolved with Ĥtot . On these dynamically accessible states, ρ̂BA , p̂ρBA , ρ̂CA , p̂ρCA and p̂uwill remain bounded and only û will be unbounded for states having support on collinear
configurations.
It must be emphasized that, clearly, one can promote other Dirac observables to selfadjoint operators on both HTI and Hphys . For example. the ‘basis set’ of Dirac observables(6) does not feature any pathologies and can be turned into self-adjoint operators, e.g.
through a symmetric factor ordering. Hence, the perspective-neutral quantum theory will
not have any issues per se. The pathologies are purely related to choosing observables
that (i) have a direct interpretation in the specific perspective of A and (ii) are canonicallyconjugate (where defined) so that later we can relate them to the reduced quantum theory
on HBC|A . As argued in sec. 2.1, better conjugate observables do not exist in this modeldue to the Gribov problem.
3.2.2

Translational reduction

As explained above, instead of directly continuing with the imposition of the rotation
constraints, we firstly include the step of translational reduction to effectively produce a
two-body problem that simplifies solving the (then transformed) constraints. This stepis the quantum analog of gauge fixing to ⃗qA = 0 in (9). Clearly, in the quantum theory
we can not gauge fix because imposing the translation constraints (30) directly produces

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

21

translation invariant states.18 In order to remove redundant degrees of freedom and reduce
the quantum theory, we thus have to proceed differently. As shown in [1], this works byfirstly trivializing the constraints and subsequently projecting onto the classical gauge
fixing conditions. Some of the below is a repetition of steps carried out in [1] and so weshall be brief.
TI
We define the constraint trivialization map, TA,BC : HTI → HA,BC, see fig. 3,
TA,BC =

Y





a a
exp i q̂A
(p̂B + p̂aC ) ,

(35)

a

which is unitary on Hkin , trivializes the translation generators into the A tensor factor†
TA,BC P̂ a TA,BC= p̂aA ,(36)

where † is defined with respect to Hkin , and maps translation invariant states toTI= |⃗p = 0⟩A ⊗|ψ⟩TI
A,BC = TA,BC |ψ⟩

Z



TI
d3 p⃗B d3 p⃗C ψBC|A(⃗
pB , p⃗C ) |⃗pB ⟩B |⃗
pC ⟩C . (37)Hence, the A tensor factor of the state carries no relevant information about the originalstate and has become redundant. We can remove it through a Page-Wootters-like [65]conditioning on the classical gauge-fixing conditions (9),19 producing states in a novel
TI
Hilbert space HBC|Ain fig. 3
3/2
q = 0 | ψ⟩TI
|ψ⟩TI
A ⟨⃗
A,BC =
BC|A := (2π)

Z

TI
d3 p⃗B d3 p⃗C ψBC|A(⃗
pB , p⃗C ) |⃗pB ⟩B |⃗
pC ⟩C .

(38)

TI
As shown in [1], this defines an isometry from HTI to HBC|A, where the inner product on
the latter is just the standard one with integration over B and C variables only.
Crucial for us is also how it transforms basic translation invariant operators†= p̂⃗i ,TA,BC p̂⃗i TA,BC†= ⃗q̂i ,TA,BC (⃗q̂i − ⃗q̂A ) TA,BC

i = B, C ,(39)

and the Hamiltonian (27), which becomes20
†
tot
d
d
ĤA,BC:= TA,BC Ĥ tot TA,BC= p̂⃗B 2 + p̂⃗C 2 + p̂⃗B · p̂⃗C + V (|⃗qB |, |⃗
qC |, |⃗qdBC |) .

(40)

The Dirac observables (34) simplify under transformation (39); we shall not spell out
their transformed form as it is clear that it amounts to simply dropping the A-label fromall expressions in (34).21 As one can easily check using (36, 39), this translation generator
trivialization map leaves the rotation generators (26b) invariant:†TA,BC R̂a TA,BC= R̂a .

(41)

18

As a consequence of the uncertainty relations, projecting onto the zero-eigenvectors of the total momentum leads to a maximal spread in gauge variables conjugate to it and thereby to translation invariance.
19

Equivalence of this reduction procedure with the Page-Wootters formalism has been demonstrated inlater work [66, 67].
20

TI
We have not written the terms which are null on HA,BC.

21

In some cases, this has to be justified by considering the operator functions as Taylor series away from
pathological configurations.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

22This implies that imposing (29) and applying the trivialization map TA,BC commutes on
HTI such that the square on the lower left of fig. 3 is indeed commutative. Noting theshape of (37), this leads to the following equivalencesR̂a |ψ⟩TI = 0

a
a
(R̂B
+ R̂C
) |ψ⟩TI
A,BC = 0

⇔

⇔

a
a
(R̂B
+ R̂C
) |ψ⟩TI
BC|A = 0 .

Hence, we lose no information by imposing two-body rotational invariance on states in
TI
HBC|A, rather than three-body rotational invariance in HTI , thereby simplifying the task.
Note how this means that we effectively turned the situation into a two-body problem.
Indeed, the use of the constraint trivialization map TA,BC mapped us to a description inwhich A’s degrees of freedom have become pure gauge, so that only B and C’s degrees of
freedom are potentially meaningful (although they still include some rotational gauge).
However, the “two bodies” here are not strictly speaking B and C; rather, they correspondto B and C’s positions relative to A. The intuition is that, as in [1], trivialising thetranslational gauge effectively amounts to eliminating one of the particles by moving toa description in which positions are defined relative to it.
3.2.3

Rotational invariance

TI
We continue with the two-body problem on HBC|A≃ L2 (R3 )⊗L2 (R3 ). In order to impose
rotational invariance, it will be more convenient to Fourier transform to configuration
space and to switch to polar coordinates (r, θ, ϕ), so we can use spherical harmonics
Y j,m (θ, φ) to define a basis for L2 (R3 )|r; j, m⟩ =ZdΩ Y j,m (θ, φ) |r, θ, φ⟩ ,(42)

where r = |⃗q |. Here, |j, m⟩ are the usual simultaneous eigenstates of the angular momentum operators R̂2 and R̂z . This construction is done more carefully in Appendix
B.
According to the Clebsch-Gordan coefficients for the decomposition of angular momentum eigenstates, we can then write an arbitrary zero-total-angular-momentum state
TI
in HBC|Aas:|ψ⟩phys
BC|A =Z2 2
drB drC rB
rC

∞
X
phys

ψBC|A (rB , rC ; j) |Φ(rB , rC ; j)⟩ ,

(43)

j=0

where
|Φ(rB , rC ; j)⟩ =X (−1)j−m

√

|m|≤j

2j + 1|rB ; j, −m⟩B |rC ; j, +m⟩C .

(44)

phys
TI
Notice that these states lie in a proper subspace of HBC|A, which we will label by HBC|A(see fig. 3), given that it now also includes rotational invariance. The transformationsof the Dirac observables (34) alluded to in sec. 3.2.2 (i.e., those in (34) with all A-labels
phys
dropped), are also observables on HBC|A. From the discussion above it follows that thisphys
Hilbert space is equivalent to H
, including the evaluation of the (transformed) Diracobservables.

3.3 Rotational reduction to A’s perspective
We have implemented translational invariance, a subsequent translational reduction androtational invariance. There is therefore still redundancy in the description which we have
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

23

to remove in order to fully construct A’s quantum frame perspective from the perspectiveneutral Dirac quantized theory and to compare the result to the reduced quantum theory
on HBC|A of sec. 3.1, In particular, we now also have to carry out a rotational reduction,⃗ = 0 in (10–12) thatwhich will be the quantum analog of the classical gauge fixing ϕfixes the rotational gauge freedom for generic configurations. Since we cannot directly
gauge fix in the Dirac quantized theory (the system is already translation and rotation
invariant after imposition of the quantum constraints), we have to proceed differently.
As explained through the translational reduction in sec. 3.2.2 (and in [1]), the quantum
analog is (1) trivialize the relevant constraints, and (2) condition on the classical gauge
fixing conditions. We will now show how this works for the rotation generators.
3.3.1

Rotational reduction

We define a rotation generator trivialization map












y
z
z
RB,C = exp i φ̂B R̂C
exp i θ̂B R̂C
exp i φ̂B R̂C,(45)

where the angular operators θ̂ and φ̂ can be defined in terms of the states |r, θ, φ⟩, see
Appendix B.3. As shown in Appendix C.1, geometrically, this sequence of rotationsof particle C effectively rotates its polar coordinate system until its polar angle θC is
measured relative to the direction of B. Hence, θC will coincide with the relative angle γ
between B and C (essentially the Dirac observable u in (7)), see also figs. 1a and 2. This
procedure is thus reminiscent of the one followed classically. Appendix C.1 also provesthat RB,C maps |ψ⟩phys
BC|A as defined in (43) tophys|ψ⟩phys
B,C|A = RB,C |ψ⟩BC|A =Z

2 2
drB drC rB
rC

∞
X

phys
(rB , rC ; j) |rB ; 0, 0⟩B |rC ; j, 0⟩C ,(−1)j ψBC|A

j=0

(46)
i.e. to a state of zero total angular momentum in B and zero angular momentum around
phys
phys
z in C, and that this is an isometry from HBC|A
to a new Hilbert subspace HB,C|A
:=
phys
phys
TI
is spanned by states of the) ⊂ HBC|A(see fig. 3). In other words, HB,C|A
RB,C (HBC|A
form |rB ; 0, 0⟩B |rC ; j, 0⟩C .
As can be expected from the shape of (46) and shown in Appendix C.2, RB,C indeed⃗⃗
trivializes the rotation constraints with RB,C (R̂B + R̂C )2 R−1
B,C being equivalent to

⃗
phys
z
(R̂B )2 |ψ⟩phys
B,C|A = R̂C |ψ⟩B,C|A = 0 .

(47)

It is also necessary to check how the Dirac observables (34) transform under RB,C .
To this end, recall that we have carried out a translational reduction in sec. 3.2.2, whichcorresponded to simply dropping the A labels from all expressions in (34). In Appendix
C.3, we show that (these transformations are only valid on states without support on

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

24

pathological configurations):22
RB,C ρ̂B R−1
B,C= ρ̂B ,

ρ
RB,C p̂ρB R−1
B,C = p̂B ,RB,C ρ̂C R−1
B,C= ρ̂C ,

ρ
RB,C p̂ρC R−1
B,C = p̂C ,RB,C ûC R−1
B,C

dθC ,
= −cot1 d d ⃗ ⃗dddθC sin θC q̂ C · p̂C + r̂C sin
θC p̂zC − p̂⃗C · ⃗q̂C cos= − −cosθC sinθC2d+p̂zC r̂C sinθC .

RB,C p̂uC R−1
B,C

(48)

The key point of those transformations is that γb gets mapped to θ̂C :RB,C γb R−1
B,C = θ̂C .

