Hierarchical Observations

Competition means poor competence.

As long as I do what I think is right, why should I care for anybody else's opinion? And as soon as I get engaged in a discussion, I am certain to partially accept the opponent's standpoint; otherwise, we would have no points of contact. Well, nothing criminal in a partial engagement. It is only taking it for serious that makes trouble.

It is especially dangerous to talk to mystically inclined former professionals who sacrificed their professionalism to a philosophy of a stinky kind. Trying to do philosophy without enough competence beyond science is as vicious as doing physics on an entirely philosophical grounds. The two areas can efficiently collaborate sometimes; this is no reason for an official marriage.

Scientific theories (provided they describe at least something) do not oppose each other; rather, they refer to the complementary aspects of the same reality. Some techniques are useful here, somebody else's approach work elsewhere. Why not admit that both contain a grain of truth? Truth is many-faceted, and nobody can be true in any respect at all.

The foundations of science are outside science. A scientist does not need to justify the choice of a particular model. All we are expected to do is indicate the observable effect and suggest some ways to check. When things start behaving in a predictable manner, just toss the tool to engineers and applied scientists, and proceed with another model, for a different case. Fundamental theories are never invented just for fun; normally, they grow from a difficult problem that cannot be solved with the existing conceptual luggage and hence needs an advanced formalism. The technical details would promptly get in place, and let philosophers habituate the public to the basic facts.

There is, however, a minor issue of reflexivity. In a way, scientists are too a part of the public, and they like being given a treat. Moreover, (high) school teachers need a number of mental hooks for the students to overcome the first shock and move on to the technical knowledge without too much pondering on methodology. Philosophy is too circumstantial and time consuming, while practical needs cannot wait. So, let us take the first affordable explanation and stop considering any alternatives. This is how quantum mechanics got stuck to the Copenhagen interpretation, however absurd and incoherent. The thing's done, we can proceed with science, as our how's do not depend on why's.

Nature reveals itself to scientists through all kinds of invariants. In a sense, a fundamental theory is an invariant of many philosophies: no revision of the foundations is acceptable unless it conforms with the already established standard. So looser the grounds for philosophical debate, unless we are going to (indirectly) support certain political movements that have nothing to do with science.

Among others, Einstein did not much praise the philosophical concoction by Bohr and Co. Wise enough, he did not try to beat Bohr on their quantum field; he only indicated that the present recipes must be either nonlocal or incomplete (how do you do, Mr. Gödel?). Any attempt to go further in this shaky area would be a concession, and Einstein wanted no trade-off.

Much later, Hugh Everett III approached the foundations of quantum mechanics from a different angle, which automatically put him off any academic career (though, possibly, he did not ever consider becoming a physics guru). Did it really change anything? Arguing against Bohr, Everett took exactly the same position, and the difference between the two interpretations is mainly terminological. Thus, both considered measurement as a kind of interaction between the quantum system and the classical observer; however, for Bohr, the observer was a mystical force that makes a superposition state "collapse" into one of the observable outcomes, while Everett pretended to describe the observer and the object as parts of the same world, with every act of measurement splitting the world into several clones, with each cloned observer registering a specific quantum state, and this lead to a mystical picture of a cloud of non-interacting worlds. Later, for this separation, the followers of Everett coined the term "decoherence", which means exactly the same as Copenhagen-style "collapse". All the other seekers retain the same core discrepancy between the quantum and classical levels, and one can read in a standard theoretical physics course (Landau):

[...] for a system entirely composed of quantum objects, it would not be possible to construct any logically consistent mechanics. [...] This [...] is logically related to the fact that the dynamical characteristics of an electron can only come as a result of measurement [...]

From this stand, one can never guess how a classical system differs from a quantum system. There are speculations on the "correspondence principle"; still, they only demonstrate that classical behavior can sometimes be obtained from quantum equations as a limit case, but the possibility and nature of "mixed" interactions (between the quantum and classical levels) remains sheer metaphor.

