An Axiomatization Proposal and a Global Existence Theorem for Strong Emergence Between Parameterized Lagrangian Field Theories
aa r X i v : . [ m a t h - ph ] S e p An Axiomatization Proposal and a Global Existence Theorem forStrong Emergence Between Parameterized Lagrangian FieldTheories
Yuri Ximenes Martins ∗ and Rodney Josu´e Biezuner † September 22, 2020
Departamento de Matem´atica, ICEx, Universidade Federal de Minas Gerais,Av. Antˆonio Carlos 6627, Pampulha, CP 702, CEP 31270-901, Belo Horizonte, MG, Brazil
Abstract
In this paper we propose an axiomatization for the notion of strong emergence phenomenonbetween field theories depending on additional parameters, which we call parameterized fieldtheories. We present sufficient conditions ensuring the existence of such phenomena between twogiven Lagrangian theories. More precisely, we prove that a Lagrangian field theory dependinglinearly on an additional parameter emerges from every multivariate polynomial theory evaluatedat differential operators which have well-defined Green functions (or, more generally, that has aright-inverse in some extended sense). As a motivating example, we show that the phenomenonof gravity emerging from noncommutativy, in the context of a real or complex scalar field theory,can be recovered from our emergence theorem. We also show that, in the same context, we couldalso expect the reciprocal, i.e., that noncommutativity could emerge from gravity. Some otherparticular cases are analyzed.
The term emergence phenomenon has been used for years in many different contexts. In eachof them, Emergence Theory is the theory which studies those kinds of phenomena. E.g, we haveversions of it in Philosophy, Art, Chemistry and Biology [1, 49, 15]. The term is also often used inPhysics, with different meanings (for a review on the subject, see [13, 18]. For an axiomatizationapproach, see [19]). This reveals that the concept of emergence phenomenon is very general andtherefore difficult to formalize. Nevertheless, we have a clue of what it really is: when looking at allthose instances of the phenomenon we see that each of them is about describing a system in termsof other system, possibly in different scales . Thus, an emergence phenomenon is about a relationbetween two different systems, the emergence relation , and a system emerges from another whenit (or at least part of it) can be recovered in terms of the other system, which is presumably morefundamental, at least in some scale. The different emergence phenomena in Biology, Philosophy, ∗ [email protected] (corresponding author) † [email protected] scale . In Mathematics, scales are better known as parameters .So, emergence phenomena occur between parameterized systems . This kind of assumption (that inorder to fix a system we have to specify the scale in which we are considering it) is at the heart ofthe notion of effective field theory, where the scale (or parameter) is governed by RenormalizationGroup flows [12, 31, 26, 22]. Notice, in turn, that if a system emerges from another, then the secondone should be more fundamental, at least in the scale (or parameter) in which the emergencephenomenon is observed. This also puts Emergence Theory in the framework of searching forthe fundamental theory of Physics (e.g Quantum Gravity), whose systems should be the minimalsystems relative to the emergence relation [13, 18]. The main problem in this setting is then theexistence problem for the minimum. A very related question is the general existence problem: giventwo systems, is there some emergence relation between them? One can work on the existence problem at different levels of depth. Indeed, since the systemsis question are parameterized one can ask if there exists a correspondence between them in somescales or in all scales . Obviously, requiring a complete correspondence between them is muchstronger than requiring a partial one. On the other hand, in order to attack the existence problemwe also have to specify which kind of emergence relation we are looking for. Again, is it a fullcorrespondence, in the sense that the emergent theory can be fully recovered from the fundamentalone, or is it only a partial correspondence, through which only certain aspects can be recovered?Thus, we can say that we have the following four versions of the existence problem for emergencephenomena: weak weak-scale weak-relation strongrelation partial full partial full scales some some all allTable 1: Types of EmergenceIn Physics one usually works on finding weak emergence phenomena. Indeed, one typicallyshows that certain properties of a system can be described by some other system at some limit,corresponding to a certain regime of the parameter space. These emergence phenomena are stronglyrelated with other kind of relation: the physical duality , where two different systems reveal the samephysical properties. One typically builds emergence from duality. For example, AdS/CFT dualityplays an important role in describing spacetime geometry (curvature) from mechanic statisticalinformation (entanglement entropy) of dual strongly coupled systems [42, 39, 46, 8, 10, 9].There are also some interesting examples of weak-scale emergence relations, following again fromsome duality. These typically occur when the action functional of two Lagrangian field theories areequal at some limit. The basic example is gravity emerging from noncommutativity following fromthe duality between commutative and noncommutative gauge theories established by the Seiberg-Witten maps [43]. Quickly, the idea was to consider a gauge theory S [ A ] and modify it into twodifferent ways:1. by considering S [ A ] coupled to some background field χ , i.e, S χ [ A ; χ ];2. by using the Seiberg-Witten map to get its noncommutative analogue S θ [ ˆ A ; θ ].2oth new theories can be regarded as parameterized theories: the parameter (or scale) of thefirst one is the background field χ , while that of the second one is the noncommutative parameter θ µν . By construction, the noncommutative theory S θ [ A ; θ ] can be expanded in a power series on thenoncommutative parameter, and we can also expand the other theory S χ [ A ; χ ] on the backgroundfield, i.e, one can write S χ [ A ; χ ] = ∞ X i =0 S i [ A ; χ i ] = lim n →∞ S ( n ) [ A ; χ ] and S θ [ ˆ A ; θ ] = ∞ X i =0 S i [ A ; θ i ] = lim n →∞ S ( n ) [ A ; θ ] , where S ( n ) [ A ; χ ] = P ni =0 S i [ A ; χ i ] and S ( n ) [ A ; θ ] = P ni =0 S i [ A ; θ i ] are partial sums. One then triesto find solutions for the following question: Question 1.
Given a gauge theory S [ A ] , is there a background version S χ [ A ; χ ] of it and a number n such that for every given value θ µν of the noncommutative parameter there exists a value ofthe background field χ ( θ ) , possibly depending on θ µν , such that for every gauge field A we have S ( n ) [ A ; χ ( θ )] = S ( n ) [ A ; θ ] ? Notice that if rephrased in terms of parameterized theories, the question above is preciselyabout the existence of a weak-scale emergence between S χ and S θ , at least up to order n . Thiscan also be interpreted by saying that, in the context of the gauge theory S [ A ], the backgroundfield χ emerges in some regime from the noncommutativity of the spacetime coordinates. Sincethe noncommutative parameter θ µν depends on two spacetime indexes, it is suggestive to considerbackground fields of the same type, i.e, χ µν . In this case, there is a natural choice: metric tensors g µν . Thus, in this setup, the previous question is about proving that in the given gauge context,gravity emerges from noncommutativity at least up to a perturbation of order n . This has beenproved to be true for many classes of gauge theories and for many values of n [40, 52, 4, 30, 16].On the other hand, this naturally leads to other two questions:1. Can we find some emergence relation between gravity and noncommutativity in the nonper-turbative setting? In other words, can we extend the weak-scale emergence relation above toa strong one?2. Is it possible to generalize the construction of the cited works to other kinds of backgroundfields? In other words, is it possible to use the same idea in order to show that differentfields emerge from spacetime noncommutativity? Or, more generally, is it possible to build aversion of it for some general class of field theories?The first of these questions is about finding a strong emergence phenomena and it has a positiveanswer in some cases [51, 5, 44, 41]. The second one, in turn, is about finding systematic andgeneral conditions ensuring the existence (or nonexistence) of emergence phenomena. At least tothe authors knowledge, there are no such general studies, specially focused on the strong emergencebetween field theories. It is precisely this point that is the focus of the present work. Indeed wewill:1. based on Question 1, propose an axiomatization for the notion of strong emergence betweenfield theories;2. establish sufficient conditions ensuring that a given Lagrangian field theory emerges from eachtheory belonging to a certain class of theories.3e will work on the setup of parameterized