Operational verification of the existence of a spacetime manifold

Let us introduce the following gedanken-experiment. We are given a swinging pendulum, denoted $A$. Without any assumptions about Newtonian mechanics (since it relies on the a priori notion of time), our objective is to describe the motion of the pendulum, as precisely as we can. The measurement apparatus we have available is a camera that can take still photos of the pendulum, along with a ruler that can measure the distances on the photos.

Start by taking $N$ photos of the pendulum, completely randomly, and use a ruler to measure the signed distance $a_{k}$ of the pendulum from its vertical axis, for every photograph $k\in\{1,\dots,N\}$. The distance is signed in the sense that if the pendulum is left of its axis we consider the distance to be negative, while if the pendulum is right of its axis, the sign of distance is positive (the choices of left and right are merely a convention and do not influence any conclusions). Moreover, we want to eliminate any information about the “time order” in which the photos might have been taken, so we remember only the following unordered set of measurements, describing what we can tell about the motion of pendulum $A$,

where

Here $a_{min}$ and $a_{max}$ denote the left and right amplitudes of the pendulum, respectively. In usual circumstances one should obtain $a_{min}=-a_{max}$, but for our purposes this equality does not really matter. Also, we assume that no two measurements produce exactly equal values of $a$, so that $a_{i}\neq a_{j}$ for all measurements $i$ and $j$. This assumption is just for convenience, and we will discuss later the conceptual case in which the measurement results are repeating themselves, and always falling into a discrete set of outcomes.

Given the measurement dataset $A$, one can draw it as a one-dimensional scatter plot on the $a$-axis, which looks similar to the plot in FIG. 1.

If we take additional $N$ measurements (i.e., a total of $2N$), the scatter plot will look similar, only with twice as many points in the segment $S_{A}$. In the limit $N\to\infty$, the scatter plot will become dense, and practically fill the whole segment $S_{A}$, given that we can measure distances $a_{k}$ with an arbitrarily small but still nonzero precision. The notion of arbitrary small distance is not in conflict with quantum mechanics, since we are measuring only the position of the pendulum, and not its momentum. With sufficiently many measurements, we can describe the motion of the pendulum using the whole set $S_{A}$ of possible positions it might be in, and we can call this set the configuration space of the pendulum.

However, our naive intuition suggests that such a description of pendulum’s motion is not completely satisfactory, since each photo demonstrates that the pendulum is at some particular distance $a$, while the configuration space of the pendulum describes the pendulum merely “in all positions”, suggesting that we are missing some extra information. We therefore ask for a more precise description of the pendulum’s motion. The precise formalization of this “intuition” will be given below, but at this stage let us appeal to the ideas of relationalism, and try to obtain the missing information by comparing the motion of the pendulum against some other physical system.

For lack of a better idea, let us introduce another pendulum, denoted $B$, in addition to $A$. In a generic situation, the pendulum $B$ may have different length and other properties compared to $A$. We put the two pendulums side by side, and randomly take photos of the whole system. However, in order to eliminate any notion of “simultaneity” of measurements between $A$ and $B$, we cut each photo into two pieces, such that each pendulum is displayed separately on its half of the photo. As before, we now shuffle all photos to erase the order in which the photos were taken, as well as the relation which photo of $A$ is paired to which photo of $B$. We end up with the following unordered datasets for the two pendulums:

After sufficiently many measurements, we can establish the individual configuration spaces for each pendulum like before,

Again, we have an intuition that this is not a complete description of the motion of the two pendulums. But now we can formalize this intuition as follows. Given the individual configuration spaces $S_{A}$ and $S_{B}$, a priori one can say that the joint configuration space will be the set $S_{A}\times S_{B}$. One can visualize this experimental result within this set graphically, as follows. Pick a random permutation $\pi$ of $N$ elements, and construct the following $2\times N$ matrix,

which describes an arbitrary “pairing” of each measurement $a_{k}$ to some measurement $b_{\pi(k)}$. These pairings define $N$ points on a scatter plot in the space $S_{A}\times S_{B}$, which will typically look like the one in FIG. 2.

One can draw a similar scatter plot for every choice of the permutation function $\pi$. Since for $N$ measurements there are $N!$ possible permutations, there are also $N!$ scatter plots. But, as an experimental result of our thought experiment, among all those scatter plots, there will be one special plot, corresponding to the permutation denoted as $\tilde{\pi}$, which is visually very distinguishable from the rest, and looks like the one in FIG. 3.

