Spacetime-Free Quantum Theory and Effective Spacetime Structure

Spacetime-Free Approach to Quantum Theory
and Effective Spacetime Structure

Matti RAASAKKA

M. Raasakka

Näyttelijänkatu 25, 33720 Tampere, Finland<EMAIL_ADDRESS>https://sites.google.com/site/mattiraa/

Received May 13, 2016, in final form January 17, 2017; Published online
January 24, 2017

Motivated by hints of the effective emergent nature of spacetime structure, we
formulate a spacetime-free algebraic framework for quantum theory, in which no
a priori background geometric structure is required. Such a framework is
necessary in order to study the emergence of effective spacetime structure in
a consistent manner, without assuming a background geometry from the outset.
Instead, the background geometry is conjectured to arise as an effective
structure of the algebraic and dynamical relations between observables that
are imposed by the background statistics of the system. Namely, we suggest
that quantum reference states on an _extended_ observable algebra, the free
algebra generated by the observables, may give rise to effective spacetime
structures. Accordingly, perturbations of the reference state lead to
perturbations of the induced effective spacetime geometry. We initiate the
study of these perturbations, and their relation to gravitational phenomena.

algebraic quantum theory; quantum gravity; emergent spacetime

81T05; 83C45; 81P10; 81R15; 46L09; 46L53

_Dedicated to the memory of Ukki (1920–2015)._

##  1 Introduction

The human species has evolved to thrive in the low-energy regime of the
Universe, where the environment can be very effectively described by classical
geometry. Our brains are thus hard-wired for geometrical thinking, which is
strongly reflected in the historical development of mathematics and physics.
Indeed, our best understanding of spacetime today, Einstein’s general theory
of relativity, is entirely based on a geometric description of its structure.
However, for more than a century now, indications have kept emerging that
geometry may not be a suitable framework for describing the behavior of
spacetime at very high energies – quantum effects seem to undermine the
geometric description of spacetime. This poses a deep challenge for
theoretical physics, not least because we may no longer be able to rely on our
innate geometrical intuition.

Quantum theory imposes fundamental limitations on the accuracy of spacetime
measurements. For example, we cannot approximate with arbitrary precision free
test point particles, with which spacetime structure may be measured according
to general relativity [75]. On the other hand, considerations of quantum
mechanical clocks reveal fundamental limitations to measurements of duration
and distance [36, 37, 55, 61, 65, 83]. Similarly, quantum field theory and
gravitation together prevent the exact localization of events [30]. Therefore,
the physical meaning of a spacetime point, and accordingly that of a spacetime
manifold, is seriously undermined [23]. Notably, such considerations also seem
to imply that a quantum theory fully incorporating gravity should not follow
by directly quantizing the inherently classical manifold structure
[75].111Notice, however, that quantizing perturbations of spacetime geometry
can still make sense as an effective field theory with a limited range of
validity, in the same way that, e.g., quantizing density perturbations (i.e.,
phonons) is sensible for some condensed matter systems, even if spacetime
structure were not fundamentally quantum.

But if geometry cannot be utilized for the fundamental description of
spacetime, what can we substitute in its place? Many suggestions have been
made since the problem of quantum gravity was first considered in the 1930s
[18] – too many to list here (however, see, e.g., [57]). Here we wish to
explore a different approach, in which we try to avoid substituting any
additional structure.

Let us begin by considering what we fundamentally mean by the notion of
spacetime structure. Physics is always tied to what can be observed, and
therefore we wish to take the operational approach to the question: How do we
measure spacetime structure? In fact, we never directly observe spacetime, but
we deduce its structure indirectly by studying the propagation of matter and
radiation. Accordingly, we are led to suspect that operational spacetime
structure may already be encoded into the structure of the theory that
describes the latter, namely, quantum field theory (QFT). Several hints and
suggestions to this effect have already appeared in the past literature, on
which our present work is partly based (see, in particular, [1, 4, 10, 11, 29,
48, 51, 78]).

We should note that the argument above could be said of any other field just
as well, as the description of physical measurements can always be cast in
terms of interactions between different physical systems. However, there are
at least two important aspects distinguishing the gravitational field: (1) We
have been able to give a succesful quantum description to the other fields –
operational information about them is already encoded into the structure of
QFT or, more specifically, the Standard Model. (2) The gravitational field
affects the behavior of matter universally, by affecting spacetime structure
itself, thus being completely independent of the probe we use for measuring
it.222As is well-appreciated, the universality of gravity is exactly what
enables us to describe it in terms of spacetime structure in the first place,
and not as just another field in spacetime [89]. In any case, the above
observation simply aims to make plausible the expectation that spacetime
geometry could be encoded into the structure of QFT, thus potentially removing
the need for an explicit quantization of gravity. It should not be taken as an
argument against other possibilities. Also, we do not mean to imply that the
other fields could not be understood in a more operational way.

Let us recall some results supporting the idea that gravity may emerge as an
effective phenomenon from a more fundamental quantum description. The first
glimpses of the emergent nature of gravitational phenomena go all the way back
to the following realization by Sakharov in 1968 [64, 86]: Consider QFT on a
spacetime with an arbitrary but fixed metric structure coupled to the field
via the covariant derivative (plus possibly a non-minimal term). Then,
generically, the Einstein–Hilbert action of general relativity along with a
cosmological constant and some higher order corrections can be seen to arise
from the one-loop contribution to the effective action. The derivation is far
from unproblematic, since the effective couplings diverge in the absence of a
UV regulator, and the cosmological term comes out far too large even with a
regulator in place. Nevertheless, it does strongly suggest the possibility of
an effective gravitational dynamics arising from quantum corrections. We may
hope that by formulating a better-behaved framework for quantum theory,
perhaps with some natural cut-off to the degrees of freedom, gravity may
emerge from effective quantum dynamics.

Another closely related approach to emergent gravity originates from the
remarkable thermodynamical properties of black holes that were discovered in
the 1970’s by Bekenstein [6, 7, 8] and Hawking [42]. QFT calculations on a
black hole background revealed that the event horizon emits thermal radiation,
which gives a temperature and an associated entropy to the black hole. Since
then, it has been further understood that such thermodynamical properties can
be associated not only to black holes but to generic causal horizons in
spacetime. As was first discovered by Unruh [84], even a uniformly
accelerating (Rindler) observer in Minkowski spacetime experiences the thermal
behavior of her apparent causal horizon. The origin of these thermodynamical
properties of causal horizons continues to be under vigorous debate. However,
some evidence indicates that the horizon entropy may be most naturally
understood as the entanglement entropy of the quantum fields arising from the
correlations between matter degrees of freedom separated by the causal horizon
– especially if one expects the full gravitational action to be induced by the
matter fields [13, 28, 47, 70, 73]. It was discovered by Jacobson that the
Einstein field equation of general relativity can be obtained as an equation
of state for the thermal properties of local Rindler horizons [46]. This
relation appears to be a generic feature of gravitational theories [25, 59].
More recently, Jacobson also showed that the semiclassical Einstein equation
can be derived in an elegant way from the hypothesis that the QFT vacuum state
locally maximizes the von Neumann entropy [48]. Also in this case it is most
natural for the Einstein equation to arise solely from the entanglement of
matter degrees of freedom. Therefore, there seems to be no need for
incorporating gravitons in the theory – in fact, the introduction of gravitons
into QFT is actively discouraged, as it would lead to a double-counting of
energy in Jacobson’s derivation [48].

Let us emphasize that our brief review above of mechanisms for the emergence
of gravity from QFT is far from exhaustive. (See, e.g., [31, 55, 60, 69, 79]
and references therein for some of the other approaches.) These mechanisms for
the emergence of gravity from quantum field theory may not all be mutually
exclusive, although the relations between them are not well understood at the
moment, as far as we know. Nevertheless, we may argue that none of these
mechanisms, or indeed any mechanism based on QFT, can present a logically
coherent explanation for the emergence of gravity as long as the fundamental
quantum theory is built on a background spacetime: According to general
relativity, gravity is an inherent property of spacetime [89]. Therefore, in
order to provide a consistent description of the emergence of gravity,
spacetime itself must be emergent, and not appear as a fundamental ingredient
in the construction of quantum theory.

Our work aims to provide a concrete mathematical framework, a spacetime-free
formulation of quantum theory, in which questions of spacetime emergence can
be directly and explicitly addressed.333Perhaps it would be more accurate to
call our approach ‘background-geometry-free’ instead of ‘spacetime-free’,
since there exist QFT models for quantum gravity (e.g., group field theories
[58]), which are formulated on auxiliary spaces not directly related to
spacetime, whereas we want to fully remove the geometric background manifold,
on which QFT is formulated. However, we opt for the ‘spacetime-free’
terminology for the sake of compactness. In order to have a chance of success,
the framework we wish to develop should satisfy at least the following three
requirements:

  1. 1.

Spacetime structure should not enter the theory as a fundamental ingredient.
Instead, we should be able to recover spacetime as an effective description of
the dynamical organization of the degrees of freedom in some regime of the
theory.

  2. 2.

Despite the lack of background spacetime, the theory should have a clear
operational interpretation in terms of (idealized) experimental arrangements
and observations.

  3. 3.

When an effective geometric background _can_ be recovered, the theory should
reproduce in the appropriate regimes our current models, general relativity
and quantum field theory.

The development of the spacetime-free framework for quantum physics in Section
2 is guided by these three requirements. We will also assume the general
validity of abstract algebraic quantum theory [2, 38, 74], which we will
consider _for all practical purposes_ as a theory of knowledge, although the
framework itself is independent of the interpretation of the quantum
state.444However, see [43, 45] for a reconstruction of qubit quantum theory
from operational considerations that are rather reminicent of our development
of the spacetime-free framework in Section 2.

The following diagram depicts the traditional logic of constructing a quantum
field theoretic model, according to which we first postulate a background
spacetime, and then build the QFT model on top of it:

$\displaystyle\textrm{spacetime}\
\stackrel{{\scriptstyle\textrm{locality}}}{{\Longrightarrow}}\
\textrm{dynamics}\ \stackrel{{\scriptstyle\textrm{KMS
cond.}}}{{\Longrightarrow}}\ \textrm{equilibrium/vacuum state}$

The arrows in this diagram should be understood as one-to-many
correspondencies. For example, the notion of locality provided by the
background geometry does not completely fix the dynamics, but strongly
restricts its choice for the physically relevant local QFT models, such as the
Standard Model. Obviously, there exist many local QFT models corresponding to
the same background geometry. Similarly, the same dynamics give rise to
several different equilibrium states parametrized by temperature.555Let us
neglect for the moment the fact that not all spacetimes allow for equilibrium
states. We will later define the concept of a reference state that need not be
an equilibrium/vacuum state. As an example, the locally covariant QFT
framework [20, 34], which is arguably the most general formulation of ordinary
QFT today, follows this logic in defining QFT models. Following ideas in
earlier works [1, 4, 10, 11, 29, 51, 78], we suggest trying to invert the
arrows leading from the spacetime to the equilibrium state in order to recover
spacetime geometry from the equilibrium state.666During the revision of this
manuscript, we became aware of the interesting paper [66] by Salehi, which
also explores the idea of state dependent dynamics for quantum systems. The
purpose of this reversal of logic is that a state of the system can be
determined and represented algebraically without referring to any spacetime
structure. Hence, our diagram would look more like this:

$\displaystyle\textrm{spacetime}\
\stackrel{{\scriptstyle\textrm{???}}}{{\Longleftarrow}}\ \textrm{dynamics}\
\stackrel{{\scriptstyle\textrm{T-T theory}}}{{\Longleftarrow}}\
\textrm{equilibrium/vacuum state}$

The second arrow from the dynamics to the equilibrium state is already known
(in some cases) to be inverted by the Tomita–Takesaki modular theory, as
reviewed in Section 2.4. The first arrow leading from the dynamics to the
spacetime is more enigmatic, although some ideas and hints on how to achieve
the inversion have appeared in the literature (see, e.g., [4, 10, 11, 24, 51,
78]). In Section 3 we will develop certain physically motivated ideas and
methods for studying the effective spacetime structure induced by the
dynamics.

The rest of this paper is organized as follows. The Sections 2.1 and 2.2 form
the core content of the paper, in which we give a mathematically precise
definition for the spacetime-free quantum theory. The ability of the
formulation to describe general quantum systems, even in the absence of
spacetime structure, relies on the universality property of the free product
of algebras, as explained in Section 2.1. In the rest of the Section 2 we
review several canonical operator algebraic structures appearing in the
spacetime-free framework that should play a key role in the extraction of
dynamics and effective spacetime structure from the organization of the
quantum statistics of observations. In Section 3 we further offer some ideas
on how to actually recover information on the effective spacetime geometry.
This section mainly serves the purpose of making it at least plausible to the
reader that such a recovery is possible, even though we do not yet have solid
results to offer on this aspect of the approach. Section 4 provides a summary
of the results, and points out some of the challenges and future prospects for
the approach. This paper also contains several appendices that offer further
details to the presentation.

##  2 Spacetime-free framework for quantum physics

In relativistic quantum field theory, the causal structure of spacetime is
encoded into the algebraic structure of the observable algebra – in
particular, commutators of spacelike separated observables vanish [38]. On the
other hand, from general relativity we know that the matter distribution of a
system has an influence on the causal structure through gravity [89]. In
quantum field theory the matter distribution is described by the quantum
state. Therefore, in order to introduce gravitational effects into quantum
field theory, we must be able to modify the formalism so that the quantum
state may influence the algebraic relations of the observables. To this end,
we first introduce the free observable algebra, which will allow us to give an
operational definition of a quantum system in the absence of a background
spacetime.

###  2.1 Free observable algebra

To define an experimental arrangement we introduce the set
$\mathcal{M}:=\\{M_{i}\colon i\in I\\}$ of measurements that can be performed
in the experiment, where $I$ is an arbitrary index set.777The measurements can
also be understood as partial observables of the system under study, as
defined in [63]. To any measurement $M\in\mathcal{M}$ we associate a spectrum
$\textrm{Spec}(M)$, which is the topological (compact Hausdorff) space of
possible values that the measurement can take.888Typically,
$\textrm{Spec}(M)\subset\mathbb{R}$ of course, but we need not to make such
restriction here. However, we do assume for simplicity that $\textrm{Spec}(M)$
is compact for all $M\in\mathcal{M}$. If $\textrm{Spec}(M)$ is not compact, we
can restrict to functions that vanish at infinity in the following, and the
construction goes through more or less the same way. We consider the spectrum
$\textrm{Spec}(M)$ to fully characterize the measurement $M\in\mathcal{M}$. We
may then associate to the measurement a unital abelian $C^{*}$-algebra
$\mathfrak{M}:=C(\textrm{Spec}(M),\mathbb{C})$ that is the algebra of
continuous complex-valued functions on its spectrum. Continuous real-valued
functions on $\textrm{Spec}(M)$ correspond to self-adjoint operators in
$\mathfrak{M}$, which represent the observables that are accessible through
the measurement $M\in\mathcal{M}$. By the Gelfand duality, $\mathfrak{M}$ is
uniquely determined by $\textrm{Spec}(M)$, and vice versa. (See, e.g., [85]
for an elementary exposition of the Gelfand duality.) In order to describe
projective measurements, we may complete $\mathfrak{M}$ in the weak operator
topology to obtain the abelian von Neumann algebra $\mathfrak{W}$, which
contains the spectral projections $\\{P_{O}\\}_{O\subset\textrm{Spec}(M)}$,
where $O\subset\textrm{Spec}(M)$ are open sets. An abelian von Neumann algebra
can be identified as a (classical) probability space, where the projection
$P_{O}$ corresponds to the proposition that the value of the random variable
resides in the open set $O\subset\textrm{Spec}(M)$. By extension, non-abelian
von Neumann algebras are often considered to be non-commutative probability
spaces. (The books [49, 50, 80, 81, 82] offer extensive references for
operator algebra theory, and [2, 38, 74] give excellent accounts of algebraic
quantum theory.)