field theories , which are given by families S [ ϕ ; ε ] offield theories depending on a fundamental parameter ε . In the situations described above, ε is thethe noncommutative parameter θ µν or the background field χ µν . In the cases where the emergencewas explicitly obtained, it was of summary importance that the parameters θ µν and χ µν belongs tothe same class of fields and that the corresponding action functions are defined on the same field A µ . Thus, given two parameterized theories S [ ϕ ; ε ] and S ′ [ ϕ ; ε ′ ] we always assume that they aredefined on the same fields and that the parameters ε and ε ′ belong to the same space. KeepingQuestion 1 as a motivation, let us say that S [ ϕ ; ε ] emerges from S ′ [ ϕ ; ε ′ ] if there is a map F on thespace of parameters such that, for every ε and every field ϕ we have S [ ϕ ; ε ] = S ′ [ ϕ ; F ( ε )]. We call F a strong emergence phenomenon between S and S ′ .Our main result states that for certain S [ ϕ ; ε ] and S ′ [ ϕ ; ε ′ ], depending on ε and ε ′ in a suitableparameter space, in the sense that it has some special algebraic structure, then these strong emer-gence phenomena exist. The formal statement of this result will be presented in Section 3 aftersome technical digression. But, as a motivation, let us state a particular version of it and showhow it can be used to recover an example of emergence between gravity and noncommutativity.First of all, recall that the typical field theory has a kinetic part and an interacting part.The kinetic part is usually quadratic and therefore of the form L knt ,i ( ϕ i ) = h ϕ i , D i ϕ i i i , with i =1 , ..., N , where N is the number of fields, D i are differential operators and h· , ·i i are pairings in thecorresponding space of fields. Summing the space of fields and letting ϕ = ( ϕ , ..., ϕ N ), D = ⊕ i D i and h· , ·i = ⊕ i h· , ·i i , one can write the full kinetic part as L knt ( ϕ ) = h ϕ, Dϕ i . On the other hand,the interacting part is typically polynomial, i.e., it is of the form L int ( ϕ ) = h ϕ, p l [ D , ..., D N ] ϕ i ,where l ≥ p l [ D , ..., D N ] = P | α |≤ l f α · D α . Since the space ofdifferential operators constitutes an algebra, it follows that p l [ D , ..., D N ] is a differential operatortoo, so that both the kinetic and the interacting parts (and therefore the sum of them, whichconstitute the typical lagrangians) are of the form L ( ϕ ) = h ϕ, Dϕ i .Notice, furthermore, that the pairing h· , ·i is typically induced by fixed geometric structures(such as metrics) in the spacetime manifold M and in the field bundle E . Thus, in the parameterizedcontext the natural dependence on the parameter is on the differential operator, i.e., the typicalparameterized Lagrangian theories are of the form L ( ϕ ; ε ) = h ϕ, D ε ϕ i . In the case of polynomialtheories (e.g., those describing interations), it is more natural to assume that the dependenceon D ε is actually on the coefficient functions f α , i.e., p lε [ D , ..., D N ] = P | α |≤ l f α ( ε ) D α . Thesecoefficient functions could be scalar functions or, more generally, parameter-valued functions ifwe have an action of parameters in differential operators. In this last case, the polynomial is p lε [ D , ..., D N ] = P | α |≤ l f α ( ε ) · D α , where the dot denotes the action of parameters in differentialoperators. Emergence Theorem (rough version)
Let M be a spacetime manifold and E → M a field bundlewhich define a pairing h· , ·i in E . Let L ( ϕ ; ε ) = h ϕ, D ε ϕ i be an arbitrary parameterized theory and L ( ϕ ; δ ) = h ϕ, p lδ [ D , ..., D N ] ϕ i a polynomial theory, e.g., an interaction term. Suppose that:1. the parameters ϕ are nonnegative real numbers or, more generally, positive semi-definiteoperators or tensors acting on the space of differential operators;2. the dependence of D ε on ε is of the form D ε ϕ = ε · Dϕ , where D is a fixed differential operator;3. the coefficient f α ( δ ) of p lδ [ D , ..., D N ] are nowhere vanishing scalar or parameter-valued func-tions, and the differential operators D , ..., D N have well-defined Green functions. hen the theory L ( ϕ ; ε ) emerges from the theory L ( ϕ ; ε ) . Now, looking at [40], consider again the question of emergent gravity in a four dimensional space-time. The first order perturbation of a real scalar field theory ϕ , coupled to a semi-Riemannian gravitational field g = η + h , in an abelian gauge background, is given by equation (10) of [40] : L grav ( ϕ ; h ) = ∂ µ ϕ∂ µ ϕ − h µν ∂ µ ϕ∂ ν ϕ (1)= −h ϕ, (cid:3) ϕ i + h ϕ, ( h · D ) ϕ i (2)= L ( ϕ ) + L ( ϕ ; h ) . (3)where in the second step we did integration by parts and used the fact that the spacetime manifoldis boundaryless. Furthermore, the pairing is just h ϕ, ϕ ′ i = ϕϕ ′ , while D = ∂ µ ∂ ν and for every2-tensor t = t µν , we have the trace action t · D = t µν ∂ µ ∂ ν . In particular, η · D = ∂ µ ∂ ν = (cid:3) . On theother hand, the first order expansion in θ of the Seiberg-Witten dual of a real scalar field theory,in an abelian gauge background, is given by (equation (7) of [40]): L ncom ( ϕ ; θ ) = ∂ µ ϕ∂ µ ϕ + 2 θ µα F ακ η κν ( − ∂ µ ϕ∂ ν ϕ + 14 η µν ∂ ρ ϕ∂ ρ ϕ ) (4)= h ϕ, ( − (cid:3) ) ϕ i + h ϕ, ( f ( θ ) · D ) ϕ i (5)= h ϕ, p θ [ (cid:3) , D ] ϕ i , (6)where in the first step we again integrated by parts, and used the fact that p θ [ x, y ] is the first orderpolynomial ( − · x + f ( θ ) · y , where f ( θ ) = θ and, in coordinates, D = 2 F ακ η κν ( ∂ µ ∂ ν − η µν (cid:3) ).Notice that if h µν is a positive-definite tensor (which is the case in Riemannian signature ),then L ( ϕ ; h ) in (3) satisfies the hypotheses of the emergence theorem. On the other hand, sincein the Riemannian setting (cid:3) is the Laplacian and D is essentially a combination of generalizedLaplacians, both of them have Green functions . Thus, if we forget the trivial case θ µν = 0, then L ncom ( ϕ ; θ ) also satisfies the hypotheses of the emergence theorem. Thus, as a consequence we seethat the gravitational term L ( ϕ ; h ) emerges from the noncommutative theory L ncom ( ϕ ; θ ), as alsoproved in [40]. In other words, there is a function F on the space of 2-tensors, such that for every h µν we have L ( ϕ ; h µν ) = L ncom ( ϕ ; F ( h µν )) . Some comments.1. In [40] the author proved explicitly that gravity emerges from noncommutativy. More pre-cisely, he gave an explicit expression for the function F ( h µν ). Here, however, our result isonly about the existence of such function,2. While above we had to assume Rimemannian signature, our main result (Theorem 3.1) appliesequally well to the Lorentzian signature. It is different, however, of the rough version statedabove. In [40] the author considered only Lorentzian spacetimes. However, we notice that in order to get the emergencephenomena the Lorentzian signature was not explicitly used, so that the same holds in the semi-Rimennanian case. In [40] an expansion of the form g µν = η µν + h µν + hη µν was considered, where h is some function, but the authorconcluded that the emergence phenomenon exists only if h = 0. Thus, we are assuming this necessary condition fromthe beginning. Actually, they are elliptic and thus have Fredholm inverses; this is enough for us.
5. In [40] analogous emergence phenomena were established in the case of complex scalar fields.Our main theorem also holds for complex fields.4. Notice that the scalar theory in the gravitational background (3) can also be written as L grav ( ϕ ; h ) = h ϕ, p h [ (cid:3) , D ] ϕ i , where p h [ x, y ] is the first order polynomial ( − · x + f ( h ) · y ,where, again f ( h ) = h . Thus, we can also see the gravitational theory as a polynomialtheory, defined by the same polynomial, but evaluated in different differential operators. Onthe other hand, (6) can be written as L ( ϕ ) + L ( ϕ ; θ ), where L ( ϕ ; θ ) = h ϕ, ( θ · D ) ϕ i . Since (cid:3) and D are generalized Laplacians, they have Green functions. Thus, if one restricts to thecase of noncommutativity parameters θ which are positive definite (say, e.g., positive definitesymplectic forms), then one can again use our emergence theorem to conclude the reciprocalfact: that L ( ϕ ; θ ) emerges from L grav ( ϕ ; h ), i.e., that noncommutativity may also emergesfrom gravity .This work is organized as follows. In Section 2 we propose, based on the lines of this introduc-tion, a formal definition for the notion of strong emergence between parameterized field theories,and we introduced the emergence problem which is about determining if there is some emergencephenomenon between such two given theories. We discuss why it is natural (or at least reasonable)to restrict to a certain class of parameterized field theories, defined by certain “generalized oper-ators”. In Section 3 our main theorem is stated. We begin by introducing its “syntax”, leavingthe precise formal statement (or “semantic”) to Section 3.1. Before giving the proof, which is a bittechnical and based on induction arguments, in Section 3.2 we analyze some particular cases of themain theorem trying to emphasize its real scope. In Section 4 the main result is finally proved. Remark.
