Preprint: In this article we describe the properties of objects that we shall call ubiquitous or with ubiquitous properties. Ubiquitous objects are such that they have the property of being simultaneously in several distinct spatial positions. Although these properties are not those of our familiar universe, such objects can nevertheless be conceived, and we strive to give a more precise characterization, in particular within the conceptual framework of n-universes. We then study the properties of such objects, in particular their relational dispositions, as they result from their very definition. Finally, we aim to show how the ubiquitous properties of objects are likely to have an application in our physical universe, in particular by allowing us to provide a specific interpretation of quantum entanglement, which avoids the use of a principle of non-locality.
In the following, we shall
describe the properties of objects that we shall call, for the sake
of this discussion, with ubiquity property (for
short, ubiquitous). To express it simply and in a preliminary
way, ubiquitous objects can be defined as objects that have the
property of being simultaneously in several distinct spatial
positions. Although such properties are not those of our familiar
universe, such objects can nevertheless be conceived, and we will
endeavour to give a more precise modelling, in particular within the
conceptual framework of n-universes (Franceschi 2001). We will
then study the properties of such objects, as they result from their
very definition. Finally, we will strive to show how the ubiquitous
properties of objects are likely to have an application in our
physical universe, in particular by allowing us to provide a
particular interpretation of quantum entanglement.
1. Modelling ubiquitous objects
In our familiar universe, an object can only occupy one spatial position at any given time. However, conceptually, we can imagine universes where objects can occupy several spatial positions at the same time. It is useful at this stage to focus on describing more accurately the properties of ubiquitous objects. For this purpose, we will use the conceptual framework of the n-universes, a methodological tool introduced in Franceschi (2001), for the purpose of analysing Goodman’s paradox. Applying Occam’s razor, the n-universes aim in particular to allow the modelling of often complex situations implemented in thought experiments and thus to simplify the related reasoning. Hence, by virtue of Occam’s razor, we strive to model a given situation by using the simplest universe model, in a way that is compatible, however, with the preservation of the essential elements of the corresponding situation. In addition, it appears that the conceptual framework of n-universes is also suitable for modelling universes with properties that are different from those of our familiar universe, as this conceptual framework is also a methodological tool for creating and defining ontologies. The n-universes thus make it possible to model objects whose properties are completely different from those of our familiar universe. In particular, the n-universes can be used to model universes where objects have the property of being able to occupy several distinct spatial positions at the same time.
Before modelling the ubiquitous properties of objects in n-universes, it is worth recalling the fundamental elements of the latter. The essential elements that make it possible to model the properties of objects in the n-universes are as follows:
▪ constants and variables: a given n-universe includes a number of criteria, which are either constants or variables. These include temporal, spatial, colour, shape, temperature, polarization, etc. criteria. Examples of n-universes are as follows:
– a n-universe with a time constant T_{0} and a location constant L_{0} ^{1}
– a n-universe with a colour variable C, a time constant T_{0} and a location variable L ^{2}
– a n-universe with a colour constant C_{0}, a time variable T and a location constant L_{0} ^{3}
▪ n-universe with a single object or multiple objects: a n-universe can have a single object (o) or multiple objects (α). Consider for example:
– a n-universe with a single object o, a time variable T and a location constant L_{0} ^{4}
– a n-universe with multiple objects α_{1}, …, α_{n}, a time variable T and a location variable L ^{5}
▪ one-one or one-many relationship of the objects with a given criterion-variable: such a distinction only concerns n-universes with criteria-variables. It can be applied to n-universes with a single object or multiple objects. When an object can exemplify several taxa of a given criterion-variable, it is in relation one-many with the latter criteria. Conversely, when an object can only exemplify one taxon of a given criterion-variable, the objects are in a one-one relationship with it. For example:
– a n-universe with a single object o, a time variable T and a location constant L_{0}. The single object o can thus exemplify several temporal positions and is therefore in a one-many relationship with respect to the temporal criterion.^{6} In our physical universe, objects possess a property of temporal persistence, which constitutes a particular case of one-many relationship of an object with respect to the temporal criterion-variable. Thus, in our universe an object o that exists in T_{1} also exists in T_{2}, …, T_{n}.
