Towards Relational Foundations for Spacetime Quantum Physics ^†

Abstract

Rovelli’s relational interpretation of quantum mechanics tells us that the description of a system in the formalism of quantum mechanics is not an absolute but is relative to the observer itself. The interpretation goes further and proposes a set of axioms. In standard non-relational language, one of them states that an observer can only retrieve a finite amount information from a system by means of measurement. Our contribution starts with the observation that quantum mechanics, i.e., quantum field theory (QFT) in dimension 1, radically differs from QFT in higher dimensions. In higher dimensions, boundary data (or initial data) cannot be characterized by finitely many measurements. This calls for a notion of measuring scale, which we provide. At a given measuring scale, the observer has partial information about the system. Our notion of measuring scale generalizes the one implicitly used in Wilsonian QFT. At each measuring scale, there are effective theories, which may be corrected, and if the theory turns out to be renormalizable, the mentioned corrections converge to determine a completely corrected (or renormalized) theory at the given measuring scale. The notion of a measuring scale is the cornerstone of Wilsonian QFT; this notion tells us that we are not describing a system from an absolute perspective. An effective theory at that scale describes the system with respect to the observer, which may retrieve information from the system by means of measurement in a specific way determined by our notion of measuring scale. We claim that a relational interpretation of quantum physics for spacetimes of dimensions greater than 1 is Wilsonian.

1. Foundations for Spacetime Quantum Physics

A spacetime context is universally accepted as the most convenient arena for the foundations of classical physics. As far as foundations are concerned, the setting in which time is the only independent variable is considered to be only an auxiliary setting that may be technically useful. There seems to be no reason to go back to a “time-only” context to address the foundations of quantum physics. We observed, however, that in the literature about the foundations of quantum physics, there is a remarkable elusion of aspects associated with the spatiotemporal nature of our world. The hypothesis behind this paper is that the foundations of quantum physics should not be confined to having only time as an independent variable.

Our particular goal is to contribute to the export of ideas of relational quantum mechanics (RQM) [1,2], a relational interpretation of quantum mechanics, and to complement them as appropriate to give a foundational interpretation that applies to spacetime systems: fields. Is it a trivial task? Is the essence of quantum physics captured in one-dimensional spacetimes, or are there key features of quantum physics that can only be appreciated in higher-dimensional spacetimes? The contribution of this paper is a “finite resolution postulate” for RQP, a relational interpretation of quantum physics (quantum field theory), constructed as a higher-dimensional counterpart of RQM, which brings the Wilsonian interpretation of QFT [3,4] to RQP.

Wilsonian QFT relies on the concept of scale: At a given scale (energy scale or length scale), a model with a cutoff (using a finite dimensional space of states) can describe the system up to some error. For some systems, the error may be reduced to zero by introducing more degrees of freedom and complicating the model, and the resulting “completely renormalized model” correctly describes the system at the given scale. Wilsonian renormalization is equivalent to the “systematic killing of infinities” [5,6,7], but, conceptually, it is much more enlightening. See Refs. [8,9,10] for an elucidative explanation on how the apparently ad hoc prescription for curing divergences in particle physics is justified from first principles within the Wilsonian reformulation of the renormalization group.

Renormalization is not included in quantum mechanics textbooks because, in most situations, one can make sense of quantum mechanics without appealing to a renormalization process. However, QFT without renormalization only describes free theories or exactly solvable models. The reason for this sharp difference is topological: The system may be characterized by initial data given over surfaces of codimension 1. In dimension 1, where time is the only independent variable, the “initial data surface” is a single point $t_0\in {\mathbb{R}}_t$. Thus, the space of boundary data for the path integral, which may be classical configuration space, is finite dimensional. Conversely, in dimension 2 and higher, the initial data surface has infinite cardinality, and the resulting space of boundary data for the path integral is infinite dimensional. In other words, in dimension 2 and higher, evaluating a finitely generated algebra of kinematical observables leads to an incomplete description of the boundary state. Wilsonian renormalization is a procedure that (when successful) constructs an incrementally better description of a system from which we have partial information. Thus, renormalization is relevant in dimension 2 and higher, and it is not relevant in dimension 1.