Although we used the example of gravity emerging from noncommutativity as a mo-tivating context, we would like to emphasize that this paper is not intended to focus on it or inbuilding concrete examples. Indeed, our aim is to propose a formalization to the notion of strongemergence, at least in the context of Lagrangian field theories, and a general strategy to investigatethe existence problem for such phenomena. With this remark we admit the need of additionalconcrete applications of the methods proposed here, which should appear in a future work. Inparticular, the case of gravity emerging from noncommutative in its difference incarnations, is toappear in a work in progress.
Recall that a field theory on a n -dimensional manifold M (regarded as the spacetime) is givenby an action functional S [ ϕ ], defined in some space of fields (or configuration space) Fields( M ),typically the space of sections of some real or complex vector bundle E → M , the field bundle .A parameterized field theory consists of another bundle P → M (the parameter bundle ), a subsetPar( P ) ⊂ Γ( P ) of global sections (the parameters ) and a collection S ε [ ϕ ] of field theories, onefor each parameter ε ∈ Par( P ). A more suggestive notation should be S [ ϕ ; ε ]. So, e.g, for thetrivial parameter bundle P ≃ M × K we have Γ( P ) ≃ C ∞ ( M ; K ) and in this case we say that wehave scalar parameters. If we consider only scalar parameters which are constant functions, then aparameterized theory becomes the same thing as a 1-parameter family of field theories. Here, andthroughout the paper, K = R or K = C depending on whether the field bundle in consideration isreal or complex. 6e will think of a parameter ε as some kind of “physical scale”, so that for two given parameters ε and ε ′ , we regard S [ ϕ ; ε ] and S [ ϕ ; ε ′ ] as the same theory in two different physical scales. Noticethat if P has rank l , then we can locally write ε = P ε i e i , with i = 1 , ..., l , where e i is a local basisfor Γ( P ). Thus, locally each physical scale is completely determined by l scalar parameters ε i whichare the fundamental ones. In terms of these definitions, Question 1 has a natural generalization: Question 2.
Let S [ ϕ ; ε ] and S [ ψ ; δ ] be two parameterized theories defined on the same spacetime M , but possibly with different field bundles E and E , and different parameter bundles P and P . Arbitrarily giving a field ϕ ∈ Γ( E ) and a parameter ε ∈ Par ( P ) , can we find some field ψ ( ϕ ) ∈ Γ( E ) and some parameter δ ( ε ) ∈ Par ( P ) such that S [ ϕ ; ε ] = S [ ψ ( ϕ ); δ ( ε )] ? In moreconcise terms, are there functions F : Par ( P ) → Par ( P ) and G : Γ( E ) → Γ( E ) such that S [ ϕ ; ε ] = S [ G ( ϕ ); F ( ε )] ? We say that the theory S [ ϕ ; ε ] emerges from the theory S [ ψ ; δ ] if the problem above has apositive solution, i.e, if we can fully describe S in terms of S . Notice, however, that as statedthe emergence problem is fairly general. Indeed, if P and P have different ranks, then, by theprevious discussion, this means that the parameterized theories S and S have a different numberof fundamental scales, so that we should not expect an emergence relation between them. Thisleads us to think of considering only the case in which P = P . However, we could also consider thesituations in which P = P , but P = f ( P ) is some nice function of P , e.g, P = P × P × ... × P .In these cases the fundamental scales remain only those of P , since from them we can generatethose in the product. Throughout this paper we will also work with different theories defined onthe same fields, i.e, E = E . This will allow us to search for emergence relations in which G is theidentity map G ( ϕ ) = ϕ .Hence, after these hypotheses, we can rewrite our main problem, whose affirmative solutionsaxiomatize the notion of strong emergence we are searching for: Question 3.
Let S [ ϕ ; ε ] and S [ ϕ ; δ ] be two parametrized theories defined on the same spacetime M , on the same field bundle E and on the parameter bundles P and P = f ( P ) , respectively.Does there exists some map F : Par ( P ) → Par ( f ( P )) such that S [ ϕ ; ε ] = S [ ϕ, F ( ε )] ? Our plan is to show that the problem in Question 3 has an affirmative solution for certainclass of parameterized field theories, which by the absence of a better name we call generalizedparameterized field theories . This will be obtained from the class of differential parameterizedtheories by means of allowing dynamical operators which are not necessarily differential, but somegeneralized version of them.
In order to motivate the need for looking at a special class of field theories, we begin bynoticing that typically the field theories are local , in the sense that they are defined by meansof integrating some Lagrangian density L ( x, ϕ, ∂ϕ, ∂ ϕ, ... ), i.e, S [ ϕ ] = R M L ( j ∞ ϕ ) dx n , where j ∞ ϕ = ( x, ϕ, ∂ϕ, ∂ ϕ, ... ) is the jet prolongation. On the other hand, a quick look at the stan-dard examples of field theories shows that, when working in a spacetime without boundary, afterintegration by parts and using Stoke’s theorem, those field theories can be stated, at least locally,in the form L ( x, ϕ, ∂ϕ ) = h ϕ, Dϕ i , where h ϕ, ϕ ′ i is a nondegenerate pairing on the space of fieldsΓ( E ) and D : Γ( E ) → Γ( E ) is a differential operator of degree d , which means that it can be locally7ritten as Dϕ ( x ) = P | α |≤ d a α ( x ) ∂ α ϕ , where α = ( α , ..., α r ) is some mult-index, | α | = α + ... + α r is its degree and ∂ α = ∂ α ◦ ... ◦ ∂ α r r , with ∂ li = ∂ l /∂ l x i . This is the case, e.g, of ϕ and ϕ scalar field theories, the standard spinorial field theories, Einstein-Hilbert-Palatini-Type theories[33, 34] as well as Yang-Mills-type theories and certain canonical extensions of them [36, 37]. Moregenerally, recall that the first step in building the Feynman rules of a field theory is to find the(kinematic part of the) operator D and take its “propagator” .In these examples, the pairing h ϕ, ϕ ′ i is typically symmetric (resp. skew-symmetric) and theoperator D is formally self-adjoint (resp. formally anti-self-adjoint) relative to that pairing. Fur-thermore, h ϕ, ϕ ′ i is usually a L -pairing induced by a semi-Riemannian metric g on the field bundle E and/or on the spacetime M , while D is usually a generalized Laplacian or a Dirac-type operatorrelative to g [17]. For example, this holds for the concrete field theories (scalar, spinorial and Yang-Mills) above. The skew-symmetric case generally arises in gauge theories (BV-BRST quantization)after introducing the Faddeev-Popov ghosts/anti-ghosts and it depends on the grading introducedby the ghost number [17].Another remark, still concerning the concrete situations above, is that if the metric g induc-ing the pairing h ϕ, ϕ ′ i is actually Riemannian (which means that the gravitational background isEuclidean), then h ϕ, ϕ ′ i becomes a genuine L -inner product and D is elliptic and extends to abounded self-adjoint operator between Sobolev spaces [20, 23]. Working with elliptic operators isvery useful, since they always admits parametrices (which in this Euclidean cases are the propa-gators) and for generalized Laplacians the heat kernel not only exists, but also has a well-knownasymptotic behavior [50], which is very nice in the Dirac-type case [11].From the discussion above, it is natural to focus on parameterized theories such that eachfunctional S [ ϕ ; ε ] is local and defined by a Lagrangian density of the L ( ∞ ϕ ) = h ϕ, D ε ϕ i , i.e, aredetermined by a single nondegenerate pairing h ϕ, ϕ ′ i in Γ( E ), fixed a priori by the nature of M and E , and by a family of differential operators D ε ∈ Diff( E ), one for each parameter ε ∈ Par( P ),where Diff( E ) = L d Diff d ( E ; E ) denotes the K -vector space of all differential operators in E (which is actually a K -algebra which the composition operation) and Diff d ( E ; E ) is the space ofthose operators of degree d . Thus: Definition 2.1.
