“We wish to make statements about ´trajectories´ of observers.
However, for us a trajectory is constantly branching (transforming from state
to superposition) with each successive measurement”
Hugh Everett III, 1957
The Far Reaching Consequences of a PhD Dissertation
When treating the revolutionary ideas of Relativity, we saw the Relativity Principle implying infinite 3-D spaces associated to each instant of time, which in turn implies that the world is at least 4-dimensional. This explained why Michelson and Morley's experiments failed to determine absolute movements with respect to an Ether, supposed the spatial absolutely steady reference system. This idea about the existence of an infinity of 3-dimensional spaces associated to a single instant of time we have to keep present when thinking to the organization of whatever into systems and subsystems. Also, the classic Principle of Superposition treated elsewhere in this website, is still presently being presented in the elementary textbooks of Physics and Electrotechnics in the high-schools and in the University courses for future Electronics and Telecommunications' Engineers. What only a few, mainly Physicists and Mathematicians, know is that the Classic Principle underwent a radical revision after 1957.
What changed the course of history was a decision, took during a meeting (circa 1955), who saw the participation of some of the top American physicists. The key point, yesterday as today, the necessity to bridge General Relativity and Quantum Mechanics theories. The dream of the participants to the meeting, to reach a recipe to quantize gravitation, thus having a unique theory encompassing all physical phenomena, whatever their scale, from the subatomic to the cosmological. This happened when the majority of the American physicists were yet doubting about some ideas, of the named Copenhagen’s Interpretation of Quantum Mechanics. A graduate student particularly brilliant in Mathematics at Princeton University, Hugh Everett III, received by his mentor prof. John Archibald Wheeler, this task of the maximum level of difficulty. It was passed to a graduate student a research activity that no one of the late professors was capable to accomplish…
Hugh Everett, III, physicist and mathematician
Albert Einstein, Hideki Yukawa and John Archibald Wheeler at the Princeton’s Institute for Advanced Studies, 1953. Three years later, Wheeler passed to Everett the task to find a bridge between General Relativity and Quantum Mechanics, allowing their unification (H. Schrader, courtesy Princeton University)
And that youngster found, deeply buried in the successful original formulation of the Quantum Mechanics, much more. He encountered a wave function for the entire Universe, including new:
- Principle of Superposition;
- Theory of Measurement.
Everett attacked the problem considering the entire Universe as Superposition of all the superpositions (or, grand total of all the possible sums), thus discovering the wave function of the Universe.
Niels Bohr in 1960. Richard Feynman can be recognised first at right side in the photo.
Everett was the first to fully recognise the centrality of the Principle of Superposition for Physics and its Technological applications. This work was kept in the Universities' files intentionally away from the limelight. This mainly because it efficiently countered Quantum Mechanics' standard interpretation, then centred around Niels Bohr’s Institute of Advanced Studies at Copenhagen, Danmark. Dissertation kept hidden until the 1970s [Note of the Author: in 1986 the Author of these notes had to ask availability of a copy of the book to the twelve main Universities of his own country. Later encountering just one, the Physics’ Department of Padua University at Padua, Italy, having a copy], when it started to be dug up in a desperate attempt to reconcile General Relativity and Quantum Mechanics. Following Everett, we have just to interpret what Quantum Mechanics really tells them, without trying to change its rules to suit their own prejudices.
To understand how many ideas exist in the word Superposition, we suggest to observe a liquid wave motion. Liquids’ wave motions, e.g. sea waves, are a particularly complex application of the Principle of Superposition. Not an exaggeration, when considering that each one frame of the video in its native high definition took 8 hours of computation time. Computational complexity is a keyword of the Binary Classifiers. Also the Machine Vision systems inside the automated analytic equipment show similar levels of computational complexity (video authored by Jamie German, 2014)
The first step lies in the admission of all of the implications of the Principle of Superposition. Principle of Superposition implying existence of macroscopic superpositions as a reality, and not just useful temporary mathematical fictions. All liquid waves, as titanic as the Oceans' waves used by surfers (see video below) or as small as those in the sink of your home, are some of the examples of the reality of the macroscopic superpositions. What follows is referred to all of the scales, and not to the subatomic. Quantum Mechanics describes an ensemble of many coexisting branches of the Multiverse, branches whose difference is reduced to 1 bit. Therefore, the discovery of Quantum Mechanics one century ago started a scientific revolution where a colossal Multiverse replaced a yet huge Universe.
