Time In Quantum Cosmology

The Problem of Time in Quantum Cosmology and Non-chronometric Temporality by Carlos Pedro Gonçalves Mathematics researcher at UNIDE-ISCTE, in the areas of quantum computation, quantum formal systems theory, quantum game theory, quantum cosmology and general risk science [email protected] (primary); [email protected]; [email protected] Maria Odete Madeira Interdisciplinary researcher in philosophy of science, systems science, complexity sciences, neurocognition, semiotics, ontology and cosmology [email protected] (primary), [email protected]

Abstract We review two lines of argument regarding the problem of time in quantum cosmology and in quantum gravity, one that invokes the path integral formalism for quantum gravity to state the absence of time between two three-geometries, and another that defends the absence of time, as a fundamental notion in physics, in terms of: (a) the configuration space argument , put forward by Barbour, Smolin and Kauffman, and (b) the Wheeler-DeWitt equation. We argue that although being correct with respect to a space-time dependent physical chronometrizable clock-time frame, both of these lines of argument fail with respect to a general sense 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 subject address 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 a general quantum gravity computation, that is expressed, in the theoretical discourse of quantum gravity, through the usage of the concepts of: (1) propagation of a wave functional in superspace, as addressed by Wheeler; (2) transition amplitudes of three-geometries and (3) the pathintegral formalism, used to calculate such amplitudes, as addressed by Hartle and Hawking. While the first temporality (space-time dependent temporality) disappears from the theory, the second plays a fundamental role, not only in the several aspects of the theory’s construction, but in the clock-time independence as well, as Wheeler showed. Given this notion of time, different from a chronometrizable, space-time geometry internal notion, we search for a general mathematical and logical structure that is capable of addressing it from a formal point of view. This is done through a family of mathematical structures that is more general than the mathematical category. These structures not only will allow us to address the 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 static clock-time-independent quantum state, can be put into a non-clock-time processual expression in terms of fine-grained computational histories, obtained from the relations between different observable’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 time in quantum cosmology (Smolin, 2001). The arguments underlying this claim can be generally split in two lines of argument that arise from the formalisms used in addressing quantum cosmology and quantum gravity. One of these lines comes from a path-integral and propagator approach to quantum gravity, where the transition amplitudes, evaluated between two three-geometries, are obtained by summing over all four-geometries that aggree with the initial and final three-geometries at the border (Hartle and Hawking, 1983). The second of these two 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 configuration space argument. An example of such argument can be found, for instance, in Barbour (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 with respect 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 spacetime internal temporal chronometrizable frames. Regarding the first line of argument, the discourse itself of the works that addressed the propagator and the path integral formalism in quantum gravity trivially shows, upon closer inspection, the presence of this other temporality, restricting the validity of the claim, for the absence of time, to chronometrizable temporal frames, internal to a spacetime geometry. For, otherwise, contradictory elements would be present in the physical discourse about the theory, leading to problems when addressing the formalism. These “apparent contradictions” already appear in the early works within quantum cosmology and quantum gravity, including Wheeler’s 1968 article “Superspace and the Nature 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 concluding about this non-existence, Wheeler uses, as a supporting argument, the behavior of the propagation of a wave packet in superspace, which immediately becomes problematic with the general statement about the non-existence of time in quantum cosmology, as a quantum propagation exemplifies a temporality property that comes from a quantum gravity computational process, expressed in terms of a unitary transition. This “apparent contradiction”, as we argue in the current work (section 2.), is only an 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 refer only to a notion of a physical clock time, internal to a space-time geometry, this physical clock time is undetermined because the space-time geometry is undetermined. However, not all temporality is removed from the description, since there is still a nonchronometric temporality associated with the basic notions of input and output of a unitary transformation of a quantum state defined over the superspace, a temporality that is of a different nature than that of the internal chronometrizable temporal frame, that depends upon the space-time geometry for its chronometrization. 2

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 two three-geometries, and work with the path integral formalism to calculate such amplitude, without incurring into the fundamental problem of using theoretically inconsistent terminology. As we argue, in greater detail, in section 2., several of the statements and basic formalism, worked out by Hartle and Hawking (1983), can only be consistent with a quantum cosmology if we accept that chronological or space-time dependent clock time is undetermined in the theoretical description, and, simultaneously, understand that quantum gravity introduces a second type of temporality different from the clock-time that is internal to a fixed space-time. This second temporality, with which we are confronted, through these early works on quantum cosmology and quantum gravity, is not chronometrizable with respect to a pre-given space-time, and has a nature only dependent upon a notion of a before and an after associated with an input and an output of a general quantum gravity computation. Indeed, in order for this temporality to be chronometrizable one would have to have a clock-time frame, which is a notion internal to a pre-given space-time geometry, which is not well defined here. Between two three-geometries we cannot measure how temporally separated 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 more primitive temporal notions of a before and an after , since one three-geometry appears before the other in the (reversible) propagator formalism, hence the reason for the terms initial and final three-geometries used by Hartle and Hawking (1983). Understanding the presence of this temporality within the theoretical description, and showing it to be present in some of the foundational works on quantum cosmology and quantum 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 relational 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 consider the binary relational structure as simply composed of a collection of objects and a collection 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 the relations considered in these structures. The collection of relations, in the binary relational structures, are not necessarily settheoretical binary relations. Indeed, as we show in the main text and in the appendix to the present work, the mathematical notion of category is a particular case of these structures. 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 formalize the fundamental non-chronometrizable temporality that is found within the works on quantum cosmology and quantum gravity. Afterwards, we apply the formalism to address the second line of arguments regarding the absence of time in quantum cosmology. In particular, we show how, even a time independent quantum state, can be ascribed a computational description with respect to 3