We use the notion of measuring scale proposed in [11], which is appropriate in a relational context. In a laboratory situation, the experimentalists set up an array of measuring devices, filters, and control elements that determine the algebra of observables accessible to the experimentalist. That algebra can be seen as a subalgebra of the algebra of observables that could be measured in principle. That subalgebra is the measuring scale [11]. Measuring scales are a partially ordered set, and coarse-graining operations are injective algebra morphisms associated with the inclusion of observables from coarser to finer scales. A continuum limit is also defined in this implementation of Wilsonian QFT.

The finite resolution postulate has origins in Wilsonian QFT and in RQM. We first noticed that the cornerstone of the Wilsonian view of QFT appeared to be compatible with one of the central elements of RQM. Then we brought in the appropriate notion of scale [11] that smoothly brings them together.

Section 2 gives a brief description of Rovelli’s relational quantum mechanics. Section 3 states the problem of finding a relational interpretation of quantum field theory and provides the initial ingredients. In particular, the finite resolution postulate is introduced. In Section 4, the problem of measurement/interaction is examined from a relational perspective following the formalism of Fewster and Verch [12]. A section providing a summary and outlooks concludes this article.

2. A Brief Description of RQM

The relational interpretation of quantum mechanics (RQM) proposed by Rovelli in 1996 [1] refines the Copenhagen interpretation of QM, as presented in textbooks, to (i) emphasize the fact that the key concept of the observable of the Copenhagen interpretation corresponds to an action on the system by another system, which is also quantum, and (ii) to consider the possibility of having a system described with respect to different reference systems. Paraphrasing Rovelli, RQM allows for different perspectives, whereas the standard Copenhagen interpretation implicitly considers only one.

RQM also democratizes the status of “the observer”. This does not mean that an atom knows quantum physics and can describe another atom in a certain language. RQM abandons the word “observable” with the purpose of not restricting the agents that can act on the system to agents with consciousness; the term that replaces it is “relational variable”. It is a function that describes a property of system $\mathcal{S}$ with respect to system ${\mathcal{S}}^{\prime }$, and it also describes a mutual action of systems ${\mathcal{S}}^{\prime }$ and $\mathcal{S}$ on each other.

Two interacting systems $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ do not act on each other continuously; instead, they do it in a discrete set of quantum events, and, when the action takes place, it is, in a certain sense, brusque. Relational variables play a dual role. On the one hand, their possible values describe system $\mathcal{S}$ with respect to system ${\mathcal{S}}^{\prime }$. On the other hand, they describe the interaction between $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$. As usual in quantum mechanics, relative variables do not assume definite values; instead, all the possible evaluations coexist until a quantum event associated with that relative variable occurs. In that moment, the relative variable takes one of its possible values. This central difference between classical and quantum mechanics is emphasized by RQM. In classical mechanics, one pretends that the measurement of a system can be performed with infinite care so as not to disturb the system. Quantum mechanics drops that assumption and assumes the consequences. The result is a formalism that more accurately describes “light systems”: those that are more sensitive to a brusque action. This phenomenon is described by Gamow in the dreams of Mr. Tompkins in a world with a very large ℏ. In that world, you cannot pat your pet cat: you either miss it or break its neck. This image is appropriate for a description of RQM; it emphasizes that it is not about the nature of measurement but about the nature of interaction between two systems.

In [2], the properties described in the previous paragraph furnish what is called a “sparse event ontology”.

According to RQM, QM is about describing interactions between systems in the events when they act on each other. More precisely, given two interacting systems $\mathcal{S}$ and $S’$, the goal of QM is to predict the probability distribution that values of relative $S-S’$ variables will take at future interaction events given the history of previous values of relative $S-S’$ variables.

An important element of RQM is that the above paragraph talks only about $S-S’$ relative variables and not about variables describing S with respect to any other system. For a discussion of Wigner’s friend scenario according to RQM, see [1]. For a discussion regarding the coexistence of different perspectives in the description of a system, see [2,13] and the references therein. For an implementation of a perspective-neutral framework, see [14].

The history of evaluations of relative $S-S’$ variables in previous events may be stored in a state describing $\mathcal{S}$ with respect to $S’$. It is important to note that, according to RQM, the state is a useful bookkeeping device, but it is not an ontological element of quantum mechanics. From this perspective, it is not necessary to include a “collapse of the wave function” in the foundations of QM: the element of abrupt change is associated with the “sparse event ontology”. When system ${\mathcal{S}}^{\prime }$ acts on system $\mathcal{S}$ leading to a quantum event, say at time t, in which a relative $\mathcal{S}-{\mathcal{S}}^{\prime }$ variable, say V, takes a given value, say $\lambda$, the bookkeeping device stores that information. In future events, the probabilities for the evaluation of relative variables are conditional to the fact that, at time t, variable V evaluated $\lambda$. This point of view does not solve all philosophical problems associated with measurement in QM, but it brings a new angle into the discussion. For more details, see [2] and the references therein.