A Branched Reality
Events and Eventualities
In the following we’ll adopt Dirac’s bra-ket notation of 1939, applied to John von Neumann's QM perspective. It is a common habit to refer to Events, when meaning happened facts. A broader idea is that of eventuality, where the Event may or may not happen. An example of eventualities below in a common roulette, a “Montecarlo randomizer-device” well emulating nearly all kinds of human and machine measurements. 38 numbered slots, but also the possibility that the ball disappears and other baffling cases.
Interestingly, all of the possible position states of the ball spinning around the roulette wheel, superimpose to produce an interference pattern. Before to observe (or, measure) the outcome, we have to consider the roulette system as being in a superposition of all its possible states. What players intuitively know be true, when waiting for the stabilisation in a single value of the precedently superimposed (summed) outcomes. The grand total of all the eventualities associated to a physical system, is the system's state space.
Observables
Another fundamental idea of Physics is that of observable. [ In the following, the symbol “ ⊕ ” identifies a vectorial sum. When referred to bi-dimensional or, higher dimensional vectors, it identifies tensorial sums ]. The notion of observablesis used to describe a set {e} of eventualities, subject to the conditions:
- mutual exclusivity. A pair of eventualities e1 and e2 are mutually exclusive if they satisfy the condition of incompatibility e1 ∩ e2 = ∅, this last a necessary condition for the Events’ observability;
- if e1 and e2 are the subspaces of the Hilbert space representing a couple of admissible eventualities, their set intersection e1 ∩ e2 will also be a Hilbert subspace. It'll represents the conjoint eventuality;
- the set union e1 ∪ e2 will in general not have the structure of a Hilbert subspace then not representing an eventuality;
- the eventualities' additive structure is an effect of the Hilbert space structure. The sum of eventualities e1 ⊕ e2 is the Hilbert subspace spanned by the separate Hilbert subspaces e1 and e2;
- the state |Ψ⟩ ∈ e1 ⊕ e2 if |Ψ⟩ is a Hilbert space vector whose form is:
|Ψ⟩ = |Ψ1⟩ + |Ψ2⟩
for a couple of Hilbert space vectors |Ψ1⟩ and |Ψ2⟩, named wave functions or state functions, satisfying the condition:
|Ψ1⟩ ∈ e1
|Ψ2⟩ ∈ e2
Orthogonality and Observables
Orthogonality is what characterizes an observable in the Quantum Theory. In the Quantum Theory, an observable for a system A consists of a complete set {e} of mutually orthogonal Hilbert subspaces:
e1 ⊕ e2 · · · eN-1 ⊕ eN
spanning the entire Hilbert space I{A}. As an example, such that their sum equals the complete set I whose probability P{I} = 1 and:
e1 ⊕ e2 · · · eN-1 ⊕ eN = I.
Measurements as Branches
The original idea of branching is not ascribed to Hugh Everett, III. Rather, it dates back to before John von Neumann, when it was assumed that the probability distribution would be provided by a pure state, specified by a unit Hilbert space vector, represented as a sum:
Ψ⟩ = Σi |Ψi⟩
where i = 1, 2, …, N - 1, N, of the eigenvectors |Ψi⟩ ∈ ei of the observable {e} being considered.