the unitary transformation between two different bases, which results in a processual and historical nature for the quantum state, that is independent of clock time, but that can be formulated in terms of a fine-grained quantum computational history, introducing 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 in quantum 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 problems with respect to its foundational notions, along with the consequences of the application of the mathematics of the binary relational structures to the formalism underlying the argument. Central to the present work is, thus, the concept of binary relational structure, where a notion of temporality emerges from a partition of a relational nexus of any two of the structure’s objects, which introduces the notions of a before and an after that result from the positions of the objects in the relations that connect them. In section 4. we conclude with some final reflections upon the main results, and address what could be considered as open issues that may arise with respect to the mathematics of binary relational structures, in connection to physics and quantum gravity.

2. A brief reflection on the conceptions of time in physics With Galileo the time was conceptualized as a fundamental physical quantity, measurable for a whole series of physical systems and, consequently, susceptible of regulating experiments 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 absolute and to flow uniformly, independently of the reference frame (Klein, 1995). In special relativity, these two notions were understood as relative. Apparently, nature seems to be such that the flow of time depends upon the relative motion of the inertial reference frames. Which led Einstein to place the event (that, one should add, is simultaneous 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 time between its arithmetizable chronometric nature, which is relative, and its fundamental temporal sense linked to causality, which is not relative, for the causality horizon of each event is a relativistic invariant (Einstein, 1953, [2004]). In accordance with special relativity, when we try to find a causal explanation of an event we must seek it in its anteriority, that is, in that which took place before the taking place of the event, that which is to the causal past of the event, which means that we can associate a relativistic invariant sense of what came before, as that which lies in the past light cone of the event, which forms the past causality horizon of the event. 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 between what comes before and what comes after. 4

We are in the horizon of causality of the dinosaurs, because the information and records of their existence, the whole line of biological, evolutionary and geological effects of their existence lie in our causal past. Also, in the auto-biographical record of an individual, this sense is present, of what came before and what may come after, of what took place in the past, and what may take place in the future. According to Einstein (1953, [2004]) the states of consciousness of an individual appear, to that individual, pictured in a series of events in which each particular state, accessible to the individual’s memory, appears to be placed in accordance with an irreducible criterion of a “before” and an “after”. There is, thus, to each individual, a sense of personal time, a subjective time that is not, in itself, and, still according to Einstein (1954, [2004]), measurable, although one can associate numbers to the different events, such that what takes place after something else, always gets a higher number, even if only the ordering of events matters, for the subjective time, and not the particular distance between the two numbers, which means that this time is not metrizable, but, solely expressibe upon an ordinal scale. Einstein (1953, [2004]) added that physical time could be considered as a physical reality since, even if its passage was relative, that passage of time was measurable by a clock moving with the reference frame, this measurability making its measurable nature objectively relative. Any physical system with a periodic motion can be considered to constitute a physical clock, albeit the pure notion of a mechanical clock is an idealized one, the clock becoming a conceptual device to express an objectivity of a time that is part, in special relativity, of a space-time continuum (Einstein, 1953, [2004]). In this way, special relativity allows one to think about time as a component part of the topos (in the original sense of the Greek term as place of the body) itself of the events. As Weyl (1952) stressed, the change, the motion and the transformations exist in time itself. The world is active, according to Weyl, in the sense that its phenomena are related by a causal connection. The relativistic invariance of causality in Minkowski’s space-time, and the conceptual importance of the event, influenced a line of thinking and interpretations of quantum theory and quantum cosmology where these two notions play a fundamental role. Markopoulou’s proposal of quantum causal histories, as a basis for quantum cosmology (Markopoulou, 2000), finds its support in the argument that the standard approach to quantum cosmology suffers from the problem of following a tradition of addressing quantum mechanical problems where a single wave function describes the entire system, pressupposing a reasoning of an observer who has access to the whole system. The idea of a local quantum evolution, and that, at the fundamental level, there may be no such thing as a quantum state of the universe, is a distancing from the usual approaches to quantum cosmology that work with the Wheeler-DeWitt equation (Markopoulou, 2000). Smolin (2003; 2006), who defends Markopoulou’s approach, which fits well with loop quantum gravity, considers that the universe may be thought of in terms of relations between events. The notion of causal relation, in the histories of physical processes, is justifiable within the quantum unitary evolution, which is deterministic. The notion of causality, as Smolin and Markopoulou use it, has, however, a double sense. In a direct and immediate sense, the causality can be understood in terms of a set of events that constitute a necessary condition for the occurrence of another event, in that sense, we have that an event E(n) is in a relation of causality with a set of events E(1), E(2),  , E(n − 1) , if it is physically necessary that: E(n) → (E(1) ∧ E(2) ∧  ∧ E(n − 1)) 5

