# Quantum reference frames: derivation of perspective-dependent descriptions
via a perspective-neutral structure

Viktor Zelezny  Institute for Quantum Optics and Quantum Information, Austrian
Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

###### Abstract

In standard quantum mechanics, reference frames are treated as abstract
entities. We can think of them as idealized, infinite-mass subsystems which
decouple from the rest of the system. In nature, however, all reference frames
are realized through finite-mass systems that are subject to the laws of
quantum mechanics and must be included in the dynamical evolution. A
fundamental physical theory should take this fact seriously. In this paper, we
further develop a symmetry-inspired approach to describe physics from the
perspective of quantum reference frames. We find a unifying framework allowing
us to systematically derive a broad class of perspective dependent
descriptions and the transformations between them. Working with a
translational-invariant toy model of three free particles, we discover that
the introduction of relative coordinates leads to a Hamiltonian structure with
two non-commuting constraints. This structure can be said to contain all
observer-perspectives at once, while the redundancies prevent an immediate
operational interpretation. We show that the operationally meaningful
perspective dependent descriptions are given by Darboux coordinates on the
constraint surface and that reference frame transformations correspond to
reparametrizations of the constraint surface. We conclude by constructing a
quantum perspective neutral structure, via which we can derive and change
perspective dependent descriptions without referring to the classical theory.
In addition to the physical findings, this work illuminates the interrelation
of first and second class constrained systems and their respective
quantization procedures.

###### Contents

  1. 1 Introduction
    1. 1.1 On quantum reference frames
    2. 1.2 On perspective neutral structures
    3. 1.3 Summary of results
  2. 2 Classical Theory
    1. 2.1 Mathematical preliminaries: constrained Hamiltonian systems
      1. 2.1.1 Constraints through the Legendre transformation
      2. 2.1.2 Computing the consistency conditions
      3. 2.1.3 Gauge transformations by first-class constraints
    2. 2.2 Transformation to relative coordinates
  3. 3 Reduced quantization of second class constraints
    1. 3.1 Mathematical preliminaries: Dirac bracket and Darboux coordinates
    2. 3.2 The classical perspective-neutral structure
    3. 3.3 Perspective-dependent frames
      1. 3.3.1 The relative position frame
      2. 3.3.2 The relative momentum frame
    4. 3.4 Changing quantum reference frames
      1. 3.4.1 Relative position to relative position
      2. 3.4.2 Relative momentum to relative momentum
      3. 3.4.3 Relative position to relative momentum
  4. 4 Towards a first-class theory
    1. 4.1 Mathematical preliminaries: The abelian conversion method
    2. 4.2 The extended phase-space
    3. 4.3 Halfway down: The intermediate phase-spaces
  5. 5 Dirac Quantization of first class constraints
    1. 5.1 Mathematical preliminaries
    2. 5.2 The perspective-neutral structure in the quantum picture
    3. 5.3 Relative position and momentum frame revisited
      1. 5.3.1 The intermediate and relative position frames
      2. 5.3.2 The intermediate and relative momentum frames
    4. 5.4 Changing quantum reference frames via the perspective neutral structure
  6. 6 Recap and Comparison
  7. 7 Limitations and Outlook
  8. A Lagrangian in relative coordinates
  9. B Derivation of consistency conditions
  10. C The unitary position-to-momentum map

##  1 Introduction

###  1.1 On quantum reference frames

In this work, we follow the paradigm that the properties of physical systems
$S$ have no absolute meaning but are only defined relatively to some other
content of the universe. In theoretical classical mechanics and special
relativity the role of this ”other content of the universe” is usually played
by abstract frames of reference. As we treat these frames as external
entities, they must be non-dynamical and sufficiently decoupled from $S$.
Upholding the paradigm, let us think of reference frames as abstractions of
physical systems, for example ideal rigid bodies with imprinted rulers
defining orientations and distances. For such reference frames to not be
affected by $S$, their mass must be much larger than that of $S$, at best
infinite.

Of course, in reality, all reference frames are realised by finite-mass
systems, for example some measurement devices in a laboratory. A fundamental
approach should take the fact seriously that those real reference frames are
subject to the laws of physics. This means that we cannot simply conjure up
new frames, like we do in Galilean or Lorentz transformations. Instead, all
reference frames must already be explicitly included in the dynamical
evolution. Following this line of thought, reference frame transformations can
be naively understood as ”jumping” between distinct parts of our system.

As a second paradigm, we assume the universal validity of quantum mechanics.
Now, if reference frames are realised through physical systems they must also
be subject to the laws of quantum theory (see [13] or [22]). One can for
instance imagine transforming to another laboratory being in superposition to
or entangled with the present one. In accepting both paradigms, we need to
face the question of how to describe physics from the perspective of _quantum
reference frames_ (QRFs) and how to relate between such descriptions.

QRFs have been extensively discussed in literature, with the greatest share of
papers examining topics in quantum information and in finite dimensional
Hilbert spaces [10, 9, 11, 14, 18, 25, 31, 34, 35, 36, 37]. The main focus
lies here on operational aspects of QRFs, such as how to communicate quantum
information without a shared frame of reference. In [2, 3, 8] the relation of
QRFs to superselection rules is discussed. Finally, the role of QRFs in
infinite-dimensional Hilbert spaces and in the context of quantum foundations
is investigated [1, 7, 6, 23, 42].

###  1.2 On perspective neutral structures

Our departing point is the paper ”Quantum mechanics and the covariance of
physical laws in quantum reference frames” [23]. Giacomini, Castro-Ruiz and
Brukner derived the transformations between reference frames attached to
quantum particles without referring to an absolute background. It is shown
that there is no unique way of defining such a transformation but that its
form depends on a choice of preferred coordinates. The findings shine light on
what a theory of ”quantum general covariance” could look like, a notion
relevant in the context of quantum gravity.

The paper ”A change of perspective: switching quantum reference frames via a
perspective-neutral framework” [42] aims to rederive the outcomes of [23] from
first principles, embedding them in a structure better suited for
generalization. Inspired from general relativity and quantum gravity the
mathematical framework of constrained Hamiltonian systems is applied. The
starting point is a Lagrangian with translational invariance. The Legendre-
transformation of this Lagrangian fails to be surjective, resulting in a
Hamilton theory which features a constraint of the total momentum and contains
redundant variables. The resulting structure is thus interpreted as an
observer-independent meta-theory which contains, so to say, all perspectives
at once but is void of direct operational meaning.

Operationally meaningful observer-dependent descriptions can then be obtained
by fixing the redundancies, which classically amounts to a choice of gauge. It
is shown that canonical transformations between two observer-dependent
descriptions can be understood as gauge-transformations. To obtain the
associated quantum picture the authors follow two distinct paths. In the
_reduced quantization_ , the authors fix the redundancies classically before
quantizing the system. This yields the quantum physics as seen from a specific
observer. In the _Dirac quantization_ it is the other way around: One
quantizes first, then fixes the redundancies by a projection on the
constraint-eigenstates. This leads to a perspective-neutral quantum theory.
Vanrietvelde et al. show that one can re-obtain the lot of observer-dependent
descriptions by unitarily rotating the system prior to projecting such that
the constraint acts only on a selected degree of freedom.

###  1.3 Summary of results

Figure 1: A schematic summary of the reduced and the Dirac quantization
scheme. Here, PNS stands for ”perspective-neutral structure” and PDD for
”perspective-dependent description”.

In the opinion of the present author the approach introduced in [42] has one
major downside: the unsymmetrical treatment of positions and momenta. While
the theory features a constraint for the total momentum, there is none for the
center-of-mass. A consequence of this privileged role of the momenta is that
only a fraction of all perspective dependent descriptions can be obtained.
Likewise, only the reference-frame transformations switching between those
frames, constituting only a small subset of all possible canonical
transformations, can be derived. How to embed the remaining ones in this
framework is left unclear.

To address this problem I propose a modified approach, starting with a
transformation to relative coordinates. This leaves us only with gauge-
independent coordinates and eliminates all references to a Newtonian
background already at the Lagrangian level. The associated Hamiltonian theory
is more symmetrical and features two constraints for both relative positions
and momenta. Unfortunately, the constraints do not commute, thus preventing
direct quantization. We tackle this difficulty following two paths.

At first, we employ the reduced quantization scheme. Here we make use of the
so-called _Dirac-bracket_ , which can be understood as the Poisson bracket
restricted to the constraint surface. Its core advantage is that the
constraints Dirac-commute, paving the way for quantization. However, this
comes at a price: We will see that the Dirac bracket is not of canonical form,
i.e. that phase space coordinates referring to different particles do not
commute along the constraint surface in general. In the style of Vanrietvelde
et al. we interpret the set of Dirac-commuting, redundant coordinates as the
classical perspective-neutral structure.

It can then be shown that the correct observer-dependent descriptions are
given by those intrinsic coordinates of the constraint surface, whose Dirac
bracket satisfies the canonical commutation relations. We will focus on two
such descriptions: In the relative position frame two selected position
coordinates match those seen from an external infinite mass-observer. In the
relative momentum frame this holds for two momentum coordinates. While the
former is included in [42], the latter is a new feature of the theory. The
relative frames can now easily be quantized via the standard recipe. But since
the quantization takes place only after having resolved the redundancies one
cannot obtain a full quantum perspective neutral structure via the reduced
quantization scheme.

To fix this, we follow a second path: We further expand the phase space and
modify the constraints to make them commute. It will be shown that partially
gauge fixing this extended space leaves us with a class of symplectomorphic
_auxiliary spaces_. One of them is identical to the perspective-neutral
structure found by Vanrietvelde et al., thus proving that the original theory
is fully embedded in our new framework. Quantizing the extended structure then
yields the sought after quantum perspective-neutral structure.

To re-obtain the perspective-dependent quantum descriptions we carry out a
quantum symmetry reduction. It consists of two unitary transformations, each
followed by a projection on either constraint. We will discover that with the
first unitary we decide between relative position and relative momentum
description, while the second unitary determines which particle will serve as
the reference frame. Finally, we will see how the quantum perspective neutral
structure can be employed to change between quantum reference frames. Figure 1
shows the connections between the various structures in a simplified scheme.

##  2 Classical Theory

###  2.1 Mathematical preliminaries: constrained Hamiltonian systems

In this section we will compactly review the mathematical formalism of
constrained Hamiltonian systems, following largely [26]. The well-informed
reader may skip this part.

####  2.1.1 Constraints through the Legendre transformation

Let us write the Euler-Lagrange-equations for a given Lagrangian $L$ in their
expanded form

$\ddot{q}_{n^{\prime}}J_{n^{\prime}n}=\frac{\partial L}{\partial
q_{n}}-\dot{q}_{n^{\prime}}\frac{\partial^{2}L}{\partial
q_{n^{\prime}}\partial\dot{q}_{n}},$ (1)

where

$J_{n^{\prime}n}=\frac{\partial^{2}L}{\partial\dot{q}_{n^{\prime}}\partial\dot{q}_{n}}.$
(2)

We observe that this linear equation determines the accelerations
$\ddot{q}_{n}$ uniquely only when the Jacobian is invertible, that is when the
determinant

$det\left(\frac{\partial^{2}L}{\partial\dot{q}_{n^{\prime}}\partial\dot{q}_{n}}\right)$
(3)

does not vanish. This translates directly into the Hamilton formalism since
$J_{n^{\prime}n}$ can be written as

$J_{n^{\prime}n}=\frac{\partial p_{n^{\prime}}}{\partial q_{n}}.$ (4)

We see that the non-vanishing of the determinant 3 is the condition for the
invertibility of the velocities as function of the coordinates and momenta.

If the rank of $J_{n^{\prime}n}$ is equal to $N-M$ then the Legendre-
transformation maps onto a $N-M$-dimensional submanifold of the phase-space.
Only configurations contained in this subspace are physically meaningful. The
constraint surface can be described by $M$ constraints

$\varphi_{m}(q,p)\simeq 0,\quad m=1...M.$ (5)

We have introduced the _weak equality symbol_ $\simeq$ to emphasize that the
quantity $\varphi_{m}$ is numerically restricted to be zero, but does not
vanish throughout phase space [26]. This means that the derivatives of
$\varphi_{m}$ do not vanish on the constraint surface in general. Therefore,
the $\varphi$’s have in general nonzero Poisson brackets with other phase-
space functions or the canonical coordinates and we must be cautious not to
solve the constraints before calculating the Poisson brackets. An equation
that holds everywhere in phase-space is called _strong_ and will be denoted by
the regular equality sign ”$=$”.