The “sparse event ontology” of RQM leads to the following physical question: when do quantum events take place? In [15], the problem is solved: quantum events are sharply located in time—they happen at specific times. QM can be used to calculate the probability density in the real line for an event to happen at each given time. In Section 4, we argue that, in higher-dimensional spacetimes, the picture changes, making events fuzzy instead of sharply located.

3. Relational Quantum Physics of Spacetime Systems Is Wilsonian

Relational quantum physics (RQP) is a relational interpretation of quantum field theory (QFT). It relies on a Lorentzian spacetime background M; Minkowski spacetime is a good example to have in mind. The independent variables live in M, and the notions of locality and causality are stated with respect to M.

RQM focuses on relative variables describing systems $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ with respect to each other, and, as in standard quantum mechanics, the same variables also describe the mutual action between ${\mathcal{S}}^{\prime }$ and $\mathcal{S}$. In a spacetime setting, it is more natural to focus on the local action of a subsystem on the rest of the system. We may consider two coupled fields and describe their interaction, which would be described by relative variables locally depending on the two fields. Another situation to consider is a single nonlinear field restricted to a pair of intersecting spacetime regions. Each region determines a subsystem, and the subsystems have a local interaction at the intersection of the regions.

Many systems of physical interest are naturally described in frameworks with gauge freedom. For system $\mathcal{S}$, let us also call $\mathcal{S}$ its space of histories according to the framework with gauge freedom. Therefore, the space of physically distinguishable histories is not $\mathcal{S}$ but a quotient space that we will call $\mathcal{S}/\mathcal{G}$, with $\mathcal{G}$ being the group of gauge transformations. It happens that, when gauge systems, say $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$, couple to each other, the framework that appropriately describes their coupling is a new gauge system with a total space of histories of ${\mathcal{S}}^T=\mathcal{S}⨂{\mathcal{S}}^{\prime }$,1 and the space of physically distinguishable histories is ${\mathcal{S}}^T/{\mathcal{G}}^T$, which is not constructed by a product of $\mathcal{S}/\mathcal{G}$ and ${\mathcal{S}}^{\prime }/{\mathcal{G}}^{\prime }$. This has been known for a long time, but a very nice argument for the physical reason behind this fact can be found in [16]. It turns out that nature is relational, and systems that are relational have natural mathematical descriptions in terms of frameworks with gauge freedom coupling to each other following those rules.

With the goal of being precise, we will follow the lead of algebraic quantum field theory (AQFT); see, for example, [17]. AQFT captures a spacetime version of the Heisenberg picture of observables by associating each spacetime region with a local algebra of observables. These associations obey a set of axioms stating that local and causal relations between different spacetime regions are mapped to the algebraic properties of the corresponding local algebras of observables; the resulting structure is called a causally local net of observables. Below, we face the task of expressing these ideas in a relational language.

We will present two independent points of view, starting with a deconstruction point of view. Consider $S\left(U\right)$ the local algebra associated with spacetime region U describing system $\mathcal{S}$. Its elements are the observables localizable at U. As we mentioned before, observables play the dual role of describing the properties of a system and the actions on a system. In AQFT, there is no subject. The system is acted upon, but the acting agent is never mentioned. The only thing that AQFT records is when–where $\mathcal{S}$ received the action. It may be a useful abstraction to think that it does not matter who acted on the system as long as it has the same effect, but we will not take that point of view. Additionally, the physical context should clarify the issue and tell us who acted on the system. We will appeal to contextuality. In a situation in which we have two coupled fields $(\varphi ,\psi )$, we may assume that $\varphi$ was locally acted upon by $\psi$, and vice versa. This will be carried out within a context described in “the interaction picture” following [12]; we give a more detailed explanation in Section 4. Also consider a situation where there is a single field $\varphi$, and a spacetime region V is used to define ${\varphi |}_{\overline{V}}$, the system of interest, which is described with respect to ${\varphi |}_{\overline{M\setminus V}}$, the rest of the system. In that situation, we may assume that ${\varphi |}_{\overline{V}}$ is being locally acted upon by the rest of the system in $\partial V$, their interaction region. Thus, when they write $S\left(U\right)$ in AQFT, we will interpret it as an abstraction for $\mathcal{S}-{\mathcal{S}}^{\prime }\left(U\right)$, the algebra of relational $\mathcal{S}-{\mathcal{S}}^{\prime }$ observables localizable at U.