▪ one-one or many-one relationship of the objects with a given criterion-variable: this type of relationship applies only to n-universes with multiple objects and leads to a distinction between two cases. When several objects can exemplify a given taxon from a certain criterion-variable, the objects are in relation many-one with the latter criterion-variable. On the other hand, when one single object can exemplify a taxon of a given variable criterion, the objects are in a one-one relationship with the variable criterion. For example:
– a n-universe with multiple objects, a time variable T and a spatial variable L. The multiple objects are in a many-one relationship with the time-variable criterion, so that several objects can exist at a given time position. In addition, the objects are in one-one relationship with the location criterion, so that there can only be one object at a given spatial position.^{7}
At this point, we are in a position to define the ubiquitous properties of objects in the n-universe framework. An ubiquitous object is thus an object that can occupy at the same time T_{0} several different spatial positions: L_{1}, L_{2}, …, L_{n}. Applying Occam’s razor in order to reason on the simplest universe model, we will first consider a single object o that occupies the distinct spatial positions L_{1} and L_{2} at time T_{0}. The corresponding n-universe thus includes a single object o, a time constant T_{0} and a location variable L with two taxa : L_{1} and L_{2}. In addition, the single object o is in a one-many relationship with the spatial variable. To fix ideas, one can imagine such a n-universe as a ball that occupies at the single time T_{0} the two spatial positions L_{1} and L_{2}.^{8}
2. Properties of ubiquitous objects
The conceptual framework that has just been described now allows us to consider, in a more precise way, the properties of ubiquitous objects. Some of these properties derive directly from the very definition of ubiquitous objects. A first property concerns the very existence of ubiquitous objects. Let us consider, for the purposes of this discussion, an ubiquitous object o that has two spatial occurrences, respectively at positions A and A’. Suppose that we come to delete the occurrence of o which is in A: in this case, it turns out that the occurrence of o which is in A’, is also deleted. This follows from the very definition of an ubiquitous object, since the same object o is found in both A and A’. For it is necessary to imagine that the object o has two occurrences in A and A’, but that the term “occurrence” reflects the viewpoint which is that of our familiar universe, because from the standpoint of the ubiquitous universe, it consists indeed of one single and unique object. In a reciprocal way, let us suppose that the occurrence of o which is in A’ is deleted. In this case, it also follows, by the very definition of the ubiquity property, that the occurrence of o which is in A, is also deleted at the same time.
Such a property relating to the very existence of the object o can also be transposed to the colour property, for example. Let us suppose that the object o has a colour property, for example red or green. Let us consider that the two occurrences of object o, which are respectively at spatial positions A and A’, are both red. What happens if the occurrence in A turns green? It follows, by the very definition of an ubiquitous object, that the occurrence in A’ also becomes green. Indeed, since it concerns the same object, it follows that the change of colour that occurs in A also takes place at the same time in A’.
In order to better study the properties of objects with ubiquitous properties, it is also useful to compare and distinguish the n-universe just described with another type of universe with two distinct balls that evolve in a non-ubiquitous n-universe. In the latter case, the situation corresponding to two distinct balls is then modelled in a n-universe comprising multiple objects (α_{1} and α_{2}), a time constant (T_{0}) and a location variable L with two taxa (L_{1} and L_{2}). In such a n-universe, objects are in a one-one relationship with the spatial variable, because only one object can occupy a single spatial position at any given time. In addition, the objects are in a many-one relationship with the time constant, so that several objects can coexist at time T_{0}. To fix ideas, we then have two balls which occupy respectively, at time T_{0}, the spatial positions L_{1} and L_{2}.^{9}
The situation just described corresponds to a thought experiment described by Max Black (1952), featuring two red balls that are identical in every respect. The universe described by Black thus features two perfectly identical spheres, which evolve in a symmetrical universe. In such a universe, a given object – a sphere – has two occurrences, whose properties are quite identical, in two symmetrical parts of the same universe.^{10} Black’s thought experiment relates to the principle of the identity of indiscernibles, whereby two objects that possess exactly the same properties are identical. But how do we decide, as Black suggests, whether it is a single ball in a ubiquitous universe or two separate balls in a non-ubiquitous universe? Can we not imagine a test that would make it possible to decide such a situation and determine the true nature of the universe in which the objects are located? It turns out here that such a test can be envisaged, considering what can be called the relational dispositions of objects A and B. Consider the following test: let A and B be the two observed balls,^{11} or both spatial occurrences of the same object.^{12}
Let us suppose now that we change the colour of the ball located at spatial position A. If there are two distinct objects in a non-ubiquitous n-universe, the change in colour of the A-ball cannot affect the colour of the ball located in A’.