(49)

This is in harmony with the above observation that the transformation RB,C has theproperty of transforming the description of particle C so that its polar angle now physicallycorresponds to the relative angle that it forms with the direction of B. Under thesetransformations, the Hamiltonian (40) is mapped to23
tot
tot
ĤB,C|A:= RB,C ĤA,BC
R−1
B,C



3



b
= e−2ρ̂B p̂ρB 2 + e−2ρ̂C p̂ρC 2 + (e−2ρ̂B + e−2ρ̂C )(1 + û2C )2 + e−ρ̂B −ρ̂C ûC (1+ u2C ) 2 p̂uC 2− e−ρ̂B −ρ̂C
−i

!

q

b
√ u p̂ρ p̂ρ
1 + u2 (p̂u p̂ρ + p̂u p̂ρ ) + b
C

C B

C C

c

1+u2C B C

+ 2ie−2ρ̂B p̂ρB + 2ie−2ρ̂C p̂ρC!

q4(ûC + û3C )(e−2ρ̂B + e−2ρ̂C ) + 2e−ρ̂B −ρ̂C (1 + 2û2C )b
1 + u2 p̂u
C

C

q
9
9
b
− (e−2ρ̂B + e−2ρ̂C )(1 + û2C ) − e−ρ̂B −ρ̂C ûC 1 + u2C + V (ρ̂B , ρ̂C , ûC ) .
4
4

(50)
We now wish to complete the rotational reduction, removing – in analogy to thetranslational reduction in sec. 3.2.2 (and in [1]) – any redundant information from ourstates (46) and recovering the reduced quantum theory on HBC|A in A perspective fromsec. 3.1. This second step will, again, be achieved by a conditioning on the classical gauge
fixing conditions and a variable change.
To this end, it is convenient to rewrite the states (46) in polar coordinates as follows
Z|Ψ⟩phys
B,C|A =phys22
drB rBdΩB drC rC
dΩC ψBC|A(rB , rC , θC ) |rB , θB , φB ⟩B |rC , θC , φC ⟩C ,(51)
where, using the ingredients of Appendix C.1, we have defined√∞X
2j + 1phys
j physψBC|A (rB , rC , θC ) :=(−1) ψBC|A (rB , rC ; j)Pj (cos θC ) ,
4π
j=0

(52)

22

Notice that these observables are, in fact, somewhat better behaved as those in (34) because theoperator θC appearing here is fully defined on all states, thanks to the conditions in Appendix B.3 which
we had not applied similarly to γ appearing in (34).
phys
We have not written the terms which are null on HB,C|A. Furthermore, for brevity we have chosen afactor ordering where the momenta always stand on the right. Despite its appearance, this Hamiltonianis actually a symmetric operator.
23

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

25

and Pj is the Legendre polynomial of degree j. As a result, we now have a wave functionof the polar coordinates and we see that it depends only on the physically meaningfulrB , rC , θC that also survived the classical gauge fixing (see fig. 2). In contrast, the threeother configuration degrees of freedom θB , φB , φC are pure gauge, and thus the distribution of any |ψ⟩phys
B,C|A is uniform with respect to each of them; these degrees of freedom
are hence now redundant and we can remove them without loss of information. This iscompletely analogous to the translational case in sec. 3.2.2 and we shall now condition on⃗ = 0 (in polar coordinates) in (9–11)
the classical gauge fixing conditions ϕ|ψ⟩BC|A := ⟨θB = 0, φB = 0, φC = 0 | ψ⟩phys
B,C|A
Z

=

phys
2
2
drB rBdrC rCdθC sin θC ψBC|A(rB , rC , θC ) |rB ⟩ |rC ⟩ |θC ⟩ , (53)(where we have made use of the normalization (101) in Appendix B). This is an isometryphysbecause, as one can easily check, phys
BC|A ⟨ϕ|ψ⟩BC|A ≡ BC|A ⟨ϕ|ψ⟩BC|A . However, we recall
that the gauge fixing conditions θB = 0, φB = 0, φC = 0 are not globally valid. While thisis a problem for distributional states with support on pathological configurations, this is
not one for generic Hilbert space states since the troublesome configurations comprise aset of measure zero.
We shall now show that the resulting symmetry reduced quantum state lies in HBC|A ,i.e. in the Hilbert space corresponding to directly quantizing A’s classical perspective. To
this end, we switch variables (rB , rC , θC ) 7→ (ρB , ρC , uC ) in line with sec. 3.1, producing|ψ⟩BC|A =Z ∞−∞dρB dρC duC e3ρB e3ρC1  32
1 + u2C

phys(ρB , ρC , uC ) |rB (ρB )⟩ |rC (ρC )⟩ |θC (uC )⟩ .
× ψBC|A(54)

Due to (101) in Appendix B, we have the normalization′ )δ(rB − rB
,
2
rB
′ )
δ(θC − θC,sin θC′⟨rB |rB⟩ =′⟨θC |θC⟩ =′⟨rC |rC⟩=′ )δ(rC − rC
,
2
rC

(55)

which differs from that in (24). Hence, redefining
ψBC|A (ρB , ρC , uC ) :=e3ρB e3ρC1  32
1 + u2C

! 12
phys
ψBC|A
(ρB , ρC , uC ) ,(56)|rB (ρB )⟩ |rC (ρC )⟩ |θC (uC )⟩ ,

(57)

and
|ρB ⟩ |ρC ⟩ |uC ⟩ :=e3ρB 3ρCe

1  32
1 + u2C! 21

we recover exactly the shape of the reduced states (24) with correct normalization|ψ⟩BC|A =Z ∞−∞dρB dρC duC ψBC|A (ρB , ρC , uC ) |ρB ⟩ |ρC ⟩ |uC ⟩ .

(58)

This redefinition is not a surprise because the Dirac quantization brought us to a kinematiQ
cal Hilbert space with measure dµ = i d3 ⃗qi , from which the measure in polar coordinates
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

26

phys
on HB,C|A
is directly inherited, whereas the reduced quantization started with a measuredµ = dρB dρC duC on HBC|A .
We need to check whether also the basic observables generating the commutator algebra behave correctly. It is clear that the translationally and rotationally reduced Diracobservables (48) are unaffected by the projection (53). The variable shift (rB , rC , θC ) 7→
(ρB , ρC , uC ) could have been carried out prior to the projection. However, this shift willaffect the precise representation of these observables. Indeed, the configuration Diracphysdobservables on HB,C|A, ρ̂B , ρ̂C and RB,C ûC R̂−1
B,C = −cot θC , will act simply as multiplication operators both before and after the projection and regardless of the wave
function redefinition (56). They just directly become the observables of the reduced theory. On the other hand, as can be checked, the momentum Dirac observables, p̂ρB , p̂ρC
phys
and RB,C p̂uC R−1
B,C are represented in terms of derivatives on wave function ψBC|A in (54)as:243p̂ρB = −i∂ρB − i ,
(59a)
2
3
(59b)
p̂ρC = −i∂ρC − i ,
2
3 uC
.
(59c)
RB,C p̂uC R−1
B,C = −i∂uC + i
2 1 + u2C

It is straightforward to prove that, due to the redefinition of the wave function normalization, this is equivalent to the following action on ψBC|A in (58)3ρB 3ρCe

e3ρB e3ρCee3ρB e3ρC1  23
1 + u2C

1  23
1 + u2C! 12
3ρB 3ρCe

e3ρB e3ρC3ρB 3ρC
RB,C p̂uC R̂−1
e
B,C e



1  32
1 + u2C

p̂ρB

e

! 12

p̂ρC

! 12

1  23
1 + u2C
1  23
1 + u2C
1  23
1 + u2C!− 12= −i∂ρB ,!− 12= −i∂ρC ,!− 12= −i∂uC .

We have thereby proved that our basic Dirac observables (34) ultimately transform correctly under the various reduction maps to the observables π̂µ of the reduced quantizedtheory in A perspective. In particular, given that we have carried out a sequence ofisometries, the expectation values of these observables will be equivalent to those of the
reduced theory in sec. 3.1.
Crucially, notice that the wave function redefinition in (56) takes care of the fact that,since total collisions and collinearity happen as ρB , ρC → −∞ and uC → ±∞, statesin the reduced theory of sec. 3.1 have a vanishing support there, while the Dirac theory
does admit states with support on such configurations. Indeed, the rescaling factor in(56) vanishes on approach of such pathological configurations and thus extinguishes any
support physical states might have there.
This completes our reduction procedure in the quantum theory (see fig. 3 for a summary), the result of which we take as A’s quantum frame perspective.
24
This ultimately follows from their representation in terms of Cartesian coordinates ⃗qi on Hkin in thestandard measure of sec. 3.2.1 and the subsequent transformations.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

27

3.3.2

Relation with reduced quantization of A’s classical perspective

Our complete quantum reduction map, Φ′A : HTI → HBC|A , given byaaΦ′A := ⟨θB = 0, φB = 0, φC = 0| RB,C δ(R̂B
+ R̂C
) ⟨⃗
χA = 0| TA,BC(60)

from the translation invariant Hilbert space HTI to what we identify as A’s quantum
frame perspective thus yields the Hilbert space HBC|A and the basic operators of the
reduced quantized theory of sec. 3.1. We recall that this procedure is equivalent to thequantum symmetry reduction, i.e. the ‘quantum coordinate map’ ΦA : Hphys → HBC|A ,ΦA := ⟨θB = 0, φB = 0, φC = 0| RB,C ⟨⃗χA = 0| TA,BC(61)

from the perspective-neutral Hilbert space to HBC|A (see fig. 3). We also emphasize that
ΦA will in fact be invertible on its domain in Hphys despite the appearance of conditionings
in its construction. A conditioning such as ⟨⃗χA = 0| would indeed not be invertible on thekinkinematical Hilbert space H
as this operation would project away non-trivial and ingeneral independent information about A. Thanks to the redundancy in the descriptionof Hphys (in terms of kinematical variables) though, a conditioning such as ⟨⃗χA = 0| only
removes redundant information from an invariant state. For example, the factor |⃗pA = 0⟩in (37) carries no more independent information. This operation will thus be invertible forphysical states. The analogous argument holds for the other conditioning in ΦA . Hence,
ΦA will be invertible on its domain and this is crucial for transforming observables from
Hphys to HBC|A which requires a conjugation with this map. We emphasize that ΦAprovides a general method for comparing Dirac with reduced quantization.
At first sight it thus seems as though ΦA just recovered the direct quantization of
A’s classical frame perspective from the perspective-neutral quantum theory on Hphys .
However, this is not the case because composite operators constructed from the elementary
ones in Dirac quantization do not in general map under ΦA to the corresponding composite
operators constructed on HBC|A in reduced quantization. In particular, as we explain
in appendix D, the Hamiltonians of the two quantizations are inequivalent; the total
Hamiltonian Ĥtot in (27), which recall is an observable on Hphys , does not reduce underredΦA to the Hamiltonian (25) of reduced quantization, i.e. ΦA Ĥtot Φ−1
A ̸= ĤBC|A , whereΦA Ĥtot Φ−1
A is simply equal to expression (50) with the p̂’s replaced by π̂’s. The difference
between the two only involves configuration variables. Both the dynamics and spectra
will thus differ.
This is in line with previous reports in the literature that the Hamiltonians in Diracand reduced quantization are typically inequivalent, e.g. see [45,52,55,56,68–70]. This discrepancy can ultimately be traced back to an ambiguity in the ordering of gauge-invariant
operators in Dirac quantization, arising due to the contribution of the volume of gauge
orbits to the Hilbert space measure [53]. As demonstrated in [53, 68], for systems withconstraints linear in momenta as in our model, one can always modify Dirac quantization through factor ordering in such a way that the Hamiltonians will agree. While thisyields a well-defined procedure, it is somewhat artificial from the point of view of Diracquantization. We will briefly illustrate this procedure in appendix D and show that it
leads to a Hamiltonian on Hphys that in fact no longer treats all degrees of freedom on
an equal footing, but is special to a frame choice. Hence, in order to ensure equivalence
between Dirac and reduced quantization, one will have to modify Dirac quantization according to the reduced theory one is interested in and different frame choices will therefore
lead to inequivalent modifications; the reduced quantizations of different classical frame
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

