Материал: искусственный интеллект

impact to appear in the central vertical sector of the screen (the central fringe) is one-third.

The double-slit experiment does not allow to determine if an electron that leaves a trace of impact on the detection screen has passed through the left slit or the right slit. This means that the properties Òpassing through the left slitÓ and Òpassing through the right slitÓ remain potential properties during the experiment, i.e., alternatives that are not resolved and therefore (as we are going to see) can give rise to interference effects [24]. Let however write PAB as the sum of two projectors: PAB = PA + PB , where PA corresponds to the property of Òpassing through the left slitÓ and PB to the property of Òpassing through the right slit.Ó Note that there is no unique way to deÞne these properties, and the associated projections, as is clear that electrons are not corpuscles moving along spatial trajectories. A possibility here is to further partition the half-space deÞned by PAB into two sub-half-spaces, one incorporating the left slit, deÞned by PA and the other one incorporating the right slit, deÞned by PB , so that PAPB = PB PA = 0. For symmetry reasons, we can assume that the electron has no preferences regarding passing through the left or right slits (this will be the case if the source is placed symmetrically with respect to the two slits),

According to the above deÞnitions, |ψA and |ψB can be interpreted as the states describing an electron passing through the left and right slit, respectively.6 In other words, in accordance with the quantum mechanical superposition principle, we have expressed the electron state in the double-slit situation as a (uniform) superposition of one-slit states. Inserting (5) in (4), now omitting the argument in the brackets to simplify the notation, we thus obtain:

μAB = ψAB |M|ψAB = 1 ( ψA| + ψB |)M(|ψA + |ψB )

2

=1 ( ψA|M|ψA + ψB |M|ψB + ψA|M|ψB + ψB |M|ψA )

2

IntAB

6Note however that, as we mentioned already, it is not possible to unambiguously deÞne the two projection operators PA and PB , for instance, because of the well-known phenomenon of the spreading of the wave-packet. In other words, there are different ways to decompose |ψAB as the superposition of two states that can be conventionally associated with the one-slit situations, as per (5).

when there are indistinguishable alternatives in an experiment, as is the case here, since we can only observe the traces of the impact in the detection screen, without being able to tell through which slit the electrons have passed, states are typically expressed as a superposition of the states describing these alternatives, and because of that a deviation from the classical probabilistic average (1) will be observed, explaining in particular why an interference-like fringe-like pattern can form.7

3 Interrogative Processes

We now want to provide a cognitivistic/conceptualistic interpretation of the doubleslit experiment, describing it as an interrogative process [11, 14]. It is of course well understood that measurements in physicsÕ laboratories are like interrogations. Indeed, when we want to measure a physical observable on a given physical entity, we can always say that we have a question in mind, that is: ÒWhat is the value of such physical observable for the entity?Ó By performing the corresponding measurement, we then obtain an answer to the question. More precisely, the outcome of the measurement becomes an input for our human mind, which attaches to it a speciÞc meaning, and it is only when such mental process has been completed that we can say to have obtained an answer to the question that motivated the measurement. In other words, there is a cognitive process, performed by our human mind, and there is a physical process, which provides an input for it.

All this is clear, however, we want to push things further and consider that a measurement can also be described, per se, as an interrogative process, independently of a human mind possibly taking knowledge of its outcome. In other words, we also consider the physical apparatus as a cognitive entity, which answers a question each time it interacts with a physical entity subjected to a measurement, here viewed as a conceptual entity carrying some kind of meaning. This means that two cognitive processes are typically involved in a measurement, one at the level of the apparatus, and another one at the level of the mind of the scientist interacting with it. The latter is founded on human meaning, but not the former, which is the reason why we have to make as humans a considerable effort to understand what is going on. In that respect, we can say that the construction of the theoretical and conceptual ediÞce of quantum mechanics has been precisely our effort in the attempt to understand the non-human meaning that is exchanged in physical processes, for instance, when an electron interacts with a detection screen in a double-slit experiment.