Figure 2: Illustration of the relation between velocities and momenta in a
constrained system. The figure shows the Legendre transformation of a
2-dimensional system with the Lagrangian $L=1/2(\dot{q}_{1}-\dot{q}_{2})^{2}$.
The momenta are $p_{1}=\dot{q}_{1}-\dot{q}_{2}$ and
$p_{2}=\dot{q}_{2}-\dot{q}_{1}$. Since $p_{1}=-p_{2}$ the tuple
$(p_{1},p_{2})$ describes only a 1-dimensional subspace of the
$(\dot{q}_{1},\dot{q}_{2})$-space and all the $\dot{q}$’s on the line
$\dot{q}_{2}-\dot{q}_{1}=c$ are mapped on the same point $p_{2}=-p_{1}=c$. We
thus need additional parameters $y_{m}(q,\dot{q})$ to render the
transformation invertible.

In the presence of constraints the variations $\delta q_{n}$ and $\delta
p_{n}$ are no longer independent as we only allow for variations that are
tangent to the constraint surface. We can equally state that all linear
combinations of constraint variations must be weakly zero, i.e.

$y_{m}\delta\varphi_{m}=y_{m}\frac{\partial\varphi_{m}}{\partial p_{n}}\delta
p_{n}+y_{m}\frac{\partial\varphi_{m}}{\partial q_{n}}\delta q_{n}\simeq 0.$
(6)

Comparing this equation with

$\left(\frac{\partial H}{\partial q_{n}}+\frac{\partial L}{\partial
q_{n}}\right)\delta q_{n}+\left(\frac{\partial H}{\partial
p_{n}}-\dot{q}_{n}\right)\delta p_{n}=0.$ (7)

and making use of the Euler-Lagrange equations leads to the Hamilton equation
of motion for a constrained system

$\displaystyle\begin{split}\dot{q}_{n}&=\frac{\partial H}{\partial
p_{n}}+y_{m}\frac{\partial\varphi_{m}}{\partial p_{n}}=\frac{\partial
H_{T}}{\partial p_{n}}\\\ \dot{p}_{n}&=-\frac{\partial H}{\partial
q_{n}}-y_{m}\frac{\partial\varphi_{m}}{\partial q_{n}}=\frac{\partial
H_{T}}{\partial q_{n}}.\end{split}$ (8)

Here we have introduced the _total Hamiltonian_

$H_{T}=H(q,p)+y_{m}(q,p,t)\varphi_{m}(q,p)$ (9)

as the original Hamiltonian plus linear combinations of the constraints.

$H_{T}$ is only well defined on the physical constraint surface and can be
extended arbitrarily in the extended space. Choosing a specific extension
corresponds to choosing the free auxiliary parameters $y_{m}$. As all
physically sensible configuration must lie on the constraint surface this
extension does not change their respective energy values.

With the total Hamiltonian we can write the total time derivative of any
function as

$\displaystyle\begin{split}\dot{F}(q,p)&=[F,H_{T}]=[F,H]+[F,y_{m}\varphi_{m}]\\\
&=[F,H]+[F,y_{m}]\varphi_{i}+y_{m}[F,\varphi_{m}]\\\
&\simeq[F,H]+y_{m}[F,\varphi_{m}].\end{split}$ (10)

####  2.1.2 Computing the consistency conditions

Consistency requires that the constraints be preserved in time, i.e that their
total time derivative vanishes on the constraint surface. So plugging them
into equation 10 gives rise to the _consistency conditions_

$\displaystyle\begin{split}\dot{\varphi}_{m}=[\varphi_{m},H]+y_{m^{\prime}}[\varphi_{m},\varphi_{m^{\prime}}]\simeq
0,\end{split}$ (11)

with

$m,m^{\prime}=1,\dotso,M.$ (12)

These equations reduce to one of the following three types [19]:

  * Cycle:

If $\varphi_{m}$ commutes with all $\varphi_{m^{\prime}}$ but not with $H$, we
obtain a relation of the form $\varphi_{k}(p,q)\simeq 0$. If linearly
independent of the former constraints, this relation consitutes a further
constraint on the system. It is evident that this new constraint must be
conserved as well, leading to a new consistency condition and so forth. The
cycle continues until we reach one of the stops below.

  * Stop 1:

If $\varphi_{m}$ does not commute with at least one constraint, we do not
obtain a new constraint but a relation for the $y$’s.

  * Stop 2:

The equation reduces to either $0=0$ or $1=0$. The first case is automatically
satisfied and leads to no further constraint, while the second one points to
an inconsistency in the equations of motions. Such theories start from self-
contradictory Lagrangians and are of no interest.

Having exhausted all equations we are left with a complete set of $J$
constraints. We can add them to the total Hamiltonian and let the consistency
conditions run now over all constraints

$[\varphi_{j},H]+y_{j^{\prime}}[\varphi_{j},\varphi_{j^{\prime}}]\simeq
0,\quad j,j^{\prime}=1,\dotso,J.$ (13)

This set of linear equation is over-complete, since a part of it has already
been solved to derive the secondary constraints. We see that the solutions
depend crucially on the matrix of commutators

$\Delta_{jj^{\prime}}=[\varphi_{j},\varphi_{j^{\prime}}].$ (14)

If it is invertible then we can find unique solutions for all $y_{j}$ and thus
the Hamiltonian and the time evolution of the system is unique too. If,
however, $\Delta_{jj^{\prime}}$ has a vanishing determinant, then there are
some zero eigenvectors and not all $y_{m}$ are uniquely determined.

Let us quickly introduce some new terminology. In [20] Dirac referred to any
function on phase-space as _first class_ , whose Poisson bracket with every
constraint is weakly equal zero

$[F,\varphi_{j}]\simeq 0\quad j=1...J.$ (15)

The Hamiltonian, for example, is first class by design. A function which has
at least one non-vanishing Poisson bracket with any constraint is called
_second class_.

We now perform linear transformations $\varphi_{j}\rightarrow
a_{jj^{\prime}}\varphi_{j^{\prime}}$ on the constraints. Our aim is to split
the such transformed constraints into a subset $\gamma_{\alpha}$ of _first
class constraints_ and a subset $\Phi_{\alpha}$ of _second class constraints_.
As combinations of weakly vanishing quantities vanish again weakly, the linear
transformation does not change the theory. After having successfully carried
out the transformation we can write the matrix of commutators in the form

$\displaystyle\Delta=\begin{pmatrix}[\gamma_{a},\gamma_{b}]&[\Phi_{\alpha},\gamma_{b}]\\\\[4.0pt]
[\gamma_{a},\Phi_{\beta}]&[\Phi_{\alpha},\Phi_{\beta}]\end{pmatrix}=\begin{pmatrix}0&0\\\
0&C_{\alpha\beta}\end{pmatrix}$ (16)

The first class constraints $\gamma_{a}$ do not impose any condition on their
respective $y_{a}$, which means that $y_{a}$ are arbitrary functions of the
phase-space coordinates and time. The second class constraints
$\Phi_{\alpha}$, on the contrary, have well defined $y_{\alpha}$, which we can
compute by means of the equations 13. Finally we can write our new total
Hamiltonian as the sum of the original Hamiltonian plus linear combinations of
all constraints

$\displaystyle\begin{split}H_{T}&=H+y_{\alpha}\Phi_{\alpha}+y_{a}(p,q,t)\gamma_{a}\\\
&=H^{\prime}+y_{a}(p,q,t)\gamma_{a},\end{split}$ (17)

where

$H^{\prime}=H+y_{\alpha}\Phi_{\alpha}.$ (18)

####  2.1.3 Gauge transformations by first-class constraints

Note that the above total Hamiltonian $H_{T}$ decomposes into two parts. While
$H^{\prime}$, containing the original Hamiltonian plus the second-class
constraints, generates a deterministic evolution, the contribution from
$y_{a}(p,q,t)\gamma_{a}$ is completely arbitrary.

Let us take e.g. $f(q,p)=q$ and compute its value after an infinitesimal time
span for distinct values of the $y_{a}$. We will obtain two different
outcomes, their difference being

$\Delta q(t_{0}+\delta t)=\delta t(y_{a}-y^{\prime}_{a})[q,\gamma_{a}].$ (19)

As the time-dependence of the functions $y_{a}$ is not accessible, we have no
means of saying anything about the evolution of $q$. This is of course
incompatible with the deterministic character of classical mechanics. If we
want unique solutions, it must be that operationally we cannot distinguish
between two configurations $(q_{i},p_{i})$ and
$(q^{\prime}_{i},p^{\prime}_{i})$ connected by a so-called _gauge
transformation_

$\displaystyle\begin{split}q_{i}\rightarrow
q^{\prime}_{i}&=q_{i}+y_{a}[q_{i},\gamma_{a}]\\\ p_{i}\rightarrow
p^{\prime}_{i}&=p_{i}+y_{a}[p_{i},\gamma_{a}].\\\ \end{split}$ (20)

This means that even though a physical state is fully determined by the
variables $(q,p)$, the converse is not true: A state does not uniquely
determine a point in phase-space of $q$’s and $p$’s but a whole set of points.
From equation 20 we learn that the gauge transformations connecting these
points are generated by the first class constraints, which is in accordance
with Noether’s theorem [30].

Furthermore, not all functions on phase-space correspond to physically
measurable quantities. It is only the gauge invariant functions, that are
those whose Poisson bracket with every first-class constraint vanishes weakly,
which can be regarded as physically meaningful observables.

Second class constraints do not act as gauge generators, as they would map
states out of the constraint surface. This can be seen by taking any second
class constraint $\Phi_{\alpha}$ and checking the transformation induced by
any other second class constraint $\Phi_{\beta}$

$\Phi_{\alpha}\rightarrow\Phi_{\alpha}^{\prime}=\Phi_{\alpha}+\epsilon[\Phi_{\alpha},\Phi_{\beta}]\not\simeq
0.$ (21)

###  2.2 Transformation to relative coordinates

In this section, we will work out the central structure of our paper, the
second-class constrained Hamiltonian system. Departing point is a toy-model
Lagrangian of three free particles in one dimension, reading

$L(x,\dot{x})=\frac{1}{2}\sum_{i=1}^{3}m_{i}\dot{x}_{i}^{2}-\frac{M}{2}\dot{x}^{2}_{cm},$
(22)

where

$x_{cm}=(\sum_{i=1}^{3}m_{i}x_{i})/M$

is the center-of-mass-position and

$M=\sum_{i=1}^{3}m_{i}$

the total mass of the system. By subtracting the center-of-mass-momentum we
have rendered the Lagrangian invariant under global translations

$(q_{i},\dot{q}_{i})\rightarrow(q_{i}+f(t),\dot{q}_{i}+\dot{f}(t))$

where $f(t)$ is an arbitrary function of time. Because of this gauge symmetry
the physical description depends only on relative degrees of freedom but not
on the choice of external reference frame [42].

Figure 3: The figure shows the transformation 23 schematically. We notice that
after the transformation $q_{cm}$ remains the only coordinate with respect to
an external observer and that the $q_{i}$ are redundant.

The starting idea of this paper is to make the reference-frame independence
explicit by going over to the coordinates

$\displaystyle\begin{split}x_{1}-x_{2}&=q_{3}\\\ x_{3}-x_{1}&=q_{2}\\\
x_{2}-x_{3}&=q_{1}\\\ \frac{1}{M}\sum_{i=1}^{3}m_{i}x_{i}&=q_{cm}.\end{split}$
(23)

The three relative coordinates $q_{i}$ are invariant under spatial
translations, while the center-of-mass coordinate remains gauge dependent and
still refers to the Newtonian background.

In geometrical terms the transformation 23 corresponds to a mapping from
$\mathbb{R}^{3}$ into $\mathbb{R}^{4}$, reaching only points on the
3-dimensional hypersurface

$q_{1}+q_{2}+q_{3}\equiv Q=0.$ (24)

It follows that the $q_{i}$ are redundantly defined; we can always express one
of the relative distances by means of the other two.111In classical mechanics
constraints are often introduced to approximate very strong and short-ranged
forces (see for example [27]). Not so in this case, where the constraint is
truly a logical necessity and any deviation from the constraint surface is
strictly prohibited by the definition of the coordinates.

The Lagrangian 22 expressed in the new coordinates reads

$L^{\prime}(q,\dot{q})=\frac{1}{2}\sum_{i=1}^{3}\mu_{i}\dot{q}_{i}^{2}+q_{4}(q_{1}+q_{2}+q_{3})$
(25)

where we have introduced the, let us call them, reduced masses

$\mu_{i}=\frac{m_{j}m_{k}}{M},\quad i\neq j\neq k.$ (26)

and added the constraint $Q$ together with a Lagrange multiplier $q_{4}$. The
calculation is given in appendix A.