Let M be a compact and oriented manifold. A background for doing emergencetheory (or simply background ) over M is given by the following data:1. a K -vector bundle E → M (the field bundle);2. a pairing h ϕ, ϕ ′ i in Γ( E );3. a parameter bundle P → M and a set of parameters Par( P ) ⊂ Γ( P ) . Definition 2.2. A differential parameterized theory in a background over M is a collection ofdifferential operators D ε ∈ Diff( E ) with ε ∈ Par( P ). The parameterized Lagrangian density is givenby L ( ϕ ; ε ) = h ϕ, D ε ϕ i . The parameterized action functional is the integral of the parameterizedLagrangian in M . Sometimes working on the narrow class of field theories defined by differential operators isnot enough. For instance, notice that in building the “propagator” of a Lagrangian field theory8 ( ∞ ϕ ) = h ϕ, Dϕ i we are actually finding some kind of “quasi-inverse” D − for the differentialoperator D . For example, in the Riemannian case, where differential operators D ∈ Diff d ( E )extend to bounded operators ˆ D ∈ B ( W d, ( E )) in the Sobolev space, if D is elliptic, then buildingthe propagator is equivalent to building parametrices, which in turn can described in terms ofFredholm inverses for ˆ D [17]. On the other hand, in the Lorentzian setting (where the spacetimemanifold is assumed globally hyperbolic and the typical differential operators are hyperbolic),building the propagator is about finding its advanced and retarded Green functions [7, 6].Independently of the case, it would be very useful if the quasi-inverse D − could exist as adifferential operator, i.e., if D − ∈ Diff( E ). Indeed, in this case D − would also define a differentialfield theory. In particular, in the parameterized case, if each D ε has a “quasi-inverse” describedby a differential operator D − ε , the collection of them define a parameterized differential theoryover the same background. In turn, from the physical viewpoint, it would be very interesting ifthe “quasi-inverse” D − of the differential operator D was a genuine inverse for D in the algebraDiff( E ). Indeed, in this case one could use the relations D ◦ D − = I and D − ◦ D = I in order toget global solutions for the equation of motion Dϕ = 0 [45].Notice, however, that the equations D ◦ D − = I = D − ◦ D typically has no solutions in Diff( E ).Indeed, recall that Diff( E ) = L d Diff d ( E ; E ) is a Z ≥ -graded algebra with composition, so that if D − was a left or right inverse for D , then deg( D − ) = − deg D , which has no solution if deg D = 0.This forces us to search for extensions of the algebra Diff( E ) such that left and/or right inversesof differential operators may exist. The obvious idea is to consider extensions by adding a negativegrading to Diff( E ) getting the structure of a Z -graded K -algebra, and the natural choice is the Z -graded algebra Psd( E ) of pseudo-differential operators, which are defined via symbol-theoretic(i.e., microlocal analysis) approach [32, 38].For constant coefficient operators, the right-inverse really exist in Psd( E ). In the case of non-constant coefficients, for Riemannian spacetimes, elliptic operators satisfying specific ellipticityconditions admit right-inverse in Psd( E ) [28, 14, 47, 48, 2, 29, 3, 25]. For globally hyperbolicLorentzian spacetimes, at least when restricted to fields with support in the causal cones, Greenhyperbolic operators (in particular normally hyperbolic operators) have both left and right inversesin Psd( E ) [7, 6]. For flat spacetimes, more abstract examples exist [27]. On the other hand, if adifferential operator is not right-invertible in Psd( E ) it could be invertible in another extension ofPsd( E ), such as in the class of Fourier-type operators [45].Our main result does not depend on the extension of Diff( E ); the only thing we need is theexistence of a right-inverse for a differential operator as some kind of operator, i.e., we only needsome K -algebra Op( E ) such that Diff( E ) ⊂ Op( E ) ⊂ End(Γ( E )) and such that the subset ofdifferential operators Op( E ) contains nontrivial elements. Definition 2.3.
Let M be a compact and orientable manifold. A generalized background over M , denoted by GB( M ) is given by the same data in Definition 2.1 and, in addition, a K -algebraOp( E ) ⊂ End(Γ( E )) extending Diff( E ) and such that Diff( E ) ∩ R Op( E ) contains non-multipliesof the identity, where R Op( E ) is the set of right-invertible elements of Op( E ). Definition 2.4. A generalized parameterized theory (GPT) in a generalized background GB( M ) isa collection of generalized operators Ψ ε ∈ Op( E ), with ε ∈ Par( P ). The corresponding Lagrangiandensity action funcional are defined analogously to Definition 2.2, just replacing D ε with Ψ ε . Definition 2.5.
We say that a GPT over M is right-invertible if the generalized operators Ψ ε areright-invertible, i.e, if they belongs to R Op( E ) for every ε ∈ Par( P ).9 .3 Polynomial Parameterized Theories Until this moment we considered theories which are defined by a 1-parameter family of differ-ential (or more general) operators. As discussed, these provide a description of free theories and oftheir parameterized version. We will show, however, that the concept in broad enough to describeinteracting theories as well. In order to do this, recall that the classical examples of interactingfield theories has interacting terms given by multivariate polynomials with variables correspondingto operators of different theories at interaction, as discussed at the introductory section.But, in order to talk about polynomials we need a ring of coefficients. In our parameterizedcontext, the natural idea is to consider coefficients depending on the parameters. This leads us tolook at the ring Map(Par( P ); K ) of scalar functions f : Par( P ) → K on the space of parameters.For a given l ≥ x , ..., x r , let Map l (Par( P ); K )[ x , ..., x r ] be the corresponding K -vector space of multivariate polynomials of degree l in the given variables. Thus, an element ofit can be written as p l [ x , ..., x r ] = P | α |≤ l f α · x α , where f α ∈ Map(Par( P ); K ) and α is a multi-index. If Ψ , ..., Ψ r ∈ Op( E ) are fixed generalized operators and p l [ x , ..., x r ] is a polynomial asabove, recalling that Op( E ) is a K -algebra, by means of replacing the formal variables x i with theoperators Ψ i we get a family of operators p l [Ψ , ..., Ψ r ]( ε ) = P | α |≤ l f α ( ε )Ψ α , with ε ∈ Par( P ). Thefollowing definitions are then natural. Definition 2.6.
Let M be a compact and oriented manifold. A polynomial parameterized theory ( PPT ) of degree l ≥ in r variables , defined on a generalized context GB( M ), is given by anelement p l of Map l (Par( P ); K )[ x , ..., x r ] and by operators Ψ i ∈ Op( E ), with i = 1 , ..., r . Theparameterized Lagrangian and the parameterized action functional are L ( ∞ ϕ ; ε ) = h ϕ, p l [Ψ , ..., Ψ r ]( ε ) ϕ i and S [ ϕ ; ε ] = ≪ ϕ, p l [Ψ , ..., Ψ r ]( ε ) ϕ ≫ , while the extended Lagrangian and the extended action functional are defined analogously by meansof replacing ψ i with f Ψ i .Now, to check that the definition above really captures the standard examples of interactingterms, if the interaction to be described is of j > j variables and such that E ≃ ⊕ i E i , with i = 1 , ..., j , and Ψ i = (0 , ..., Φ i , ..., i ∈ Γ( E )and Φ i ∈ Γ( E i ). Here, E i is the field bundle of the i th field theory . Remark . Every PPT of arbitrary degree l and in arbitrary number r of variables is a GPT,on the same generalized background, with parameterized operator Ψ ε = p l [Ψ , ..., Ψ r ]( ε ), so thatthe concept of GPT is really broad enough to describe both free and interacting terms of a typicalLagrangian field theory.Closing this section, notice that a priori we have two possible definitions of right-invertibilityfor a PPT. Since by the above every PPT is a GPT, we could say that a PPT is right-invertible if it is as a GPT, i.e., if for every ε the generalized operator p l [Ψ , ..., Ψ r ]( ε ) is right-invertible.But we could also define a right-invertible PPT as such that each generalized operator Ψ i , with i = 1 , ..., r , is right-invertible. These conditions are very different. Indeed, the first one is aboutthe invertibility of multivariate polynomials in noncommutative variables, while the second one isabout the invertibility of the variables of a multivariate polynomial. Thus, the first one relies on Gauge field with gauge group G are incorporated as follows. Recall that they are principal G -connections, whichcan be regarded as 1-forms in M taking values in the adjoint G -bundle E g . Thus, we just take E i = T M ∗ ⊗ E g . p l , while the second one is a condition only on the operators Ψ i .Luckly, we will need only the second condition, leading us to the define: Definition 2.7.