where → is he symbol for the material implication. In this case, provided with the information that the conjunction (E(1) ∧ E(2) ∧  ∧ E(n − 1)) holds (in the sense that it is true that each event in the conjunction has occurred), then, E(n) must also have occurred. Given a network of causal relations, defined by implications of the kind introduced in the previous paragraph, and given the truth value of the necessary conditions of the material implications, we know all of the network’s narrative, in terms of what occurred and 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 thought of as taking the information of the other events in its causal past and performing a computation (or a quantum computation in the quantum theoretical setting), sending the result of such a computation to its future through the (quantum) causal network. This computational notion, associated with causal histories, is such that we can consider, in a more general sense, the causal past of an event as the set of events from which it can receive information, and the causal future of an event can be considered to be the set of events that can process information from that event. The special theory of relativity establishes a limit to causality, as it distinguishes between timelike, spacelike and lightlike separations. Quantum theory, on the other hand, places us before the problem of nonlocal connections, which are not causal but spacelike correlational, in the sense that we need to consider the entire spacelike surface, where two or more entangled events occur, to be an interconnected whole, described by a quantum state. This configures a local arrow of time in the form of local temporal directedness due to entanglement-induced local decoherence. Information is “exported” to the whole, and only the whole has the whole information, so to speak. In general relativity, and in the standard approach to quantum cosmology, the concept of time becomes more problematic, though less so, as we saw, in the quantum causal histories approach. The first thing to notice, as Wheeler (1968) stressed, regarding classical geometrodynamics, is that the concept of event becomes less primitive and less significant. One may choose to consider space-time to be comprised of elementary objects or points called events, or, one may, instead, choose to consider the three-geometry to be the primary concept, and the event the secondary concept, an event lying at the intersection of such and such three-geometry. Under this last perspective, the temporal relation between two three-geometries would be determined by the structure of the four-geometry, which in turn derives from the inter-crossings of all the other three-geometries. In classical geometrodynamics, whether one started with the three-geometry as the primitive 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 difference when one turns to quantum geometrodynamics, for there is no such thing as a welldefined four-geometry, because, as far as we know, and still according to Wheeler, no probability amplitude function can propagate through superspace as an indefinitely sharp wave packet, the wave packet spreads. 6

The conclusion, following Wheeler’s argument, is that the space-time, the time, the notion of before and after do not exist as primitive notions within the theory, which is a problematic statement, from a theoretical discursive point of view, given the fact that even if a four-geometry is undetermined, and even if the before and after of a space-time geometry 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 propagation, commits us to a temporal sense present in superspace, different from the internal times definable and chronometrizable within each four-geometry. It is important to review Wheeler’s argument, with respect to this matter, in order to better understand this issue. The first thing to notice is that the spreading of the wave packet means that it associates a finite probability to a domain of superspace of finite measure, this domain encompassing a set of three-geometries far too numerous to accommodate in any one four-geometry. One can express this situation, according to Wheeler (1968), by stating that the propagation takes place in superspace, not by following any one classical history of space but by summation of contributions from an infinite variety of such histories. It is noticeable here the unavoidance of the apparent “discursive trap” that talks about quantum evolution 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 propagator 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 two temporal senses that must be conceptually disentangled. On the one hand, we have the chronological time that is an internal definition to each four-geometry, and, on the other hand, we have a temporal sense, distinct and not ontologically committed to an internal measurable chronometrics, but, instead, associated with the more elementary notion of a temporal property simply exemplified by an order of a before and an after of a computation, 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 discursive bias due to unexpunged terminology, habituated to a regular usage of quantum mechanics. Indeed, the unitary connection between two three-surfaces introduces a temporal 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 multitude of probable chronometrics, each associated with different (to be summed-over) histories 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 that building 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 organize the three-geometries of significance into a definite relationship, one to another, even the 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 that of a unitary connection that connects an input and output of a (general1) cosmological quantum 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 not primary, but secondary in the structure of a cosmological physical theory. Although Wheeler may be right about the first (space-time), there are some problems with the conclusion about the second (time), when it is stated categorically or in a general sense of the term time. The time, to which Wheeler is referring, is a four-geometry-dependent chronometrizable temporal frame, this time loses meaning within a quantum cosmological setting, since there is no well-defined geometrical structure upon which one may define notions, internal to that structure, such as that of before, after, present, past or future, where each event occupies a position in a “grand catalog called space-time”, to borrow Wheeler’s expression. Nonetheless, as we stated above, we still have to deal with quantum unitarity, and the path-integral formulation still brings with it a notion of a temporality associated with a notion of a quantum superspace history, a temporality that is distinct from the arithmetizable, chronometrizable time that may be defined with respect to a four-geometry. This distinction between the two temporalities, the internal space-time and the quantum evolution temporality, becomes more explicit in later works on quantum cosmology, 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′ , we must sum over all the internal fourgeometry hij , to another three-geometry hij ′ on a final surface, the two surgeometries that match hij on an initial surface and hij faces being connected by this sum. One can see here that, through the sum, we are obtaining, in a first, and crude, approximation of the quantum cosmological problem, the transition function (Hartle and Hawking, 1983): ′ |hij i = hhij