Given only the Lagrangian 25, the role of $q_{4}$ as a Lagrange multiplier is
not apparent. It follows that $q_{4}=q_{4}(t)$ can be treated as just another
dynamical variable and that it enters the formalism on equal footing with the
first three $q_{i}$ [38]. We observe that the Lagrangian, as expected, does
not depend on the gauge-dependent center-of-position $q_{cm}$. This degree of
freedom is of no physical relevance and will be discarded.

Let us now apply the methods established throughout section 2.1 to obtain the
Hamilton theory. We start by computing the conjugate momenta

$\displaystyle\begin{split}p_{i}&=\frac{\partial
L}{\partial\dot{q}_{i}}=\mu_{i}\dot{q}_{i}\quad i=1,2,3\\\ p_{4}&\simeq
0.\end{split}$ (27)

There is one constraint $p_{4}\simeq 0$, which is added with yet another
Lagrange multiplier $y_{i}$ to the standard Hamiltonian $H^{\prime}$ yielding

$\displaystyle\begin{split}H&\simeq H^{\prime}(q,p)+y_{i}\Phi_{i}\\\
&\simeq\sum_{i}p_{i}\dot{q}_{i}(q,p)-L(q,\dot{q}_{i}(q,p))+y_{i}\Phi_{i}\\\
&\simeq\frac{1}{2}\sum_{i=1}^{3}\left(\frac{p_{i}^{2}}{\mu_{i}}-q_{4}q_{i}\right)+yp_{4}.\end{split}$
(28)

Starting from Hamiltonian 28 we need to work through the consistency scheme
discussed in section 2.1.2. The calculation, done in detail in appendix B,
yields the Hamiltonian

$H=\frac{1}{2}\sum_{i}\frac{p_{i}^{2}}{\mu_{i}}.$ (29)

together with 2 effective constraints

$\Phi_{1}=\sum_{i}q_{i},\quad\Phi_{2}=\sum_{i}\frac{p_{i}}{\mu_{i}}.$ (30)

In the _original phase-space_ $\mathcal{P}_{orig}\sim\mathbb{R}^{6}$,
coordinatized by the $q_{i}$ and $p_{j}$, they define a 4-dimensional
constraint surface $\Phi_{1}\simeq\Phi_{2}\simeq 0$, on which all physically
sensible configurations must lie. Clearly, they are second-class as

$[\Phi_{1},\Phi_{2}]=\sum_{i}\frac{1}{\mu_{i}}.$ (31)

##  3 Reduced quantization of second class constraints

###  3.1 Mathematical preliminaries: Dirac bracket and Darboux coordinates

If there are no constraints present and the phase-space is linear there is a
standard quantization recipe (see e.g. Dirac [21], p. 84ff). It tells us to
assign to each function an operator acting on some Hilbert space $\mathcal{H}$
such that the commutator of quantum operators equals $i\hbar$ times the
operator of the Poisson bracket

$[\hat{F},\hat{G}]=i\hbar\widehat{[F,G]}.$ (32)

In the presence of second-class constraints the quantization procedure is more
involved. Classically, the constraints restrict the system to a submanifold of
the original phase space. We would like to mirror this behaviour in the
quantum theory by imposing the restrictions

$\hat{\Phi}_{\alpha}\ket{\phi}_{phys}=0\quad\forall\;\alpha$ (33)

on the state vectors $\ket{\phi}_{phys}$ which span the physical Hilbert space
$\mathcal{H}_{phys}$. From the consecutive application of constraints
$\hat{\Phi}_{\alpha}\hat{\Phi}_{\beta}\ket{\phi}_{phys}=0$ and
$\hat{\Phi}_{\beta}\hat{\Phi}_{\alpha}\ket{\phi}_{phys}=0$ it follows readily
that

$[\hat{\Phi}_{\alpha},\hat{\Phi}_{\beta}]\ket{\phi}_{phys}=0.$ (34)

The commutator of second class constraints however is not a constraint itself
so that equation 34 is only satisfied when $\ket{\phi}_{phys}=\ket{0}$. But
then $\mathcal{H}_{phys}=\\{\ket{0}\\}$ and the theory is trivial [28].

The difficulties originate already in the classical theory: While the
constraints vanish on the constraint surface, their first derivatives in
general do not, leaving the Poisson bracket ambiguous. This means, for a
function $f$ on the phase space and a constraint $\Phi\simeq 0$ we have
$f+\Phi\simeq f$, but in general

$[f,g]\not\simeq[f+\Phi,g].$ (35)

A solution to both the problems in classical and quantum theory was developed
by P.A.M. Dirac in 1950 (see [19] and for a detailed derivation [20]). He
generalized the original Poisson bracket to the _Dirac bracket_ , given by

$\\{F,G\\}=[F,G]-[F,\Phi_{\alpha}]C^{\alpha\beta}[\Phi_{\beta},G]$ (36)

where $C^{\alpha\beta}$ is the inverse of the second-class bracket-matrix
$C_{\alpha\beta}$ appearing in 16. The core advantage is that the Dirac
bracket of an arbitrary function with any second class constraint vanishes:

$\\{\Phi_{\alpha},F\\}=0.$ (37)

From a geometrical viewpoint, the Dirac bracket is related to the pull-back of
the symplectic two-form onto the constraints surface and picks up only those
variations tangent to it [26]. As a consequence, there are in general always
contributions from non-Poisson-commuting observables. This entails that
Poisson-commuting observables are in general not Dirac-commuting. We will see
this concretely in the next section.

By virtue of 37, the Dirac bracket, unlike the Poisson bracket, is
unambiguous, i.e.

$f^{\prime}\simeq f\rightarrow\\{f^{\prime},\cdot\\}\simeq\\{f,\cdot\\},$

such that we can set the constraints to zero before computing the bracket. To
quantize a second-class system, we simply replace the Poisson bracket by the
Dirac Bracket in the correspondence rule

$[\hat{F},\hat{G}]=i\hbar\widehat{\\{F,G\\}}.$ (38)

The constraint operators can now be given any c-values, in particular
$\Phi_{\alpha}=0$, enforcing the restriction 33. When we have found a
representation of 38 we have found the quantum theory.

  * Caveat:

This is a highly nontrivial problem which has no general solution. It can be
solved only in special instances, for example when the Dirac bracket amounts
to c-numbers. Luckily, this is exactly the case for our system at hand.

There is yet an alternative to trying to solve 38 directly. With the Dirac
bracket, the second-class constraints can be treated as strong identities
expressing some canonical coordinates in terms of others. This allows us to go
over to _intrinsic_ (local) coordinates
$u_{i}(q,p),\pi_{i}(q,p),i=1,\dots,N-M$ spanning the constraint surface.
Expressed in the new coordinates the constraints
$\Phi_{\alpha}(q_{n}(u_{i},\pi_{i}),p_{n}(u_{i},\pi_{i}))$ vanish identically
[29]. Of special interest are those intrinsic coordinates which Dirac-commute
according to the canonical commutation relations

$\displaystyle\\{u_{i}(q,p),\pi_{j}(q,p)\\}$ $\displaystyle=\delta_{ij}$ (39)
$\displaystyle\\{u_{i}(q,p),u_{j}(q,p)\\}$
$\displaystyle=\\{\pi_{i}(q,p),\pi_{j}(q,p)\\}=0.$ (40)

They are referred to as _Darboux coordinates_ [29]. When they are found, we
have obtained a _reduced_ theory, with no constraints present and no
redundancies left. The Darboux coordinates are nicely canonical so that from
this point on we can follow the standard quantization procedure to obtain the
quantum theory.

  * Caveat:

Darboux coordinates are only defined up to a canonical transformation, but
canonical quantization and canonical transformations are generally non-
commutative operations. They do commute only for linear and point
transformations (in both $q$ and $p$) [15].

Last but not least it is worth noticing that the Dirac bracket with any first
class constraint is weakly equal to the Poisson bracket. Thus the equations of
motions remain invariant by the replacement

$\dot{F}=\\{F,H\\}\simeq[F,H].$ (41)

###  3.2 The classical perspective-neutral structure

We will now apply the reduced quantization scheme to our theory at hand. At
first, let us compute the Dirac-bracket. The bracket-matrix 14 reads

$\displaystyle C=\begin{pmatrix}0&\mu\\\ -\mu&0\\\ \end{pmatrix}$ (42)

where we have introduced

$\mu=\sum_{i}\frac{1}{\mu_{i}}.$ (43)

Since either $C^{-1}$ or the Poisson bracket of coordinates and constraints
vanish, most of the terms in 36 drop out and we obtain

$\displaystyle\begin{split}\\{q_{i},q_{j}\\}&=0\\\ \\{p_{i},p_{j}\\}&=0\\\
\\{q_{i},p_{j}\\}&=\delta_{ij}-\frac{1}{\mu\mu_{i}}.\end{split}$ (44)

In the style of Vanrietvelde et al. I now propose to interpret the Dirac
bracket 44 along with the constraints 30 and the Hamiltonian 29 as the
_classical perspective-neutral structure_ , living in the 6-dimensional phase
space $\mathcal{P}_{orig}.$ As all relative variables are treated on equal
footing it contains all perspectives at ones while the inherent redundancies
prevent an immediate operational interpretation.222Note that position and
momentum observables of different particles do NOT Dirac-commute. Because as
we restrict to the constraint surface, a variation of one of the $q_{i}$
necessarily entails some variation of the other relative position coordinates.
This holds as well for the momenta. This effect might come unexpected, but can
actually be easily observed in nature when we use our own body as reference
frame. If we push planet earth downward by muscle force, we must accelerate in
the opposite direction to satisfy conservation of momentum (a process more
commonly referred to as ”jumping”). We can now see the stars in the zenith
moving towards us, though they are not affected by any force.

How do we continue from here to find the perspective-dependent descriptions?
It seems plausible to assume that choosing a certain perspective must somehow
relate to getting rid of the redundancies. We know that we can do so by going
over to intrinsic coordinates $u_{i}(q,p),\pi_{i}(q,p)$, spanning the
4-dimensional $\mathcal{P}_{persp}$. Since there is no unique way, let us try
the most obvious and define

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\begin{pmatrix}u_{2}-u_{3}\\\ u_{3}\\\
-u_{2}\end{pmatrix}\\\ \begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&=\begin{pmatrix}-\mu_{1}\left(\frac{\pi_{2}}{\mu_{3}}-\frac{\pi_{3}}{\mu_{2}}\right)\\\
\pi_{3}\\\ -\pi_{2}\end{pmatrix},\end{split}$ (45)

according to figure 4. Observe the changed numbering of the intrinsic
coordinates and the additional minus sign.

The Dirac bracket can now be expressed as

Figure 4: The figure shows a naive transformation to internal coordinates 45.
The position vectors $q_{i}$ on the left hand side live in the configuration
space. Displayed on the right hand side are the momentum vectors $p_{j}$,
living in momentum space. Observe that we have changed the numbering and
direction of the intrinsic coordinates, such that $u_{i},\pi_{i}$ refer to the
i’th particle.

$\displaystyle\begin{split}\\{u_{i},u_{j}\\}&=0\\\
\\{\pi_{i},\pi_{j}\\}&=0\qquad\qquad\qquad i,j=2,3.\\\
\\{u_{i},\pi_{j}\\}&=\delta_{ij}-\frac{1}{\mu\mu_{i}}\end{split}$ (46)

These intrinsic observables $u_{i}$ and $\pi_{j}$ do still not commute for
$i\neq j$. From a mathematical viewpoint, the impossibility to find
simultaneous eigenvectors of the $\hat{u}$’s and $\hat{p}$’s even for distinct
particles follows. Therefore, from the perspective of the QRF attached to
particle 1 the Hilbert space of the two other particles can not be partitioned
as $H_{2,3}=H_{2}\otimes H_{3}$ and the standard quantization recipe is bound
to fail. Thinking in physical terms, the commutation relations 46 mean that
$u_{i}$ and $\pi_{j}$ cannot be measured simultaneously and that there exist
uncertainty relations between those variables. Have we thus finally
encountered a paradox? Why should a position measurement of one particle
affect the momentum of another, completely unrelated particle? The solution to
this alleged contradiction lies again in the finite mass $m$ of the system
serving as reference frame and has already been presented in [1, 7, 6]. From
the uncertainty principle it follows that if one wants to measure the position
of a particle with an accuracy $\Delta x$ a finite amount of momentum

$\Delta p>\frac{\hbar}{\Delta x}$ (47)

has to be exchanged. The system serving as reference frame is therefore
boosted by an uncertain amount $\frac{\Delta p}{m}$. In classical mechanics
one can always keep the amount of exchanged momentum arbitrary small, such
that $\Delta p<<m$. In the quantum regime, however, this approximation is no
longer valid as masses are small and $\Delta p$ increases with measurement
accuracy. Here, one can always envisage a measurement where $\Delta p\sim m$,
resulting in a non negligible kickback of the reference system [1].