On the other hand, if it consists of two spatial occurrences of the same object in a ubiquitous n-universe, the change in colour of the ball located in A must lead at the same time to the change in colour of the ball located in A’. For this results from the fact that it consists, by definition, in the ubiquitous n-universe, of a single object. The determination test could thus take the following form: if the colour of the ball located in A (and conversely the ball located in A’) changes from C_{1} to C_{2} at time T1, then the colour of the ball located in A’ (and conversely the ball located in A) must also change from C_{1} to C_{2} at time T_{1}.
An additional clarification can be added here regarding the criterion that has just been defined to distinguish an ubiquitous object from two distinct objects in a classical universe. Indeed, the change in property that has been described must occur at exactly the same time. In other words, if a given property (e. g. colour) of the ball A changes at time T_{1}, then the same property of the ball located in A’ must also change very exactly at time T_{1}. Here, it is not enough that the change occurs at a time position slightly later than T_{1}, but it must occur at exactly the same time, so that the previous criterion is verified.
At this stage, it appears that a second type of ubiquitous property can also be considered. Thus, an object can also be considered ubiquitous, as long as its two occurrences have properties that are not identical, but complementary. Let us consider, to fix ideas, a discrete property of polarization (positive or negative). Let us consider an ubiquitous object such as a ball, which simultaneously occupies two spatial positions L_{1} and L_{2}. We also consider that if the polarization of the ball located in L_{1} becomes negative (respectively positive) at time T_{1}, then the polarization of the second occurrence placed in L_{2} also becomes positive (respectively negative) at time T_{1}. It appears that the polarizations of the two occurrences of the ball are complementary. In such a universe, as we can see, ubiquitous objects are characterized by relational dispositions of polarization that are complementary. Such properties also correspond, as we can see, in an intuitive sense, to those of a universe with ubiquitous objects.
It can also be observed here that the execution of the test just described presupposes n-universes which are an extension of the two competing n-universes previously described. Indeed, the execution of such a test presupposes that the corresponding n-universe includes a temporal variable, instead of a temporal constant. Indeed, the execution of the test requires at least two time positions (T_{1} and T_{2}), since it is necessary to make a comparison between the state of the n-universe before the execution of the test (at T_{1}) and after its execution (at T_{2}). Thus, it is appropriate to place oneself in a n-universe with a time variable. In addition, it also proves necessary to introduce a colour variable, since a colour change must be tested. It will therefore be necessary to place oneself in a n-universe with a colour variable, with two taxa: C_{1} and C_{2}.^{13}
It can be seen at this stage that the test just described can be more generally defined. It suffices indeed to take into account in a broader sense the relational dispositions of objects. The previous test concerning colour can thus be extended to the effect of a change in shape, dimension, spatial position, etc., and generally to all the properties of the object in question. In general, we thus consider the effect of the modification of any property of a ball’s occurrence on the corresponding property of the second occurrence of this ball. In this context, the previous test can be described in the following general form: if the <property_{1}> of the ball A changes from Taxon_{1} to Taxon_{2} at time T_{k}, then the <property_{1}> of the ball B also changes from Taxon_{1} to Taxon_{2} at time T_{k} (where <property_{1}> denotes a given taxon among the criteria of colour, shape, size, etc.).
