_To be or not to be_ , but where?

Guilherme Franzmann1,2,3<EMAIL_ADDRESS>
1Nordita, KTH Royal Institute of Technology and Stockholm University,
Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

2Department of Philosophy, Stockholm University,
Universitetsvägen 10D, 114 18 Stockholm, Sweden

3Basic Research Community for Physics e.V.,
Veddeler Brückenstrasse 122, 20539 Hamburg, Germany

Abstract

The identification of physical subsystems in quantum mechanics as compared to
classical mechanics poses significant conceptual challenges, especially in the
context of quantum gravity. Traditional approaches associate quantum systems
with classical ones localized in spacetime, using either Hilbert space factors
for finite-dimensional systems or local operator algebras in algebraic quantum
field theory. These methods ensure statistical independence for state
preparations and measurements. However, canonical linearized quantum gravity
disrupts this framework by preventing the formation of gauge-invariant local
algebras, thereby undermining the statistical independence required in
measurements. This presents a major obstacle for modeling early universe
cosmology, gravity-induced-entanglement experiments, and poses a significant
roadblock toward a comprehensive theory of quantum gravity. A pivotal shift is
proposed: the identification of classical and quantum systems should be
dynamically evolving rather than static, opening the possibility of a single-
world unitary quantum mechanics. This perspective aligns with the broader aim
of understanding how classical spatiotemporal existence emerges from quantum
mechanics and connects the measurement problem with quantum gravity.

###### Contents

  1. 1 The mysterious box
  2. 2 How space hides in quantum mechanics
  3. 3 A Tale of Emergence
    1. 3.1 The Structure of Physical Theories
    2. 3.2 Emergence - Level 0: Classical Mechanics and the Center of Mass
    3. 3.3 Emergence - Level 1: Classical Mechanics and Coarse Graining
    4. 3.4 Emergence - Level 2: Quantum Mechanics and Classical Mechanics
  4. 4 To be or not to be, but where?
    1. 4.1 Where does the ontology reside?
    2. 4.2 What are the possibilities?
    3. 4.3 Classical Ontology
    4. 4.4 Quantum Ontology
    5. 4.5 How do we decide?
  5. 5 Status-of-art, _or crash course on algebraic quantum field theory_
    1. 5.1 What could we have hoped for?
    2. 5.2 The Algebraic Approach
    3. 5.3 Trapped in Spacetime
    4. 5.4 More ambiguities!
  6. 6 What about gravity?
    1. 6.1 Operational Measurement Limitation
    2. 6.2 _You can’t always get what you want_
  7. 7 Why should we care?
    1. 7.1 Early Universe Cosmology
    2. 7.2 Gravitationally-Induced-Entanglement (GIE) Experiments
  8. 8 Single-world unitary quantum mechanics
    1. 8.1 Quantum subsystems are not time-invariant
    2. 8.2 Comparison with general relativity
    3. 8.3 Discussion
  9. 9 We are inside the box!
  10. 10 Acknowledgements

##  1 The mysterious box

One day, you arrive at your lab and find a mysterious box. The box is
completely black except for a small display where it says “ENERGY
SPECTROMETER”. You try to open the box but cannot quite figure out how. So,
you decide to interact with the display, and write down all the possible
energy values of whatever is inside the box. As you near the end of this
tedious task, all the years spent modeling physical systems give you a hunch
that perhaps whatever is inside the box is actually a simple physical system.

You decide to model it by using a lattice of spin$-\frac{1}{2}$ sites, or
_qubits_. Among all the different terms you can add to the Hamiltonian of this
system, you start by considering that each site couples to their neighbor’s
spin in a given direction with some coupling strength $J$ and that they might
be under the influence of some external magnetic field $h$. Given all your
experience, and the fact that you know how many energy levels there
are222Knowing how many energy levels there are, you also know the minimum size
of the Hilbert space needed to model the system assuming the absence of energy
degeneracies., you propose the following model333We suppress identity
operators for simplicity unless needed to make a point.:

$H=J\sum_{i=1}^{3}\sigma_{i}^{z}\sigma_{i+1}^{z}+h\sum_{i=1}^{4}\sigma_{i}^{z}\,.$
(1.1)

You believe there are 4 qubits, each coupled to its neighbor in the
$z$-direction, with a magnetic field in the same direction. You compute the
energy spectrum of this system, and, to your surprise, there is a pair of $J$
and $h$ for which the spectrum matches exactly with the values displayed on
the black box!

Excited, you storm out of the lab and call a friend to see the black box and
share your discovery. When you both return to the lab and you are about to
show your model, your friend stops you, wanting to figure out the mystery
themselves. After some time, they return, having found their own model for
what is inside the box, described by the following Hamiltonian:

$H=J\sum_{i=1}^{4}\mu^{x}_{i}+h\sum_{i=1}^{3}\mu_{i}^{z}\mu_{i+1}^{z}-J\mu_{4}^{x}+h\mu_{1}^{z}\,.$
(1.2)

It is also a model of spin sites! It looks quite similar to your own, except
that the coupling between their spin sites is not the same as yours, but what
you had as the magnetic field. And their magnetic field is on the
$x$-direction, and not over all their spins… also, their first spin is under
another magnetic direction on the $z$-direction. How is that possible?

You discuss what is going on trying to understand how you both arrived at
different models of what is inside the box that nonetheless matched exactly
the energy spectrum. Until your friend has an idea: what if what you both
called qubits are somehow related to each other? After some thought, you
arrive at the following identification:

$\mu_{i}^{z}=\prod_{j\leq
i}\sigma_{j}^{x},\quad\mu_{i}^{x}=\sigma_{i}^{z}\sigma_{i+1}^{z},\quad\mu_{4}^{x}=\sigma_{4}^{z}\,,$
(1.3)

which, together with the swapping of the coupling strength and magnetic field,
maps one model into the other exactly.

At first, you are not very surprised, as it is quite common in physics to find
different models to describe the same phenomenology. However, as you both
ponder about this map, you start to become suspicious about what is going on.
For instance,

$\mu^{z}_{4}=\sigma_{1}^{x}\sigma_{2}^{x}\sigma_{3}^{x}\sigma_{4}^{x}\,,$
(1.4)

tells you that the spin site 4 in the $\mu$-model in the $z$-direction is
somehow mapped into the product of all the spin sites in the $x$-direction in
the $\sigma$-model. Meanwhile,

$\mu_{1}^{x}=\sigma_{1}^{z}\sigma_{2}^{z}\,,$ (1.5)

relates the $\mu$-sites in the $x$-direction to pairwise neighbors in the
$\sigma$-model in the $z$-direction. The problem is that both you and your
friend used the numbers ${1,2,3,4}$ to label spin _sites_ , which are
localized in spacetime. However, the spin sites in each description are non-
locally related. Each model assumes a different notion of the system,
interactions, and spatial distribution of the physical subsystems. But,
surely, one model must be the _real_ one, or so you hope.

As you both start to lose patience and commit to discovering what the box
hides, you decide to follow the example of particle physicists and grab a
hammer to smash it open. Just as you are about to strike the first blow, the
box disappears. Nonetheless, the question remains: how can you make sense of
what was inside the box?

##  2 How space hides in quantum mechanics

The gedanken experiment above illustrates that quantum systems only described
by their global Hilbert space, $\mathcal{H}$, a state vector, $\ket{\Psi}$,
and a Hamiltonian, $H$, might not fully characterize its classical properties,
such as the notion of a “site” - a location in space. After all, we only had
access to the overall energy associated with each of the global states of the
system444Once again, we considered a simplified scenario where each energy
value displayed in the spectrometer is associated with a non-degenerated
energy eigenstate, thus also fixing the size of the Hilbert space. All the
eigenvalues together can be used to reconstruct the Hamiltonian matrix in its
diagonal form.. This is puzzling because typically, the situation is reversed:
we impart classical notions of location into our quantum mechanical
descriptions. For example, for every spin site located at position $i$, we
assign a $\mathbb{C}^{2}$-Hilbert space factor, such that the global Hilbert
space is

$\mathcal{H}=\bigotimes_{i=1}^{n}\mathbb{C}^{2}\,.$ (2.1)

This is just a simple example of a general attitude555We can speculate whether
this attitude is intrinsically derived from the quantization program, where
classical observables are promoted to quantum operators, and among them,
position and momentum take a special role. Had the program been
_classicalization_ instead, then the emphasis could have been rather
different.: we typically assign a Hilbert space $\mathcal{H}_{\rm system}$ to
each classical subsystem in a given spatial (or more generally, spacetime)
region666As usual, this story is fairly more complicated, in particular in the
context of infinite-dimensional systems and constrained theories, where only
the notion of localizability for the observables is guaranteed [1]. We will
come back to that in Sec. 5 and 6. $U$. Thus, some notion of space, or at very
least, some notion of location in space seems to leak into our quantum-
mechanical descriptions. Let’s make that explicit.

Back at your lab, you consider an extended quantum mechanical system divided
into two parts, $A$ and $B$, with global Hilbert space $\mathcal{H}$. They can
be interacting or not, very much like the spin sites from above. Each part is
surrounded by measuring devices, defining the sets of controlled observables
$\\{\mathcal{O}^{A}\\}$ and $\\{\mathcal{O}^{B}\\}$, which belong to the
algebra of operators on $\mathcal{H}$, $L(\mathcal{H})$, while a clock fixed
against the wall sets up a reference frame for the entire lab. Thus, at any
fixed time in the lab, these parts are spacelike separated, and you expect
measurements on $A$ to not affect measurements on $B$, at least within the
time interval light takes to travel between the systems. Essentially, you aim
to ensure that the statistics of these parts are independent, implementing an
operational notion of macrocausality. For example, the statistics associated
with any observable in part $A$ are obtained by $\tr[\rho\mathcal{O}^{A}]$,
where $\rho$ is the state of the system and belongs to the state space
$\mathcal{S}(L(\mathcal{H}))$. Now consider two measurements, one in each part
of the system, defining two spacetime events that are spacelike separated.
Each measurement updates the state of the system, but their statistics should
be the same whether we consider the original state, $\rho$, or the updated
state corresponding to one of the measurements being performed,
$\rho^{\prime}$ since they were performed at spacelike separations. Therefore,
we demand $\tr[\rho\mathcal{O}^{A}]=\tr[\rho^{\prime}\mathcal{O}^{A}]$, and
similarly for $B$. It turns out that this is possible if, and only if, the set
of observables for each part commute [2, 3], i.e.
$[\mathcal{O}^{A},\mathcal{O}^{B}]=0$. This argument can be generalized to
subalgebras associated with arbitrary causally disconnected spacetime regions,
where then goes by the name _microcausality_ , or Einstein separability. We
will discuss this in Sec. 5.

Thus, we have two commuting sets of observables. If we also require that they
together span all the possible operators associated with the system777Rmember
that any operator can be decomposed into a complex sum of two self-adjoint
operators, i.e., observables., i.e.,
$\mathcal{O}^{A}\otimes\mathcal{O}^{B}\cong L(\mathcal{H})$, then it follows
[4] that the Hilbert space of the quantum system can be factorized,
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Thus, our local
assignment of Hilbert space factors to parts of a bigger system is born out of
macrocausality, operationally implemented in a way that leads to
microcausality, and finally leading to the factorization of the Hilbert space.

Now that the physical intuition is properly encoded in the math, let’s be more
precise and general. Following [4], we consider a finite-dimensional Hilbert
space $\mathcal{H}$ and a collection of subalgebras of observables
corresponding to each subsystem satisfying the following properties:

  * •

Local acessibility: Each $\\{\mathcal{O}^{I}\\}$ corresponds to a set of
controllable observables, where $I$ labels the different systems;

  * •

Subsystem independence: $[\mathcal{O}^{I},\mathcal{O}^{J}]=0$, where $I\neq
J$, which we now understand as a direct consequence of operational
macrocausality;

  * •

Completeness: $\bigvee_{I}\mathcal{O}^{I}\cong L(\mathcal{H})$, there is a
minimal inclusion of the subalgebras that is isomorphic to the whole algebra
of operators.

Once these properties are in place, then the set of subalgebras induces a
tensor product structure (TPS) [5]:

######  Definition 1 (TPS)

: A TPS $\mathcal{T}$ of Hilbert space $\mathcal{H}$ is an equivalence class
of isomorphisms $T:\mathcal{H}\rightarrow\otimes_{i}\mathcal{H}_{i},$ where
$T_{1}\sim T_{2}$ whenever $T_{1}T_{2}^{-1}$ can be written as a product of
local unitaries $\otimes_{i}U_{i}$ and permutations of subsystems.

By choosing a TPS, we determine how the Hilbert space of the entire quantum
mechanical system (like the box in the example above) will be partitioned.
However, now we understand that this assignment is intrinsically related to
our classical notion of spacetime. On the other hand, if nature is
fundamentally quantum mechanical, it might make more sense to start with the
global Hilbert space and then find an appropriate decomposition888There are
recent works in this direction, for instance [6, 7, 5]. \- particularly one
that aligns with our classical observables, such as the notion of _site 1 is
here, site 2 is there_. The question, however, is whether there are multiple
classical possibilities associated with the same quantum system, as seen in
the different _site_ attributions in our gedanken experiment above, and vice-
versa.