Note that there is no configuration of the masses which reproduces the
standard bracket structure for all three coordinate pairs. If, however, one of
the particles masses is much larger than the others, this particle can serve
as a classical reference frame. For example, for $m_{1}>>m_{2}\sim m_{3}$ the
mass term $\frac{m_{i}}{M}=\frac{1}{\mu\mu_{i}}$ converges to zero and we re-
obtain the canonical commutation relations. Further, $\mu_{1}\rightarrow 0$
and $\mu_{2}\rightarrow m_{3},\mu_{3}\rightarrow m_{2}$ such that the
perspective-neutral Hamiltonian 29 converges to the standard Hamiltonian for a
two-particle system. This is an important result. It shows that the standard
way of describing physics from an infinite-mass frame is fully embedded in our
newly developed framework.

###  3.3 Perspective-dependent frames

From the failing of the our first attempt we learn that not all sets of
intrinsic coordinates are physically useful. If we want observer-dependent
descriptions which can be canonically quantized for arbitrary mass
configurations, we must search for the Darboux coordinates. To keep it simple
and not encounter any mathematical pitfalls we restrict to those which are
linear in and do not mix the the original positions and momenta. Their most
general form is given by

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}=\begin{pmatrix}m_{1A}u_{A}+m_{1B}u_{B}\\\
m_{2A}u_{A}+m_{2B}u_{B}\\\ m_{3A}u_{A}+m_{3B}u_{B}\end{pmatrix}\\\
\begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}=\begin{pmatrix}n_{1A}\pi_{A}+n_{1B}\pi_{B}\\\
n_{2A}\pi_{A}+n_{2B}\pi_{B}\\\
n_{3A}\pi_{A}+n_{3B}\pi_{B}\end{pmatrix}.\end{split}$ (48)

The 12 coefficients must satisfy 4 constraints

$\displaystyle\sum_{i}m_{iA}$ $\displaystyle=0$
$\displaystyle\qquad\sum_{i}\frac{n_{iA}}{\mu_{i}}$ $\displaystyle=0$ (49)
$\displaystyle\sum_{i}m_{iB}$ $\displaystyle=0$
$\displaystyle\qquad\sum_{i}\frac{n_{iB}}{\mu_{i}}$ $\displaystyle=0$

as well as 9 Darboux conditions

$m_{iA}n_{jA}+m_{iB}n_{jB}=\delta_{ij}-\frac{1}{\mu\mu_{i}}$ (50)

with

$i,j=1,2,3.$ (51)

The Darboux conditions enforce the canonical commutation relations between the
intrinsic coordinates. Of these 13 constraints only 8 are linearly
independent, leaving us with 4 free coefficients.

Still, we are left with a plethora of possible descriptions. How do we choose
among them? I propose to coordinatize the constraint surface by two of the
$q_{i}$ or $p_{j}$ and then to complete the remaining coordinates according to
49 \- 50. Following this line of thought, I want to lay emphasis on two
internal descriptions. For $m_{1}\rightarrow\infty$ both of them will converge
to the ”classical” description from an external infinite-mass frame.

####  3.3.1 The relative position frame

Let us choose $m_{2A}=-m_{3B}=1$ and $m_{2B}=m_{3A}=0$ and set the
placeholders to $A=3,B=2$, such that the position coordinates are named
according to figure 4. By applying the constraints and working through the
Darboux conditions we arrive at

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\begin{pmatrix}u_{2}-u_{3}\\\ u_{3}\\\
-u_{2}\end{pmatrix}\\\ \begin{pmatrix}p_{1}\\\\[4.0pt] p_{2}\\\\[4.0pt]
p_{3}\end{pmatrix}&=\frac{1}{\mu}\begin{pmatrix}(\frac{\pi_{2}}{\mu_{3}}-\frac{\pi_{3}}{\mu_{2}})\\\\[4.0pt]
\frac{1}{\mu_{3}}\pi_{2}+(\mu-\frac{1}{\mu_{2}})\pi_{3}\\\\[4.0pt]
-(\mu-\frac{1}{\mu_{3}})\pi_{2}-\frac{1}{\mu_{2}}\pi_{3}\end{pmatrix}.\end{split}$
(52)

In this description we have a direct expression for the distances between
particle 1 & 3 and 1 & 2, respectively, while the distance between 2 & 3 can
only be indirectly computed through $u_{2}-u_{3}$. It seems therefore
reasonable to interpret 52 as the physics seen from the perspective of
particle 1. The internal position observables are in accordance with those
seen from an external infinite-mass frame, while the internal momentum
observables are linear combinations thereof. By such reasoning we name this
set of variables the _relative position frame of particle 1_.

Plugging 52 into the perspective-neutral Hamiltonian 29 yields

$H_{q1}=\frac{1}{2\mu}\Big{(}(\frac{\mu}{\mu_{3}}-\frac{1}{\mu_{3}^{2}})\pi_{2}^{2}+(\frac{\mu}{\mu_{2}}-\frac{1}{\mu_{2}^{2}})\pi_{3}^{2}+2\frac{\pi_{2}\pi_{3}}{\mu_{2}\mu_{3}}\Big{)}$
(53)

which looks tidier when expressed in the original masses

$H_{q1}=\frac{1}{2}\left(\frac{m_{1}+m_{2}}{m_{1}m_{2}}\pi_{2}^{2}+\frac{m_{1}+m_{3}}{m_{1}m_{3}}\pi_{3}^{2}+2\frac{\pi_{2}\pi_{3}}{m_{1}}\right).$
(54)

The index q next to $H$ indicates that we are we are in the relative position
representation and the 1 that we are using particle 1 as reference frame.
Observe that apart from the potential energy, which we have assumed to be
zero, our outcome is identical to the findings of [42]. The modification to
other particles follows readily.

####  3.3.2 The relative momentum frame

Let us now choose $n_{2A}=-n_{3B}=1$ and $n_{2B}=n_{3A}=0$ and set the
placeholders again to $A=3,B=2$, such that the momentum coordinates are named
according to the right side of figure 4. In this setup we coordinatize the
constraint surface by

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\frac{1}{\mu}\begin{pmatrix}\frac{1}{\mu_{1}}(v_{3}-v_{2})\\\
\frac{v_{2}}{\mu_{2}}+(\mu-\frac{1}{\mu_{2}})v_{3}\\\
-(\mu-\frac{1}{\mu_{3}})v_{2}-\frac{v_{3}}{\mu_{3}}\end{pmatrix}\\\
\begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&=\begin{pmatrix}-\mu_{1}(\frac{\rho_{3}}{\mu_{2}}-\frac{\rho_{2}}{\mu_{3}})\\\
\rho_{3}\\\ -\rho_{2}.\end{pmatrix}\end{split}$ (55)

Observe that now only the momenta of particle 2 & 3 correspond to the external
view, while the remaining momentum and all positions are expressed through
linear combinations of the $v$’s and $\rho$’s. We will call this set of
internal observables the _relative momentum frame of particle 1_. In this
frame the Hamiltonian reads

$H_{p1}=\frac{1}{2}\Big{(}\frac{\mu_{1}+\mu_{2}}{\mu_{2}^{2}}\rho_{3}^{2}+\frac{\mu_{1}+\mu_{3}}{\mu_{3}^{2}}\rho_{2}^{2}-\frac{2\mu_{1}}{\mu_{2}\mu_{3}}\rho_{2}\rho_{3}\Big{)}.$
(56)

The relative momentum frame cannot be derived directly via the framework of
[42]. This shows that while encompassing its precursor’s finding, the approach
proposed in this thesis shines further light on the theory of QRFs.

###  3.4 Changing quantum reference frames

Our next objective is to find the connection between canonical QRF-
transformations and the perspective neutral structure. In stark contrast to
the work of Vanrietvelde et al., in our framework, reference frame
transformations cannot amount to gauge transformations, as there is no gauge-
freedom in purely second-class systems. In exchange, we have the freedom to
choose any Darboux-parametrization of the constraint surface, each of which is
corresponding to a perspective dependent description. As all sets of Darboux
coordinates are connected via a canonical transformation, we can conclude: To
canonically transform between two perspective dependent descriptions means to
Darboux-reparametrize the constraint surface.

Finding the (quantum) canonical transformation connecting two descriptions is
a three-step-process:

  1. 1.

We write the original, relative phase space variables $(q,p)$ in terms of the
two sets of Darboux coordinates $(u_{j},\pi_{j})$ and
$(\bar{u}_{j},\bar{\pi}_{j})$.

  2. 2.

We use this embedding in the perspective neutral structure to find the
expression of one set of coordinates in terms of the other, i.e.

$\displaystyle\bar{u}_{i}$ $\displaystyle=a_{ij}u_{j}$ (57)
$\displaystyle\bar{\pi}_{i}$ $\displaystyle=b_{ij}\pi_{j}.$

  3. 3.

To obtain the transformation in the quantum picture we compute the unitary
representation of the given canonical transformation, e.g. by the methods
derived in [33] or by means of appendix C.

Let us make this procedure explicit in three examples, where we set $m_{i}=1$
for simplicity.

####  3.4.1 Relative position to relative position

If we want to compute the transformation between two relative position frame,
for instance of particle 1 and 2, we write

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\begin{pmatrix}u_{2}-u_{3}\\\ u_{3}\\\
-u_{2}\end{pmatrix}=\begin{pmatrix}-\bar{u}_{3}\\\ \bar{u}_{3}-\bar{u}_{1}\\\
\bar{u}_{1}\end{pmatrix}\\\ \begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&=\frac{1}{3}\begin{pmatrix}\pi_{2}-\pi_{3}\\\
\pi_{2}+2\pi_{3}\\\
-2\pi_{2}-\pi_{3}\end{pmatrix}=\frac{1}{3}\begin{pmatrix}-\bar{\pi}_{1}-2\bar{\pi}_{3}\\\
-\bar{\pi}_{1}+\bar{\pi}_{3}\\\
2\bar{\pi}_{1}+\bar{\pi}_{3}\end{pmatrix}.\end{split}$ (58)

From this we can read off the canonical transformation

$\displaystyle u_{2}$ $\displaystyle=-\bar{u}_{1},$ $\displaystyle\qquad
u_{3}$ $\displaystyle=\bar{u}_{3}-\bar{u}_{1}$ (59) $\displaystyle\pi_{2}$
$\displaystyle=-(\bar{\pi}_{1}+\bar{\pi}_{3})$ $\displaystyle\qquad\pi_{3}$
$\displaystyle=\bar{\pi}_{3}.$

Giacomini et.al [23] found that the transformation 59 can be represented by
the unitary operator

$\displaystyle H_{2}\otimes H_{3}\rightarrow H_{3}\otimes H_{1}$ (60)
$\displaystyle\hat{S}_{q1\rightarrow
q2}=\hat{\mathcal{Q}}_{2,1}e^{i\hat{u}_{2}\hat{\pi}_{3}}$

where $\hat{\mathcal{Q}}_{A,B}$ is the position parity-swap operator, acting
on vectors in $H_{A}\otimes H_{B}$ as

$\hat{\mathcal{Q}}_{A,B}\ket{x}_{A}\otimes\ket{\Psi}_{B}=\ket{\Psi}_{B}\otimes\ket{-x}_{C}.$
(61)

####  3.4.2 Relative momentum to relative momentum

We write out the relative momentum representations of particle 1 and 2:

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\frac{1}{3}\begin{pmatrix}u_{2}-u_{3}\\\ u_{2}+2u_{3}\\\
-2u_{2}-u_{3}\end{pmatrix}=\frac{1}{3}\begin{pmatrix}-\bar{u}_{1}-2\bar{u}_{3}\\\
-\bar{u}_{1}+\bar{u}_{3}\\\ 2\bar{u}_{1}+\bar{u}_{3}\end{pmatrix}\\\
\begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&=\begin{pmatrix}\pi_{2}-\pi_{3}\\\ \pi_{3}\\\
-\pi_{2}\end{pmatrix}=\begin{pmatrix}\bar{\pi}_{3}\\\
\bar{\pi}_{3}-\bar{\pi}_{1}\\\ \bar{\pi}_{1}\end{pmatrix}.\end{split}$ (62)