A central mathematical construct for the development of our framework is the
free product of algebras999Throughout this paper, we refer by the term ‘free
product’ to what is more precisely called the _reduced_ free product, where
unit elements of the component algebras are identified., which plays a
fundamental role in the non-commutative probability theory [72, 87]. The free
product $\mathfrak{A}_{1}\star\mathfrak{A}_{2}$ of two unital $*$-algebras
$\mathfrak{A}_{i}$, $i=1,2$, is linearly generated by finite sequences
$x_{1}x_{2}\cdots x_{n}$, where $x_{k}\in\mathfrak{A}_{1}$ or
$x_{k}\in\mathfrak{A}_{2}$ for each $k\in 1,\ldots,n$. The product of two
elements $x_{1}\cdots x_{m},y_{1}\cdots
y_{n}\in\mathfrak{A}_{1}\star\mathfrak{A}_{2}$ is given by the concatenation
of sequences, i.e.,

$\displaystyle(x_{1}\cdots x_{m})\star(y_{1}\cdots y_{n})=x_{1}\cdots
x_{m}y_{1}\cdots y_{n}.$

Moreover, the involutive ∗-operation is given by $(x_{1}\cdots
x_{m})^{*}=x_{m}^{*}\cdots x_{1}^{*}$. Finally, we impose the following
equivalence relations on the elements of
$\mathfrak{A}_{1}\star\mathfrak{A}_{2}$:

  1. 1.

$\mathbbm{1}_{\mathfrak{A}_{1}}\sim\mathbbm{1}_{\mathfrak{A}_{2}}\sim\mathbbm{1}_{\mathfrak{A}_{1}\star\mathfrak{A}_{2}}$,
the unit element (i.e., the empty sequence) in
$\mathfrak{A}_{1}\star\mathfrak{A}_{2}$, where
$\mathbbm{1}_{\mathfrak{A}_{i}}$ denotes the unit element in
$\mathfrak{A}_{i}$, and $\sim$ implies equivalence.

  2. 2.

If $x_{k},x_{k+1}\in\mathfrak{A}_{i}$ for the same $i=1,2$, we set
$x_{1}\cdots x_{k}x_{k+1}\cdots x_{n}\sim x_{1}\cdots(x_{k}\cdot
x_{k+1})\cdots x_{n}$, where in the latter we denote by $(x_{k}\cdot
x_{k+1})\in\mathfrak{A}_{i}$ the product of $x_{k}$ and $x_{k+1}$ in
$\mathfrak{A}_{i}$.

The free product of $*$-algebras is an associative and commutative operation
in the category of $*$-algebras. The free product $*$-algebra
$\mathfrak{A}_{1}\star\mathfrak{A}_{2}$ is always non-abelian and infinite-
dimensional unless one of the factors is trivial (i.e., isomorphic to
$\mathbb{C}$). Moreover, it has the following important _universality
property_ : $\mathfrak{A}_{1}\star\mathfrak{A}_{2}$ is the unique unital
$*$-algebra, from which there exists a unital $*$-homomorphism to any other
$*$-algebra generated by $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ as unital
$*$-subalgebras [87]. In this precise sense, the free product does not impose
any relations between the elements of its factors except for the
identification of units. Accordingly, it is the canonical structure to start
with, when we wish to impose arbitrary relations between unital subalgebras.

Let us then define the _free observable algebra_ $\mathfrak{F}$ associated
with an experimental arrangement. It is the unital $*$-algebra given by the
free product $\mathfrak{F}:=\star_{i\in I}\mathfrak{W}_{i}$ of the abelian von
Neumann algebras $\mathfrak{W}_{i}$ associated with the measurements
$M_{i}\in\mathcal{M}$ available in the experimental arrangement. In
particular, we interpret an element of the form
$P_{O_{1}}^{(i_{1})}P_{O_{2}}^{(i_{2})}\cdots
P_{O_{n}}^{(i_{n})}\in\mathfrak{F}$, i.e., a sequence of spectral projections
$P_{O_{k}}^{(i_{k})}\in\mathfrak{W}_{i_{k}}$,
$O_{k}\subset\textrm{Spec}(M_{i_{k}})$, to represent a sequence of measurement
outcomes associated with the measurements. Importantly, the set of finite
sequences of spectral projections linearly generate the free observable
algebra, since each of the factors $\mathfrak{W}_{i}$ is (the norm completion
of the algebra) linearly generated by its spectral projections. The unit
element $\mathbbm{1}_{\mathfrak{F}}\in\mathfrak{F}$ represents the case of no
measurements.

Let us emphasize that the ordering of spectral projections refers to the
order, in which the measurement results are recorded by the experimenter – not
(necessarily) the temporal order of the measurement events. Thus, no global
time flow is needed for the definition of this ordering, but only the
assumption that the experimenter can perform sequences of measurements. The
ordering of the measurements is relevant for quantum systems due to quantum
uncertainty relations: Recording the value of one observable quantity may
influence the results of other measurements.

It may be helpful to contrast the free product with the tensor product to
better understand the physical significance of the free observable algebra.
The usual way to combine the observable algebras of individual quantum systems
to form the observable algebra of the composite system is the tensor product.
However, the tensor product already requires certain assumptions on the
relations between the component systems (e.g., that their observable algebras
commute mutually), which are equivalent to the operational independence of the
systems [77]. Indeed, usually the quantum systems that are put together by
tensor product are considered spacelike separated and/or causally independent.
The free product, on the other hand, does not impose any such relations a
priori. By the universality property, we may recover any possible algebraic
relations between the individual systems via unital $*$-homomorphisms of the
free product of their observable algebras. The tensor product structure
corresponding to operational independence of the systems is but only one of
the vast array of possible relations between the component algebras that can
be obtained in this way. Other possibilities include the many ways in which
the systems may not be independent or mutually commuting, and thus one can see
how these homomorphisms may naturally introduce temporal/causal structure on
the composite system. Above, we have restricted to consider the situation in
which the individual systems correspond to abelian observable algebras
generated by single observables, because we wish not to assume anything about
the spatiotemporal relations of the observables a priori, whereas non-abelian
algebras may already contain information on the temporal structure or the
dynamics of the system.101010It is, of course, possible to generalize the
construction of the free observable algebra also to the case where the
individual algebras are non-abelian, because the free product can be defined
for any family of $*$-algebras. The tensor product of abelian algebras is
again an abelian algebra, and therefore would not give us any interesting
algebraic structure in any case, whereas the free product is always an
infinite-dimensional non-abelian algebra, if the component algebras are non-
trivial (i.e., not isomorphic to $\mathbb{C})$.

###  2.2 Reference states and the physical observable algebra

In addition to the set of possible measurements, an experiment is always
characterized by the statistical background to the measurements, by which we
refer specifically to the probabilities and correlations of measurement
outcomes in the absence of any additional external interference with the
measured system. In ordinary QFT in Minkowski spacetime, for example, the
statistical background for the usual particle scattering experiments carried
out in a vacuum environment is represented by the vacuum state.
Experimentally, the statistical background for an experiment is obtained via
the calibration of the measurement instruments prior to any actual
measurements. In order to obtain good statistics for the background noise, one
must repeat the same calibration experiment a large number of times with a
collection of systems prepared in exactly the same way – the statistical
ensemble. In any realistic situation the experimental background statistics
are, of course, always limited in accuracy due to the finite ensemble size.
However, in most cases we may believe that the background statistics converges
to a limit as the cardinality of the ensemble is increased. If the calibration
experiments are performed on the same system but at separate intervals of
external time as experienced by the experimenter, while letting the system to
‘relax’ between the experiments, the statistical background must be in
equilibrium with respect to the external time flow. In a controlled
measurement scheme, the measurement results are compared with the statistical
background in order to differentiate the effects of external influence on the
system. It is a characteristic property of quantum systems that the
statistical background can never be completely trivial, since quantum effects
produce fluctuations and correlations in measurement outcomes even in the pure
vacuum.

As usual in algebraic quantum theory, we will use states to represent
information about the experimental system. (See, e.g., [2, 38, 74] for
comprehensive accounts of the algebraic formulation of quantum theory.) A
_state_ $\omega$ on the free observable algebra $\mathfrak{F}$ is a linear
functional $\omega\colon\mathfrak{F}\rightarrow\mathbb{C}$, which is111111Note
that not all of these requirements are completely independent: For example,
the positivity of a state is enough to guarantee self-adjointness and
continuity in the case of $C^{*}$-algebras.

  1. 1)

positive: $\omega(a^{*}a)\geq 0$ for all $a\in\mathfrak{F}$,

  2. 2)

self-adjoint: $\omega(a^{*})=\overline{\omega(a)}$ for all $a\in\mathfrak{F}$,

  3. 3)

normalized: $\omega(\mathbbm{1}_{\mathfrak{F}})=1$, and

  4. 4)

independently continuous in each of the elements $x_{1},\ldots,x_{n}$ for any
sequence $x_{1}\cdots x_{n}\in\mathfrak{F}$.

Let us denote the space of states on $\mathfrak{F}$ by
$\mathcal{S}(\mathfrak{F})$. Since the free observable algebra is generated by
finite sequences of spectral projections, the values on these elements
determine the state. For any state $\omega\in\mathcal{S}(\mathfrak{F})$ we
interpret

$\displaystyle\omega\Big{(}\big{(}P_{O_{1}}^{(i_{1})}P_{O_{2}}^{(i_{2})}\cdots
P_{O_{n}}^{(i_{n})}\big{)}^{*}\big{(}P_{O_{1}}^{(i_{1})}P_{O_{2}}^{(i_{2})}\cdots
P_{O_{n}}^{(i_{n})}\big{)}\Big{)}\in[0,1]$ (2.1)

as the probability for the sequence of measurement events corresponding to the
sequence of spectral projections $P_{O_{1}}^{(i_{1})}P_{O_{2}}^{(i_{2})}\cdots
P_{O_{n}}^{(i_{n})}$. This probability interpretation is standard in quantum
theory (see Appendix A). However, here we propose to generalize equation (2.1)
to the case in hand, where no global time parameter or evolution is available
a priori. In fact, the generalization asks to distinguish between the time-
ordering of events as recorded by the experimenter, which is represented by
the ordering of the spectral projections, and the time-evolution of the
experimental system. From the point of view of general relativity, it is
natural and expected that such a distinction should be made due to the lack of
a global time-ordering of events that would tie together the time of the
experimenter and that of the system.121212The distinction seems also natural
from the point of view of the Bayesian interpretation of the quantum state
(see, e.g., [19, 35]), according to which the state represents the subjective
knowledge of the experimenter about the quantum system.

Next we review very briefly the Gelfand–Naimark–Segal (GNS) construction for a
state $\omega\in\mathcal{S}(\mathfrak{F})$. (See, e.g., [67] for a more
complete exposition.) Let us denote the _null ideal_ of $\omega$ by

$\displaystyle\mathcal{N}_{\omega}:=\big{\\{}a\in\mathfrak{F}\colon\omega(a^{*}a)=0\big{\\}},$

which is a left ideal in $\mathfrak{F}$. Thus, we may consider the quotient

$\displaystyle\mathfrak{F}/\mathcal{N}_{\omega}:=\big{\\{}a+\mathcal{N}_{\omega}\colon
a\in\mathfrak{F}\big{\\}}$

as a linear space. Denote by
$|a\rangle_{\omega}\in\mathfrak{F}/\mathcal{N}_{\omega}$ the equivalence class
containing the element $a\in\mathfrak{F}$. Then, $\omega$ provides a non-
degenerate inner product

$\displaystyle\langle
a|b\rangle_{\omega}:=\omega(a^{*}b)\qquad\forall\,|a\rangle_{\omega},|b\rangle_{\omega}\in\mathfrak{F}/\mathcal{N}_{\omega}$

on $\mathfrak{F}/\mathcal{N}_{\omega}$. We may now complete
$\mathfrak{F}/\mathcal{N}_{\omega}$ in the norm induced by this inner product
to obtain the GNS Hilbert space
$\mathcal{H}_{\omega}:=\overline{\mathfrak{F}/\mathcal{N}_{\omega}}$. Setting

$\displaystyle\pi_{\omega}(a)|b\rangle_{\omega}\stackrel{{\scriptstyle!}}{{=}}|ab\rangle_{\omega}\in\mathfrak{F}/\mathcal{N}_{\omega}$

for all $a\in\mathfrak{F}$ and
$|b\rangle_{\omega}\in\mathfrak{F}/\mathcal{N}_{\omega}$ gives rise to the GNS
representation $\pi_{\omega}\colon\mathfrak{F}\rightarrow
B(\mathcal{H}_{\omega})$ of $\mathfrak{F}$ on the whole of
$\mathcal{H}_{\omega}$ by continuity. We have by construction that the unit
vector
$|\mathbbm{1}_{\mathfrak{F}}\rangle_{\omega}\in\mathfrak{F}/\mathcal{N}_{\omega}$
satisfies
$\langle\mathbbm{1}_{\mathfrak{F}}|\pi_{\omega}(a)|\mathbbm{1}_{\mathfrak{F}}\rangle_{\omega}=\omega(a)$
for all $a\in\mathfrak{F}$. It is also cyclic in $\mathcal{H}_{\omega}$ with
respect to $\pi_{\omega}(\mathfrak{F})$, i.e., the subspace

$\displaystyle\big{\\{}\pi_{\omega}(a)|\mathbbm{1}_{\mathfrak{F}}\rangle_{\omega}\in\mathcal{H}_{\omega}\colon
a\in\mathfrak{F}\big{\\}}\subset\mathcal{H}_{\omega}$

is norm dense in $\mathcal{H}_{\omega}$. By the von Neumann double-commutant
theorem, we may complete $\pi_{\omega}(\mathfrak{F})\subset
B(\mathcal{H}_{\omega})$ in the weak operator topology on
$B(\mathcal{H}_{\omega})$ by taking the double-commutant
$\mathfrak{A}_{\omega}:=\pi_{\omega}(\mathfrak{F})^{\prime\prime}$, which is
therefore a von Neumann algebra.131313The commutant of a subalgebra
$\mathfrak{A}\subset B(\mathcal{H})$ is defined as
$\mathfrak{A}^{\prime}:=\\{a^{\prime}\in B(\mathcal{H})\colon[a,a^{\prime}]=0\
\forall a\in\mathfrak{A}\\}$. The canonical extension $\tilde{\omega}$ of the
state $\omega\in\mathcal{S}(\mathfrak{F})$ onto $B(\mathcal{H}_{\omega})$ (and
thus onto $\mathfrak{A}_{\omega}$) is obtained as
$\tilde{\omega}(A):=\langle\mathbbm{1}_{\mathfrak{F}}|A|\mathbbm{1}_{\mathfrak{F}}\rangle_{\omega}$
for all $A\in B(\mathcal{H}_{\omega})$.

Notice that, by the completion procedure, the GNS Hilbert space
$\mathcal{H}_{\omega}$ also contains any vector of the form
$|x_{1}x_{2}\cdots\rangle_{\omega}\in\mathcal{H}_{\omega}$, consisting of an
infinite number of elements $x_{k}\in\cup_{i}\mathfrak{W}_{i}$
$\forall\,k\in\mathbb{N}$, associated with the limit of a Cauchy sequence of
vectors $(|x_{1}x_{2}\cdots x_{n}\rangle_{\omega})_{n\in\mathbb{N}}$. A
similar statement applies to $\mathfrak{A}_{\omega}$ as a completion of
$\pi_{\omega}(\mathfrak{F})\subset B(\mathcal{H}_{\omega})$ in the weak
operator topology on $B(\mathcal{H}_{\omega})$.