By the way, why should we take the correspondence principle as a one-way road? Yes, quantum systems may reach a classical limit (to comply with Everett's conjectures); but classical systems, too, may develop quantum behavior under certain conditions (as Einstein believed). The industry of quantum computing seem to perfectly support such an extension. Similarly, the relations between Newtonian and relativistic mechanics are far from mere low-speed (or low-energy) reduction.

Returning to the Bohr-Everett controversy, one is somewhat surprised to encounter all those "collapses" and "decoherence" in an apparently linear theory. We perfectly know that any furcation is due to inherent nonlinearity. Any dynamics will be quite smooth until we explicitly introduce nonlinear terms in the equations of motion, initial and boundary conditions, or as an outer constraint. A closer examination reveals the machinery of magic: the very distinction of subsystems within the whole leads to a significant modification of the original theory. This is a very general statement, but here, a simple quantum-mechanical illustration is enough.

A system W with the Hamiltonian H evolves in compliance with the standard equation of motion:

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with the overall state vector being an integrity comprising anything at all. All such vectors constitute the configuration space of this all-embracing system. So far, everything is linear and smooth (assuming a regular Hamiltonian). Now, let the system W contain a subsystem P; the rest of the system will then be treated as its complement Q. Formally, this can be represented by splitting the whole configuration state into a direct (Cartesian) product of subspaces pertaining to the subsystems P and Q respectively:

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where the projection operators P and Q are defined as follows:

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Any projection is an essentially singular operation (similar to the well known Heaviside step function). Our theory would still remain linear, if the subsystems had no relation to each other. Unfortunately, this not always so; moreover, most often, this is not what we really want! For interacting subsystems, the complete Hamiltonian takes a block-matrix form:

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The original regular Hamiltonian is thus expanded into a sum of essentially nonlinear terms, and one can expect almost anything! Just split the world W into the observer P and a quantum system Q, and get all kinds of dynamical peculiarities, from topological intricacies to violent furcation. The other time you might wish to consider a different splitting, and you'll get an alternative picture of the same one and only world. There is no need to collapse or clone: all the possibilities are already in there, as the complementary aspects of the whole.

Initially unstructured, our universe has unfolded into a two-level hierarchical system: an integrity on the higher level, interacting subsystems in the background. In the same manner, any subsystem can develop into a hierarchy, so that both the observer and the object are hierarchical. Suppose that the object (a classical or quantum subsystem Q) is unfolded in a number of sub-objects, and some of them may refer to higher-level dynamics, up to the dynamic of the observer. Then a trivial tree-like structure will twist into an intricate topology, and lower-level contributions will no longer be just minor corrections. Modern physics has developed extensive regularization techniques to cope with this problem.

Once again: it does not really matter, which language we employ to explicate this two-level scheme (the level of integrity + the level of interacting subsystems). The conclusion will be the same. Normally, we engage much more components, with the resulting sequences of subspaces being discrete, continuous, or even higher-cardinality "vectors". No wonder that almost any desirable behavior can be reconciled with the same fundamental theory (or, rather, a paradigm). When people badly want something, they'll find the way to get it.

Just to hint to a possible branch of discourse, consider the highly-degenerate limit H_P = 0. Isn't it much like a classical observer that can influence a quantum system's behavior, with no back effect on the observer's structure and dynamics? The dimensionality of such an observer's configuration space will determine the possible outcomes of any measurement, so that the observer's interaction with the quantum system is to result in some statistical distribution. This limit could hence be called space-like observer, as thus we get the idea of the overall organization of nature. Yet another illustrative example is provided by the limit H_P ~ ħω ≠ 0, which (according to the equation of motion) makes the observer a kind of clock, the frequency ω determining the "time pace" for the system observed (the scale of consideration). Finally, the doubly degenerate limit H_P = 0 and H_Q = 0 is a good model of purely classical measurement.