Once a TPS is introduced, we examine the 3-tuple $(\mathcal{H},H,\ket{\Psi})$
through the composition of each factor $\mathcal{H}_{i}$ (or local
subalgebras, $\\{\mathcal{O}_{i}\\}$). This is exactly what happened in our
box experiment, where you and your friend tacitly assumed different TPSs. Each
$\mathcal{T}$ considered is not related by local unitaries or permutations of
subsystems; instead, the set of variables are non-locally related as shown in
(1.3), amounting to a global unitary transformation. Thus, the TPSs are not
equivalent. That can also be checked by fixing the factorization of the
Hilbert space, $\bigotimes_{i}\mathcal{H}_{i}$ and showing that for each TPSs
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, the operators $T_{1}HT_{1}^{-1}$ and
$T_{2}HT_{2}^{-1}$ are not the same up to conjugation by local unitaries,
permutations of subsystems, and transposition. Therefore, a change of TPS has
an equivalent perspective where the TPS is fixed but the Hamiltonian is
unitarily transformed, keeping its spectrum.

Now, although the TPSs implicitly assumed for each description of the box are
not equivalent, the associated Hamiltonians present the same level of
locality, being the sum of $1$\- and $2$-point terms. Both Hamiltonians are
said to be $2$-local. This is a specific case of a duality [5]:

######  Definition 2 (Duality)

Two TPSs are dual if they are not equivalent and their associated
Hamiltonians, which have the same spectra, are $k$-local.

You might expect that the issue of identifying the appropriate classical
limit, understood as the assignment of Hilbert space factors to
spatiotemporally localized systems, arises only when dual descriptions exist.
After all, if the box cannot be opened (i.e., we cannot extend its Hamiltonian
[8]), we have incompatible classical descriptions of what is in the box.
However, the problem runs deeper. Even without dualities, a generic Hilbert
space with a generic Hamiltonian will have multiple tensor factorizations,
only constrained by the dimension of the Hilbert space. And without further
specification, all these decompositions are equally reasonable representations
of the quantum-mechanical system.

This quantum democracy has led to recent efforts to identify criteria for
singling out a preferred factorization of the Hilbert space [6, 7]. In
particular, [5] suggests a criterion based on the fact that for a TPS with $n$
subsystems, whenever $n\gg k$, the set of $k$-local Hamiltonians has measure
zero. If one can then prove that dualities are not present, this would imply
an almost unique TPS for a $k$-local Hamiltonian. Thus, by considering
$k$-locality of the Hamiltonian as an additional structure to
$(\mathcal{H},H,\ket{\Psi})$, a preferred factorization is selected. This also
determines a preferred way to map quantum subsystems, as factors in Hilbert
space, into classical ones, as systems localized in spacetime.

The keyword here is _map_. As we have discussed, there are ways to connect
aspects of the classical world to its underlying quantum structure. For
example, we have established the connection between operational macrocausality
and microcausality, and discussed how classical systems are associated with
algebraic structures in quantum mechanics. However, we have also encountered
dualities that hinder this direct connection. To further explore how the
classical world relates to the quantum one, we will consider their connection
through the lens of emergence.

##  3 A Tale of Emergence

In this section, we will discuss the concept of emergence, start to clarify
the distinction between classical and quantum systems and establish
foundations to better understand our mysterious box.

We start by providing our working definition of emergence. Following [9, 10,
11], we consider:

######  Definition 3 (Emergence)

Emergence is understood as the appearance, in a certain description of a
physical system, of properties that are novel with respect to a different
(more “fundamental”) description of the same system, and robust, thus stable
enough to represent a characterization of the new description and to form part
of new predictions stemming from it. Emergence, in this understanding, usually
requires the use of some limiting procedure and of a number of (possibly)
drastic approximations, to allow the novel properties to become visible in the
new description.

Key to this definition are the terms _novel_ , _robust_ , and _new
predictions_. When we have a description that is novel and robust, in the
sense that it stands on its own and provides a way to predict the behavior of
physical systems, we typically call it a physical theory. Of course, this also
underlies what we mean by a more fundamental description. Therefore, perhaps
we should stop here and scrutinize the structure of what we mean by physical
theories to make the different flavors of emergence more explicit.

###  3.1 The Structure of Physical Theories

Most physical theories can be broken into four different categories. Briefly,

  * •

Underlying state space ($\Omega$): This represents the space of all possible
states of a physical system. For example, in Lagrangian mechanics, this is the
configuration space, while in Hamiltonian mechanics, it is the phase space. In
quantum mechanics, we typically consider the Hilbert space999See footnote 20,
though.;

  * •

Degrees of freedom ($\Phi$): For the same type of underlying space, there can
be different degrees of freedom. For instance, in phase space, one can
describe a particle or a field. In quantum mechanics, we typically consider
the wavefunction (state vector) or, more generally, the density matrix;

  * •

Observables ($\mathcal{O}$): These are the quantities that can be measured,
typically represented as maps from the state space to the set of real
numbers101010This is the naive definition. The more general definition flips
the role of states and observables, and it is the state that is a linear
functional of the observables into the real numbers, as introduced later in
the paper.;

    * –

In classical mechanics, observables are typically functions
$f:\Omega\rightarrow\mathbb{R}$ , where $\Omega$ is the phase space, for
example.

    * –

In quantum mechanics, observables are self-adjoint (Hermitian) operators
$\mathcal{O}$ on a Hilbert space $\mathcal{H}$, ensuring real eigenvalues that
are associated with measured values.

  * •

Dynamical equation ($U$): Once a state is known for a given degree of freedom,
the dynamical equation determines the evolution of the state, defining a
trajectory in the state space. In classical mechanics, we have Newton’s,
Euler-Lagrange or Hamiltonian equations111111In this section, we will assume
classical mechanics is governed by Newton’s equations, as this is sufficient
to make our point., while in quantum mechanics, we have the Schrödinger or
Heisenberg equation.

In the language of [12], these elements are reduced to a triple $T=(S,Q,D)$:
the state-space, the quantities, and the dynamics. The reason why we say
_most_ physical theories, is because some do not fit this decomposition: e.g.,
in thermodynamics, the second law does not take the form of a dynamical
equation. Nonetheless, to convey our points in the next sections, we will
ignore these subtleties.

We will also consider another simplification: instead of discussing the state
space, we will focus on the notion of _stage space ( $\mathcal{S}$)_, which is
the space implicitly or explicitly associated with some notion of existence,
serving as the background against which a theory is defined. Thus, to discuss
emergence, we will rely on the scheme defined by the quartet
$(\mathcal{S},\Phi,\mathcal{O},U)$.

Now that we have at hands the definition of emergence and the underlying
structure of physical theories, we can establish different levels of emergence
by considering some examples.

###  3.2 Emergence - Level 0: Classical Mechanics and the Center of Mass

Our first example involves the center of mass of a system. The center of mass
is an important concept in classical mechanics because it simplifies the
analysis of many-body systems. By reducing the complexity of the system, we
can describe the motion of the entire system using a single point that
represents the average position of all the particles weighted by their masses.
This simplification is useful in various applications, such as understanding
the motion of celestial bodies, analyzing the stability of structures, and
simplifying collision problems.

We begin by defining the center-of-mass particle’s position as a map $X_{\rm
CM}$,

$\displaystyle X_{\rm CM}:\left\\{\begin{aligned}
&\mathds{R}^{3N}\rightarrow\mathds{R}^{3}\\\ &\vec{x}_{\rm
CM}\equiv\frac{1}{M}\sum_{I=1}^{N}m_{\rm I}\vec{x}_{\rm I}\,,\quad\quad
M=\sum^{N}_{I=1}m_{\rm I}\,,\end{aligned}\right.$

which maps position vectors in spacetime to a single position vector in
spacetime121212If we wanted to focus on state space instead of spacetime, we
could say that this map collapses the $N$-particle configuration space into a
1-particle configuration space.. This map is surjective but not injective, so
it is not invertible. In other words, it is a _projection map_. This means
that if we only had access to the center of mass data of a system, there would
be many inequivalent possible lower-level arrangements of it that would
correspond to that information.

Thus, we can consider the center of mass as a new description of the system,
even though the type of degree of freedom remains the same. The system is now
described by the position and velocity of the center-of-mass particle. Since
the type of degree of freedom remains the same in the new description, the
type of observables also remains the same, namely differentiable functions,
${f_{a}}$, that map points in phase space to real numbers (or probability
distributions, more generally). Thus, we can also think of the center-of-mass
map as inducing a projection map in the space of observables, which changes
the codomain of the observables from $\mathds{R}^{3N}\times\mathds{R}^{3N}$ to
$\mathds{R}^{3}\times\mathds{R}^{3}$. However, the type of observables does
not change, meaning we can still ask the same questions about the center of
mass as we could about any of the particles of the system before the mapping.
The same applies to the equation of motion, which remains Newton’s second law
for a particle with total mass $M$, resulting from the linearity of the
equation.

We summarize all these points in Table 1 and take it to represent a mild form
of emergence, referred to _Emergence - Level 0_.

|  |  Fundamental Description |  |  Emergent Description
---|---|---|---|---
Stage Space |  |  $\mathcal{M}$ |  |  $\mathcal{M}$
|  |  |  |
Degrees of Freedom |  |  $10^{51}\text{ of }(\vec{x},\vec{v})^{\prime}s$ |  |  $(\vec{x}_{\rm CM},\vec{v}_{\rm CM})$
|  |  |  |
Observables |  |  $f_{a}(\vec{x}_{\rm I},\vec{v}_{\rm I})$ |  |  $f_{a}(\vec{x}_{\rm CM},\vec{v}_{\rm CM})$
|  |  |  |
Dynamical Equations |  |  $\vec{F}(\vec{x}_{\rm I})=m_{\rm I}\frac{d\vec{v}_{\rm I}}{dt}$ |  |  $\vec{F}(\vec{x}_{\rm CM})=M\frac{d\vec{v}_{\rm CM}}{dt}$
Table 1:  Summary of the fundamental and emergent descriptions for the center-
of-mass of an object in classical mechanics. Fundamentally, there is a large
number of degrees of freedom for individual particles - for instance, Earth
alone has approximately $10^{51}$ particles. The emergent description retains
the same type of observables and equations of motion but reduces the
complexity by aggregating the system into a single center-of-mass particle.
The stage space remains the same.

#### 3.2.1 Does the center of mass exist?

As we aim to build our intuition around ontology as emergence unfolds, we can
begin by asking some fundamental questions. There is plenty of literature
around this topic [13, e.g. and references therein], but we just want to
highlight a few things to slowly build our intuition for later on.

First, in some situations, the center of mass defines a point that lies within
the composite object it represents, making it materially instantiated. In such
cases, there is literally a material point with its own position and velocity
vector that follows the center-of-mass trajectory. However, this is not always
the case. Often, the center of mass does not correspond to any actual material
point within the defining object. From a substantivalist perspective, all
points in space exist in an absolute sense, so the center of mass trajectory
simply labels some of these points.

Rather than focusing on material instantiation or substantivalism to argue for
the existence of the center of mass, we can adopt the perspective supported by
our definition of emergence. As long as new entities are novel and stable
enough to yield reliable predictions, such as the case for the center of mass,
we can attribute emergent ontological status to them, similar to Wallace’s
implementation [13] of Dennett’s _real patterns_ [14] in physics.

###  3.3 Emergence - Level 1: Classical Mechanics and Coarse Graining

Our next example is a well-established one: how thermodynamics emerges from
statistical mechanics. We will simplify the discussion by considering only an
ideal gas in equilibrium, focusing on how the different elements of the
physical structure change from one theory to the other.

Thermodynamics also applies to systems of many particles, but instead of
averaging over the positions of the particles, we average over the square of
their velocities. We define the mean map $M$ as follows:

$\displaystyle M:\left\\{\begin{aligned}
&\mathds{R}^{3N}\longrightarrow\mathds{R}\\\ &\langle
v^{2}\rangle\equiv\frac{1}{N}\sum_{I=1}^{N}v_{I}^{2}\,,\end{aligned}\right.$
(3.1)

from which energy, temperature, and pressure for an ideal gas are defined as,

$E=\frac{3}{2}k_{B}\langle v^{2}\rangle\,,\quad\quad T=\frac{m}{3k_{B}}\langle
v^{2}\rangle\,,\quad\quad P=\frac{Nm}{3V}\langle v^{2}\rangle\,.$ (3.2)

Here, $V$ is the volume of the container where the particles can move, and $m$
is their mass.

Together, the equations above give rise to the ideal gas law, $PV=k_{B}nT$,
where $n$ is the number of moles, which is a significant departure from the
underlying Newton’s equation governing each of the particles in the gas. In
the equilibrium case, the situation is quite drastic as all the degrees of
freedom are reduced to three, but in reality, two, since the ideal gas law is
a constraint equation that establishes the temperature and pressure of a
uniform gas in a box of volume $V$. Thus, all the physical observables
associated with the box are now functions of these macroscopic quantities,
$F_{a}(T,V,P)$, which are quite different from the underlying particle
ontology, since we now talk about entropy, heat capacities, etc.