We will mathematically model the statistical background to an experiment by a
special state, the _reference state_ $\Omega\in\mathcal{S}(\mathfrak{F})$.
Namely, $\Omega$ encodes the probabilities of measurement outcomes in the
absence of any additional external perturbations to the experimental system
via equation (2.1).141414As an example, for particle scattering experiments in
a vacuum environment in Minkowski spacetime, the reference state would be
given by the vacuum state (or its restriction to a local subsystem). The
reference state $\Omega$ gives rise to the $*$-representation
$\pi_{\Omega}\colon\mathfrak{F}\rightarrow B(\mathcal{H}_{\Omega})$ of the
free observable algebra on the GNS Hilbert space $\mathcal{H}_{\Omega}$, and
to the von Neumann algebra
$\mathfrak{A}_{\Omega}:=\pi_{\Omega}(\mathfrak{F})^{\prime\prime}\subset
B(\mathcal{H}_{\Omega})$. We will assume that the restriction of $\Omega$ on
each subalgebra $\mathfrak{W}_{i}\subset\mathfrak{F}$ is faithful, so that
$\mathfrak{W}_{i}$ are represented faithfully in $\mathfrak{A}_{\Omega}$. We
call $\mathcal{H}_{\Omega}$ and $\mathfrak{A}_{\Omega}$ the _physical Hilbert
space_ and the _physical observable algebra_ of the experimental arrangement,
respectively. The algebraic structure of the physical observable algebra
$\mathfrak{A}_{\Omega}$ encodes the causal and dynamical properties of the
observables under consideration. The free observable algebra together with the
reference state form a tuple $(\mathfrak{F},\Omega)$, which we take to fully
determine the experimental arrangement.

Crucially for the above, the GNS representation $\pi_{\Omega}$ as a
$*$-homomorphism may impose non-trivial algebraic relations between the factor
algebras $\mathfrak{W}_{i}$ of the free observable algebra
$\mathfrak{F}\equiv\star_{i}\mathfrak{W}_{i}$, and thus induce non-trivial
relations among the observables in $\mathfrak{A}_{\Omega}$. For example, two
observables or, more generally, two subalgebras of observables
$\mathfrak{B}_{1},\mathfrak{B}_{2}\subset\mathfrak{A}_{\Omega}$ are _jointly
measurable_ if they commute in $\mathfrak{A}_{\Omega}$:
$\mathfrak{B}_{1}\subset\mathfrak{B}_{2}^{\prime}$. Moreover, two subalgebras
$\mathfrak{B}_{1},\mathfrak{B}_{2}\subset\mathfrak{A}_{\Omega}$ are
_operationally independent_ if they satisfy the _split property_ : There
exists a type I von Neumann factor algebra $\mathfrak{C}\subset
B(\mathcal{H}_{\Omega})$ such that
$\mathfrak{B}_{1}\subset\mathfrak{C}\subset\mathfrak{B}_{2}^{\prime}$. For
operationally independent subalgebras there exist arbitrary normal product
states, so the subsystems represented by the two subalgebras can be
decorrelated and prepared independently [77].151515Clearly, operational
independence implies joint measurability. For finite-dimensional algebras the
two notions coincide. In this sense, there exists a complete set of operations
(represented mathematically by completely positive maps) that affect
expectation values of observables only in one of the subalgebras but not the
other. Therefore, operational independence is strongly related to causal
independence of subsystems – indeed, subsystems that are spatially separated
by a finite distance are known to be operationally independent in physical QFT
models [77]. In the spacetime-free framework, we could even take operational
independence as the rigorous definition of causal independence. Notice that
operational independence does not imply that there are no statistical
correlations between the two subalgebras of observables in an arbitrary state,
only that it is possible to remove all correlations.

###  2.3 Covariance and symmetries of the experimental arrangement

Let $\alpha\in\textrm{Aut}(\mathfrak{F})$ be a $*$-automorphism of the free
observable algebra, and define
$\alpha\triangleright(\mathfrak{F},\Omega):=(\alpha(\mathfrak{F}),\alpha^{*}(\Omega))$,
where $\alpha^{*}(\Omega):=\Omega\circ\alpha^{-1}$. We call this action by
automorphisms on the tuple a  _covariant transformation_. The covariantly
transformed tuple $(\alpha(\mathfrak{F}),\alpha^{*}(\Omega))$ corresponds to
the same experimental arrangement as $(\mathfrak{F},\Omega)$, because it leads
to the same expectation values and thus the same algebraic relations for the
physical observable algebra. Therefore, there is a one-to-one correspondence
between experimental arrangements and the equivalence classes of tuples by
covariant transformations.161616Here, in the definition of the free observable
algebra $\mathfrak{F}$, we implicitly include the identification of the
subalgebras $\mathfrak{W}_{i}$ corresponding to the initial set of physical
observables giving rise to $\mathfrak{F}=\star_{i}\mathfrak{W}_{i}$.

For a given reference state $\Omega\in\mathcal{S}(\mathfrak{F})$, there is a
special subgroup

$\displaystyle\textrm{Sym}_{\Omega}(\mathfrak{F}):=\big{\\{}\alpha\in\textrm{Aut}(\mathfrak{F})\colon\alpha^{*}(\Omega)=\Omega\big{\\}}\subset\textrm{Aut}(\mathfrak{F})$

of automorphisms of $\mathfrak{F}$, the _symmetry group_ of the experimental
arrangement. Only the free observable algebra transforms under the covariant
action by symmetries,
$\alpha\triangleright(\mathfrak{F},\Omega)=(\alpha(\mathfrak{F}),\Omega)$. Any
symmetry $\alpha\in\textrm{Sym}_{\Omega}(\mathfrak{F})$ can be represented on
the GNS Hilbert space $\mathcal{H}_{\Omega}$ by a unitary operator
$U_{\alpha}\in B(\mathcal{H}_{\Omega})$, which satisfies
$U_{\alpha}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}$
[49, 50]. On the other hand, we may more generally consider the group of
unitaries $\mathcal{U}_{\mathfrak{F}}:=\\{U\in B(\mathcal{H}_{\Omega})\colon
U\mathfrak{A}_{\Omega}U^{*}=\mathfrak{A}_{\Omega}\\}$ fixing the physical
observable algebra $\mathfrak{A}_{\Omega}$, and its subgroup
$\mathcal{U}_{\mathfrak{F}}^{\Omega}:=\\{U\in\mathcal{U}_{\mathfrak{F}}\colon
U|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}\\}$
that leaves $\tilde{\Omega}$ invariant, which is the group of _physical
symmetries_ of the system. $\textrm{Sym}_{\Omega}(\mathfrak{F})$ is isomorphic
to a subgroup of $\mathcal{U}_{\mathfrak{F}}^{\Omega}$ through its unitary
representation on $\mathcal{H}_{\Omega}$. Then, a _continuous physical
symmetry_ is given by a (strongly) continuous one-parameter group of
automorphisms $\alpha\colon
s\mapsto\alpha_{s}\in\textrm{Aut}(\mathfrak{A}_{\Omega})$ that are induced by
some unitaries in $\mathcal{U}_{\mathfrak{F}}^{\Omega}$. By Stone’s theorem,
the group of unitaries $s\mapsto U_{s}$ representing the continuous symmetry
$\alpha$ on $\mathcal{H}_{\Omega}$ is generated as $U_{s}=e^{isL_{\alpha}}$ by
a (possibly unbounded but densely defined) self-adjoint operator $L_{\alpha}$
affiliated with $B(\mathcal{H}_{\Omega})$ that satisfies
$L_{\alpha}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=0$ [49, 50].

###  2.4 Equilibrium condition and thermal dynamics

The reference state $\tilde{\Omega}\in\mathcal{S}(\mathfrak{F})$ is faithful
on $\pi_{\Omega}(\mathfrak{F})$ if the null ideal
$\mathcal{N}_{\Omega}=\\{a\in\mathfrak{F}\colon\Omega(a^{*}a)=0\\}$ satisfies
$\mathcal{N}_{\Omega}=\ker(\pi_{\Omega}):=\\{a\in\mathfrak{F}\colon\pi_{\Omega}(a)=0\\}$.
This is equivalent with $\mathcal{N}_{\Omega}$ being two-sided and therefore
self-adjoint (i.e., if $\Omega(a^{*}a)=0$ for $a\in\mathfrak{F}$, then also
$\Omega(aa^{*})=0$). In terms of measurement probabilities the self-
adjointness of $\mathcal{N}_{\Omega}$ corresponds to the following statistical
property of the reference state: If any sequence of spectral projections
$P_{O_{1}}^{(i_{1})}P_{O_{2}}^{(i_{2})}\cdots P_{O_{n}}^{(i_{n})}$ has a
vanishing probability via equation (2.1), then also the probability for the
reversed sequence $P_{O_{n}}^{(i_{n})}P_{O_{n-1}}^{(i_{n-1})}\cdots
P_{O_{1}}^{(i_{1})}$ vanishes. Accordingly, the requirement
$\mathcal{N}_{\Omega}=\ker(\pi_{\Omega})$ is implied by the detailed balance
condition, which states that the probabilities for any process and its reverse
are the same in equilibrium, and actually implies that $\Omega$ may represent
an equilibrium state.

In particular, when the extended reference state $\tilde{\Omega}$ is faithful
on the physical observable algebra
$\mathfrak{A}_{\Omega}:=\pi_{\Omega}(\mathfrak{F})^{\prime\prime}\subset
B(\mathcal{H}_{\Omega})$, it gives rise to the following two canonical
operators on the GNS Hilbert space $\mathcal{H}_{\Omega}$ by Tomita–Takesaki
modular theory [76, 81]:

  1. 1.

The _modular operator_ $\Delta_{\Omega}$ is a (possibly unbounded) positive
operator affiliated with $B(\mathcal{H}_{\Omega})$, which satisfies
$\Delta_{\Omega}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}$
and
$\Delta_{\Omega}^{it}\mathfrak{A}_{\Omega}\Delta_{\Omega}^{-it}=\mathfrak{A}_{\Omega}$
for all $t\in\mathbb{R}$. The modular operator induces a strongly continuous
one-parameter group of automorphisms of $\mathfrak{A}_{\Omega}$, the _modular
flow_ , $\sigma^{\Omega}\colon
t\mapsto\sigma^{\Omega}_{t}\in\textrm{Aut}(\mathfrak{A}_{\Omega})$ through the
unitary action

$\displaystyle\Delta_{\Omega}^{it}a\Delta_{\Omega}^{-it}\equiv\sigma^{\Omega}_{t}(a)\qquad\forall\,a\in\mathfrak{A}_{\Omega},\
t\in\mathbb{R}.$

The extended reference state $\tilde{\Omega}$ satisfies the
Kubo–Martin–Schwinger (KMS) equilibrium condition with respect to the modular
flow $\sigma^{\Omega}$, and the modular flow is the unique one-parameter group
of automorphisms of $\mathfrak{A}_{\Omega}$ with this property, up to
rescaling $t\mapsto\lambda t$, $\lambda\in\mathbb{R}_{+}$ of the flow
parameter [16, 17]. (This rescaling freedom means that the temperature of the
equilibrium state is indetermined.)

  2. 2.

The _modular involution_ $J_{\Omega}$ is an anti-linear involutive operator
(i.e., $J_{\Omega}^{*}=J_{\Omega}^{-1}=J_{\Omega}$), which satisfies
$J_{\Omega}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}$
and
$J_{\Omega}\mathfrak{A}_{\Omega}J_{\Omega}=\mathfrak{A}_{\Omega}^{\prime}$.
Accordingly, $\mathfrak{A}_{\Omega}$ and $\mathfrak{A}_{\Omega}^{\prime}$ are
(anti) isomorphic von Neumann algebras. Moreover, we have
$J_{\Omega}\Delta_{\Omega}J_{\Omega}=\Delta_{\Omega}^{-1}$.

These two operators arise from the polar decomposition of the operator
$S_{\Omega}\equiv
J_{\Omega}\Delta_{\Omega}^{\frac{1}{2}}\colon\mathcal{H}_{\Omega}\rightarrow\mathcal{H}_{\Omega}$
defined through its action
$S_{\Omega}|a\rangle_{\Omega}=|a^{*}\rangle_{\Omega}$ for all
$a\in\mathfrak{F}$, and are therefore completely canonical. We may also
consider the _modular generator_ $D_{\Omega}:=-\ln\Delta_{\Omega}$, which is a
self-adjoint operator affiliated with $B(\mathcal{H}_{\Omega})$. The modular
generator annihilates the cyclic vector,
$D_{\Omega}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=0$, and therefore
generates a continuous physical symmetry. In particular, $D_{\Omega}$
generates the one-parameter group represented by the unitaries
$\Delta_{\Omega}^{it}\equiv e^{-itD_{\Omega}}$. The spectrum of $D_{\Omega}$
is always symmetric with respect to zero, and therefore it cannot be directly
interpreted as the Hamiltonian of the system. For the GNS representation
induced by a thermal state of a finite-dimensional quantum system,
$D_{\Omega}$ in fact corresponds to the Liouville operator
$L_{H}(A)\equiv[H,A]$, where $H$ is the Hamiltonian of the system. On the
other hand, $D_{\Omega}$ is not always affiliated with the observable algebra
for infinite-dimensional systems, because the unitaries
$\Delta_{\Omega}^{it}\in B(\mathcal{H}_{\Omega})$ do not belong to
$\mathfrak{A}_{\Omega}$ for all $t\in\mathbb{R}$ (i.e., they induce outer
automorphisms of $\mathfrak{A}_{\Omega}$), and therefore cannot be
approximated by physical measurements.171717Note that this is, however, in a
qualitative agreement with general relativity, where there does not exist a
general globally defined observable for the total energy of a system [89].

Connes and Rovelli [27] have suggested to consider the one-parameter group of
automorphisms given by the modular flow
$\sigma^{\Omega}\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A}_{\Omega})$
as the physical time-evolution for a background-independent quantum system –
the so-called _thermal time hypothesis_. Indeed, $\sigma^{\Omega}$ gives the
unique dynamics on $\mathfrak{A}_{\Omega}$ with respect to which the extended
reference state $\tilde{\Omega}$ is in equilibrium. We will apply this idea in
the spacetime-free framework in order to recover ‘thermal’ unitary dynamics
for a quantum system in the case that the extended reference state
$\tilde{\Omega}$ may represent equilibrium, i.e., when it is faithful on the
physical observable algebra $\mathfrak{A}_{\Omega}$.

There are a few apparent challenges to the thermal time hypothesis: (1) To
begin with, a pure state does not induce a non-trivial modular structure, and
therefore the vacuum state of QFT cannot give rise to global dynamics. This
problem can be overcome by noting that the restriction of the vacuum onto a
subregion of spacetime gives rise to a thermal state that does induce non-
trivial modular dynamics. For example, the restriction of the Minkowski vacuum
of a free neutral scalar field onto a half-space is well-known (by the
Bisognano–Wichmann theorem) to give rise to a modular flow that is given by
the one-parameter group of Lorentz boosts that preserve the corresponding
Rindler wedge [15]. In this case, the integral curves of the flow correspond
to the worldlines of accelerated observers, for whom the boundary of the
Rindler wedge is an apparent causal horizon. Correspondingly, the modular
generator is the generator of the proper time evolution of these observers, so
the modular flow is indeed seen to give the time-evolution of a particular
class of observers. In the universe we inhabit, no localized observer can
access all the degrees of freedom in the universe, but only her causal past
(the ‘observable universe’), so the restriction of the global pure state even
for an inertial observer can be physically justified in this way. In addition,
our universe is filled with thermal cosmic background radiation. Remarkably,
Rovelli [62] has shown that the thermal dynamics induced by the statistical
state describing cosmic background radiation in the Robertson–Walker spacetime
agrees with the usual cosmological dynamics with respect to the
Robertson–Walker time. (2) Secondly, for QFT on curved spacetime there do not
exist any equilibrium states, unless the background spacetime is static [71,
90], implying that for the majority of spacetimes we cannot obtain the
spacetime structure from a thermal state. Actually, this seemingly problematic
point is consistent with our point of view, according to which the spacetime
structure is determined by the reference state: Clearly, for the effective
spacetime geometry to be static, the reference state must be suitably
invariant under the dynamics.181818The exact form of the invariance depends on
the way that the effective spacetime geometry depends on the reference state.
However, the spacetime-free framework also applies to the case where the
reference state cannot represent equilibrium, although in this case we cannot
recover thermal dynamics, and we must use other properties of the system to
extract the effective spacetime geometry. (3) Thirdly, the modular flow for
QFT states does not in fact correspond generically to the time-evolution of
the system [15]. In a few cases, such as the vacuum state restricted to a
Rindler wedge in Minkowski spacetime, the modular flow is seen to be related
to time-evolution, but in most known cases the action of the modular flow is
non-local with respect to the background geometry. We might interpret this as
signaling the incompatibility of such a state to act as a reference state for
the particular background spacetime geometry, as in the spacetime-free
formulation the state should probably give rise to a different effective
background geometry (if any), with respect to which the dynamics are local. In
fact, in Section 3 we will explore the idea that a notion of locality for the
effective spacetime structure can be _defined_ by the requirement that the
thermal dynamics induced by the reference state are local.