In this view, the incoherent Bohr-Everett debate collapses to nothing. There are different ways to introduce nonlinearity in the scheme of measurement, and each solution is good for something, provided we do not forget that no differences can be definable except within the same, and any specific hierarchical structure assumes other structures to unfold where appropriate. Neither of such "biased" representations can exhaust the diversity of the real world, which may occasionally manifest itself as a quite different system of layers, with some peculiar definition of the observer. Since the distinction between the object and the observer is relative (so that the object can as well be said to observe the observer), one could fancy entirely observerless hierarchies, the world as it is, which serves as an objective background for any cultural reality. It is important that every time we find something in the world, much more is yet left to find. We cannot expect nature to always follow out caprices.

To be more specific, let us recall the universal prejudice that quantum measurement reveals one of the possible states of the object. Not at all! All we can affirm is that, after the act of measurement, the observer will be found in one of the possible observer's states (for instance, the positions of the pointer on a dial, or the sets of spectral intensities). In our quantum example, we refer to the eigenstates of the observer Hamiltonian H_P. Nothing can be said about the resulting state of the object (as described by the residual Hamiltonian H_Q). Moreover, if some classically observable effect required a definite state of a quantum system (which is a very strong demand, indeed), the system would be certain to leave that state in the course of measurement, and hence what we measure can never refer to how the object currently goes, but rather to the way we employ the outcome, the trace of our interaction with the object.

Popular writers readily draw pictures like

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meaning that, when (say) an electron in a superposition state impacts the observer in some neutral "vigilant" state φ, we get a superposition of two possible outcomes, each referring to a "correct" measurement, the perfect correspondence of the experimental value to the real nature. Then they either invoke "collapse" of the superposition into one of the possibilities, or admit a parallel evolution of the non-intersecting "worlds" after the rite of decoherence... The logical version of the same would rather picture the final state as ; that is, a direct product of a superposition state of the observer (obviously identifiable with the experimentally observed spectrum) and an undefined state of the quantum system (which may occasionally run into another observer and show up as anything at all). All the "entanglement" talk is then sheer advertising. We cannot keep a quantum system in a definite state after measurement (or any other creative act). Even worse, any interaction with a classical system cannot lead to pure (superposition) states: it will always result in a mixed state (for instance, as described with a density matrix). The system's "preparation" to measurement is an indispensable part of any experiment. This is how we suppress all interactions outside the scope of interest (thus unfolding the object's hierarchy into a specific configuration space). For repetitive observations of the same kind, we need to physically bring the system back to the incoming configuration, "preparing" it again and again.

Speaking of the structure of the object's configuration space, we already assume an unfolded hierarchy far beyond the primary separation of the observer and the object. That is, to build a meaningful physical theory, we need explore the structure of the object, splitting it into all kinds of interacting components and assuming that only a few of them participate in the intended measurement. Some parts may be classical; some other parts will demonstrate quantum behavior. In any case the number of projectors involved can be huge; therefore, any linearized theory is nothing but a practically acceptable approximation.

This, once again, puts us in the context of philosophical controversy. Under-philosophers with a physical background appeal to "reality" to substantiate their speculative constructs. Philosophizing Everett did not see that his objections to Bohr merely revived the struggle of medieval "realists" against as medieval "nominalists". The both parties are wrong, since, primarily, they reduce all the human activity to cognition, and then admit that our knowledge is comprehensive and perfect, identical to the world. In any case, representing the integrity of the world with the equation P + Q = 1, we can only focus on the observer part P, as we cannot know for sure the inner organization of the object of study. To account for the possible alternatives, we approach the object from many angles, measuring all kinds of complementary characteristics:

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With each choice, something (and, in fact, the major part) is left beyond science. Now look at the naive Everett's statement that it is only the complete wave function of the world that deserves the name of physical reality. Leaving aside the extremely narrow vision of the world understood as only one of the possible representations of the observer + object type, one could note that wave functions are no physical entities at all: they entirely belong to the observer as handy shortcuts for a specific attitude to the world. This is the way we see the world, and in no way the world as it is. Eventually, the same world has many other aspects that must be treated differently, possibly with no recourse to the quantum paradigm. In other words, our knowledge of any individual physical system is essentially incomplete, nothing to say about the knowledge of whole world. In physics, we are interested with a tiny portion of the possibilities pertaining to our practical needs; in the context of a different activity, we may describe the same with yet another physics, as incomplete, and as effective just because of that incompleteness.