28

perspectives are unitarily inequivalent. This is untenable in light of intending to construct a notion of quantum frame covariance. Instead, we propose to embrace the generalinequivalence of Dirac and reduced quantization and accept its consequences.
The conclusion we draw from this is that it is generally not equivalent to construct
the quantum frame perspective of A from the perspective-neutral quantum theory, or to
directly quantize A’s classical frame perspective. So which version should one identify as
A’s quantum frame perspective? Our proposal is to give primacy to the more fundamental perspective-neutral structure in both the classical and quantum theory from which
the internal perspectives are derived. In the context of quantum reference frames, thismeans giving Dirac quantization primacy over the reduced quantization method. We
therefore identify the quantum symmetry reduced theory obtained from the perspectiveneutral structure via ΦA as A’s internal quantum frame perspective and not the direct
quantization of A’s classical frame perspective.
Our reasoning for this is as follows: Dirac quantization treats all degrees of freedom,including whatever we may choose as a reference frame, on an equal footing and in particular translates the classical redundancy in the description of physics into the quantum
theory. This redundancy is the prerequisite for being able to describe the same physicalsituation in many different ways and therefore for establishing a notion of quantum frame
covariance. Such a covariance is a necessity in the presence of many physically equally
good choices of quantum reference frames within a given problem, and in reduced quantization one generally lacks the linking structure between the quantizations of the different
classical frame perspectives which, as mentioned, are unitarily inequivalent. By contrast,
in Dirac quantization one obtains a perspective-neutral structure by construction whichlinks the different possible internal frame perspectives in the form of quantum coordinatetransformations, as we shall illustrate in the next section, extending [1]. This is the natural quantum analog of classical frame changes through coordinate transformations and inline with quantum reference frame transformations being merely changes of description.
In short, quantum covariance is in our view the reason one should give Dirac quantization
precedence.
This upgrades the arguments for preferring Dirac over reduced quantization usuallyfound in the literature according to which the former should be considered more generalbecause in the latter the reference system has already been removed prior to quantization such that it can never feature any quantum fluctuations. While it is correct thatthe reference frame is absent from reduced quantization, this reasoning is somewhat misleading as in Dirac quantization too the reference frame no longer features independent
quantum fluctuations after its degrees of freedom have been selected as the redundantones. This is ultimately the reason why in the quantum symmetry reduced theory theframe degrees of freedom too are absent just like in reduced quantization. However, what
our framework underlines is that, while the reference system does not feature them in
its own perspective, it can feature independent quantum fluctuations in the perspectiveof a different quantum reference frame, while maintaining a link between the different
perspectives through the perspective-neutral Hilbert space. We will illustrate this next.
In summary, although experiments will ultimately have to decide which quantization
method is the appropriate one,25 our conceptual proposal in the context of quantum
reference frames is to retain only the Dirac quantized theory (and the various symmetry
reduced theories obtained from it), while discarding the reduced quantizations as unitarily
25

This requires better understanding measurements in the context of quantum frames, e.g. by building
up on [71].

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

29

inequivalent theories altogether.26

3.4 Transformations between relative states
In this section, we give the transformation of the quantum state when changing from theperspective of A to the perspective of C, expanding the framework of [1]. It is clear that,starting from the perspective-neutral theory, we could equally well have chosen to go into
the perspective of system C, and that this can be achieved by following the same steps
explained previously, but swapping all A and C labels. Once in a specific perspective,
we can transform all the way back to the physical Hilbert space Hphys , exploiting theinvertibility of ΦA (cf. sec. 3.3.2), as shown in the following diagram:
−1
TA,BC

Hphys
TC,AB

phys
HA,BC

phys
HC,AB

|⃗
p=0⟩A ⊗(·)phys
BC|A
phys
HBC|A

C ⟨χ=0|
T
SA→C

R−1
B,C

RB,A

phys
HB,C|Aphys
HB,A|C|jA =0;mB =0,mC =0⟩⊗(·)BC|A

HBC|A

phys
HAB|C

⟨θB ,φB ,φA =0|
SA→C

HBA|C

where (·)BC|A means inserting the input state from HBC|A into the brackets, upon which
the new tensor factor equipping the arrow is appended to the input state. The quantum
reference frame transformation from A’s to C’s perspective thus takes the form of aquantum coordinate transformation, SA→C : HBC|A → HAB|C ,SA→C := ΦC ◦ Φ−1A(62)

which passes through the perspective-neutral Hilbert space Hphys . Note, however, thatin practice we never need to explicitly go back to Hphys , which we also never fully constructed. Instead, we can sidestep it, as we now explain.
phys
It is worth stressing that, in order to transform from HBC|A to HB,C|A, one needs
1 R√
to tensor by |jA = 0; mB = 0, mC = 0⟩ ⊗ (·)BC|A = 2 2π dΩB dϕC |θB ; ϕB ; ϕC ⟩ ⊗ (·)BC|A ,which corresponds to averaging over the classical gauge conditions. This operation inverts
the conditioning on the classical gauge fixing conditions, and restores the gauge invarianceof the model.
26

In particular, it is not a problem that the acts of (i) reducing perspective-neutral states via theinstantaneous map (61) (cf. sec. 3.3.1) to ones on HBC|A (the Hilbert space of both reduced quantizationand symmetry reducing the Dirac quantized theory), and (ii) time evolving with either Ĥtot on Hphys
red
or HBC|A
on HBC|A do not commute, as a result of the two Hamiltonians being unitarily inequivalent.
This is just a manifestation of the dynamical inequivalence of Dirac and reduced quantization and wehenceforth only work with the former. Indeed, in our proposal, the correct Hamiltonian on HBC|A isred
the quantum symmetry reduced one ΦA Ĥtot Φ−1
A , and (i) and (ii) do commute when replacing HBC|A
with it. The quantum frame perspectives are thus dynamically consistent with the perspective-neutral
descriptions.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

30

Informally, the steps from HBC|A to HAB|C can be written as
T
SA→C = ⟨θB = 0; ϕB = 0; ϕC = 0| RB,C SA→C
R−1
B,A |jA = 0; mB = 0, mC = 0⟩ ⊗ (·)BC|A .
(63)
phys
phys
T
Here, SA→C
: HBC|A→ HAB|Cis the three-dimensional analog of the relative statetransformation in the translational case of [1, 2], which provides a shortcut, sidestepping
Hphys , in the above diagram and mapping directly between the Hilbert spaces that havebeen reduced with respect to translation invariance only:
T
SA→C:= PCA exp(i ⃗q̂C · p̂⃗B ) ,(64)

where PCA is now a three-dimensional parity swap:
PCA |⃗p⟩A = |−⃗
p⟩C .

(65)

Explicitly evaluated on some arbitrary initial state in HBC|A|ψ⟩BC|A =Z

2 2
drB drC dθC sin(θC )rB
rC ψBC|A (rB , rC , θC ) |rB ⟩B |rC ⟩C |θC ⟩C(66)

this yields |ψ⟩BA|C = SA→C |ψ⟩BC|A , where|ψ⟩BA|C =Z

2 2
drA drB dθA sin(θA )rBrA ψBC|A (dAB , rA , γB|C ) |rA ⟩A |rB ⟩B |θA ⟩A ,(67)

and where dAB is the distance between A and B and γB|C is the old angle θC , both
expressed in terms of the variables relative to C as
dAB =

q

2 + r 2 − 2r r cos θ ,
rA
A B
A
B

cos γB|C = q

rA − rB cos θA
2 + r 2 − 2r r cos θ
rA
A B
A
B

.

(68)
(69)

This is a result which has to be expected geometrically. It corresponds to replacing therelative distances between A and B by that between C and B and having now the angle
θA as the one between A and B as seen from C.
It is easy to show that, similarly to what happens in the one-dimensional case exploredin [2] and [1], entanglement and superposition are QRF-dependent features. Here, we
illustrate this operational consequence of the framework on the angular degrees of freedom,
whose treatment is a novel aspect of this work. For the purpose of illustrating thiseffect, it is enough to consider, in the initial QRF of A, a (non-normalised) separable
state consisting in a superposition of two angles, and sharply localised in the two radial
components of B and C. For example, assuming θi ̸= 0 and ri ̸= 0, i = 1, 2, so that weare not dealing with total collisions or collinearity,1|ψ⟩BC|A = r12 r22 |r1 ⟩B |r2 ⟩C √ (sin θ1 |θ1 ⟩C + sin θ2 |θ2 ⟩C ) .
2(70)

From the perspective of C, it is straightforward to show that, via Eq. (67)1|ψ⟩AB|C = r1 r22 |r2 ⟩A √ (q1 cos θ1 |q1 ⟩B |ϕ1 ⟩A + q2 cos θ2 |q2 ⟩B |ϕ2 ⟩A ) ,2

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

(71)

31

where, for i = 1, 2, we haveqi =q

r12 + r22 − 2r1 r2 cos θicos ϕi = q

r2 − r1 cos θi

.

(72)

r12 + r22 − 2r1 r2 cos θi

From the previous expression, it is clear that entanglement is QRF-dependent, thus generalising the result of [2] and [1] to the three-dimensional case.

3.5 Remarks on the unitarity of the quantum frame transformations
Let us now come to the important question of whether our quantum frame transformationsin (63) are unitary. We will be somewhat informal in our answer, in line with the previous
discussion, as a rigorous statement would require substantially more efforts that go beyond
the scope of this manuscript. Our answer has two parts, one pertaining to proper Hilbert
space states, the other to distributional states.
For proper Hilbert space elements, the transformation will, informally at least, beTunitary in the sense of being both invertible and an isometry. Firstly, the operator SA→Cin (64) is clearly unitary (see also [1, 2]). The more subtle part are the transformations RB,C and RB,A and the conditioning on the classical gauge fixing condition, suchas ⟨θB = 0; ϕB = 0; ϕC = 0|, which is not globally defined on the classical configuration
space, owing to the mechanical analog of the Gribov problem. However, as noted before,
the pathological configurations – total collisions and collinearity – comprise a set of measure zero in the gauge invariant configuration space [60]. It follows from appendix C.1
that RB,C is invertible and isometric on its domain and, given that the pathological
phys
such
configurations are a measure zero set, this should be a dense subspace of HBC|A
that it is unitary (similarly with A, C interchanged). The above conditioning, on theother hand, is well-defined on states of the form |jA = 0; mB = 0, mC = 0⟩ ⊗ |ψ⟩BC|A ,phys(cf. the expressions in equations (51,52)), where
which comprise the Hilbert space HB,C|A
1 R√|jA = 0; mB = 0, mC = 0⟩ = 2 2π dΩB dϕC |θB ; ϕB ; ϕC ⟩. As observed in secs. 3.3.1 and3.3.2, the conditioning is an isometry and invertible, as it only removes redundant information; indeed, as discussed above, the inverse operation is given by appending the tensor
factor |jA = 0; mB = 0, mC = 0⟩ ⊗ (·)BC|A . We therefore conclude that, for Hilbert spaceelements, the quantum frame transformation SA→C in (63) is unitary. While we haveargued here for three particles, the same will hold for arbitrary N , as the pathologicalconfigurations always constitute a measure zero set in the gauge-invariant configuration
space.
What about distributional states with support on the classically pathological configurations? For those states, the quantum frame transformation SA→C in (63) is not defined,
in particular, because the rotational trivializations RB,C and RB,A are not defined (cf.
appendix C.1). Note, however, that the classical gauge fixing conditions associated with
A’s perspective will fail if and only if the gauge fixing conditions of C’s (and B’s) perspective fail; for three particles collinearity and total collision mean the failure of either
internal perspective, see fig. 2. All the valid configurations of A’s perspective can therefore be mapped into valid configurations in C’s perspective and vice versa; in this sense
the quantum frame transformations SA→C is invertible for distributional states for threeparticles too.
The situation for distributional states will, however, be different in general with N > 3.
In that case, it will generally no longer be true that the gauge associated with one frame’s

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

32

perspective fails if and only if the gauge associated with the second frame’s perspective
fails. The two frames may use non-overlapping subsets of particles to set up there local
orientation relationally. When the particles used by the first frame are collinear or collide, the particles employed by the second frame may be in a generic configuration. In
such a case, the transformation between the first and second frame will only be definedon those distributional states which correspond to global particle configurations that are
not pathological for both frame perspectives. This means that, in those cases, the quantum reference frame transformation will only be invertible on a subset of distributional
states that are valid in either perspective. The absence of globally valid classical frame
perspectives thus only plays a significant role for distributional states.