Despite the change in the type of degrees of freedom, equations, and
observables, the stage space remains physical space. To make this more
explicitly, we can relax the condition of global thermal equilibrium and
consider only local thermal equilibrium. In this case, the ideal gas law holds
locally, and the temperature and pressure become scalar fields that vary from
point to point in space. In this scenario, there are also dynamical equations,
known as general transport equations, that explicitly incorporate time.

We summarize this example in Table 2 to illustrate what we call _Emergence -
Level 1_.

|  |  Fundamental Description |  |  Emergent Description
---|---|---|---|---
Stage Space |  |  $\mathcal{M}$ |  |  $\mathcal{M}$
|  |  |  |
Degrees of Freedom |  |  $n\times 10^{23}\text{ of }(\vec{x},\vec{v})^{\prime}s$ |  |  $(T,V,P)$
|  |  |  |
Observables |  |  $f_{a}(\vec{x}_{\rm I},\vec{v}_{\rm I})$ |  |  $F_{a}(T,V,P)$
|  |  |  |
Dynamical Equations |  |  $\vec{F}(\vec{x}_{\rm I})=m_{\rm I}\frac{d\vec{v}_{\rm I}}{dt}$ |  |  $PV=k_{\rm B}nT$
Table 2: Summary of the fundamental and emergent descriptions in the context
of equilibrium thermodynamics. Once again, fundamentally there is a large
number of degrees of freedom for individual particles (of order $n$ times
Avogrado’s number in a room), while the emergent description simplifies the
system by focusing on macroscopic thermodynamic variables like temperature,
volume, and pressure. The emergent description retains the same stage space
but significantly reduces the complexity by aggregating the behavior of
numerous particles into a few macroscopic observables and a single equation.

#### 3.3.1 Does temperature exist?

We can now pose the same type of question we asked about the center of mass,
but this time for the temperature field. If the ontological status of the
center of mass was unclear but did not prevent us from using emergence to
assign it a sense of reality at the emergent level, the same applies to the
temperature field.

In fact, the more _drastic approximations_ we impose to transition from the
base theory to the emergent description, the more inclined we might be to
establish the independent existence of entities at the emergent level, as long
as they help provide reliable predictions. This is certainly the case for the
temperature field, as it is a key variable in equations of state, such as the
ideal gas law, and in determining phase transitions and thermal properties.
Moreover, temperature is a directly measurable quantity; it can be empirically
determined using thermometers and other temperature-sensitive devices. These
measurements consistently align with theoretical predictions, reinforcing the
reality of temperature as an emergent property. Thus, while temperature is not
a fundamental property at the microscopic level, its emergent ontological
status is well-supported by its predictive power and experimental
verification.

###  3.4 Emergence - Level 2: Quantum Mechanics and Classical Mechanics

Just a year after Boltzmann equations were derived to describe the statistical
behavior of out-of-equilibrium thermodynamical systems, Riemann [15] was
already anticipating what would be required for the next level of emergence:

> The questions about the infinitely great are for the interpretation of
> nature useless questions. But this is not the case with the questions about
> the infinitely small. It is upon the exactness with which we follow
> phenomena into the infinitely small that our knowledge of their causal
> relations essentially depends.
> (…)
> Now it seems that the empirical notions on which the metrical determinations
> of space are founded, the notion of a solid body and of a ray of light,
> cease to be valid for the infinitely small. We are therefore quite at
> liberty to suppose that the metric relations of space in the infinitely
> small do not conform to the hypotheses of geometry, and we ought in fact to
> suppose it, if we can thereby obtain a simpler explanation of phenomena.
> (Riemann, 1873)

And about three decades later, quantum mechanics was developed, and the stage
space itself would no longer be based on spacetime, but rather on quantum
space, called Hilbert space.

Quantum mechanics also follows the structure of physical theories discussed
above. Quantum mechanical degrees of freedom are typically represented as
vectors in Hilbert space131313In this section, this simplification suffices to
the points we want to make., evolving according to the Schrödinger equation.
Physical observables are represented by self-adjoint operators instead of
smooth measurable functions. However, at face value, the theory introduces
other structures, such as the Born rule and the (real or apparent) collapse of
the quantum state: the Schrödinger equation is linear, it allows for
superpositions of different states, but it seems that measurements unavoidably
reduce the superimposed quantum state to a single one. In other words,
measurements seemingly give out single outcomes. This is also known as the
_measurement problem_ 141414Given the extensive literature, among personal
favorites are [16, 17, 18, 19] and will be further discussed in Sec. 8.

Now, it is generally believed that the world is fundamentally quantum
mechanical. After all, it rules the small, microscopic world, and the
macroscopic world is composed of these small entitites, so it seems that
classical mechanics has lost its status as a base theory, and should be seen
as the emergent description of the macroscopic world instead. As this time
around the state space also changes, we call this _Emergence - Level 2_. We
summarize the example of the classical emergent description from quantum
mechanics in Table 3.

|  |  Fundamental Description |  |  Emergent Description
---|---|---|---|---
Stage Space |  |  $\mathcal{H}$ |  |  $\mathcal{M}$
|  |  |  |
Degrees of Freedom |  |  Q-states $|\psi\rangle$ |  |  $(\vec{x}_{\rm I},\vec{v}_{\rm I})$
|  |  |  |
Observables |  |  $\hat{\mathcal{O}}_{A}^{\dagger}=\hat{\mathcal{O}}_{A}$ |  |  $f_{a}(\vec{x}_{\rm I},\vec{v}_{\rm I})$
|  |  |  |
Dynamical Equations |  |  $i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}|\psi\rangle$ |  |  $\vec{F}(\vec{x}_{\rm I})=m_{\rm I}\frac{d\vec{v}_{\rm I}}{dt}$
Table 3: Summary of the fundamental and emergent descriptions in the context
of quantum and classical mechanics. At the quantum level, the stage space is
represented by the Hilbert space, and self-adjoint operators act as
observables on quantum states that follow the Schrödinger equation. Meanwhile,
the classical stage space is spacetime, where we define positions and
velocities and measurable functions of them as observables, and the dynamical
equation is Newton’s equation.

##  4 To be or not to be, but where?

If we had doubts about the existence of emergent entities in the previous
examples of lower levels of emergence, it is now time to question or reaffirm
the existence of classical degrees of freedom, and even their stage space,
through the lens of emergence. Unlike the previous examples, degrees of
freedom and observables associated with space and time, and space and time
themselves, are deeply rooted in our physical intuition about the world.

This is evident from our language. In fact, ontology, understood as the study
of what there is and their relations [20], relies on the verb _to be_ for its
definition. This verb is inherently intransitive and can stand alone to define
existence. However, it reflects the physical intuition that entities exist in
space and time, as no other type of stage space was considered until the
advent of quantum mechanics, at least in physics151515In the philosophy of
mind, non-physicalist philosophers have introduced mental states that do not
have the material world as their stage space. The ontological challenges of
reconciling the classical and quantum stage spaces can draw parallels from
that literature, as explored in [21, 22], where the _hard problem of
spacetime_ is discussed.. Thus, we are now faced with two possibilities: i) to
move from a classical ontology (where the verb _to be_ is defined in
spacetime) to a quantum ontology (where the verb _to be_ is defined in Hilbert
space), or ii) to allow for a multi-level ontology [11], reassured by the
notion of classical real patterns emerging from the underlying quantum world
[13].

We could summarize the second perspective by promoting the verb “to be” from
an intransitive verb to a mandatory prepositional verb, where things are _in
spacetime_ or _in Hilbert space_. By establishing this nuance in language, it
becomes clearer what work needs to be done to connect the entities defined in
each space. In particular, as seen in the previous examples of lower levels of
emergence, lower-level entities can typically be grouped in many different
ways to give rise to entities in the emergent description (e.g., many
configurations of masses can give rise to the same center of mass position).
Thus, the typical identification of classical systems with their quantum
counterparts seems to be a limitation of the quantization program, which
starts from classical systems and attempts to derive their quantum
description, instead of starting from the base level where quantum systems are
defined and allowing a considerable amount of ambiguity in how their classical
counterparts are obtained in the emergent level. It is this latter program,
motivated by insights from quantum gravity, that we will pursue in Sec. 8.

###  4.1 Where does the ontology reside?

Returning to our initial example, we find ourselves in a conundrum: is the
system inside the box a spin chain with interacting sites represented by
$\sigma_{i}$ or $\mu_{i}$, or perhaps something else entirely? We might hope
to break the box and directly observe its contents, thus extending the initial
Hamiltonian and coupling the system to something else. However, if the box
cannot be broken—similar to the lone harmonic oscillator and string theory
discussed in [8]—we face a choice: where should the ontology reside?

Before exploring the options, it is important to emphasize that we have
identified two different notions of systems. Classical systems occupy regions
of spacetime and are described by degrees of freedom in spacetime, while
quantum systems are described by degrees of freedom in Hilbert space. While
classical systems typically interact locally in spacetime, meaning their
interactions are point-like in the Hamiltonian (the systems need to make
contact), quantum systems interact locally when the interaction terms in the
Hamiltonian can be factorized as the product of operators acting independently
on each Hilbert space factor associated with each subsystem (for a review, see
[3]). Thus, both the notion of subsystems and the notion of locality in the
Hamiltonian are fundamentally different161616The mismatch between quantum-
mechanical locality and spatiotemporal locality has been discussed in the
context of loop quantum gravity, referred to as disordered locality [23], and
even considered as a possible explanation for dark energy [24]..

###  4.2 What are the possibilities?

As ontology can be quickly summarized as the study of what there is, it is
crucial to clarify what is meant by _what_ and _is_. As we discussed above, we
now see that _what_ can refer to two very different types of things: classical
or quantum systems. Similarly, _being_ can be defined in different senses,
either in terms of the classical or quantum stages. Moreover, the discussion
also expands to the possible maps that could be established between these
different modes of existence, which unavoidably leads to the discussion about
the quantum-classical limit.

The project of ontology extends beyond simply defining what exists. Once we
establish (or rather, posit) what things exist, the project then explores the
most general features and relations among these entities. Therefore, our
discussion about locality, and more broadly physical interactions, is also
directly related to ontological matters.

Another important aspect of this discussion is the expected plurality of
entities. If we argue for the universe as a whole, either classically or
quantum-mechanically, we talk about _things_ in the singular and focus on the
holistic aspects of either spacetime or Hilbert space, their global state, and
Hamiltonian as what ultimately exists. The partition of these spaces into what
we typically see as classical and quantum subsystems is viewed as emergent,
despite the persistence of _real patterns_ associated with emergent entities
[14, 13]. From this perspective, the second part of the ontological project
becomes superfluous, as there is no meaningful notion of _relations_ for a set
composed of a single element, except the trivial comparison with the empty
set171717An instance of this comparison in physics can lead to questions such
as _Why is there something rather than nothing?_ [25].. A recent defense of
this position can be found in [26]. This holistic approach will be contrasted
with the _atomic_ one where there exists more than one single irreducible
entity and relationships among them.

Thus, we end with four different possibilities: {holism, atomism} $\times$
{classical, quantum}. We have identified at least one relevant category to
guide the atomistic approach, which is locality (see Table 4). There is
extensive literature discussing all these different possibilities, so we will
not attempt this here, but just focus on crucial aspects involving these
different ontologies that will set the stage for the rest of the paper.

| Quantum | Classical
---|---|---
Holism | Holistic Quantum Ontology | Holistic Classical Ontology
Atomism | Atomistic(k) Quantum Ontology | Atomistic Classical Ontology
Table 4: Here we show the different possibilities where the ontology of the
world can be placed. The index $(k)$ represents the degree of quantum-
mechanical locality. Note that any degree of spatiotemporal nonlocality is
seen as action-at-a-distance, while quantum-mechanical $k$-locality simply
means that the interactions in the Hamiltonian involve terms relating up to
$k$-subsystems, as discussed in Sec. 2.

###  4.3 Classical Ontology

Why should we even discuss classical ontology? After all, isn’t the world
fundamentally quantum mechanical? This is precisely why we start with simpler
questions about the ontological status of the center of mass and temperature.
Similarly, we could invoke emergence to justify why classical mechanics should
be granted ontological status, following the reasoning of [13]. However, as we
will discuss further in Sec. 5 and Sec. 8, quantum mechanics, as it stands,
appears unavoidably anchored in the classical world.

Moreover, we seem to exist in spacetime, so it is natural to align our
ontological intuitions with this crucial aspect of our pursuit of increasingly
better descriptions of the world. Ontology, after all, is meant to connect
with our intuitive understanding of the world and it is reasonable to assign
scientific reality (in the sense of scientific realism [27]) to entities that
take on clearly defined spatiotemporal regions. Let’s consider each option
carefully.

##### Classical Holistic Ontology

views the classical world as one. This perspective contrasts with the
atomistic view, which holds that the world consists of independent parts. In
classical physics, the theory that allows us to consider the world and its
overall evolution is general relativity. In this framework, we can write
equations, such as the Friedmann equations, that are meant to describe the
entire Universe. It is by attributing reality to the solutions of these
equations that we are led into discussions about _eternalism_ and the block
universe [28], as well as the cosmological multiverse [29], for example.

From an epistemological standpoint, classical holistic ontology challenges
reductionism, which seeks to explain complex systems purely by understanding
their individual parts [30, 13]. The properties of a whole system often emerge
from the collective dynamics of its parts and cannot be predicted in practice
solely from the behavior of individual components. Examples include the type
of mechanical and thermodynamical emergent behaviors described in the last
section.