In this way, we respond to one of Einstein's criticisms. Yes, quantum mechanics is indeed incomplete, just like any other physical theory is (including special and general relativity). The illusion of classical completeness is due to the spatial treatment of the observer as a non-changing background in which all the physical systems are embedded. Since classically moving objects exhibit the same structural invariance, the correspondence between the dynamical variables of the object and the quantities observed is basically a straightforward translation. On the contrary, quantum objects do not exhibit their inner motion to the observer, and no quantum theory can be complete in the classical sense.

Are there macroscopic systems that cannot be described within the classical paradigm? Lots. Any classical system at all will show an analog of quantum behavior in the presence of interactions that cannot be exactly specified or controlled. Nonlinear dynamics pictures that as chaos; in hierarchical systems, lower-level interactions will influence higher-level dynamics in a quantum-like manner. Consider the well-known example of Brownian motion, with the average distance from the origin increasing as the square root of (macroscopic) time. This means that a combination of several independent processes of that type must be described by the superposition of the corresponding virtual amplitudes, while the observable intensities are to be obtained as squared amplitudes, according to the quantum rule.

This is a right moment to touch the issue of probabilistic methods in quantum physics, with their numerous mystical explanations. In fact, we can never observe any probabilities. Statistics pertains to our ways of data processing, and never to physical systems of any kind. There are physical events. If we do not discern individual events, we measure some aggregate features (intensities). The only difference between the classical and quantum pictures is in the manner of aggregation. With no interference between the individual inner events, the sum of intensities will represent the total intensity. When some interactions have to be considered as partially intersecting in time, quantum interference takes place. The propagation of a quantum particle is then pictured as a kind of flow, with the corresponding variation of intensities. Depending on detector thresholds (yet another nonlinearity!), the observer will either get a sequence of "random" pulsations, or a full-fledged spectrum. Virtually, the whole bulk of quantum physics (as well as classical statistical physics and thermodynamics) could be rewritten in a probability-free language; we do not need this notion in physics. While a working physicist is quite comfortable with the probabilistic slang devoid of any subjective connotations, politically engaged philosophers substitute physical notions for all kinds of spiritualistic ideas to justify social discrepancies with pseudoscientific blather.

It is exactly the introduction of unphysical notions that allows philosophical prestidigitators screw physics into whatever they like. It does not matter whether we adopt classical or quantum paradigm. Popular books never miss the opportunity to mention the mental experiment with electron spin states and their superposition, with the resulting "entanglement". Let us stage the same trick in a purely classical way. Suppose that we write characters 0 and 1 on two sheets of paper and put them in sealed envelopes, which are then delivered to the opposite points of the globe (or the Galaxy). For observers located in the destination points, very far from each other, the probability of finding 0 or 1 after the seal is broken equals 1/2. However, when one of the observers breaks the seal and sees the label 0, the far-away observer is certain (with 100% probability) to have the label 1; in the slang of the official quantum philosophy, the wave function has collapsed in no time, despite the relativistic restriction on the speed of signal propagation. We do not need any microscopic particles, or quantum fields; mere paper and pencil are enough.