4 Conclusions
In this article, we have expanded on the systematic approach to quantum reference frametransformations started in [1], by applying it to the relational N -body problem in threedimensional space with translational and rotational invariance. As discussed in detail,this model is particularly interesting because it is subject to a mechanical analog of the
Gribov problem in gauge field theories and thereby does not permit globally valid reference
frame perspectives. Since this is a property of generic systems with gauge symmetry, thisexample serves as an illustration how our approach can also handle challenges that arise
in generic situations.
In particular, the N -body problem features a number of technical challenges – all ofwhich are a consequence of the compactness of the rotational gauge orbits –, such as aconstraint surface that is foliated by gauge orbits of different dimension and an absence of
globally valid pairs of canonically conjugate Dirac observables. As a result, our descriptions relative to specific frame perspectives are not valid on all states; in particular, theinvariant observables which we choose to describe the physics relative to specific choicesof frames become unbounded on collisional or collinear configurations (a set of measurezero). But given the absence of global perspectives, this is inevitable and will appear, in
one form or another, in most interesting systems. These challenges notwithstanding, we
demonstrate how one can use the quantum reduction method, consisting of a constraint
trivialization and a conditioning on the classical gauge fixing conditions, to systematicallyconstruct transformations between different quantum reference frame perspectives in thethree-body problem. In particular, for Hilbert space states, these transformations areunitary, despite the absence of classically global internal perspective. This is because thepathological configurations comprise a set of measure zero in the gauge-invariant configuration space. For distributional states, on the other hand, this transformation will not be
invertible in general situations with N > 3. We have also shown that the transformations
lead to a quantum frame dependence of entanglement. This work thereby generalizes theapproach to constructing quantum reference frame transformations of both [1, 2], which
only considered one-dimensional particle models, to a three-dimensional setting including
rotations.
This article also substantiates our conceptual argumentation in [1]. Specifically, itshows that also in this more general setting, and despite the absence of global relational
perspectives, our interpretation of the classical constraint surface and the Dirac quantized physical Hilbert space as perspective-neutral structures remains valid. Similarly,
our interpretation of (specifically) gauge-fixed reduced phase spaces and quantum symmetry reduced Hilbert spaces as the physics seen from a particular frame perspective is
supported by the N -body problem. In particular, our work also sheds novel light on
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

33

the relation between Dirac and reduced quantization from the perspective of quantum
reference frames. As we have seen, quantizing a classical frame perspective is generally
inequivalent to constructing the quantum frame perspective by symmetry reducing theperspective-neutral structure obtained through Dirac quantization. The general inequivalence of Dirac and reduced quantization [17, 18, 45–56] can be phrased more physicallyas ‘frame associated symmetry reduction and quantization don’t commute’. Based on
quantum frame covariance, we have argued that precedence should be given to Diracquantization and hence the perspective-neutral structure from which internal frame perspectives are derived.
As an outlook, our method is further developed by applying it to quantum cosmologicalmodels in [72, 73] to demonstrate how one can also witch temporal reference systems, i.e.
relational clocks, in a quantum theory. This, in fact, also inspires a new perspective on
the ‘wave function of the universe’. As will be exhibited in [74], our frame transformation
method is actually equivalent to that originally developed in [14–16], provided one restricts
to a semiclassical regime within which the latter was constructed. Finally, an extension of
the original quantum reference frame approach of [2] to relativistic systems with internal
spin degrees of freedom is presented in [75] together with an analysis of the operational
consequences of such extended quantum frame transformations.

Acknowledgments
It is a pleasure to thank Časlav Brukner and Esteban Castro-Ruiz for many discussions
on quantum reference frames. PH is grateful to Tim Koslowski for originally proposing
the model employed in this article and many discussions on the N -body problem. PH
also thanks Flavio Mercati for correspondence on Dirac observables in the three-bodyproblem. The project leading to this publication has initially received funding from theEuropean Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 657661 (awarded to PH). PH also acknowledges
support through a Vienna Center for Quantum Science and Technology Fellowship. F.G.
acknowledges support from the John Templeton Foundation, Project 60609, “Quantum
Causal Structures”, from the research platform “Testing Quantum and Gravity Interface
with Single Photons” (TURIS), the Austrian Science Fund (FWF) through the projectI-2526-N27, the doctoral program “Complex Quantum Systems” (CoQuS) under Project
W1210-N25, and from Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department
of Innovation, Science and Economic Development and by the Province of Ontario throughthe Ministry of Colleges and Universities. This publication was made possible throughthe support of a grant from the John Templeton Foundation. The opinions expressed inthis publication are those of the authors and do not necessarily reflect the views of theJohn Templeton Foundation.

A The rotation and translation invariant N -body problem
A.1 The rotation and translation invariant Lagrangian
We shall take the N particles to be of unit mass and the configuration manifold as
Q = R3N . The tangent bundle is therefore T Q ≃ R6N . We define the positions relative

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

34

to the center of mass (which are translation invariant),




N
1 X
a
i
= qai −
q0i
= qai − qCM
qi  ,
N j=1 a(73)

a . We also define the total angular momentum,
as well as their derivatives q̇0i

⃗ =
L

N
X

⃗q0i × ⃗q˙0i ,

(74)

i=1

and the moment of inertia 3 × 3 matrix,
M ab =

NX

a b|⃗q0i |2 δ ab − q0i
q0i .

(75)

i=1

We then define the Lagrangian as27
L =

N


1 X
⃗ · M −1 · L⃗ − V {|⃗qi − ⃗qj |}N⃗q̇0i 2 − 1 L
i,j=1
2 i=1
|2
{z
}
rot
Ekin

=

N
1 X
⃗q̇i 2 − 1
2 i=1
2N

|

N
X

!2

⃗q̇i

−

i=1

{z

cm
Ekin

}



1⃗
⃗ − V {|⃗qi − ⃗qj |}N
L · M −1 · L
i,j=1 , (76)
|2
{z
}
rot
Ekin

where V is invariant under translations and rotations, and thus depends only on thecm
absolute distances between the particles. We have subtracted the kinetic energy Ekinof the center of mass, so that only the motion relative to the latter contributes to therot , so that the global state of rotation of theenergy; and the rotational kinetic energy Ekinsystem does not contribute to the energy. In consequence, this Lagrangian is singular andfeatures gauge symmetries. It is easy to see that L is invariant under global translations
since ⃗q0i is translation invariant,(qia , q̇ia ) 7→ (qia + f a (t), q̇ia + f˙a (t)),(77)

where the f a (t) are arbitrary functions of time that do not depend on particle i. In
addition, L is invariant under rotations, that is(⃗qi , ⃗q˙i ) 7→ (R(t) · ⃗qi , R(t) · ⃗q˙i + Ṙ(t) · ⃗qi ),(78)

where R(t) is a rotation matrix dependent on time. Indeed, if we look at infinitesimalrotations,(⃗qi , ⃗q˙i ) 7→ (⃗qi + ϵ ω⃗ (t) × ⃗qi , ⃗q˙i + ϵ ω⃗ (t) × ⃗q˙i + ϵ ω⃗˙ (t) × ⃗qi ),

(79)

27

Of course the definition of this Lagrangian crucially depends on the invertibility of the inertia matrix
M . Interestingly, the subset of the tangent bundle on which M is not invertible is not exactly equivalent
to the subset of the cotangent bundle on which the canonical constraints become dependent. For example,
when all the positions of the particles are aligned but not their momenta, the canonical constraints arestill independent but M is not invertible.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

35

then we have the transformations



M

−1

ab



→ M

N
N
1 X
1 X
2
cm
cm
⃗,
q̇ − Ekin →q̇ 2 − Ekin+ ϵω⃗˙ · L2 i=1 i2 i=1 i

(80a)

⃗ → L
⃗ + ϵM · ω⃗,L⃗˙ + ϵ ω
⃗ ×L

(80b)

−1

ab



−ϵ M

−1

ac

N
X

!

d
c
(⃗
ω × ⃗q0i )c q0i+ q0i(⃗
ω × ⃗q0i )d



M −1

db

, (80c)i=1

which, when combined, give L → L.
This symmetry becomes even more explicit in the canonical formulation. Indeed, theLegendre transformation to (⃗qi , p⃗i ) ∈ T ∗ Q ≃ R6N is given by
pai =



a
∂L
a
−1 ⃗
,
=
q̇
+
⃗
q
×
M
·
L
0i
0i
∂ q̇ia(81)

and those momenta can be checked to verify the constraints (1). The Hamiltonian thenobtained is (3); indeed, the easiest way to see this is by applying the Legendre transformation (81) to (3) to reconstruct the Lagrangianp⃗i · ⃗q˙i − H

L̃ =

X

=

X

i

i

=

⃗ · ⃗q˙i − 1⃗q˙0i + ⃗q0i × (M −1 · L)22⃗
⃗q˙0i + ⃗q0i × (M −1 · L)

X



(82)

i

2
1
1X 2
cm
⃗
q̇i − Ekin−.
⃗q0i × (M −1 · L)
2 i
2Using the formula for the squared norm of a cross product, the last term can be comrot , thus proving that L̃ = L. Note that this reconstruction of theputed to be exactly Ekin
Lagrangian does not mean inverting the Legendre transformation (81) which defines a
map from T Q to the constraint surface C ⊂ T ∗ Q and which thus is not invertible [41].

A.2 Dependence of constraints on total collisions and collinearity
Here, we show that the six constraints (1) become linearly dependent on total collisions
and when all particles are collinear. Moreover, it will follow that the gauge orbits containing these pathological configurations are only three- and five-dimensional, respectively.
The simplest (and most extreme) situation is the N particle collision at the origin⃗qi = p⃗i = 0, i = 1, . . . , N , where the (phase space) gradients of the rotation generators
Ra vanish and only the translation generators P a define independent flows. The origin
does not rotate and this means the gauge orbit in which it lies is only three-dimensional.
Indeed, neither the rotation (1b) nor translation generators (1a) can change p⃗i = 0, ∀ i (see(4, 5)), and only the translation generators can change the location of the total collision,
however, not the fact that ⃗qi = ⃗q, ∀ i. For p⃗i = 0 and ⃗qi = ⃗q, ∀ i, one has dRa = ϵabc q b dP c
and so the gradients of the rotation generators will always be linearly dependent on thegradients of the translation generators. Such a gauge orbit is then three-dimensional.
Similarly, if the N particles are collinear and the momenta are also aligned with theaxis of collinearity, only two of the three gradients of Ra are linearly independent because
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

36

the rotation around the axis of collinearity acts trivially on the particles. More precisely,
collinearity means that ⃗qi − ⃗qj = qij ⃗v , p⃗i = pi ⃗v , i, j = 1, . . . , N , for some ⃗v ∈ R3 . On any
such configurationv a dRa = v a ϵabc v b
= −

X
X
1 X
1
qij dpci −v a ϵabc qjb dP c + v a ϵabc v cpi dqib
N − 1 i,j
N −1 jiX
1v a ϵabc qjb dP c ,N −1 jwhich defines a linear dependence among the six constraints (1). Note that any suchconfiguration can be mapped to ⃗qi = qi ⃗v and p⃗i = pi ⃗v , where even v a dRa = 0, through a
global translation that puts the configuration origin on the axis of collinearity. Since noother special direction exists for such configurations, this is the only linear dependence
and so two of the three rotation generator gradients are linearly independent. Neither
rotations nor translations can change the collinearity of the particles in both configurations
and momenta and so on every point of such a gauge orbit there will always be justfive independent constraints. This means that the gauge orbits in which such collinear
scenarios reside are only five-dimensional.
Since the translation generators are always independent and it is not possible that
only one rotation generator is independent (it would have to leave two linearly independent vectors invariant), there are no other types of lower dimensional orbits. Hence, theconstraint surface is foliated by six-, five- and three-dimensional gauge orbits.