Finally, let us also mention that the modular involution $J_{\Omega}$ for the
vacuum state restricted to Rindler wedges in Minkowski spacetime is known to
have a physical interpretation as a combination of the CPT operator and a
spatial rotation for relativistic QFT models that satisfy certain general
algebraic requirements [15]. Accordingly, the existence of the modular
involution is strongly related to Lorentz invariance through the CPT theorem
and the symmetry between matter and antimatter (i.e., retarded and advanced
solutions to the equations of motion), since the CPT transformation in QFT
maps particles to antiparticles, and vice versa [38]. Indeed, taking advantage
of the relation between the modular involutions and CPT transformations, it
has been shown in [22, 78, 91] that the modular involutions induced by the QFT
vacuum restricted to Rindler wedges can be used to generate a representation
of the proper Poincaré group, thus recovering spacetime structure from the
purely algebraic data of a vacuum state and a suitable family of subalgebras
of observables. However, it is unclear whether this method for deriving
spacetime from algebraic data can be extended to less symmetric situations.

###  2.5 Perturbations of the reference state

The extended reference state $\tilde{\Omega}$ may be perturbed by a finite
collection of operators $b_{k}\in B(\mathcal{H}_{\Omega})$ by defining the
perturbed reference state as

$\displaystyle\tilde{\Omega}^{\prime}(a):=N^{-1}\sum_{k}\tilde{\Omega}(b_{k}^{*}ab_{k})$

for all $a\in\mathfrak{A}_{\Omega}$, where
$N:=\sum_{k}\tilde{\Omega}(b_{k}^{*}b_{k})\in\mathbb{R}_{+}$ is a
normalization constant (assumed to be nonzero). Such a perturbed state lies in
the folium of the reference state as it is represented by the density
operator191919The folium of a state $\omega$ consists of those states, which
can be represented by density operators in $B(\mathcal{H}_{\omega})$ [38].

$\displaystyle\rho_{\tilde{\Omega}^{\prime}}:=N^{-1}\sum_{k}b_{k}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}\langle\mathbbm{1}_{\mathfrak{F}}|b_{k}^{*}\in
B(\mathcal{H}_{\Omega}).$

The perturbed extended reference state $\tilde{\Omega}^{\prime}$ gives rise to
a perturbed state $\Omega^{\prime}$ on the free observable algebra
$\mathfrak{F}$ by its restriction onto $\pi_{\Omega}(\mathfrak{F})\subset
B(\mathcal{H}_{\Omega})$.

We may then consider the GNS representation
$\pi_{\Omega^{\prime}}\colon\mathfrak{F}\rightarrow
B(\mathcal{H}_{\Omega^{\prime}})$ of $\mathfrak{F}$ induced by such a
perturbed reference state that gives rise to the perturbed physical observable
algebra
$\mathfrak{A}_{\Omega^{\prime}}\\!:=\\!\pi_{\Omega^{\prime}}(\mathfrak{F})^{\prime\prime}$.
In this way, perturbations of the state of the system may affect the algebraic
as well as the statistical relations of the physical observables, which we
suspect may be associated with the change of the effective spacetime structure
and thus gravitational effects. Importantly, we always have
$\ker\pi_{\Omega}\subset\ker\pi_{\Omega^{\prime}}$ for perturbations of
$\tilde{\Omega}$ by elements in $B(\mathcal{H}_{\Omega})$, which implies that
such perturbations cannot destroy the joint measurability of observables in
$\mathfrak{F}$: If $[a,a^{\prime}]\in\ker\pi_{\Omega}$ for some
$a,a^{\prime}\in\mathfrak{F}$, then also
$[a,a^{\prime}]\in\ker\pi_{\Omega^{\prime}}$. This is very important from the
physical point of view, because otherwise perturbations to the system could
completely change the physical properties (e.g., the causality structure) of
the system. Notice that $\ker\pi_{\Omega}\subset\ker\pi_{\Omega^{\prime}}$ can
be a proper inclusion only if $\pi_{\Omega}$ is reducible. Also, the perturbed
state $\tilde{\Omega}^{\prime}$ cannot be faithful on the unperturbed physical
observable algebra $\mathfrak{A}_{\Omega}$. (However,
$\tilde{\Omega}^{\prime}$ may still be faithful on
$\mathfrak{A}_{\Omega^{\prime}}$.) We take the physical states of the original
system to inhabit the folium of the reference state. Accordingly, we will call
$B(\mathcal{H}_{\Omega})$ the _perturbation algebra_ , as it induces
perturbations of the reference state. In ordinary QFT, the quantum field
operators belong to the perturbation algebra.202020Actually, the smeared field
operators in QFT should correspond to unbounded operators affiliated with
$B(\mathcal{H}_{\Omega})$, but we may consider the exponentiated (Weyl) field
operators, which are bounded. The original unbounded field operators are
recovered as generators of strongly continuous one-parameter groups of
unitaries in $B(\mathcal{H}_{\Omega})$. See [38, 41] for the algebraic
construction of quantum fields from the representations of the observable
algebra. As for the field operator algebra in algebraic QFT, the perturbation
algebra provides an extension of the physical observable algebra.

The action of any physical symmetry implemented by a unitary
$U\in\mathcal{U}_{\mathfrak{F}}^{\Omega}$ on the density operators as
$\rho_{\tilde{\Omega}^{\prime}}\mapsto U\rho_{\tilde{\Omega}^{\prime}}U^{*}\in
B(\mathcal{H}_{\Omega})$ maps the folium of the reference state to itself.
Therefore, the perturbations carry a representation theory of the group of
physical symmetries (although it may very complicated in general).

In Appendix B we propose a definition for the mass of a static perturbation
(based on the Connes cocycle derivative), and argue that perturbations whose
mass is positive according to this definition always render some new
observables jointly measurable (i.e., mutually commutative), which we suggest
is reminicient of lightcone focusing by gravity. We have delegated this rather
technical and provisional discussion to an appendix in order to make the
presentation more streamlined for the benefit of the reader.

This finishes our sketch of a spacetime-free framework for quantum physics. In
the next section we will consider some ways to recover information about the
possible effective spacetime structure induced by the reference state and its
perturbations. However, before that we will briefly discuss the connection of
the spacetime-free framework to the usual (algebraic) formulation of quantum
field theory. Also, for the simplest concrete examples of the above
construction, see Appendix C.

###  2.6 Relation of the spacetime-free framework to quantum field theory

The above formulation of the spacetime-free framework in terms of the free
observable algebra is rather reminicent of the construction of the
Borchers–Uhlmann algebra in algebraic quantum field theory [14]. In short, the
Borchers–Uhlmann algebra is obtained as the free tensor algebra over the
vector space of Schwarz functions on Minkowski spacetime. Then, a state
specified by the Wightman functionals imposes non-trivial algebraic relations
between the elements of the Borchers–Uhlmann algebra, which encode the
dynamics of the model. There are at least two important differences between
the two constructions:

  * •

Unlike the free observable algebra in the spacetime-free framework, the
Borchers–Uhlmann algebra is constructed on top of a fixed background spacetime
geometry, which is obviously antithetical to the goals of our approach.

  * •

In the case of the Borchers–Uhlmann algebra, the elements of the free tensor
product are functions in spacetime, whereas in our case we constructed the
free product of abelian von Neumann algebras generated by self-adjoint
operators. This reflects a difference in the physical interpretation of the
elements of the construction. Namely, the elements of the Borchers–Uhlmann
algebra are not necessarily physical observables, but correspond to localized
fields in the absence of dynamical relations, i.e., they are ‘kinematical’
operators. In our case, the requirement that the initial algebras are
faithfully represented through the GNS representation implies that they
correspond to physical observables.

The common key idea between the two formalisms is, however, that the quantum
state imposes dynamical relations on the algebra of observables. Indeed, in
some cases the formalisms seem to be closely related. In particular, we could
restrict to consider some subset of operators of the Borchers–Uhlmann algebra
that is generated by functions localized to Cauchy surfaces, so that they are
in fact physical observables. Then, the Wightman functionals would provide the
reference state encoding the statistics of observations for these observables.
However, the exact details of the relationship remain to be worked out.

We would also like to point out one possible confusion that may arise from
quantum field theory concerning the choice of the reference state. In ordinary
QFT most states on the observable algebra are considered unphysical – for
example, those not satisfying the Hadamard condition [5]. However, such
criteria for physical states usually rely on the background spacetime
structure, and in effect guarantee that the state is compatible in some way
with the fixed background geometry. In the absence of a background geometry,
however, these criteria are inapplicable, so how are we able to distinguish
the physical states from the unphysical ones? We would like to point out that
a state that appears wildly unphysical with respect to some fixed background
geometry, may in fact be well-behaved with respect to the effective background
geometry, to which it gives rise. On the other hand, when the background
spacetime is not restricting the symmetries of the system, the equivalence
classes of states describing the same physics should be much larger than with
a fixed background spacetime. In particular, as we observed above, any
covariant transformation
$(\mathfrak{F},\Omega)\mapsto(\alpha(\mathfrak{F}),\alpha^{*}\Omega)$ given by
a $*$-automorphism $\alpha\in\textrm{Aut}(\mathfrak{F})$ leads to the same
physical description of the quantum system in the spacetime-free formulation,
whereas if the observables were labeled by some form of background spacetime
information (e.g., spacetime regions) from the beginning, such transformations
would not in general correspond to geometric transformations of the background
spacetime structure. Therefore, the physics described by the spacetime-free
framework may be more unique than that of ordinary QFT.

##  3 Recovering effective spacetime structure

In this section we will develop some ideas and methods for the extraction of
spacetime structure from the spacetime-free framework, which is clearly
necessary in order to connect the theory with known physics and experiments.
In fact, the reconstruction of spacetime has been considered before in the
context of algebraic quantum field theory in the literature. The earliest
works we have found addressing the issue are [4, 51], where the inverse
problem of recovering spacetime topology and causal structure from the net of
local observable algebras is considered. On the other hand, [22, 78, 91] show
that it is possible to recover the symmetry group of spacetime from the
modular structure of subalgebras in some highly symmetric cases. These results
already show that quite a lot of information about spacetime structure is
encoded into quantum field theory. But they also carry the contradiction
within them that the original formulation of quantum field theory relies on a
background spacetime, which is what we try to remedy in this work.

###  3.1 Locality from the dynamical properties of subalgebras

By locality we refer to the existence and the identification of local
subsystems. We may distinguish the following two notions of locality that are
_a priori_ independent212121Similar distinction between notions of locality
has been made before, e.g., in [34].:

  * •

_Locality with respect to a background geometry._ The topology of the
background manifold gives rise to a geometrical notion of local spacetime
regions and the corresponding localized subsystems in the usual formulations
of QFT.

  * •

_Locality with respect to the dynamics._ The dynamics may give rise to an
operational notion of locality, which can be determined by studying the
evolution of matter systems, e.g., the propagation of excitations over the
vacuum.

The choice of dynamics in field theory is usually guided by the principle of
locality, which can be understood as the requirement that local subsystems
interact with the rest of the system only at the boundary of the corresponding
local spacetime region. The requirement of locality for interactions with
respect to the background geometry ensures that the dynamical notion of
locality agrees with the geometrical notion of locality. In the absence of a
background geometry, one must solely rely on the dynamical notion of locality,
and thus base the definition of locality on the dynamical properties of
subsystems. In particular, we suggest to study the propagation of causal
influences as encoded into the commutation relations between
observables.222222On the other hand, let us emphasize that there may also
exist more algebraic procedures to identify local subalgebras of observables,
such as the method of modular localization [21] for free quantum field theory
on Minkowski spacetime. However, modular localization relies heavily on the
symmetry properties of spacetime and the Fock space structure, and therefore
is not directly applicable to generic spacetimes or interacting theories.

In algebraic QFT, observable algebras $\mathfrak{A}(\mathcal{O})$ are
associated to spacetime regions $\mathcal{O}$. Typically, the observable
algebras associated with local spacetime regions are von Neumann subfactors
(i.e., $\mathfrak{A}(\mathcal{O})^{\prime\prime}=\mathfrak{A}(\mathcal{O})$
and
$\mathfrak{A}(\mathcal{O})\cap\mathfrak{A}(\mathcal{O})^{\prime}\cong\mathbb{C}$)
of the total observable algebra. Moreover, many of the models satisfy the
_Haag duality_ :
$\mathfrak{A}(\mathcal{O}_{c})=\mathfrak{A}(\mathcal{O})^{\prime}$, where
$\mathcal{O}_{c}$ denotes the causal complement to the region $\mathcal{O}$
[38]. These properties can be physically motivated by the joint measurability
of commuting observables. Specifically, the commutant
$\mathfrak{A}(\mathcal{O})^{\prime}$ contains all the observables that are
jointly measurable with all the observables in $\mathfrak{A}(\mathcal{O})$
and, therefore, describes the rest of the degrees of freedom in addition to
those in $\mathfrak{A}(\mathcal{O})$ that are necessary to completely specify
the quantum state of the system. As an example, in QFT for the free scalar
field, the field values on any Cauchy surface determine a pure state of the
system, as the corresponding field operators form a maximal abelian subalgebra
of the physical observable algebra, i.e., a complete set of jointly measurable
observables. We may then think of $\mathfrak{A}(\mathcal{O})$ and
$\mathfrak{A}(\mathcal{O})^{\prime}$ as splitting some Cauchy surface in two
parts, as any maximal abelian subalgebra is correspondingly split between the
two algebras, each of them describing the degrees of freedom associated to one
of the two complementary parts of the Cauchy surface. As the joint
measurability of observables and the maximal abelian subalgebras retain their
physical interpretation even in the absence of a background spacetime, we will
require any subalgebra of observables
$\mathfrak{B}\subset\mathfrak{A}_{\Omega}$ corresponding to a subsystem to be
a von Neumann subfactor, and we take the relative commutant
$\mathfrak{B}^{\prime}=\\{b^{\prime}\in\mathfrak{A}_{\Omega}\colon[b,b^{\prime}]=0\,\forall\,b\in\mathfrak{B}\\}$
to represent the causal complement, or the environment, to the subsystem
represented by $\mathfrak{B}$.

The propagation of causal influences manifests in how the thermal dynamics
induced by the modular flow
$\sigma^{\Omega}\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A}_{\Omega})$
affects the commutators between different subalgebras of observables. In
particular, the evolution
$\mathfrak{B}\mapsto\sigma^{\Omega}_{t}(\mathfrak{B})$ tends to ‘scramble’ the
local subalgebras of observables $\mathfrak{B}\subset\mathfrak{A}_{\Omega}$.
Let $\mathfrak{B}_{1},\mathfrak{B}_{2}\subset\mathfrak{A}_{\Omega}$ be two
subfactors such that $\mathfrak{B}_{1}\subset\mathfrak{B}_{2}^{\prime}$. We
may then consider the commutators $[\sigma^{\Omega}_{t}(b_{1}),b_{2}]$ for
different $t\in\mathbb{R}$ and $b_{1}\in\mathfrak{B}_{1}$,
$b_{2}\in\mathfrak{B}_{2}$, and their (normalized weak operator) norms

$\displaystyle\frac{\|[\sigma^{\Omega}_{t}(b_{1}),b_{2}]\|}{2\|b_{1}\|\|b_{2}\|}\in[0,1].$

Clearly, at $t=0$ the norm vanishes, and its growth away from $t=0$ for some
elements $b_{1}\in\mathfrak{B}_{1}$ and $b_{2}\in\mathfrak{B}_{2}$ indicates
the necessity of some causal relations forming between the degrees of freedom
associated to the two subfactors. In local QFT, for two field subalgebras that
are local and finitely spacelike separated, the commutator for any pair of
elements remains zero for a finite interval around $t=0$, if the Einstein
causality property is satisfied (i.e., propagation happens inside
lightcones).232323A model in algebraic QFT is said to satisfy Einstein
causality if $\mathfrak{B}_{1}\subset\mathfrak{B}_{2}^{\prime}$ whenever
$\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$ are subalgebras associated with
spacelike separated spacetime regions [38]. Even if Einstein causality is
violated by the model (e.g., for non-relativistic systems such as spin
chains), the derivative

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}[\sigma^{\Omega}_{t}(b_{1}),b_{2}]=i[[D_{\Omega},b_{1}],b_{2}]$

still vanishes, if the thermal dynamics does not directly couple
$\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$. Therefore, we see that the behavior
of the quantity

$\displaystyle
C^{\Omega}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2}):=\sup\left\\{\frac{\|[\sigma^{\Omega}_{t}(b_{1}),b_{2}]\|}{2\|b_{1}\|\|b_{2}\|}\in[0,1]\colon
b_{1}\in\mathfrak{B}_{1},\ b_{2}\in\mathfrak{B}_{2}\right\\}\in[0,1]$

as a function of $t$ encodes information on the dynamical coupling of
subfactors $\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$. In particular, its
growth can in some cases be used to estimate the strength of direct dynamical
coupling between $\mathfrak{B}_{1}$ and $\mathfrak{B}_{2}$. Our primary
suggestion for a strategy to recover the operational topology of the
experimental system is based on studying the magnitude and growth of such
commutators to estimate the strength of causal dependence between different
subsystems.