Looking at the plot, it is straightforward to interpret it as a set of points on a $1$-dimensional curve, denoted ${\cal C}$. The experimental fact that the joint configuration space $S_{A\cup B}$ is equal to the curve ${\cal C}$ has one important consequence — it captures our intuition of a more precise description of the motion of pendulums $A$ and $B$. This is because the curve ${\cal C}$ is equivalent to a functional dependence (a correlation) between observables $a$ and $b$, of the general form

The equation (6) actually defines the curve ${\cal C}$ as a set of all pairs $(a,b)\in S_{A}\times S_{B}$ which satisfy (6). At this point we can spell out the formalization of our intuition for the results of the thought experiment, in the form of the following conjecture.

Note that, while visual distinguishability is appealing, it is a particular pattern-recognition trait of a human brain. In order to have a more formal definition of why this particular diagram is “special”, without appealing to our eyes and brains, we can resort to the statistical analysis described in detail in the Appendix A. This can be implemented as a computer algorithm operating on dataset (3), eliminating any reliance on visual inspection of scatter plots. Also, the formal statistical analysis has an advantage of being applicable to higher dimensions, i.e., beyond the $2$- and $3$-dimensional scatter plots, in contrast to visual inspection by a human. This property will become important in Section III.

The “special” permutation $\tilde{\pi}$, corresponding to the “special” plot above, has several crucial properties:

Once the existence and properties of the $1$-dimensional curve ${\cal C}$ have been established, one may be tempted to call it the “time manifold”. However, there are two problems associated with that. First, in our case of the two pendulums the curve ${\cal C}$ is almost certainly self-intersecting, preventing it from being a manifold in the proper sense. Second, if we denote the length of the pendulums $A$ and $B$ as $l_{A}$ and $l_{B}$ respectively, it may happen that their square-roots are incommensurate,

Note that if we recall Newtonian mechanics, (10) in fact means that the periods of the two pendulums are incommensurate. Of course, in order to be consistent, we have to assume that we have no knowledge about either Newtonian mechanics nor about the concept of a “period”, so we refrain from this interpretation. Nevertheless, we are still allowed to measure the lengths $l_{A}$ and $l_{B}$ using a ruler, calculate their square roots, and discuss the validity of (10) in a given experiment. And if it happens that (10) holds, in the limit $N\to\infty$ the curve ${\cal C}$ becomes a space-filling curve (passing through almost every point in the rectangle $S_{A}\times S_{B}$), invalidating the statement that it is $1$-dimensional, in the sense of the analysis discussed in Appendix A.

In order to work around these two issues, it is useful to enlarge our physical system yet again.

In addition to pendulums $A$ and $B$, let us introduce yet another pendulum, denoted $C$, and repeat the whole analysis, taking photos of all three pendulums side-by-side. This time we end up with the dataset

After sufficiently many measurements, we can establish the individual configuration spaces for each pendulum, as before,

We can now expect that the joint configuration space will be a measure-zero subset of the $3$-dimensional box $S_{A}\times S_{B}\times S_{C}$. To see this, introduce two arbitrary permutations of $N$ elements, $\pi$ and $\rho$, and construct the following $3\times N$ matrix,

which describes an arbitrary ordering of our dataset into triplets $(a_{k},b_{\pi(k)},c_{\rho(k)})$, $k=1,\dots,N$. We use these triplets as data points in the $3$-dimensional scatter plot whose domain is the box $S_{A}\times S_{B}\times S_{C}$. One can construct $(N!)^{2}$ such scatter plots, one for each choice of the permutations $\pi$ and $\rho$. A typical plot will have data points randomly distributed and filling up the whole box, as in FIG. 4.

However, either by using visual inspection over $(N!)^{2}$ such diagrams, or by using the data-analysis approach from Appendix A, one can again establish the existence of the “special” diagram, see FIG. 5,

corresponding to the “special” permutations $\tilde{\pi}$ and $\tilde{\rho}$.