A.3 Independent Dirac observables on generic N particle configurations
There are various ways to convince oneself geometrically that only 3N − 6 of the N2
relative distances |⃗qi − ⃗qj | are independent. Our entire discussion below is for genericconfigurations, i.e. away from pathologies such as total collision or collinearity of all
N particles. For example, for N = 3, the particles form a triangle and all three relative
distances are independent. Owing to the gauge symmetry, the orientation and localizationof this triangle in Newtonian background space is unphysical. Now add a fourth particle;
evidently, this particle requires three additional relative distances to be fully localized inrelation to the other three particles in the triangle. The physically relevant configurational
information of thefour particles is thus now contained in the edge lengths of a tetrahedron
and again all 42 = 6 relative distances are independent. Redundancy now comes in if one
adds a fifth particle: in order to fully localize it with respect to the other four, only the
relative distances to three of the four particles, forming a triangle, are required, see fig.
1. The fifth particle forms another tetrahedron with the three particles relative to which
it has been localized. The relational information in the configurations is thus stored in
two tetrahedra that are glued together along a common triangle. There are thus nine
independent edge lengths and the tenth relative distance (the one between the fifth andfourth particle) is automatically determined from the others, no matter how the glued
tetrahedra are embedded in the ambient three-dimensional Newtonian background space.
This construction generalizes to an arbitrary number N of particles: for each further
particle, only three independent relative distances are needed to fully localize it relative
to the previous ones and all other relative distances between it and the rest are redundant.
The independent relational information in the configurations can thus be encoded in a
3D triangulation with N vertices in Euclidean space and this triangulation has 3N − 6edges. There are thus 3N − 6 independent absolute relative distances for N ≥ 3 particles.
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

37

Likewise, it is not hard to see that indeed only 3N − 6 of the N (N2+1) momentum
Dirac observables p⃗i · p⃗j are independent. Thanks to (1a), there are (at most) two linearly
independent momenta for N = 3 and their three inner products are independent. For
N = 4, one has (at most) three linearly independent momenta, thus forming a basis,and their six inner products again are independent. Now for any additional particle k:
the three inner products of its momentum vector p⃗k with the three basis vectors are
independent and clearly determine the inner product of p⃗k with any other momentum
vector. Hence, there are 3N − 6 independent Dirac observables among the p⃗i · p⃗j .

A.4 Absence of global canonically conjugate Dirac observables
Recall from sec. 2.1 that any Dirac observable is a function of the basic gauge invariant
combinations
fijkl := (⃗qi − ⃗qj ) · (⃗qk − ⃗ql ),gijk := (⃗qi − ⃗qj ) · p⃗k ,hij := p⃗i · p⃗j .

(83)

It follows from Appendix A.3 that, on generic configurations, there will be 3N − 6 independent configuration and 3N − 6 momentum Dirac observables among this set. Therecannot be more independent ones because the phase space is 6N -dimensional and, on
generic points of C, all six first class constraints (1) are independent, i.e. their gradientsare linearly independent and tangential to C.
To analyze what happens for N particle collisions and collinearity, we need to be
more precise. (In)dependence of phase space functions is defined in terms of linear(in)dependence of their phase space gradients. Clearly, at the origin of phase space,⃗qi = p⃗i = 0, ∀ i, which is contained in C and lies on a three-dimensional gauge orbit (see Appendix A.2), all gradients of the basic Dirac observables above vanish, i.e.
dfijkl = dgijk = dhij = 0. In this sense, there are no independent Dirac observables
among the set (83) at the origin; the phase space flows they generate (and which arealways tangential to C) vanish at this point on C. This will already be enough to argue
that there can be no global canonically conjugate Dirac observables on C.
Nevertheless, it is also interesting to firstly consider what happens when all N particles
are collinear and also their momenta are aligned along this axis, i.e. ⃗qi −⃗qj = qij ⃗v , p⃗i = pi ⃗v ,∀ i, j, and some ⃗v ∈ R3 . We noted in Appendix A.2 that such configurations reside in
five-dimensional gauge orbits. The differentials of the basic Dirac observables (83) readon such configurations
dfijkl = qij ⃗v · (d⃗qk − d⃗ql ) + qkl ⃗v · (d⃗qi − d⃗qj ) ,dgijk = pi ⃗v · (d⃗qi − d⃗qj ) + qij ⃗v · d⃗
pk ,
dhij

(84)

= pi ⃗v · d⃗
pj + pj ⃗v · d⃗pi .

Hence, the only differentials appearing in these equations are ⃗v · (d⃗qi − d⃗qj ) and ⃗v · d⃗pi .
Now there are only N − 1 linearly independent differences ⃗v · (d⃗qi − d⃗qj ) as ⃗v · d⃗qi − d⃗qj =P⃗v · (d⃗qk − d⃗qj ) − ⃗v · (d⃗qk − d⃗qi ). Similarly, we have ⃗v · dP⃗ = i ⃗v · d⃗pi . Thus, altogether, onsuch collinear configurations there will only be 2(N −1)+1 linearly independent gradientsof the observables (83), of which 2(N −1) are non-trivial observables (instead of 6N −12))and the additional one corresponds to a linear combination of the momentum constraints⃗v · dP⃗ . That is, 4N − 10 otherwise independent flows generated by the Dirac observables(83) vanish for such collinear scenarios.
These observations imply that there can be no global canonically conjugate Diracobservable pairs, for if there were, their gradients would have to be non-vanishing on
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

38

all of C. In fact, suppose there were a full set of global canonically conjugate Diracobservables Qα , Πβ , α, β = 1, . . . , 3N − 6. They would define a matrix of their gradientswith rank equal to 6N − 12 everywhere on C, thanks to!

{Qα , Πβ } = Ωµν ∂µ Qα ∂ν Πβ ≈ δ αβ ,(85)

where Ω is an antisymmetric, contravariant tensor whose components in our coordinates
read
Ω

µν

µ

0 1
−1 0

ν

= {z , z } =!

,and the z µ , µ = 1, . . . , 6N , label the canonical phase space coordinates q1x , q1y , q1z , q2x , . . . ,px1 , py1 , pz1 , px2 , . . ., while ∂µ := ∂z∂µ . Indeed, given that Ωµν is invertible, validity of (85) on
all of C would imply that ∂µ Qα , ∂ν Πβ constitute 6N − 12 linearly independent gradients
everywhere on C. But this is in conflict with our observation that all Dirac observables arefunctions of fijkl , gijk , hij in (83) and that the latter have identically vanishing gradients atthe origin of phase space, which lies on C, or only 2(N − 1) linearly independent gradientson totally collinear configurations. More precisely, summarizing fijkl , gijk , hij in one label
set OΓ , where Γ runs over as many values as there are functions in (83), we can write′
Qα (OΓ ), Πβ (OΓ ). Note that these will not be unique functional dependences because thebasic Dirac observables (83) have some obvious dependences among them. However, for
any consistent such choice of functional dependence, one finds, using the chain rule,′!

{Qα , Πβ } = Ωµν ∂µ OΓ ∂ν OΓ ∂Γ Qα ∂Γ′ Πβ ≈ δ αβ ,(86)where ∂Γ := ∂/∂OΓ . Given that Ω is invertible and ∂µ OΓ vanishes at the origin anddefines a matrix of rank equal to 2(N − 1) for total collinearity, the right hand side is
impossible to achieve with non-singular ∂Γ Qα , ∂Γ Πβ . But if the latter become singular,then also the gradients ∂µ Qα = ∂µ OΓ ∂Γ Qα , etc., become ill-defined and so Qα , Πβ ,despite possibly being non-singular as functions (such as, e.g., the relative distances |⃗qi −⃗qj |), cannot be conjugate on such pathological configurations.
In summary, there are no global canonically conjugate Dirac observable pairs on C andthe same, in fact, immediately implies to Dirac observable pairs with affine conjugation
relations, where, instead of {Qα , Πβ } ≈ δ αβ , one would have {Qα , Πβ } ≈ δ αβ Qα (no sum
over α). The global absence of either makes reduced quantization a priori challenging as
neither canonical nor affine quantization [59] can be applied without imposing additionalboundary conditions.

A.5 Proof of the three-body gauge-fixing in the 3D caseLet us formally prove that the gauge is indeed totally fixed. One way is to consider the
Poisson-bracket Matrix C of all our constraints and show that it is invertible. Concatenating all our constraints in a 12-list (Λα ) = (P 1 , P 2 , P 3 , R1 , R2 , R3 , χx , χy , χz , ϕ1 , ϕ2 , ϕ3 ),C is a 12 × 12 antisymmetric matrix defined by Cαβ = {Λα , Λβ }. Computation of C gives(with all 12 constraints imposed):

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

39





















C=
































.
















0
1 0 0
0 1 0
0 0 1

0

0

z
qB
0
0
z
z
0 −qB
qC
x
0
0
−qC

0
z
−qB
0
z
0
qB
z
0
−qC

0

−1 0
0
0 −1 0
0
0 −1

0

0
0
x
qC

(87)

C is then invertible, with its inverse being:

















(C −1 ) = 


















1 0 0
0 1 0
0 0 1

0

0

0
− q1z
B
0
0

−1 0
0
0 −1 0
0
0 −1

0

0
1
z
qB
qz
− qz Cqx
B C

0
0
1
x
qC

0
1
z
qB

0

0
0

− q1z
B
0

0

0
z
qC
z qx
qB
C

− q1x

C

















 . (88)
















Note that this matrix is indeed only invertible when also (12) are imposed.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

40

A.6 Solving constraints for the redundant momenta
Using the constraints (1), we can express the momentum degrees of freedom that we are
getting rid of in terms of the six surviving phase space degrees of freedom:pyB = 0 ,
1 z x
1 y
x z
= − z (qCp C − qC
pC ) ,
pxB = − z RC
qB
qB
pyC = 0 ,1 z xx z
pxA = −pxB − pxC = −pxC + z (qC
pC − qC
pC ) ,qB
pyA = −pyB − pyC = 0(89)

pzA = −pzB − pzC .
Note that this constitutes a (non-global) constraint abelianization [41].

A.7 Switching internal perspectives classically
Using (89), the canonical embedding map of the reduced phase space in A perspective
into C reads
ιBC|A : PBC|Azzxz(qB > 0, pB , qC > 0, pxC , qC
, pzC ),→ Czxz
7→ (qB> 0, pzB , qC
> 0, pxC , qC
, pzC , ⃗qA = 0, p⃗A = −⃗pB − p⃗C ,

1 z x
y
y
x z
x
= pyB = pyC = 0, pxB = − z (qC= qC
p C − qC
pC )
qB
= qB
qB
and its image is precisely C ∩ GBC|A . Conversely, the projection
πBC|A : C ∩ GBC|A → PBC|A
does precisely the opposite, dropping all redundant information, so that πBC|A ◦ ιBC|A =IdPBC|A . Clearly, the same structures can be constructed for C perspective.
Let us construct the gauge transformation αA→C , taking us from C ∩ GBC|A to C ∩GAB|C , where GAB|C is defined by (9–12), in four steps. The flow on C generated bys , where s is the flow parameter. The gaugesome constraint C will be denoted by αC
transformation of some phase space function F simply drags its argument along the flows · F (X) = F (αs (X)), where X ∈ C, Written explicitly,
αC
C
s
αC
· F (X) =

∞ k
X
s
k=0

k!

{F, C}k (X) ,(90)

where {F, C}k = {. . . {{F, C}, C}, . . . , C} is the k-nested Poisson bracket of F with C.
1. First, we translate B along z to the origin. Using (4), the corresponding gaugetransformation is easy to evaluate on the canonical variables, starting at some X0 ∈C
(91)
αPs1z · qiz (X0 ) = qiz (X0 ) + s1 ,−q z (X0 )leaving all other variables invariant. Clearly, αP zB
does the job, so we flow withz (X ), where q z (X ) is the actual value of the relative
parameter distance s1 = −qB00B
distance of B from A before the translation at X0 .
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

41!

