What is Emergence?

The term emergence phenomenon has been used in very different contexts at least since 1800’s. In each of them, Emergence Theory is the theory which studies those kind phenomena. E.g, we have version of it in Biology, Philosophy, Art and Chemistry [1,2]. The term is also used for many times in Physics, with different meanings (for a review on the subject, see [3,4]. For an axiomatization approach, see [5]).

The above reveals that the concept of emergence phenomenon is at the same time very general and difficult to formalize. Nevertheless, we have a clue of what it really is: when looking at all those presentations we see that each of them is about describing a system in terms of some other system, possible in different scales. Thus, an emergence phenomenon is about a relation between two different systems: the emergence relation, and a system emerges from other when it (or at least part of it) can be recovered in terms of other system, presumably more fundamental, at least in some scale. The different emergence phenomena in Biology, Philosophy, Physics, and so on, are obtained by fixing in the above abstract definition a meaning for system, scale, etc.

Notice that, in this approach, in order to talk about emergence we need to assume that to each systems of interest we have assigned an scale. In Mathematics, scales are more known as parameters. So, emergence phenomena occurs between some kind of parameterized systems. This kind of assumption (that in order to fix a system we have to specify the scale in which we are considering it) is the heart of the notion of effective field theory, where the scale (or parameter) is governed by Renormalization Group flows [6-9].

Notice, in turn, that if a system emerges from some other, then the second one should be more fundamental, at least in the scale (or parameter) in which the emergence phenomenon is observed. This also put Emergence Theory in the framework of searching for the fundamental theory of Physics (e.g Quantum Gravity), whose systems should be the minimal systems relative to the emergence relation [3,4]. The main problem in this setting is thus then the existence problem for the minimum. A very related question is the general existence problem: given two systems, there exists some emergence relation between them?

The Emergence Problem

One can work on the existence problem is different depth levels. Indeed, since the systems is question are parameterized one can ask if there exists an emergence correspondence between them in some scale or in every scale. Obviously, by requiring a complete correspondence is much more strong than requiring a partial correspondence. On the other hand, in order to attack the existence problem we have to specify which kind of emergence relation we are looking for. Again, is it a full correspondence, in the sense that the emergent theory can fully reconstructed from the fundamental one, or is it only a partial correspondence, through which only certain aspects can be recovered? Thus, we can say that we have four versions of the existence problem for emergence phenomena, as below.

weak version weak-scale

version weak-relation version strong version correspondence partial full partial full scales some some all all

Towards Strong Version Results

At least in Physics (only of those described field where I have a few of acknowledgement), the people are usually working on finding weak emergence phenomena. Indeed, they usually show that certain properties of a system can be described by some other system in a limit case, corresponding to certain regime on the parameter space.

These emergence phenomena are strongly related to other kind of correspondence between systems: the physical duality, where two different theories exibes same properties (this is different from mathematical duality, where concepts/theories have a dual if they have some analogous new incarnation). Indeed, one typically build emergence from duality.

For example, AdS/CFT duality plays an important role in describing spacetime geometry (curvature) from mechanic statistical information (entanglement entropy) of dual strongly coupled systems [10-15].

There are also some interesting examples of weak-scale emergence relations. These typically occur when the action functional of two Lagrangian field theories are equal at some limit. A basic example is gravity emerging from noncommutativity, again using a physical duality.

More precisely, in [16] Nathan Seiberg and Edward Witten suggested the existence of a map (the Seiberg-Witten map) assigning to each gauge theory \( S[A] \) a noncommutative analogue \( S_{\theta}[A;\theta] \) (in the sense of noncommutative geometry), which is some kind of non-perturbative version of a sequence of theories depending on a noncommutative parameter \( \theta ^{\mu

u} \) that measures the noncommutativity of the spacetime local coordinates, i.e, \( [x^{\mu},x^{

u}]=\theta ^{\mu

u} \). In other words, \[S_{\theta}[A;\theta] =\sum_{i=0}^{\infty}S_i[A;\theta ^i] = \lim_{n \rightarrow \infty} S_{(n)}[A;\theta], \] where \(S_{(n)}[A;\theta]=\sum_{i=0}^{n}S_i[A;\theta ^i]\). Furthermore, they argue that there exists a physical duality between a gauge theory and its noncommutative analogue.

Many people then started to use this map to build emergence phenomena. The idea was to consider a gauge theory and modify it in two different ways:

by introducing a background field \( \chi \); by using the Seiberg-Witten map to get its noncommutative analogue.