Z

δg µν exp(iSE [g µν ])

(1)

where SE is the classical action for gravity, including a cosmological constant Λ. As Hartle and Hawking (1983) noticed, when addressing the above formula, one cannot specify the time in these transition amplitudes. Which is indeed true, in the sense of a chronometric time defined within the four-geometry. This time, or rather, this notion of time is dependent upon the four-geometry, so that it is undetermined. However, we also see the usage, by the authors, of the term propagator to address the above definition (Hartle and Hawking, 1983), which is, by definition, a theoretical notion riddled with temporal connotations, that form part of its fundamental semantics. Although this might seem to introduce a contradiction, or an ambiguated discursive problem, it is in fact  a correct statement, as long as we understand that the transition ′ amplitude 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 even local, time direction, definable within the context of a space-time geometry, which is a notion 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) quantum computation. 8

In this sense, we are confronted with another notion of a time which is different from the physical, arithmetizable, chronometrizable time, and which is a more primitive notion associated with a before and after of a computation, where the how much before and the how much after are senseless statements since we have no way of assigning numbers to measure a chronological distance between input and output, because that chronological 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 that was associated to the notion of time in physics. We provide for a contribution for such a reflexive inquiry, in the next section, from a mostly mathematical point of view, by addressing a primitive family of mathematical structures that are close to the notion of mathematical category, and in which primitive non-chronometrizable notions of time are obtained.

3. Primitive non-chronometrizable temporalities and relational 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 mathematically addressable problems where the notion of number and the notion of quantity are 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 to be 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 orderable quantity in terms of mathematical generality. Indeed, although an orderable quantifiable set can be numerically labeled with respect to the quantifiable property that defines 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 numerically labeled in terms of any significant orderable quantity property, that is, one can numerically label a non-quantifiable set, but those numbers are, in general, arbitrary assignments. We already saw an example of this, when we addressed Einstein’s views on the subjective time. In statistics this is especially pertinent, since a qualitative ordinal variable can be numerically labeled in an arbitrary way, so long as the order of the objects in the ordinal scale is reflected in the order of the numeric assignments. Without a theoretically justified interval distance, any numeric interval between levels in the ordinal scale is valid. With the development of mathematics and metamathematics we came to realize that some mathematical notions and structures are very general, primitive and, thus, fundamental. As a consequence of these findings, we find ourselves before a major theoretical change, coming from the displacement of the focus from the notions of quantity and number 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: 9

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 relations 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 formula 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 binary relations one works with the extension, namely, with the set of objects that satisfy the relation, 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 is defined in its extension. This process leads to a level of analysis where the relation is reduced 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 set identity, if their truth sets are identical, even if, they are indeed different with respect to their intension. In the above structures this is not so. We are working with the notion of logical intension with respect to the relations. Which means that, even if two relations have the same truth-set extension, they are still considered to be different, if they have different intensions. For instance, if a married couple, Fred and Wilma, owns a restaurant, then the set theoretical expression would be {(Fred, Wilma), (Wilma, Fred)} for both the business partner relation and for the relation of marriage. However, in terms of the above relational structures the relations “being married to” and “being a business partner of ” would constitute two different relations with respect to the universe of discourse composed 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, means that one should not confuse the relational structures as mathematical objects with the category 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 ordered pairs. Indeed, we show in the appendix to the present work, that the mathematical category is a particular family of binary relational structures. We also address, in the appendix, the category Rel and the category Mag to make clearer the difference between the binary relational structures defined above and the category Rel. Furthermore, 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 above definition, it is important to understand each of the four conditions. Since each of the four conditions that define the binary relational structure can be extended to n-ary relational structures, we discuss the role and meaning of each condition in a general sense. The first condition tells us that all the relations between any two objects (or n-tuples of objects for the general case) are in R, which means that R is an 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 each member of R must be different from the empty set. The third condition restricts the relations in R to be solely relations between objects of 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 satisfied for an object and itself. This relation of an object to itself is called an identity. The identity is always stated in terms of a binary relation, and it is assumed to be included in every definition of n-ary relational structures. Indeed, we have to assume the relation of identity to be able to work with the notion of object, because otherwise the object would lose its integrity (Madeira, 2008b). Any structure of individuated objects must be such that each object is maintained and sustained 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 in the sense of not being generally measurable in terms of any kind of clock-time, arises from an analysis of the notion of relational nexus. This notion can be built in stages. First, we consider, for any object X ∈ O, the subset 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 be the 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 relations, 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 element, 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 relational 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 ) 4. We stress that N (X ֋ Y ) = N (Y ֌X).