A trajectory is constantly branching, transforming from state to superposition of states, with each successive measurement. The Hilbert space represents a system before the measurement, when all of the states of the system exist in superposition. What visible is Everett's original idea, where Time has the direction of the branching. Also the opposite happens, where different branches join themselves. Modern meaning associated to the classic term interference
“Quantum Theory is predicting the probabilities of sequences of alternatives at a series of times”
The summation symbol above Σ referred to a vectorial or tensorial sum, an alias of superposition. We are touching this point to remark that the Principle of Superposition played a central role yet in the original Quantum Mechanics. It was considered necessary by its many developers, like John von Neumann, Max Planck, Niels Bohr, Werner Heisenberg, Wolfgang Pauli, Max Born, Erwin Schroedinger, Paul Langevin, etc., before Hugh Everett, III was born. Yet before Everett, the Copenhagen's viewpoint described the observation (or, measurement) process as having a first step consisting of splitting |Ψ⟩ into the set of alternative projections:
|Ψi⟩ = ei |Ψ⟩ [1]
onto the relevant eigenspaces, named branches.
Different Histories
Doing this Quantum Theory is predicting the probabilities of sequences of alternatives at a series of times. Then, histories. As an example, the sequence of ranges of centre-of-mass position of a container at a series of times, giving a coarse-grained description of its motion when transported by a Conveyor. Sequences of sets of alternatives at a series of times, specify a set of different histories of the model. An individual history in the set corresponds to a particular sequence of alternatives and is represented by the corresponding chain of projection operators. To prevent the disturbing implication of the many alternative histories experienced by a single object, the creators of Quantum Mechanics added a second step to the observation process, cleared by the figure below. Here, a projection of a quantum state-vector | ψ⟩ into a vector subspace S by a projector P(S ). Here shown the projection of | ψ⟩ onto a ray corresponding to | ψm⟩, with which it makes an angle θ. The probability for this transition to occur results the square-cosine of the angle θ between the subspace S and the vector | ψ⟩, thus illustrating von Neumann’s concept of probabilities evaluated by the measurement of angles.
![Projection of a quantum vector on a subspace. John von Neumann: Logic and Space of Information
Signature of John Von Neumann, who created much of the Quantum logic and terminology
Oscillation characteristic values (eigenvalues) are modes of vibration, each one oscillating its own frequency, directly visible in this image of a drumhead (image credit J. Kreso, 1998-2010)
Basic Quantum Terminology
Coherent, typically applied to a system, in modern Physics is a synonimous of superimposed. In this sense, superimposed (or, coherent) are the orthonormal eigenvectors constituting a base for the wave function Ψ, representing an object, however complex or massive may it be. Then, decoherent means reduced by a measurement to one definite value (eigenvalue of the eigenstate), the only one we perceive in the multitude of alternatives.
How was it possible for the classic idea of superposition to generate the non-classic kind of logic of observables, input currents, output currents, valence band and gaps, underlying all designs based on semiconductor junctions ? We’ll answer quoting at length from the the creator of the Quantum Logic. The Hungarian physicist and mathematician John von Neumann, delivered as an address September 2–9, 1954, where he expressed the spirit of his creation. A fruitful spirit, after considering that all the Computers and Smartphones used to read this text, and not only all of the Industrial Machinery and Equipments, exist and are as performant as we know they are, because abandoned the old classic Logic and embraced John von Neumann's Logic:
“If you take a classical mechanism of logics, and if you exclude all those traits of logics which are difficult and where all the deep questions of the foundations come in, so if you limit yourself to logics referred to a finite set, it is perfectly clear that logics in that range is equivalent to the theory of all sub-sets of that finite set, and that probability means that you have attributed weights to single points, that you can attribute a probability to each event, which means essentially that the logical treatment corresponds to set theory in that domain and that a probabilistic treatment corresponds to introducing measure. I am, of course, taking both things now in the completely trivialized finite case.
But it is quite possible to extend this to the usual infinite sets. And one also has this parallelism that logics corresponds to set theory and probability theory corresponds to measure theory and that given a system of logics, so given a system of sets, if all is right, you can introduce measures, you can introduce probability and you can always do it in very many different ways.
In the quantum mechanical machinery the situation is quite different. Namely instead of the sets use the linear sub-sets of a suitable space, say of a Hilbert space. The set theoretical situation of logics is replaced by the machinery of projective geometry, which in itself is quite simple.