We will not enter here into the details of this conceptuality interpretation of quantum mechanics, and simply refer to the review article [14] and to the references

7Of course, to characterize in detail such pattern one should explicitly solve the Schršdinger equation, which however would go beyond the scope of the present text.

cited therein; this not only for understanding the genesis of this interpretation, but also for appreciating why it possibly provides a deep insight into the nature of our physical word. In the following, we limit ourselves to describing the double-slit experiment in a cognitivistic way, as this will be useful when we transpose the approach to an IR-like ambit. So, we start from the hypothesis that the electrons emitted by the electron gun are Ômeaning entitiesÕ, i.e., entities behaving in a way that is similar to how human concepts behave. And we also consider the detection screen to be a Ôcognitive entityÕ, i.e., an entity sensitive to the meaning carried by the electrons and able to answer questions by means of a written (pointillistic) language of traces of impact on its surface. We are then challenged as humans to understand the meaning of this language, and more precisely to guess the query that is answered each time, and then see if the collection of obtained answers is consistent with the logic of such query.

There are of course different equivalent ways to formulate the question answered by the screen detectorÕs mind. A possible formulation of it is the following: ÒWhat is a good example of a trace of impact left by an electron passing through the left slit or the right slit?Ó This way of conceptualizing the question is of course very Òhuman,Ó being based on the prejudice that the electron would be an entity always having spatial properties, which is not the case (this depends on its state). But we can here understand the Òpassing throughÓ concept as a way to express the fact that the probability of detecting the electron by the Þnal screen is zero if both slits are closed. An alternative way of formulating the same question, avoiding the Òpassing throughÓ concept could be: ÒWhat is a good example of an effect produced by an electron interacting with the barrier having both the left and right slits open?Ó However, we will use in our reasoning the previous formulation of the question, as more intuitive for our spatially biased human minds. What we want is to explain the emergence of the fringe pattern by understanding the process operated by the detection screen, when viewed as a cognitive entity answering the above question.

The Þrst thing to observe is that such process will be generally indeterministic. Indeed, when we say Òpassing through a slit,Ó this is not sufÞcient to specify a unique trajectory in space for an electron (when assumed to be like a spatial corpuscle). This means that, if the screen cognitive entity thinks of the electron as a corpuscle, there are many ways in which it can pass through a slit, so, it will have to select one among several possibilities, which is the reason why, every time the question is asked, the answer (the trace of the impact on the screen) can be different (and cannot be predicted in advance), even though the state of the electron is always the same. The same unpredictability will manifest if the screen cognitive entity does not think of the electron as a spatial entity, but as a more abstract (non-spatial) conceptual entity, which can only acquire spatial properties by interacting with it. Indeed, also in this case the actualization of spatial properties will be akin to a symmetry breaking process, whose outcomes cannot be predicted in advance.

To understand how the cognitive process of the screen detector entity might work, let us Þrst concentrate on the central fringe, which is the one exhibiting the higher density of traces of impact and which is located exactly in between the two slits. It is there that the Òscreen mindÓ is most likely to manifest an answer. To understand

the reason of that, we observe that an impact in that region elicits a maximum doubt as regard the slit the electron would have taken to cross the barrier, or even that it would have necessarily passed through either the left or the right slit, in an exclusive manner. Thus, an impact in that region is a perfect exempliÞcation of the concept Òan electron passing through the left slit or the right slit.Ó Now, the two regions on the screen that are exactly opposite the two slits, they have instead a very low density of traces of impact, and again this can be understood by observing that an answer in the form of a trace of impact there would be a very bad exempliÞcation of the concept Òan electron passing through the left slit or the right slit,Ó as it would not make us doubt much about the slit taken by the electron. Moving from these two low-density regions, we will then be back in situations of doubt, although less perfect than that of the central fringe, so we will Þnd again a density of traces of impact, but this time less important, and then again regions of low density will appear, and so on, explaining in this way the alternating fringe pattern observed in experiments [11, 14].