Finally, another important property of ubiquitous objects deserves to be mentioned. It refers to the geometry of the space where ubiquitous objects evolve. An interesting question arises in this context: what is the measurement of the distance from A to A’, which correspond to the two occurrences of the ubiquitous object? It turns out that such a length results from the very definition of an ubiquitous object. Indeed, since it is the same object, which has a location in A and another in A’, the distance AA’ can only be zero, as it is the distance from an object to itself. From the viewpoint of our familiar universe, such an object appears as a segment, but from the ubiquitous point of view, it is a point. Let us also consider two ubiquitous objects A and B, whose occurrences are respectively in A and A’, as well as in B and B’ : from the point of view of our familiar universe, these four points form a rectangle, but from a ubiquitous point of view, it is a segment, since the distance AA’, as well as the distance BB’ are nil.^{14}
Figure
4 : Geometry of ubiquitous objects: the distance AA’
3. Modelling quantum entanglement in a ubiquitous n-universe
The above considerations describe theoretical universes in which objects with ubiquitous properties evolve. Conceptually, such universes do not pose any particular problems. But we can ask ourselves, at this stage, if these last universes with their specific properties are not suitable for modelling objects in our physical universe. In the following, we will try to show how such a modelling could be applied to entangled photons, and thus constitute an adequate interpretation for the phenomenon of quantum entanglement.
The phenomenon of quantum entanglement finds its historical starting point in the famous thought experiment described by Einstein, Podolsky and Rosen (1935), where the authors stage a system composed of two quantum particles. The thought experiment highlights a conflict with a fundamental physical principle: the principle of locality. According to this principle, a measurement made on a given physical system cannot have an effect on a second system, isolated from the first. For this reason, the conclusions that result from this experiment constitute what has been called the EPR paradox. According to Einstein, Podolsky and Rosen, the EPR paradox reveals that the theory that describes quantum mechanics is incomplete. Although the resulting conclusion is counter-intuitive, the thought experiment underlying the EPR paradox has subsequently led to strong experimental confirmations. One of them (Aspect et al. 1982) included the elements of the EPR paradox. This experiment staged photons and produced exactly the results predicted by quantum mechanics: the measurement performed showed that the polarization state of the photons was correlated. Experiments confirming the EPR paradox data have generally been interpreted as highlighting a principle of non-locality: particles interact when they are separated by a distance such that no remote action is possible, at a speed not exceeding the speed of light. This principle of non-locality, anticipated by Einstein, proves to be prima facie counter-intuitive.
While the phenomenon of quantum entanglement has given rise to multiple experimental confirmations, its interpretation itself has posed a problem. Several interpretations of quantum mechanics have thus been proposed, in order to solve the EPR paradox. Among these latter interpretations, some are based on multiple worlds, on the possibility of retrograde causality (De Beauregard 1977, 1979), or on the consideration of a speed of light c in vacuum greater than that (2.99792458 x 108 m/s) resulting from the theory of relativity, etc. We shall focus on showing how the ubiquitous n-universe model could be an adequate alternative for the interpretation of quantum entanglement. The corresponding interpretation takes into account the ubiquitous properties of objects, the main one being that a given object can exemplify several spatial positions at the same time. Modelling with ubiquitous n-universes is likely to have advantages, in particular the fact that it does not require the use of a non-locality principle. Indeed, under ubiquitous modelling, the two entangled photons constitute the two occurrences of one single object. And the properties resulting from the relational dispositions of photons make it possible to account for the fact that the change in polarization of one of the entangled photons (A_{1}) instantly causes the change in polarization of the photon that is correlated to it (A_{2}).
Indeed, as mentioned above, the relational dispositions of ubiquitous objects in the n-universes can be applied to a colour criterion, but also to other criteria, such as polarization properties. Entangled photons A_{1} and A_{2} thus present polarizations that are complementary. Thus, if one of the photons has a vertical polarization, the other photon has a horizontal polarization. Let us consider the dispositional relationships at the level of photon polarization. In a non-ubiquitous model, the change in polarization of one of the photons must have no effect on the polarization of the other. Conversely, in ubiquitous modelling, the change in polarization of one of the entangled photons must result in the correlative change in polarization of the other photon at the same moment. For the two photons constitute, in ubiquitous modelling, two spatially distant occurrences of one single object. As a result, the distance A_{1}A_{2} between the two photons A_{1} and A_{2} is equal to 0, which results from the fact that the two distant spatial positions are instantiated by a single ubiquitous object. And the distance from an object to itself can only be zero. This has the effect of reconciling the properties of the correlated photons with the locality principle.