To some extent, this perspective was fully justified in Newtonian mechanics,
where anything was non-locally connected to everything else. After special
relativity, we learn that events can be spacelike separated, but as long as
the background spacetime was Minkowsky, any two spacelike-separated events
would always eventually find a common past. It is really with general
relativity that this holistic ontology could face some challenges, as the very
solutions of the theory that give rise to the cosmological multiverse describe
distinct spacetime regions that are never in causal contact. This defines
_independent parts_ of the Universe from a global perspective. Let’s use this
as a cue to move on to the next possibility.

##### Atomistic Classical Ontology

posits that the world is composed of distinct, independent entities that
interact in spacetime. This perspective aligns with the traditional view in
classical physics, where systems are considered as collections of discrete
parts, each occupying a specific region in spacetime and interacting either
locally or non-locally with other systems. Typically, we pursue classical
theories that describe phenomena through localized interactions between
particles or fields, emphasizing the independence of parts when regions are
spacelike separated.

As discussed above, in general relativity there are solutions that establish
clear global distinctions between different patches of spacetime. However, we
often assume this to be possible, at least approximately, on a local scale.
One can imagine a thick Cauchy slice of the entire spacetime and then consider
any two spacelike-separated regions within this slice as independent. However,
it is important to note that while events within these regions in a thick
Cauchy slice cannot causally interfere with each other, the content of these
regions might still be causally related through their past lightcone
overlap181818Here, we see the first hints of the separation between
independent measurements and independent state preparations. This joint
independence leads to the concept of statistical independence in classical and
quantum mechanics [31, 3].. It is important to note that such a procedure is
only possible by first determining the spacetime metric, which involves
solving the Einstein equations with global initial conditions and integrating
over the entire spacetime. Therefore, this local approximation is only
feasible by considering the global structure of spacetime. This will be
further discussed in Sec. 8.

###  4.4 Quantum Ontology

The context of quantum mechanics makes everything much more interesting. In
quantum mechanics, the primary elements of reality are not particles localized
in spacetime but quantum states described by vectors in Hilbert space. These
states can exist in superpositions and exhibit spatiotemporal non-local
correlations, challenging the classical notion of independence. Adopting a
quantum ontology thus requires rethinking the independence and interaction of
parts. This perspective forces a reconsideration of how we define and measure
the components of a system, as well as how we understand their
interactions—topics we will explore in Sec. 7 and Sec. 8. Quantum mechanics
seems to suggest a more holistic view, where the properties of the whole
system cannot always be decomposed into the properties of its parts. This
reasoning also justifies approaches to quantum cosmology and views where
reality is seen as an evolving vector in Hilbert space [26].

##### Quantum Holistic Ontology

proposes that the fundamental nature of reality is best understood as an
interconnected whole rather than as a collection of independent parts. This
view is deeply rooted in the principles of quantum mechanics, where
entanglement and superposition create correlations between systems that defy
classical notions of separability. Unlike classical systems, quantum systems
do not merely consist of discrete entities with localized interactions;
instead, they are described by global quantum states in Hilbert space, which
capture the complex, non-local relationships between parts of the system.

From my perspective, the main reason to consider this ontology is to attribute
a sense of reality to entities that satisfy the Schrödinger equation, in
contrast to quantum subsystems that seemingly demand some form of non-unitary
evolution, apparent or not. Thus, the measurement problem will play a crucial
role in how we approach this topic in Sec. 8.

##### Atomistic(K) Quantum Ontology

explores the possibility of maintaining a form of atomism within the framework
of quantum mechanics, where the universe is composed of distinct quantum
subsystems. Unlike classical atomism, which relies on spatial localization and
independence, each quantum subsystem is associated either with factors in
Hilbert space for finite-dimensional systems (Sec. 2) or via local operator
algebras in algebraic quantum field theory (Sec. 5). However, the existence of
entanglement undermines the statistical independence of quantum systems, at
least as long as they are understood as invariant under the unitary evolution
of the global Hilbert space to which they belong. We will revisit this in Sec.
8, where we will offer an alternative way to bridge holistic and atomistic
quantum ontology, providing a new perspective on the measurement problem.

###  4.5 How do we decide?

As mentioned briefly above, we seem to be trapped in spacetime with classical
measurements. Thus, given our invocation of scientific realism, which employs
rationales such as the no-miracles argument [27] to attribute a sense of
reality to theoretical terms in our theories, we soon are led into problems,
such as our mysterious box in Sec. 1. There, we saw that under our limited
measurements—only the energy displayed by the box—we could not infer the
nature of the quantum system inside the box.

As discussed in Sec. 2, atomistic classical and quantum ontologies are
intrinsically interconnected by the requirement that local measurements can be
performed independently, establishing a direct connection between macro- and
microcausality. Now, if there is ambiguity at the quantum level about how the
Hilbert space is partioned and how local the quantum subsystems are
interacting, we could imagine that further criteria should be imposed to
select a preferred factorization of the Hilbert space, or assignment of local
algebras. This project is pursued to some extent191919Other ideas for
selecting a preferred factorization of the Hilbert space have been recently
pursued in [6, 7, e.g.], with general discussions about decompositions of
Hilbert space found in [32]. in [5] where they argue that imposing a quantum
mereology resulting in the most possible quantum-local Hamiltonian is (almost
always) sufficient to select a particular TPS, which then leads to a specific
classical limit. From the perspective of classical ontology, we can invert
this picture and view their program (which could be thought of as establishing
the grounds for discussing quantum mereology from a Hilbert space
fundamentalist perspective) as recognizing that we indeed live in a specific
classical limit and then finding the necessary criteria at the quantum level
aligned with this classical limit.

It is in this same vein that our approach to the measurement problem will be
pursued in Sec. 8. But first, we need to understand how the current state-of-
the-art connects the quantum and classical realms, and then show how a quantum
theory of gravity jeopardizes such a picture.

##  5 Status-of-art, _or crash course on algebraic quantum field theory_

At the end of the day, we are seemingly trapped in spacetime. By this, I mean
that the _real patterns_ [14] surrounding us are best described by spacetime
physics, which includes all the possible measurements we make [33]. In fact,
despite their complexity, measurements still amount to simply counting—such as
the number of clicks in a detector or spots on a screen—from which we can
infer the state of our physical systems.

Returning to the language introduced in Sec. 3.1, the expectation values
$\langle\mathcal{O}\rangle$ of our measurements are a combination of a certain
quantity $\mathcal{O}$ given some state $s$ of the physical system:

$\langle\mathcal{O}\rangle\equiv\langle s|\mathcal{O}|s\rangle\,.$ (5.1)

You can think of each quantity as representing a real physical apparatus that
is meant to make a specific measurement, with its display providing the range
of possible values it can take once a system is measured.

The question is: how do we connect the fact that we are in spacetime, with our
measurement devices and physical systems in our labs, to the more abstract
formalism of quantum theory? In other words, how should we map the more
fundamental quantum world into the realm of classical spacetime?

To understand this, we will qualitatively describe the state-of-the-art in our
algebraic understanding of quantum theory. We will keep things as simple as
possible to build the necessary intuition, allowing us to appreciate where
things might go wrong as we approach the regime of quantum gravity.
Additionally, we will rely on Sec. 2 to discuss the interplay between the
concepts of locality and physical systems across classical spacetime and
quantum Hilbert space. By the end of our discussion, the mystery of the box
from Sec. 1 should become clearer, and we will have introduced the necessary
tools to consider a new approach to the measurement problem in Sec. 8.

###  5.1 What could we have hoped for?

We operate under the belief that the world is fundamentally quantum-mechanical
and that the entire spacetime is associated with an underlying global Hilbert
space. Here we see the relevance, and direct connection between the holistic
ontologies discussed in Sec. 4.2. A crucial aspect of our activities in
spacetime is the idea that it can be divided into independent parts. This
concept underlies the separation between a system to be measured, our devices,
and even ourselves as observers. The question then becomes what is needed to
ensure that the whole can be broken into independent pieces, and to what
extent this can be realized in Hilbert space.

Unexpectedly, this is a deep philosophical and physical problem that also
appears in many other disciplines. We do not intend to provide an exhaustive
discussion of the matter, but we will highlight at least one criterion that we
already met that is needed classically to define any reasonable notion of
independence: _causal independence_. If we know that two systems cannot
exchange any information in a way that all possible observers would agree on,
this is a strong criterion for considering them as independent entities.
Special and general relativity provide the framework where this can be
formulated. We will refer to this causal independence as _spatiotemporal
locality_ , as it establishes the condition under which we can securely say
that something is independently located relative to something else (if they
were not causally disconnected, there would be observers for whom the systems
could be arbitrarily close to each other).

Let’s be more precise. What we have in mind are two small regions of
spacetime, $U$ and $V$, which are spacelike separated and define local Cauchy
surfaces. This means that there is a spacelike curve inside each region that
contains all the necessary information to describe anything inside those
regions.

Thus, what we could have hoped for is an analog of them in Hilbert space. That
is, the entire Hilbert space can also be broken down into independent pieces,
with each piece mapping directly into these regions in spacetime. This seems
like a natural expectation: we have independent regions in spacetime
fundamentally emerging from independent patches in Hilbert space, allowing us
to precisely relate the prepositional verb _to be in spacetime_ to the verb
_to be in Hilbert space_. It turns out that this is not immediately possible,
and the situation is aggravated when we attempt to include gravity in the
picture.

If we recall our initial mysterious box, perhaps this should not come as a
surprise. Although quite often locality in Hilbert space, in the sense of
local subsystems with commuting algebras, is a direct consequence of
spatiotemporal locality (as discussed in Sec. 2), this is mostly our spacetime
parochialism. As emphasized in recent works [34, 5, 6, 7, e.g.], the notion of
spatiotemporal locality should be seen as emergent from the quantum formalism.
So what can be done instead?

###  5.2 The Algebraic Approach

There is, however, a direct mapping of spatiotemporal locality to another
space: the _space of observables_. Instead of trying to define smaller patches
of Hilbert space associated with causally independent spacetime regions, we
can associate a local algebra of observables with each region. These algebras
commute with each other if the regions are causally independent,

$[\mathcal{R}(U),\mathcal{R}(V)]=0\,,$ (5.2)

where $U$ and $V$ are the two spacelike-separated regions. This is a defining
property in algebraic quantum field theory (AQFT). AQFT starts by associating
algebras of observables with regions of spacetime. For an open, finitely
extended region $U$ in Minkowski space, there is an associated algebra
$\mathcal{R}(U)$, where the self-adjoint elements of $\mathcal{R}(U)$ are
interpreted as observables measurable within $U$. These algebras must satisfy
several properties that capture the essence of locality and causality, among
them [35]:

  * •

Isotony: If $U\subset V$, then $\mathcal{R}(U)\subset\mathcal{R}(V)$;

  * •

Microcausality: If $U$ and $V$ are spacelike separated, then all observables
in $\mathcal{R}(U)$ commute with all observables in $\mathcal{R}(V)$, i.e.,
$[\mathcal{R}(U),\mathcal{R}(V)]=0$;

  * •

Causality Structure: Let $\widehat{U}$ denote the causal completion of $U$.
The algebra associated with $\widehat{U}$ is the same as the algebra
associated with $U$, i.e., $\mathcal{R}(\widehat{U})=\mathcal{R}(U)$.

These properties ensure that the theory respects the causal structure of
spacetime, corresponding to the hyperbolic propagation character of fields.
This is very similar to the algebraic structure present in finite-dimensional
quantum systems reviewed in Sec. 2. To some extent, this is quite intuitive.
Imagine you are inside one of these regions with your set of measurement
devices. You know that they cannot influence the measurement devices of
another causally disconnected region, as discussed earlier. Therefore, it
seems reasonable to imagine that the operators associated with your
measurement devices commute with those from other causally disconnected
spacetime regions.

However, note that the operators in the equation above are not defined over
smaller patches of the Hilbert space; instead, they are global operators. So,
flipping things around, imagine the entire Hilbert space associated with the
entire spacetime202020This is a possibly contentious statement. In AQFT,
different vacua lead to inequivalent representations of the algebra of
observables, resulting in different Hilbert spaces [36]. In fact, these
inequivalent representations arise from various superselection sectors and
symmetry-breaking patterns, fundamental for understanding phenomena such as
spontaneous symmetry breaking in physics. An example is the different vacuum
states in gauge theories, leading to distinct physical sectors that cannot be
coherently superposed. However, we expect that this is not a problem from the
point of view of a final, unified theory of nature.. Then, consider all the
possible observables in it and start separating them into commuting
subalgebras. So far, this has nothing to do with spatiotemporal locality.
However, once you start labeling these subalgebras with $U$, $V$, etc., you
can use that as a condition to establish spatiotemporal locality as described
before. Thus, spatiotemporal locality might emerge from commuting subsets of
observables defined across the entire Hilbert space.

But with what states should these observables be paired? As we discussed in
Sec. 3.1, relevant quantities yield values that can be related to measurements
only when combined with states. Similarly, we can define a state space over
the whole algebra, thus they are global states, but that can have their domain
restricted to the local subalgebras. This means that when combined with the
local algebra of observables $\mathcal{R}(U)$, they generate values
corresponding to local measurements. It is important to distinguish this from
the concept of a smaller Hilbert space, and its respective local states, given
the obstruction discussed in Sec. 5.1.