2. Next, we rotate around the y-axis until C lies on the z-axis. The necessary angle isthe one between A and C, as seen from B and can be found from gauge-invariant
quantities by using:
2 − l2l2 + lABACcos ΘAC|B = BC,
(92)
2lAB lBC
2 = ⃗where lijqij 2 is the squared distance between particles i and j. Using (5, 90),
this transformation is easy to evaluate on the canonical variables. Writing X1 =−q z (X0 )(X0 ), we find
αP zB
s2
x
= cos s2 qix (X1 ) + sin s2 qiz (X1 ) ,αRy · qi (X1 )

(93)

s2
x
αR
y · pi (X1 )s2zαRy · qi (X1 )s2zαRy · pi (X1 )(94)

cos s2 pxi (X1 ) + sin s2 pzi (X1 ) ,cos s2 qiz (X1 ) − sin s2 qix (X1 ) ,cos s2 pzi (X1 ) − sin s2 pxi (X1 ) ,

=
=
=

(95)
(96)

−Θ

leaving all other variables invariant. Evidently, αRy AC|B achieves the desired transformation.
3. Now that C lies on the z-axis, we translate it to the origin.
−Θ
αRy AC|B (X1 ), it is clear that

Writing X2 :=−q z (X2 )

αP zC

(97)

accomplishes the desired transformation.
x < 0 and so we
4. However, after this sequence of transformations, we will have qA
π , so that now q x > 0.
finally rotate once more by an angle π around the z-axis αRzA

In conjunction, jumping from the reference frame of A to the reference frame of C canbe achieved through the sequence of gauge transformations−q z (X2 )πCαA→C := αRz ◦ αP z−Θ−q z (X0 )◦ αRy AC|B ◦ αP zB

,

(98)

SA→C := πAB|C ◦ αA→C ◦ ιBC|A : PBC|A → PAB|C .

(99)

which completes the map, depicted in the diagram of sec. 2.3,

Recalling that one has to swap some redundant and non-redundant Dirac observables (inline with the A, C label exchange), it reads in coordinates:
z
x
z
(qB
> 0, pzB , qC
> 0, pxC , qC
, pzC )

7→

′x
qA
=

z qx
2 − qz qz
x
qB
rC
qC
x
x
C
B C
, p′x
=
(p
+
p
)
−
(pz + pzC ),
B
C
rBC A
rC rBCrBC B

2 − qz qz
2 − qz qz
rC
rC
qx
B C
B C
, p′z
(pzB + pzC ) − C (pxB + pxC ) ,A =−rBCrC rBC
rBC
!
2
z
z
x
rC − qB qC z
qC
′z
′z
z x
x z
qB = rBC , pB =p −z (qC pC − qC pC ) ,rC rBC B rBC qB′zqA=

where the primed and unprimed variables
q are the ones after and before
q the total transforxzx )2 + (q z − q z )2 .
22mation SA→C , respectively, and rC = (qC ) + (qC ) and rBC = (qC
B
C
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

42

B Using spherical coordinates
B.1 Spherical coordinates
In L2 (R3 ), we define spherical-coordinates eigenstates by:|r, θ, φ⟩ = |x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ⟩ ,∀ r > 0, θ ∈ [0, π], φ ∈ [0, 2π[ .
(100)We have to remember that those states are normalized slightly differently,⟨r′ , θ′ , φ′ |r, θ, φ⟩ =1
r2 sin θδ(r − r′ ) δ(θ − θ′ ) δ(φ − φ′ ) ,(101)

but they define a basis for L2 (R3 ), except for the fact that there are pathological situations
at r = 0 and θ = 0, π. One way to see this situation is that the use of spherical coordinatescorresponds to an isometry between L2 (R3 ) and Hsphe , the subspace of L2 (R+ ×S 2 ) definedby the following conditions on a wave function gg(r = 0, θ, φ) = α ,(102a)

g(r, θ = 0, φ) = β(r) ,(102b)

g(r, θ = π, φ) = γ(r) ,(102c)

and where the integration measure is dµ = r2 dΩ = r2 dθ sin θ dφ.
A state |ϕ⟩ associated to the density g in spherical coordinates (verifying conditions(102)) should be written:
|ϕ⟩ =

Z

dr r2 dΩ g(r, θ, φ) |r, θ, φ⟩ .

(103)

dr r2 dΩ g1 (r, θ, φ)∗ g2 (r, θ, φ)(104)

In this way one has⟨ϕ1 |ϕ2 ⟩ =Z

and| ⟨r, θ, φ|ϕ⟩ |2 = |g(r, θ, φ)|2 ,(105)

so one sees that g(r, θ, ϕ) can physically be considered as the density of probability that|ϕ⟩ is measured at the position (r, θ, φ) in spherical coordinates, as should be expected.

B.2 Spherical harmonics states
Let us restrict ourselves for a moment to CL∞ (R3 ) = C ∞ (R3 ) ∩ L2 (R3 ), which is dense in
L2 (R3 ). It is spanned by the:|r; j, m⟩ =ZdΩ Y j,m (θ, φ) |r, θ, φ⟩ ,(106)

where the Y j,m are the usual spherical harmonics. Those states are built on the usual
basis |j, m⟩ (where j ≥ 0 and |m| ≤ j) of common eigenstates of R̂z and R̂2 .
A state in CL∞ (R3 ) can then be decomposed as:
|ϕ⟩ =

Z

dr r2

∞
X
X

f (r; j, m) |r; j, m⟩ ,(107)

j=0 |m|≤j

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

43where the only remaining condition on f , stemming from (102a), is:f (r = 0; j, m) = α δj,0 δm,0(108)

As CL∞ (R3 ) is dense in L2 (R3 ), decomposition (107) (where f verifies (108)) is also
valid for any |ϕ⟩ ∈ L2 (R3 ).

B.3 Spherical coordinates operators
Lastly, let us define operators r̂, θ̂ and φ̂. A natural way to do so would be to define iton the orthogonal basis defined in (100):r̂ |r, θ, φ⟩ = r |r, θ, φ⟩ ,(109a)

θ̂ |r, θ, φ⟩ = θ |r, θ, φ⟩ ,(109b)

φ̂ |r, θ, φ⟩ = φ |r, θ, φ⟩ .

(109c)

Yet, θ̂ and φ̂ defined in this way are not well-defined everywhere, as the action of bothof them on states with r = 0, as well as the action of φ̂ on states with θ = 0 and θ = π,would then map on states which do not verify conditions (102). This is why we will saythat definition (109) is assumed only out of these pathological cases, on which we willcomplete it by the convention:θ̂ |r = 0, θ, φ⟩ = 0 ,(110a)

φ̂ |r = 0, θ, φ⟩ = 0 ,(110b)

φ̂ |r, θ = 0, φ⟩ = 0 ,(110c)

φ̂ |r, θ = π, φ⟩ = 0 .

(110d)

It is important to note that the choice of definition (110) will have practical consequences
only when we consider states comprising a Dirac distribution at r = 0, θ = 0, or θ = π;otherwise they do not matter as they only impact states with a zero measure. Note thatr = 0 will correspond to collisions in our model, which we have ruled out dynamically.

C Rotational reduction in the quantum theory
C.1 The rotation generator trivialization map and its action on states
We had defined the transformation (45)












y
z
z
exp i θ̂B R̂C
exp i φ̂B R̂C.
RB,C = exp i φ̂B R̂C(111)

It is instructive to understand the construction of RB,C by first having a look at thefollowing unitary transformation on L2 (R3 ):










R(θ, φ) = exp i φR̂z exp i θR̂y exp i φR̂z



(112)

Let us look at its effect on a state |⃗q0 ⟩ where ⃗q0 = (r0 sin θ0 cos φ0 , r0 sin θ0 sin φ0 , r0 cos θ0 ).
We know that R̂z and R̂y are the generators of rotations around the z- and y-axes,

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

44

respectively, so:
























R(θ, φ) |⃗q0 ⟩ = exp i φR̂z exp i θR̂y exp i φR̂z |⃗q0 ⟩= exp i φR̂z exp i θR̂y |⃗q1 ⟩(113)= exp i φR̂z |⃗q2 ⟩ = |⃗q3 ⟩ ,where (we only give the values of the important coordinates):⃗q1 = (r0 sin θ0 cos(φ0 − φ), r0 sin θ0 sin(φ0 − φ), r0 cos θ0 )(114a)

q2z = q3z = r0 cos θ0 cos θ + r0 sin θ0 cos(φ0 − φ) sin θ .

(114b)⃗q3 can then itself be rewritten in spherical coordinates as ⃗q3 = (r0 sin γ cos η, r0 sin γ sin η, r0 cos γ),
where the new angular coordinates γ ∈ [0, π], η ∈ [0, 2π[ are defined by:cos(γ(θ, θ0 , φ − φ0 )) = q3z /r0 = cos θ cos θ0 + sin θ sin θ0 cos(φ − φ0 ) ,sin(η(θ, θ0 , φ − φ0 )) sin(γ(θ, θ0 , φ − φ0 )) = q3y /r0 .

(115a)(115b)The geometrical interpretation of (115a) is that γ(θ, θ0 , φ − φ0 ) is the angle between the
directions (θ, φ) and (θ0 , φ0 ), or equivalently the polar coordinate of the direction (θ0 , φ0 )if one takes direction (θ, φ) to be the new z-axis; η is not relevant to us.
Going back to RB,C , one can now see that:
RB,C |rB , θB , φB ⟩B |rC , θC , φC ⟩C= |rB , θB , φB ⟩B ⊗ (R(θB , φB ) |rC , θC , φC ⟩C )(116)

= |rB , θB , φB ⟩B ⊗|rC , γ(θB , θC , φB − φC ), η(θB , θC , φB − φC )⟩C ,so we see that RB,C acts as a rotation on the C-factor of the tensor product, mapping
it to a description in spherical coordinates where the polar angle γ is now relative to thedirection of B. The azimuth angle η after transformation RB,C is not relevant for ouranalysis, as we shall see shortly.
phys
HBC|A
is spanned by the states |Φ(rB , rC ; j)⟩ as defined in (43, 44). Let us therefore
determine the action of RB,C on the |Φ(rB , rC ; j)⟩. Using 116, we find:
RB,C |Φ(rB , rC ; j)⟩ = (−1)jZdΩB dΩC(−1)m j,−m√Y(θB , φB )Y j,m (θC , φC )
2j
+
1
|m|≤j
X

|rB , θB , φB ⟩B |rC , γ(θB , θC , φB − φC ), η(θB , θC , φB − φC )⟩C .
(117)
The Legendre addition theorem states that:(−1)m j,−m√Y(θB , φB )Y j,m (θC , φC ) =
2j
+
1
|m|≤j
X

√

2j + 1
Pj (cos γ) ,
4π

(118)

where γ is defined as in (115a) and Pj is the Legendre polynomial of degree j. The changeof variables (θC , φC ) → (γ, η) leaves the element of integration invariant and yields:jRB,C |Φ(rB , rC ; j)⟩ = (−1)

Z1dΩB dγ sin γdη √2 π

s2j + 1
Pj (cos γ) |rB , θB , φB ⟩B |rC , γ, η⟩C .
4π
(119)

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

45

One can recognize the spherical harmonics Y 0,0 (θB , φB ) = 2√1 π and Y j,0 (γ, η) =and thus finally rewrite:
RB,C |Φ(rB , rC ; j)⟩ = (−1)j |rB ; 0, 0⟩B |rC ; j, 0⟩C

q2j+1
4π Pj (cos γ),

(120)

phys
Hence, RB,C maps a given |ψ⟩phys
BC|A ∈ HBC|A as decomposed in (43) to:phys|ψ⟩phys
B,C|A := RB,C |ψ⟩BC|A =Z

2 2
drB drC rB
rC

∞
X

phys
(−1)j ψBC|A(rB , rC ; j) |rB ; 0, 0⟩B |rC ; j, 0⟩C .

j=0

(121)
phys
TI
Recall that HBC|A
is a proper subspace of HBC|A
and so inherits its inner productphys
TI
from the latter. Now RB,C leaves HBC|A invariant and so just rotates HBC|A
into aphys
phys
TI(see fig. 3). It can be easily
new Hilbert subspace HB,C|A := RB,C (HBC|A ) ⊂ HBC|A
checked by direct calculation that RB,C leaves the inner product invariant and so it
phys
phys
defines an isometry from HBC|A
to HB,C|A. Notice also that it is invertible (away fromthe pathological states (110)) with












y
z
z
R−1
B,C = exp − i φ̂B R̂C exp − i θ̂B R̂C exp − i φ̂B R̂C .

(122)

C.2 Trivializing the rotation constraints
phys
TI
as a subspace of HBC|A,
Let us look at how the constraints (26b), which define HBC|Atransform under RB,C . We will give the results of the calculations, which were done using
the usual commutation relations between the R̂a and using their representation in polar
coordinates:R̂x = i(sin φ ∂θ + cot θ cos φ ∂φ ) ,(123a)

R̂y = i(− cos φ ∂θ + cot θ sin φ ∂φ ) ,(123b)

R̂x = −i∂φ .