It is possible that for some relativistic infinite-dimensional systems
$C^{\Omega}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2})$ only takes values 0 and 1.
In this case, we may extract information on the spatial distance between
subsystems from the value of the parameter $t$, for which the value of
$C^{\Omega}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2})$ changes from 0 to 1. In
the following, however, we will consider the case that
$C^{\Omega}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2})$ behaves continuously in
$t$ (e.g., for finite-dimensional systems). In this case, we would like to
identify the local subsystems as those represented by subfactors
$\mathfrak{B}\subset\mathfrak{A}_{\Omega}$, which are most weakly coupled to
their commutant $\mathfrak{B}^{\prime}$, representing the causal complement to
the subsystem. This definition is motivated by the principle of locality as
interpreted above: Local subsystems interact with their environment only
through their boundary. In this sense, local subsystems are the most robust
against influences from their environment. The strength of the dynamical
coupling between a subsystem represented by $\mathfrak{B}$ and its causal
complement $\mathfrak{B}^{\prime}$ can be estimated by the growth of
$C^{\Omega}_{t}(\mathfrak{B},\mathfrak{B}^{\prime})$ at $t=0$. In particular,
we expect
$\frac{\textrm{d}}{\textrm{d}t}C_{t}^{\Omega}(\mathfrak{B},\mathfrak{B}^{\prime})$
at $t=0$ to be minimized by the subset of _local spherical_ subfactors in any
set of subfactors
$\\{\alpha(\mathfrak{B}_{0})\subset\mathfrak{A}_{\Omega}\colon\mathfrak{B}_{0}\subset\mathfrak{A}_{\Omega},\,\alpha\in\textrm{Aut}(\mathfrak{A}_{\Omega})\\}$
connected by automorphisms of $\mathfrak{A}_{\Omega}$. By the term ‘spherical’
we take into account that the magnitude of
$\frac{\textrm{d}}{\textrm{d}t}C_{t}^{\Omega}(\mathfrak{B},\mathfrak{B}^{\prime})$
may also depend on the size of the boundary of the spatial region associated
with $\mathfrak{B}$. The totality of local subfactors can then be obtained as
the net of subfactors generated by the local spherical ones. The local
subfactors are determined only up to the symmetry group of the reference
state, since $C_{t}^{\Omega}(\mathfrak{B},\mathfrak{B}^{\prime})$ is invariant
under symmetry transformations. In other words, a local subfactor is mapped to
a local subfactor by the symmetry transformations.

Ultimately, to justify the above definition of local subsystems, we should be
able to show that the growth of
$C^{\Omega}_{t}(\mathfrak{B},\mathfrak{B}^{\prime})$ (or a quantity similar to
it) is indeed minimized by a class of local subsystems in the usual
formulation of local QFT (or some finite-dimensional regularization thereof).
Unfortunately, $C^{\Omega}_{t}(\mathfrak{B},\mathfrak{B}^{\prime})$ is rather
difficult to compute explicitly due to the supremum over algebra elements, and
thus this definition remains largely a hypothesis or a suggestion for now.
However, in Appendix D we study the behavior of a similar quantity measuring
the magnitude of commutators for finite spin lattices, which is straighforward
to compute. We have verified that the growth of this quantity is indeed
minimized for certain local subalgebras, which offers at least some
plausibility for the above definition. Nevertheless, the full verification of
the hypothesis is left for future work.

###  3.2 Metric information from the readings of quantum clocks

In the previous subsection, we proposed how to identify the local perturbation
subalgebras, and thus define a notion of locality for the system. In this
subsection, we will briefly consider some preliminary ideas on how to recover
metric data. As the starting point of our considerations, we take the view
that the operational geometry is determined by the readings of
clocks.242424The recent work [24] by Cao, Carroll and Michalakis nicely
complements our approach by showing that one can reconstruct spatial metric
information from the mutual information shared by different subsystems.

Quantum clock systems have been considered before mainly in the context of
non-relativistic quantum mechanics (see, e.g., [36, 61, 65, 83]). It was found
that quantum effects impose inherent restrictions on the accuracy of
measurements of duration and distance. Our treatment of quantum clocks differs
quite significantly from the earlier works, since we work in the spacetime-
free framework for quantum physics as formulated in Section 2. Nevertheless,
similar restrictions are expected to be valid due to uncertainty relations,
which still arise from the non-commutativity of observables.

Let us also mention, although we will not explore this option any further
here, that it may be possible to apply non-commutative geometric methods à la
Connes to recover metric information [26, 85]. Indeed, the modular structure
of von Neumann algebras is remarkably similar to the definition of a spectral
triple, which is a natural non-commutative generalization of a metric space
[9, 10]. However, the modular generator, unlike the usual Dirac operator,
operates in the ‘quantum phase space’ and not on spacetime, _per se_.
Therefore, we expect the spectral geometry induced by the modular structure to
describe the non-commutative phase space geometry rather than directly
spacetime. The relation between the two can be quite complicated in field
theory. Accordingly, we consider it a more feasible first take on the problem
of obtaining spacetime metric data to study the evolution of quantum clock
systems.

Let us again assume that the reference state $\tilde{\Omega}$ is faithful on
the physical observable algebra $\mathfrak{A}_{\Omega}$, and therefore gives
rise to the thermal dynamics. Since there is no physical evolution in
equilibrium, we must consider perturbations to the reference state. A
perturbation $\tilde{\Omega}^{\prime}$ of the reference state $\tilde{\Omega}$
is localized to a local subsystem represented by a local observable subalgebra
$\mathfrak{B}\subset\mathfrak{A}_{\Omega}$, if it satisfies
$\tilde{\Omega}^{\prime}(b^{\prime})=\tilde{\Omega}(b^{\prime})$ for all
$b^{\prime}\in\mathfrak{B}^{\prime}$. A simple clock system can be taken to
consist of a (possibly approximately) local perturbation
$\tilde{\Omega}^{\prime}$ of the reference state and an associated observable
(i.e., a self-adjoint operator) $\gamma\in\mathfrak{A}_{\Omega^{\prime}}$
representing the reading of the clock. The evolution of a perturbation
$\tilde{\Omega}^{\prime}$ relative to the reference state $\tilde{\Omega}$ is
given by the modular flow
$\sigma^{\Omega}\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A}_{\Omega})$
as explained in Section 2.4. We may then require that the expectation value

$\displaystyle\langle\gamma\rangle_{\Omega^{\prime},t}:=\tilde{\Omega}^{\prime}\big{(}\sigma^{\Omega}_{t}(\gamma)\big{)}\in\mathbb{R}$

of the clock reading increases monotonically during the evolution of the
system, which suggests the requirement
$i\tilde{\Omega}^{\prime}\big{(}[D_{\Omega},\gamma]\big{)}>0$ for the pair
$(\tilde{\Omega}^{\prime},\gamma)$ specifying the clock. If such an observable
$\gamma$ is found, we may always further scale $\gamma\mapsto r\gamma$ by a
positive real number $r\in\mathbb{R}_{+}$, so that
$i\tilde{\Omega}^{\prime}\big{(}[D_{\Omega},\gamma]\big{)}=1$. Adopting the
relativistic terminology, we may say that such an observable $\gamma$ measures
the _proper time_ of the perturbation $\tilde{\Omega}^{\prime}$. For the clock
to be of practical use, we should also require the distribution of the value
of $\gamma$ to be peaked around its expectation value.

Now, let $\mathfrak{B}_{1},\mathfrak{B}_{2}\subset B(\mathcal{H}_{\Omega})$ be
two subalgebras of perturbations localized in different local subsystems. To
measure the proper time that it takes for a clock to travel between the two
subsystems, we must find a clock system that propagates from one to the other.
Let $\tilde{\Omega}^{\prime}$ be a perturbation induced from the reference
state $\tilde{\Omega}$ by an operator $b\in B(\mathcal{H}_{\Omega})$ such that
$\Delta_{\Omega}^{-it_{1}}b\Delta_{\Omega}^{it_{1}}\in\mathfrak{B}_{1}$ and
$\Delta_{\Omega}^{-it_{2}}b\Delta_{\Omega}^{it_{2}}\in\mathfrak{B}_{2}$ for
some $t_{1},t_{2}\in\mathbb{R}$, $t_{1}<t_{2}$. Then, we could define for the
clock pair $(\tilde{\Omega}^{\prime},\gamma)$ the expectation value of the
measured time between the two local spacetime regions as the difference
$\tilde{\Omega}^{\prime}\big{(}\sigma^{\Omega}_{t_{2}}(\gamma)-\sigma^{\Omega}_{t_{1}}(\gamma)\big{)}$
in the clock readings at the two times.

We suggest that it is possible to recover the effective metric relationships
between local spacetime regions defined by the local perturbation subalgebras
by studying the evolution of such quantum clock systems. Of course, in
general, one must use several clock systems simultaneously, in which case one
needs to find several local perturbation operators $b_{i}\in
B(\mathcal{H}_{\Omega})$, $i=1,\ldots,n$ for some $n\in\mathbb{N}$, which
evolve approximately independently from each other, propagating between local
subalgebras of $B(\mathcal{H}_{\Omega})$, and a corresponding set of
(approximately) mutually commuting observables
$\gamma_{i}\in\mathfrak{A}_{\Omega^{\prime}}$, $i=1,\ldots,n$, where
$\tilde{\Omega}^{\prime}$ is the reference state perturbed by each of
$b_{i}\in B(\mathcal{H}_{\Omega})$. Finding a family of such perturbations,
which determines the spacetime geometry to a satisfying accuracy, is
undoubtedly a highly non-trivial task. Whether such families of quantum clock
systems can actually be found for some systems remains an open theoretical
question at the moment, although it appears to us that we use exactly such
systems to determine spacetime structure in practice. We will leave the
further development of these ideas to future work.

###  3.3 Perturbations of the effective spacetime structure and gravity

Finally, we wish to mention a couple of ideas concerning the relationship
between the perturbations of the reference state and gravitational phenomena.
We saw in Section 2 how perturbations of the reference state may change the
causal relations of physical observables by altering the GNS representation of
the free observable algebra. In Section 2 (and Appendix B) we argued that
perturbations with positive mass always causally decouple some observables,
which seems at least suggestive of gravitational phenomena. It is evident that
perturbations of the reference state also alter the effective geometric
properties of the quantum system as defined in this section, but we have not
studied the exact nature of such perturbations so far.

Let us also mention the recent work [48] by Jacobson, in which he derives the
semiclassical Einstein equation from the hypothesis that the QFT vacuum state
restricted locally to small causal diamonds maximizes the (von Neumann)
entanglement entropy. The derivation points to another way of relating
perturbations of the reference state (in this case the vacuum) to gravity in
terms of quantum statistics instead of representation theory. However, in
Jacobson’s derivation the division of entropy into UV (high energy) and IR
(low energy) parts plays an important role: The entropy associated to the UV
degrees of freedom leads via the area law to the geometric part of the
Einstein equation, while the IR entropy is related to the matter energy-
momentum tensor. It is argued that the simultaneous perturbation of the two
vanishes for local subsystems due to the maximization of the total entropy by
the vacuum state, and out comes the Einstein equation. In our formalism it is
not clear how to divide the degrees of freedom into the UV and IR parts. In
the presence of an equilibrium state we do have a notion of energy provided by
the eigenvalues of the modular operator, but the observable algebra does not
generally factorize into high and low energy parts. We could require of the
reference state for such factorization to hold, so that low and high energy
degrees of freedom are (approximately) decoupled. However, we are also able to
do without such an assumption if the total entanglement entropy for the
restriction of the vacuum onto local subsystems satisfies the area law, i.e.,
its leading contribution is proportional to the area of the boundary of the
local subsystem252525The entanglement entropy is usually divergent in QFT. To
make it finite, some kind of a regularization must be introduced.: The ‘first
law of entanglement entropy’ [12] implies for first order variations of a
thermal equilibrium state that $\delta S=\delta\langle H_{\Omega}\rangle$,
where $S$ is the entanglement entropy and $\langle H_{\Omega}\rangle$ is the
expectation value of the (modular) Hamiltonian induced by the reference state
$\Omega$.262626Here, the modular Hamiltonian is obtained as the generator of
the modular flow when the modular flow is given by inner automorphisms, i.e.,
can be induced by the adjoint action of unitaries in the physical observable
algebra. We are not aware of the extension of the entanglement first law to
the general case, where the generator of the time-evolution does not belong to
the observable algebra. From the entanglement first law we may derive the
semiclassical (linearized) Einstein equation for ball shaped spatial regions
by relating the entropy to the spacetime geometry via the area law and the
modular Hamiltonian to the energy-momentum tensor exactly as in the derivation
[48] of Jacobson. This idea has also been considered before in the context of
AdS/CFT correspondence (see, e.g., [53]). To apply it in the spacetime-free
framework, we should require that the reference state is in thermal
equilibrium when restricted to local subsystems. Then, if we can relate
spacetime geometry to the entanglement entropy via the area law, this
derivation shows that Einstein gravity could naturally arise from the quantum
statistics, at least in its linearized form.

Unfortunately, this is as far as our understanding of the exact relationship
between perturbations and gravity extends at the moment. Indeed, showing that
the perturbations to the effective spacetime structure induced by
perturbations of the reference state (e.g., a local restriction of Minkowski
vacuum or the cosmological thermal state) correspond to gravitational effects
is perhaps the most important and exciting prospect of our current research.

##  4 Conclusion

###  4.1 Summary of results

In Section 2 we formulated a spacetime-free algebraic framework for describing
experimental arrangements with quantum systems. The key motivation for the
construction was the observation that the quantum state should be able to
alter the causal relations of observables in order to incorporate
gravitational phenomena into quantum (field) theory. Accordingly, the starting
point for the formulation was taken to be the _free observable algebra_
obtained as the free product $*$-algebra of the abelian von Neumann algebras
associated to individual measurements. The other component required to define
an experimental arrangement was identified as the _reference state_ describing
the statistical background to measurements. We then explained how the
reference state may impose causal relations and, in some cases, also time-
evolution on the observables through the GNS representation of the free
observable algebra and its modular structure. In particular, the physical
observable algebra embodying the dynamical relations was obtained as (the norm
completion of) the image of the GNS representation of the free observable
algebra induced by the reference state. We also defined the concepts of
covariance and symmetry in the framework. Finally, we considered the
description of perturbations to the reference state and, in particular,
defined the algebra of operators that may be used to induce perturbations. We
observed that perturbations of the reference state may causally decouple
observables, which was argued to be suggestive of gravitational effects.

In Section 3 we considered some methods to extract effective topological and
geometric structure from the dynamics of perturbations induced by the
reference state. We suggested basing the operational definition of locality on
the algebraic and dynamical properties of subalgebras of observables. In
particular, local subalgebras were defined as those that are most weakly
coupled with their causal complements. Moreover, we argued operational metric
information to be obtainable from the readings of quantum clock systems, which
we defined. Unfortunately, our discussion had to remain mostly at the level of
hypotheses, although these hypotheses seem rather compelling and physically
well-motivated to us. In any case, we would like to emphasize that there must
undoubtedly exist _some_ way to recover topological and geometric information
about spacetime from the dynamics of a quantum (field) system, since it is
through the dynamical behavior of matter systems that we determine the
structure of spacetime in practice. In the present work, we have proposed only
a couple of methods, but other more appropriate and practical ones may exist.

