Time in Quantum Cosmology

8/14/2019 Time in Quantum Cosmology

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The Problem of Time in Quantum Cosmology

and Non-chronometric Temporality

by

Carlos Pedro GonçalvesMathematics researcher at UNIDE-ISCTE, in the areas of quantum computation,

quantum formal systems theory, quantum game theory, quantum cosmology and generalrisk science

[email protected] (primary); [email protected]; [email protected]

Maria Odete MadeiraInterdisciplinary researcher in philosophy of science, systems science, complexity sci-

ences, neurocognition, semiotics, ontology and [email protected] (primary), [email protected]

Abstract

We review two lines of argument regarding the problem of time in quantum cosmology and inquantum gravity, one that invokes the path integral formalism for quantum gravity to state theabsence of time between two three-geometries, and another that defends the absence of time, asa fundamental notion in physics, in terms of: (a) the configuration space argument , put forwardby Barbour, Smolin and Kauffman, and (b) the Wheeler-DeWitt equation.

We argue that although being correct with respect to a space-time dependent physicalchronometrizable clock-time frame, both of these lines of argument fail with respect to a generalsense of temporality, expressed in terms of the more elementary notions of a before and an after of a quantum computation.

With respect to the first line of argument, it is shown that the early works on the subjectaddress two kinds of temporalities, one that is the space-time geometric dependent temporality,which coincides with the usual definition of a space-time dependent physical chronometrizable

clock-time frame, the other is a temporality associated to the notions of input and output of ageneral quantum gravity computation, that is expressed, in the theoretical discourse of quantumgravity, through the usage of the concepts of: (1) propagation of a wave functional in super-space, as addressed by Wheeler; (2) transition amplitudes of three-geometries and (3) the path-integral formalism, used to calculate such amplitudes, as addressed by Hartle and Hawking.While the first temporality (space-time dependent temporality) disappears from the theory, thesecond plays a fundamental role, not only in the several aspects of the theory’s construction, butin the clock-time independence as well, as Wheeler showed.

Given this notion of time, different from a chronometrizable, space-time geometry internalnotion, we search for a general mathematical and logical structure that is capable of addressingit from a formal point of view. This is done through a family of mathematical structures that ismore general than the mathematical category . These structures not only will allow us to addressthe nature of the temporality present in the transition amplitudes between two three-geometries,but they will also allow us to refute the configuration space argument and to show how a staticclock-time-independent quantum state, can be put into a non-clock-time processual expressionin terms of fine-grained computational histories, obtained from the relations between differentobservable’s bases.

Keywords: Quantum cosmology, time, relational structures, relational nexus

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1. Introduction

One of the major open problems, within quantum cosmology, is the so-called problem of time , which is usually expressed in terms of a general statement of the absense of timein quantum cosmology (Smolin, 2001).

The arguments underlying this claim can be generally split in two lines of argumentthat arise from the formalisms used in addressing quantum cosmology and quantum

gravity. One of these lines comes from a path-integral and propagator approach toquantum gravity, where the transition amplitudes, evaluated between two three-geome-tries, are obtained by summing over all four-geometries that aggree with the initial andfinal three-geometries at the border (Hartle and Hawking, 1983). The second of thesetwo lines comes from the analysis of the Wheeler-DeWitt equation, and it can be called,to a good approximation regarding the underlying argument, the time-independent con-

figuration space argument . An example of such argument can be found, for instance, inBarbour (1994), in Smolin (2001) and in Kauffman and Smolin (1997).

We address, in this work, each of these two lines of argument, and show that,although the claim for the non-existence of time, at a fundamental level, is correct withrespect to a space-time internal temporal chronometrizable frame, it is not correct as a

general statement with respect to the presence of a temporality independent from space-time internal temporal chronometrizable frames.

Regarding the first line of argument, the discourse itself of the works that addressedthe propagator and the path integral formalism in quantum gravity trivially shows, uponcloser inspection, the presence of this other temporality, restricting the validity of theclaim, for the absence of time, to chronometrizable temporal frames, internal to a space-time geometry. For, otherwise, contradictory elements would be present in the physicaldiscourse about the theory, leading to problems when addressing the formalism.These “apparent contradictions” already appear in the early works within quantum cos-mology and quantum gravity, including Wheeler’s 1968 article “Superspace and theNature of Quantum Geometrodynamics” and Hartle and Hawking’s 1983 article “The

Wave Function of the Universe”.In Wheeler’s article, the conclusion that there is no sense of time in quantum

geometrodynamics is defended and sustained by a set of arguments. However, in con-cluding about this non-existence, Wheeler uses, as a supporting argument, the behaviorof the propagation of a wave packet in superspace, which immediately becomes problem-atic with the general statement about the non-existence of time in quantum cosmology,as a quantum propagation exemplifies a temporality property that comes from aquantum gravity computational process, expressed in terms of a unitary transition.

This “apparent contradiction”, as we argue in the current work (section 2.), is onlyan apparent one, due to an ambiguated phrasing of the statement regarding the absense

of time in the theory of quantum gravity. Indeed, the statement should be taken to referonly to a notion of a physical clock time, internal to a space-time geometry, this phys-ical clock time is undetermined because the space-time geometry is undetermined. How-ever, not all temporality is removed from the description, since there is still a non-chronometric temporality associated with the basic notions of input and output of a uni-tary transformation of a quantum state defined over the superspace, a temporality thatis of a different nature than that of the internal chronometrizable temporal frame, thatdepends upon the space-time geometry for its chronometrization.

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This is what allows Wheeler to speak of a propagation of a wave packet in superspace,and what allows Hartle and Hawking to speak of a transition amplitude between twothree-geometries, and work with the path integral formalism to calculate such ampli-tude, without incurring into the fundamental problem of using theoretically inconsistentterminology.

As we argue, in greater detail, in section 2., several of the statements and basic for-malism, worked out by Hartle and Hawking (1983), can only be consistent with aquantum cosmology if we accept that chronological or space-time dependent clock time

is undetermined in the theoretical description, and, simultaneously, understand thatquantum gravity introduces a second type of temporality different from the clock-timethat is internal to a fixed space-time. This second temporality, with which we are con-fronted, through these early works on quantum cosmology and quantum gravity, is notchronometrizable with respect to a pre-given space-time, and has a nature only depen-dent upon a notion of a before and an after associated with an input and an output of ageneral quantum gravity computation .

Indeed, in order for this temporality to be chronometrizable one would have to have aclock-time frame, which is a notion internal to a pre-given space-time geometry, which isnot well defined here. Between two three-geometries we cannot measure how temporallyseparated they are, because that measurement depends upon the four-geometries that

are being summed over in the path integral. All that remains, therefore, are the moreprimitive temporal notions of a before and an after , since one three-geometry appearsbefore the other in the (reversible) propagator formalism, hence the reason for the termsinitial and final three-geometries used by Hartle and Hawking (1983).

Understanding the presence of this temporality within the theoretical description, andshowing it to be present in some of the foundational works on quantum cosmology andquantum gravity are the main objectives of section 2., where we provide for a brief review on the conception of time within physics.

In section 3., we propose the usage of a mathematical structure, called a binary rela-tional structure . These structures should not be confused with the usual mathematical

set theoretical structures known as binary relations that belong to the category Rel.Indeed, the general family of structures called mathematical categories is a sub-family of these binary relational structures . In an intuitive, first approximation, one may considerthe binary relational structure as simply composed of a collection of objects and a collec-tion of arrows or morphisms , along with the identity morphism . However, this would be

just a first approximation, since even the notion of morphism is a particular case of therelations considered in these structures.