Here we arrive at a crucial question: how do we know that the values generated
by combining these global operators and states have anything to do with the
values we actually measure in labs in spacetime? Well, that is exactly what
modeling entails. We theorize quantum systems to reproduce what we observe in
the laboratory. However, we remain trapped at the upper level in spacetime,
and it is worth asking whether there could be other quantum models that would
lead to the same measured values in spacetime, much like our mysterious box.

By analogy with coarse-grained emergent descriptions discussed in Sec. 3.2 and
3.3, in the space of quantum models aimed at reproducing classical spacetime
measurements, is there a notion of an ‘ensemble’ of models that all converge
to the same set of numbers one writes down in the lab? Is there a notion of
model coarse-graining that needs to be considered because, similar to our
accessible macrostates in thermodynamics, we only have access to classical
measurements in the end? We will return to these questions in Sec. 8, but we
have already seen that in the case of dualities, this is exactly the
situation: different sets of commuting subalgebras can be used to arrive at
the same classical measurements, even though they give rise to different
spatiotemporal local structures, as observed in Sec. 2.

This highlights the subtleties involved in defining quantum subsystems and the
emergent nature of spatiotemporal locality from a more fundamental quantum
framework. And now, at least we finally understand why the maps between the
different sets of spin sites were non-locally related since the choice of
commuting subalgebras is one of the criteria that leads to the notion of
independent subsystems.

###  5.3 Trapped in Spacetime

In the previous section, we discussed how the algebraic approach in AQFT
allows us to map spatiotemporal locality to the space of observables. This
highlights how our classical understanding of locality and independence can
emerge from a more fundamental quantum framework. However, this reliance on
classical concepts is not merely a convenient tool; it is deeply ingrained in
the very formulation of quantum theory.

It has long been realized by some of the founders of quantum mechanics that
the theory is rather particular, especially in the context of “quantization,”
where we impose our classical preconceptions, such as spatiotemporal locality,
onto the quantum world. In fact, Bohr viewed this as an unavoidable necessity:

> ‘It is decisive to recognize that, however far the phenomena transcend the
> scope of classical physical explanation, the account of all evidence must be
> expressed in classical terms. The argument is simply that by the word
> experiment we refer to a situation where we can tell others what we have
> done and what we have learned and that, therefore, the account of the
> experimental arrangements and of the results of the observations must be
> expressed in unambiguous language with suitable application of the
> terminology of classical physics.’ (Bohr, 1949, p. 209 [37])

Similarly, Heisenberg says:

> ‘The Copenhagen interpretation of quantum theory starts from a paradox. Any
> experiment in physics, whether it refers to the phenomena of daily life or
> to atomic events, is to be described in the terms of classical physics. The
> concepts of classical physics form the language by which we describe the
> arrangement of our experiments and state the results. We cannot and should
> not replace these concepts by any others.’ (Heisenberg 1958, p. 44 [38])

Both argue prescriptively that classical physics is not going anywhere, but
rather is fundamentally necessary in the context of measurements in quantum
mechanics. A few decades later, Landau and Lifshitz summarize the situation
as:

> “Thus quantum mechanics occupies a very unusual place among physical
> theories: it contains classical mechanics as a limiting case, yet at the
> same time it requires this limiting case for its own formulation.” (Landau &
> Lifshitz, 1977, p.3 [39])

These quotes have been beautifully compiled in the introduction chapter of
[19], where the author himself provides a more rigorous formulation of such a
prescription in terms of our modern understanding of the algebraic approach to
quantum mechanics:

> The physically relevant aspects of the non-commutative operator algebras of
> quantum-mechanical observables are only accessible through commutative
> algebras.
> (Landsman, 2017, p. 10)

Now that we understand how classical notions of spacetime are intrinsically
embedded in quantum theory, it is time to extend our discussion to a dynamical
spacetime and its interaction with the quantum framework. As general
relativity can be seen as a field theory with long-range interactions, we will
first review how gauge theories are treated in the context we have been
discussing to ease our way into this complex discussion.

###  5.4 More ambiguities!

We have been discussing the quantum-mechanical observables associated with
microcausality since the beginning of this paper, but we have not yet
addressed what they are in the context of field theory. Unlike in finite-
dimensional quantum mechanics, where the simple assumption that two quantum
subsystems are spacelike separated is enough to represent their set of
possible observables as two commuting sets of local algebras, and where their
interacting Hamiltonian can be thought of as something we switch on and off in
the lab, in quantum field theory, we do not have such luxury. Instead, we have
long-range interactions that are ever-present and are modeled by gauge
theories, where the observables associated with the free theory typically
transform under gauge transformations. Thus, even if the free theory would
allow us to define commuting sets of subalgebras associated with spacelike-
separated regions, thereby satisfying (5.2), once the gauge interaction is
taken into account, that microcausal relationship is no longer gauge
invariant.

Since we do not observe violations of macrocausality in the lab, and
macrocausality is intrinsically connected to microcausality, the question
becomes whether our quantum field theories allow the construction of both
gauge-invariant and microlocal observables. The expectation is that those are
the ones actually measured in the lab. Let’s explore how this works in quantum
electrodynamics (QED) [40] as an example.

#### QED couple with scalar field

Let’s consider QED coupled with a scalar field $\phi$ with charge $q$,

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\alpha}(\partial_{\mu}A^{\mu})^{2}-|(\partial_{\mu}-iqA_{\mu})\phi|^{2}-\frac{1}{2}m^{2}|\phi|^{2},$
(5.3)

where different choices of $\alpha$ fix the gauge: for instance,
$\alpha\rightarrow 0$ corresponds to Lorenz gauge. Being a gauge interaction,
the gauge field transforms, as well as any other charged field under it,

$\displaystyle A_{\mu}\to
A_{\mu}-\partial_{\mu}\Lambda\quad\text{and}\quad\phi(x)\to
e^{-\mathrm{i}q\Lambda(x)}\phi(x)\,,$ (5.4)

such that the Lagrangian is invariant, as gauge transformations are
ambiguities in the description of the gauge interaction, and physics should
not depend on them.

If we could turn off the gauge interaction, or effectively deactivate the
scalar field charge, we could consider the scalar field itself as a local
observable (assuming some smearing) associated with different spacetime
regions, such that for any two spacelike-separated regions $x\in U$ and $y\in
V$, microcausality holds:

$[\phi(x),\phi(y)]=0.$ (5.5)

This makes sense since the field itself, acting on the vacuum, can create
local excitations associated with particles, which certainly seem to be
observed in the lab. However, since we cannot simply turn off the interaction,
this field operator is not gauge-invariant, raising the question of whether
the particle content observed in the lab is well-posed by the theory.

Fortunately, we can _dress_ the field observable in a way that makes it gauge-
invariant by defining a different observable:

$\Phi(x)=V(x)\phi(x)\,,$ (5.6)

where the dressing $V(x)$ is defined such that it takes the inverse
transformation of the field, $V(x)\to e^{iq\Lambda(x)}V(x)$, making $\Phi(x)$
gauge-invariant. Such a dressing is found to be:

$V(x)=\exp\left(iq\int
d^{4}x^{\prime}f^{\mu}(x,x^{\prime})A_{\mu}(x^{\prime})\right)\,,\quad\text{where}\quad\partial_{\mu}^{\prime}f^{\mu}(x,x^{\prime})=\delta^{4}(x-x^{\prime})\,,$
(5.7)

assuming the gauge transformations vanish at infinity. There is considerable
freedom in choosing the function $f^{\mu}$, and different choices lead to
different dressings. The physical intuition behind this dressed observable is
that it creates particles of the scalar field along with a _cloud_ of photons
surrounding each particle. Different dressings correspond to different
configurations of these clouds.

Having found gauge-invariant observables, the question turns to whether they
are microcausal. Upon quantizing this theory, one imposes canonical
commutation relationships for both fields. For spacelike-separated regions, we
have:

$[\Phi(x),\Phi(y)]=\phi(x)\phi(y)[V(x),V(y)]+V(y)V(x)[\phi(x),\phi(y)]\,,$
(5.8)

which vanishes despite the non-local nature of $V(x)$. This can be most easily
seen when one picks a frame where it becomes an equal-time commutator, and
then any pair of commutators among the fields vanishes212121We refer the
reader to [40] for more details.. Despite $\Phi(x)$ being gauge-invariant and
defining local algebras that satisfy microcausality, it is intrinsically a
non-local operator as the integration extends to infinity. However, another
algebra of operators can be constructed in which we have local operators
instead, where the electromagnetic field is screened by introducing a second
operator creating an oppositely charged particle [41]:

$D(x,x^{\prime})=\phi(x)e^{iq\int_{\Gamma}dx^{\mu}A_{\mu}}\phi^{*}(x^{\prime}),$
(5.9)

where $\Gamma$ is a curve connecting $x$ and $x^{\prime}$ and it represents a
Wilson line connecting two filed excitations of opposite charges. Then, as
long as $D(x,x^{\prime})$ can be localized in $U$, it will commute with
analogous operators defined in other spacelike-separated regions, and once
again we find a gauge-invariant algebra of observables that is microcausal.

##  6 What about gravity?

Any good story starts with a mystery and at some point introduces a villain.
Our understanding of gravity has dramatically changed since Newton’s theory.
Initially, space and time were deemed absolute, and gravity was yet another
force acting on this absolute stage. However, we have since learned with
Einstein that the stage itself is dynamical, changing depending on the matter-
energy act at play, and this very dynamics is responsible for gravitational
phenomena.

One of the key principles guiding Einstein’s development of his theory of
gravity was _relationalism_ , which can be understood as [42]:

> Space and time do not exist as independent substances, there is only the
> material content of the universe. Space and time are merely defined through
> spatiotemporal relations among the material objects in the universe (and
> their parts). Spacetime is thus a (constructed) relational complex of
> relative positions of material objects. (Wüthrich, 2024)

We can directly appreciate his take on it when reflecting on the development
of relativity [43]:

> The systematic decoupling of basic geometrical concepts (straight line,
> distance, etc.) from the bodies of experience, of which they are abstract
> representations, must not let us forget that in the end geometry is supposed
> to tell us about the behavior of the bodies of experience. If there were no
> practically rigid bodies that can be brought into congruence with one
> another, we would not speak of the congruence of distances, triangles, etc.
> It is clear that for the physicist geometry becomes meaningful only as he
> associates bodies of experience with those basic concepts, for example by
> associating the concept of distance with a practically rigid body with two
> markings. Vice versa, this association makes Euclidean geometry a science of
> experience in the truest sense, just like mechanics. The sentences of
> geometry can then be confirmed or falsified, just like the sentences of
> mechanics. (Einstein, 1919-20)

Thus, it is quite clear that Einstein’s perspective does not take space, and
perhaps time, as independent substances. However, Einstein’s general
relativity did not quite perfectly achieve this strict version of
relationalism as it does allow vacuum solutions, but it was a significant step
in that direction. Regardless, general relativity is extremely successful and
underlies both astrophysics and modern cosmology, each presenting a large body
of evidence towards its validity across different physical regimes.

However, it is not clear whether general relativity applies to all regimes. In
fact, most theoretical evidence points to its breakdown, including the
theory’s own solutions in the context of spacetime singularities, such as
black holes or anything resembling the beginning of the Universe. On one hand,
physicists are not overly concerned, as it is believed that general relativity
is simply a limit of an underlying quantum theory of gravity where such issues
would not be present; on the other hand, we are approaching the turn of the
century since the first attempts to quantize gravity [44], yet there is no
clear consensus on the correct path to take. One reason for this lies in the
very different nature of quantum mechanics and general relativity, as
highlighted here [45]:

> (Quantum) physics lies in the relation between the observer and observed
> systems; one can regard quantum theory as a law book which governs the
> observer’s acquisition of information from physical systems in terms of how
> and how much information is accessible. In quantum theory the information
> flow is physical, encoded in physical systems that can undergo their own
> dynamics, such as electrons. This physical information flow is under the
> focus of the theory but the spatiotemporal stage on which such information
> flows is externally given.
>
> By contrast, (…) much of the physics of general relativity lies in the
> communication relations among observers (or more generally systems); general
> relativity can be understood as a law book governing where and when
> information can be communicated. In general relativity the information flow
> among idealised observers, at least in terms of light signals, is, strictly
> speaking, external, while the stage on which this information flows, i.e.
> spacetime, is dynamical and the focus of the theory. Alice can send a light
> signal to Bob, depending on their causal relations, but the influence of
> this light signal on the dynamical spacetime they live in is ignored, no
> matter how energetic it is. (Höhn, 2017)

Nonetheless, many lessons have been learned so far. But we will focus on one
lesson that I take to be underappreciated, and will lead us to reconsider
several approaches to important problems in different areas of physics in Sec.
7. This is the problem of defining local gauge-invariant observables in
linearized quantum gravity.