Again, we read off the canonical transformation

$\displaystyle u_{2}$ $\displaystyle=-(\bar{u}_{1}+\bar{u}_{3})$
$\displaystyle\qquad u_{3}$ $\displaystyle=\bar{u}_{3}$ (63)
$\displaystyle\pi_{2}$ $\displaystyle=-\bar{\pi}_{1},$
$\displaystyle\qquad\pi_{3}$ $\displaystyle=\bar{\pi}_{3}-\bar{\pi}_{1}.$

and the associated unitary

$\displaystyle H_{2}\otimes H_{3}\rightarrow H_{3}\otimes H_{1}$ (64)
$\displaystyle\hat{S}_{p1\rightarrow
p2}=\hat{\mathcal{P}}_{2,1}e^{i\hat{\pi}_{2}\hat{u}_{3}}$

where $\hat{\mathcal{P}}_{A,B}$ is now the momenta parity-swap operator,
acting as

$\hat{\mathcal{P}}_{A,B}\ket{p}_{A}\otimes\ket{\Psi}_{B}=\ket{\Psi}_{B}\otimes\ket{-p}_{C}.$
(65)

####  3.4.3 Relative position to relative momentum

This time, we compute the transformation between the relative position and
relative momentum of particle 1. We write

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&=\begin{pmatrix}u_{2}-u_{3}\\\ u_{3}\\\
-u_{2}\end{pmatrix}=\frac{1}{3}\begin{pmatrix}\bar{u}_{2}-\bar{u}_{3}\\\
\bar{u}_{2}+2\bar{u}_{3}\\\ -2\bar{u}_{2}-\bar{u}_{3}\end{pmatrix}\\\
\begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&=\frac{1}{3}\begin{pmatrix}\pi_{2}-\pi_{3}\\\
\pi_{2}+2\pi_{3}\\\
-2\pi_{2}-\pi_{3}\end{pmatrix}=\begin{pmatrix}\bar{\pi}_{2}-\bar{\pi}_{3}\\\
\bar{\pi}_{3}\\\ -\bar{\pi}_{2}\end{pmatrix}\end{split}$ (66)

Again, we read off the canonical transformation

$\displaystyle u_{2}$ $\displaystyle=\frac{1}{3}(2\bar{u}_{2}+\bar{u}_{3})$
$\displaystyle\qquad u_{3}$
$\displaystyle=\frac{1}{3}(\bar{u}_{2}+2\bar{u}_{3})$ (67)
$\displaystyle\pi_{2}$ $\displaystyle=2\bar{\pi}_{2}-\bar{\pi}_{3},$
$\displaystyle\qquad\pi_{3}$ $\displaystyle=-\bar{\pi}_{2}+2\bar{\pi}_{3}$

The associated unitary representation is

$\displaystyle H_{1}\otimes H_{2}\rightarrow H_{1}\otimes H_{2}$ (68)
$\displaystyle\hat{S}_{q1\rightarrow
p1}=e^{-i\frac{\log{3}}{2}(u_{2}-u_{3})(\pi_{2}-\pi_{3})},$

see appendix C for the detailed derivation. Since we do not change particle,
no parity-swap operator appears.

##  4 Towards a first-class theory

###  4.1 Mathematical preliminaries: The abelian conversion method

In the reduced quantization approach only the intrinsic variables spanning the
reduced phase-spaces $\mathcal{P}_{persp}$ are quantized. Thus, to obtain the
perspective dependent quantum descriptions and the transformations between
them we had to resort to the the classical perspective neutral structure. This
is somewhat unfortunate. We would rather have a _quantum perspective neutral
structure_ , from which we could derive the perspective-dependent descriptions
without ever referring to the classical theory.

Remember, that the non-commutativity of the second-class constraints was the
main obstruction to a direct quantization of $\mathcal{P}_{persp}$. If we
could somehow turn them first-class, most of our problems would be solved.
This is the basic idea behind the _abelian conversion_ scheme, first
introduced in [12]. Instead of passing on to the Dirac bracket and reducing
the Phase space, we take the opposite direction and extend it even further by
introducing new sets of canonical coordinates. On this extended phase-space we
then convert the original second class to first class constraints. Canonical
quantization of the extended phase is straightforward and will yield the
sought-after perspective-neutral structure in the quantum picture. Let us
quickly summarise the main ideas.

When converting, we have to keep the physical degrees of freedom constant. But
contrary to second class constraints, first class constraints entail a gauge
symmetry. Thus, we have to add $2M$ coordinates $\psi_{a}$ for every $2M$
second class constraint [4]. We demand that they fulfill the canonical
symplectic structure

$\displaystyle[\psi_{a},\psi_{b}]=\omega_{ab},\quad\omega=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}$ (69) $\displaystyle[y_{a},\psi_{b}]=0\quad\forall\;a,b,$
(70)

where we have condensed the old positions and momenta into a single variable,
writing

$q_{i}=y_{n},\;n=1\;...\,N,\qquad p_{i}=y_{n},\;n=N+1\;...\;2N.$

On the extended phase space $\mathcal{P}_{ext}$, coordinatized by
$(y_{n},\psi_{m})$, the original second class constraints are converted into
abelian333The condition that the new constraints need to have an abelian
structure is not mandatory. In the non-abelian case, equation 71 can have the
more general form
$[\tilde{\Phi}_{\alpha},\tilde{\Phi}_{\beta}]=c_{\alpha,\beta}^{\gamma}\tilde{\Phi}_{\gamma}$.
first class constraints

$\tilde{\Phi}_{\alpha}=\tilde{\Phi}_{\alpha}(y,\psi)$

by requiring that they fulfill the differential equations

$[\tilde{\Phi}_{\alpha},\tilde{\Phi}_{\beta}]=0\quad\forall\;\alpha,\beta$
(71)

with the initial conditions

$\tilde{\Phi}_{\alpha}(y,\psi=0)=\Phi(y).$ (72)

We must further ensure that the extended gauge-system is dynamically
equivalent to the original second-class system. This is achieved by a specific
extension of the dynamical phase-space functions
$F(y)\rightarrow\tilde{F}(y,\psi)$ [4]. The abelianized functions
$\tilde{F}(y,\psi)$ have to be gauge invariant with respect to all converted
constraints

$[\tilde{\Phi}_{\alpha},\tilde{F}]=0$ (73)

and for $\psi=0$ have to coincide with the old ones

$\tilde{F}(y,\psi=0)=F(y).$ (74)

The initial conditions 72 and 74 ensure that by gauge fixing the system with
$\psi_{a}=0$ we regain the equations of motion of the original theory.

Finding solutions to the equations 71 \- 74 turns out to be highly nontrivial.
Fortunately, we do not need to discuss the general case. Amorim and Das showed
in [5] that when the original constraints are linear in the phase space
coordinates the abelianized constraints can be written as

$\tilde{\Phi}_{\alpha}=\Phi_{\alpha}+X_{\alpha a}\psi^{a},$ (75)

where

$X_{\alpha a}X_{\beta b}\omega^{ab}=-C_{\alpha\beta}.$ (76)

In this case, the computation of the converted phase-space functions is much
simpler as well. We find that a set of gauge-independent Dirac observables is
given by

$\tilde{y}^{c}=y^{c}-\psi^{b}\omega_{ab}X^{\alpha
a}[\Phi_{\alpha},y^{c}]_{y}=y^{c}-\psi^{b}B^{c}_{b}.$ (77)

Due to the linearity of the coordinate functions, the matrix $B$ is constant.
The $\tilde{y}^{c}$ commute by construction with all abelianized constraints

$\displaystyle\begin{split}[\tilde{\Phi}_{\alpha},\tilde{y}^{c}]=&\;[\Phi_{\alpha},y^{c}]-[\Phi_{\alpha},\psi^{b}]B^{c}_{b}-X_{\alpha
a}B^{c}_{b}[\psi^{a},\psi^{b}]\\\ =&\;[\Phi_{\alpha},y^{c}]-X_{\alpha
a}B^{c}_{b}\omega^{ab}\\\ =&\;0\end{split}$ (78)

Any observable, e.g. the Hamiltonian, defined by the replacement

$\tilde{H}(y,\psi)=H(\tilde{y})$

inherits this property and becomes also first class. The total Hamiltonian is
obtained by adding the first-class constraints to $\tilde{H}$, yielding

$\tilde{H}_{T}(y,\psi)=H(\tilde{y})+\sum_{i}\lambda_{i}\tilde{\Phi}_{i}.$

###  4.2 The extended phase-space

Let us employ the Abelianization scheme for our system at hand. We start with
relation 76, which in matrix form can be written as

$X\omega X^{T}=-C$ (79)

Carrying out the matrix multiplication we obtain the following equations for
the elements of X

$\displaystyle\begin{split}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}&\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\\\ &=\begin{pmatrix}0&-cb+da\\\
-ad+cb&0\end{pmatrix}=\begin{pmatrix}0&-\mu\\\ \mu&0\end{pmatrix}.\end{split}$
(80)

So the general form of $X$ is given by

$X=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\quad\text{where}\quad cb-da=\mu$
(81)

This is just a rescaling of the defining condition $da-cb=1$ for 2D canonical
transformations [33]. We denote the new set of canonical variables extending
the original phase-space by $(p,q)$. With 75 we obtain the transformed
constraints

$\displaystyle\tilde{\Phi}_{1}$ $\displaystyle=\sum_{i}q_{i}+aq+bp$ (82)
$\displaystyle\tilde{\Phi}_{2}$
$\displaystyle=\sum_{i}\frac{p_{i}}{\mu_{i}}+cq+dp.$ (83)

acting on the extended 8-dimensional phase space. They are proofed to be first
class when the condition $cb-da=\mu$ holds. Now we compute the functions
$\tilde{y}$ via the equation 77. $B$ is given in matrix form by

$\displaystyle B$
$\displaystyle=\omega^{-1}X^{-1}\norm{[\Phi_{\alpha},y^{c}]}$ (84)
$\displaystyle=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\frac{1}{-\mu}\begin{pmatrix}d&-b\\\
-c&a\end{pmatrix}\begin{pmatrix}0&0&0&1&1&1\\\
-{\mu_{1}}^{-1}&-{\mu_{2}}^{-1}&-{\mu_{3}}^{-1}&0&0&0\end{pmatrix}$ (85)
$\displaystyle=-\frac{1}{\mu}\begin{pmatrix}a\,{\mu_{1}}^{-1}&a\,{\mu_{2}}^{-1}&a\,{\mu_{3}}^{-1}&c&c&c\\\\[4.0pt]
b\,{\mu_{1}}^{-1}&b\,{\mu_{2}}^{-1}&b\,{\mu_{3}}^{-1}&d&d&d.\end{pmatrix}$
(86)

where $\norm{a_{ij}}$ denotes the matrix with components $a_{ij}$. We can thus
compute the Dirac observables

$\displaystyle\tilde{q}_{i}$ $\displaystyle=q_{i}+\frac{1}{\mu\mu_{i}}(aq+bp)$
$\displaystyle i$ $\displaystyle=1,2,3$ (87) $\displaystyle\tilde{p}_{i}$
$\displaystyle=p_{i}+\frac{1}{\mu}(cq+dp)$ $\displaystyle i$
$\displaystyle=1,2,3.$ (88)

The Hamiltonian becomes

$\tilde{H}=\frac{1}{2}\sum_{i}\frac{(p_{i}+\nicefrac{{(cq+dp)}}{{\mu}})^{2}}{\mu_{i}}+\lambda_{1}\tilde{\Phi}_{1}+\lambda_{2}\tilde{\Phi}_{2}.$
(89)

This is the most general form of the extended structure. However, the explicit
representation of the extended phase space does not matter, as
$[aq+bp,cq+dp]=ad-bc=-\mu$. So before we continue, let us choose
$a=1,d=-\mu,b=c=0$ to simplify the notation. With this, the extended
constraints read

$\displaystyle\tilde{\Phi}_{1}$ $\displaystyle=\sum_{i}{q}_{i}+q\simeq 0$ (90)
$\displaystyle\tilde{\Phi}_{2}$
$\displaystyle=\sum_{i}\frac{{p}_{i}}{\mu_{i}}-\mu
p=\sum_{i}\frac{{p}_{i}-p}{\mu_{i}}\simeq 0,$ (91)

the Dirac observables

$\displaystyle\tilde{q}_{i}$ $\displaystyle=q_{i}+\frac{q}{\mu\mu_{i}}$ (92)
$\displaystyle\tilde{p}_{i}$ $\displaystyle=p_{i}-p$ (93)

and the Hamiltonian

$\displaystyle\tilde{H}$
$\displaystyle=\frac{1}{2}\sum_{i}\frac{(p_{i}-p)^{2}}{\mu_{i}}+\lambda_{1}\tilde{\Phi}_{1}+\lambda_{2}\tilde{\Phi}_{2}.$
(94)

###  4.3 Halfway down: The intermediate phase-spaces

The 8 dimensional extended phase space $\mathcal{P}_{ext}$ was explicitly
constructed so that the original phase space is reestablished by gauge fixing
$q\simeq p\simeq 0$. There is no need, however, to do this simultaneously. We
can fix either one of the gauges and define a Dirac bracket on the such
obtained 6-dimensional second-class constraint surface. Passing on to
intrinsic coordinates then yields an symplectomorphic set of intermediate
phase spaces $\mathcal{P}_{int}$ which still feature one first-class
constraint.