The collection of relations , in the binary relational structures , are not necessarily set-theoretical binary relations. Indeed, as we show in the main text and in the appendix tothe present work, the mathematical notion of category is a particular case of these struc-tures. In the appendix, we address the examples of Rel and Mag, in order to make

clear the distinction between the binary relational structures and Rel.In section 3. we use the binary relational structures in order to understand and for-

malize the fundamental non-chronometrizable temporality that is found within theworks on quantum cosmology and quantum gravity.

Afterwards, we apply the formalism to address the second line of arguments regardingthe absence of time in quantum cosmology. In particular, we show how, even a timeindependent quantum state, can be ascribed a computational description with respect to

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the unitary transformation between two different bases, which results in a processualand historical nature for the quantum state, that is independent of clock time, but thatcan be formulated in terms of a fine-grained quantum computational history, intro-ducing a consistent histories’ type of formalization for a clock-time static quantum state.

This allows us to address the second line of arguments for the absence of time inquantum cosmology, based upon the Wheeler-DeWitt equation and upon the notion of configuration space . In section 3. we review this line of argument, and address its prob-lems with respect to its foundational notions, along with the consequences of the appli-

cation of the mathematics of the binary relational structures to the formalism underlyingthe argument.

Central to the present work is, thus, the concept of binary relational structure , wherea notion of temporality emerges from a partition of a relational nexus of any two of thestructure ’s objects, which introduces the notions of a before and an after that resultfrom the positions of the objects in the relations that connect them.

In section 4. we conclude with some final reflections upon the main results, andaddress what could be considered as open issues that may arise with respect to themathematics of binary relational structures , in connection to physics and quantumgravity.

2. A brief reflection on the conceptions of time inphysics

With Galileo the time was conceptualized as a fundamental physical quantity, measur-able for a whole series of physical systems and, consequently, susceptible of regulatingexperiments and relating them mathematically, thus, the time became the measure of the motion (Klein, 1995).

In Newton, this arithmetizable, physically measurable time was considered to be abso-

lute and to flow uniformly, independently of the reference frame (Klein, 1995). In spe-cial relativity, these two notions were understood as relative. Apparently, nature seemsto be such that the flow of time depends upon the relative motion of the inertial refer-ence frames. Which led Einstein to place the event (that, one should add, is simulta-neous with itself) as the central element of the physical theory.

Special relativity places us before a different perspective, regarding the nature of time.Indeed, accepting special relativity entails accepting a distinction in the cosmic timebetween its arithmetizable chronometric nature, which is relative, and its fundamentaltemporal sense linked to causality, which is not relative, for the causality horizon of eachevent is a relativistic invariant (Einstein, 1953, [2004]).

In accordance with special relativity, when we try to find a causal explanation of anevent we must seek it in its anteriority, that is, in that which took place before thetaking place of the event, that which is to the causal past of the event, which meansthat we can associate a relativistic invariant sense of what came before, as that whichlies in the past light cone of the event, which forms the past causality horizon of theevent. In the same way, each event can be considered to be at the cause of other events,which means that each event has a future causality horizon, making it a border betweenwhat comes before and what comes after.

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where → is he symbol for the material implication. In this case, provided with the infor-mation that the conjunction (E (1) ∧ E (2) ∧

∧ E (n − 1)) holds (in the sense that it istrue that each event in the conjunction has occurred), then, E (n) must also haveoccurred.

Given a network of causal relations, defined by implications of the kind introduced inthe previous paragraph, and given the truth value of the necessary conditions of thematerial implications, we know all of the network’s narrative, in terms of what occurredand what did not occurr. This fact constitutes a central feature of the relation of

causality expressed by the above implications, which is, thus, a deterministic relation.Smolin (2003), however, presents an additional interpretation of a causal network,

considering it in terms of information processing, such that, each event can be thoughtof as taking the information of the other events in its causal past and performing a com-putation (or a quantum computation in the quantum theoretical setting), sending theresult of such a computation to its future through the (quantum) causal network.

This computational notion, associated with causal histories, is such that we can con-sider, in a more general sense, the causal past of an event as the set of events fromwhich it can receive information, and the causal future of an event can be considered tobe the set of events that can process information from that event.

The special theory of relativity establishes a limit to causality, as it distinguishesbetween timelike, spacelike and lightlike separations.

Quantum theory, on the other hand, places us before the problem of nonlocal connec-tions, which are not causal but spacelike correlational, in the sense that we need to con-sider the entire spacelike surface, where two or more entangled events occur, to be aninterconnected whole, described by a quantum state.

This configures a local arrow of time in the form of local temporal directedness due toentanglement-induced local decoherence. Information is “exported” to the whole, andonly the whole has the whole information, so to speak.

In general relativity, and in the standard approach to quantum cosmology, the con-

cept of time becomes more problematic, though less so, as we saw, in the quantumcausal histories approach.

The first thing to notice, as Wheeler (1968) stressed, regarding classical geometrody-namics, is that the concept of event becomes less primitive and less significant. One maychoose to consider space-time to be comprised of elementary objects or points calledevents, or, one may, instead, choose to consider the three-geometry to be the primaryconcept, and the event the secondary concept, an event lying at the intersection of suchand such three-geometry.

Under this last perspective, the temporal relation between two three-geometries wouldbe determined by the structure of the four-geometry, which in turn derives from theinter-crossings of all the other three-geometries.

In classical geometrodynamics, whether one started with the three-geometry as theprimitive concept, or with the event as the primitive concept makes little difference,according to Wheeler. However, still according to Wheeler (1968), it makes all the differ-ence when one turns to quantum geometrodynamics, for there is no such thing as a well-defined four-geometry, because, as far as we know, and still according to Wheeler, noprobability amplitude function can propagate through superspace as an indefinitelysharp wave packet, the wave packet spreads.

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The conclusion, following Wheeler’s argument, is that the space-time, the time, thenotion of before and after do not exist as primitive notions within the theory, which is aproblematic statement, from a theoretical discursive point of view, given the fact thateven if a four-geometry is undetermined, and even if the before and after of a space-timegeometry are internal to that geometry, there is still the problem of the propagation of the wave packet in superspace, that, from the moment in which we use the term propa-gation, commits us to a temporal sense present in superspace, different from the internaltimes definable and chronometrizable within each four-geometry.

It is important to review Wheeler’s argument, with respect to this matter, in order tobetter understand this issue. The first thing to notice is that the spreading of the wavepacket means that it associates a finite probability to a domain of superspace of finitemeasure, this domain encompassing a set of three-geometries far too numerous toaccommodate in any one four-geometry.

One can express this situation, according to Wheeler (1968), by stating that the prop-agation takes place in superspace, not by following any one classical history of space butby summation of contributions from an infinite variety of such histories. It is noticeablehere the unavoidance of the apparent “discursive trap” that talks about quantum evolu-tion in superspace, which installs, in the discourse, a quantum temporality that is proper

to the notion of unitary quantum evolution of a wave function, to the notion of a propa-gator in superspace, and to the notion of sum-over-histories of space-time geometries.

However, this “discursive trap” is only an apparent one, due to an ambiguation of twotemporal senses that must be conceptually disentangled. On the one hand, we have thechronological time that is an internal definition to each four-geometry, and, on the otherhand, we have a temporal sense, distinct and not ontologically committed to an internalmeasurable chronometrics, but, instead, associated with the more elementary notion of atemporal property simply exemplified by an order of a before and an after of a computa-tion, an input and an output .

Thus, in this case, Wheeler can talk about a unitary connection, and use temporal

discursive elements with respect to it, without incurring into any problem of discursivebias due to unexpunged terminology, habituated to a regular usage of quantummechanics. Indeed, the unitary connection between two three-surfaces introduces a tem-poral nature of its own, the temporality associated to a notion of a before and an after ,an input and an output of a non-chronometric connection, which, itself, contains a multi-tude of probable chronometrics, each associated with different (to be summed-over) his-tories of space-time geometries (the internal times being a part of these histories).