The other point of view considers two coupled spacetime systems, $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$, which, together, make system $\mathcal{C}$. The algebraic description of the coupled system would determine a net of observables ${\left\{\mathcal{C}\left(U\right)\right\}}_{U\subset M}$. Relational variables describing $\mathcal{S}$ with respect to ${\mathcal{S}}^{\prime }$ are self-adjoint elements of $\mathcal{C}\left(U\right)$. One may consider that relational observables are only those that are invariant under isometries of the background spacetime M and that non-invariant information is regarded as gauge2. We may consider whether variables that are invariant under isometries are necessarily relational. The point of view now described starts with the definition that relational variables are those that are invariant under the appropriate group of gauge symmetries. This elegant and powerful point of view is used in the Perspective-Neutral framework for Quantum Reference Frames [14].

Wilsonian renormalization is not easy to find in the principles of AQFT. The physical interpretation of theories that fulfill AQFT’s axioms is that they correspond to completely renormalized theories. That is, they could be free theories, exactly solvable models, or they may be constructed following the continuum limit of Wilsonian renormalization. In the latter case, they correspond to the organization of completely corrected effective theories at all possible measuring scales. More precisely, the limit involves an inverse limit (i.e., with arrows going in the opposite direction to the coarse-graining maps) of effective theories and rescaling. Since all possible measuring scales are considered, the complete algebra associated with a nonempty spacetime region would not be finitely generated. For renormalizable theories, however, the Wilsonian continuum limit tells us that the algebra of observables can be approximated by finitely generated algebras after an appropriate rescaling and limiting procedure. This consequence of Wilsonian QFT is behind the results stating that the algebras of AQFT are hyperfinite von Neumann type $II{I}_1$ factors [18]; for an overview and more references, see [19].3

Now we will argue that extending the ideas of RQM to a spacetime setting naturally leads to a Wilsonian point of view on QFT. Moreover, the operational philosophy of the relational point of view fits directly with the description of effective theories that have not yet been renormalized. The specific proposal that we justify below will take the form of a “finite resolution postulate” in RQP.

In the original version of RQM [1], the Postulate of Limited Information stated, “There is a maximum amount of relevant information that can be extracted from a system”.

In the more recent presentation [2] that we follow in Section 2, the same proposal is presented with a change of emphasis. In that reference, the Postulate of Limited Information is described saying that “relevant information is finite for a system with compact phase space”.

Since phase spaces in field theory are always infinite dimensional, it is clear that the idea behind the Postulate of Limited Information must be appropriately generalized before applying it to spacetime systems.

The appropriate generalization is suggested by Wilsonian QFT. Let us start by considering a laboratory situation where the system of interest $\mathcal{S}$ is a field. The experimentalist sets up a collection of measuring devices, filters, and controllers. The measurements taking place during the experiment are the result of the actions of system $\mathcal{S}’$ on $\mathcal{S}$ controlled by the experimentalist. The controlled environment crafted by the experimentalist may be mathematically described as a set of observables $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$, where U is the spacetime region in which the experiment takes place, and $\lambda$ is a label that lets us distinguish one laboratory setup from another one. The observables in $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$ correspond to the self-adjoint elements of a star algebra $\widetilde{\mathcal{S}-{\mathcal{S}}^{\prime }}(U,\lambda )$. The physical situation demands two important properties in an acceptable algebra $\widetilde{\mathcal{S}-{\mathcal{S}}^{\prime }}(U,\lambda )$: (i) It needs to be finitely generated. (ii) Furthermore, its elements need to reflect the fact that the experimentalist resources are finite; $\mathcal{S}’$ does not act on $\mathcal{S}$ with infinite energy, or, equivalently, $\mathcal{S}’$ cannot act on $\mathcal{S}$ sharply measured properties at points.