We say that a PPT in r variables is right-invertible if the defining operators Ψ i ,with i = 1 , ..., r , are right-invertible. Notice that, as in Definitions 2.2, 2.5 and 2.6, a parameterized field theory has as parametersa single element ε ∈ Par( P ). On the other hand, as discussed at the beginning of Section 2, aphysical theory may depend on many fundamental scales. This leads us to consider parameterizedtheories depending on a list ε ( ℓ ) = ( ε , ..., ε ℓ ) of elements of Par( P ), i.e, such that ε ( ℓ ) ∈ Par( P ) ℓ .In this case, we say that the number ℓ ≥ fundamental degree of the parameterized theory.We will also use the convention that Par( P ) is a singleton, whose element we denote by ε (0). Moreprecisely, we have the following definition. Definition 2.8.
Let M be a compact and oriented manifold. A GPT of degree ℓ ≥ M ) is given by a collection of generalized operators Ψ ε ( ℓ ) ∈ Op( E ). Particularly,a PPT of degree ( l, ℓ ) in r variables is given by an element p lℓ ∈ Map l (Par( P ) ℓ ; K )[ x , ..., x r ] andby operators Ψ i ∈ Op( E ), with i = 1 , ..., r . After the previous digression we are now ready to state our main theorem. The syntax is thefollowing:
Main Theorem Syntax.
Let M be a compact and oriented manifold with a fixed generalizedbackground GB( M ) . Let S be a GPT of degree ℓ and S a PPT of degree ( l, ℓ ′ ) in r variables.Suppose that:1. S depends on the fundamental parameters ε ( ℓ ) in a “linear” way;2. the PPT theory S is right-invertible and its coefficients f α are “suitable” functions in a sensethat depends on r, l and ℓ ′ .Then S emerges from S .In order to turn this syntax into a rigorous statement, let us described what we meant bya “linear” dependence on the parameters and by “suitable” functions. Since the term “linear”typically means “preservation of some algebraic structure”, it is implicit that we will need toassume some algebraic structure on the space of parameters Par( P ) ℓ . To begin, we will require anassociative and unital K -algebra structure, whose sum and multiplication we will denote by “+ ℓ ”and “ ∗ ℓ ”, respectively, or simply by “+” and “ ∗ ” when the number ℓ of fundamental parameters isimplicit. Thus, Definition 3.1.
A GPT of degree ℓ in a generalized background GB( M ) is linear (or homomorphic )if Par( P ) ℓ has a K -algebra structure and the rule ε ( ℓ ) Ψ ε ( ℓ ) is a K -algebra homomorphism.11ome properties of the emergence phenomena (as those proved in Subsection 4.1) depends onlyon the preservation of sum “+”, scalar multiplication, or product “ ∗ ”. This motivates the followingdefinition: Definition 3.2.
Under the same notations and hypotheses of the last definition, we say that a GPTis additive (resp. multiplicative ) if the rule ε ( ℓ ) Ψ ε ( ℓ ) is K -linear (resp. a multiplicative monoidhomomorphism). If it is only required Ψ cε ( ℓ ) = c Ψ ε ( ℓ ) we say that the GPT is scalar invariant .On the other hand, notice that in a PPT the generalized operators Ψ always appear multipliedby a number f ( ε ( ℓ )) ∈ K , which depends on the parameters ε ( ℓ ). However, if one recalls that theparameters are interpreted as fundamental scales, it should be natural to consider parameters ε ( ℓ )(instead of numbers f ( ε ( ℓ )) ∈ K assigned to them) multiplying the operators Ψ. Thus, we need anaction · ℓ : Par( P ) ℓ × Op( E ) → Op( E ). But, since by Definition 2.3 and by the above assumptionboth Op( E ) and Par( P ) ℓ are K -algebras, it is natural to require some compatibility between theaction · ℓ and these K -algebra structures. More precisely, we will assume that · ℓ is K -bilinear andthat the following condition are satisfied for every Ψ , Ψ ′ ∈ Op( E ) and every ε ( ℓ ) ∈ Par( P ) ℓ :( ε ( ℓ ) · ℓ Ψ) ◦ Ψ ′ = ε ( ℓ ) · ℓ (Ψ ◦ Ψ ′ ) and ( ε ( ℓ ) ∗ δ ( ℓ )) · ℓ Ψ = ε ( ℓ ) · ℓ ( δ ( ℓ ) · ℓ Ψ) . (7) Remark . If not only conditions (7) are satisfied, but alsoΨ ◦ ( ε ( ℓ ) · ℓ Ψ ′ ) = ε ( ℓ ) · ℓ (Ψ ◦ Ψ ′ ) (8)is satisfied for every ε ( ℓ ) and every Ψ , Ψ ′ , then the algebra Par( P ) ℓ must be commutative. SeeComment 3. On the other hand, if (7) is satisfied for every ε ( ℓ ) and every Ψ , Ψ ′ , but (8) is satisfiedfor every ε ( ℓ ) and a single Ψ ′ = Ψ, then Par( P ) ℓ need not be commutative.Now, to make a rigorous sense of Condition 2 in the previous syntactic statement, let us clarifywhat we mean by a “suitable” function f ∈ Map(Par( P ) ℓ ; K ). In few words, a function f is“suitable” if it belongs to some functional calculus. For us, a functional calculus of degree ℓ is asubset C ℓ ( P ; K ) of Map(Par( P ) ℓ ; K ) endowed with a function Ψ ℓ − : C ℓ ( P ; K ) → Op( E ) assigningto each function f ∈ C ℓ ( P ; K ) a generalized operator Ψ ℓf , which is compatible with the action · ℓ ,in the sense that Ψ ℓf ◦ [ f ( ε ( ℓ ))Ψ] = ε ( ℓ ) · ℓ Ψ . (9) Definition 3.3.
A functional calculus is unital if C ℓ ( P ; K ) contains the constant function f ≡ Lemma 3.1.