11

(2)

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 fundamental notion of a before and an after in a relation. Thus, for instance, in the relational 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, in the sense that if we begin in X , then, we see Y coming after X , while that, if we begin in Y , then, we see X coming after Y . It is noticeable that we can break up each relation in N (X ↔ Y ) into two unidirectional 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, their mathematical nature is not fundamentally restricted by the imposition of an underlying chronometrics. In the above definition, one does not impose any underlying fundamental numeric property, or general family of properties that would introduce, foundationally, a temporal metric, allowing the measurement of the separation between the two terms in the relation. 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 might exemplify, fundamentally and foundationally a numeric property that could form the basis 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 from the exemplification of an underlying numeric property. Another restriction could be obtained through an appropriate axiomatic system. It is known, for instance, from expected utility theory, that an ordinal system can be transformed into a numerical scale if that ordinal system satisfies certain properties with respect to combination with probabilities. We shall return to this issue at the beginning of section 4., when we present a final reflection upon the main results of the present work and address, in that reflection, some of Barbour’s claims and the nature of a physical 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 mathematically 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. Nonetheless, 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 the order of the relation, without the ability to state how much before or how much after, or even 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 more structured sense of temporality, other than the simple before and after . Indeed, in two cases the before and after are not interchangeable, while in the last case they are interchangeable. This makes evident that the roles of what comes before and what comes after are frozen in the first two elements of the partition, and we have an arrow of time and 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 any chronometrics. The problem of an undefined metric for time, therefore, does not exclude the existence of a temporality, or even an ability to address such a temporality within a formal, mathematical theoretical setting. Besides providing for an intuition and a formalism that is able to address the pathintegral and propagator extra-clock-time temporalities of quantum gravity, these mathematical structures bring to light a problem in how one conceptually addresses configuration 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 the partition of a relational nexus, contrasting with what appeared to be a timeless structure of relations, is sufficient to raise debatable issues with respect to the time-independent configuration space argument for the elimination of time as a concept, when it is taken 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 cosmology, it has been argued with respect to classical cosmology as well, and, even, to physics 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 notion of space; the notion of configuration and the category of possibility. The notion of configuration, in quantum mechanics, is a familiar one, in a more general sense, however, which is the one assumed in the general argument, the term configuration must be taken with respect to its etymological genetics in the Latin term cum+figurare, which means to shape together , where the action of forming or of making shape – figurare – has the corresponding noun figura which means shape, form, or pattern. 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 system takes 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 their underlying etymological structure and semantics. The fact that these terms have a discursive and conceptual pre-existence, rooted in both a linguistic and philosophical tradition that has addressed them, is a first problem that introduces one element of illegitimacy and error in the line of argument. This is not the only problem with the configuration space argument for the non-existence 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 committed to a notion of temporality, and if that terminology plays a constructive role in the theory, then, the statement of no-temporality becomes inconsistent with respect to the theory 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 different alternative configurations which it can assume. Along with the statement of possibility, 13

comes the final element which introduces a geometrization of the different possible configurations in the form of a space of possible configurations, where each point is, as Barbour (1994) put it, a distinct structured whole, this means that we can indeed consider each configuration as an object of a relational system. Then, we have the collection of objects C, which is the configuration space, as our fundamental building block. The proponents of the configuration space argument, and, in particular, Barbour (1994), assume that physics must be built in terms of such a collection 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, one loses any sense of a clock time and one has a simple spatial curve without any reference to time. From this point to the next step that states that there is no temporality, may seem as a legitimate direction. However, it is not. Indeed, one loses any track of any kind of clock time and one has a curve in a space, without any reference to a physical time, which makes the temporal labelling a secondary aspect of the world. However, the fact that the curve is a relational object imposes a temporality of its own, which may be chronometrizable with respect to physical criteria but not chronometric with respect to its fundamental temporal nature. Indeed, we may work with the relational structure (C , R), where the relational nexus between 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 two points/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, the configuration x appearing before (input) the configuration y (output). The second nexus 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, is irrelevant with respect to the fundamental temporality expressed in the fact that the non-oriented curves connecting two objects can be considered conceptually akin to a reversible computation, and broken down into two oriented curves, each one implementing 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 fundamental 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 spatial geometry. Any spatial relation, or figure contains the temporality of its spatiality which is 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 configuration 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 is an important argument as one moves from the general configuration space argument to quantum theory. In this case, the argument is linked to the Wheeler-DeWitt equation, that effectively introduces a time-independent wave functional for the geometry plus field configurations, with compact spatial sections (D’Eath, 1996). Smolin (2001) addressed the general wave functional of the universe in terms of a set of assumptions that support a general theory of quantum cosmology, under the WheelerDeWitt framework. We can express these main assumptions as follows: •

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

•

The wave functional of the universe exists and is defined to be a normalizable complex functional upon an expansion in terms of R ΨUniverse that is normalizable 2 CUniverse (that is C dµ|ΨUniverse| = 1) and the normalizable states define a Universe space with a Hilbert space structure, with inner product given by: hΦ|Ψi = R dµΦ∗Ψ. C Universe

•

The wave functional of the universe satisfies the momentum constraints along with the Wheeler-DeWitt equation.