Undoubtedly, the most significant aspect of the present work is the
formulation of the spacetime-free framework for quantum theory that led to the
realization of a mechanism for the quantum state to influence the causal
relations and the time-evolution of a quantum system in the spacetime-free
framework. The mechanism may allow for the appearance of gravitational
phenomena associated to perturbations of the reference state, such as the
vacuum, although the exact nature of these effects remains to be studied more
carefully.

###  4.2 Challenges to the approach

Perhaps the most immediate and fundamental challenge for the physical
interpretation of the spacetime-free framework for quantum physics, as we have
presented it in Section 2, is the question of how to determine the reference
state. We explained the operational meaning of the reference state in terms of
the statistical background to measurements, but in most situations it is
clearly not possible to determine the reference state of a system
experimentally with sufficient accuracy. The problem is most obvious in the
case of cosmology, which is partly why we restricted our language to the
description of controlled laboratory experiments (the other reason being
conceptual clarity). We would like to emphasize that the reference state
completely determines the model we have of a system. Therefore, in a sense,
the question is like asking for the origin of the gauge group or the equations
of motion in field theory. It may be that the reference state simply has to be
a theoretical input to the model in the cases, where it cannot be
experimentally deduced. On the other hand, there may exist some universality
argument imposing restrictions on the reference state for cosmology arising
from, for example, a renormalization property. In particular, it appears
reasonable to expect that the ‘universal’ reference state should converge to a
pure vacuum state in the limit of the whole universe, whereas in the opposite
limit of Planckian systems it should presumably converge to a trace,
describing a maximally symmetric infinite temperature state. There may exist a
natural choice of a flow from one to the other. Maybe it is also worth noting
that no physical observer actually has an access to all the observables of the
universe, but different observers can access different subsets of the set of
all observables. (See, e.g., [39] for a realization of such a situation in
ordinary QFT.) The restriction of the universal quantum state onto the
subalgebra of observables accessible to a single observer will introduce non-
trivial evolution on the subalgebra, whose properties may be relevant for
identifying the appropriate universal state.

Another obvious challenge concerns the recovery of spacetime structure and
gravity. As we already emphasized, on physical grounds we expect that there
should exist an operational way to recover spacetime structure from the
dynamics of a quantum system. Whether the ideas we put forward in this work
are up to the task remains to be verified. Obviously, to determine if the
perturbations of the effective spacetime structure induced by the
perturbations of the reference state give rise to gravitational phenomena, one
must first understand how the effective spacetime structure can be properly
studied. However, let us also point out that the universality of the free
algebra construction seems to imply that _some_ class of perturbations must
lead to the right kind of deformation of the effective spacetime structure.

###  4.3 Outlook

It is clear that a lot of work remains to be done to verify the physical
relevance of the spacetime-free approach to quantum physics presented here. In
particular, the following two important tasks would bring the goal within
reach:

  * •

Show explicitly how the ordinary formulation of quantum field theory on a
background spacetime (e.g., flat Minkowski spacetime) can be related to the
spacetime-free framework.

  * •

Show that spacetime structure can be recovered in the framework to a
sufficient degree, and find concrete useful methods to accomplish the
recovery.

In this work, we have touched upon both of the tasks, but without conclusive
results. However, if these challenges can be positively tackled, then we would
be able to study the exact relation between the perturbations of the reference
state (e.g., the vacuum) and gravitational phenomena. This could offer a
conceptually coherent and elegant explanation for the emergence of gravity
from quantum physics, and open up a vast array of further questions about the
physical implications of the framework.

A further important question concerns the emergence of Minkowski spacetime:

  * •

Show that the theory leads generically to an approximately flat 4-dimensional
effective local spacetime structure in some appropriate regime.

The solution to this problem should probably follow from the statistical
properties of quantum states on some free observable algebras, which consist
of measurements able to discern spacetime structure. In [22] it has been shown
that the Poincaré symmetry group of flat spacetime is generated in algebraic
QFT by the modular involutions associated to subalgebras corresponding to
Rindler wedges, but it is not clear at the moment how to translate this result
to the spacetime-free framework. Other ideas worth exploring, which have not
been mentioned in this work, include the intriguing relation of half-sided
modular inclusions to the Lorentz group [15], which might explain the
emergence of local spacetime symmetries in more generic situations. Lorentz
symmetry has also recently been derived from quantum informational principles
[44]. On the other hand, we have not even touched on the question of spacetime
dimensionality. The dimension four has many special properties (see, e.g.,
[68]), and we cannot help wondering whether these are important. At the same
time, the idea that the 3-dimensionality of space is related to the
3-dimensionality of the Bloch ball is an old one (see, e.g., [56, 88]), and
worth exploring further in our framework.

Another interesting topic of future research is the exploration and
classification of physical observable algebras obtained from a finite set of
binary measurements (and their limits):

  * •

Classify states on the free observable algebras
$\star_{i=1}^{n}\mathbb{C}^{2}$ for $n\in\mathbb{N}$ according to the kind of
causal structures and evolution that they impose on the observables.

(In Appendix C we covered some examples with two binary measurements.) This
elementary class of free observable algebras describes experimental
arrangements with a finite number of binary ‘yes/no’ questions, which covers a
wide range of experimental situations. We could even argue that they cover all
the practically realizable situations, since in a realistic measurement we
always obtain only a finite amount of information [88].

All in all, the spacetime-free approach to quantum theory introduced in the
current work appears to us as a promising (if still rather hypothetical) new
candidate in the collection of attempts to explain the origin of spacetime and
gravity. We kindly invite anyone interested to take part in the future work to
further elucidate its viability.

##  Appendix A Probability formula for a sequence of measurements

In the standard Schrödinger picture quantum mechanics the state of a closed
quantum system is described by a unit state vector in some Hilbert space of
vector states $\mathcal{H}$. The time-evolution of the state vector is
generated by the Hamiltonian operator $H$ as
$|\psi(t+s)\rangle=U(s)|\psi(t)\rangle\in\mathcal{H}$, where $U(s):=e^{isH}$
is the one-parameter group of unitaries implementing the time-evolution, and
$|\psi(t)\rangle\in\mathcal{H}$ represents the state of the system at time
$t\in\mathbb{R}$. The probability amplitude for a transition from a vector
state $|\psi\rangle\in\mathcal{H}$ at time $t$ to another vector state
$|\xi\rangle\in\mathcal{H}$ at time $s$ is expressed in the standard way
$\langle\xi|U(s-t)|\psi\rangle\in\mathbb{C}$.

Let us consider further the case, where after transitioning to
$|\xi\rangle\in\mathcal{H}$ at time $s$ (e.g., through a projective
measurement), we again let the system to evolve freely, until transitioning
yet again to another vector state $|\phi\rangle\in\mathcal{H}$ at time $r$. By
standard quantum mechanics, the probability amplitude for this process may be
expressed as the product of the two transition amplitudes

$\displaystyle\langle\phi|U(r-s)|\xi\rangle\langle\xi|U(s-t)|\psi\rangle\in\mathbb{C}.$

In general, we may write as

$\displaystyle\mathcal{A}(|\psi_{i}\rangle,t_{i}):=\prod_{i=1}^{n}\langle\psi_{i}|U(t_{i}-t_{i-1})|\psi_{i-1}\rangle\in\mathbb{C}$

the probability amplitude for a process consisting of an initial vector state
$|\psi_{0}\rangle\in\mathcal{H}$ at time $t_{0}\in\mathbb{R}$ followed by
projective measurements onto $|\psi_{i}\rangle\in\mathcal{H}$ at times
$t_{i}$, $i=1,\ldots,n$.

The probability for the process is obtained as the squared norm of the
probability amplitude

$\displaystyle\mathcal{P}(|\psi_{i}\rangle,t_{i}):=|\mathcal{A}(|\psi_{i}\rangle,t_{i})|^{2}=\left(\prod_{i=1}^{n}\langle\psi_{i-1}|U(t_{i-1}-t_{i})|\psi_{i}\rangle\right)\left(\prod_{j=1}^{n}\langle\psi_{j}|U(t_{j}-t_{j-1})|\psi_{j-1}\rangle\right).$

Let $P_{i}:=|\psi_{i}\rangle\langle\psi_{i}|$ denote the projection onto the
subspace of $\mathcal{H}$ spanned by the unit vector
$|\psi_{i}\rangle\in\mathcal{H}$, which mediates the associated projective
measurement. We then obtain

$\displaystyle\mathcal{P}(|\psi_{i}\rangle,t_{i})=\langle\psi_{0}|U(t_{0})\left(\overrightarrow{\prod_{i=1}^{n}}U(t_{i})^{*}P_{i}U(t_{i})\right)\left(\overleftarrow{\prod_{i=1}^{n}}U(t_{i})^{*}P_{i}U(t_{i})\right)U(t_{0})^{*}|\psi_{0}\rangle,$

where an arrow on top of a product sign determines the direction, in which the
product index increases along the factors. Here, each of the factors
$P_{i}^{\prime}:=U(t_{i})^{*}P_{i}U(t_{i})=U(t_{i})^{*}|\psi_{i}\rangle\langle\psi_{i}|U(t_{i})$
is a projection corresponding to the state vector
$U(t_{i})^{*}|\psi_{i}\rangle$, i.e., the intermediate state
$|\psi_{i}\rangle$ translated in time from the transition time $t_{i}$ to
$t=0$. Likewise, $|\psi_{0}^{\prime}\rangle:=U(t_{0})^{*}|\psi_{0}\rangle$ is
the initial state vector translated from the initial time $t_{0}$ to $t=0$.
Accordingly, we may further rewrite

$\displaystyle\mathcal{P}(|\psi_{i}\rangle,t_{i})=\langle\psi_{0}^{\prime}|\left(\overrightarrow{\prod_{i=1}^{n}}P_{i}^{\prime}\right)\left(\overleftarrow{\prod_{i=1}^{n}}P_{i}^{\prime}\right)|\psi_{0}^{\prime}\rangle.$

Finally, allowing for an arbitrary (possibly mixed) initial state
$\omega_{0}$, we obtain the expression

$\displaystyle\omega_{0}\big{(}\big{(}P_{1}^{\prime}P_{2}^{\prime}\cdots
P_{n}^{\prime}\big{)}\big{(}P_{n}^{\prime}P_{n-1}^{\prime}\cdots
P_{1}^{\prime}\big{)}\big{)}$ (A.1)

for the probability of a process consisting of projective
measurements.272727This standard formula for the probability of a history of
events is also the starting point of the consistent histories approach to
quantum mechanics (see, e.g., [40]). Notice that if two projective
measurements corresponding to the projections $P_{1}$, $P_{2}$ are performed
at two times $t_{1}$, $t_{2}$, which are related through the time-evolution
$P_{2}=U(t_{2}-t_{1})P_{1}U(t_{2}-t_{1})^{*}$, then
$P_{1}^{\prime}=P_{2}^{\prime}$ in (A.1), since they correspond to the same
operator, when translated to $t=0$:

$\displaystyle
P_{2}^{\prime}=U(t_{2})^{*}P_{2}U(t_{2})=U(-t_{1})P_{1}U(-t_{1})^{*}=U(t_{1})^{*}P_{1}U(t_{1})=P_{1}^{\prime}.$

In other words, in equation (A.1) operators at different times are identified
through the time-evolution. Operators at different times can also always be
compared through the identity map if the time-evolution is an isomorphism, but
the ‘same’ measurement (as identified through the identity map) at two
different times corresponds (in general) to two different projections in
(A.1).

Importantly, in equation (A.1) the time parameter does not appear explicitly,
but the time-evolution is already accounted for in the definitions of the
initial state and the projective measurements as being translated to a common
instant of time $t=0$. Ultimately, the time-evolution is hidden in the
algebraic relations between the projections that are induced by the time-
translations. Moreover, we did not impose any ordering of the measurement
times $t_{i}$, so the ordering of the projections in (A.1) refers to the order
in which the experimenter records the measurement outcomes, and not
(necessarily) the temporal order of the measurement events themselves.
Therefore, equation (A.1) appears promising for generalization to the case,
where no global time parameter or evolution is assumed. Indeed, in the main
text we will see how a state can be used to impose algebraic relations between
spectral projections, and thus possibly give rise to a notion causal ordering
and evolution.

##  Appendix B Positive energy perturbations and gravity

In this Appendix, we consider a definition of mass for static perturbations,
and study if their effect on the structure of the observable algebra resembles
in any sense that of gravity.

We assume in this Appendix that the extended reference state $\tilde{\Omega}$
is faithful on the physical observable algebra $\mathfrak{A}_{\Omega}$, and
thus induces the modular flow
$\sigma^{\Omega}\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A}_{\Omega})$
as described in Section 2.4. If the perturbed reference state
$\tilde{\Omega}^{\prime}$ is faithful on the perturbed physical observable
algebra $\mathfrak{A}_{\Omega^{\prime}}$, it likewise induces the perturbed
modular flow
$\sigma^{\Omega^{\prime}}\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A}_{\Omega^{\prime}})$
on $\mathfrak{A}_{\Omega^{\prime}}$. In this case we may write
$\mathfrak{A}_{\Omega^{\prime}}=P_{\Omega^{\prime}}\mathfrak{A}_{\Omega}P_{\Omega^{\prime}}\subset\mathfrak{A}_{\Omega}$,
where $P_{\Omega^{\prime}}\in\mathfrak{A}_{\Omega}^{\prime}$ is the central
projection such that
$\\{A\in\mathfrak{A}_{\Omega}\colon\tilde{\Omega}^{\prime}(A^{*}A)=0\\}=\mathfrak{A}_{\Omega}(\mathbbm{1}-P_{\Omega^{\prime}})$
(i.e., the support projection of $\tilde{\Omega}^{\prime}$). The perturbed
modular flow $\sigma^{\Omega^{\prime}}$ is related to the original unperturbed
one $\sigma^{\Omega}$ by a strongly continuous one-parameter family of partial
isometries
$t\in\mathbb{R}\mapsto\delta^{\Omega^{\prime}\Omega}_{t}\in\mathfrak{A}_{\Omega}$
called the _Connes cocycle derivative_ of $\tilde{\Omega}^{\prime}$ with
respect to $\tilde{\Omega}$, which satisfies

$\displaystyle\sigma^{\Omega^{\prime}}_{t}(a)=\delta^{\Omega^{\prime}\Omega}_{t}\sigma^{\Omega}_{t}(a)\big{(}\delta^{\Omega^{\prime}\Omega}_{t}\big{)}^{*}\qquad\text{and}\qquad\delta^{\Omega^{\prime}\Omega}_{s+t}=\delta^{\Omega^{\prime}\Omega}_{s}\sigma^{\Omega}_{s}\big{(}\delta^{\Omega^{\prime}\Omega}_{t}\big{)}$

for all $t,s\in\mathbb{R}$ and
$a\in\mathfrak{A}_{\Omega^{\prime}}\subset\mathfrak{A}_{\Omega}$ [81]. The
cocycle derivative always belongs to the physical observable algebra
$\mathfrak{A}_{\Omega}$, unlike the modular operators, and is therefore
(approximately) measurable according to the operational interpretation of our
formalism. Physically speaking, the cocycle derivative reflects the difference
in equilibrium dynamics of the two states. In the following, we would like to
suggest that (the generator of) the cocycle derivative provides a measure of
energy difference, or mass, for a perturbed state that is stationary under the
thermal (modular) dynamics. However, first we need to review a few facts about
the structure of the GNS Hilbert space.