The three-geometries that occur with significant probability amplitudes do not fit,according to Wheeler, and cannot be fitted into any single four-geometry. Without thatbuilding plan, what we call the internal cosmic time, that might be definable within a

fixed four-geometry, at least locally, is undefined and, without a building plan to orga-nize the three-geometries of significance into a definite relationship, one to another, eventhe internal geometric-specific notion of a time ordering of events is devoid of meaning.One must stress that this time ordering, and this before and after, is distinct from thatof a unitary connection that connects an input and output of a (general1) cosmologicalquantum computation.

1. General in the sense that it does not necessarily involve a qubit .

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From this, Wheeler concludes that the concepts of space-time and time itself are notprimary, but secondary in the structure of a cosmological physical theory. AlthoughWheeler may be right about the first (space-time), there are some problems with theconclusion about the second (time), when it is stated categorically or in a general senseof the term time . The time , to which Wheeler is referring, is a four-geometry-dependentchronometrizable temporal frame, this time loses meaning within a quantum cosmolog-ical setting, since there is no well-defined geometrical structure upon which one maydefine notions, internal to that structure, such as that of before, after, present, past orfuture, where each event occupies a position in a “grand catalog called space-time”, toborrow Wheeler’s expression.

Nonetheless, as we stated above, we still have to deal with quantum unitarity, and thepath-integral formulation still brings with it a notion of a temporality associated with anotion of a quantum superspace history, a temporality that is distinct from the arithme-tizable, chronometrizable time that may be defined with respect to a four-geometry.

This distinction between the two temporalities, the internal space-time and thequantum evolution temporality, becomes more explicit in later works on quantum cos-mology, and, in particular, in Hartle and Hawking’s work (Hartle and Hawking, 1983).

As stressed by Hartle and Hawking, when we define the amplitude to go from a three-

geometry hij , to another three-geometry hij′

, we must sum over all the internal four-geometries that match hij on an initial surface and hij

′ on a final surface, the two sur-faces being connected by this sum. One can see here that, through the sum, we areobtaining, in a first, and crude, approximation of the quantum cosmological problem,the transition function (Hartle and Hawking, 1983):

hij′ |hij =

δgµν exp(iS E [gµν ]) (1)

where S E is the classical action for gravity, including a cosmological constant Λ.

As Hartle and Hawking (1983) noticed, when addressing the above formula, onecannot specify the time in these transition amplitudes. Which is indeed true, in thesense of a chronometric time defined within the four-geometry. This time, or rather, thisnotion of time is dependent upon the four-geometry, so that it is undetermined. How-ever, we also see the usage, by the authors, of the term propagator to address the abovedefinition (Hartle and Hawking, 1983), which is, by definition, a theoretical notion rid-dled with temporal connotations, that form part of its fundamental semantics.

Although this might seem to introduce a contradiction, or an ambiguated discursiveproblem, it is in fact a correct statement, as long as we understand that the transitionamplitude

hij′hij is, from a mathematical point of view, a relation between an input

state and an output state , which expresses a primitive notion of temporality, neither

committed nor inseparable from a chronometrics based upon a space-time geometry.Thus, we have a temporality that is different, in its nature, from any global, or evenlocal, time direction , definable within the context of a space-time geometry , which is anotion internal to the quantum theory underlying Eq.(1).

Thus, the unitary connection, expressed mathematically by the integral on the right of Eq.(1), can be thought of in light of an input and an output of a (general) quantumcomputation.

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In this sense, we are confronted with another notion of a time which is different fromthe physical, arithmetizable, chronometrizable time, and which is a more primitivenotion associated with a before and after of a computation, where the how much beforeand the how much after are senseless statements since we have no way of assigning num-bers to measure a chronological distance between input and output , because thatchronological distance is part of what is the object of probabilization in the theory,being summed-over in the computation istelf.

Such a realization leads to the need to find, within the mathematical and logical

inquiry about the time, a more primitive structure than the temporal arithmetic thatwas associated to the notion of time in physics. We provide for a contribution for such areflexive inquiry, in the next section, from a mostly mathematical point of view, byaddressing a primitive family of mathematical structures that are close to the notion of mathematical category , and in which primitive non-chronometrizable notions of time areobtained.

3. Primitive non-chronometrizable temporalities andrelational structures

In defining physical and mathematical notions, intended for applied science, we have

been largely influenced by a fundamental role of the notions of quantity and number.From a mathematical point of view, this raises the issue that we may run into mathe-

matically addressable problems where the notion of number and the notion of quantityare not fundamental. In that sense, even though a mathematically inclined science of something that is not measurable can be developed, that science cannot be considered tobe x-metrics (where x stands for whatever area of application one is addressing).

It is a known mathematical fact, that the notion of order precedes the notion of order-able quantity in terms of mathematical generality. Indeed, although an orderable quan-tifiable set can be numerically labeled with respect to the quantifiable property thatdefines it, and, thus, ordered in terms of the numeric values assumed by the members of the set with respect to that underlying property, not all orderable sets can be numeri-cally labeled in terms of any significant orderable quantity property , that is, one cannumerically label a non-quantifiable set , but those numbers are, in general, arbitraryassignments. We already saw an example of this, when we addressed Einstein’s views onthe subjective time. In statistics this is especially pertinent, since a qualitative ordinalvariable can be numerically labeled in an arbitrary way, so long as the order of theobjects in the ordinal scale is reflected in the order of the numeric assignments. Withouta theoretically justified interval distance, any numeric interval between levels in theordinal scale is valid.

With the development of mathematics and metamathematics we came to realize thatsome mathematical notions and structures are very general, primitive and, thus, funda-mental. As a consequence of these findings, we find ourselves before a major theoreticalchange, coming from the displacement of the focus from the notions of quantity andnumber to the notions of relation and of morphism (the last in the context of category theory ).

A most primitive formal structure, within mathematics, is the relational structure ,which is nothing more than a collection of objects along with a collection of relations.We can define the most simple case of binary relational structure , in a structured form,as follows:

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Definition 1. A binary relational structure R = (O , R) is composed of a collection of objects O, whose members are called the objects of R, and a collection of binary rela-tions R satisfying the following four conditions:

1. For every related pair of objects X , Y ∈ O there is a non-empty set of relations R′ ⊆ R, with cardinality at least one, such that, for every R ∈ R′, the well formed

formula R(X , Y ), which reads X is related to Y, under R, holds.

2. For each relation R in R, there is at least one pair of objects X,Y in O, such that, it holds that R(X , Y ).

3. For any R ∈ R, its truth set 2 is composed only of pairs of objects in O.

4. There is a relation I in R such that, for every object X in O, the well formed for-mula I (X , X ) holds, and I (X , Y ) if, and only if, X and Y are the same object in O.

The extension to n-ary relations follows immediately.

It is important to stress that we are defining relations in terms of their intension 3

rather than in terms of their extension . Usually, in the mathematical theory of binaryrelations one works with the extension , namely, with the set of objects that satisfy therelation, an approach which abstracts away the meaning and definition of the relation ,

i.e., its intension , to focus solely upon the truth-set representation , where this set isdefined in its extension . This process leads to a level of analysis where the relation isreduced to a set theoretical structure and, in particular, to the composition of the set,which means that two relations become identical, in accordance with the notion of setidentity, if their truth sets are identical , even if, they are indeed different with respect totheir intension .