4 Modeling the QWeb

Having analyzed the double-slit experiment, and its possible cognitivistic/conceptualistic interpretation, we are now ready to transpose its narrative to the modeling of the meaning entity associated with the Web, which we have called the QWeb. Our aim is to provide a rationale for capturing the full meaning content of a collection of documental entities, which in our case will be the webpages forming the Web, but of course all we are going to say also works for other corpora of documents. As we explained in Sect. 1, there is a universal line for going from abstract concepts to more concrete ones, which is the one going from concepts indicated by single words (or few words) to those that are complex combinations of large numbers of concepts, which in our spatiotemporal theater can manifest as full-ßedged stories, and which in our case we are going to associate to the different pages of the Web. Assuming they would have been numbered, we denote them Wi , i = 1, . . . , n. The meaning content of the Web has of course been created by us humans, and each time we interact with the webpages, for instance, when reading them, cognitive processes will be involved, which in turn can give rise to the creation of new webpages. However, we will not be interested here in the modeling of these human cognitive activities, as well as when we model an experiment conducted in a physicsÕ laboratory we are generally not interested in also modeling the cognitive activity of the involved scientists.

As mentioned in Sect. 1, we want to fully exploit the analogy between an IR process, viewed as an interrogation producing a webpage as an outcome, and a measurement, like the position measurement produced by the screen detector in a double-slit experiment, also viewed as being the result of an interrogative process. So, instead of the n cells Ci , i = 1, . . . , n, partitioning the surface of the detection screen, we now have the n webpages Wi , i = 1, . . . , n, partitioning

the Web canvas. What we now measure is not an electron, but the QWeb meaning entity, which similarly to an electron we assume can be in different states and can produce different possible outcomes when submitted to measurements. We will limit ourselves to measurements having the webpages Wi as their outcomes. More precisely, webpages Wi will play the same role as the cells Ci of the detection screen in the double-slit experiment, in the sense that we do not distinguish in our measurements the internal structure of a webpage, in the same way that we do not distinguish the locations of the impacts inside a single cell. So, similarly to what we did in Sect. 2, we can associate each webpage with a state |ei , i = 1, . . . , n, so that {|e1 , . . . , |en } will form a basis of the n-dimensional QWebÕs Hilbert state space.

Let us describe the kind of measurements we have in mind for the QWeb. We will call them Òtell a story measurements,Ó and they consist in having the QWeb, prepared in a given state, interacting with an entity sensitive to its meaning, having the n webpages stored in its memory, as stories, so that one of these WebÕs stories will be told at each run of these measurements, with a probability that depends on the QWebÕs state. The typical example of this is that of a search engine having the n webpages stored in its indexes, used to retrieve some meaningful information, with the QWeb initial state being an expression of the meaning contained in the retrieval query (here assuming that the search engine in question would be advanced enough to also use indeterministic processes, when delivering its outcomes).

If the state of the QWeb is |ei , associated with the webpage Wi , then the Ôtell a story measurementÕ will by deÞnition provide the latter as an outcome, with probability equal to one. But the states |ei , associated with the stories written in the webpages Wi , only correspond, as we said, to the more concrete states of the QWeb, according to the deÞnition of concreteness given in Sect. 1, and therefore only represent the tip of the iceberg of the QWebÕs state space, as it would be the case for the position states of an electron. Indeed, the QWebÕs states, in general, can be written as a superposition of the webpagesÕ basis states:

We can right away point out an important difference between (7) and what is usually done in IR approaches, like the so-called vector space models (VSM), where the states that are generally written as a superposition of basis states are those associated with the index terms used in queries (see, for instance, [30, p. 5], and [27, p. 19]). Here it is exactly the other way around: the dimension of the state space is determined by the number of available documents, associated with the outcome-states of the Ôtell a story measurementsÕ, interpreted as stories, i.e., as the more concrete states of the QWeb entity subjected to measurements. This also means that (as we will explain in the following) the states associated with single terms will not necessarily be mutually orthogonal, i.e., will not generally form a basis. Of course, another important difference with respect to traditional IR approaches is that the latter are built upon real vector spaces, whereas our quantum