However, all quantum mechanical probabilities are defined by inner products of vectors. Essentially if a state of a system is given by one vector, the transition probability in another state is the inner product of the two which is the square of the cosine of the angle between them. In other words, probability corresponds precisely to introducing the angles geometrically. Furthermore, there is only one way to introduce it. The more so because in the quantum mechanical machinery the negation of a statement, so the negation of a statement which is represented by a linear set of vectors, corresponds to the orthogonal complement of this linear space.
And therefore, as soon as you have introduced into the projective geometry the ordinary machinery of logics, you must have introduced the concept of orthogonality. This actually is rigorously true and any axiomatic elaboration of the subject bears it out. So in order to have logics you need in this set of projective geometry with a concept of orthogonality in it.
In order to have probability all you need is a concept of all angles, I mean angles other than 90º. Now it is perfectly quite true that in a geometry, as soon as you can define the right angle, you can define all angles. Another way to put it is that if you take the case of an orthogonal space, those mappings of this space on itself, which leave orthogonality intact, leave all angles intact, in other words, in those systems which can be used as models of the logical background for quantum theory, it is true that as soon as all the ordinary concepts of logics are fixed under some isomorphic transformation, all of probability theory is already fixed.
What I now say is not more profound than saying that the concept of a priori probability in quantum mechanics is uniquely given from the start. You can derive it by counting states and all the ambiguities which are attached to it in classical theories have disappeared. This means, however, that one has a formal mechanism, in which logics and probability theory arise simultaneously and are derived simultaneously. I think that it is quite important and will probably [shed] a great deal of new light on logics and probably alter the whole formal structure of logics considerably, if one succeeds in deriving this system from first principles, in other words from a suitable set of axioms.
All the existing axiomatisations of this system are unsatisfactory in this sense, that they bring in quite arbitrarily algebraical laws which are not clearly related to anything that one believes to be true or that one has observed in quantum theory to be true. So, while one has very satisfactorily formalistic foundations of projective geometry of some infinite generalizations of it, of generalizations of it including orthogonality, including angles, none of them are derived from intuitively plausible first principles in the manner in which axiomatisations in other areas are.”](../../../_Media/projection_of_a_quantum_vec_med-2.png)
Projection of a quantum state-vector | ψ⟩ into a vector subspace S by a projector P(S ). Here shown the projection of | ψ⟩ onto a ray corresponding to | ψm⟩, with which it makes an angle θ. The probability for this transition to occur is cos2 θ, thus illustrating von Neumann’s concept of probabilities evaluated by the measurement of angles. A von Neumann measurement corresponds to the set of possible such projections onto a complete orthogonal set rays of the Hilbert space being measured (abridged by Jaeger/2009)
The “disturbing implication” of the many actual histories experienced by a single object after an interaction (or, measurement) is deeply ingrained in the Principle of Superposition. Single object coexisting in each one of the histories. It is exactly because Quantum Mechanics makes since the start of the Principle of Superposition its corner stone, that the fathers of Quantum Mechanics encountered since the start those also the disturbing implications. Second step that the Theory itself, resumed in the superposition [1], does not encompass. A second step famously named collapse, whereby the set would be replaced by a single renormalized branch vector:
that would turn up with the corresponding conditional probability:
Pi = ⟨Ψi |Ψi⟩
Branches' Measures
To make sense of the everettian perspective about branching, in the following we'll be using some pretty decent though imperfect analogies. The wave or state function branches, in general, have different measures or, in the word used by Everett, weights. The concept of branch own weight is hinted by the figure at left side, representing a cone. The base of a physical object shaped following the geometry of a cone, results its most stable and extended (and, related to others) surface. Visibly more stable and extended that its vertex. The cone’s base represents its unique stable position. In this perspective, a branch of the wave function due to the projection of the vertex has smaller measure in the Hilbert space than another deriving by the many more eigenvectors of the base. Also, an object lasting a long time like, as an example, a neutron with its mean lifetime > 1029 years is an eventuality reproposed in many individual Events. Thus having a measure in the Hilbert space greater than, i.e., the subatomic particle named neutral pion, whose mean lifetime is just ~10-17 seconds. Events which, in the relativistic viewpoint, are 3-dimensional leaves composed of all what exists and tagged by different instants of Time.