Let us at first partially gauge fix the extended system by $\Phi_{3}=p\simeq
0$. We now have a mixed theory featuring one first-class constraint $\Phi_{2}$
and two second class constraints $\Phi_{1}$ & $\Phi_{3}$. With
$[\Phi_{1},\Phi_{3}]=1$ the Dirac-bracket reads

$\displaystyle\\{q_{i},p_{j}\\}$ $\displaystyle=\delta_{ij}$
$\displaystyle\qquad\\{q,p\\}$ $\displaystyle=0$ (95)
$\displaystyle\\{q_{i},p\\}$ $\displaystyle=0$
$\displaystyle\qquad\\{q,p_{j}\\}$ $\displaystyle=0.$

Again, we can go over to intrinsic Darboux coordinates defined by
$q=-\Phi_{1}$ and $p=0$. Both $q$ and $p$ drop out and we are left with the
_intermediate momentum space_ $\mathcal{P}_{int,p}$, characterised by

$\displaystyle\tilde{q}_{p,i}$
$\displaystyle=q_{i}-\frac{\Phi_{1}}{\mu\mu_{i}}$
$\displaystyle\qquad\Phi_{2}$ $\displaystyle=\sum\frac{p_{i}}{\mu_{i}}\simeq
0$ (96) $\displaystyle\tilde{p}_{p,i}$ $\displaystyle=p_{i}$
$\displaystyle\qquad H_{p}$
$\displaystyle=H(p_{i})+\lambda^{\prime}_{2}\Phi_{2}.$

This structure is identical to the classical perspective neutral structure
established in [42]. We can see that when we set $\mu_{i}=1/3$. Then we have
$\Phi_{2}\simeq\sum p_{i}$ and we can define new Dirac observables by
$\tilde{q}_{p,i}-\tilde{q}_{p,j}=q_{i}-q_{j}$. However, the interpretation is
different, as the variables $q_{i}$ and $p_{i}$ now acquire meaning as
relative coordinates. Thus fixing the gauge with $q_{i}\simeq 0$ is
unjustifiable from a physical viewpoint. It seems that $\Phi_{1}\simeq 0$,
bringing us back to $\mathcal{P}_{orig}$, is indeed the only admissible gauge.
We see that the alleged gauge-freedom exploited in [42] is replaced by the
freedom to choose internal Darboux-coordinates.

If we instead gauge fix by $q\simeq 0$ first, we obtain the _intermediate
position space_ $\mathcal{P}_{int,q}$

$\displaystyle\tilde{q}_{q,i}$ $\displaystyle=q_{i}$
$\displaystyle\quad\Phi_{1}$ $\displaystyle=\sum{q}_{i}\simeq 0$ (97)
$\displaystyle\tilde{p}_{q,i}$ $\displaystyle=p_{i}-\frac{\Phi_{2}}{\mu}$
$\displaystyle\quad H_{q}$
$\displaystyle=H(p_{i})-\frac{1}{2\mu}\Phi_{2}^{2}+\lambda_{1}^{\prime}\Phi_{1}.$

It is easy to see that gauge-fixing the system a second timeby $\Phi_{2}\simeq
0$ leads us back to original phase space.

It remains to check if the spaces 97 and 96 are symplectomorphic. Indeed, we
find that they are connected by the transformation

$\displaystyle S_{q\rightarrow p}:$
$\displaystyle\mathcal{P}_{int,q}\rightarrow\mathcal{P}_{int,q}$ (98)
$\displaystyle
p_{i}-\frac{1}{\mu}\sum\frac{p_{j}}{\mu_{j}}\rightarrow\bar{p}_{i}$
$\displaystyle
q_{i}\rightarrow\bar{q}_{i}-\frac{1}{\mu\mu_{i}}\sum\bar{q}_{j},$

which can be shown to be canonical by

$\displaystyle\begin{split}\frac{\partial\bar{p}_{i}}{\partial
p_{j}}&=\delta_{ij}-\frac{1}{\mu\mu_{j}}=\frac{\partial
q_{i}}{\partial\bar{q}_{j}}\\\ \frac{\partial\bar{p}_{i}}{\partial
q_{j}}&=\delta_{ij}-\frac{1}{\mu\mu_{j}}=\frac{\partial
p_{i}}{\partial\bar{q}_{j}}.\end{split}$ (99)

##  5 Dirac Quantization of first class constraints

###  5.1 Mathematical preliminaries

Having worked through the Abelianization scheme we have obtained a pure first-
class Hamiltonian system. Various methods of quantizing such a system are
known (see [26], chapter 13, for an overview). An alternative can be found in
the _Dirac quantization_. Here one quantizes the full extended phase-space
including redundant degrees of freedom and solves the constraints in the
quantum theory. To this end we promote all coordinates on $\mathcal{P}_{ext}$
to operators acting on an _extended Hilbert Space_ $\mathcal{H}_{ext}$ and
require that the physical states are zero eigenstates of the constraints

$\hat{\Phi}_{\alpha}\ket{\phi}_{phys}=0.$ (100)

This condition is tantamount to requiring that physical states are invariant
under gauge transformations

$e^{i\epsilon_{\alpha}\hat{\Phi}_{\alpha}}\ket{\phi}_{phys}=\ket{\phi}_{phys}\quad\forall\alpha\;\;\text{(no
sum)}$ (101)

generated by the constraints [26]. We can find solutions by _group averaging_
[32] via the projector

$\displaystyle\begin{split}\delta_{\Phi}:\frac{1}{4\pi^{2}}\underbrace{\int_{-\infty}^{+\infty}e^{is_{1}\Phi_{1}}ds}_{=2\pi\delta_{\Phi_{1}}}\underbrace{\int_{-\infty}^{+\infty}e^{it\Phi_{2}}dt}_{=2\pi\delta_{\Phi_{2}}}\ket{\chi}_{ext}\ket{\chi}_{phys},\end{split}$
(102)

where we used that when $[\hat{A},\hat{B}]=0$ the identity
$e^{\hat{A}\hat{B}}=e^{\hat{A}}e^{\hat{B}}$ holds such that we can project on
each constraint individually by $\delta_{\Phi_{i}}$.

This projective method is also used in quantum information where it is usually
referred to as ”G-twirling [34]. In QI one deals mostly with compact groups,
in which case the projector is well defined. However, the group associated
with the constraint 100 is non-compact and the (improper) projector 102 does
not converge on $\mathcal{H}_{ext}$. In other words, the physical states are
not actually contained in $L^{2}(\mathbb{R}^{8})$ as they are not square-
integrable. A ”home” for the projector and the physical state can be found in
the so-called rigged Hilbert space [16, 17, 40], with the hermitian and
(presumably) positive definite scalar product [32]

$\displaystyle\begin{split}\left(\delta_{\Phi}f,\delta_{\Phi}g\right)_{phys}&=[\delta_{\Phi}(g)](f)\widehat{=}\bra{f}\delta_{\Phi}\ket{g}\\\
\forall f,&g\in\mathcal{D}_{ext}.\end{split}$ (103)

###  5.2 The perspective-neutral structure in the quantum picture

Putting hats on the observables, equations 90 \- 94 comprise what we call the
quantum perspective-neutral structure. A quantum state in the extended Hilbert
space $\mathcal{H}_{ext}$, for example given in momentum representation by

$\ket{\chi}_{ext}=\int\chi(\vec{p},p)\ket{\vec{p}}\ket{p}d\vec{p}dp.$ (104)

has 4 degrees of freedom, twice as many as physically observable. Again, these
redundancies must be fixed to re-obtain the perspective-dependent
descriptions. Here we follow largely the quantum symmetry reduction scheme of
Vanrietvelde et al.: We define a unitary trivialization map $\hat{T}$ that
transforms one constraint in such a way that it acts only on one variable.
After an (improper) projection on this trivialized constraint we can discard
the now redundant degrees of freedom. As we are dealing we two constraints we
need two trivialization maps, each followed by a projection.

It seems sensible to get rid of the additional 4th degree of freedom at the
very beginning, as how to physically interpret it remains - at least to the
author - rather obscure. To this end, we can choose to project on any linear
combination of the constraints, given that it is rotated to act only one the
$(q,p)$-slot. We thus obtain one of many _intermediate Hilbert spaces_
$\mathcal{H}_{int}$, which are the quantum analogues of the intermediate phase
spaces. By repeating the procedure - trivialization followed by projecting -
for a coordinate of our liking we arrive at the perspective dependent
description. We will now have a look at the symmetry reduction scheme in
detail.

###  5.3 Relative position and momentum frame revisited

####  5.3.1 The intermediate and relative position frames

Let us start with an initial projection on $\hat{p}=0$. We therefore need to
define the trivialization map

$\hat{T}_{p}=e^{-\frac{i}{\mu}\hat{q}\hat{\Phi}_{2}}$ (105)

which rotates the constraints to

$\displaystyle\hat{T}_{p}\hat{\tilde{\Phi}}_{1}\hat{T}^{\dagger}_{p}$
$\displaystyle=\hat{\Phi}_{1}=0,$
$\displaystyle\hat{T}_{p}\hat{\tilde{\Phi}}_{2}\hat{T}^{\dagger}_{p}$
$\displaystyle=\hat{p}=0.$ (106)

Acting with $\hat{T}_{p}$ on a general state in $\mathcal{H}_{ext}$ following
up with the projector $\delta_{T{\tilde{\Phi}}_{2}T^{\dagger}}=\delta_{p}$
yields

$\delta_{p}\hat{T}_{p}\ket{\chi}_{ext}=\\\
\int\chi(\vec{p},\sum_{i}\frac{p_{i}}{\mu_{i}\mu})\ket{p_{1}}\ket{p_{2}}\ket{p_{3}}d\vec{p}\otimes\ket{p=0}.$
(107)

We can use this result to compute the action of $\delta_{p}\hat{T}_{p}$ on the
Dirac observables and on the Hamiltonian 94. From
$\hat{p}\,\delta_{p}\hat{T}_{p}\ket{\chi}_{ext}=0$ it follows that

$\displaystyle\begin{split}{}_{ext}\bra{\chi}(T_{p})^{\dagger}(\delta_{p})^{\dagger}\hat{p}\,\delta_{p}\hat{T}_{p}\ket{\chi}_{ext}=0\\\
\quad\rightarrow\quad(\delta_{p})^{\dagger}\hat{p}\delta_{p}=0\end{split}$
(108)

So after acting with $T_{p}$ we can discard all terms containing $\hat{p}$,
which leaves us with

$\displaystyle\begin{split}\hat{\tilde{q}}_{i}&\quad\rightarrow\quad\hat{q}_{q,i}=\hat{q}_{i}\\\
\hat{\tilde{p}}_{i}&\quad\rightarrow\quad\hat{p}_{q,i}=\hat{p}_{i}-\frac{\hat{\Phi}_{2}}{\mu}\\\
\hat{\tilde{H}}&\quad\rightarrow\quad\hat{H}_{q}=\hat{H}(p_{q,i})\\\ &\hskip
49.0pt=\hat{H}(p_{i})-\frac{1}{2\mu}\hat{\Phi}_{2}^{2}+\hat{\lambda}_{q}\hat{\Phi}_{1}\end{split}$
(109)

Now all information is stored in the first three slots and as in the classical
case $\hat{q}$ and $\hat{p}$ have dropped entirely from the formalism. It is
therefore permissible to discard the 4th degree of freedom altogether by
projecting onto the classical gauge fixing condition

$\displaystyle\ket{\chi}_{q}$
$\displaystyle=\sqrt{2\pi}\bra{q=0}\delta_{p}\hat{T}_{p}\ket{\chi}_{ext}=\int
dp\braket{p}{\,\delta_{p}\hat{T}_{q}}{\chi}_{ext}$ (110)
$\displaystyle=\int\chi_{q}(\vec{p})\ket{p_{1}}\ket{p_{2}}\ket{p_{3}}d\vec{p}.$

where

$\chi_{q}(\vec{p})=\chi(\vec{p},\sum_{i}\frac{p_{i}}{\mu_{i}\mu}).$

The structure above is the quantum version of 97. We therefore speak of the
_intermediate position space_ $\mathcal{H}_{int,q}$.