###  6.1 Operational Measurement Limitation

Let’s start intuitively [46, see interview with Nima Arkani-Hamed]. Imagine
you are back in your lab, and you want to make a perfectly sharp measurement,
or at least as sharp as possible. The problem is how to store that information
in your apparatus. Quantum mechanically, you could imagine having an infinite
chain of qubits to record any measurement using binary digits. This would take
a lot of space, so perhaps you have access to a quantum field that has an
infinite-dimensional Hilbert space at each point in space. Either way, you
need an infinite-dimensional Hilbert space for a perfectly sharp measurement.
While this may not be feasible in practice, there is nothing in quantum
mechanics that prevents you from imagining it in principle.

However, once gravity is considered, all this stored information will
eventually start to backreact on the geometry of the lab, until it reaches a
threshold and transforms it into a black hole. This is not surprising, as
black holes are the most information-dense objects according to the Bekenstein
bound [47, 48, 49]. Since the entropy of a black hole scales as
$S=\frac{A}{G}$, where $A$ is its area, and it is a maximally entropic state,
we can infer that the Hilbert space associated with it has a dimension of
approximately $\mathcal{H}\sim e^{S}$ [50, 51]. Thus, the error in the
experiment will scale as $e^{-A/G}$. Therefore, there is no precise observable
one can associate with any measurement performed in a finite-sized room.

Is there a way out of this? One approach to bypass these constraints is to
imagine a lab that can extend indefinitely instead of having a fixed size. As
you record more digits from your measurements, you avoid saturating the local
bound on information density and prevent creating a black hole. However, to
keep sharpening the measurement, you will eventually need to push your
apparatus all the way to the boundary of spacetime. This is why the only sharp
observables in gravity are those associated with the boundary. In
asymptotically flat spacetime, it is the S-matrix that defines such
observables, with in and out states defined at infinity. Similarly, in black
hole physics, significant quantities are often defined globally.

Now, let’s explore more concretely what happens when we try to define local
gauge-invariants observables in gravity.

###  6.2 _You can’t always get what you want_

Let’s now consider general relativity coupled with a scalar field and pursue a
similar strategy as we did in Sec. 5.4 for QED coupled with a scalar field
[40]. Thus, our starting point is

$\displaystyle\mathcal{L}=\frac{2}{\kappa^{2}}R-\frac{1}{2}\left(g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+m^{2}\phi^{2}\right)$
(6.1)

where $\kappa=\sqrt{32\pi G}$. This theory is diffeomorphic invariant We will
consider a fixed flat background with perturbations on top, so we expand
$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$, where $h_{\mu\nu}$ is a
canonically normalized field. Under infinitesimal linearized diffeomorphisms,
the fields transform as

$\displaystyle\delta\phi=-\kappa\xi^{\mu}\partial_{\mu}\phi+\mathcal{O}(\kappa^{2})\quad\text{and}\quad\delta
h_{\mu\nu}=-2\partial_{(\mu}\xi_{\nu)}+\mathcal{O}(\kappa)$ (6.2)

for an arbitrary vector field $\xi^{\mu}$. Once again, the scalar field alone
is not invariant and cannot be used to define local-diff invariant
observables. But looking at its transformation, in a fixed background, the
linear diff transformation of the scalar field corresponds to a local
infinitesimal translation. As we know how to represent translations of any
observable by a unitary transformation, the operator that we seek takes the
form

$\displaystyle\Phi(x)=e^{\mathrm{i}V^{\mu}P_{\mu}}\phi(x)e^{-\mathrm{i}V^{\mu}P_{\mu}}=\phi(x^{\mu}+V^{\mu})$
(6.3)

where $P^{\mu}=-i\partial_{\mu}$ is the translation operator and $V^{\mu}$ is
a functional of the metric perturbation, which at linear order we require to
transform as

$\displaystyle\delta V^{\mu}=\kappa\xi^{\mu}\,.$ (6.4)

Using (6.2) and (6.4) it is straightforward to show the invariance of $\Phi$
at leading order in $\kappa$. As we will discuss more in Sec. (7), we expect
that any final theory of quantum gravity will have an effective perturbative
regime that falls into the theory being studied here, and we argue that much
of the current physical phenomenology is modeled, tacitly or explicitly, under
this linear perturbative regime.

The real question is what $V^{\mu}$ is and how it depends on the metric. We
could imagine that we should seek its form similar to what we did in (5.7),

$V^{\mu}(x)=\kappa\int
d^{4}x^{\prime}f^{\mu\nu\lambda}(x,x^{\prime})h_{\nu\lambda}(x^{\prime})\,\quad\text{where}\quad
2\partial_{\nu}^{\prime}f^{\mu\nu\lambda}(x,x^{\prime})=\delta^{4}(x-x^{\prime})\eta^{\mu\lambda}\,,$
(6.5)

as it does guarantee that $V^{\mu}(x)$ transforms as (6.4). However, it turns
out that $f^{\mu\nu\lambda}$ depends non-locally on the metric. And that
should have been expected, because in (5.7), it was the gauge field that was
used for the dressing, and the gauge field in GR is not the metric but the
connection. To see how this impacts the dressing, let’s take a look at a
single example of $V^{\mu}$ associated with the gravitational Wilson line [40,
52, 53]:

$\displaystyle
V_{\mu}=\frac{\kappa}{2}\int_{x}^{\infty}dx^{\prime\nu}\left[h_{\mu\nu}(x^{\prime})+2\int_{x^{\prime}}^{\infty}dx^{\prime\prime\lambda}\partial_{[\mu}h_{\nu]\lambda}(x^{\prime\prime})\right]\,,$
(6.6)

where the integration is along an arbitrary curve from $x$ to infinity. In
physical terms, the dressing corresponds to a gravitational string extending
to infinity. The crucial difference between the gravitational dressing and the
one in QED, and more generally Yang-Mills theories, is the presence of the
derivatives of the metric.

Now that we have obtained dressed gravitational operators that are invariant
under linear diffs, do they satisfy microcausality? Unfortunately, not. As we
had before, the commutator of $\Phi$ between two spacelike regions depends on
$[V(x),V(y)]$ as in (5.8), but inside of $V^{\mu}$ there is the metric and its
derivatives, including its time-derivative. What was crucial for the QED
example was that the dressing was made with the gauge field, which is the same
object that was quantized, but that is not the case for gravity. Typically, in
canonical quantization of gravity, both non-perturbative and linearized
gravity, we quantize the metric instead, which its necessity can be traced
back to the introduction of the Ricci scalar (both the Riemann tensor and
Ricci tensor can be fully written only in terms of the connection)222222Here,
we see a very clear motivation to consider Loop Quantum Gravity where the
connection, expressed in terms of Ashtekar variables, is quantized instead of
the metric. For a recent discussion, [54, and references therein].. You might
wonder about the other type of dressed observable (5.9), but those are not
available due to the absence of negatively charged particles under gravity.

The evidence against the existence of gauge-invariant local observables in
quantum gravity is compelling. Already three decades ago, [55] demonstrated
that in closed universes, invariant observables cannot be constructed as local
functions of the canonical variables, linking the absence of local observables
to topological symmetries. More recently, [56] argues that while classical
gravitational observables can be microcausal, this property is limited at
quantum scales, with significant implications for black hole information.
Additionally, [41] shows that any operator with nonzero Poincaré charges must
be gravitationally dressed, implying nonlocality and the failure of defining
local subsystems in gravity and [53] highlights the conflict of defining
mutually independent subsystems via standard tensor products or operator
algebras in gravitational contexts. For more references, see [57, section
Observables and references therein] and [58].

##  7 Why should we care?

Despite all the candidates, we still have not reached a consensus regarding
the theory of quantum gravity. However, it is widely accepted that any theory
of quantum gravity must have a low-energy quantum gravity (LEQG) limit, which
at energies well below the Planck scale, and recovers the effective field
theory derived from general relativity by quantizing metric perturbations on a
classical background. This argument is extensively developed in [59], where it
is also shown that the body of evidence from astrophysics and cosmology, which
requires a simultaneous treatment of quantum-mechanical and gravitational
effects, necessitates such a theory.

More recently, the prospect of probing the quantum nature of gravity in the
laboratory has been proposed [60, 61]. It is claimed that detecting
entanglement between two mass probes, mediated by gravitational interaction,
would demonstrate the quantum nature of gravity [62]. The modeling of such
experimental proposals makes direct use of LEQG, to the extent that some argue
that the aspects of quantization these experiments can probe are the same ones
already probed in cosmological observations [63].

However, in light of our discussion so far, we should not expect to be able to
measure gauge-invariant observables in LEQG that also respect microcausality.
We have seen that microcausality underpins our operational notion of
macrocausality. Thus, in regimes where LEQG is expected to play a major role,
both microcausality and macrocausality violations might occur. For instance,
macrocausality in the early universe is a key assumption to introduce the
horizon problem that led to the inflationary paradigm. Meanwhile, in Sec. 2,
we established that microcausality, and more generally having commuting sets
of observables for independent quantum systems, is a necessary condition to
have a well-defined tensor product structure. This structure underlies the
notion of quantum-mechanical systems and is tacitly assumed in the formulation
of the measurement problem in quantum mechanics, and indeed in any
measurement, including those trying to probe the quantum nature of gravity in
the lab.

In this section, we briefly review physical regimes where LEQG is significant,
highlight assumptions of microcausality and macrocausality, and discuss
potential future directions.

###  7.1 Early Universe Cosmology

We do not have direct access to the very early universe, but among all the
different cosmological observations that offer us a window into its early
stages, the Cosmic Microwave Background (CMB) radiation takes the main role.
Spread out throughout the whole universe, it provides an aged snapshot of the
Universe approximately 380 000 years after the Big Bang. It is a snapshot in
the sense that this background radiation was produced during a short phase
where the light in the early Universe went from propagating in an opaque
medium, where the photons were constantly interacting with electrons and
baryons, to a transparent one where they could freely propagate until reaching
our current detectors. It is aged because the Universe has evolved since this
transition, and the detected radiation distribution today carries memories of
the Universe’s history when compared to its expected profile back when it was
emitted. For example, one of the memories impinges upon the whole radiation
spectrum, which has been redshifting since its emission, and today is detected
at a temperature of $2.7$ K as opposed to its initial temperature of around
$3000$ K.

As we discussed in Sec 3.3, if we are talking about temperature it must mean
that we are close to thermal equilibrium. That is the case for the CMB, which
presents an overall spectrum very close to that of a black body, with small
temperature fluctuations of order $10^{-5}$ K. These fluctuations are also
part of the spectrum’s memory, as they reflect small density variations in the
early Universe. One of the initial surprising features of the CMB was exactly
this proximity to thermal equilibrium. The medium transition from opaque to
transparent happens as the local temperature decreases enough such that, on
average, photons cannot excite electrons out of proton’s electric potential
anymore, allowing them to form stable hydrogen. This local temperature depends
on the amount of local energy densities. Observing thermal equilibrium in the
CMB indicates that the energy density was very evenly distributed both
homogeneously and isotropically, otherwise different parts of the Universe
would reach this transition at different times, leading to larger variations
of the temperature of the detected photons depending on which part of the
Universe we are probing (or, literally, looking at). This even distribution of
energy density is particularly puzzling because, under the standard Big Bang
evolution, the CMB corresponds to causally disconnected patches of the
universe. Therefore, we need to explain how the Universe’s energy distribution
was so even at the time the CMB was emitted232323For more quantitative
details, see [64, 65]..

Moreover, even if we could argue that the initial state of the Universe
corresponded to perfect homogeneity due to unknown properties of quantum
gravity, we would still need to explain why the fluctuations in the CMB are
not random, but instead have correlations over very large distances that were
not causally connected, so the light did not have enough time to travel that
much since the beginning of the Big Bang. In fact, without assuming any new
physics between the Big Bang and the emission of the CMB, we can show that
points separated by more than two degrees on the sky seem to not have been in
causal contact.

Therefore, to explain this amount of uniformity in the CMB many different
ideas have been introduced, and today the early universe’s paradigm is based
on cosmic inflation: an early-on accelerated phase of expansion of the
Universe that effectively blows up the size of the past causal horizon so that
there would be enough time for light to propagate over large distances and
create the sort of correlations observed in the CMB. The success of inflation
comes from providing a mechanism for the generation of primordial density
perturbations: they are quantum fluctuations that during inflation are
stretched out to macroscopic scales, and they match the statistical properties
of the temperature fluctuations in the CMB.

Thus, currently, the early universe paradigm is based on apparent violations
of macrocausality due to a high degree of homogeneity of the radiation in the
CMB together with seemingly super-causal correlations. Meanwhile, the current
paradigm offers a solution based on inflation, which is typically modeled by a
scalar field sourcing the fast-expanding cosmological background, while
combined quantization of the field and spacetime fluctuations are ultimately
responsible for the primordial density fluctuations. Hence, the paradigm falls
into LEQG.

This can potentially introduce some internal inconsistency. The reason is that
the very reason to introduce new physics in the early universe is based on
assumptions (macrocausality) that are not a given within the background
theoretical framework (LEQG).