In the above structures this is not so. We are working with the notion of logical inten-sion with respect to the relations. Which means that, even if two relations have thesame truth-set extension , they are still considered to be different, if they have differentintensions . For instance, if a married couple, Fred and Wilma, owns a restaurant, thenthe set theoretical expression would be {(Fred, Wilma), (Wilma, Fred)} for both thebusiness partner relation and for the relation of marriage. However, in terms of theabove relational structures the relations “being married to ” and “being a business partner of ” would constitute two different relations with respect to the universe of discoursecomposed of {Fred, Wilma}, even if each of these two relations has the (extensionally same ) truth set , given by: {(Fred, Wilma), (Wilma, Fred)}.

This, and the fact that the objects that we work with are not necessarily sets, meansthat one should not confuse the relational structures as mathematical objects with thecategory Rel, which is the category of relations that has sets for the class of objects and, as the morphisms , binary relations, defined extensionally as subsets of orderedpairs. Indeed, we show in the appendix to the present work, that the mathematical cate-gory is a particular family of binary relational structures . We also address, in theappendix, the category Rel and the category Mag to make clearer the difference

between the binary relational structures defined above and the category Rel. Further-more, we address the role played by Rel in the nature and properties of the truth sets of the above binary relational structures .

2. We call the model of the relation, that is, the set of pairs of objects for which the relation holds, the truth

set. The truth set establishes a connection between the binary relational structures and the category of the binary

relations over the class of sets, even though the binary relational structures are more general than the notion of

mathematical category. In the appendix we address these two issues.

3. This is a logical notion not to be confused with the notion of intention with a t .

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Besides this point, that should be taken into account when working with the abovedefinition, it is important to understand each of the four conditions.

Since each of the four conditions that define the binary relational structure can beextended to n-ary relational structures , we discuss the role and meaning of each condi-tion in a general sense. The first condition tells us that all the relations between anytwo objects (or n -tuples of objects for the general case) are in R, which means that R isan exhaustive set of binary (n-ary ) relations with respect to the underlying collection of objects O. The second condition limits each relation to be exemplified by at least a pair

of objects in O, or, in a more general sense (for n-ary relations ), the truth set of eachmember of R must be different from the empty set.

The third condition restricts the relations in R to be solely relations between objectsof O, which means that R is exclusive with respect to the underlying collection of objects. Finally, the fourth condition introduces a reflexive relation that is only satisfiedfor an object and itself. This relation of an object to itself is called an identity . Theidentity is always stated in terms of a binary relation, and it is assumed to be includedin every definition of n-ary relational structures .

Indeed, we have to assume the relation of identity to be able to work with the notionof object, because otherwise the object would lose its integrity (Madeira, 2008b). Anystructure of individuated objects must be such that each object is maintained and sus-

tained in its identity, while it remains an object of the relational structure (Madeira,2008b).

Now, as it is defined, a binary relational structure does not include, in its definition,an explicit temporality of any sense, other than the eternal return of an object to itself via the permanent coincidence of the object with itself, in the identity relation I .

A primitive sense of temporality associated with a before and an after , primitive inthe sense of not being generally measurable in terms of any kind of clock-time, arisesfrom an analysis of the notion of relational nexus .

This notion can be built in stages. First, we consider, for any object X ∈ O, thesubset of objects with which X is in relation, and write O(X ). We know that this set is

non-empty, since each object is at least in relation with itself.

We can first define an identity nexus of an object N (X &X ) as the singleton {I }.Now, for any Y ∈ O(X ), distinct from X , define the relational nexus of X and Y to bethe subset of R of the relations that X and Y exemplify, and denote this nexus by

N (X &Y ). Given this relational nexus one defines the partition between: (a) those rela-tions, if there are any, in N (X &Y ) whose truth sets contain (X ,Y ) as element but not(Y , X ); (b) those relations, if there are any, whose truth sets contain ( Y , X ) as ele-ment, but not (X ,Y ); (c) those relations, if there are any, whose truth sets contain both(X ,Y ) and (Y , X ) as elements.

The first element of the partition, that corresponds to the case (a), as defined above,

is the relational nexus denoted by N (X Y ), the second case (case (b)), is the rela-tional nexus denoted by N (X Y ),4 and the third case (case (c)) is the relational nexus denoted by N (X ↔ Y ).

The union of these three cases recovers the full relational nexus of X and Y , that is:

N (X Y ) ∪ N (X Y ) ∪ N (X ↔ Y ) = N (X &Y ) (2)

4. We stress that N (XY )= N (Y X).

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Once we partition the relational nexus in terms of the above sequence of disjunctions,we obtain a directional sense for the relations, this directional sense introduces the fun-damental notion of a before and an after in a relation. Thus, for instance, in the rela-tional nexus N (X Y ), Y comes after X , while in the relational nexus N (X Y ) X

comes after Y , finally, in the relational nexus N (X ↔ Y ) there is a bidirectionality, inthe sense that if we begin in X , then, we see Y coming after X , while that, if we beginin Y , then, we see X coming after Y .

It is noticeable that we can break up each relation in N (X ↔ Y ) into two unidirec-

tional sub-relations, one leading from X to Y and another from Y to X .Considering the general partition of Eq.(2), we see the emergence of a notion of before

and a notion of after , as a consequence of the configuration of the relational nexus of two objects. These notions appear in these structures as primitive ordinal notions, theirmathematical nature is not fundamentally restricted by the imposition of an underlyingchronometrics.

In the above definition, one does not impose any underlying fundamental numericproperty, or general family of properties that would introduce, foundationally, a tem-poral metric, allowing the measurement of the separation between the two terms in therelation.

In order for a temporal metrics to emerge, one would have to assume an axiomatics

that restricted the formalism to a subset of relational structures whose objects mightexemplify, fundamentally and foundationally a numeric property that could form thebasis for a constructible chronometrizable relational nexus . But this would be a subset of the above relational structures , a subset for which the relational nexus is obtained fromthe exemplification of an underlying numeric property.

Another restriction could be obtained through an appropriate axiomatic system. It isknown, for instance, from expected utility theory, that an ordinal system can be trans-formed into a numerical scale if that ordinal system satisfies certain properties withrespect to combination with probabilities. We shall return to this issue at the beginningof section 4., when we present a final reflection upon the main results of the presentwork and address, in that reflection, some of Barbour’s claims and the nature of a phys-

ical clock time in relation to the nature of the temporality of the relational structures ,introduced in this section.

Now, so far, we have the emergence of a notion of before and and a notion of after ,not necessarily chronometrizable with respect to some underlying physical or mathemat-ically significant quantifiable property, since we cannot, in general, assign, in a relational structure , a chronometric sense of how separated two objects are in a relation. Nonethe-less, we see a before and an after and the emergence of a general temporal sense,expressible solely in terms what takes place before and what takes place after , in theorder of the relation, without the ability to state how much before or how much after, oreven how late. This makes the temporality of the above partition more general, but,even so, present in the formalization.

A further inspection of the above partition, however, shows the presence of a morestructured sense of temporality, other than the simple before and after . Indeed, in twocases the before and after are not interchangeable, while in the last case they are inter-changeable. This makes evident that the roles of what comes before and what comesafter are frozen in the first two elements of the partition, and we have an arrow of timeand an irreversibility. Indeed, the full relational nexus partitions in two temporally asymmetric relational nexus ( N (X Y ) and N (X Y )), and a temporally symmetric one ( N (X ↔ Y )).

12



These results show how a temporal logic may be present without reference to anychronometrics. The problem of an undefined metric for time, therefore, does not excludethe existence of a temporality, or even an ability to address such a temporality within aformal, mathematical theoretical setting.