(123c)

phys, the constraints get mapped to:
In terms of action on the coefficient function ψB,C|A
physxx
RB,C (R̂B
+ R̂C
) R−1
B,C ψB,C|A(124a)= (i sin φB ∂θB + i cot θB cos φB (∂φB − ∂φC ) − i

cos φB∂φ )ψ phys = 0
sin θB B B,C|A

y
y
phys
RB,C (R̂B
+ R̂C
) R−1
B,C ψB,C|A(124b)= (i cos φB ∂θB + i cot θB sin φB (∂φB − ∂φC ) − i

sin φB∂φ )ψ phys = 0
sin θB B B,C|A

phys
phys
z
z
RB,C (R̂B
+ R̂C
) R−1
B,C ψB,C|A = −i(∂φB − ∂φC )ψB,C|A = 0(124c)

This can also be written in terms of the R̂s acting on the state |ψ⟩phys
B,C|A :
physxxRB,C (R̂B
+ R̂C
) R−1
B,C |ψ⟩B,C|Ayx= (sin φ̂B (sin φ̂B R̂B− cos φ̂B R̂B)zz+ cot θ̂B cos φB (R̂C
− R̂B
)+

(125a)dcos
φB zR̂ ) |ψ⟩phys
B,C|A = 0 .
sin θB B

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

46

y
y
phys
RB,C (R̂B
+ R̂C
) R−1
B,C |ψ⟩B,C|Ayx= (cos φ̂B (sin φ̂B R̂B− cos φ̂B R̂B)zz+ cot θ̂B sin φB (R̂C
− R̂B
)+

(125b)dsin
φB z
R̂B ) |ψ⟩phys
B,C|A = 0 .
sin θB

phys
−1
z
z
z
z
RB,C (R̂B+ R̂C) RB,C|ψ⟩phys
B,C|A = (R̂C − R̂B ) |ψ⟩B,C|A = 0(125c)

Inserting (124c) in (124a) and (124b), and then rotating them by angle φB , one finds that
(124) implies:phys∂θB ψB,C|A=0(126a)
phys∂φB ψB,C|A=0(126b)

phys∂φC ψB,C|A=0(126c)

which is equivalent to the system of constraints:
y
phys
phys
phys
x
z
z
R̂B
|ψ⟩phys
B,C|A = R̂B |ψ⟩B,C|A = R̂B |ψ⟩B,C|A = R̂C |ψ⟩B,C|A = 0 .

(127)This is, of course, exactly what one would expect from the shape of (121). In turn,it is easy to see that (127) implies (124); thus (127) can indeed be taken as the set ofphys
TI
as a subset of HBC|A.
constraints which defines HB,C|A

C.3 Dirac observables and rotational reduction
Next, we have to check how our Dirac observables, written in (34) equivalently for Hphysphys, we have to checkand HTI , transform under RB,C . Since we apply this map to HBC|A
how the translationally reduced Dirac observables transform under RB,C . Recall fromsec. 3.2.2 that the translational reduction of the Dirac observables in (34) amounts tosimply dropping the A labels on all sides.
It will also be convenient to write the operator corresponding to the angle γ between
phys
as follows:
B and C as seen from A (essentially û in (34)) on HBC|A
γb = arccos cos θ̂B cos θ̂C + sin θ̂B sin θ̂C cos(φ̂C − φ̂B )= arccos



(128)

⃗qBd
· ⃗qC rB rC

To compute how the observables transform, one can remember that RB,C acts on theaa as a sequence of rotations (whose ‘parameters’ are the operators θ̂ and φ̂ ).
p̂C and q̂CBB
Indeed, if s ∈ {p, q}:
RB,C ŝxC R−1
B,C= cos φ̂B (sin θ̂B ŝzC + cos θ̂B (cos φ̂B ŝxC − sin φ̂B ŝyC ))− sin φ̂B (sin φ̂B ŝxC + cos φ̂B ŝyC )RB,C ŝyC R−1
B,C= sin φ̂B (sin θ̂B ŝzC + cos θ̂B (cos φ̂B ŝxC − sin φ̂B ŝyC ))+ cos φ̂B (sin φ̂B ŝxC + cos φ̂B ŝyC )RB,C ŝzC R−1
B,C= cos θ̂B ŝzC − sin θ̂B (cos φ̂B ŝxC − sin φ̂B ŝyC ))The key point of those transformations is that γb in (128), gets mapped to θ̂C :RB,C γb R−1
B,C = θ̂C .
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

(130)
47

Moreover, the calculations give that the forms of ρ̂B , ρ̂C , p̂ρB and p̂ρC remain invariant
under RB,C ; only ûC and p̂uC have non trivial transformations (these transformations are
only valid on states without support on pathological configuratoins):
RB,C ûC R−1
B,C = −√b(q ) +(q )dθ= −cot
x 2
C
z
qC

y 2
C

C

RB,C p̂uC R−1
B,C
=−

q
q
1
b
z b
z )2 + (q x )2 + (q y )2 )−1 ⃗x )2 + (q y )2 ((qx )2 + (q y )2 p̂z
⃗
b
− q̂C
(qC
q̂
·
p̂
+
(qC
C
C
C
C
C
C
C
C
2

z
− p̂⃗C · ⃗q̂C q̂C

=−

1
2

q

q

b
z )2 + (q x )2 + (q y )2 )−1 + p̂z bb(q x )2 + (q y )2 ((q(q x )2 + (q y )2
C

C

C

C

C

C

C

C



d
d
d
d
d
d
θC .
θC + p̂zC r̂C sin
θC p̂zC − p̂⃗C · ⃗q̂C cos
θC sinθC ⃗q̂C · p̂⃗C + r̂C sin− cos
θC sinD Illustration of the Hamiltonian discrepancy between Dirac and reduced quantizationred′
It can be computed that the Hamiltonians ĤBC|Aand Φ′A Ĥtot Φ′−1
A , where ΦA is given by(60), respectively obtained through reduced quantization and through Dirac quantization
followed by our quantum reduction method, differ by a term only involving configuration
observables:red
ĤBC|A− Φ′A Ĥtot Φ′−1
A =



3
1
(1 + û2C ) 2 300 (e3ρ̂B −ρ̂C + e3ρ̂C −ρ̂B ) + 567 eρ̂B +ρ̂C ûC (1 + û2C )
16


3



+ 100 (e2ρ̂B −2ρ̂C + e2ρ̂C −2ρ̂B ) + 267 (e2ρ̂B + e2ρ̂C ) (1 + û2C ) 2
1+ 133 eρ̂B +ρ̂C û3C + 333 (e2ρ̂B + e2ρ̂C )u2C (1 + û2C ) 2

!

b

1 −2

· (e2ρB + e2ρC ) (1 + u2C ) + eρB +ρC uC (1 + u2C ) 2

.
(131)

−1
We note that Φ′A Ĥtot Φ′−1
A is equal to ΦA Ĥtot ΦA , which is the Hamiltonian reduced fromthe physical Hilbert space to HBC|A because the lower left diagram in fig. 3 commutesand Ĥtot takes the same form on HTI as it does on Hphys . However, as argued in the main
body, in practice it is easier to construct Φ′A . The discrepancy between the Hamiltonians
thus arises relative to standard choices in Dirac quantization.
There are ways to modify Dirac quantization for systems with constraints linear in
momenta, as in the present manuscript, such that it agrees with reduced quantization [53,68]. While [68] modifies the quantization of gauge degrees of freedom and only constructs
a Hilbert space for gauge invariant degrees of freedom, [53] modifies the quantization of
the gauge-invariant degrees of freedom in Dirac quantization. For example, the method
phys in such a way
in [53] constructs the Hamiltonian only on the physicalp Hilbert space H
that it is self-adjoint with respect to the measure |hγ|, where h, given in (20) is themeasure on the physical configuration space, and γ is the measure on the gauge orbits.28
28

the determinant of γαβ , the induced metric for displacements along gauge orbits, given by γαβ =P γ isaaaKα Kβ , where the Kα denote the six constraints (1).

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

48

Applied to our model, it is thus defined not in terms of the momentum operators p̂ai
on Hkin , but in terms of the momentum operators π̂µphys (where µ ∈ {ρBA , ρCA , u}) onp
Hphys , which are self-adjoint with respect to the measure |γ h|. The Hamiltonian is thenhowever given by [53]
1
1
1
1
′
Ĥtot
= |h|− 4 π̂µphys |h| 2 hµν π̂νphys |h|− 4 + V (ρ̂BA , ρ̂CA , û) .
2

(132)

Specifically,
the Laplace-Beltrami operator is now constructedwith respect to the measurepp|h| and no longer with respect to the measure |γh|. Furthermore, the Hamiltonianis already written with respect to a special choice of gauge invariant degrees of freedom,here associated with the reference frame choice A. A different reference frame choice,
say C, would yield a different Hamiltonian when written out similarly because it would
be based on different canonical pairs. As is well known through the Groenewold-vanHove phenomenon, classical canonical transformations will in general not translate into
unitary transformations in the quantum theory, especially when the variable change isnonlinear [76].
Applying our quantum reduction procedure to this modified quantization, the effect of
its successive steps will simply be to rotate the physical Hilbert space so as to replace thelabels ρBA , ρCA , u with, respectively, ρB , ρC , uC , and to modify the measure of this space
so that the π̂µphys are ultimately mapped to the π̂µ obtained through reduced quantization.
red
The Hamiltonian (132) will, by construction, be mapped precisely to ĤBC|A
as defined in(25), thus ensuring that the Dirac and reduced quantized theories are equivalent under ourquantum reduction procedure, as far as the dynamics is concerned. For other composite
operators one may similarly have to modify the quantization. While this method isinteresting for understanding the relation between Dirac and reduced quantization, themodifications in this construction are not particularly natural from the point of view ofthe former. Especially the fact that the gauge invariant Hamiltonian (132) does awaywith the redundancy inherent in Dirac quantization and is special to a choice of frame isbreaking the strength of Dirac quantization that we exploited in the main body for defining
a notion of quantum frame covariance. In particular, this means that the modification
one has to apply to Dirac quantization in order to obtain equivalence with the reduced
quantization of a classical frame perspective will depend on the choice of that frame
because the different reduced quantizations are unitarily inequivalent. This is in conflictwith quantum frame covariance.
By contrast, in our construction Ĥtot does not face any factor ordering ambiguities
on Hkin , which comes with a standard Lebesgue measure, and neither on Hphys , when
expressed in the individual kinematical particle variables. Most importantly, our construction admits the redundancy underlying quantum frame covariance and the HamiltonianĤtot treats all degrees of freedom on an equal footing.