Based on this argument, a future direction is to study the early universe
using non-local gauge-invariant observables dressed by gravitational gauge
fields. It is known that dressing a field theory creates a new Fock space from
which scattering amplitudes can be analyzed, in particular the states are
defined over a new vacuum. Thus, basically we have either a set of observables
with the undressed vacuum, or a dressed set of observables with the dressed
vacuum. This is reminiscent of the setup of the Unruh effect, or Hawking
radiation, where different observers, one inertial and one accelerating, have
access to different sets of observables and vacua, and the vacuum state of one
is seen as a thermal bath to the other. Could our observations of the CMB seen
from the local perspective242424Since gravity is not modelled in the
detection, it is as if we are framing the observations from the point of view
of the undressed theory. look thermal exactly because of the mismatch between
these vacua?252525A similar argument can be made in the context of the
cosmological constant problem, as the calculations making the problem explicit
typically rely on the Fock space defined by the free quantum field theory
[66]. It would be interesting to check whether the zero-point energies of the
dressed modes contribute differently than the ones defined by the free
theory..

###  7.2 Gravitationally-Induced-Entanglement (GIE) Experiments

The quantum nature of gravity remains a topic of debate [67, 68, and
references therein]. Thus, it seems more than justified the hype around the
possibility of probing whether gravity is non-classical in the lab [60, 61].
Originally, the experimental setup involves two masses, each in a spatial
superposition, interacting only through gravity. Due to their spatial
superposition, each pair of trajectories induces a different unitary phase due
to the interaction Hamiltonian, which depends on the relative distance of the
trajectories, different for all the pairs. Thus, if the initial state was a
product state between the superimposed spatial states of each mass, after
their interaction each term will have a different phase, and the overall state
can no longer be decomposed as a product state. Following [61], if the initial
state is considered

$|\Psi(t=0)\rangle=\frac{1}{2}(|L\rangle_{1}+|R\rangle_{1})(|L\rangle_{2}+|R\rangle_{2})\,,$
(7.1)

where $L$ and $R$ stand for left and right trajectories, then the final state
is

$|\Psi(t=t_{f})\rangle=\frac{1}{2}\left(e^{i\phi_{LL}}|L\rangle_{1}|L\rangle_{2}+e^{i\phi_{LR}}|L\rangle_{1}|R\rangle_{2}+e^{i\phi_{RL}}|R\rangle_{1}|L\rangle_{2}+e^{i\phi_{RR}}|R\rangle_{1}|R\rangle_{2}\right)\,.$
(7.2)

For example, $\phi_{LL}=Gm_{1}m_{2}t_{f}/\hbar(x_{L_{1}}-x_{L_{2}})$
represents the phase difference for the left-left trajectory. Therefore, the
final state is entangled. This is a prediction of standard quantum mechanics.

The experiment described above is known as the bipartite model, which is
typically considered a Newtonian approximation of the tripartite model. For
this model, gravity appears as a mediator subsystem, such that the masses
interact only directly with it. Thus, for instance, instead of
$|L\rangle_{1}|L\rangle_{2}$, we have
$|L\rangle_{1}|\gamma_{LL}\rangle|L\rangle_{2}$ which evolves to
$e^{i\phi_{LL}}|L\rangle_{1}|\gamma_{LL}\rangle|L\rangle_{2}$. As each of the
$|\gamma\rangle$ states represents a different classical gravitational
configuration, there is a superposition of classical gravitational states.
What is crucial to the tripartite model assuming interactions between the
masses only mediated by the subsystem $|\gamma\rangle$ is that, if the
$|\gamma\rangle$ is a classical subsystem (the set of its observables define a
commuting subalgebra), then one can show that no entanglement can be produced
[69, see for an overview]. Hence, were the experiment performed, witnessing
the entanglement between the two masses would be strong evidence of the
quantum nature of gravity.

At least that is the common lore. Critical analyses of the modeling and
implication of a positive detection have already been discussed [70]. In [63],
it has been argued that such experiments are probing the quantum nature of
gravity similarly as the quantized cosmological perturbations are being probed
by the CMB observations under the inflationary paradigm discussed above. They
highlight two key differences: i) cosmological perturbations focus on field
fluctuations, whereas GIE experiments involve cat states, which are
significantly more quantum (measured by non-classicality such as Wigner
negativity), and ii) GIE experiments explore a new, untested regime of
physics, namely, the gravitation of delocalized masses.

Here we want to call attention to yet two other aspects underlying the
modeling of such experiments that, as far as I know, have not been pointed out
so far. Due to the low energy regime and reliance on perturbative quantum
gravity, these experiments fall within the LEQG framework. However, as
discussed in Sec 2, a well-defined tensor product structure, necessary for
composing different systems, is only ensured in the presence of
microcausality. Moreover, the mass states above are also defined as
excitations of some free matter field [61, see supplementary material], which
given our discussion in Sec. 6.2, are not gauge invariant under linear
diffeormorphisms. Thus, the consistent use of LEQG to describe such
experiments, believed to be fundamental to any possible conclusion to be
derived from either positive or negative entanglement detections, seems to
require an improvement of their description both in terms of the factorization
understanding of the Hilbert space as well as the type of states that are
considered262626There is yet another issue that has to do with state
preparation. In the context of field theories, the idea behind a product state
is generalized to what is called statistical independence [31, 3], which
guarantees that any pair of state preparations made by two different
experimenters in two spacelike regions can be seen as a preparation of the
system as a whole. Achieving statistical independence requires both
microcausality and the split property, which together ensure a well-defined
tensor product structure and thus product states. Once again, LEQG might seem
to violate microcausality and it remains unclear whether quantum gravity would
satisfy anything like the split property [71, 72, 73]..

##  8 Single-world unitary quantum mechanics

Our discussion about the GIE experiments invites us towards a much broader
shift in our conceptual perspective. The breakdown of microcausality in LEQG,
and the subsequent loss of a well-defined tensor product structure, undermines
the concept of quantum-gravitational subsystems, which due to the universal
nature of gravity applies to everything. This shift may offer a new
perspective on the measurement problem, suggesting that quantum subsystems are
not invariant under measurements or generic Hamiltonian evolution272727This
idea was first explored in [74] by modeling an apparatus and a quantum system,
both as qubits. Although the mathematical implementation was limited, some of
the conceptual framework was developed there, though not in the context of
LEQG and microcausality violations. More recently, thanks to Jan Glowacki and
Tein van der Lugt, I became aware of [75], where similar ideas are pursued
over solid mathematical foundations. Perhaps, the discussion presented here
offers a possible answer to their final question concerning a unified story
between their framework and quantum gravity.. Just as the relativization of
time and space was crucial for the development of special relativity, the
relativization of quantum subsystems might be essential for resolving the
measurement problem and advancing towards a final theory of quantum gravity.

To the core of the idea of relativizing the notion of quantum subsystems
resides our discussion about emergence from Sec. 3 and Sec. 4.2, as there is a
difference between our classical systems, including our lab devices and their
single measurement outcomes, and their quantum descriptions as quantum
subsystems. While it is their classical descriptions that are anchored in
spacetime and invariant over time, their quantum representations can move
freely within the overall Hilbert space and need to be re-established at every
stage of the global Hilbert space evolution. Below, I sketch how this could
work, consider the close resemblance between this approach and the methods
used in general relativity to establish a local inertial frame, and speculate
on connections to several other research fields.

###  8.1 Quantum subsystems are not time-invariant

Let’s consider a global finite-dimensional (for simplicity) Hilbert space,
$\mathcal{H}$, the algebra of bounded operators on the Hilbert space,
$B(\mathcal{H})$, a Hamiltonian $H$, and an initial pure state $\rho_{0}$.
Then, for each observable $\mathcal{O}\in B(\mathcal{H})$, we have

$v_{\mathcal{O}}(\tau)=\tr\rho(\tau)\mathcal{O}\,,\quad\text{where}\quad\rho(\tau)=e^{iH\tau}\rho_{0}e^{-iH\tau}\,,$
(8.1)

where the unitary $U(\tau)\equiv e^{iH\tau}$ induces an inner automorphism of
the algebra by the self-adjoint Hamiltonian and parametrized by the real
parameter282828Although we typically consider the evolution parameter $\tau$
to be time, particularly in light of the experimental success of applying
Schrödinger’s equation in lab, at this level it is just a parametrization of
this inner automorphism. It is part of the work in establishing precisely the
quantum-classical limit that the relationship between $\tau$ and geometric
time $t$ should be stipulated. From this point of view, quantum mechanics is
inherently background independent. $\tau$. For any fixed $\tau_{0}$, the set
$\\{v_{\mathcal{O}}\\}_{\tau_{0}}$ fully parametrizes all the data of the
theory, as the theory evolves deterministically. Note that
$\\{v_{\mathcal{O}}\\}_{\tau_{0}}$ is invariant under the following unitary
map:

$T:\mathcal{H}\longrightarrow\bigotimes_{i}\mathcal{H}^{i}\,,$ (8.2)

that transforms states and operators as:

$\displaystyle\mathcal{O}$ $\displaystyle\longrightarrow T\mathcal{O}T^{-1}$
(8.3) $\displaystyle|\Psi\rangle$ $\displaystyle\longrightarrow
T|\Psi\rangle\,,$ (8.4)

which is the map used to define the tensor product structure in Definition 1.

Let’s now consider a subalgebra $\emph{App}\in B(\mathcal{H})$ associated with
our experimental devices: $\mathcal{O}_{I}^{\rm app}\in\emph{App}$, where
$\\{\mathcal{O}_{I}^{\rm app}\\}$ form an orthogonal basis with $d^{2}-1$
elements for the apparatus. Moreover, let’s assume, for now, that indeed our
apparatuses are independent quantum systems (microcausality holds292929The
split property would be necessary around the region where our devices are
located had we considered infinite-dimensional Hilbert spaces [31, 3].) such
that $B(\mathcal{H})$ is isomorphic to $B(\mathcal{H}^{\rm app})\otimes
B(\mathcal{H}^{\rm EE})$ (akin to _completeness_ in Sec. 2 but now only for
bounded operators), where EE stands for everything else, $\text{dim
}\mathcal{H}^{\rm app}=d$ and $\text{dim }\mathcal{H}^{\rm EE}=D$, and
$d/D\lll 1$. This is good since we typically do not have access to the whole
Hilbert space, so we want to single out our apparatus. This implies that a TPS
$\mathcal{T}$ is being considered, such that:

$\displaystyle\mathcal{H}$
$\displaystyle\overset{\mathcal{T}}{\longrightarrow}\mathcal{H}^{\rm
app}\otimes\mathcal{H}^{\rm EE}$ (8.5) $\displaystyle\mathcal{O}^{\rm
app}_{I}$ $\displaystyle\overset{\mathcal{T}}{\longrightarrow}\tilde{O}^{\rm
app}_{I}\otimes\mathds{I}^{\rm EE}$ (8.6) $\displaystyle\rho$
$\displaystyle\overset{\mathcal{T}}{\longrightarrow}\sum_{n,m}^{d}\sqrt{p_{n}p_{m}}|u_{n}^{\rm
app}\rangle\langle u_{m}^{\rm app}|v_{n}^{\rm EE}\rangle\langle v_{m}^{\rm
EE}|\,,$ (8.7)

where $\tilde{O}^{\rm app}_{I}\in B(\mathcal{H}^{\rm app})$ and we used the
Schmidt decomposition for a generic state $\rho$, so $\\{|u_{n}^{\rm
app}\rangle\\}$ and $\\{|v_{n}^{\rm EE}\rangle\\}$ form orthonormal bases for
$\mathcal{H}^{\rm app}$ and $\mathcal{H}^{EE}$, respec., and $p_{n}>0$. The
Hamiltonian also factorizes:

$H\overset{\mathcal{T}}{\longrightarrow}a_{0}\mathds{I}_{d+D}+\sum_{I=1}^{d^{2}-1}a_{I}\tilde{\mathcal{O}}^{\rm
app}_{I}+\sum_{J=1}^{D^{2}-1}b_{J}\tilde{\mathcal{O}}^{\rm
EE}_{J}+\sum_{I=1}^{d^{2}-1}\sum_{J=1}^{D^{2}-1}c_{IJ}\tilde{\mathcal{O}}^{\rm
app}_{I}\tilde{\mathcal{O}}^{\rm EE}_{J}\,,$ (8.8)

Crucially, note that from the quartet
$(\mathcal{H},\rho_{0},B(\mathcal{H}),H,)$, now we have factorized each of
these elements, and associated with the factorization $\mathcal{T}$ are the
coefficients,

$\\{p_{n},a_{0},a_{I},b_{J},c_{IJ}\\}=\\{p_{n}(\mathcal{T}),a_{0}(\mathcal{T}),a_{I}(\mathcal{T}),b_{J}(\mathcal{T}),c_{IJ}(\mathcal{T})\\}\,,$
(8.9)

which parametrize the states and the Hamiltonian, thus kinematics and dynamics
of the theory, in a given TPS. Their dependence on the TPS can be simply
understood by the fact that our starting point did not have any of these
coefficients present, thus they are contingent on the TPS chosen. On the other
hand, they are not all independent from one another, as they are all connected
to the class of unitary map $\mathcal{T}$ being considered, and its unitary
nature will impose constraints on these parameters.