There are now several new features to be discussed. To begin with, as an experimental result of our thought-experiment, we can establish that our dataset $(A,B,C)$ together with the permutations $\tilde{\pi}$ and $\tilde{\rho}$, satisfies all three previously discussed properties of existence, self-reinforcement and dimensionality. Therefore, the dataset again gives rise to a $1$-dimensional curve ${\cal C}$, as an an experimental result. In principle, the addition of the third pendulum could have rendered the data-points distributed along some $2$-dimensional surface in the box rather than a $1$-dimensional curve, while still being compatible with the $1$-dimensional curve if one only looks at the projection defined as the $(A,B)$ subset of the data. But such a scenario did not happen, and instead the data points are still aligned along a $1$-dimensional curve ${\cal C}$. Therefore, it is a genuine experimental signal that the joint configuration space has $1$-dimensional structure. Moreover, this conclusion survives if one enlarges the physical system even further, by adding additional pendulums, or any other (even aperiodic) mechanical systems. From the point of view of our thought experiment, it is a completely general result.

Second, the $1$-dimensional nature of the curve ${\cal C}$ is equivalent to the set of two correlation functions between three observables $a,b,c$,

In general, if we have a very big mechanical system of $K$ observables $a^{(1)},\dots,a^{(K)}$, we will find in total $K-1$ correlation functions

whose set of solutions describes a $1$-dimensional curve ${\cal C}$ in a $K$-dimensional configuration space.

Next, given the curve ${\cal C}$, one can project it onto the $(a,b)$ plane, simply by ignoring the value of the observable $c$, so that one recovers a curve defined by the points $(a_{k},b_{\tilde{\pi}(k)})$ in the space $S_{A}\times S_{B}$. Assuming that the datasets $A$ and $B$ are identical to those used in the previous, two-pendulum example, the “special” permutation $\tilde{\pi}$ that we have found in the previous subsection will be identical to the one we found now. In other words, the presence or absence of the dataset $C$ in the statistical analysis of the Appendix A will not change the permutation $\tilde{\pi}$, although a priori this is mathematically possible. The fact that this does not happen in our thought experiment is a consequence of the nontrivial nature of the dataset $(A,B,C)$.

Since the permutation $\tilde{\pi}$ is independent of the presence of the third dataset, we can use the third dataset to “resolve” the self-intersecting points of the projection of the curve ${\cal C}$ to the space $S_{A}\times S_{B}$. Namely, the presence of the third observable establishes that we are actually observing a projection of two different points of the curve ${\cal C}$ onto the same point in the $S_{A}\times S_{B}$ subspace. Specifically, if we have the following two points on the curve ${\cal C}$,

such that $a_{k}\approx a_{m}$, $b_{\tilde{\pi}(k)}\approx b_{\tilde{\pi}(m)}$ but $c_{\tilde{\rho}(k)}\neq c_{\tilde{\rho}(m)}$, we see that although the points ${\cal P}_{k}$ and ${\cal P}_{m}$ may belong to quite distant parts of the curve ${\cal C}$, they will both project to the “same point” in the subspace $S_{A}\times S_{B}$, leading to the apparent intersection. This is again a general feature of mechanical systems — if our curve ${\cal C}$ happens to self-intersect anywhere, we can always enlarge the physical system by adding another observable which will “distinguish” between the two appearences of the apparent intersection point, resolving it into two distinct points.

Finally, let us just shortly note that the issue of space-filling curve ${\cal C}$ can be eliminated by extending the physical system with an observable $x$ which has a noncompact domain $S_{X}$, and can be considered to be monotonically increasing, such as entropy or whatever else is at hand. This means that our big configuration space will be noncompact, and the curve ${\cal C}$ will intersect every hypersurface $x=const$ precisely once, sidestepping any issue of space-filling curves.

The thought experiment described above provides an operational protocol to establish the existence of a $1$-dimensional, non-self-intersecting, non-space-filling curve ${\cal C}$ as a joint configuration space of a given number of observable mechanical degrees of freedom. The curve ${\cal C}$ has all the hallmarks of a manifold (it is a nonempty set with a well-defined topology and dimension), and in order to properly promote it into a manifold, all we need is a set of coordinate charts, i.e., an atlas. Assuming for simplicity that ${\cal C}$ has a topology of an open line, it is enough to consider a single coordinate chart across the whole curve (as opposed to a circle which requires at least two charts), called time chart:

Being a chart, i.e., a homeomorphism, this map is invertible, giving rise to parametric equations of motion for the observables $a,b,c,\dots$,