More generally, the test for the relational dispositions of objects just described is likely to apply to different photon properties, such as polarization or wavelength. However, such a test is also likely to apply to the very existence of photons. Indeed, the dispositional relationships inherent to the very existence of photons relate to the destruction of one of the photons. If the two photons did not have a non-ubiquitous behaviour, the destruction of one should have no effect on the existence of the other. On the other hand, if the two occurrences of photons were to exhibit ubiquitous behaviour, the corresponding dispositional relationship should be reflected in the fact that the destruction of one photon leads to the destruction of the other at the same time, being one and the same object.^{15}
In this respect, another consequence of ubiquitous modelling is that it provides a unitary point of view for the phenomenon of superposition of quantum states. This is another difference with the standard interpretation of the quantum entanglement phenomenon. Indeed, we can recall that before a photon is placed in a correlated state, it is classically in a state of quantum superposition, having both horizontal and vertical polarization. In the non-ubiquitous description that results from the standard interpretation of the correlation of two photons, after the two photons have been correlated, they each have a different and complementary polarization. And at this stage, the two correlated photons are no longer in a state of quantum superposition. On the other hand, in the ubiquitous model, the two entangled photons constitute the two occurrences of the same object. Thus, while the two photons are placed in a correlated state, the object persists in the same state of superposition, since it is the same object that occupies two different spatial positions. Thus, such an ubiquitous object is always in a state of quantum superposition, either before or during correlation.
Finally, it turns out that the present interpretation of quantum entanglement is associated with a particular conception of space. Such a conception of space is essentially Leibnizian in nature. Leibniz (1956) considers that space does not possess existence or property in the absolute, but is determined by the sum of the relationships between the objects that occupy it. In this context, there is no notion of absolute space or a type of universal geometry that prevails for all physical objects. On the other hand, it results in different types of geometries, each adapted to the types of objects or particles considered. A given type of geometry then only prevails when it is associated with a given type of physical object or particle. For macroscopic objects that are familiar to us, Euclidean geometry is perfectly suitable. On a very large scale, the curvature of space associated with the presence of matter requires the use of Riemanian geometry. On the other hand, other types of objects such as photons could be governed by the geometry associated with ubiquitous n-universes. Thus, in this context, we would not have a geometry that prevails for all objects, but rather different geometries which are determined by different types of objects present in space. The geometry associated with space would not be absolute, but related to the context, and in particular to the objects that evolve within it. Thus, for the notion of distance in particular, it would follow that there is no absolute notion of distance between two given points A_{1} and A_{2} of space. For it is the type of object that occupies each of the two points that determines the calculation of the distance associated with them. Are they two familiar macroscopic objects that occupy points A_{1} and A_{2}, then the distance A_{1}A_{2} is governed by Euclidean geometry, to which our first intuition corresponds. On the other hand, if these are two entangled photons occupying the spatial positions A_{1} and A_{2}, then the non-ubiquitous geometry prevails and applies to the calculation of the distance A_{1}A_{2}, which is then equal to 0. Thus, it is only when one considers a notion of absolute distance that one obtains a contradiction. On the other hand, as soon as we consider given physical objects and calculate the distance with their own geometry, the contradiction disappears.^{16}
References
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Black, Max. 1952. The Identity of Indiscernibles. Mind 61: 153-164. |
De Beauregard, C. 1977. Time Symmetry and the Einstein Paradox. Il Nuovo Cimento, 42B: 41-64. |
De Beauregard, C. 1979. Time symmetry and the Einstein Paradox. II. Il Nuovo Cimento, 51B: 267-279. |
Einstein, Albert, Podolsky, Boris & Rosen, Nathan. 1935. Can Quantum-mechanical Description of Physical Reality Be Considered Complete?. Physical Review. 47: 777-780. |
Franceschi, Paul. 2001. Une solution pour le paradoxe de Goodman. Dialogue 40: 99-123. |
Leibniz, G. W., S. Clarke & I. Newton, 1956, The Leibniz-Clarke Correspondence, H. G. Alexander (éd.), Manchester: Manchester University Press. |
Leslie, J. 2001. Infinite Minds. Oxford & New York: Oxford University Press. |
Nogues et al. (1999) Seeing a single photon without destroying it, Nature, 400: 239 |