The logical fallacy involved is that there are no probabilities at all, and there is only one (global) observer who can detect the quite deterministic state of the two-envelope system in several ways (specifically, opening either one or another envelope; there are also other, more sophisticated techniques). If the global observer could observe the process of putting the sheets of paper in the envelopes, there would be no need for any measurement; assuming that this part is beyond the observable (or computable) scope, we switch on to an incomplete theory with hidden variables of unknown nature. However, since the possible outcomes of measurement are subject to a global constraint (the assumption that only two options are allowed), it does not matter which exactly part of the system is observed; the rest is derived from the deterministic character of the constraint. This explains the illusion of nonlocality. In fact, the both envelopes are never independent, they are components of the same system located in a single point (the position of the global observer). The volume of the system may expand, but it does not move as a whole, and there is no need to send any signals. Considering two spatially separated observers is either a metaphorical reference to the distributed measurement instrumentation, or an admission of additional data available on each side: namely, the knowledge of the form of the existing constraint. In the latter case we get a local theory with hidden variables; otherwise, there would be no means of relating one envelope to another, and even no idea about the presence of anything at all except the local experimental setup.

Such global frameworks exist in any human activity at all. We cannot operate but within the current cultural environment, which often would impose severe restrictions as to what is practically permitted. In particular, no information transfer could be possible without a commonly accepted protocol. The common idea of probability as a summary of our expectations is thus perfectly grounded. And that is why the same data can be encoded in many ways and transmitted using quite different physical media. Without the cultural preliminaries, we would only contemplate a physical process developing on itself, regardless of any interpretations. In the quantum language, to refer to this contextual dependence, we would speak of correlation, coherence, system preparation; a classical description would refer to hidden variables. This is essentially the same.

Hierarchical approach explicates the relations between the global observer and the inner space of the system to observe (the object). However huge, all the inner space is contained within a single point for the (classical or quantum) observer; moreover, the very notion and structure of the inner space is only explicable in terms of the observer, who has to decide on the mode of observation (that is, the expectable results) before any experimenting. Note that the constraints thus produced apply to the inner motion of the object rather than to the observer; to describe observer involvement, we would need a higher level of hierarchy, and a "bigger" observer. In our scheme of classical "entanglement", we could introduced an intermediate level of spatially distributed (classical) observers with the light barrier constraint. Since the outcome of the experiment is only determined in respect to the most global observer (presumably sitting in the same origin point), the lower-level measurements would have to be conveyed to that decision maker for aggregation, and the overall procedure would take just enough time to comply with the classical idea of locality. For the global observer, of course, everything would still happen within a single moment.

As mentioned above, the inter-level relations are not trivial: an additional (mediatory) level can be introduced between any other levels, and several level can be aggregated into a grosser structure. This is known as hierarchical conversion. The choice of a frame of reference, or the specification of a quantum system, can serve as two typical examples. The integrity of the hierarchy is restored on a higher level, which implies certain rules for transition from one unfolding to another.

So far classical and quantum systems did no significantly differ from each other. Now, it is high time to dwell upon the principal source of quantum nonlocality, the indistinguishability of the quantum objects. In our everyday life, we are quite accustomed to practical equivalence: if there are several ways of getting somewhere, taking any of them will do. In the quantum world, it seems like, selecting one of the possibilities, we necessarily invoke all the others. We can never tell, which individual electron has been deflected up or down after passing through the Stern-Gerlach magnet: all the electrons of the incident beam are partially present in both the upper and the lower deflected beams. Formally, this interchangeability is introduced in the equations of motion using additional (anti)symmetrizing operators, which makes the system explicitly nonlinear. The principle does not depend on the nature (and even presence) of real physical interaction; the exchange terms are present in any case, determining the topology of the configuration space. For the same reasons, no spatial separation can eliminate exchange effects; formally, it looks like a kind of infinitely fast interaction which is not subject to any relativistic restrictions and hence is essentially nonlocal. This is a real mystery, but we cannot avoid it, especially in high-energy physics.

The very independence of exchange from the physical background suggests the thought of an artificial character of quantum exchange: it does not pertain to the object, but rather to the organization of the observer (including both experimental setup and the procedures of result aggregation). If so, classical systems, too, could be treated the same way. Let us illustrate it with a real-life example.