The norm completion of the set of vectors
$\\{aJ_{\Omega}aJ_{\Omega}|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}\in\mathcal{H}_{\Omega}\colon
a\in\mathfrak{A}_{\Omega}\\}$ is called the _natural positive cone_
$\mathcal{P}_{\Omega}\subset\mathcal{H}_{\Omega}$, and it has several
important algebraic characteristics [76, 81]. In particular, for any state
$\tilde{\Omega}^{\prime}$ on $\mathfrak{A}_{\Omega}$ there exists a unique
unit vector $|\psi_{\Omega^{\prime}}\rangle_{\Omega}\in\mathcal{P}_{\Omega}$
such that
$\tilde{\Omega}^{\prime}(a)=\langle\psi_{\Omega^{\prime}}|a|\psi_{\Omega^{\prime}}\rangle_{\Omega}$
for all $a\in\mathfrak{A}_{\Omega}$. Accordingly, the norm completion of the
subspace

$\displaystyle\big{\\{}\pi_{\Omega}(a)|\psi_{\Omega^{\prime}}\rangle_{\Omega}\in\mathcal{H}_{\Omega}\colon
a\in\mathfrak{F}\big{\\}}\subset\mathcal{H}_{\Omega}$

is isomorphic to the GNS Hilbert space $\mathcal{H}_{\Omega^{\prime}}$ induced
by the state $\tilde{\Omega}^{\prime}$, since
$|\psi_{\Omega^{\prime}}\rangle_{\Omega}\equiv|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega^{\prime}}\in\mathcal{H}_{\Omega^{\prime}}$
is a cyclic vector in $\mathcal{H}_{\Omega^{\prime}}$ by definition.
Therefore, we may consider the GNS Hilbert space
$\mathcal{H}_{\Omega^{\prime}}$ of the perturbed state $\Omega^{\prime}$ as a
Hilbert subspace of the reference GNS Hilbert space $\mathcal{H}_{\Omega}$
that is invariant under the action of $\pi_{\Omega}(\mathfrak{F})$. In order
to define the mass operator, we note that the cocycle derivative
$\delta^{\Omega^{\prime}\Omega}_{t}\colon\mathcal{H}_{\Omega}\rightarrow\mathcal{H}_{\Omega}$
restricts to an isometry
$u^{\Omega^{\prime}\Omega}_{t}\colon\Delta_{\Omega}^{it}\mathcal{H}_{\Omega^{\prime}}\rightarrow\mathcal{H}_{\Omega^{\prime}}$,
since
$\ker(\delta^{\Omega^{\prime}\Omega}_{t})^{\bot}=\Delta_{\Omega}^{it}\mathcal{H}_{\Omega^{\prime}}$.

In the special case that the perturbed state $\tilde{\Omega}^{\prime}$ is
stationary282828We restrict to consider the stationary case mainly for
technical reason. It may be possible to extend the definition of the mass
operator to a more general situation, but we have not explored this
possibility so far. under the thermal dynamics given by the modular flow
$\sigma^{\Omega}$ induced by the reference state $\tilde{\Omega}$ (i.e.,
$\tilde{\Omega}^{\prime}(\sigma^{\Omega}_{t}(a))=\tilde{\Omega}^{\prime}(a)$
for all $a\in\mathfrak{A}_{\Omega}$ and $t\in\mathbb{R}$), we have that
$\Delta_{\Omega}^{it}|\psi_{\Omega^{\prime}}\rangle_{\Omega}=|\psi_{\Omega^{\prime}}\rangle_{\Omega}$
and thus
$\Delta_{\Omega}^{it}\mathcal{H}_{\Omega^{\prime}}=\mathcal{H}_{\Omega^{\prime}}$
for all $t\in\mathbb{R}$. Accordingly,
$u^{\Omega^{\prime}\Omega}_{t}\colon\mathcal{H}_{\Omega^{\prime}}\rightarrow\mathcal{H}_{\Omega^{\prime}}$
are unitary, and form a one-parameter group (i.e.,
$u^{\Omega^{\prime}\Omega}_{t}u^{\Omega^{\prime}\Omega}_{s}=u^{\Omega^{\prime}\Omega}_{t+s}$
for all $t,s\in\mathbb{R}$). We may then consider the generator
$h^{\Omega^{\prime}\Omega}$ of the unitaries
$u^{\Omega^{\prime}\Omega}_{t}\equiv\exp(-ith^{\Omega^{\prime}\Omega})$, which
is a densely defined self-adjoint operator affiliated with
$\mathfrak{A}_{\Omega^{\prime}}\subset B(\mathcal{H}_{\Omega^{\prime}})$. We
suggest to interpret $h^{\Omega^{\prime}\Omega}$ as measuring the physical
energy/mass content associated with the perturbation. In particular,
$\Omega^{\prime}$ is considered to represent a positive mass perturbation with
respect to the background defined by the reference state $\Omega$, if
$h^{\Omega^{\prime}\Omega}$ is a positive operator. Notice that stationary
perturbations are mapped to stationary perturbations by the symmetries of the
reference state $\tilde{\Omega}$. If $\tilde{\Omega}^{\prime}$ is faithful on
the unperturbed observable algebra $\mathfrak{A}_{\Omega}$, then
$h^{\Omega^{\prime}\Omega}$ simply gives the difference of the generators of
the two dynamics (i.e., Hamiltonians), with respect to which the two states
are in equilibrium. However, we must allow $\tilde{\Omega}^{\prime}$ not to be
faithful in order for its GNS representation to have a non-trivial kernel, and
thus alter the algebraic structure of the physical observable algebra.

We conjecture the following physically interesting feature of the above
mathematical structure: If the mass operator $h^{\Omega^{\prime}\Omega}$ is
positive, then $\ker\pi_{\Omega}\subset\ker\pi_{\Omega^{\prime}}$ is a proper
inclusion. Below, we will provide a sketch for a proof in the finite-
dimensional case, but a rigorous proof is left for future work. If this
property holds true, it implies physically that perturbations with positive
mass always turn some new measurements jointly measurable. This seems
analogous of the property of gravity that a positive mass perturbation focuses
lightcones, and therefore always makes some previously timelike separated (not
jointly measurable) local subsystems spacelike separated (jointly measurable).

Let the perturbed state $\Omega^{\prime}$ be stationary under the reference
modular flow $\sigma^{\Omega}$. Here we provide a sketch of a proof in the
finite-dimensional case for the fact that
$\ker\pi_{\Omega}\subset\ker\pi_{\Omega^{\prime}}$ must be a proper inclusion
if the mass operator $h^{\Omega^{\prime}\Omega}$ is strictly positive.

Let $\rho_{\Omega},\rho_{\Omega^{\prime}}\in B(\mathcal{H}_{\Omega})$ be the
density operators corresponding to the reference state $\Omega$ and its
perturbation $\Omega^{\prime}$ on $\mathfrak{A}_{\Omega}$, respectively. Since
$\Omega^{\prime}$ is stationary under $\sigma^{\Omega}$, which is represented
by the adjoint action of $\rho_{\Omega}^{it}$, the density operators
$\rho_{\Omega}\in B(\mathcal{H}_{\Omega})$ and $\rho_{\Omega^{\prime}}\in
B(\mathcal{H}_{\Omega})$ must have a common basis of eigenvectors in
$\mathcal{H}_{\Omega}$. In fact, we have
$u^{\Omega^{\prime}\Omega}_{t}=\rho_{\Omega^{\prime}}^{it}\rho_{\Omega}^{-it}$.
For the generator $h^{\Omega^{\prime}\Omega}$ of
$u^{\Omega^{\prime}\Omega}_{t}$ to be a strictly positive operator, each non-
vanishing eigenvalue of $\rho_{\Omega^{\prime}}$ must be larger than the
corresponding eigenvalue of $\rho_{\Omega}$, because the eigenvalues of
$h^{\Omega^{\prime}\Omega}$ are logarithms of ratios of the non-zero
eigenvalues of the two density operators. On the other hand, both $\Omega$ and
$\Omega^{\prime}$ are normalized, i.e., the eigenvalues of both
$\rho_{\Omega}$ and $\rho_{\Omega^{\prime}}$ sum up to $1$. Therefore, some of
the eigenvalues of $\rho_{\Omega^{\prime}}$ must vanish, i.e., the support of
$\Omega^{\prime}$ is non-trivial. Accordingly,
$\ker\pi_{\Omega}\subset\ker\pi_{\Omega^{\prime}}$ is a proper inclusion.

##  Appendix C Experiments with two binary measurements

Let us consider the simplest non-trivial example of the framework we have
presented in Section 2: an experimental arrangement, where we have access to
only two measurements, which can both take only two values each. The
measurements are thus represented by two abelian von Neumann algebras
$\mathfrak{W}_{i}\cong\mathbb{C}^{2}$, $i=1,2$, each linearly spanned by two
spectral projections $P^{(i)}_{k}\in\mathfrak{W}_{i}$, $k=1,2$, which satisfy
$P^{(i)}_{k}P^{(i)}_{l}=\delta_{kl}P^{(i)}_{k}$ and
$\sum_{k}P^{(i)}_{k}=\mathbbm{1}$. The free product $*$-algebra
$\mathfrak{F}\cong\mathbb{C}^{2}\star\mathbb{C}^{2}$ is linearly spanned by
finite sequences of spectral projections

$\displaystyle P^{(i_{1})}_{k_{1}}P^{(i_{2})}_{k_{2}}\cdots
P^{(i_{n})}_{k_{n}}\in\mathfrak{F},$

where $i_{m}\neq i_{m+1}$ for all $m=1,\ldots,n-1$, $k_{m}=1,2$ for all
$m=1,\ldots,n$, and $n\in\mathbb{N}$, which represent sequences of measurement
outcomes. Actually, due to the relation $P^{(i)}_{2}=\mathbbm{1}-P^{(i)}_{1}$
for $i=1,2$, only the sequences of the form

$\displaystyle P^{(i_{1})}_{1}P^{(i_{2})}_{1}\cdots
P^{(i_{n})}_{1}\in\mathfrak{F}$

are linearly independent in $\mathfrak{F}$.

So far we are considering abstract elements of an abstract algebra. The
reference state gives the physical meaning to the formalism by assigning the
observational probabilities to different measurement outcome sequences.
Therefore, it is interesting to consider different kind of states on
$\mathfrak{F}$ and the kinds of physical observable algebras that they may
induce.

### States on $\boldsymbol{\mathbb{C}^{2}\otimes\mathbb{C}^{2}}$

As the simplest example, a state $\omega$ on the tensor product
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ can be lifted onto $\mathfrak{F}$ via
the pull-back of the algebra homomorphism
$\phi\colon\mathfrak{F}\cong\mathbb{C}^{2}\star\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
defined by

$\displaystyle\mathbbm{1}_{\mathfrak{F}}\stackrel{{\scriptstyle\phi}}{{\mapsto}}\mathbbm{1}^{(1)}\otimes\mathbbm{1}^{(2)}\equiv\mathbbm{1}_{\mathbb{C}^{2}\otimes\mathbb{C}^{2}},$
$\displaystyle
P^{(1)}_{1}\stackrel{{\scriptstyle\phi}}{{\mapsto}}P^{(1)}_{1}\otimes\mathbbm{1}^{(2)},$
$\displaystyle
P^{(2)}_{1}\stackrel{{\scriptstyle\phi}}{{\mapsto}}\mathbbm{1}^{(1)}\otimes
P^{(2)}_{1},\qquad\text{and}$ $\displaystyle P^{(i_{1})}_{1}\cdots
P^{(i_{n})}_{1}\stackrel{{\scriptstyle\phi}}{{\mapsto}}P^{(1)}_{1}\otimes
P^{(2)}_{1}$

for any sequence such that $i_{m}\neq i_{m+1}$ for all $m=1,\ldots,n-1$ and
$n>1$. Then, the pull-back state
$\Omega:=\omega\circ\phi\colon\mathfrak{F}\rightarrow\mathbb{C}$ satisfies

$\displaystyle\Omega(\mathbbm{1}_{\mathfrak{F}})=1,$
$\displaystyle\Omega\big{(}P^{(1)}_{1}\big{)}\in[0,1],$
$\displaystyle\Omega\big{(}P^{(2)}_{1}\big{)}\in[0,1],$
$\displaystyle\Omega\big{(}P^{(i_{1})}_{1}\cdots
P^{(i_{n})}_{1}\big{)}=\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}\in[0,1]$
(C.1)

for any sequence such that $i_{m}\neq i_{m+1}$ and $n>1$, so the expectation
values on the basis elements of $\mathfrak{F}$ form a very simple pattern. In
particular, all the expectation values are uniquely determined in terms of
$\Omega\big{(}P^{(1)}_{1}\big{)}$, $\Omega\big{(}P^{(2)}_{1}\big{)}$ and
$\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}$.

Let us assume that the state $\omega$ is faithful on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. However, $\Omega$ is clearly _not_
faithful on the free algebra $\mathbb{C}^{2}\star\mathbb{C}^{2}$. Indeed,
$\big{[}P^{(1)}_{k},P^{(2)}_{l}\big{]}\in\mathcal{N}_{\Omega}:=\\{a\in\mathfrak{F}\colon\Omega(a^{*}a)=0\\}$
for any $k,l=1,2$ as is easily verified by using the expectation values of
$\Omega$ above, and therefore we have
$\big{|}P^{(2)}_{l}P^{(1)}_{k}\big{\rangle}_{\Omega}\sim\big{|}P^{(1)}_{k}P^{(2)}_{l}\big{\rangle}_{\Omega}$
in the GNS Hilbert space $\mathcal{H}_{\Omega}$. It is also straightforward to
verify, e.g., that

$\displaystyle\big{|}P^{(1)}_{k_{1}}P^{(2)}_{l_{1}}P^{(1)}_{k_{2}}P^{(2)}_{l_{2}}\cdots
P^{(1)}_{k_{n}}P^{(2)}_{l_{n}}\big{\rangle}_{\Omega}\sim\big{|}P^{(1)}_{k_{1}}P^{(2)}_{l_{1}}\big{\rangle}_{\Omega}$

in $\mathcal{H}_{\Omega}$ if $k_{m}=k_{m+1}$ and $l_{m}=l_{m+1}$ $\forall\,m$,
and $\sim 0$ otherwise. Similar equivalence relations apply between other
sequences of length $>2$ and sequences of length $2$ (or the null vector).
Accordingly, we obtain the GNS Hilbert space
$\mathcal{H}_{\Omega}\cong\mathbb{C}^{4}$ with the orthonormal basis of
vectors

$\displaystyle
e_{kl}:=\frac{1}{\sqrt{\Omega\big{(}P^{(1)}_{k}P^{(2)}_{l}\big{)}}}\big{|}P^{(1)}_{k}P^{(2)}_{l}\big{\rangle}_{\Omega}\in\mathcal{H}_{\Omega},\qquad
k,l=1,2.$

The cyclic vector is given by

$\displaystyle|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=\sum_{k,l}\big{|}P^{(1)}_{k}P^{(2)}_{l}\big{\rangle}_{\Omega}=\sum_{k,l}\sqrt{\Omega\big{(}P^{(1)}_{k}P^{(2)}_{l}\big{)}}e_{kl}\in\mathcal{H}_{\Omega}$

in terms of the orthonormal basis. We have likewise
$\pi_{\Omega}(\mathbb{C}^{2}\star\mathbb{C}^{2})\cong\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
for the GNS representation. Thus, the infinite-dimensional non-abelian free
observable algebra $\mathbb{C}^{2}\star\mathbb{C}^{2}$ is reduced to the
finite-dimensional abelian physical observable algebra
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ by the statistics (C.1) of the reference
state $\Omega$. The tensor product structure of the physical observable
algebra indicates the operational independence of the two measurements.

The modular structure induced by the reference state $\Omega$ is rather
trivial, because the physical observable algebra is abelian: We have
$\Delta_{\Omega}=\mathbbm{1}$ and $J_{\Omega}=C$, the complex conjugation
operator. The perturbation algebra is given by $B(\mathbb{C}^{4})\cong
B(\mathbb{C}^{2})\otimes B(\mathbb{C}^{2})$. Perturbations of the reference
state $\Omega\mapsto\Omega^{\prime}$ such that
$\Omega^{\prime}(a)=\sum_{n}\Omega(b_{n}^{*}ab_{n})$, $b_{n}\in
B(\mathbb{C}^{4})$, for all
$a\in\mathfrak{A}_{\Omega}\cong\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ may alter
the statistics (C.1) by changing the expectation values
$\Omega\big{(}P^{(1)}_{1}\big{)}$, $\Omega\big{(}P^{(2)}_{1}\big{)}$ and
$\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}$, but they cannot introduce non-
trivial commutation relations between the observables. Therefore, also all
physical perturbations have trivial modular structure. We must proceed to more
involved examples in order to recover non-trivial thermal dynamics.