We propose to condense both required properties into one saying that “the relational information contained in a quantum event is finite”. Let us explain. In the laboratory situation described above, quantum events result from the interaction between ${\mathcal{S}}^{\prime }$ and $\mathcal{S}$, as described by some element of $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$. The defining property of the event is that the “responsible variable” has a definite evaluation. Thus, the event contains the following relational information: (I) the element of $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$ that caused the event and (II) the value that the variable takes at the event. By requiring that the relational information contained in a quantum event is finite, we constrain the amount of information of types (I) and (II), which implies that the required properties hold.

Let us clarify what we mean by “relational information” by using an example. Consider that $\mathcal{S}$ is electromagnetic radiation and that ${\mathcal{S}}^{\prime }$ is a field describing a (monochromatic) photographic plate. A monochromatic photographic plate could be seen as a device measuring the position within a screen, but it would be more accurate to describe it as an array of detectors asking the electromagnetic radiation, “is your energy $h\nu$?”. When the energy has that value, the detector has a certain probability of detecting it producing a quantum event. After the plate detects a photon, the position in the screen is recorded in two ways. First, the particular detector of the array can be identified. We consider this relational information because it refers to the relation between $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$. Independently, the location of the detector can be given with arbitrary precision, but we do not consider that information to be relational. It refers to the relation between (the state of) ${\mathcal{S}}^{\prime }$ and spacetime. The initial preparation of the state of ${\mathcal{S}}^{\prime }$ is what keeps the information of exactly where the experimentalist placed the photographic plate (the array of detectors).4 Since it is not information directly associated with the $\mathcal{S}-{\mathcal{S}}^{\prime }$ interaction leading to the quantum event, we do not classify it as relational information contained in the event. We consider that the relational information contained in the event includes the element of the set of observables $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$ (up to rescaling) that triggered the event and the evaluation obtained, which, in this example, was only a yes or no value.

It seems that the laboratory situation described above is too restrictive to extract lessons from it for QFT in general situations, but the continuum limit is the tool that lets us extrapolate. In the continuum limit of Wilsonian QFT, as implemented in [11], the observables of any possible laboratory setup are considered. A “laboratory setup” is modeled by a subalgebra of kinematical observables $\mathcal{S}-{\mathcal{S}}^{\prime }(U,\lambda )$ called a measuring scale. A preferred, partially ordered, and directed set of measuring scales is introduced. The mentioned properties of the set of measuring scales are necessary for the continuum limit to make sense. Two basic ingredients are involved in the continuum limit: The first is a set of regularization maps bringing any possible laboratory setup to each of the preferred measuring scales. The second is a set of coarse-graining maps that take observables from finer to coarser scales. For the continuum limit to work, two limiting processes need to converge. First, an arbitrary observable has a collection of substitutes, one for each measuring scale. There is a theory at each scale, and, if the predictions determined by those theories regarding the substitute observables converge, we have a prediction for the original observable. Second, the theory corresponding to a given measuring scale is constructed as a limit of effective theories. Those effective theories use only finitely many degrees of freedom; in other words, a cutoff is placed, making their space of histories finite dimensional. Each effective theory has adjustable parameters (“coupling constants”), which are fine-tuned according to a renormalization prescription that intends to “make them all model the same physics at large scales”. It is expected that the expectation value of an observable is calculated with greater accuracy if it is calculated using the effective theory corresponding to a finer measuring scale, with a cutoff that uses a larger set of degrees of freedom. Then, in [11], the pull-back of the coarse-graining maps is responsible for removing the cutoff and correcting effective theories. This process simultaneously allows for more and more degrees of freedom in the description and eliminates the arbitrariness in the construction of effective theories. If the descriptions converge as the cutoff is removed, the limit defines a “completely corrected theory”, which is the one used in the first limiting process of the continuum limit. For details and an example of an interacting relativistic QFT constructed following this procedure, see [11].

Does the statement “the relational information contained in a quantum event is finite” hold true in the continuum limit? At first glance, the phrase “continuum limit” might suggest that an infinite number of degrees of freedom must be considered, implying that a real event inherently contains infinite information. However, this intuition is misleading. According to the Wilsonian perspective on the continuum limit, any physical situation can indeed be represented by a finitely generated algebra. Furthermore, the finite resolution postulate implies that the allowed observables must have a finite set of possible evaluations, and this condition also survives the continuum limit.