Besides providing for an intuition and a formalism that is able to address the path-integral and propagator extra-clock-time temporalities of quantum gravity, these mathe-matical structures bring to light a problem in how one conceptually addresses configura-tion spaces and the relational nature of configuration spaces . The natural emergence of

a temporality, not necessarily defined in terms of a clock temporality, but present in thepartition of a relational nexus , contrasting with what appeared to be a timeless struc-ture of relations, is sufficient to raise debatable issues with respect to the time-indepen-dent configuration space argument for the elimination of time as a concept, when it istaken in its full generality, and not in terms of the chronometrizable clock-time frame,internally definable within a pre-given space-time geometry.

Although this line of argument can be seen as naturally arising within quantum cos-mology, it has been argued with respect to classical cosmology as well, and, even, tophysics in general (Kauffman and Smolin, 1997; Smolin, 2001; Barbour, 1994).

Indeed, the argument applies to a general system, placing, as a fundamental concept,

the configuration space , which can be defined as the space of possible configurations of that system (Barbour, 1994). Three elements are essencial to this definition – the notionof space ; the notion of configuration and the category of possibility .

The notion of configuration , in quantum mechanics, is a familiar one, in a more gen-eral sense, however, which is the one assumed in the general argument, the term configu-ration must be taken with respect to its etymological genetics in the Latin termcum + figurare , which means to shape together , where the action of forming or of makingshape – figurare – has the corresponding noun figura which means shape, form, or pat-tern.

A configuration is not only a shape, form or pattern, it possesses an active sense of patternization or making pattern , this active sense is committed, in its conceptual and

etymological genetics, to a temporalization , present in the process by which the systemtakes on a shape, or form, or produces a certain pattern. The denial of the existence of such a semantic connection is an error, proceeding from uprooting the terms from theirunderlying etymological structure and semantics. The fact that these terms have a dis-cursive and conceptual pre-existence, rooted in both a linguistic and philosophical tradi-tion that has addressed them, is a first problem that introduces one element of illegiti-macy and error in the line of argument.

This is not the only problem with the configuration space argument for the non-exis-tence of time, but it is a sufficiently important one, since the proponents of this line of argument argue against the existence of time in a general sense, that is, any kind of temporality is considered not to be fundamental. This being so, then, one should be

careful in the choice of the key terminology, if that terminology is somehow committedto a notion of temporality, and if that terminology plays a constructive role in thetheory, then, the statement of no-temporality becomes inconsistent with respect to thetheory itself.

The second important element to the notion of configuration space is the category of possibility , that, in this case, introduces a physical openness of the system to differentalternative configurations which it can assume. Along with the statement of possibility,

13



comes the final element which introduces a geometrization of the different possible con-figurations in the form of a space of possible configurations , where each point is, as Bar-bour (1994) put it, a distinct structured whole , this means that we can indeed considereach configuration as an object of a relational system.

Then, we have the collection of objects C, which is the configuration space, as our fun-damental building block. The proponents of the configuration space argument, and, inparticular, Barbour (1994), assume that physics must be built in terms of such a collec-tion of objects and relations between the elements of such a collection, without reference

to time.And, indeed, one can do so. In succint terms if one does not consider the temporal

physical clock chronometrized labels, associated to a curve in a configuration space, oneloses any sense of a clock time and one has a simple spatial curve without any referenceto time. From this point to the next step that states that there is no temporality, mayseem as a legitimate direction. However, it is not. Indeed, one loses any track of anykind of clock time and one has a curve in a space, without any reference to a physicaltime, which makes the temporal labelling a secondary aspect of the world. However, thefact that the curve is a relational object imposes a temporality of its own, which may bechronometrizable with respect to physical criteria but not chronometric with respect toits fundamental temporal nature.

Indeed, we may work with the relational structure (C , R), where the relational nexusbetween two objects x, y ∈ C can be, naturally defined as:

N (x↔ y) (3)

where each relation in N (x ↔ y) is a curve in configuration space connecting the twopoints/configurations. It is noticeable that this nexus introduces a reversible connection,and a reversible computation, that can be broken down in two temporal directions:

N (x y) ∪ N (x y) (4)

The first nexus expresses the result of “travelling” along each curve in C from x to y, theconfiguration x appearing before (input ) the configuration y (output ). The secondnexus expresses the result of the inverse computation where y appears before (input )and x after (output ).

The passage of time along each curve, defined by some temporal parametrization, isirrelevant with respect to the fundamental temporality expressed in the fact that thenon-oriented curves connecting two objects can be considered conceptually akin to areversible computation, and broken down into two oriented curves, each one imple-menting a computation that is the inverse computation of the other.

Thus, (all) time is not removed from a configuration space definition. Even if one

cannot label a curve with respect to a pre-given time frame, one still finds the more fun-damental temporality expressed by the notions of input and output of a computation.

In general terms, one can understand the consequence of these results for a spatialgeometry. Any spatial relation, or figure contains the temporality of its spatiality whichis the spatiality necessary for the figure to be configured.

If, in general, one cannot speak of a time independence with respect to a curve in con-figuration space, what can one state or know about a static quantum state?

14



In principle, a time-independent, essencially static quantum state is timeless. That isan important argument as one moves from the general configuration space argument toquantum theory. In this case, the argument is linked to the Wheeler-DeWitt equation,that effectively introduces a time-independent wave functional for the geometry plusfield configurations, with compact spatial sections (D’Eath, 1996).

Smolin (2001) addressed the general wave functional of the universe in terms of a setof assumptions that support a general theory of quantum cosmology, under the Wheeler-DeWitt framework. We can express these main assumptions as follows:

• The configuration space for the universe, associated with some relevant cosmolog-ical variable that is expressible as a physically observable quantity, is knowablefor technologized theorizing observers inside the universe, that is, there is aCUniverse that is knowable, from a theoretical and empirical point of view.

• The wave functional of the universe exists and is defined to be a normalizablecomplex functional ΨUniverse that is normalizable upon an expansion in terms of CUniverse (that is

CUniverse

dµ|ΨUniverse|2 = 1) and the normalizable states define a

space with a Hilbert space structure, with inner product given by: Φ|Ψ = CUniverse

dµΦ∗Ψ.

• The wave functional of the universe satisfies the momentum constraints alongwith the Wheeler-DeWitt equation.

Under this framework, we are effectively working with a time-independent wave func-tional, and time seems to disappear from the theory.

Smolin (2001) reviewed several arguments against this approach to quantum cos-mology and against the argument for timelessness, on several grounds, including:

• The limits in the ability to build a formal mathematical theory that incorporatesthe observables for quantum cosmology using the above approach, linked with theproblem that the Hamiltonian constraint observables are extremely difficult toconstruct in real field theories of gravitation, made more severe by the need toconsider the conjecture regarding the presence of chaotic behavior in gravitationalsystems;

• The possible unobservability and unkowability of the configuration space of theuniverse, including the problem of the computational complexity for constructinga mathematical representation of complex configuration spaces.

It is about the limits to our ability to theorize that Smolin’s objections, against theargument for the absence of time, ultimately rest.

Smolin’s arguments emphasize the role played by the requirement that a theory of cosmology must be falsifiable in the usual way that ordinary classical and quantum theo-ries are. This, in turn, leads to the requirement that a sufficient number of observablescan be determined by information that reaches an observer inside the universe, allowingthat observer to know the quantum state of the universe. Only if this is the case can webuild a quantum theory of cosmology, based upon the above set of assumptions, suchthat this theory obeys the standard methodological and epistemological scientificrequirements that any scientific theory must obey.

It is around our ability to build a theory of quantum cosmology that these argumentsorbit, however, the arguments presented above do not, by themselves, refute the argu-ment for the elimination of time, as Smolin (2001) admits it.

15



The problem of time is addressed by Smolin as a human theorization problem, thetemporality being the result of the human inability to access a timeless quantum state of the universe.