### States on $\boldsymbol{\mathbb{M}_{2}(\mathbb{C})}$

We may consider a homomorphism
$\phi\colon\mathbb{C}^{2}\star\mathbb{C}^{2}\rightarrow\mathbb{M}_{2}(\mathbb{C})$,
the algebra of 2-by-2 complex-valued matrices, identifying the projections
$P^{(i)}_{k}\in\mathbb{C}^{2}\star\mathbb{C}^{2}$ with two arbitrary pairs of
complementary projections given by
$P^{(i)}_{1}=\big{|}u^{(i)}\big{\rangle}\big{\langle}u^{(i)}\big{|}\in\mathbb{M}_{2}(\mathbb{C})$
and $P^{(i)}_{2}=\mathbbm{1}_{2}-P^{(i)}_{1}$, where
$\big{|}u^{(i)}\big{\rangle}:=\big{(}u^{(i)}_{1},u^{(i)}_{2}\big{)}\in\mathbb{C}^{2}$
is an arbitrary vector of unit norm. We then simply find

$\displaystyle\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}=\big{|}\big{\langle}u^{(1)}|u^{(2)}\big{\rangle}\big{|}^{2(n-1)}P^{(1)}_{1}P^{(2)}_{1},$
$\displaystyle\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}P^{(1)}_{1}=\big{|}\big{\langle}u^{(1)}|u^{(2)}\big{\rangle}\big{|}^{2n}P^{(1)}_{1},$
$\displaystyle
P^{(2)}_{1}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}=\big{|}\big{\langle}u^{(1)}|u^{(2)}\big{\rangle}\big{|}^{2n}P^{(2)}_{1}.$

We then consider a state $\omega$ on $\mathbb{M}_{2}(\mathbb{C})$ given by
$\omega(a)=\operatorname{tr}(\rho a)$, where
$\rho\in\mathbb{M}_{2}(\mathbb{C})$ is a positive matrix with unit trace, and
again set
$\Omega=\omega\circ\phi\colon\mathbb{C}^{2}\star\mathbb{C}^{2}\rightarrow\mathbb{C}$
as our reference state. By the above, we find

$\displaystyle\Omega\big{(}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}\big{)}=\lambda^{n-1}\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}\qquad\forall\,n=1,2,\ldots,$
$\displaystyle\Omega\big{(}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}P^{(1)}_{1}\big{)}=\lambda^{n}\Omega\big{(}P^{(1)}_{1}\big{)}\qquad\forall\,n=0,1,\ldots,$
$\displaystyle\Omega\big{(}P^{(2)}_{1}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}\big{)}=\lambda^{n}\Omega\big{(}P^{(2)}_{1}\big{)}\qquad\forall\,n=0,1,\ldots,$

where

$\displaystyle\lambda=\big{|}\big{\langle}u^{(1)}|u^{(2)}\big{\rangle}\big{|}^{2}\in[0,1],$
$\displaystyle\Omega\big{(}P^{(i)}_{1}\big{)}=\big{\langle}u^{(i)}|\rho|u^{(i)}\big{\rangle}\in[0,1],\qquad
i=1,2,$
$\displaystyle\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}=\big{\langle}u^{(1)}|u^{(2)}\big{\rangle}\big{\langle}u^{(2)}|\rho|u^{(1)}\big{\rangle}\in\mathbb{C}.$
(C.2)

Thus, all the expectation values are determined by
$\Omega\big{(}P^{(i)}_{1}\big{)}$,
$\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}$ and the parameter $\lambda$,
which are related through (C.2).

It is interesting to study the possible equivalence relations that we may have
in this case for the GNS Hilbert space
$\mathcal{H}_{\Omega}\equiv\overline{\mathfrak{F}/\mathcal{N}_{\Omega}}$. For
example, we find by a direct calculation that

$\displaystyle\Omega\Big{(}\big{|}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}-P^{(1)}_{1}P^{(2)}_{1}\big{|}^{2}\Big{)}=\lambda\big{(}\lambda^{n-1}-1\big{)}^{2}\Omega\big{(}P^{(2)}_{1}\big{)},$

where we introduced the notation $|a|^{2}:=a^{*}a$. We have that
$\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}-P^{(1)}_{1}P^{(2)}_{1}\in\mathcal{N}_{\Omega}$,
and therefore
$\big{|}\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}^{n}\big{\rangle}_{\Omega}\sim\big{|}P^{(1)}_{1}P^{(2)}_{1}\big{\rangle}_{\Omega}$
in $\mathcal{H}_{\Omega}$ if the above expression vanishes. But this happens
only if either (i) $\lambda=0$, which implies
$P^{(1)}_{1}=\mathbbm{1}-P^{(2)}_{1}\equiv P^{(2)}_{2}$, or (ii) $\lambda=1$,
which implies $P^{(1)}_{1}=P^{(2)}_{1}$. Both of the cases reduce the
observable algebra into the abelian $\mathbb{C}^{2}$, and therefore are rather
trivial. On the other hand, we may compute for a constant
$\alpha\in\mathbb{C}$

$\displaystyle\Omega\Big{(}\big{|}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}-\alpha
P^{(1)}_{1}\big{|}^{2}\Big{)}=|\lambda-\alpha|^{2}\Omega\big{(}P^{(1)}_{1}\big{)},$

so we see that
$\big{|}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}\big{\rangle}_{\Omega}\sim\lambda\big{|}P^{(1)}_{1}\big{\rangle}_{\Omega}$
in $\mathcal{H}_{\Omega}$, and similarly
$\big{|}P^{(2)}_{1}P^{(1)}_{1}P^{(2)}_{1}\big{\rangle}_{\Omega}\sim\lambda\big{|}P^{(2)}_{1}\big{\rangle}_{\Omega}$.
Actually, we further have that all sequences of projections are equivalent to
linear combinations of sequences of length $\leq 2$, since
$\mathcal{N}_{\Omega}$ is a left-ideal. Moreover, sequences of length $<2$ may
be expressed as linear combinations of sequences of length $2$ (e.g.,
$P^{(1)}_{1}=P^{(1)}_{1}P^{(2)}_{1}+P^{(1)}_{1}P^{(2)}_{2}$), and therefore
the elements
$\big{|}P^{(i)}_{k}P^{(j)}_{l}\big{\rangle}_{\Omega}\in\mathcal{H}_{\Omega}$
span the GNS Hilbert space. The cyclic vector is given in terms of these
vectors as

$\displaystyle|\mathbbm{1}_{\mathfrak{F}}\rangle_{\Omega}=\sum_{k,l}\big{|}P^{(i)}_{k}P^{(j)}_{l}\big{\rangle}_{\Omega}$

for any $i,j=1,2$, $i\neq j$. However, not all of them are linearly
independent, because we have the relations

$\displaystyle\sum_{l}\big{|}P^{(i)}_{k}P^{(j)}_{l}\big{\rangle}_{\Omega}=\big{|}P^{(i)}_{k}\big{\rangle}_{\Omega}=\sum_{l}\big{|}P^{(j)}_{l}P^{(i)}_{k}\big{\rangle}_{\Omega}$

for all $i\neq j$. Let us assume that $\lambda\in(0,1)$ and
$\Omega\big{(}P^{(i)}_{1}\big{)},\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}\in(0,1)$
for $i=1,2$, i.e., $\omega$ is faithful on $\mathbb{M}_{2}(\mathbb{C})$. Then,
starting for example with the orthonormal vectors
$\big{|}P^{(1)}_{k}\big{\rangle}_{\Omega}\in\mathcal{H}_{\Omega}$, $k=1,2$,
and applying the Gram–Schmidt method to find the rest of the basis will leave
us with four orthonormal basis vectors, and accordingly
$\mathcal{H}_{\Omega}\cong\mathbb{C}^{4}$, as for the usual GNS construction
with respect to a faithful state on $\mathbb{M}_{2}(\mathbb{C})$. The modular
operator $S_{\Omega}$ defined by
$S_{\Omega}|a\rangle_{\Omega}=|a^{*}\rangle_{\Omega}$ gives rise to the
modular structure. We have that, given an orthonormal basis, $S_{\Omega}$ may
be represented by a non-singular matrix in $\mathbb{M}_{4}(\mathbb{C})$
composed with a complex conjugation, which takes care of the anti-linearity of
the map $|a\rangle_{\Omega}\stackrel{{\scriptstyle
S_{\Omega}}}{{\mapsto}}|a^{*}\rangle_{\Omega}$. The polar decomposition of
this matrix will give us representations of the modular operators
$\Delta_{\Omega}$ and $J_{\Omega}$ acting on
$\mathcal{H}_{\Omega}\cong\mathbb{C}^{4}$, which are generically non-trivial
in this case.

### Free product states

The free product of states plays an important role in the study of free
product algebras and free independence in non-commutative probability theory
[87]. In our case, a free product state $\Omega$ is defined by the following
centralizing property:

$\displaystyle\Omega\big{(}\big{(}P^{(i_{1})}_{1}-\Omega\big{(}P^{(i_{1})}_{1}\big{)}\big{)}\big{(}P^{(i_{2})}_{1}-\Omega\big{(}P^{(i_{2})}_{1}\big{)}\big{)}\cdots\big{(}P^{(i_{n})}_{1}-\Omega\big{(}P^{(i_{n})}_{1}\big{)}\big{)}\big{)}=0$
(C.3)

for all sequences such that $i_{m}\neq i_{m+1}$ for all $m=1,\ldots,n-1$ and
$n\in\mathbb{N}$. Let us call $n$ the order of the expectation value
$\Omega\big{(}P^{(i_{1})}_{1}P^{(i_{2})}_{1}\cdots P^{(i_{n})}_{1}\big{)}$.
Since we may write (C.3) as

$\displaystyle\Omega\big{(}P^{(i_{1})}_{1}P^{(i_{2})}_{1}\cdots
P^{(i_{n})}_{1}\big{)}+\sum(\textrm{products of expectation values of
order}<n)=0,$

it is actually possible to solve for the expectation values recursively to all
orders in terms of the first order expectation values
$\Omega\big{(}P^{(i)}_{1}\big{)}$, $i=1,2$. However, the explicit expression
in terms of $\Omega\big{(}P^{(i)}_{1}\big{)}$ grow rapidly in complexity as a
function of the order [87]. For the few lowest orders we find

$\displaystyle\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}\big{)}=\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)},$
$\displaystyle\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}\big{)}=\Omega\big{(}P^{(2)}_{1}P^{(1)}_{1}P^{(2)}_{1}\big{)}=\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)},$
$\displaystyle\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}P^{(2)}_{1}\big{)}=\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)}\big{[}\Omega\big{(}P^{(1)}_{1}\big{)}+\Omega\big{(}P^{(2)}_{1}\big{)}-\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)}\big{]},$
$\displaystyle\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}\big{)}=\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)}\big{[}\Omega\big{(}P^{(1)}_{1}\big{)}+\Omega\big{(}P^{(2)}_{1}\big{)}-\Omega\big{(}P^{(1)}_{1}\big{)}^{2}$
$\displaystyle\hphantom{\Omega\big{(}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}P^{(2)}_{1}P^{(1)}_{1}\big{)}=}{}-4\Omega\big{(}P^{(1)}_{1}\big{)}\Omega\big{(}P^{(2)}_{1}\big{)}+4\Omega\big{(}P^{(1)}_{1}\big{)}^{2}\Omega\big{(}P^{(2)}_{1}\big{)}\big{]}.$

Despite their complexity, the free product states are known to satisfy a few
important properties292929For the proofs of these and other interesting
properties, see, e.g., [3, 32, 33] and the references therein.:

  * •

If the restrictions of a free product state $\Omega$ onto the free product
factors are faithful, $\Omega$ is faithful on the free product. Therefore,
$\Omega$ gives rise to a faithful GNS representation of
$\mathbb{C}^{2}\star\mathbb{C}^{2}$, and the measurements are ‘freely
independent’.

  * •

If the restrictions of a free product state $\Omega$ onto the free product
factors are tracial, $\Omega$ is tracial on the free product. Since all states
on $\mathbb{C}^{2}$ (or any other abelian algebra) are tracial, a free product
state $\Omega$ provides a normalized trace on
$\mathbb{C}^{2}\star\mathbb{C}^{2}$. Consequently, the physical observable
algebra is of type II1 according to the classification of von Neumann
algebras.

  * •

Since a free product state $\Omega$ on $\mathbb{C}^{2}\star\mathbb{C}^{2}$ is
a trace, its modular automorphism group is trivial. However, non-free
perturbations of $\Omega$ may induce non-trivial dynamics.

Interestingly, tensor product states and free product states are opposite
extremes in the continuum of states on $\mathbb{C}^{2}\star\mathbb{C}^{2}$ in
the following sense: The first ones lead to a fully commutative algebra, while
the second ones lead to a maximally non-commutative algebra of physical
observables.

##  Appendix D Identification of local subsystems in spin lattices

The dynamics of a closed quantum system described by the observable algebra
$\mathfrak{A}$ is given by a one-parameter family of automorphisms
$\sigma\colon\mathbb{R}\rightarrow\textrm{Aut}(\mathfrak{A})$. The magnitude
of correlations between two subalgebras
$\mathfrak{B}_{1},\mathfrak{B}_{2}\in\mathfrak{A}$ at times $0$ and $t$,
respectively, can be quantified by

$\displaystyle\mathcal{C}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2}):=\sup_{\begin{subarray}{c}b_{1}\in\mathfrak{B}_{1}\\\
b_{2}\in\mathfrak{B}_{2}\end{subarray}}\frac{\|[\sigma_{t}(b_{1}),b_{2}]\|}{2\|b_{1}\|\|b_{2}\|}\in[0,1].$

The Lieb–Robinson theorem [52, 54] sets a bound on the magnitude of
correlations for a certain class of quantum systems, whose dynamics are
generated by time-independent Hamiltonians, which are approximately local with
respect to some background geometry, such as a lattice. A simple form of the
bound reads

$\displaystyle\mathcal{C}_{t}(\mathfrak{B}_{1},\mathfrak{B}_{2})\leq
Ce^{-\mu(d_{12}-v|t|)},$

where $d_{12}$ is the spatial distance between the two subsystems represented
by $\mathfrak{B}_{1,2}$, and $C,\mu,v{\in}\mathbb{R}_{+}$ are finite
constants, whose values depend on the dynamics. In particular, $v$ is called
the Lieb–Robinson velocity, which is the (approximate) maximum speed for
information transfer in the system, analogous to the speed of light in
relativistic systems. Due to the Lieb–Robinson bound, the correlations between
subsystems decay exponentially outside the effective ‘lightcone’.

The correlations between a subsystem represented by $\mathfrak{B}$ and its
environment $\mathfrak{B}^{\prime}$ in the total system are measured by
$\mathcal{C}_{t}(\mathfrak{B},\mathfrak{B}^{\prime})$. Thus, the growth of
correlations induced by the dynamics may be quantified by

$\displaystyle\mathcal{D}(\mathfrak{B}):=\left.\frac{\textrm{d}}{\textrm{d}t}\right|_{t=0}\mathcal{C}_{t}(\mathfrak{B},\mathfrak{B}^{\prime}).$

Let $\mathfrak{B}\subset\mathfrak{A}$ be any subfactor of the physical
observable algebra. We then compare the values of
$\mathcal{D}(\alpha(\mathfrak{B}))$ for different
$\alpha\in\textrm{Aut}(\mathfrak{A})$. We conjecture that a subalgebra
$\alpha(\mathfrak{B})\subset\mathfrak{A}$, for which this quantity acquires a
minimum should correspond to a local spherical subsystem.

Taking advantage of the Lieb–Robinson theorem, it is not difficult to see why
$\mathcal{D}(\mathfrak{B})$ is minimized by local subsystems for at least some
simple non-relativistic quantum systems with local Hamiltonians, although a
more rigorous proof could certainly be deviced: A global automorphism