Initially, situations are tentatively described using effective theories that involve a cutoff. As this cutoff is progressively removed, these tentative descriptions may converge towards a stable and consistent representation. The information content of a quantum event itself remains fixed; what changes is the complexity of the computational resources required to produce an accurate description, potentially growing without bound as the cutoff diminishes. This happens because the removal of the cutoff in an effective theory does not change the algebra of observables that it describes. The intrinsic informational content of an event is tied to the algebra of observables available in the context in which the event takes place.

A crucial test for the convergence of the continuum limit is the renormalizability of the theory in the Wilsonian sense. Renormalizability ensures that, as the cutoff is removed, both the qualitative behavior and numerical predictions of the theory stabilize and converge. Thus, provided that this convergence occurs, statements such as “the relational information contained in a quantum event is finite” indeed remain valid in the continuum limit.

Therefore, we propose that, for spacetime systems, the Postulate of Limited Information of RQM should be generalized as follows:

• Finite Resolution Postulate:
  “The relational information contained in a quantum event is finite.”

Now that we stated an element of RQP as a postulate, it seems appropriate to include the cornerstone of the whole approach also as a postulate, which would be written beforehand in a logical presentation.

• Sparse Event Ontology Postulate:
  “The set of quantum events associated with the interaction between two systems is a discrete set.”

Note that, in laboratory situations, the locus of interaction can be confined to a compact set K. Therefore, the Sparse Event Ontology Postulate implies that the set of events associated with the $\mathcal{S}-{\mathcal{S}}^{\prime }$ interaction is a finite set. Then the finite resolution postulate implies that the entire set of events that can take place in an experiment contains finite relational information.

4. “Measurement”/Interaction According to RQP

A laboratory situation in which a measurement takes place can be described by using a relational language. In QFT, the literature dealing with the fundamental interpretational aspects of measurement is not broad. In the first part of this section, we will describe the measurement framework presented by Fewster and Verch [12] from the point of view of RQP (from the perspective of the experimentalist). We will close this section with remarks regarding a possible parallel study of the same situation describing the system of interest directly with respect to the probe system, ignoring the experimentalist. We will focus on two issues: First, we will discuss the nature of quantum events in a purely relational description. We will then argue that the finite resolution postulate favors a relational description with a Wilsonian point of view.

Consider the situation described in [12], where there is $\mathcal{A}$, a system of interest; $\mathcal{B}$, a probe system; and an experimentalist who, after the $\mathcal{A}-\mathcal{B}$ interaction takes place, measures an aspect of $\mathcal{B}$ (for example, the position of the pointer of the apparatus) in a quantum mechanical sense. The assumptions about the systems are that $\mathcal{A}$ and $\mathcal{B}$ are fields following the axioms of AQFT,5 which interact only within a compact region K. Another assumption is that the experimentalist independently prepared systems $\mathcal{A}$ and $\mathcal{B}$ before their interaction in states $\omega ,\sigma$, respectively.

The point of view in Ref. [12] regarding states coincides with that of RQP, as described in Section 2: The state is a bookkeeping device. The bookkeeping log is used to calculate the probabilities of possible evaluations as conditional to be consistent with the log (and according to the known states $\omega ,\sigma$ in the past of K). Note that, when time is the only independent variable, as in QM, a bookkeeping device is simply an ordered list, and, for spacetime systems, the bookkeeping needs to associate events with spacetime loci.

Let us be more precise when we talk about the observables that describe the measurement. Systems $\mathcal{A}$ and $\mathcal{B}$ interact inside K and behave as independent systems outside K. One option is to describe the interaction using the relational variables of the coupled system $\mathcal{C}$. This is not only difficult but is also not readily useful in the given laboratory situation. Instead, Ref. [12] describes the interaction using the variables in $\mathcal{U}$, a hypothetical uncoupled system that behaves exactly the same as the coupled system within spacetime regions that do not intersect K. This point of view is adequate for describing $\mathcal{A}$ with respect to $\mathcal{B}$, as if it were not coupled to $\mathcal{A}$. At the mathematical level, for regions V that do not intersect K, there is a canonical isomorphism between the algebras $\mathcal{C}\left(V\right)$ and $\mathcal{U}\left(V\right)=\mathcal{A}\left(V\right)⨂\mathcal{B}\left(V\right)$. The measurement of a property of $\mathcal{B}$ by the experimentalist is taken as being associated with the variable $1⨂B\in \mathcal{U}\left(V\right)$ for some region V in the future of K. Note that the same variable has a corresponding element $O_B={\gamma }^{+}(1⨂B)\in \mathcal{C}\left(V\right)$, the coupled system, where ${\gamma }^{+}$ is the canonical isomorphism between the uncoupled and coupled systems for regions in the future of K. We then see that $O_B$ is the Heisenberg evolution of a variable describing the interaction between $\mathcal{A}$ and $\mathcal{B}$ at the interaction region K. The evaluation $\lambda$ that the experimentalist reads and writes in the log corresponds to the observable $O_B$ of the coupled system that interacted at K and provides information about system $\mathcal{A}$ described with respect to its interaction with $\mathcal{B}$ (from the perspective of the experimentalist).