In every generalized background
GB( M ) , for every ℓ ≥ there exists a unique func-tional calculus of degree ℓ such that C ℓ ( P ; K ) is the set of nowhere vanishing functions f : Par( P ) ℓ → K if ℓ > , and the constant function f ≡ if ℓ = 0 .Proof. First, assume existence for every ℓ ≥
0. Uniqueness in the case ℓ = 0 is obvious. In the case ℓ >
0, from the existence hypothesis we have Ψ ℓf ◦ [ f ( ε ( ℓ ))Ψ] = ε ( ℓ ) · ℓ Ψ for every f , Ψ and ε ( ℓ ), sothat Ψ ℓf ◦ Ψ = ( ε ( ℓ ) · ℓ Ψ) /f ( ε ( ℓ )). In particular, for Ψ = I , we get Ψ ℓf = ( ε ( ℓ ) · ℓ I ) /f ( ε ( ℓ )), provinguniqueness. In order to prove existence, for each ℓ ≥ ℓf = ( ε ( ℓ ) · ℓ I ) /f ( ε ( ℓ )), so thatΨ ℓf ◦ [ f ( ε ( ℓ ))Ψ] = ( ε ( ℓ ) · ℓ I ) ◦ Ψ = ε ( ℓ ) · ℓ ( I ◦ Ψ) = ε ( ℓ ) · ℓ Ψ , where in the last step we used the compatibility between · ℓ and ◦ , as described in (7).12inally, let us state a few more necessary technical assumptions:1. The K -algebra of fundamental parameters Par( P ) ℓ is required to have square roots. Thismeans that the function ε ( ℓ ) ε ( ℓ ) ∗ ε ( ℓ ) is surjective, i.e, for every ε ( ℓ ) we there is another p ε ( ℓ ) ∈ Par( P ) ℓ such that ( p ε ( ℓ )) = p ε ( ℓ ) ∗ p ε ( ℓ ) = ε ( ℓ ) . Right multiplication by I is injective . More precisely, for every ε ( ℓ ) , δ ( ℓ ) ∈ Par( P ) ℓ , if ε ( ℓ ) · ℓ I = δ ( ℓ ) · ℓ I , then ε ( ℓ ) = δ ( ℓ ), i.e., ϕ i = δ i for i = 1 , ..., ℓ .Some comments concerning these technical conditions:1. The square roots p ε ( ℓ ) in condition 1. need not be unique.2. Condition 2 above and compatibility conditions (7) imply that the right multiplication r ℓ Ψ :Par ℓ ( P ) → Op( E ) is injective for every right-invertible operator Ψ ∈ R Op( E ). Thus, forevery such Ψ, the map r ℓ Ψ is actually an isomorphism Par( P ) ℓ ≃ Par( P ) ℓ · ℓ Ψ between itsdomain and its image.3. If both compatibility conditions (7) and (8) are satisfied, then Condition 2 above implies thatthe K -algebra (Par( P ) ℓ , ∗ ℓ ) is commutative. This follows basically from the Eckmann-Hiltonargument [21, 24].Now, suppose that Par( P ) ℓ has a K -algebra structure. Then Par( P ) kℓ has an induced K -algebra structure, with k >
0, given by componentwise sum and multiplication. If Par( P ) ℓ hassquare roots, then Par( P ) kℓ has too, given by p ε ( kℓ ) = ( p ε ( ℓ ) , ..., p ε ( ℓ k )), where ε ( kℓ ) =( ε ( ℓ ) , ..., ε ( ℓ k )) and ε ( ℓ i ) = ( ε i, , ..., ε i,ℓ ). On the other hand, since Par( P ) lℓ embeds in Par( P ) kℓ as ε ( ℓ ) ( ε ( ℓ ) , ..., ε ( ℓ l ) , , ..., l ≤ k , it follows that every action · kℓ of Par( P ) kℓ can be pulledback to an action · lℓ of Par( P ) lℓ .If C kℓ ( P ; K ) ⊂ Map(Par( P ) kℓ ; K ) is a set of functions, pullback by the inclusion ı : Par( P ) lℓ ֒ → Par( P ) kℓ above defines a new set of functions C lℓ ; k ( P ; K ) ⊂ Map(Par( P ) lℓ ; K ). Furthermore, if C kℓ ( P ; K ) is actually a functional calculus defined by a map Ψ kℓ − : C kℓ ( P ; K ) → Op( E ), we have aninduced functional calculus in C lℓ ; k ( P ; K ), defined by the function Ψ lℓ ; k − : C lℓ ; k ( P ; K ) → Op( E ) suchthat Ψ lℓ ; kı ◦ f = Ψ kℓf . Notice that if C kℓ ( P ; K ) is unital and/or satisfies the third technical conditionsabove, then C lℓ ; k ( P ; K ) does. We can now rigorously state the main result of this paper. First, notice that the three technicalconditions of last section are about algebraic properties of the space Par ℓ ( P ) of fundamental pa-rameters and on its action on the algebra Op( E ) of generalized operators. On the other hand, recallfrom Definition 2.3 that both Par ℓ ( P ) and Op( E ) are part of the data defining a generalized back-ground. Thus, those conditions are actually conditions on the underlying generalized backgroundGB( M ). This motivates the following definition. Definition 3.4.
Let M be a compact and oriented smooth manifold and let ℓ ≥ k > M ) is of ( ℓ, k ) -type if:13. the space of fundamental parameters Par ℓ ( P ) has an structure of K -algebra with square roots;2. the induced K -algebra Par kℓ ( P ) acts in Op( E ) by an action · kℓ which is injective at theidentity operator I and compatible with a functional calculus C kℓ ( P ; K ).Our main theorem is then the following: Theorem 3.1 (Emergence Theorem) . Let M be a compact and oriented manifold and let GB( M ) be a generalized background of ( ℓ, k ) -type, with k > . Let S be a GPT of degree ℓ and let S be aPPT of degree ( l, ℓ ′ ) in r variables, where ℓ ′ = k ′ ℓ , with < k ′ ≤ k , defined on GB( M ) . Assume:1. S is homomorphic;2. S is right-invertible and the coefficient functions f α : Par( P ) k ′ ℓ → K belongs to the functionalcalculus C k ′ ℓ ; k ( P ; K ) .Then S emerges from S . Before proving the emergence theorem, let us say that the proof which will be given here canbe generalized, with basically the same steps and arguments, in some directions (for more details,see [35]):1. the spacetime manifold M need not be compact nor orientable . Notice that these conditionswere used only to define integrals. Thus, instead, one could only assume integrability condi-tions on global sections (such as compact supportness) and consider Lagrangians as takingvalues on general densities.2. the space of fundamental parameters need not be a K -algebra, but a more lax algebraic entity .As will be clear during the proofs, we only need the “nonnegative” part of a K -algebra.More precisely, what we really need is that the set Par( P ) ℓ can be regarded as the subsetof a K -algebra A , which is closed by sum, multiplication, and scalar multiplication by R ≥ .Notice that if P is a K -algebra bundle, then Par( P ) ℓ can always be realized as a subset ofthe K -algebra Γ( P ) ℓ .3. the coefficient functions f α need not be scalar functions, but actually maps f α : Par( P ) ℓ ′ → Par( P ) ℓ , so that the scalar multiplication f α ( ε ( ℓ ))Ψ is replaced by the action · ℓ . The notionsof functional calculus, etc., can be defined in an analogous way such that the syntax of thetheorem remains the same . Let M be a compact and orientable manifold, K = C an E = M × C the complex trivial linebundle, regarded as a field bundle with space of fields given by complex scalar functions C ∞ ( M ; C ),endowed with the pairing h ϕ, ψ i = ϕψ . In addition, let P = M × C , viewed now as the parameterbundle, and take the constant functions as parameters, so that Par( P ) ≃ C . This data clearlydefines a background over M . Let Op( E ) be the complex algebra Psd( M × C ) of pseudo-differential Recall from the discussion at Introduction that parameter-valued coefficient functions plays an important role inthe description of emergent gravity in terms of emergence phenomena. Further explanations will appear in a work inprogress. P ) ≃ C is a C -algebra with square roots and with their action in Psd( M × C ) via scalarmultiplication, if z · I = z ′ · I , then clearly z = z ′ . Finally, let C ( P ; C ) be the unital functionalcalculus given by nowhere vanishing functions f : Par( P ) ≃ C → C , as in Lemma 3.1, defining ageneralized background over M of (1 , Corollary 3.1.
Let M be a compact and oriented manifold and let GB( M ) be the generalizedbackground of (1 , -type defined above. Let D ∈ Diff( M × C ) be any idempotent differential operator,i.e., there exists n > such that D n = D n . For given r > and l ≥ , let p l [ x , .., x r ] bea polynomial of degree l in r variables and whose coefficients f α : C → C are nowhere vanishingfunctions. Let D , ..., D r ∈ Diff( M × C ) other differential operators and assume one of the followingconditions:1. the operators D i , with i = 1 , ..., r are of constant coefficient;2. there exists a Riemannian metric in M such that D i , with i = 1 , ..., r are strongly elliptic inthe sense of any of references [28, 14, 47, 48, 2, 29, 3, 25] ;3. there exists a Lorentzian metric in M such that M is globally hyperbolic and each D i , with i = 1 , ..., r , is Green hyperbolic.Then theory L ( ∞ ϕ ; ε ) = ϕεD n ϕ emerges from theory L ( ∞ ϕ ; δ ) = ϕp l [ D , ..., D r ] ϕ = X | α |≤ l ϕf α ( δ ) D α ϕ. Proof.