However, before excluding any temporality with respect to a static clock-time inde-pendent quantum state, we should begin by looking at the mathematical structure of aHilbert space, as a geometric space, raising the problem of how one should interpret aclock-time independent quantum state.

If the universe can be thought of as being a quantum system with a clock-time inde-pendent quantum state (corresponding mathematically to a clock-time independentwave functional), then, even if we are unable to address this state, from a theoreticalpoint of view, we must consider the nature of static clock-time independent quantumstates, and their relation with temporality before arguing about the problem of time.

At this point, one should look in more deeply at the relational structures of the basesof the Hilbert spaces, as vector spaces, and at the consequences of these relational struc-tures for the mathematical nature of the space of normalized kets .

Indeed, a Hilbert space can be considered in terms of the above relational structures .To understand how this is so, let us consider a single Hilbert space H and the set of alternative bases for this space B, which, for simplicity’s sake, we take to be such that

each basis in B is discrete.Now, we can build an example of a relational structure that, although being indepen-

dent of any physical clock time, it still produces temporalities that come from temporally symmetric relational nexus .

Let us, then, consider the relational structure (B , U ), where U is the set of unitarytransformations associated with the change of basis.

Then, we can see that, given any two bases:

B1 = {|φn}

B2 = {|ψn}

we can expand each element of the second basis in terms of the elements of the firstbasis as:

|ψn =m,n

U mn |φm (5)

The expansion coefficients U mn can, then, be expressed as:

φm

ψn

= U mn (6)

which can be interpreted as the amplitudes of the transitions of the initial ket (input )|ψn to the final ket (output ) |φm. Alternatively, we can interpret each |ψn as thetransformed state of an initial state |φn under an appropriate unitary transformation.Indeed, the coefficients U mn form the entries of a unitary matrix that is the matrix rep-resentation of a unitary operator U ̂(2, 1) ∈ U , such that:

U ̂(2, 1)|φn =m,n

U mn |φm = |ψn (7)

16



thus:

φm|U ̂(2, 1)|φn = φm|ψn = U mn (8)

Since the unitary transformation is reversible we can see that, while U ̂(2, 1) establishes a

connection from the first to the second basis, its conjugate transpose U ̂(1, 2) = U ̂(2, 1)

establishes the inverse connection. We, therefore, have the relational nexus :

N (B

1 ↔B

2) = {U ˆ

(2, 1), U ˆ

(1, 2)} (9)

which is typical of the reversible logic implemented by a unitary transformation.

One may notice that although no effective temporal physical clock frame was defined,and although we are dealing with chronologically spatial relations, there is a temporalityexpressed in the transition from one basis to another (Eq.(6)), and in the unitary trans-formation of each basis element of one basis in the basis element of another basis(Eq.(7)). These relations possess a temporality , independently of the definition of anactual physical clock, internally defined within some space-time metrics.

This temporal sense “leaks out” to the quantum formalism. Indeed, given the expan-sion of a quantum state in either of the two basis:

|Ψ =n

cn|ψn (10)

|Ψ =m

cm′ |φm (11)

we have that:

(12)

Ψ(φm) = φm|Ψ = cm′ =

n

φm|ψncn =n

φm|U ̂(2, 1)|φncn (13)

Ψ(ψn) = ψn|Ψ = cn =m

ψn|φmcm′ =m

ψn|U ̂(1, 2)|ψmcm′ (14)

Therefore, in Eq.(13) we can see that the amplitudes Ψ(φm) can be considered as asuperposition of quantum computational histories with quantum logical gate U ̂(2, 1) ini-tial kets ranging over the basis B1 and final ket |φm. Each computational history isweighted by the amplitude cn with the index n ranging over the basis B1, expressing anuncertainty associated with the input state .

Therefore, we can see that the static expansion expressed by Eq.(11) can be put into aprocessual form, where the amplitudes Ψ(φm) correspond to the probability amplitudesof the output of the computation being |φm in a quantum computation with general

propagator expressed by φm|U ̂(2, 1)|φn, and where the cn correspond to the ampli-tudes associated with the input states . A similar reasoning can be applied to the expan-sion of Eq.(10) and to Eq.(14).

We can also introduce the projector chains:

P mn = |φmφm||ψnψn| (15)

P nm′ = |ψnψn||φmφm| (16)

17



and write:

Ψ(φm) =n

φm|P mn |Ψ (17)

Ψ(ψn) =m

ψn|P nm′ |Ψ (18)

These projector chains can be interpreted in light of the projector chains of the consis-tent histories’ formalism (Omnès, 1988, 1992; Griffiths, 1993, 1994; Gell-Mann andHartle 1993, 1994, 1996, 1998; Hartle, 2007), the P mn representing a history of aquantum system, since we have:

P mn = U ̂(1, 2)|ψmψm|U ̂(1, 2)|ψnψn| = U ̂(2, 1)|ψmψm|U ̂(2, 1)|ψnψn| (19)

P nm′ = U ̂(2, 1)|φnφn|U ̂(2, 1)|φmφm| = U ̂(1, 2)|φnφn|U ̂(1, 2)|φmφm| (20)

which completes the connection to the projector chains of the consistent histories’s for-mulation, since Eqs.(19, 20) can be considered to be analogous to projector chains forfine-grained histories of a quantum computational network, with the correspondingquantum logical gates U ̂(2, 1) and U ̂(1, 2), respectively5 (Hartle, 2007).

This deepens the result about the temporality expressed by the relation between thetwo bases, that can be considered as a quantum computation. The unitary transforma-tion that implements this computation is such that the elements one of the basis takethe role of inputs and the elements of the other basis take the role of outputs .

In this way, a single time independent state of superposition, can be put into a tem-poral expression in terms of a set of quantum computational histories, fine-grained withrespect to the quantum logical gate, which is the unitary transformation that imple-ments the change of basis.

This makes the result more compelling, as it shows how a temporal sense, present inthe relational nexus , that is different from a clock time, may turn a representation of aclock-time independent quantum state, that represents a single static object, into a pro-cessual representation, in terms of a quantum computational history, with a formalismanalogous to the one used in the consistent histories’ approach to quantum theory.

Thus, a problem is raised in regards to the statement made with a character of gener-ality with respect to the timelessness of the quantum state of a system, described by astationary quantum state. Such a state indeed should satisfy a clock-time independentSchrödinger-like equation, and, thus, be effectively timeless with respect to a physicalclock time, however, the mathematical processual nature of such a state, uncovered bythe relational structure of the observables’ eigenbases, precludes the generalization of thestatement to any kind of temporality.

Ultimately, results such as those of Eqs.(13, 14, 19 and 20), raise the more generalissue of the terminology used to address our physical theories of the universe, and, inparticular, the legitimacy of using the concept of state , rather than the term process , torefer to a ket , which resends to a snapshot-like staticness influenced by a classicalphysics’ tradition (Baugh et al., 2003).

5. Looking at (13) and (19) and at (14) and (20) we see that the second projector in each chain is analogous

to a projector in the Heisenberg picture.

18



4. Final reflections and open issues

Considering the general question placed by Barbour (1994): is time a basic concept? Theresult of the analysis of arguments and of theoretical discourse developed in the previoustwo sections, as well as the mathematical results obtained in the previous section, leadsto an answer to the above question, with another question: what time?

Indeed, the result of the work developed above shows that temporality and the notionof time go beyond the more restrictive chronometrizable notion of time, that is inter-

nally definable with respect to a space-time geometry.

The arguments of Barbour (1994), of Kauffman and Smolin (1997), and Smolin(2001), along with the Wheeler-DeWitt equation show how a chronometrizable physicalclock time may indeed not be a basic concept .