Fewster and Verch also find an “induced variable” $A\in \mathcal{A}\left(V^{\prime }\right)$ (for a region $V’$ in the past of K) that has the property of having the same expectation value (in state $\omega$) that $O_B={\gamma }^{+}(1⨂B)$ has in the state of $\mathcal{C}$ corresponding to the uncoupled state $\omega ⨂\sigma$ of $\mathcal{U}$. They do not claim that $O_B={\gamma }^{-}(A⨂1)$. This helps them in the construction of a “state update rule”.

The mathematical setting of Fewster and Verch gives us the information that the interaction took place within K. In [12], they show that, if a set of measurement variables of the same type are considered (i.e., observable $O’$ takes value $\lambda ’$ due to interaction within a compact region $K’$, etc.), the corresponding expectation values calculated as conditional to the other measurements in the log satisfy the properties that one could expect from the causal structure in M. For details, see [12].

Let us now give a list of remarks regarding the measurement framework according to RQP. We will focus on issues that would be critical in describing the same “measurement”/interaction situation from a purely relational point of view. This exercise will help us refine this interpretation of QFT before we extend its principles to more general situations.

(i) The event triggered by $O_B$. As mentioned above, in [12], it is proven that the measurement described only affects other measurements of the same type if the other coupling region is causally in the future of K (unless there is a correlation of the induced observables according to $\omega$). Thus, we do not know the precise location of the event triggered by $O_B$, but we know that it takes place inside K. We could try to find a probability distribution for its location inside K following a spacetime analog of the procedure used by Rovelli to solve the problem of finding the moment in time when an event in RQM takes place [15]. The procedure in [15] can be followed to find the moment in time when an event takes place, but the issue of location in spacetime requires further localization: Recall the example of the photographic plate described in Section 3. There, we argued that the relational information contained in the event could be used to determine exactly which one of the detectors that conform the photographic plate detected the photon. Independently, the precise location of the plate could be determined; this was not considered to be relational information contained in the event. Therefore, the objective of calculating a probability distribution for the location of the event from relational information makes sense only within that context. In our example, we could use relational information to calculate the probability that a given detector is activated, but we could not find its precise spacetime location from that information. Additionally, if we know what detector was activated, as well as the exact position of the plate, would that provide a sharp location for the event? No. Even if the detector in the photographic plate is very small, it is not a mathematical point. Extrapolating from the example to the general situation requires an interpretation of the relational variables available at a given interaction scale. Their interpretation provides coarse-grained information about system $\mathcal{S}$ with respect to system ${\mathcal{S}}^{\prime }$. Even when coarse graining is modeled by decimation, the interpretation of the variables provides a representative sample. Thus, with relational information at a given interaction scale, the event can only be located in a coarse-grained context: its location is fuzzy (and determinable only at the probabilistic level as usual in quantum physics).

From the point of view where “relational” is taken to mean invariant with respect to the relevant automorphism group [14], which, in this situation, is the Poincaré group, the position of the plate is clearly not relational information.

(ii) The complete relational information content of the experiment and Wilsonian QFT. The Sparse Event Ontology Postulate, the finite resolution postulate, and the compactness of K imply that the set of events that take place due to the interaction at K contains a finite amount of relational information. Thus, the interaction scale $\mathcal{A}-\mathcal{B}(K,\lambda )$ obtained as the largest common subalgebra of the set of algebras associated with regions containing K is finitely generated, and the set of evaluations of any of its variables is finite. It is then natural to describe this situation using an effective theory using an algebra that is a type I factor. It may be the case that the effective theory can be completely corrected and that a continuum limit yields a concrete procedure to approach the type $II{I}_1$ factor, which describes the continuum limit by means of type I factors. We only want to point out that following this procedure is natural in a purely relational framework since $\mathcal{A}-\mathcal{B}(K,\lambda )$ happens to be finitely generated in all laboratory situations.