To dispose of the remaining redundancy and obtain the perspective-dependent
quantum state we must carry out the symmetry reduction once more. Contrary to
the first projection, we are now free to choose the slot we want to get rid
of. This amounts to defining another trivialization map, such that the
transformed constraint acts only on the d.o.f. in question. For example, to
get rid of the first slot we apply

$\hat{T}_{q1}=e^{-i\frac{\pi}{2}(\hat{q}_{2}\hat{p}_{3}-\hat{q}_{3}\hat{p}_{2})}e^{-i\hat{p}_{1}(\hat{q}_{2}+\hat{q}_{3})}$
(111)

While the second exponential is the actual trivialization operator, the first
one induces the canonical transformation

$\displaystyle q_{2}$ $\displaystyle\rightarrow q_{3}$ $\displaystyle p_{2}$
$\displaystyle\rightarrow p_{3}$ (112) $\displaystyle q_{3}$
$\displaystyle\rightarrow-q_{2}$ $\displaystyle p_{3}$
$\displaystyle\rightarrow-p_{2}.$ (113)

This 90°-rotation in the $(q_{2},q_{3})$\- and $(p_{2},p_{3})$-subspaces is
needed to bring the coordinate names in accordance with those used in the
reduced quantization scheme. $\hat{T}_{q1}$ rotates the remaining constraint
to

$\hat{T}_{q1}\hat{\Phi}_{1}\hat{T}_{q1}^{\dagger}=\hat{q}_{1}$ (114)

Let us apply $\hat{T}_{q1}$ followed by the projection
$\sqrt{2\pi}\bra{p_{1}=0}\delta_{q_{1}}$ to the Dirac-observables. We thereby
obtain a new description in $\mathcal{H}_{persp}$:

$\displaystyle\begin{split}\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}&\rightarrow\begin{pmatrix}q_{2}-q_{3}\\\ q_{3}\\\
-q_{2}\end{pmatrix}\\\ \begin{pmatrix}p_{q,1}\\\ p_{q,2}\\\
p_{q,3}\end{pmatrix}&\rightarrow\frac{1}{\mu}\begin{pmatrix}\frac{p_{2}}{\mu_{3}}-\frac{p_{3}}{\mu_{2}}\\\
\frac{p_{2}}{\mu_{3}}+p_{3}(\mu-\frac{1}{\mu_{2}})\\\
-p_{2}(\mu-\frac{1}{\mu_{3}})-\frac{p_{3}}{\mu_{2}}\end{pmatrix}.\end{split}$
(115)

Plugging this into the Hamiltonian 109 we arrive at

$\hat{H}_{q}\rightarrow\hat{H}_{q1}=\frac{1}{2\mu}\Big{(}(\frac{\mu}{\mu_{3}}-\frac{1}{\mu_{3}^{2}})\hat{p}_{2}^{2}+(\frac{\mu}{\mu_{2}}-\frac{1}{\mu_{2}^{2}})\hat{p}_{3}^{2}+2\frac{\hat{p}_{2}\hat{p}_{3}}{\mu_{2}\mu_{3}}\Big{)}.$
(116)

This is equal to the relative position Hamiltonian 53 from the perspective of
particle 1, which we obtained through the reduced-quantization method. There
are no constraints or redundancies left. We can summarise the whole process by

$\ket{\chi}_{qi}=\bra{p_{i}=0}\bra{q=0}\delta_{q_{i}}\delta_{p}T_{q_{i}}T_{p}\ket{\chi}_{ext}.$
(117)

####  5.3.2 The intermediate and relative momentum frames

Now let us project on $\hat{q}=0$ first. In this case we apply the
trivialization map

$\hat{T}_{q}=e^{-i\hat{p}\hat{\Phi}_{1}}$ (118)

which rotates the constraints to

$\displaystyle\hat{T}_{q}\hat{\tilde{\Phi}}_{1}\hat{T}_{q}^{\dagger}$
$\displaystyle=\hat{q}=0$
$\displaystyle\hat{T}_{q}\hat{\tilde{\Phi}}_{2}\hat{T}_{q}^{\dagger}$
$\displaystyle=\hat{\Phi}_{2}=0.$ (119)

This rotation, followed by the projection
$\sqrt{2\pi}\bra{p=0}\delta_{\hat{q}}$ transforms the Dirac-coordinates and
the Hamiltonian to

$\displaystyle\begin{split}\hat{\tilde{q}}_{i}&\quad\rightarrow\quad\hat{q}_{p,i}=\hat{q}_{i}-\frac{\Phi_{1}}{\mu\mu_{i}}\\\
\hat{\tilde{p}}_{i}&\quad\rightarrow\quad\hat{p}_{p,i}=\hat{p}_{i}\\\
\hat{\tilde{H}}&\quad\rightarrow\quad\hat{H}_{p}=\hat{H}(p_{p,i})=\hat{H}+\lambda^{\prime}_{2}\Phi_{2}.\end{split}$
(120)

It comes as no surpise, that the above structure is the quantum version of the
classical intermediate momentum space. To obtain the perspective-dependent
description we must rotate the system once more, this time by

$\hat{T}_{p1}=e^{-i\frac{\pi}{2}(\hat{q}_{2}\hat{p}_{3}-\hat{q}_{3}\hat{p}_{2})}e^{i\mu_{1}q_{1}(\frac{\hat{p}_{2}}{\mu_{2}}+\frac{\hat{p}_{3}}{\mu_{3}})}$
(121)

and then project by $\sqrt{2\pi}\bra{p_{1}=0}\delta_{p_{1}}$. This transforms
the Dirac-coordinates to

$\displaystyle\begin{split}\begin{pmatrix}q_{p,1}\\\ q_{p,2}\\\
q_{p,3}\end{pmatrix}&\rightarrow\frac{1}{\mu}\begin{pmatrix}\frac{1}{\mu_{1}}(q_{3}-q_{2})\\\
\frac{q_{2}}{\mu_{3}}+q_{3}(\mu-\frac{1}{\mu_{2}})\\\
-q_{2}(\mu-\frac{1}{\mu_{3}})-\frac{q_{3}}{\mu_{2}}.\end{pmatrix}\\\
\begin{pmatrix}p_{1}\\\ p_{2}\\\
p_{3}\end{pmatrix}&\rightarrow\begin{pmatrix}\mu_{1}(\frac{p_{2}}{\mu_{3}}-\frac{p_{3}}{\mu_{2}})\\\
p_{3}\\\ -p_{2}.\end{pmatrix}.\end{split}$ (122)

The Hamiltonian becomes

$\hat{H}_{p}\rightarrow\hat{H}_{p1}=\frac{1}{2}\Big{(}\frac{\mu_{1}+\mu_{2}}{\mu_{2}^{2}}\hat{p}_{3}^{2}\\\
+\frac{\mu_{1}+\mu_{3}}{\mu_{3}^{2}}\hat{p}_{2}^{2}-\frac{2\mu_{1}}{\mu_{2}\mu_{3}}\hat{p}_{2}\hat{p}_{3}\Big{)}.$
(123)

We have therefore arrived at the relative-momentum description from the
perspective of particle one. We can again condense the process to

$\ket{\chi}_{pj}=\bra{q_{j}=0}\bra{p=0}\delta_{p_{j}}\delta_{q}T_{p_{j}}T_{q}\ket{\chi}_{ext}.$
(124)

###  5.4 Changing quantum reference frames via the perspective neutral
structure

Now let us see how to change reference frames via the the quantum perspective-
neutral theory. We will focus on the three cornerstone-transformation of type
LABEL:eq:_unitary_relative_position_transformation, 64 and 68, by whose
combination one can compose arbitrary maps. To transform between relative
position frames we make use of the intermediate position space. That means, to
recover transformations like
LABEL:eq:_unitary_relative_position_transformation we at first invert the
quantum symmetry reduction and re-embed $\mathcal{H}_{q1}$ into
$\mathcal{H}_{q}$, followed by a projection on $\mathcal{H}_{q2}$. Concretely,
in [42] it is proofed that this map is given by

$\hat{S}_{qi\rightarrow qj}:=\int
dq_{j}\bra{q_{j}}\hat{T}_{qj}\,(\hat{T}_{qi})^{\dagger}\ket{q_{i}=0}\otimes[\cdot]$
(125)

where the reduced state $\ket{\chi}_{q1}$ must be inserted into the empty slot
$[\cdot]$.

Likewise, a transformation between two relative momentum maps can be
implemented by

$\hat{S}_{pi\rightarrow pj}:=\int
dp_{j}\bra{p_{j}}\hat{T}_{pj}\,(\hat{T}_{pi})^{\dagger}\ket{p_{i}=0}\otimes[\cdot].$
(126)

Observe, that via the intermediate spaces we can switch between particle-
perspectives but not between relative position and momentum representation. To
make this possible we have to go all the way up and make use of the extended
Hilbert space. Transformations of the type 68 are thus implemented by

$\hat{S}_{qi\rightarrow pi}:=\int
dqdp_{j}\bra{p_{j}}\hat{T}_{pj}\bra{q=0}\hat{T}_{q}(\hat{T}_{p})^{\dagger}\ket{p=0}\otimes(\hat{T}_{pi})^{\dagger}\ket{q_{i}=0}\otimes[\cdot].$
(127)

##  6 Recap and Comparison

Let us review our framework’s mathematical structure and compare it to that of
Vanrietvelde et al.. Both frameworks originated from a translational
invariant, toy model Lagrangian. While in [42] the Legendre transformation was
performed on the original, gauge dependent variables, in this thesis we first
introduced gauge-invariant, relative variables. This mapping entailed a
constraint of the relative positions. The Legendre transformation led to the
known constraint for the sum of relative momenta, yielding a symmetrical
Hamiltonian structure living in $\mathcal{P}_{orig}$.

As the constraints were second-class, there was no gauge freedom to exploit
and we could not obtain the perspective dependent descriptions by gauge
fixing. Instead we showed that the perspective dependent descriptions can be
understood as Darboux coordinates on the constraint surface. This approach
allowed us to derive a broader class of perspective dependent frames. In
addition to the already known relative momentum frames $\mathcal{P}_{pi}$ we
derived the new relative position frames $\mathcal{P}_{qj}$.

$\mathcal{P}_{ext}\sim\mathbb{R}^{8}$$\mathcal{P}_{int,q}$$\mathcal{P}_{int,p}$$\sim$$\mathcal{P}_{orig}\sim\mathbb{R}^{6}$$\mathcal{P}_{qi}\sim\mathbb{R}^{4}\sim$$\;\mathcal{P}_{pi}$Darbouxcoordinates$q=0$$p=0$$\Phi_{2}\simeq
0$$\Phi_{1}\simeq
0$Abelianization$q_{i}=0$$\mathcal{H}_{ext}$$\mathcal{H}_{int,q}$$\mathcal{H}_{int,p}$$\sim$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathcal{H}_{pi}}\sim\mathcal{H}_{qi}$$\bra{q=0}\delta_{p}T_{p}$$\bra{p=0}\delta_{q}T_{q}$$\bra{q_{i}=0}\delta_{qi}T_{qi}$$\bra{p_{i}=0}\delta_{pi}T_{pi}$Dirac
quantizationreduced quantization Figure 5: A summary of all the spaces and
their connections making appearance in this thesis. The orange-coloured parts
of the framework appear already in [42], visually demonstrating that original
framework is fully integrated in our broader structure. The horizontal axis
indicates the number of unphysical degrees of freedom featured in the theory.
The further on top, the more redundant coordinates are present. The strong
equations $q,p,q_{i}=0$ should be understood as including the transition to
intrinsic coordinates.