Suppose we are selling oil to a number of customers who load their tankers and pay for the quantity indicated in the bill of lading. Oil is delivered to the terminals from a number of production centers with some natural logistic delays. If the tanker were to be loaded directly from the terminals, that would mean a lot of anchorage time and huge demurrage charges. So, we first accumulate oil on a hub tanker for further transshipment to the destination tanker, in bulk. Now, as we expect several customers within a certain period, we load the hub for all of them, indicating the specific quantities accumulated for each. In the quantum language, the hub is in a superposition state, with the coefficients related to the contract volumes. When a customer tanker comes, we load it with the requested volume of oil, which will drive the hub in a different superposition state ready for the next operation.

The real magic begins when we consider that one portion of oil is no different from another: Whatever we give to any customer contains a mixture of volumes originally intended for many others, and what is left after transshipment contain a contribution from the supply of already serviced tanker as well! We can calculate the exact proportions, and they will dynamically change in the course of load/unload operations. Sometimes the hub gets empty; sometimes its load goes to a single customer; but in most cases, the contributions intended for different customers will be heavily mixed, hence establishing a kind of kinship for a half of the world! Taking in account that each portion of oil is delivered by an individual price (as fixed well beforehand), we come to the necessity of reevaluating the market value of the oil supplied, and the interplay of this "quantum" prices with the documented sale price is a major source of profit for traders, just like quantum interference fires lots of domestic or industrial devices.

Of course, there is nothing special in oil trade. One could as well consider preparing for shopping and collecting money for bread, butter, cheese, and apples, on the basis of preliminary price estimates. For some reasons, you may decide to buy only bread and apples, with the expenditures proportionally redistributed. Or, some other day, you might write a paper for a journal, using the fragments intended for other publications. And so on.

The message is clear: we need exchange corrections because we describe the state of the object in terms of observer states. Quantum electrons know about our calculation schemes no more than oil knows about the final consumer. From the very beginning, they do not exist separately; there is a compound object of some peculiar nature, which does no need to comply with our limited notions. As we artificially separate the parts where they are physically inseparable, this inseparability is to be accounted for in terms of explicit (anti)symmetrizing. If some genius will suggest a new formalism for the theory of closely coupled systems, the spurious interactions will disappear in no time. For instance, instead of considering electrons (or other elementary particles), one could speak about quantum currents with different characteristics (just like we speak about electron and ion plasma in the interstellar gas). As soon as individual electrons are eliminated from theory, there is no place for indistinguishability considerations and nonlocal exchange. As indicated, most physical peculiarities are due to some nonlinear effects. However, we are to distinguish two complementary source of nonlinearity in any physical theory: the inner (objective) complexity and the (subjective) mode of interpretation. Practical experience unfolds according to both the objective aspect of the world and our habits or intentions. The emergence of a new paradigm is a result of cultural development in general.

Is that any different from the popular philosophy of emergentism? Some well-ordered structures appear in nonlinear dynamics as a natural consequence of physical laws; other structures seem to be introduced in an arbitrary manner, just like the above observer-like description of the object. As long as we keep away from the mystical emergence from nothing, the already known examples provide a sensible basis for improving physical theories (as a part of self-improvement in general). The key feature of the hierarchical approach is the relativity of any distinctions and their relatedness to a specific hierarchical structure, an instance unfolding of the same hierarchy. That is, we perceive things as they are for us within the culture, depending on everybody's social position. In particular, the limitations of our theories are not mere illusions: they reflect a certain stage of development, and primarily, the overall technological level. Scientific discoveries come in favorable conditions, changing the world for us.

To conclude, what happens to the battle of interpretations? One many-faceted world instead of the chaos of randomly proliferating worlds. Coexistence of different implementations instead of arbitrary collapse. Natural description instead of observer-imposed artefacts. The relativity of quantum-classical duality and its reproducibility on any level of the subject/object hierarchy. A hierarchical vision of relativity. And, of course, the universal complementarity of viewpoints, aimed at increasing everybody's competence, instead of dull and unrewarding competition.