The above statements may seem in conflict with describing systems whose symmetry group is a nontrivial Lie group. The key to disentangling the apparent paradox is that, above, we are talking about relational information. It is also relevant that, in the process of approaching a type $II{I}_1$ factor by means of type I factors, the symmetry of the system is only recovered in the continuum limit. An illuminating example can be found in [11].

For a particular laboratory situation, the experimentalist can tell us what set of observables they have access to; we know that it must be a subset of the set of self-adjoint elements of $\mathcal{C}\left(K\right)$ and that all of these variables are of type $O_B$ for some $B\in \mathcal{B}\left(V\right)$ for a region V in the future of K.

In the description of the interaction with respect to the probe system (where the experimentalist is not considered), the elements of $\mathcal{C}\left(K\right)$ that should be called relational variables are not of type $O_B$ described above. Instead, relational variables correspond to mutual actions between $\mathcal{A}$ and $\mathcal{B}$. If we want to describe them from the point of view of the hypothetical uncoupled system, where the interaction is still assumed to be confined to a compact set K, the variables should be a sum of terms of type $\gamma (A_i⨂B_i)$, where the pairs $(A_i,B_i)$ describe reciprocal actions.

5. Summary and Outlook

As we emphasized in the Introduction Section, for interacting QFTs, Wilsonian renormalization is necessary. This is very different from the “accidental situation of QM”, which considers a one-dimensional spacetime. Spacetimes of dimension 1 have Cauchy surfaces of dimension 0, which means that they have finite dimensional spaces of initial conditions unlike what happens in higher dimensions.

In this paper, motivated by the Wilsonian view of QFT, we propose generalizing the Postulate of Limited Information of the original formulation of RQM to apply it to spacetime systems in the form of the finite resolution postulate. The form of the postulate fits so naturally in the relational view of RQM that it is tempting to say that the relational point of view of quantum physics leads to the cornerstone of Wilsonian QFT. Our presentation in Section 3 and Section 4 provides supporting elements.

We argued in Section 4 that an event that took place at a given interaction scale cannot be sharply located by means of relational information. Similarly, the prediction of the probability distribution for the location of future quantum events can only be carried out with the resolution available at the relevant interaction scale. This is an important difference to what happens in QM, where quantum events are sharply located in time.

An important element of the measuring scheme of Fewster and Verch [12] is the consideration of a situation in which the interaction among systems $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ is confined to a compact set K. This situation permits a description of the coupled system in terms of a hypothetical uncoupled system and a scattering map. Clearly, the general situation for interacting fields is not of this type. We have plans to address this problem in the near future. As a spoiler, we are modeling the interaction region as the gluing of cells. Full cells with a compact dimension host a coupled system and a hypothetical uncoupled system. Thus, the dynamics of the coupled system can be cast in terms of a “scattering map” acting on the restriction of the uncoupled system to the boundary. The gluing of neighboring higher-dimensional cells to model larger domains of interaction is then performed at the level of the uncoupled systems by the imposition of appropriate gluing equations.

In the Introduction Section, we mentioned that gauge systems are among the most important systems in physics. We also mentioned that the coupling of these types of systems is described at the non-gauge invariant level. The reason for their importance in physics and for the way in which they couple to each other seems to be tied to physics being essentially relational [16]. Thus, work on the relational interpretation of quantum physics should address gauge systems. We plan to address this issue together with the problem of extending to unconfined interaction domains.

In the modeling of real-life laboratory situations and non-ideal general situations, the essential tool is statistical quantum field theory. This fact, together with the abovementioned ubiquitous appearance of gauge systems, suggests that, in order to extend the considerations of this paper to non-idealized situations, a framework for the quantum field theory of gauge systems at finite temperature should be considered (see, for example, [22]).

The perspective-neutral framework in [14] and the Positive Formalism of the General Boundary Formulation [23,24,25] treat quantum fields with the goal of being relational in some way. It would be interesting to take into account the finite resolution postulate and use it as the initial ingredient to incorporate Wilsonian renormalization into those frameworks.