Under this framework, we are effectively working with a time-independent wave functional, and time seems to disappear from the theory. Smolin (2001) reviewed several arguments against this approach to quantum cosmology and against the argument for timelessness, on several grounds, including: •

The limits in the ability to build a formal mathematical theory that incorporates the observables for quantum cosmology using the above approach, linked with the problem that the Hamiltonian constraint observables are extremely difficult to construct in real field theories of gravitation, made more severe by the need to consider the conjecture regarding the presence of chaotic behavior in gravitational systems;

•

The possible unobservability and unkowability of the configuration space of the universe, including the problem of the computational complexity for constructing a mathematical representation of complex configuration spaces.

It is about the limits to our ability to theorize that Smolin’s objections, against the argument 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 theories are. This, in turn, leads to the requirement that a sufficient number of observables can be determined by information that reaches an observer inside the universe, allowing that observer to know the quantum state of the universe. Only if this is the case can we build a quantum theory of cosmology, based upon the above set of assumptions, such that this theory obeys the standard methodological and epistemological scientific requirements that any scientific theory must obey. It is around our ability to build a theory of quantum cosmology that these arguments orbit, however, the arguments presented above do not, by themselves, refute the argument for the elimination of time, as Smolin (2001) admits it. 15

The problem of time is addressed by Smolin as a human theorization problem, the temporality 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 independent quantum state, we should begin by looking at the mathematical structure of a Hilbert space, as a geometric space, raising the problem of how one should interpret a clock-time independent quantum state. If the universe can be thought of as being a quantum system with a clock-time independent quantum state (corresponding mathematically to a clock-time independent wave functional), then, even if we are unable to address this state, from a theoretical point of view, we must consider the nature of static clock-time independent quantum states, 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 bases of the Hilbert spaces, as vector spaces, and at the consequences of these relational structures 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 independent 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 unitary transformations associated with the change of basis. Then, we can see that, given any two bases: B1 = {|φn i} B2 = {|ψn i} we can expand each element of the second basis in terms of the elements of the first basis as: X |ψn i = Umn |φm i (5) m,n

The expansion coefficients Umn can, then, be expressed as:

 φm ψn = Umn

(6)

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

X

Umn |φm i = |ψn i

m,n

16

(7)

thus: hφm |Uˆ (2, 1)|φn i = hφm |ψn i = Umn

(8)

Since the unitary transformation is reversible we can see that, while Uˆ (2, 1) establishes a connection from the first to the second basis, its conjugate transpose Uˆ (1, 2) = Uˆ (2, 1)z establishes the inverse connection. We, therefore, have the relational nexus: N (B1 ↔ B2) = {Uˆ (2, 1), Uˆ (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 temporality expressed in the transition from one basis to another (Eq.(6)), and in the unitary transformation 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 an actual physical clock, internally defined within some space-time metrics. This temporal sense “leaks out” to the quantum formalism. Indeed, given the expansion of a quantum state in either of the two basis: X |Ψi = cn |ψn i n X ′ |φm i |Ψi = cm

(10) (11)

m

we have that: (12) X X ′ = hφm |ψn icn = hφm |Uˆ (2, 1)|φn icn Ψ(φm) = hφm |Ψi = cm n Xn X ′ ′ Ψ(ψn) = hψn |Ψi = cn = hψn |φm icm = hψn |Uˆ (1, 2)|ψm icm m

(13) (14)

m

Therefore, in Eq.(13) we can see that the amplitudes Ψ(φm) can be considered as a superposition of quantum computational histories with quantum logical gate Uˆ (2, 1) initial kets ranging over the basis B1 and final ket |φm i. Each computational history is weighted by the amplitude cn with the index n ranging over the basis B1, expressing an uncertainty associated with the input state. Therefore, we can see that the static expansion expressed by Eq.(11) can be put into a processual form, where the amplitudes Ψ(φm) correspond to the probability amplitudes of the output of the computation being |φm i in a quantum computation with general propagator expressed by hφm |Uˆ (2, 1)|φn i, and where the cn correspond to the amplitudes associated with the input states. A similar reasoning can be applied to the expansion of Eq.(10) and to Eq.(14). We can also introduce the projector chains: Pmn = |φm ihφm ||ψn ihψn | ′ Pnm = |ψn ihψn ||φm ihφm | 17

(15) (16)

and write: Ψ(φm) = Ψ(ψn) =

X

n X

hφm |Pmn |Ψi

(17)

′ |Ψi hψn |P nm

(18)

m

These projector chains can be interpreted in light of the projector chains of the consistent histories’ formalism (Omnès, 1988, 1992; Griffiths, 1993, 1994; Gell-Mann and Hartle 1993, 1994, 1996, 1998; Hartle, 2007), the Pmn representing a history of a quantum system, since we have: Pmn = Uˆ (1, 2)|ψm ihψm |Uˆ (1, 2)z|ψn ihψn | = Uˆ (2, 1)z|ψm ihψm |Uˆ (2, 1)|ψn ihψn | ′ = Uˆ (2, 1)|φn ihφn |Uˆ (2, 1)z|φm ihφm | = Uˆ (1, 2)z|φn ihφn |Uˆ (1, 2)|φm ihφm | Pnm