One can think of both new theories as parameterized theories: the parameter (or scale) of the first one is the background field \( \chi \), while of the second one is the noncommutative parameter \( \theta ^{\mu,

u} \). By construction, the noncommutative theory \( S_{\theta}[A;\theta] \) can be expanded in a power series on the noncommutative parameter, and we can also expand the other theory \( S_{\chi}[A;\chi] \) on the background field, i.e, one can write \[S_{\chi}[A;\chi] =\sum_{i=0}^{\infty}S_i[A;\chi ^i] = \lim_{n \rightarrow \infty} S_{(n)}[A;\chi], \] where \(S_{(n)}[A;\chi]=\sum_{i=0}^{n}S_i[A;\chi ^i]\). The idea was then try to find solutions for the following question:

Question: Given a gauge theory \( S[A] \), there exists a background version \(S_{\chi}[A;\chi]\) and a number \( n \) such that for every given value \(\theta ^{\mu

u}\) of the noncommutative parameter there exists a configuration of the background field \(\chi(\theta)\), possibly depending on \(\theta ^{\mu

u}\), such that for every gauge field \(A\) we have \(S_{(n)}[A;\chi(\theta)]=S_{(n)}[A;\theta]\)?

Notice that if rephrased in terms of parameterized theories, the question above is precisely about the existence of an emergence relation between \(S_{\chi}\) and \(S_{\theta}\), at least up to order \(n\). This can also be interpreted saying that, in the context of the gauge theory \(S[A]\), the background fields \(\chi\) emerges in some regime from the noncommutativity of the spacetime coordinates. Since the noncommutative parameter \(\theta ^{\mu

u}\) depends on two spacetime indexes, it is suggestive to consider background fields of the same type, i.e, \(\chi ^{\mu

u}\). In this case, there is a natural choice: metric tensors \(g ^{\mu

u}\). Thus, in this setup we are proving that in the given gauge context, gravity emerges from noncommutativity at least up to perturbation order \(n\)!

Many people proved that this is really true for many classes of gauge theories and for many values of \(n\), which is amazing! E.g, see [17-21]. On the other hand, this fact nature leads to two other questions:

Can we find some emergence relation between gravity and noncommutativity in the nonperturbative setting? In other words, can we extend the weak-scale emergence relation above to a strong one? Is it possible to generalize the construction of the cited works to other kind of background fields? In other words, is it possible to used the idea in order to show that different fields emerge from spacetime noncommutativity?

The first question (about finding a strong emergence phenomena) has a positive answer in some cases [22-25]. The second question is about finding general emergence phenomena. Until my (small) understanding, I am unaware of studies that aim to prove general theorems on the existence of emergence relations, specially on the strong case…

Towards General Results

As discussed above, it is very desirable to find general conditions to ensure the existence (or non-existence) of strong emergence relations. This is the one of the focus of our group. Indeed, in a first work (see here) we investigated conditions under which two generic Lagrangian field theories are involved in a strong emergence phenomenon.

More precisely, we worked with parameterized Lagrangian densities of the form \(\mathcal{L}_{\varepsilon}(x,\varphi,\partial\varphi) = \langle\varphi,\Psi_{\varepsilon}\varphi\rangle \), where \(\varepsilon \) is the parameter (or scale), \(\varphi \) is a generic field (section of some field bundle \(E\) )and \(\Psi \) is some operator (possibly nonlocal) on \(E\), such as a differential operator or, more generally, a pseudo-differential operator. Furthermore, the brackets denote a pairing on the space of field, typically induced by a pairing on the manifold. We proved that:

Under certain weak algebraic assumptions on the space of parameters, if the bracket is induced by a Riemannian metric on base manifolds, i.e, if we are working in the euclidean setting, then a given Lagrangian field theory \(\mathcal{L}_{1,\varepsilon}(x,\varphi,\partial\varphi) \) strongly emerges from any other Lagrangian field theory \(\mathcal{L}_{2,\delta}(x,\varphi,\partial\varphi )\) whose operator \(\Psi _{2,\delta} \) is a multivariate polynomial \(P(\Psi _1,…,\Psi_l,\delta) \) with coefficients in nowhere vanishing functions and whose variables are right-invertible operators.

Since the typical examples of right-invertible operators are some flavors of elliptic pseudo-differential operators, the synthesis of the above result is the following:

In the euclidean case, typical Lagrangian field theories emerges from multivariate polynomial theories defined by certain elliptic pseudo-differential operators.

This result is about giving sufficient conditions to ensure existence of strong emergence. We also showed that these conditions are not necessary by means of writing down explicit examples of emerging theories which are not of the given shape. We also use our strategy to build some obstruction results.

On the other hand, in our check list is to show that all the previously mentioned examples of gravity emerging from noncommutativity can be regarded as particular cases of our theorem. We also aim to show that from our approach one can derive some new examples.

Local, But Really General

Although the sufficient conditions that we found are not strong, they have the disadvantage of demanding a right-inverse for the operators that are used as the variables of \(P(\Psi _1,…,\Psi_l,\delta) \). This means that our previous result is about strong (or global) and typical (or half-general) emergence.

We are involved in another project where we aim to ensure the existence of give generic emergence phenomena, in the sense that there exists a generic set into the space of parameterized field theories such that given a field theory, then it locally emerges from any theory on that set. But, these are scenes of next chapters…

Yuri Ximenes Martins

Co-founder of the Math-Phys-Cat Group

03/19/2020