1 Formally, a ΩT_{0}L_{0}.
2 Formally, a ΩCT_{0}L.
3 Formally, a ΩC_{0}TL_{0}.
4 Formally, a ΩoTL_{0}.
5 Formally, a Ω αTL.
6 Formally, a ΩT>o–L_{0}.
7 Formally, a ΩT< α=L.
8 Formally, the situation takes place in a ΩT_{0}–o<L, a n-universe comprising a single object (o), a time constant (T_{0}), a location variable (L) with two taxa L_{1} and L_{2}, and where the single object is in one-many relationship with the spatial variable.
9 Formally, it consists of a ΩT_{0}< α=L.
10 A similar situation is also described by John Leslie (2001, p. 153), who describes a thought experiment that involves three aligned spheres whose properties are identical in every respect, and such that the two outer spheres are located at equal distance from the central sphere: “Here is a yet greater paradox for Identity of Indiscernibles to swallow. Try to picture a cosmos consisting just of three qualitatively identical spheres in a straight line, the two outer ones precisely equidistant from the one at the centre. Aren’t there plain differences here? The central sphere must be nearer to the outer spheres than these are to each other. Identity of Indiscernibles shudders at the symmetry of the situation, however. It holds that the so-called two outer spheres must really be only a single sphere. And this single sphere, which now has all the same qualities as its sole surviving partner, must really be identical to it. There is actually just one sphere! And next, the two halves of that sphere, being once again the same in their qualities, must likewise be numerically identical so that we have only a hemisphere, which in turn becomes a quarter-sphere, and so on, until all we are left with is an infinitely thin splinter, a line, which must in the end shrink to a point. Does that make sense?” The situation described by Leslie is paradoxical, because the application of the classical version of the identity of indiscernibles leads to the conclusion that the three original spheres ultimately constitute a single and unique object.
11 In the non-ubiquitous n-universe ΩT_{0}<α=L.
12 In the ubiquitous n-universe ΩT_{0}–o<L.
13 In this context, the two alternative models are as follows. In the first (ubiquitous) modeling, we consider a red ball that exists both in time T_{1} and T_{2}, at both spatial positions L_{1} and L_{2}. The corresponding n-universe is a ΩL>o<T>o<C, a n-universe comprising a single object (o), a color variable (C), a time variable (T) and a location variable (L). Here, the single object exists at two consecutive time positions, so that this object is in a one-many relationship with the time-variable criterion (o<T). Similarly, this unique object simultaneously occupies two distinct spatial positions. Thus, such an object is also in a one-many relationship with the spatial criterion-variable (L>o). Finally, the single object is also in a one-many relationship with the color criterion-variable (o<C), since it can change color. On the other hand, in the second (non-ubiquitous) modeling, there are two red balls that occupy the spatial positions L_{1} and L_{2} respectively, at both time T_{1} and T_{2}. The situation thus takes place in a ΩL=α<T>α<C<α, a n-universe comprising multiple objects (α_{1} and α_{2}), a color variable (C), a time variable (T) and a location variable (L). In such a n-universe, objects exist at two consecutive time positions, so that the objects are in a one-many relationship with the time-variable criterion (α<T). On the other hand, these same objects can only occupy one spatial position at a given time; thus, the objects are also in a one-one relationship with the spatial criterion-variable (L=α). In addition, the objects are also in one-many relationship with the color criterion-variable (α<C), since they can change color. Finally, the objects are in a many-one relationship with the color variable (C< α), since several objects can have the same color.
14 If we generalize this model to several locations in the space where ubiquitous objects evolve, A, A’, A’, A”, A”’, … it follows that the distance AA’ A” is also zero, etc. From the viewpoint of our universe, it is a triangle, but from a ubiquitous point of view, it is a point.
15 The present modelling in the n-universes of the entanglement of two photons leads to an interpretation of quantum entanglement, which could be testable. In this context, tests on the dispositional properties of entangled photons could be implemented and provide the elements to validate or invalidate the hypothesis associated with it.
16 I thank the participants of the Congress Espace, temps et dimensions supplémentaires held in Cargese on 3/2/2005 for useful comments.