This map defines $a(t)$, $b(t)$, etc., as functions of the time coordinate $t\in\mathbb{R}$, and they satisfy the correlation functions (15). Of course, one can introduce a different chart $T^{\prime}:{\cal C}\to\mathbb{R}$, such that the composition $\phi\equiv T^{\prime}\mathop{\circ}\nolimits T^{-1}:\mathbb{R}\to\mathbb{R}$ is a homeomorphism of the real line. One can assume that $\phi$ is a smooth function, in which case it is a diffeomorphism. It defines the time reparametrization transformation, as

which keeps the correlation functions (15) invariant. This becomes obvious if we use the composition notation $F_{k}(a(t),b(t),\dots)\equiv(F_{k}\mathop{\circ}\nolimits T^{-1})(t)$, so that

obviously implies

for all $k=1,2,\dots,K$, where $\phi^{-1}$ is a coordinate transformation from $t^{\prime}$ to $t$. Since the curve ${\cal C}$ is in fact the joint configuration space of $K$ observables, and it is defined by the equations $F_{k}=0$, the invariance of the correlation functions establishes the invariance of the joint configuration space ${\cal C}$ under the action of the $1$-dimensional group of diffeomorphisms $Di\!f\!\!f(\mathbb{R})$. This feature of ${\cal C}$ is called the time reparametrization invariance, and the group $Di\!f\!\!f(\mathbb{R})$ is a subgroup of the larger group of spacetime diffeomorphisms, $Di\!f\!\!f(\mathbb{R}^{4})$, as we shall see in the next section.

Let us finish this section with a remark that the existence of the curve ${\cal C}$ in our dataset means that it is impossible to “erase” the information about coincident measurements of the observables. Recall that we were “cutting the photos” to display only individual pendulums, in order to erase the information about the pairing of experiment outcomes. The existence of the curve ${\cal C}$ fully recovers that information from our dataset, which means that this information cannot be erased. In other words, the information about “conditional measurements” — measuring the position of the pendulum $A$ under the condition that pendulum $B$ has some given position — has its own objective existence, and is encoded in the correlations present in the dataset.

In Section II we have introduced the time manifold by looking at a mechanical system, while in Section III we have introduced the spacetime manifold, looking at fields instead. However, one may ask a natural question about the notion of a space manifold, and its differences from spacetime. To that end, let us discuss another illustrative example of the application of our gedanken-eksperiment.

Consider a room with a lamp and some furniture. The experimental apparatus used to perform measurements over this system consists of two cameras separated several centimeters apart. Each camera performs a measurement that provides the following data:

• •

  the polar and azimuthal angles $\theta$ and $\varphi$ of the incident light ray,

• •

  the frequency $\nu$ and intensity $I$ of the ray.

Since we have two cameras, in total there are eight observables per measurement. As always, we collect the measured data ignoring any order, and perform the analysis of the protocol, to reach the conclusion that there are five correlation functions between observables, which means that all measurements can be arranged on a 3-dimensional manifold. Intuitively, this camera setup corresponds to the stereoscopic eyesight, that gives one a perception of depth and thus the notion of a 3-dimensional space of the room. One should note that the notion of time is absent from this description, since the scene of the room is static. Hence the resulting 3-dimensional manifold deserves the name space manifold.

Nevertheless, one can also consider a room with a lamp and furniture, and additionally a cat moving around inside. If we now perform the same type of measurements of the system with identical cameras as before, we will find only four correlation functions between eight observables, which means that all measurements can be arranged on a 4-dimensional manifold, rather than a 3-dimensional one. It is obvious that the motion of the cat renders the dataset fundamentally more complicated, with less amount of correlation. In other words, the observed 4-dimensional manifold corresponds to spacetime, since in this case the scene of the room is not static anymore.

This example illustrates the dependence of the outcome of the experiment on the properties of the system being observed. It may happen that a system has a high level of symmetry (in the example above, the time-translation invariance), which introduces additional correlations into the dataset and lowers the dimension of the resulting manifold. This is the main way to distinguish space from spacetime — space is in fact spacetime with a certain global symmetry, which renders time unobservable. Similarly, the time manifold from the pendulum example in Section II is also spacetime with a global space-translation symmetry (the pendulum swings the same way regardless of its spatial position), which renders space unobservable.