However, whenever we consider a quantum computation, when we address a time-independent ket , or the relations between different physical observables’ eigenbasis, wefind a basic temporality that is definable with respect to the configuration of the rela-tional nexus of a system of relations between objects .

In physics, as we saw in the previous section, if we accept the fundamental role of aconfiguration space , we find this temporality present, even in the absence of any kind of

fundamental clock time.

Therefore, we are led to a conceptual need of defining a relation time , as a funda-mental (ordinal) time, which is the time of the order of the objects’ positions in the rela-tion, and that ultimately proceeds from the connection of two individuations that areseparated, but linked by the relation, and, thus, are temporally connected in the tempo-rality that is the order of terms in the relation.

For relational structures such that, given any two objects X and Y , N (X &Y ) iseither N (X Y ), N (X Y ) or N (X ↔ Y ), it is possible to obtain a numeric scale forthe objects that reflects the relation time , by introducing the structure of prevalences inrelational nexus (O , N ), defined as:

X N Y if , andonlyif , N (X &Y ) = N (X Y ) ∨ N (X &Y ) = N (X ↔ Y )

X ∼Y if , andonlyif , N (X &Y ) = N (X ↔ Y )

for any X , Y ∈ O.

This order expresses the relation time that results from the peculiar order of terms inthe relational nexus N (X &Y ). Now, if we were to combine the objects with probabili-ties, and introduce von Neumann and Morgenstern’s (1953, [1990]) axiomatic forexpected utility, a chronometric could, then, be assigned to the above structures,reflecting the order N , which would result in an axiomatic for expected time . Of course, this is a special case, but it serves to show how a chronometric time may emerge

from a purely relational temporal background .

In this sense, one may be inclined to aggree with the position that a chronometric time may not be fundamental, and may emerge from a more fundamental temporalstructure that is purely ordinal. The above result is, at least, sufficient to show that anordinal temporality, that is a time that emerges within a relational structure , can bemore fundamental from a physical point of view, playing a foundational role in an emer-gence of a space-time chronometrizable physical time.

19



In this work we have solely dealt with reversible unitary transformations, which leadsto relational nexus of the kind of N (X ↔ Y ). A nexus of this kind is reversible , in thesense that we can take the temporal connection in either direction. A second kind of temporality underlies quantum mechanics, and it is specific to that theory’s logical andmathematical expression.

To understand this we may take the example of the qubit :

|ψ = ψ(0)|0 + ψ(1)|1 (21)

where the two weights ψ(i), i = 0, 1 are time independent complex numbers satisfyingthe normalization condition |ψ(0)|2 + |ψ(1)|2 = 1. In this “static” expansion, we find thatthe weights assign amplitudes to the transitions:

0|ψ = ψ(0) (22)

1|ψ = ψ(1) (23)

which, under Heisenberg’s interpretation of quantum mechanics, determine the proba-

bility of actualization of the input |ψ to an output of |0 or |1, mathematicallyexpressible through a stochastic selection of a projection.

The temporality inherent in this is a clear one, even though the category of causalitydoes not apply to the context of the notions of dynamis (potentia ) and energeia (act ),we know, from these notions, and from the above mathematical expressions of thesenotions that the actualized quantum state is preceded by a corresponding potentialreality, upon which it is founded and grounded.

Since the nature of the dynamis , or potentia , is to tend towards the act that deter-mines it, there is a temporal sense, associated with the “static” expansion of Eq.(21), andidentifiable in the physical interpretation of that equation, within Heisenberg’s interpre-

tation of quantum mechanics, as a mathematical expression of the Aristotelic notions of dynamis or potentia and energeia or act .

Understanding the process of actualization, if one accepts Heisenberg’s interpretationof quantum mechanics, leads one to a discussion about decoherence, which can andshould be extended not only to quantum cosmology6 but, also, to the mathematicalstructures introduced in section 3..

Another open problem, that may be raised, regards the nature of the relational struc-tures when, they, themselves are subject to a quantum description. Several differentissues arise in this case. For instance, a quantum causal history is such that, if we labelthe edges by quantum states and the nodes by unitary operators, we have that the

incoming and outgoing quantum states to a single node are in a relation, this relation islocally reversible (due to the unitarity), that is, we have the general relational nexus associated with the system of node + edges:

N (|ψ ↔ |ψ ′) =

U ̂, U ̂

(24)

6. See, for instance, Kiefer (2003) for an example of such a discussion.

20



However, in the directionality imposed upon the network, only one of the directions isusually chosen, that is, the local reversible nexus is partioned in terms of:

N (|ψ |ψ ′) ∪ N (|ψ |ψ ′) =

U ̂

∪

U ̂

(25)

and one of the directions N (|ψ |ψ ′) or N (|ψ |ψ ′) is assigned to the network bythe labelling of the node and the choice of the directionality given to the edges7.

Now, this assumes a fixed causal structure, already frozen by a pre-selection of one of the nexus of the partition. We may, however, consider this selection not to be a givenand assume a quantum superposition of the partitionned nexus , which would lead to thelocal network state:

|ΓLoc = Ψ1| N (|ψ |ψ ′) + Ψ2| N (|ψ |ψ ′) (26)

which incorporates a quantum extension of the theory of binary relational structures ,introduced in the previous section.

This leads to a non-fixed causal structures in the quantum causal histories approach

to quantum cosmology. An issue already raised by Hardy (2007), regarding the quantumgravity computer.

The matter of how one of the directions gets to be selected, faces us with the problemof decoherence with respect to the above local state, and the problem of decoherence inquantum binary relational structures .

In some paths of research, uncertainty regarding the choice of nexus may lead to theresearch problem of decoherence and recoherence. In other paths, we may address acomputation that takes place with respect to the whole network which transcends thelocal temporal connection, determining it.

Considering this last case, a second order kind of temporal uncertainty is produced in

Eq.(19), as what comes before and after in a computation is not fixed, since the compu-tation itself is not yet selected. An environmental decoherence mechanism would pro-duce a diagonal local network state, but this mechanism would be such that it wouldcompute the entire temporal connection, which means that it would introduce a secondorder temporality, that takes the relation itself as an ob ject of another relation.

As an example, let us consider the above local network state and define N 1 ≡ N (|ψ |ψ ′) and N 2 ≡ N (|ψ |ψ ′).

Then, we have a local basis {| N 1, | N 2}. Adding a local environment, partitioned inat least two parts8, we can introduce the computation:

|ΓLoc, A0, E 0Ψ1| N 1, A1, E 1 + | N 2, A2, E 2 (27)

implemented by the entanglement operator U ̂Ent.

7. We need the two pieces of information (labelling and directionality given to the edges) since in some cases

we may have U ̂=U ̂.

8. This is necessary in order for a unambiguous flow of information from the system to the environment to

take place, as shown and addressed by Paz and Zurek (2002).

21



Thus, considering system + environment we obtain the relational nexus :

|ΓLoc, A0, E 0 ↔ Ψ1|N 1, A1, E 1 + |N 2, A2, E 2

=

U ̂Ent, U ̂Ent

(28)

Properly considered, this is a reversible process, where U ̂Ent produces entanglement andU ̂Ent transforms the entangled state9 into an unentangled state10. However, for a large

enough environment, as discussed by Zeh (2002), the entanglement can be considered to

be almost irreversible, which means that, with respect to the local network we wouldhave the transition given by the following nexus , expressed in terms of density opera-tors:

N

|ΓLoc

ΓLoc| |Ψ1|2|N 1, A1

N 1, A1| + |Ψ2|2|N 2, A2

N 2, A2|

(29)

with the environment described by the kets |Ai playing the role analogous to a recordkeeping physical apparatus.

It is noticeable that we are dealing with a quantum computation that computes anentire local structure of quantum computations, and that the final relational nexus is thenexus of the quantum states for the relational nexus of a local quantum computation.