(19) (20)

which completes the connection to the projector chains of the consistent histories’s formulation, since Eqs.(19, 20) can be considered to be analogous to projector chains for fine-grained histories of a quantum computational network, with the corresponding quantum logical gates Uˆ (2, 1) and Uˆ (1, 2), respectively5 (Hartle, 2007). This deepens the result about the temporality expressed by the relation between the two bases, that can be considered as a quantum computation. The unitary transformation that implements this computation is such that the elements one of the basis take the 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 temporal expression in terms of a set of quantum computational histories, fine-grained with respect to the quantum logical gate, which is the unitary transformation that implements the change of basis. This makes the result more compelling, as it shows how a temporal sense, present in the relational nexus, that is different from a clock time, may turn a representation of a clock-time independent quantum state, that represents a single static object, into a processual representation, in terms of a quantum computational history, with a formalism analogous 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 generality with respect to the timelessness of the quantum state of a system, described by a stationary quantum state. Such a state indeed should satisfy a clock-time independent Schrödinger-like equation, and, thus, be effectively timeless with respect to a physical clock time, however, the mathematical processual nature of such a state, uncovered by the relational structure of the observables’ eigenbases, precludes the generalization of the statement to any kind of temporality. Ultimately, results such as those of Eqs.(13, 14, 19 and 20), raise the more general issue of the terminology used to address our physical theories of the universe, and, in particular, the legitimacy of using the concept of state, rather than the term process, to refer to a ket, which resends to a snapshot-like staticness influenced by a classical physics’ 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? The result of the analysis of arguments and of theoretical discourse developed in the previous two sections, as well as the mathematical results obtained in the previous section, leads to an answer to the above question, with another question: what time? Indeed, the result of the work developed above shows that temporality and the notion of time go beyond the more restrictive chronometrizable notion of time, that is internally 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 physical clock time may indeed not be a basic concept. However, whenever we consider a quantum computation, when we address a timeindependent ket, or the relations between different physical observables’ eigenbasis, we find a basic temporality that is definable with respect to the configuration of the relational nexus of a system of relations between objects. In physics, as we saw in the previous section, if we accept the fundamental role of a configuration 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 fundamental (ordinal) time, which is the time of the order of the objects’ positions in the relation, and that ultimately proceeds from the connection of two individuations that are separated, but linked by the relation, and, thus, are temporally connected in the temporality that is the order of terms in the relation. For relational structures such that, given any two objects X and Y , N (X&Y ) is either N (X ֌ Y ), N (X ֋ Y ) or N (X ↔ Y ), it is possible to obtain a numeric scale for the objects that reflects the relation time, by introducing the structure of prevalences in relational nexus (O, %N ), defined as: X %N Y if, and only if, N (X&Y ) = N (X ֌ Y ) ∨ N (X&Y ) = N (X ↔ Y ) X ∼ Y if, and only if, 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 in the relational nexus N (X&Y ). Now, if we were to combine the objects with probabilities, and introduce von Neumann and Morgenstern’s (1953, [1990]) axiomatic for expected 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 temporal structure that is purely ordinal. The above result is, at least, sufficient to show that an ordinal temporality, that is a time that emerges within a relational structure, can be more fundamental from a physical point of view, playing a foundational role in an emergence of a space-time chronometrizable physical time. 19

In this work we have solely dealt with reversible unitary transformations, which leads to relational nexus of the kind of N (X ↔ Y ). A nexus of this kind is reversible, in the sense 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 and mathematical expression. To understand this we may take the example of the qubit: |ψ i = ψ(0)|0i + ψ(1)|1i

(21)

where the two weights ψ(i), i = 0, 1 are time independent complex numbers satisfying the normalization condition |ψ(0)|2 + |ψ(1)|2 = 1. In this “static” expansion, we find that the weights assign amplitudes to the transitions: h0|ψ i = ψ(0) h1|ψ i = ψ(1)

(22) (23)

which, under Heisenberg’s interpretation of quantum mechanics, determine the probability of actualization of the input |ψ i to an output of |0i or |1i, mathematically expressible through a stochastic selection of a projection. The temporality inherent in this is a clear one, even though the category of causality does 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 these notions that the actualized quantum state is preceded by a corresponding potential reality, upon which it is founded and grounded. Since the nature of the dynamis, or potentia, is to tend towards the act that determines it, there is a temporal sense, associated with the “static” expansion of Eq.(21), and identifiable in the physical interpretation of that equation, within Heisenberg’s interpretation 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 interpretation of quantum mechanics, leads one to a discussion about decoherence, which can and should be extended not only to quantum cosmology6 but, also, to the mathematical structures introduced in section 3.. Another open problem, that may be raised, regards the nature of the relational structures when, they, themselves are subject to a quantum description. Several different issues arise in this case. For instance, a quantum causal history is such that, if we label the 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 is locally reversible (due to the unitarity), that is, we have the general relational nexus associated with the system of node + edges: n o N (|ψ i ↔ |ψ ′i) = Uˆ , Uˆ z 6. See, for instance, Kiefer (2003) for an example of such a discussion.