It is not completely obvious what would change regarding the outcomes of our thought experiment, if one were to take into account the effects of non-commuting observables, i.e., quantum effects. On one hand, the above analysis makes use of only mutually commuting observables (positions of several pendulums in mechanics, or values of different fields in field theory all mutually commute). This may suggest that our results should not be disturbed by the fact that there exist other observables, which fail to commute with the ones used in the experiment. In principle, one can extend the set of sampled observables up to the so-called complete set of compatible observables, without changing anything in the above analysis.

On the other hand, a very precise measurement of a coordinate of a given pendulum may uncontrollably perturb its momentum, so much that the pendulum fails to swing in the usual way, which would introduce a form of an intrinsic noise in the expermental data. In that case, subsequent measurements of the position of the pendulum may fail to be well correlated to the measurements of other observed pendulums, potentially hindering the predictions for the dimension and topology of the proper configuration space. Nevertheless, in a carefully controlled experiment, this noise can probably be kept below the expected experimental error. Additionally, care should be taken to distinguish the single-shot measurement of a single instance of a pendulum (which gives a random result, per QM), from the statistical measurement of an ensemble of identically prepared pendulums (which is probabilistically determined by QM). The uncertainty relations hold for the latter, while in our study we are interested in the former.

The proper quantum mechanical treatment is out of the scope of this paper, and we postpone it for future work. However, we would like to point the reader to a recent work PaunkovicVojinovic2020 , studying the notion of the so-called quantum switch Chiribella . There, a quantum mechanical observable was proposed which counts the number of relevant spacetime points, distinguishing 4-point optical realisations of the quantum switch from a 2-point gravitational switch (see also a related subsequent work FaleiroPaunkovicVojinovic2023 ). The quantum mechanical measurement operator constructed in PaunkovicVojinovic2020 represents another example of a property of a spacetime manifold that is observable. Specifically, despite the fact that individual points of the manifold are not observable, they are distinguishable, in the sense that one can experimentally tell apart two different spacetime points. This property is of course also diffeomorphism-invariant, since diffeomorphisms are bijective maps and can never map two different points into a single point. Therefore, there should be a corresponding physical observable that distinguishes two (or more) different spacetime points. This further supports our current result, opening up a possibility for generalisations to the quantum realm.

Throughout Section III, it was claimed that the correlations one ought to find in real experimental data will ultimately support the conclusion $D\equiv\dim{\cal M}=4$. This claim is supported by everyday experience and virtually all experiments ever performed in the history of physics so far. These experiments roughly cover the scales from $10^{-20}\,{\rm m}$ (the scale of the current LHC and LIGO experiments), up to $10^{26}\,{\rm m}$ (the scale of the observable Universe). There is a further range of scales, from $10^{-20}\,{\rm m}$ down to $10^{-35}\,{\rm m}$ (the Planck scale), and maybe even beyond that. We currently have no experimental data from this range to either support the result $D=4$ or disprove it. Eventual access to this data could in principle change the result for $D$. For example, in the context of string theory PolchinskiBook1 ; PolchinskiBook2 , one imagines spacetime to have six additional spacelike dimensions, wrapped up into a small Calabi-Yau manifold. If we were to measure various fields at the scale smaller than the size of that Calabi-Yau manifold, our analysis of the data would yield $D=10$, or $4$ plus whatever number of compactified small extra dimensions exist at that scale.

In this sense, the dimension of the spacetime manifold may be a scale-dependent quantity, like running coupling constants in QFT, having different “effective” values at different scales. So far we have no data that would indicate anything other than $D=4$, but in principle this may change. Related to this, one may ask if our analysis supports the “running” of $D$ to values smaller than $4$ at smaller scales. Values larger than $4$ are obviously possible, but smaller values are also possible in principle. Namely, with sufficient resolution, one may notice that what looks like a $4$-manifold is (for example) in fact a $1$-dimensional curve densely packed up in a space-filling fashion, like some finite iteration of the Peano curve. In this sense, a fundamentally $1$-dimensional manifold can look at large scales as a $4$-dimensional one. Something along these lines apparently happens in the context of Causal Dynamical Triangulations scenario CDTreview1 ; CDTreview2 , giving rise to $D=2$ near the Planck scale. That said, note that the CDT results discuss a different concept of a spacetime dimension, namely an effective dimension visible to a random walker. This is in general not equivalent to the notion of topological dimension that we discuss here.