Indeed, the result of Eq.(27) corresponds to the case where the computation expressedby the nexus N (|ψ |ψ ′

occurs, with probability |Ψ1|2, or the reverse computation,

expressed by the nexus N (|ψ |ψ ′) occurs, with probability |Ψ2|2.

The generalization of this type of research to quantum causal histories, and to theirapplications in quantum gravity research, may be important to understand the natureand role of time in the theory of quantum space-time, and, in particular, in the researchprogram of loop quantum gravity.

9. Producing local decoherence.

10. Producing a recoherence.

22



Appendix

We use, throughout this appendix, the notation and notions addressed in the main text,including the notions of logical intension and relational nexus .

Stated in a general sense, a relational structure is composed of a collection of objects and a collection of n-ary relations defined in terms of their intensions .

A category is a particular structure, within a binary relational structure (O , R), where

the set of relations is restricted to morphisms between objects, satisfying a number of conditions.

The first condition is that each morphism f in R relates a pair of objects in a direc-tional way, that is, f ∈ N (X Y ), where X is known as the domain of f , or source object and Y as the codomain , or target object , and one writes:

f : X

Y

The relational nexus N (X Y ), thus, corresponds to the class of all morphisms from X to Y . By definition 1., in the main text, we know that every morphism between anytwo objects must be in R. Also, we have that there has to be an identity morphism for

each object, corresponding to the identity relation , evaluated for each object:

id: X

X

The second condition is that R be closed under composition of morphisms, defined suchthat, for three objects X, Y , Z ∈ O, and two morphisms f and g , f ∈ N (X Y ) andg ∈ N (Y Z ), then, we have that the composition g ◦ f is such that:

g ◦ f ∈ N (X Z )

and:

g ◦ f = g : (f : X

Y )

Z

That is, the morphism g ◦ f from X to Z is first obtained by applying the morphism f

from X to Y , and, then, the morphism g from Y to Z .

The third condition imposes that composition of morphims is associative and thefourth that the composition is commutative with respect to the identity . A structuresatisfying these conditions is called a category .

It follows from these results that a mathematical category is a binary relational sub-structure , in the sense that the collection of relations is restricted to the collection of morphisms .

A binary relational structure is more general than a category , and contains in it struc-tures that are more general than categories . Indeed, we could simply consider a binary relational substructure composed of a collection of objects and a collection of mor-phisms , with the identity morphism included, these being the only definable restrictions.Any mathematical category is such a structure, but the fact that nothing is stated aboutcomposition means that not every collection of objects and morphisms , with the identity morphism included, constitutes a category , nonetheless, it constitutes a binary relational structure .

23



Examples of binary relational substructures that are categories include the category of binary relations Rel, which has sets by objects and, by morphisms , the binary relations ,that are defined as subsets of the set of ordered pairs A × B, for any two sets.

It is important to notice that the structure composed of the truth sets of a binary relational structure , along with the collection of objects of that structure, is, trivially, amember of Rel, since each truth set is a set of ordered pairs, that can be expressed asthe Cartesian product of two subsets of the collection of objects.

The structureMag

is another example of a binary relational substructure that is acategory , having, magmas by objects , that is, algebraic structures composed of sets witha binary operation, and morphisms given by homomorphisms of operations. It is useful,for illustration purposes, to address this structure.

First, we notice that a magma is an algebraic structure that consists of a set A

equipped with a single binary operation:

ιA: A × A

A

where we write, for any a, b ∈ A:

ιA

: (a, b) ∈ A × A

aιA

b ∈ A

Thus, we have the magma (A, ιA).

Now, it becomes important to consider the following four definitions, from universalalgebra (Burris and Sankappanavar, 1981):

Definition 2. For a nonempty set A, and a nonnegative integer n, we define A0 =∅

,

and, for n > 0, An is the set of n-tuples of elements from A. An n-ary operation (or function) on A is any function F from An to A; n is the arity (or rank) of F.

Definition 3. A finitary operation is an n-ary operation, for some n. The image of the n-tuple (a1,

, an), under an n-ary operation F, is denoted by F (a1,

, an).

Definition 4. An operation F on A is called a nullary operation (or constant) if its arity is zero, being completely determined by the image F (∅) in A of the only element ∅in A0, and, as such, one can identify it with the element F (∅). Thus, a nullary opera-tion is thought of as an element of A.

Taking into account these definitions we can define a language or type of algebras as(Burris and Sankappanavar, 1981):

Definition 5. A language (or type) of algebras is a set F of function symbols such that a nonnegative integer n is assigned to each member F of F. This integer is called the arity (or rank) of F, and F is said to be an n-ary function symbol. The subset of n-ary

function symbols in F is denoted by Fn.

From this last definition, the general definition of an algebra , within universal algebrais given by (Burris and Sankappanavar, 1981):

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Definition 6. If F is a language of algebras, then, an algebra a of type F is a structure (A, F ), where A is a nonempty set and F is a family of finitary operations on A indexed by the language F, such that, corresponding to each n-ary function symbol F in F, there is an n-ary operation F A on A. The set A is called the universe (or underlying set) of the algebra a= (A, F ), and the F A’s are called the fundamental operations of a.

If F is finite, such that F = {F 1,

, F n}, we write (A, F 1,

, F n) for the algebra a =(A, F ), with the convention that arity F 1 ≥ arity F 2 ≥

≥ arity F n.

It is straightforward to see that a magma (A, ιA), with A nonempty, is an algebra inthe above sense, that is, it is an algebra with a family of binary operations indexed by asingleton language F= {ι}.

The binary relational structure of magmas MAG = (M, RM) has by collection of objects the magmas .

Let us, then, consider the subset of the set of relations between magmas in MAGcomprised of the set of homomorphisms , defined as follows:

Definition 7. Given two magmas (A, ιA) and (B , ιB) the homomorphism from (A, ιA)

to (B , ιB) is defined as the mapping f : A B satisfying:

f (aιAb) = f (a)ιBf (b)

Thus, a homomorphism between two magmas is a mapping that preserves the binaryoperation. It is noticeable that, under this restriction, one is no longer working with thewhole relational structure MAG, since the set of relations between magmas is restrictedto operation preserving mappings. The structure within MAG, with which we areworking, contains the same collection of objects as MAG and works with the subset of homomorphisms for the set of relations. Such a structure is the category Mag.

This example not only shows how the binary relational structures are more generalthan the mathematical categories , it also helps to understand how a category is a sub-structure within the binary relational structure .

It is important to consider the relation between the truth sets of the relations inMAG and Rel, in connection with Mag.

The relation between MAG and Rel can be established through the generality of theconcept of set , which can be defined as any collection of objects. Taking this intoaccount, we can indeed consider the collection of objects of MAG as a set. On the otherhand, any binary relation R defined over pairs of magmas has, by truth set , the set of ordered pairs of magmas that exemplify it.

Since each relation in MAG is defined with respect to its intension , and not withrespect to its extension , two relations that correspond to different binary properties arenot identical, however, they may have the same truth set .

We can, therefore, define the model of RM, A(RM) as the collection of truth sets forthe relations in RM. From the previous paragraph, it follows that the mapping fromRM to A(RM) is onto, but not one-to-one.

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Given the general notion of set, it is clear that A(RM) is a class of sets of orderedpairs of magmas , each such set being a subset of the power set M2 = M × M . Whichmeans that A(RM) can be put in correspondance with a class of morphisms in Rel

defined by the general condition:

f : M

M ′

where M and M ′ are sets of magmas in M.

The truth sets corresponding to the homomorphisms in Mag is, thus, a subclass of A(RM), which can be put into correspondence with a class of morphisms in Rel,through the above scheme.

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