Under a given TPS, we can rewrite the global invariants (8.1) as

$\displaystyle v_{\mathcal{O}^{\rm app}_{I}}$
$\displaystyle=\sum_{m,n=1}^{d}\sqrt{p_{n}p_{m}}\tr_{\rm
app}\left(\tilde{\mathcal{O}}^{\rm app}_{I}|u_{n}^{\rm app}\rangle\langle
u_{m}^{\rm app}|\right)\tr_{\rm EE}\left(|v_{n}^{\rm EE}\rangle\langle
v_{m}^{\rm EE}|\right)$ $\displaystyle=\sum_{n=1}^{d}p_{n}\tr_{\rm
app}\left(\tilde{\mathcal{O}}^{\rm app}_{I}P_{n}^{\rm
app}\right)\,,\quad\text{where}\quad P_{n}^{\rm app}:=|u_{n}^{\rm
app}\rangle\langle u_{n}^{\rm app}|$
$\displaystyle=\sum_{n=1}^{d}p_{n}\tilde{v}_{\mathcal{O}^{\rm app}_{I,n}}\,,$
(8.10)

where $\tilde{v}_{\mathcal{O}^{\rm app}_{I,n}}$ are the local (in the local
unitary sense) invariants associated with the local observable
$\tilde{O}_{I}^{\rm app}$ in the state $|u_{n}^{\rm app}\rangle$. Thus,
generically, global invariants can be decomposed as a convex sum of local
invariants.

Now we introduce three crucial assumptions that will need to be further
investigated in the future. The first two are strictly mathematical and demand
a much closer study of TPS transformations, while the third one is based on
our current understanding of LEQG, and quantum gravity more generally. It
hints at the existence of a new dynamical equation in quantum mechanics that
would be responsible for the evolution of the TPS in quantum systems, very
much akin to the evolution of the metric in general relativity. They are:

  * •

For $d/D\lll 1$ (which is absolutely the case), there are many inequivalent
TPSs $\mathcal{T}_{i}$. In fact, as they are connected by global unitaries
that can typically be continuously parametrized, we will assume any two TPS
are related by continuous deformations and parametrized by $\tau$:
$\mathcal{T}(\tau)$303030Despite introducing this infinity to parameterize all
the different TPS that fix the dimensionality of the Hilbert spaces of the
apparatus and EE, it pales in comparison with all the possible different TPS
that can be generically considered, e.g., bipartite with the same
dimensionality for each subsystem or non-bipartite ones. Another way of
understanding this is by taking the Hamiltonian perspective from [5], where
one always considers a fixed TPS but then varies the Hamiltonian, akin to the
coefficients in (8.9) evolving.;

  * •

There is a set of TPS, $\\{\mathcal{T}_{M}\\}$, where the state of the system
factorizes: $\rho\overset{\mathcal{T}_{M}}{\longrightarrow}\rho^{\rm
app^{\prime}}\otimes\rho^{\rm EE^{\prime}}$, where ′ indicates that whatever
quantity being considered is in such a TPS;

  * •

Due to violations of microcausality in quantum gravity, imposing a fixed TPS
in any quantum-gravitational system is ill-posed and can only be considered an
approximation. As the system evolves under the global Hamiltonian, the
approximate TPS changes continuously and can be parametrized by time,
$\mathcal{T}(\tau)$. This can be understood in terms of the coefficients above
evolving, e.g., $a_{I}=a_{I}(\tau)$;

We will show that by putting together these assumptions we can arrive at the
following conclusion:

Single outcomes in any measurement correspond to a composition of a change of
TPS together with a local unitary transformation of the local algebra assigned
to the apparatus.

In other words, a necessary condition for a single-outcome measurement at a
given $\tau_{0}$ happens whenever
$\mathcal{T}(\tau_{0})\in\\{\mathcal{T}_{M}\\}$.

Let’s start by showing this and then we discuss the assumptions and conclusion
in light of a close analogy to finding local inertial frames in general
relativity. Whenever $\mathcal{T}(\tau_{0})\in\\{\mathcal{T}_{M}\\}$, we have
a product state between what we call apparatus and everything else, so we can
write the state as

$|\Psi(\tau_{0})\rangle\overset{\mathcal{T}_{M}}{\longrightarrow}\sum_{n=1}^{d}c_{n}|u_{n}^{\rm
app^{\prime}}\rangle\otimes\sum_{m=1}^{D}d_{m}|v_{m}^{\rm
EE^{\prime}}\rangle\,,$ (8.11)

where these are bases for the apparatus and everything else in one of the
$\mathcal{T}_{M}$ (to not get confused with the bases introduced in (8.1)).
Now, we can simply _rotate/align_ our apparatus by a local unitary
transformation, such that

$|\Psi(\tau_{0})\rangle=|o_{I,n}^{\rm
app^{\prime}}\rangle\otimes\sum_{m}^{D}d_{m}|v_{m}^{\rm
EE^{\prime}}\rangle\,\quad\text{where}\quad\bar{\mathcal{O}}^{\rm
app^{\prime}}_{I}|o_{I,n}^{\rm app^{\prime}}\rangle=o_{I,n}^{\rm
app^{\prime}}|o_{I,n}^{\rm app^{\prime}}\rangle\,,$ (8.12)

and $\bar{\mathcal{O}}^{\rm app^{\prime}}_{I}=U^{-1}\tilde{\mathcal{O}}^{\rm
app}_{I}U$. Thus, the local state is one of the rotated apparatus’ eigenstates
with eigenvalue $o_{I,n}^{\rm app^{\prime}}$. Crucially, we do not know which
one. Then, it is easy to show that

$\displaystyle v_{\mathcal{O}^{\rm app}_{I}}$ $\displaystyle=\tr_{\rm
app^{\prime}}\left(|o_{I,n}^{\rm app^{\prime}}\rangle\langle o_{I,n}^{\rm
app^{\prime}}|\bar{\mathcal{O}}^{\rm app^{\prime}}_{I}\right)\tr_{\rm
EE^{\prime}}\rho^{\rm EE^{\prime}}$ $\displaystyle=o_{I,n}^{\rm
app^{\prime}}\,.$ (8.13)

Therefore, by identifying the appropriate TPS and locally rotating/aligning
the apparatus, we see that local measurement outcomes are actually the global
invariants that we started with. Crucially, to arrive at this point, besides
our assumptions (to be scrutinized), we only relied on global and local
unitary transformations. We expect that the probabilistic nature of the theory
arises from our lack of knowledge about the specific set of unitaries required
to reach this point, such that probabilities will be epistemic. This will be
closely investigated in the future. In any case, given that all considerations
involved only unitary transformations (both for TPS updates and time
evolution) and we still arrive at a description of the apparatus with single
outcomes related to the global invariants, we term this yet-to-be-finalized
theory _single-world unitary quantum mechanics_. Under $\mathcal{T}_{M}$,
local measurements can be associated with single outcomes, even though all
transformations considered were unitary.

###  8.2 Comparison with general relativity

Let’s clarify the preceding discussion by drawing a comparison with a similar
procedure in general relativity. As one moves through spacetime and aims to
consistently compare measurements at different points along their trajectory,
it is essential to update their local inertial frames (tetrads) based on the
local metric of spacetime. To achieve this, one must first solve the Einstein
field equations for the metric, which informs the local geometry and allows
for the determination of tetrads. Tetrads provide local orthonormal bases,
with the timelike tetrad vector typically aligned with the observer’s
4-velocity to serve as the time direction in the observer’s local frame. If
the timelike tetrad vector is not aligned with the observer’s 4-velocity, it
represents a boosted frame relative to the observer’s rest frame, leading to
different local measurements of time and space, akin to special relativity.
Therefore, it is crucial for the observer to measure their relative velocity
in relation to some reference points to determine the appropriate tetrad to
use.

The core process involves three key steps to define the appropriate local
inertial frame: first, determine the spacetime metric by solving the Einstein
field equations, which, despite being local equations, often require global
boundary conditions for a solution; second, establish the family of local
inertial frames using the metric; and finally, align the timelike tetrad with
the observer’s 4-velocity, which is a local process requiring measurements of
the observer’s relative velocity to reference points.

Thus, the situation surrounding measurements in quantum mechanics seems quite
analogous. The third assumption above hints towards an analog to the Einstein
equations for the metric, but now a new dynamical equation for
$\mathcal{T}(\tau)$, involving the global evolution of states. Once that is
known, we can establish which different TPSs are temporarily available, which,
like tetrads, need to be updated over time, allowing the introduction of
approximate well-defined local quantum subsystems. Finally, one needs to make
local measurements to _orient_ their local basis such that measurements yield
single outcomes.

###  8.3 Discussion

Within this approach, we have directly connected insights from quantum
gravity, and its lack of gauge-invariant local observables, to the measurement
problem in quantum mechanics. Although much work needs to be done, if this
program succeeds, it would mean that all classical measurements are
interconnected by a web of measurement dualities. The discussion was limited
to bipartite systems, but typically we also consider an observer in a quantum
measurement, which nowadays is much better discussed and formalized in the
context of quantum reference frames313131There is a large literature on this
topic [76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, e.g.]. For a recent paper,
see [87, and references 1-23 therein].. In this program, changes of TPS appear
naturally as one considers transformations between quantum systems to select
the one that will be the reference frame, very similar to what we do in
special relativity to choose the preferred inertial frame considered at rest.
However, to the best of my knowledge, the status of the program remains mostly
kinematical, in the sense that they do not consider the change of TPS due to
time evolution for the same quantum system, similar to a classical frame
having to update their local inertial frame as they move in a dynamical
spacetime. Three recent papers that delve into dynamics are [88, 89, 90].
These works, together with the formalism introduced in [91], might allow for
an extension of this proposal to infinite-dimensional Hilbert spaces and help
better quantify the mathematical assumptions surrounding the existence of a
large and continuous set of different TPS related to classical measurements.

##  9 We are inside the box!

The most vivid image you can retrieve from this whole story is that we are
inside that mysterious box! In fact, due to the seemingly unavoidable non-
local nature of gauge-invariant observables in quantum gravity, to some
extent, we are the box. This is why holography has been one of the main
entries into quantum gravity over the last few decades.

The issue we highlight here is that, generally, we are not using holography to
solve elementary problems in physics involving gravity and quantum
phenomena323232As usual, there are counterexamples such as [92, 93, and
references therein]. Typically, we rely on perturbative quantum gravity under
standard canonical quantization. Despite preventing us from establishing
gauge-invariant local observables, we write down models in these contexts as
if that were not an issue. And, in truth, it works—to the extent that we can
conduct experiments in our labs and explain them successfully. Naturally, this
raises two questions: why does it work, and where should we expect it not to
work?

In [71], an answer to the first question is proposed. For classical gravity,
they demonstrate that certain classical configurations can be gravitationally
dressed in a way that the dressing depends only on the Poincaré charges
(energy, momentum, and angular momentum) of the configuration. Different
configurations with the same Poincaré charges are indistinguishable from
outside their region, suggesting a way to localize information classically. In
the quantum context, they argue that perturbative quantum gravity in the
presence of global symmetries allows for the definition of localized
information or “gravitational qubits,” which can be localized to regions in
such a way that their information content is not accessible outside the
region. Meanwhile, [94] seems to obtain a general and fully non-perturbative
framework for constructing dynamical reference frames in generally covariant
theories. By constructing relationally local observables throughout spacetime
that satisfy microcausality in the bulk, they establish what they call
”relational locality.”

In our attempt to answer the second question, we propose further scrutiny of
early universe models that use perturbative quantum gravity to explain the
origin of cosmological perturbations and the modeling of gravitational-
induced-entanglement experiments. We also implied that the measurement problem
might be a consequence of the global imposition of a TPS under unitary
evolution. In fact, within our approach, there also seems to exist some notion
of relational locality, as each tensor product structure temporarily induces a
notion of microcausality that is continuously updated under global unitary
evolution. It will be interesting to see whether their work can provide the
mathematical framework in which single-world quantum mechanics can be
embedded.

We end by observing that progress is typically made when obvious notions are
finally put into question. Among the historically remarkable examples, perhaps
Einstein’s scrutinizing of the nature of simultaneity leading to the theory of
relativity and Planck’s debunking of the radiation energy spectrum continuum
giving rise to quantum mechanics stand out the most. Another seemingly obvious
concept is the idea of independent physical systems. Combining relativity and
quantum mechanics challenges the immediate naive understanding of how to
independently describe different interacting physical systems, especially
across different levels of descriptions, so we hope that further analyzing
this notion will pave the way towards a better understanding of quantum
mechanics and quantum gravity.

##  10 Acknowledgements

This work is the result of numerous discussions with many people over the last
two years. I am deeply grateful to the EmerGe (Emergent Geometries)
collaboration, which includes Eugenia Colafranceschi, Andrea Di Biagio, Joakim
Flinckman, Jan Głowacki, Niels Linnemann, and Florian Niedermann.
Collaborating with them has allowed me to deepen my knowledge and interest in
classical and quantum subsystems. This interest began during the Sejny Summer
Institute in 2021, sparked by conversations with Pierre Martin-Dussaud. I have
discussed using changes in the TPS to address the measurement problem with
Andrea Di Biagio, Flavio Del Santo, Patrick Fraser, Arsalan Adil, Leon
Loveridge, Anne-Catherine de la Hamette, and Jan Głowacki. The last three
provided feedback on the discussions presented in this paper. I am also
thankful to Vasileios Fragkos for his feedback on the section on GIE
experiments, as well as to Benjamin Bose and Florian Niedermann for reviewing
parts of the draft. Finally, many ideas concerning the measurement problem
emerged from supervising the master thesis of Nicolas Boulles, with whom I had
extensive discussions on the topic.

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