20

(24)

However, in the directionality imposed upon the network, only one of the directions is usually chosen, that is, the local reversible nexus is partioned in terms of: o n o n N (|ψi ֌ |ψ ′i) ∪ N (|ψi ֋ |ψ ′i) = Uˆ ∪ Uˆ z

(25)

and one of the directions N (|ψi ֌ |ψ ′i) or N (|ψ i ֋ |ψ ′i) is assigned to the network by the 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 given and assume a quantum superposition of the partitionned nexus, which would lead to the local network state: |ΓLoci = Ψ1|N (|ψi ֌ |ψ ′i)i + Ψ2|N (|ψ i ֋ |ψ ′i)i

(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 quantum gravity computer. The matter of how one of the directions gets to be selected, faces us with the problem of decoherence with respect to the above local state, and the problem of decoherence in quantum binary relational structures. In some paths of research, uncertainty regarding the choice of nexus may lead to the research problem of decoherence and recoherence. In other paths, we may address a computation that takes place with respect to the whole network which transcends the local 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 computation itself is not yet selected. An environmental decoherence mechanism would produce a diagonal local network state, but this mechanism would be such that it would compute the entire temporal connection, which means that it would introduce a second order temporality, that takes the relation itself as an object of another relation. As an example, let us consider the above local network state and define N1 ≡ N (|ψ i ֌ |ψ ′i) and N2 ≡ N (|ψ i ֋ |ψ ′i). Then, we have a local basis {|N1i, |N2i}. Adding a local environment, partitioned in at least two parts8, we can introduce the computation: |ΓLoc, A0, E0i ֌ Ψ1|N1, A1, E1i + |N2, A2, E2i

(27)

implemented by the entanglement operator UˆEnt. 7. We need the two pieces of information (labelling and directionality given to the edges) since in some cases we may have Uˆ = Uˆ z . 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: n

o n o z |ΓLoc, A0, E0i ↔ Ψ1|N1, A1, E1i + |N2, A2, E2i = UˆEnt, UˆEnt

(28)

Properly considered, this is a reversible process, where UˆEnt produces entanglement and z transforms the entangled state9 into an unentangled state10. However, for a large UˆEnt 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 would have the transition given by the following nexus, expressed in terms of density operators:  





N |ΓLoc ΓLoc| ֌ |Ψ1|2|N1, A1 N1, A1| + |Ψ2|2|N2, A2 N2, A2| (29) with the environment described by the kets |Ai i playing the role analogous to a record keeping physical apparatus. It is noticeable that we are dealing with a quantum computation that computes an entire local structure of quantum computations, and that the final relational nexus is the nexus of the quantum states for the relational nexus of a local quantum computation. Indeed, the result of Eq.(27) the computation expressed
corresponds to the case where ′ 2 by the nexus N (|ψ i ֌ |ψ i occurs, with probability |Ψ1| , or the reverse computation, expressed by the nexus N (|ψ i ֋ |ψ ′i) occurs, with probability |Ψ2|2. The generalization of this type of research to quantum causal histories, and to their applications in quantum gravity research, may be important to understand the nature and role of time in the theory of quantum space-time, and, in particular, in the research program 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 directional 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 any two 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 such that, for three objects X , Y , Z ∈ O, and two morphisms f and g, f ∈ N (X ֌ Y ) and g ∈ 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 the fourth that the composition is commutative with respect to the identity. A structure satisfying these conditions is called a category. It follows from these results that a mathematical category is a binary relational substructure, 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 structures 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 morphisms, 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 about composition 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, a member of Rel, since each truth set is a set of ordered pairs, that can be expressed as the Cartesian product of two subsets of the collection of objects. The structure Mag is another example of a binary relational substructure that is a category, having, magmas by objects, that is, algebraic structures composed of sets with a 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

 aιAb ∈ A

Thus, we have the magma (A, ιA). Now, it becomes important to consider the following four definitions, from universal algebra (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 operation 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 algebra is given by (Burris and Sankappanavar, 1981): 24

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 FA on A. The set A is called the universe (or underlying set) of the algebra a = (A, F ), and the FA’s are called the fundamental operations of a. If F is finite, such that F = {F1,  , Fn }, we write (A, F1,  , Fn) for the algebra a = (A, F ), with the convention that arity F1 ≥ arity F2 ≥  ≥ arity Fn. It is straightforward to see that a magma (A, ιA), with A nonempty, is an algebra in the above sense, that is, it is an algebra with a family of binary operations indexed by a singleton 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 MAG comprised 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 binary operation. It is noticeable that, under this restriction, one is no longer working with the whole relational structure MAG, since the set of relations between magmas is restricted to operation preserving mappings. The structure within MAG, with which we are working, 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 general than the mathematical categories, it also helps to understand how a category is a substructure within the binary relational structure. It is important to consider the relation between the truth sets of the relations in MAG and Rel, in connection with Mag. The relation between MAG and Rel can be established through the generality of the concept of set, which can be defined as any collection of objects. Taking this into account, we can indeed consider the collection of objects of MAG as a set. On the other hand, 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 with respect to its extension, two relations that correspond to different binary properties are not 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 for the relations in RM. From the previous paragraph, it follows that the mapping from RM to A(RM) is onto, but not one-to-one. 25

Given the general notion of set, it is clear that A(RM) is a class of sets of ordered pairs of magmas, each such set being a subset of the power set M2 = M × M. Which means 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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