Following this line of thought, we also had to adapt our understanding of
reference frame transformations. While in [42] reference frame transformations
were related to gauge transformations, we showed that they can be understood
as Darboux-reparametrizations of the constraint surface. Thus, we were not
only able to recover the transformations
$\mathcal{P}_{pi}\rightarrow\mathcal{P}_{pj}$ connecting relative momentum
frames, but could derive the transformations
$\mathcal{P}_{qi}\rightarrow\mathcal{P}_{qj}$ between relative position frames
and $\mathcal{P}_{qi}\rightarrow\mathcal{P}_{pi}$ between relative position
and relative momentum frames. Canonical quantization of the perspective
dependent phase spaces then led us to the perspective dependent Hilbert spaces
$\mathcal{H}_{qi}$ and $\mathcal{H}_{pi}$.

Our next objective was to find the quantum perspective neutral structure. To
this end, we expanded the original phase space to $\mathcal{P}_{ext}$ and
modified the constraints to make them first class, a process called
Abelianization. Canonical quantization of $\mathcal{P}_{ext}$ yielded the
sought-after perspective neutral Hilbert space $\mathcal{H}_{ext}$. To return
to the perspective dependent frames, we followed an extended version of the
quantum symmetry reduction scheme employed in [42]. Having two constraints,
two trivialization maps and two projections were needed. Since it was not
clear how to interpret the additional degrees of freedom physically, we tried
to get rid of them first. We could do this by either projecting on $\hat{p}=0$
or $\hat{q}=0$ (or any linear combinations thereof), both options leading to
the unitary equivalent, intermediate Hilbert spaces $H_{int,q}$ and
$H_{int,p}$. We found that projecting on $\hat{q}=0$ resulted in a Hilbert
space equivalent to the auxiliary Hilbert space of [42]. Trivializing and
projecting on the remaining constraint then lead us back to the various
perspective dependent Hilbert spaces.

We could then make use of the quantum perspective neutral structure to switch
perspectives. Jumping between two relative momentum frames is done by first
inverting the second symmetry reduction, thus embedding $\mathcal{H}_{pi}$
into $\mathcal{H}_{p}$ and then projecting onto a different degree of freedom.
In an analogous way we could change between two relative position frames. To
switch relative position to relative momentum, however, we had to go all the
way up and and embed the perspective dependent descriptions in the extended
Hilbert space.

Still, there is one part missing. Where is the original phase space of [42]
hidden in our framework? We found that partially gauge fixing
$\mathcal{P}_{ext}$ led us to the intermediate phase spaces $P_{int,q}$ and
$P_{int,p}$. The latter could be shown to be equivalent to the auxiliary space
in [42]. But even though their form is equal, there is an important
interpretive difference. As the coordinates have relative meaning in out
framework, fixing the gauge by $q_{i}=0$ is physically unjustified. The only
permissible gauge is $\sum q_{i}\simeq 0$, leading us back to
$\mathcal{P}_{orig}$.

##  7 Limitations and Outlook

In the present thesis, we have further developed the first-principle approach
to quantum reference frames introduced by Vanrietvelde et al.. Our
construction provides a unifying framework to systematically derive a large
class of perspective dependent descriptions and the transformations connecting
them. However, it can count only as a very first step towards a truly general
theory of quantum reference frames.

Some of the necessary generalization are obvious. Our framework covered only a
toy model of three free particles in one dimension. As a first step, it should
be generalized to multiple particles in 3d space. In [41], it is shown that in
this case globally valid gauge conditions are impossible. Since we can regard
two second class constraints as mutual gauge fixes, this will likely obstruct
the definition of globally valid Darboux coordinates. Further, we should
investigate systems with non-vanishing potential energy. It is not obvious
that the Dirac bracket will remain its simple form in this case.

In our simple toy-model we only obtained a subset of all canonical
transformations between quantum reference frames. The limiting factor was the
general non-commutativity of canonical transformations and canonical
quantization. To not encounter possible mathematical pitfalls, we restricted
to linear canonical transformations which do not mix positions and momenta.
This excludes the larger part of the extended Galilean transformations, like
boosts and transformations to accelerated reference systems, as presented in
[23]. To extend the framework, a better understanding of the interplay of
canonical quantization and canonical transformations is crucial.

Finally, if our aim is to develop a theory about quantum gravity, a
generalization to relativistic systems is a plausible next step. Some research
papers advance in this direction [24, 39]. But to the author’s knowledge,
there does not yet exist a comprehensive theory describing transformations
between finite-mass reference frames moving with relativistic velocities, not
even at the classical level. Only after we have better understood the
particularities of such systems from a classical viewpoint we can start
working towards a Lorentz-invariant theory of quantum reference frames.

##  Appendix A Lagrangian in relative coordinates

Let us walk through the short computation of the Lagrangian in relative
coordinates 25. We start by expanding the center-of-mass-term and pulling it
under the sum, yielding

$L(x,\dot{x})=\frac{1}{2}\sum_{i=1}^{3}\left(m_{i}\dot{x}_{i}^{2}-\frac{m_{i}^{2}}{M}\dot{x}_{i}^{2}\right)-\frac{1}{M}\sum_{i<j}m_{i}m_{j}\dot{x}_{i}\dot{x}_{j}.$
(A1)

Let us have a closer look at the first term ($i=1$) of this expression:

$m_{1}\dot{x}_{1}^{2}-\frac{m_{1}^{2}}{M}\dot{x}_{1}^{2}=\frac{1}{M}\dot{x}_{1}^{2}\left(m_{1}(m_{1}+m_{2}+m_{3})-m_{1}^{2})\right)=\frac{m_{1}(m_{2}+m_{3})}{M}\dot{x}_{1}^{2}.$
(A2)

Expanding both sums we obtain

$L=\frac{1}{2M}(m_{1}(m_{2}+m_{3})x_{1}^{2}+m_{2}(m_{1}+m_{3})x_{2}^{2}+m_{3}(m_{1}+m_{2})x_{3}^{2}+2m_{1}m_{2}\dot{x}_{1}\dot{x}_{2}+2m_{1}m_{3}\dot{x}_{1}\dot{x}_{3}+2m_{2}m_{3}\dot{x}_{2}\dot{x}_{3}$
(A3)

which we can contract to full squares to arrive at

$L=\frac{1}{2M}\left(m_{1}m_{2}(x_{1}-x_{2})^{2}+m_{1}m_{3}(x_{1}-x_{3})^{2}+m_{2}m_{3}(x_{2}-x_{3})^{2}\right).$
(A4)

Plugging in the relative coordinates 23 and making use of the reduced masses
$\mu_{i}$ then yields the Lagrangian 25.

##  Appendix B Derivation of consistency conditions

This appendix shows the detailed computation of the consistency conditions

$\dot{\Phi}_{m}=[\Phi_{m},H_{T}]\simeq 0$ (B1)

and the derivation of the full constraint algebra starting from the
Hamiltonian

$H_{T}=\frac{1}{2}\sum_{i=1}^{3}\left(\frac{p_{i}^{2}}{\mu_{i}}-q_{4}q_{i}\right)+yp_{4}$
(B2)

and the constraint $\Phi_{1}=p_{4}\simeq 0$.

Since $p_{4}$ commutes with all coordinates but $q_{4}$, we recover easily
constraint 24 by

$\displaystyle\dot{\Phi}_{1}=[p_{4},H_{T}]=\sum_{i=1}^{3}q_{i}=\Phi_{2}\simeq
0.$ (B3)

Plugging this again in B1 yields

$\displaystyle\dot{\Phi}_{2}$
$\displaystyle=[\sum_{i=1}^{3}q_{i},\frac{1}{2}\sum_{j=1}^{3}\frac{p_{j}^{2}}{\mu_{j}}+q_{4}\sum_{i=1}^{3}q_{i}+yp_{4}]$
$\displaystyle=\frac{1}{2}\sum_{i,j}\frac{1}{\mu_{j}}[q_{i},p_{j}^{2}]=\frac{1}{2}\sum_{i,j}\frac{2}{\mu_{j}}[q_{i},p_{j}]p_{j}=\sum_{i,j}\frac{1}{\mu_{j}}\delta_{ij}p_{j}$
(B4) $\displaystyle=\sum_{i=1}^{3}\frac{p_{j}}{\mu_{j}}=\Phi_{3}\simeq 0.$
(B5)

We continue with

$\displaystyle\dot{\Phi}_{3}$
$\displaystyle=[\sum_{i=1}^{3}\frac{p_{j}}{\mu_{j}},q_{4}\sum_{i=1}^{3}q_{i}]=\sum_{i,j}q_{4}[\frac{p_{j}}{\mu_{j}},q_{i}]=\sum_{i,j}\frac{q_{4}}{\mu_{j}}\delta_{ij}$
(B6) $\displaystyle=\mu q_{4}=\Phi_{4}\simeq 0,$ (B7)
$\displaystyle\dot{\Phi}_{4}$ $\displaystyle=[\mu q_{4},yp_{4}]=\mu
y[q_{4},p_{4}]$ (B8) $\displaystyle=\mu y\simeq 0.$ (B9)

Since the last equation gives a condition for the Lagrange-multiplier, the
process terminates here. We are left with four constraints

$\begin{split}\Phi_{1}&=p_{4}\simeq 0,\\\ \Phi_{2}&=\sum_{i}q_{i}\simeq 0,\\\
\Phi_{3}&=\sum_{i}\frac{p_{i}}{\mu_{i}}\simeq 0,\\\ \Phi_{4}&=\mu q_{4}\simeq
0\end{split}$ (B10)

and one condition for the Lagrange multiplier

$y\simeq 0.$ (B11)

Note that the product of two weak equations $\Phi_{i}\simeq\Phi_{j}\simeq 0$
is a strong equation, that is its Poisson bracket

$[F,\Phi_{i}\Phi_{j}]=[F,\Phi_{i}]\Phi_{j}+\Phi_{i}[F,\Phi_{j}]\simeq 0$ (B12)

with an arbitrary function $F(q,p)$ vanishes on the constraint surface. We can
therefore discard the last two terms from the Hamiltonian B2 and simply write

$H_{T}=\frac{1}{2}\sum_{i}\frac{p_{i}^{2}}{\mu_{i}}.$ (B13)

Since both $q_{4}$ and $p_{4}$ are restricted to zero, they carry no physical
information and can be discarded from the formalism [20]. This means that the
system at hand lives in a 6-dimensional phase-space and features two
constraints.

##  Appendix C The unitary position-to-momentum map

To find the unitary operator

$\hat{S}_{q1\rightarrow p1}=e^{\lambda\hat{r}}$ (C1)

inducing the canonical transformation

$\displaystyle u_{2}$ $\displaystyle=\frac{1}{3}(2\bar{u}_{2}+\bar{u}_{3})$
$\displaystyle\qquad u_{3}$
$\displaystyle=\frac{1}{3}(\bar{u}_{2}+2\bar{u}_{3})$ (C2)
$\displaystyle\pi_{2}$ $\displaystyle=2\bar{\pi}_{2}-\bar{\pi}_{3},$
$\displaystyle\qquad\pi_{3}$ $\displaystyle=-\bar{\pi}_{2}+2\bar{\pi}_{3}$

we first compute the invariants of the transformation

$I_{1}=u_{2}+u_{3},\qquad I_{2}=\pi_{2}+\pi_{3}.$ (C3)

In the unitary representation $I_{1}$ and $I_{2}$ must also stay invariant
[43]. Using the Hadamard identity we can write

$\hat{S}\hat{I}_{i}\hat{S}^{\dagger}=\hat{I}_{i}+\lambda[\hat{r},\hat{I}_{i}]+\frac{1}{2!}\lambda^{2}[\hat{r}[\hat{r},\hat{I}_{i}]]+\frac{1}{3!}\lambda^{3}[\hat{r}[\hat{r}[\hat{r},\hat{I}_{i}]]]+\dots\stackrel{{\scriptstyle!}}{{=}}\hat{I}_{i}.$
(C4)

We thus see that we must choose $\hat{r}$ such that the commutator
$[\hat{r},\hat{I}_{i}]$ vanishes. After a little guesswork we find

$\hat{r}=(\hat{u}_{2}-\hat{u}_{3})(\hat{\pi}_{2}-\hat{\pi}_{3}).$ (C5)

It remains to calculate the phase $\lambda$. To this end we compute e.g.

$\hat{S}\hat{\pi}_{2}\hat{S}^{\dagger}=\pi_{2}+\frac{\pi_{2}-\pi_{3}}{2}(e^{2is}-1).$
(C6)

Comparing this with C2 we find $s=-i\frac{\log(3)}{2}$. This leaves us with
the sought-after unitary operator

$\hat{S}_{q1\rightarrow p1}=e^{-i\frac{\log
3}{2}(\hat{u}_{2}-\hat{u}_{3})(\hat{\pi}_{2}-\hat{\pi}_{3})}.$ (C7)

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