The result $D=4$ is also contingent on the choice of observables measured and used to conclude that $D=4$. In principle, if we were to extend our set of observables, we could find additional large dimensions. For example, so far we can detect the presence of dark matter only through its gravitational interaction with regular matter. In addition to that, one could imagine that there are also direct (contact) interactions between dark and regular matter, but that they are obscure enough not to be easily visible (like neutrinos, which interact with other matter only through short-range weak interactions and even weaker gravity). Nevertheless, if we somehow manage to measure these additional observables dependent on dark matter, they may change our statistics even in the IR regime, i.e., at large scales. It may thus turn out that our analysis yields $D=7$ or $D=12$ or any number larger than $4$. An example of this scenarios are braneworld cosmologies similar to Randall–Sundrum model RandallSundrum1 ; RandallSundrum2 and similar, where all “standard” observables we can measure happen to be nonzero only on a $4$-dimensional submanifold of some target manifold of larger dimension, while the “dark matter” observables would be nonzero even in the bulk of the target manifold, ultimately giving rise to $D>4$. So far we have not found any such observables, but this may also change in the future.

Looking at the analysis of the dataset of our thought experiment, one can raise an objection that the analysis of the correlations between observables may fail to give a unique result. Namely, in generic circumstances, any finite dataset constructed by $N$ measurements of $K$ observables can be permuted into $(N!)^{K-1}$ possible arrangements, and for each of them one can evaluate the suitable critical parameter (see Appendix A) that singles out the one “special” permutation. Since the set of permutations is finite, we end up with a finite collections of the values of the critical parameter, and any such collection contains a minimal element. This minimal parameter corresponds to a “special” permutation, since it features the strongest correlation in the dataset, ultimately describing a spacetime manifold. Given this setup, it may turn out that this minimal parameter is not unique, in the sense that several different permutations of the dataset all have this same minimal value of the parameter. In such a case, there are in principle several different possible arrangements of data featuring equally strong correlations, leading to several different possible candidate manifolds.

On one hand, it is not feasible to study this question numerically in practice, since the set of $(N!)^{K-1}$ of all possible permutations is incredibly huge, while permutations featuring the minimal parameter are likely to be a scarce subset of these. This limits us to theoretical arguments that generic datasets feature minimal critical parameter for only one permutation. The main argument provided so far is based on the self-reinforcement property, which is unlikely to hold for more than one permutation and its restrictions to arbitrary subsets of data.

On the other hand, it is easy to construct explicit examples which feature more than one “special” permutation, each corresponding to the same minimal value of the critical parameter. This is expected to happen if a physical system being measured features global symmetries. For example, pendulums are invariant with respect to a global left-right symmetry, in the sense that one can find two “special” permutations, which are mirror images of each other. While these formally give rise to two different time manifolds, these manifolds are represented by two isomorphic datasets, and therefore in fact represent the one and the same time manifold, despite the nonuniqueness. In this sense, nonuniqueness that stems from the existence of global symmetries is benign, and does not invalidate the analysis and conclusions of the thought experiment.

Also, one of the most often discussed examples is the “diagonal” permutation of the dataset, where the values for each observable are sorted in an ascending (or descending) fashion. This leads to a dataset where points are roughly aligned along a $1$-dimensional curve connecting two opposite corners of the $K$-dimensional kinematic configuration space. Being $1$-dimensional, this permutation is likely to have a very small value of the critical parameter, possibly the minimal one. Nevertheless, it is not hard to demonstrate that this permutation fails to satisfy the self-reinforcement property, since its restrictions to data subsets will fail to coincide with the full permutation. In other words, as one adds more data to the dataset, the resulting $1$-dimensional curve “wiggles”, changing its shape for every extension of the dataset. Thus such a permutation has to be excluded from consideration, regardless of the very small value of the critical parameter. We conjecture that the self-reinforcement property will be violated for all permutations featuring small critical parameter, except for one — which can thus be uniquely recongized as “special”. However, any potential proof of this conjecture is not available at this time, and requires further study.