The base of a cone is not just bigger than its vertex: it also represents its unique stable position. Its measure in the Hilbert Space is bigger than that of its vertex and of the many lateral sides over which it’d lie inclined. What does not mean that only the cone's base exists
Time and the Quantum
One of the main problems in the Copenhagen's view is related to the timing of the collapse. Relativity Theory since the start objects that a question about when something happens refers to the concept of time. A concept of time subjectively dependent on the choice of the reference system. The problem is so acute, that it was later discovered that also the classic newtonian Mechanics shows paradoxical effects of that time-ordered idea of “collapse”. Paradoxical effects hinting to an intrinsic error. Also, when Hugh Everett III was a child he exchanged mails with the creator of Relativity Theory, Albert Einstein, then at Princeton University. In some way, when adult Hugh Everett III made justice of many Einstein’s famous remarks with respect to Copenhagen's view. In opposition to Copenhagen's view, following Everett before a measurement the system A would not be in a pure state. Rather, in a mixed state, simultaneously existing in a mix of all its allowed states. Its corresponding von Neumann operator, temporarily representing the probability:
P = Σi |Ψi⟩⟨Ψi | [2]
distinguished by the pure probability operator:
P(0) = |Ψ⟩⟨Ψi | [3]
and whichever may turn out the value of a posteriori probability operator:
P[i] = 1/Pi |Ψi⟩⟨Ψi | [4]
Quantum superposition in a subset of the State space named Hilbert space. Ψi (i = 1, 2, 3) ∈ ei of the observable {e}, are orthogonal vectors named eigenvectors. Their sum Ψsuperposition spans the entire vector space, representing the State of a physical system. Everett recognized the permanent superposition of all the physical systems
Everett replaced the assumption that the system A was initially in a pure state, by the more general assumption that it was in an initial state described by an a priori Probability operator P(0), then an arbitrary sum of pure state operators. The effect of the first step of the observation process will be to provide a temporary Probability operator given no longer by the equation [2], rather by the more general:
P = Σi Pi P[i] [5]
where the operators P[i] are the a posteriori Probabilities for the outfeeding branches. An example of them, the eventualities ei. These a posteriori probability operators, and the corresponding values of probability, are given in terms of the a priori probability operator P(0) by:
P[i] = 1/Pi ei P(0) ei
where: Pi = tr{P(0) ei}
or, from the temporary probability operator given in the equation [5], by expressions like:
P[i] = 1/Pi ei P ei
where: Pi = tr{P ei}
Multiverse, in Synthesis
Bryce Seligman DeWitt, who edited the collection of papers (DeWitt, 1973) in the publication cited before, wrote a Preface we'll quote in the following. It offers a straight and clear insight on the logic machinery underlying The Theory of the Universal Wave Function, including the modern version of the Principle of Superposition:
“In 1957, in his Princeton doctoral dissertation, Hugh Everett, III, proposed a new interpretation of quantum mechanics that denies the existence of a separate classical realm and asserts that it makes sense to talk about a state vector for the whole universe. This state vector never collapses and hence reality as a whole is rigorously deterministic. This reality, which is described jointly by the dynamical variables and the state vector, is not the reality we customarily think of, but is a reality composed of many worlds.
However astounding it may appear, the modern version of the Principle of Superposition is saying that a multitude of Chess Games are being simultaneously played. Played in different 3-dimensional spaces superimposed in the 4-dimensional we inhabit. Part of the games correspond to the summation of all the possible familiar time-ordered sequences conceived by Chess Games’ top masters and by the Supercomputers programmed to play Chess. The arena where the theory started to be experimentally verified by mean of the Mach-Zehnder interferometer was that of Quantum Optics, later extended until mesoscopic-sized objects visible to the unaided eyes
By virtue of the temporal development of the dynamical variables the state vector decomposes naturally into orthogonal vectors, reflecting a continual splitting of the universe into a multitude of mutually unobservable but equally real worlds, in each of which every good measurement has yielded a definite result and in most of which the familiar statistical quantum laws hold. (…) Looked at in one way, Everett's interpretation calls for return to naive realism and the old fashioned idea that there can be direct correspondence between formalism and reality”.