Since D n = D n ◦ D n = D n , the rule ε εD n is clearly homomorphic. On the otherhand, from the discussion on Section 2.2 each of the three hypotheses above implies that D i , for i = 1 , ..., r , are right-invertible as objects of Psd( M × C ). Since the coefficient functions f α arenowhere vanishing and therefore belong to the functional calculus of GB( M ), the result followsfrom Theorem 3.1. Remark . From Comment 1, the same construction of GB( M ) holds if M is a bounded openset of some R N . From Comment 3 it remains valid if Par( P ) ≃ R ≥ are the constant non-negativereal functions and the functional calculus C ( R ≥ ; C ) consists of the nowhere vanishing functionstaking values in R ≥ . The difference, in this case, is that both ε and f α ( δ ) are real numbers, sothat the Lagrangians in the last corollary are real too. See [35] for further details.Other generalizations of the last corollary, and particular cases of Theorem 3.1, are the following:1. We can consider other kind of fields . In Corollary 3.2 we considered a generalized backgrounddefined on the trivial line bundle M × C . Notice, however, that if E is any complex bundlewith an Hermitian metric at the fibers, then the space of pseudo-differential operators Psd( E )remains well-defined as a Z -graded C -algebra, so that the same thing holds equally well ifinstead of scalar fields one considered vector fields and tensor fields.15. We can consider other kind of parameters . In Corollary 3.2 we considered ℓ = 1 and Par( P ) ≃ C (or R ≥ , due to the remark above). We could consider, more generally, Par( P ) as anycomplex algebra with square roots endowed with an action · : Par( P ) × Psd( E ) → Psd( E ),where E is a complex vector bundle (due to the last remark), such that conditions (7) andCondition 2 are satisfied. Let Ψ be such that Ψ n is idempotent and suppose that (8) aresatisfied for fixed Ψ = Ψ ′ = Ψ n . Then the rule ε ε · Ψ n is an algebra homomorphism andCorollary 3.2 holds equally well. Example 3.1 (operator parameters) . Take P = End( E ), so that Γ( P ) ≃ End(Γ( E )), which isan associative C -algebra with an obvious action in Psd( E ) by composition such that conditions(7) and Condition 2 are clearly satisfied. Let Ψ ∈ Psd( E ) be such that Ψ n is idempotent andlet Z (Ψ n ) denote its centralizer, i.e., the subalgebra of all elements σ ∈ End(Γ( E )) such that σ ◦ Ψ n = Ψ n ◦ σ , so that (8) is satisfied. Then take Par( P ) as some subalgebra of Z (Ψ n ) withsquare roots. As a concrete example, one can take Par( P ) as the subalgebra of nonnegativebounded self-adjoint operators in Γ( E ) which commutes with Ψ n .3. We can consider other kinds of parameterized operators.
In Corollary 3.2 and in the abovegeneralizations we considered only parameterized operators of the form Ψ ε = ε · Ψ, whichforced us to assume Ψ idempotent. Indeed, notice that the nilpotency condition was usedonly to ensure that ε ε · Ψ is an algebra homomorphism. More generally, let Par( P ) be somecomplex algebra with square roots endowed with a representation ρ : Par( P ) → End C (Γ( E ))and define the action of Par( P ) in Psd( E ) by ε · Ψ := ρ ( ε ) ◦ Ψ, so that the second conditionin (7) is clearly satisfied. If the action is faithful, then Condition 2 is satisfied too. Finally, ifthe action is compatible with the algebra structure of Psd( E ), i.e., if ρ ( ε ) ◦ (Ψ ◦ Ψ ′ ) = ( ρ ( ε ) ◦ Ψ) ◦ ( ρ ( ε ) ◦ Ψ ′ ) (10)for every ε ∈ Par( P ) and Ψ , Ψ ′ ∈ Psd( E ), then the first part of (7) is also satisfied and forevery fixed Ψ the rule ε ρ ( ε ) ◦ Ψ is homomorphic, so that Corollary 3.2 holds analogously.
Example 3.2.
Recall that a ring R is Boolean if each element is idempotent, i.e., if x ∗ x = x for every x ∈ R . Let Bol( P ) be a Boolean ring and take Par( P ) = Bol( P ) ⊗ Z C . Let ρ : Bol( P ) → End C (Γ( E )) be a faithful representation of this Boolean ring and notice that(10) is immediately satisfied (recall that every Boolean ring is commutative). Tensoring with C we get a faithful representation of Par( P ) = Bol( P ) ⊗ Z C . Finally, notice that every Booleanring has square roots, since for every x we have x = x , i.e., √ x = x . Example 3.3 (a more concrete case) . Let P = M × A be a trivial algebra bundle, so thatΓ( P ) ≃ C ∞ ( M ; A ). Let Bol( A ) ⊂ A be the Boolean ring of the idempotent elements of A and take Bol( P ) as the set of functions f : M → A such that f ( x ) ∈ Bol( A ) for every x ∈ M . Now, let ρ : A → End( F ) be a faithful representation of A in the typical fiber of E .It induces a faithful representation ρ : Γ( P ) → End C (Γ( E )) and, therefore, by restriction afaithful representation of Par( P ). We suspect that the same holds, more generally, for Von Neumann regular rings, but we do not have a proof ofthis. Proof of Theorem 3.1
In this section we prove our emergence theorem. The proof will be inductive on the number r ofvariables of S . In order to prove the base case, i.e., the emergence theorem when S is a univariatepolynomial, we will need to use some additivity and multiplicativity properties of the emergencephenomena. In turn, the induction step will be based on a technical lemma.In order to better understand the whole proof, this section will be organized as follows. InSubsection 4.1 we prove the basic properties of the emergence phenomena needed to prove thebase step. In Subsection 4.2 this base step is proved. In Subsection 4.4, Theorem 3.1 is finallydemonstrated, with the technical lemma used for the induction step being presented before inSubsection 4.3. • In the following discussion, when the degree of a GPT does not matter it will be made implicitin order to simplify the notation. In these cases we will also write ε instead of ε ( ℓ ). Thus,from now on, by saying “ let Ψ ε be a GPT over GB( M )” we mean that it is any GPT of anydegree ℓ . Definition 4.1.
Let Ψ ε and Ψ ′ ε ′ be GPT over the same generalized background GB( M ). The sum and the composition between them are the GPT over GB( M ) given by Ψ +( ε,ε ′ ) = Ψ ε + Ψ ′ ε ′ andΨ ◦ ( ε,ε ′ ) = Ψ ε ◦ Ψ ′ ε ′ . Notice that the sum and the composition between GPT of degrees ℓ and ℓ ′ hasdegree ℓ + ℓ ′ . Lemma 4.1.
Let Ψ ,ε , Ψ ,δ and Ψ ,κ be three GPT over the same generalized background GB( M ) with fundamental parameter algebra Par( P ) ℓ , where ℓ is the degree of the first GPT, such that:1. Ψ ,ε is multiplicative;2. Ψ ,ε emerges from both Ψ ,δ and Ψ ,κ .Then Ψ ,ε emerges from the compositions S ,δ ◦ S ,κ and S ,κ ◦ S ,δ .Proof. From the second hypothesis we conclude that Ψ ,ε = Ψ ,F ( ε ) and Ψ ,ε = Ψ ,G ( ε ) for certainfunctions F, G . Composing them and using the first hypothesis, we findΨ ,ε = Ψ ,ε ◦ Ψ ,ε = Ψ ,F ( ε ) ◦ Ψ ,G ( ε ) = Ψ ,G ( ε ) ◦ Ψ ,F ( ε ) . Let √− : Par( P ) ℓ → Par( P ) ℓ be a function selecting to each fundamental parameter ε ′ a squareroot √ ε ′ , which exists by hypothesis. Then, for every ε ′ one getsΨ ,ε ′ = Ψ ,F ( √ ε ′ ) ◦ Ψ ,G ( √ ε ′ ) = Ψ ,G ( √ ε ′ ) ◦ Ψ ,F ( √ ε ′ ) = Ψ ◦ H ( ε ′ ) , finishing the proof.In a completely analogous way one proves the following. Lemma 4.2.
Let Ψ ,ε , Ψ ,δ and Ψ ,κ be three GPT over the same generalized background GB( M ) ,such that:1. Ψ ,ε is scalar invariant;2. Ψ ,ε emerges from both Ψ ,δ and Ψ ,κ .Then Ψ ,ε also emerges from the sum Ψ ,δ + Ψ ,κ . .2 Base of Induction In this subsection, using the additivity and multiplicativity properties of last section, we willprove the following lemma, which will be the base of the induction step in the proof of Theorem3.1:
Lemma 4.3.
Let
GB( M ) be generalized background of ( ℓ, k ) -type. Let Ψ ,ε ( ℓ ) be a GPT of degree ℓ and let Ψ ,δ ( ℓ ′ ) a PPT of degree ( l, ℓ ′ ) in r = 1 variables, defined on GB( M ) and such that ℓ ′ = k ′ ℓ ,with < k ′ ≤ k . Suppose that:1. Ψ ,ε ( ℓ ) is homomorphic;2. Ψ ,δ ( ℓ ′ ) is right-invertible and the coefficient functions f α : Par( P ) k ′ ℓ → R of the polynomial p lℓ ′ defining Ψ ,δ ( ℓ ′ ) belongs to the functional calculus C k ′ ℓ ; k ( P ; K ) .Then Ψ ,ε ( ℓ ) emerges from Ψ ,δ ( ℓ ′ ) . We begin with another lemma.