As we noted at the end of Section III, in principle one can imagine a theoretical model which does not feature anything like a $4$-manifold in its postulated structure. Further, such a model may give us predictions for the values of all possible observables, and we can use it to generate a dataset, study it using our methods, and end up with correlations among observables which give rise to a $4$-manifold. If such a scenario happens, one says that spacetime emerges from the theoretical model, and that the model predicts the values of spacetime dimension and topology.

Despite many hopeful attempts (usually in the context of quantum gravity), there are no particular theoretical models that have managed to achieve this, even with a wrong prediction for the dimension and topology. The reason for this are two important criteria that such a model must satisfy in order to make a legitimate prediction:

• (1)

  The information about the $4$-manifold must not be present in the axiomatic structure of the model. Namely, if we include the $4$-manifold structure as an input in the construction of the theory, it should come as no surprise that one can recover that information from the model later on, in various different forms. However, this can be tricky to test, since the information about a $4$-manifold can be encoded in a non-obvious, cryptic fashion, and it may be hard to prove that some of the axioms of the model are indeed equivalent to the assumption that there exists a $4$-manifold in the theory.

• (2)

  One must demonstrate that the actual dimension of the would-be manifold is explicitly computable from the model. For example, one can try to evaluate a bunch of observables using the model, and then apply our analysis onto that data, in order to obtain $D=4$. In this sense, the dimension of spacetime would be an explicit consequence of the dynamics of the observables in the model. Alternatively, one may use some other way to calculate the spacetime dimension, but this again needs to be a consequence of dynamics of the observables, giving rise to appropriate correlations in the dataset that would ultimately lead to a $4$-manifold. However, it is not enough to handwavingly claim that the model “in principle” predicts these correlations, because they cannot be a generic feature of the model. The correlations must be explicitly calculated, or otherwise rigorously proved to exist, and in addition it must be also rigorously proved that there cannot be any further correlations beyond these, since any further correlations would lower the dimension below $4$, falsifying the model.

For example, a typical spinfoam model of quantum gravity is constructed as a state sum over the values of the fields living on a $2$-complex (the spin foam), providing one with a way to calculate (expectation) values of observables EPRL ; FK . However, usually by construction, the $2$-complex which is used is dual to a triangulation of a $4$-manifold. Because of this property, the spinfoam model fails to satisfy the criterion (1) — the information about a $4$-manifold is already integrated into the model as an axiom, rather than being a property of the dynamics of the observables. Moreover, if one manages to “fix” this by generalizing the $2$-complex structure somehow, so that it fails to be dual to a manifold (for example, by arranging that each dual cell has different topological dimension), there is criterion (2) — one must use the model to explicitly demonstrate that the observables will always feature appropriate correlations (as a consequence of the dynamics of the model) so that they give rise to a $4$-manifold, at least in some large-distance limit. No such model has ever been proposed, and no such calculation has ever been performed.

Another example would be bosonic string theory, which is formulated by explicitly assuming the existence of a $D$-manifold, albeit with an unspecified value for $D$ PolchinskiBook1 ; PolchinskiBook2 . Then one uses the dynamics of the model to evaluate $1$-loop beta-functions, and from the requirement that these vanish, one obtains a consistency requirement $D=26$. The explicit assumption of a $D$-dimensional manifold is an input to the model, and thus already violates criterion (1). Criterion (2) comes very close to being satisfied, but unfortunately it is contingent on the particular choice of $\zeta$-function regularization, making use of the famous “identity”

obtained by analytic continuation of the Riemann $\zeta$-function, i.e., postulated as an axiom of the model. This ultimately renders the value of $D$ to be a part of the definition of the theory, completing the argument that criterion (1) is indeed violated.

One can naturally expect to find many other proposals for spacetime emergence from various theoretical models throughout the literature. However, in order to take any of these proposals seriously, one must first demonstrate that criteria (1) and (2) are fulfilled, which is highly nontrivial, and likely has not been done for any of the existing proposals.

Let us note that one might also consider another type of emergence, which stems from raw experimental data, as opposed to emergence from a theoretical model, which was discussed so far in this Subsection. In fact, the approach taken in this work represents an example of this second type of spacetime emergence. In that sense, the spacetime emergence from raw experimental data is precisely a synonym for the notion of the verification of the existence of spacetime, as used in the title. It is important to emphasise that the term “emergence” can thus have quite different meanings, depending on the context, and one should take care in its use.