References[1] A. Vanrietvelde, P. A. Höhn, F. Giacomini, and E. Castro-Ruiz, “A change ofperspective: switching quantum reference frames via a perspective-neutral
framework,” Quantum 4 (2020) 225, arXiv:1809.00556 [quant-ph].
[2] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, “Quantum mechanics and thecovariance of physical laws in quantum reference frames,” Nature Commun. 10
no. 1, (2019) 494, arXiv:1712.07207 [quant-ph].
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

49

[3] B. S. DeWitt, “Quantum theory of gravity. I. The canonical theory,” Phys.Rev. 160(1967) 1113–1148.
[4] C. Rovelli, Quantum Gravity. Cambridge University Press, 2004.
[5] C. Rovelli, “Quantum reference systems,” Class.Quant.Grav. 8 (1991) 317–332.
[6] C. Rovelli, “What is observable in classical and quantum gravity?,”Class.Quant.Grav. 8 (1991) 297–316.
[7] K. Kuchař, “Time and interpretations of quantum gravity,”
Int.J.Mod.Phys.Proc.Suppl. D20 (2011) 3–86. Originally published in the Proc. 4th
Canadian Conf. on General Relativity and Relativistic Astrophysics, eds. G.
Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).
[8] C. J. Isham, Canonical Quantum Gravity and the Problem of Time, pp. 157–287.
Springer Netherlands, Dordrecht, 1993.
[9] J. D. Brown and K. V. Kuchař, “Dust as a standard of space and time in canonicalquantum gravity,” Phys. Rev. D 51 (May, 1995) 5600–5629.
[10] B. Dittrich, “Partial and complete observables for Hamiltonian constrained
systems,” Gen.Rel.Grav. 39 (2007) 1891–1927, arXiv:gr-qc/0411013 [gr-qc].
[11] B. Dittrich, “Partial and complete observables for canonical General Relativity,”
Class.Quant.Grav. 23 (2006) 6155–6184, arXiv:gr-qc/0507106 [gr-qc].
[12] J. Tambornino, “Relational Observables in Gravity: a Review,” SIGMA 8 (2012)
017, arXiv:1109.0740 [gr-qc].
[13] T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge
University Press, 2007.
[14] M. Bojowald, P. A. Höhn, and A. Tsobanjan, “An Effective approach to the
problem of time,” Class. Quant. Grav. 28 (2011) 035006, arXiv:1009.5953[gr-qc].
[15] M. Bojowald, P. A. Höhn, and A. Tsobanjan, “Effective approach to the problem of
time: general features and examples,” Phys.Rev. D83 (2011) 125023,
arXiv:1011.3040 [gr-qc].
[16] P. A. Höhn, E. Kubalova, and A. Tsobanjan, “Effective relational dynamics of anonintegrable cosmological model,” Phys.Rev. D86 (2012) 065014,
arXiv:1111.5193 [gr-qc].
[17] B. Dittrich, P. A. Höhn, T. A. Koslowski, and M. I. Nelson, “Can chaos be
observed in quantum gravity?,” Phys. Lett. B769 (2017) 554–560,
arXiv:1602.03237 [gr-qc].
[18] B. Dittrich, P. A. Höhn, T. A. Koslowski, and M. I. Nelson, “Chaos, Diracobservables and constraint quantization,” arXiv:1508.01947 [gr-qc].
[19] Y. Aharonov and L. Susskind, “Charge Superselection Rule,” Phys. Rev. 155 (1967)
1428–1431.
[20] Y. Aharonov and L. Susskind, “Observability of the sign change of spinors under 2πrotations,” Phys. Rev. 158 (Jun, 1967) 1237–1238.
[21] Y. Aharonov and T. Kaufherr, “Quantum frames of reference,” Phys. Rev. D 30
(Jul, 1984) 368–385.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

50

[22] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection
rules, and quantum information,” Rev. Mod. Phys. 79 (2007) 555–609,
arXiv:quant-ph/0610030.
[23] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner, “Quantum
communication using a bounded-size quantum reference frame,” New Journal of
Physics 11 no. 6, (2009) 063013, arXiv:0812.5040.
[24] G. Gour and R. W. Spekkens, “The resource theory of quantum reference frames:
manipulations and monotones,” New Journal of Physics 10 no. 3, (Mar, 2008)
033023.
[25] M. C. Palmer, F. Girelli, and S. D. Bartlett, “Changing quantum reference frames,”
Phys. Rev. A89 no. 5, (2014) 052121, arXiv:1307.6597 [quant-ph].
[26] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner, “Degradation of aquantum reference frame,” New Journal of Physics 8 no. 4, (2006) 58,
arXiv:quant-ph/0602069.
[27] A. R. Smith, M. Piani, and R. B. Mann, “Quantum reference frames associatedwith noncompact groups: The case of translations and boosts and the role ofmass,” Physical Review A 94 no. 1, (2016) 012333, arXiv:1602.07696.
[28] D. Poulin and J. Yard, “Dynamics of a quantum reference frame,” New J. Phys. 9
no. 5, (2007) 156–156, arXiv:quant-ph/0612126.
[29] M. Skotiniotis, B. Toloui, I. T. Durham, and B. C. Sanders, “Quantum Framenessfor CPT Symmetry,” Phys. Rev. Lett. 111 no. 2, (2013) 020504, arXiv:1306.6114[quant-ph].
[30] L. Loveridge, P. Busch, and T. Miyadera, “Relativity of quantum states andobservables,” EPL (Europhysics Letters) 117 no. 4, (2017) 40004,
arXiv:1604.02836.
[31] J. Pienaar, “A relational approach to quantum reference frames for spins,”
arXiv:1601.07320 [quant-ph].
[32] R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and P. Skrzypczyk, “Physics
within a quantum reference frame,” Journal of Physics A: Mathematical andTheoretical 44 no. 14, (2011) 145304, arXiv:1007.2292.
[33] P. A. Höhn and M. P. Müller, “An operational approach to spacetime symmetries:
Lorentz transformations from quantum communication,” New J. Phys. 18 no. 6,(2016) 063026, arXiv:1412.8462 [quant-ph].
[34] P. A. Guérin and Č. Brukner, “Observer-dependent locality of quantum events,”
New J. Phys. 20 no. 10, (2018) 103031, arXiv:1805.12429 [quant-ph].
[35] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal
order,” Nature Commun. 3 (2012) 1092, arXiv:1105.4464 [quant-ph].
[36] L. Hardy, “The Construction Interpretation: Conceptual Roads to Quantum
Gravity,” arXiv:1807.10980 [quant-ph].
[37] J. Barbour and B. Bertotti, “Mach’s principle and the structure of dynamical
theories,” Proceedings of the Royal Society of London A: Mathematical, Physicaland Engineering Sciences 382 no. 1783, (1982) 295–306.
[38] F. Mercati, Shape Dynamics: Relativity and Relationalism. Oxford University
Press, 2018.
Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

51

[39] P. A. Höhn, “Reflections on the information paradigm in quantum andgravitational physics,” J. Phys. Conf. Ser. 880 no. 1, (2017) 012014,arXiv:1706.06882 [hep-th].
[40] P. A. Dirac, Lectures on Quantum Mechanics. Yeshiva University Press, 1964.
[41] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton
University Press, 1992.
[42] L. Hardy, “Implementation of the Quantum Equivalence Principle,” in Progress andVisions in Quantum Theory in View of Gravity: Bridging foundations of physics
and mathematics. 3, 2019. arXiv:1903.01289 [quant-ph].
[43] M. Zych, F. Costa, and T. C. Ralph, “Relativity of quantum superpositions,”
arXiv:1809.04999 [quant-ph].
[44] C. Rovelli, “Why Gauge?,” Found. Phys. 44 no. 1, (2014) 91–104,
arXiv:1308.5599 [hep-th].
[45] A. Ashtekar and G. t. Horowitz, “On the canonical approach to quantum gravity,”
Phys. Rev. D26 (1982) 3342–3353.
[46] V. Guillemin and S. Sternberg, “Geometric quantization and multiplicities of grouprepresentations,” Inventiones mathematicae 67 (10, 1982) 515–538.
[47] Y. Tian and W. Zhang, “An analytic proof of the geometric quantization conjectureof guillemin-sternberg,” Inventiones mathematicae 132 no. 2, (1998) 229–259.
[48] P. Hochs and N. Landsman, “The guillemin–sternberg conjecture for noncompact
groups and spaces,” Journal of K-theory 1 no. 3, (2008) 473–533,
arXiv:math-ph/0512022.
[49] M. J. Gotay, “Constraints, Reduction, and Quantization,” J. Math. Phys. 27 (1986)
2051–2066.
[50] K. V. Kuchař, “Covariant Factor Ordering of Gauge Systems,” Phys. Rev. D 34
(1986) 3044–3057.
[51] A. Ashtekar, Lectures on Nonperturbative Canonical Gravity, vol. 6 of Advances
series in astrophysics and cosmology. World Scientific.
[52] K. Schleich, “Is reduced phase space quantization equivalent to Diracquantization?,” Class. Quant. Grav. 7 (1990) 1529–1538.
[53] G. Kunstatter, “Dirac versus reduced quantization: A Geometrical approach,”Class. Quant. Grav. 9 (1992) 1469–1486.
[54] P. Hajicek and K. V. Kuchar, “Constraint quantization of parametrized relativistic
gauge systems in curved space-times,” Phys. Rev. D41 (1990) 1091–1104.
[55] J. D. Romano and R. S. Tate, “Dirac Versus Reduced Space Quantization of Simple
Constrained Systems,” Class. Quant. Grav. 6 (1989) 1487.
[56] R. Loll, “Noncommutativity of constraining and quantizing: A U(1) gauge model,”Phys. Rev. D41 (1990) 3785–3791.
[57] J. Barbour, T. Koslowski, and F. Mercati, “Identification of a gravitational arrowof time,” Phys. Rev. Lett. 113 no. 18, (2014) 181101, arXiv:1409.0917 [gr-qc].
[58] J. Barbour, T. Koslowski, and F. Mercati, “Entropy and the Typicality of
Universes,” arXiv:1507.06498 [gr-qc].

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

52

[59] C. Isham, “Topological and global aspects of quantum theory,” in Relativity,
Groups and Topology II, Les Houches Summer School, 1983, B. DeWitt and
R. Stora, eds., p. 1062. North Holland, Amsterdam, 1984.
[60] R. G. Littlejohn and M. Reinsch, “Internal or shape coordinates in the n-bodyproblem,” Physical Review A 52 no. 3, (1995) 2035.
[61] R. G. Littlejohn and M. Reinsch, “Gauge fields in the separation of rotations andinternal motions in the n-body problem,” Reviews of Modern Physics 69 no. 1,(1997) 213.
[62] D. Marolf, “Refined algebraic quantization: Systems with a single constraint,”
arXiv:gr-qc/9508015 [gr-qc].
[63] D. Marolf, “Group averaging and refined algebraic quantization: Where are wenow?,” arXiv:gr-qc/0011112 [gr-qc].
[64] A. Kempf and J. R. Klauder, “On the implementation of constraints throughprojection operators,” J. Phys. A34 (2001) 1019–1036, arXiv:quant-ph/0009072[quant-ph].
[65] D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamics
described by stationary observables,” Phys. Rev. D27 (1983) 2885.
[66] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, “Trinity of relational quantum
dynamics,” Phys. Rev. D 104 no. 6, (2021) 066001, arXiv:1912.00033[quant-ph].
[67] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, “Equivalence of approaches torelational quantum dynamics in relativistic settings,” Front. in Phys. 9 (2021) 181,
arXiv:2007.00580 [gr-qc].
[68] K. Kuchař, “Covariant factor ordering of gauge systems,” Phys. Rev. D 34 (Nov,
1986) 3044–3057.
[69] D. McMullan and J. Paterson, “Covariant Factor Ordering of Gauge Systems Using
Ghost Variables. 1. Constraint Rescaling,” J. Math. Phys. 30 (1989) 477.
[70] B. P. Dolan, “BRST symmetry and quantum mechanics on homogeneous spaces,”
Journal of Physics A: Mathematical and General 23 no. 20, (Oct, 1990) 4439–4453.
[71] J. M. Yang, “Switching quantum reference frames for quantum measurement,”
Quantum 4 (2020) 283, arXiv:1911.04903 [quant-ph].
[72] P. A. Höhn and A. Vanrietvelde, “How to switch between relational quantum
clocks,” New J. Phys. 22 no. 12, (2020) 123048, arXiv:1810.04153 [gr-qc].
[73] P. A. Höhn, “Switching Internal Times and a New Perspective on the ‘Wave
Function of the Universe’,” Universe 5 no. 5, (2019) 116, arXiv:1811.00611[gr-qc].
[74] J. De Vuyst, P. A. Höhn, and A. Tsobanjan to appear .
[75] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, “Relativistic Quantum Reference
Frames: The Operational Meaning of Spin,” Phys. Rev. Lett. 123 no. 9, (2019)
090404, arXiv:1811.08228 [quant-ph].
[76] V. Guillemin and S. Sternberg, Symplectic techniques in physics. Cambridge
University Press, Cambridge, 1990.

Accepted in Quantum 2023-05-09, click title to verify. Published under CC-BY 4.0.

53