Lemma 4.4.
Let Ψ ,ε ( ℓ ) be a GPT of degree ℓ defined on a generalized background GB( M ) of ( ℓ, k ) -type, with k > . Then Ψ ,ε ( ℓ ) emerges from every GPT Ψ ,δ ( ℓ ′ ) over GB( M ) , which hasdegree ℓ ′ = k ′ ℓ for some < k ′ ≤ k , and such that Ψ l ,δ ( ℓ ′ ) = g ( δ ( ℓ ′ ))Ψ l , with l ≥ , where Ψ isright-invertible and g ∈ C k ′ ℓ ; k ( P ; K ) .Proof. Since Ψ = I is right-invertible, the case l = 0 is a particular setup of case l = 1. Further-more, if l > l is right-invertible too, so that the case l > l = 1 case. Thus, it is enough to work with l = 1. Thus, let R Ψ ∈ Op( E )be a right-inverse for Ψ and notice that to find an emergence from Ψ ,ε ( ℓ ) to Ψ ,δ ( ℓ ′ ) is equivalentto building a function F : Par( P ) ℓ → Par( P ) ℓ ′ such that Ψ ,ε ( ℓ ) ◦ R Ψ = g ( F ( ε ( ℓ ))) I . From (9) andfrom the fact that the right multiplication by right-invertible operators is injective, last condition isin turn equivalent to the existence of F such that Ψ ℓ ′ g ◦ Ψ ,ε ( ℓ ) ◦ R Ψ = F ( ε ( ℓ )) · ℓ ′ I , but this actuallydefines F via Par( P ) ℓ ′ · ℓ ′ I ≃ Par( P ) ℓ ′ . Sketch of proof of Lemma 4.3.
Given a PPT Ψ ,δ ( ℓ ′ ) = P i f i ( δ ( ℓ ′ ))Ψ i in the hypothesis, for each j = 1 , ..., l let Γ j = P li = j f i ( δ ( ℓ ′ ))Ψ i − and notice thatΨ ,δ ( ℓ ′ ) = ( l X i =1 f i ( δ ( ℓ ′ ))Ψ i − ) ◦ Ψ = Γ ( δ ( ℓ ′ )) ◦ Ψ . Since Ψ is right-invertible and GB( M ) is of ( ℓ, k )-type, with k >
0, from Lemma 4.4 it follows thatΨ ,ε ( ℓ ) emerges from 1 · Ψ . Thus, if S ε ( ℓ ) itself emerges from Γ one can use Lemma 4.1 to concludethat it actually emerges from Γ ◦ Ψ. In turn, notice that Γ = f · I + Γ ◦ Ψ = Γ ◦ Ψ + f · I . But,since I is right-invertible and since f ∈ C ℓ ′ ; k ( P ; K ), from Lemma 4.4 we get that S ε ( ℓ ) emergesfrom the theory defined by f · I , while by the same argument we see that Γ ◦ Ψ emerges from f · I . Therefore, if Ψ ,ε ( ℓ ) emerges from Γ ◦ Ψ we will be able to use Lemma 4.2 to concludethat it emerges from Γ , finishing the proof. It happens that, as done for Γ ◦ Ψ, we see that Γ Here we are using explicitly that the functional calculus is unital. ,ε ( ℓ ) emerges from Ψ. Thus, our problem is to provethat Ψ ,ε ( ℓ ) emerges from Γ instead of from Γ . A finite induction argument proves that if Ψ ,ε ( ℓ ) emerges from Γ l , then it emerges from Γ j , for each j = 1 , ..., l . Recall that Γ l = f l · Ψ l − . Since f l ∈ C ℓ ′ ; k ( P ; K ) we can use Lemma 4.4 to see that S ε ( ℓ ) really emerges from Γ l . Also as a consequence of the properties of the emergence phenomena, we can now prove thefollowing technical lemma, which will be used in the induction step of Theorem 3.1.
Lemma 4.5.
Let Ψ ,ε be a GPT over a generalized background GB( M ) . Given l ≥ , let Ψ j ,δ j and Ψ s ,κ s , with ≤ j, s ≤ l be two families of GPT, also defined over GB( M ) . Assume that:1. Ψ ,ε is homomorphic;2. Ψ ,ε emerges from Ψ j ,δ j and from Ψ s ,κ s for every j, s .Then Ψ ,ε emerges from Ψ sδ J ,κ J = P sj =1 Ψ j ,δ j ◦ Ψ j ,κ j , for every s = 1 , ..., l Proof.
We proceed by induction in l . First of all, notice that from the first two hypotheses andfrom Lemma 4.1 we see that Ψ ,ε emerges from the composition Ψ j ,δ j ◦ Ψ j ,κ j for every j = 1 , ..., l .In particular, it emerges from Ψ δ ,κ = Ψ ,δ ◦ Ψ ,κ , which is the base of induction. For every m = 1 , ..., l − m +1 ,δ m +1 ◦ Ψ m +1 ,κ m +1 . For the induction step, suppose thatΨ ,ε emerges from Ψ mδ J ,κ J = P mj =1 Ψ j ,δ j ◦ Ψ j ,κ j for every 1 ≤ m ≤ l − m +1 δ J ,κ J . Notice thatΨ m +1 δ J ,κ J = m +1 X j =1 Ψ j ,δ j ◦ Ψ j ,κ j = m X j =1 (Ψ j ,δ j ◦ Ψ j ,κ j ) + Ψ m +1 ,δ m +1 ◦ Ψ m +1 ,κ m +1 = Ψ mδ J ,κ J + (Ψ m +1 ,δ m +1 ◦ Ψ m +1 ,κ m +1 ) . From the induction hypothesis Ψ ,ε emerges from Ψ mδ J ,κ J , while by the above it also emerges fromΨ m +1 ,δ m +1 ◦ Ψ m +1 ,κ m +1 . The result then follows from Lemma 4.2. We can finally proof our emergence theorem. For the convenience of the reader, we state itagain.
Theorem 3.1 (Emergence Theorem)
Let M be a compact and oriented manifold and let GB( M ) be a generalized background of ( ℓ, k ) -type, with k > . Let Ψ ,ε ( ℓ ) be a GPT of degree ℓ and let Ψ ,δ ( ℓ ) be a PPT of degree ( l, ℓ ′ ) in r variables, where ℓ ′ = k ′ ℓ , with < k ′ ≤ k , defined on GB( M ) .Suppose that:1. Ψ ,ε ( ℓ ) is homomorphic;2. Ψ ,δ ( ℓ ) is right-invertible and the coefficient functions f α : Par( P ) k ′ ℓ → K belongs to thefunctional calculus C k ′ ℓ ; k ( P ; K ) . hen Ψ ,ε ( ℓ ) emerges from Ψ ,δ ( ℓ ) .Proof. The proof will be done by induction in r . The base of induction is Lemma 4.3. Sup-pose that the theorem holds for each r = 1 , ..., q and let us show that it holds for r = q +1. Let p lℓ ′ ; q +1 [ x , ..., x r +1 ] = P | α |≤ l f α · x α be a multivariate polynomial with coefficients inMap(Par( P ) ℓ ′ ; K ), which actually belong to C k ′ ℓ ; k ( P ; K ). Since for every commutative ring R wehave R [ x , ..., x q +1 ] ≃ R [ x , ..., x q ][ x q +1 ], given right-invertible generalized operators Ψ , ..., Ψ q +1 ∈ Op( E ) one can writeΨ ,δ ( ℓ ) = p lℓ ′ ; q +1 [Ψ , ..., Ψ r +1 ] = X j p l j ℓ ′ ; q,j [Ψ , ..., Ψ q ] · Ψ jq +1 , where each p l j ℓ ′ ; q,j [ x , ..., x q ] ∈ Map l j (Par( P ) ℓ ′ ; K )[ x , ..., x q ] has coefficients which belongs to belongsto C k ′ ℓ ; k ( P ; K ). Thus, by the induction hypothesis, Ψ ,ε ( ℓ ) emerges from p l j ℓ ′ ; q,j [ x , ..., x q ]. Since Ψ q +1 is right-invertible and since the functional calculus is unital, from Lemma 4.4 we see that Ψ ,ε ( ℓ ) emerges from Ψ jq +1 . The result then follows from Lemma 4.5. Acknowledgments
The first author was supported by CAPES (grant number 88887.187703/2018-00).
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