The Debate
Bryce G. DeWitt, the physicist who published Hugh Everett III’s PhD dissertation, including the Principle of Superposition in its modern quantum version (
Larry Murphy)
What before DeWitt wrote, hints to a fact: Everett’s interpretation of Quantum Mechanics (QM) is the simplest and straightest statement about what QM is saying us. Everett's understanding of QM formalism was such to determine the first real step by step analysis of what really a Measurement is. In the debate about the interpretation of QM, Hugh Everett III was a follower of Albert Einstein's position. To have an idea of the kind of ‘philosophy’ accompanying the “shut up and calculate” imposed credo, Copenhagen's interpretation was insisting that reality is attributed to objects by the observation. Einstein and Everett, on the opposite, are convinced defenders of the idea that Reality exists unrelated to Measurements and existed before any human started to observe Nature.
A historically new and deeper operative meaning had been established for the Principle of Superposition, based on the Schroedinger equation [6]:
[6]
where:
- i imaginary defined in complex numbers’ set ℂ { a + i b; a,b ∈ ℝ }, say √-1;
- V ( r, t ) potential energy influencing the particle;
- m mass of the particle;
- ħ Planck constant divided by 2π, equal to 1.05459 x 10-34 J s
- ψ( r, t ) wave function, defined over space and time;
- ∇2 Laplacian operator:
A meaning where the wave function Ψ is the fundamental entity, obeying at all times a deterministic wave equation. A Schroedinger’s equation which is simply the equation of the Irish mathematician William Rowan Hamilton in disguise, thus bridging the classic mechanics’ discoveries to the extremely fine-detailed quantum scales. This picture makes sense only when the observation processes are treated within the theory. It is only in this manner that the apparent existence of definite macroscopic objects, as well as localized phenomena (such as tracks in cloud chambers), can be satisfactorily explained in a wave theory where the waves are continually diffusing.
After 1990 it became clear that choices in the way-outs are apparent. All of them are explored. Following modern insights of String Theory and Quantum Mechanics, to that instant of time are referred many more results than the 3 here visible. Each result an Event itself, triggered in correspondance to different properties. In the last years they had been experimentally observed on macroscopic scales the less intuitive results
A deduction of this theory is that phenomenas will appear to observers to be subject to the discontinuities which are everywhere observed. The quantum-jumps exist as relative phenomena (e.g., the states of an object-system relative to chosen observer states show this effect), while the absolute states change quite continuously. And that’s why one of the names of the theory is Relative State formulation. Every individual result is not less true and real than the others.
We are indicating in the figure above this multitude of coexisting, rather than alternative, histories by mean of the three ways branching out of a single. The new Principle is capable to include the entire step-by-step evolution of a system composed of subsystems. And, what is most important, this modern version holds for any system of Quantum Mechanics for which the classic version of the Superposition Principle holds and is applicable to all physical systems, regardless of size. As a consequence, what in these pages is being presented is the most modern conception of Measurement.
Principle of Superposition.
Quantum Version.
In the following the text of the modern version of the Principle of Superposition due to Everett, where:
- A is a quantity with eigenfunctions ΦiS1 measured in a system S1 by an Apparatus;
- ai = (ΦiS1, ΨS1) are the projections on the eigenspaces of the different eigenvalues of A, after the measurement;
- the brackets [ … ] denote values recorded in the memory of the measurement apparatus. Exactly what invariably happens to the measurements performed by the nonclassic subsystems (namely, detectors’ semiconductors, part of the inspections) of other subsystems of those apparatuses we name Electronic Inspectors.
“.…For any situation in which the existence of a property Ri for a subsystem S1 of a composite system S will imply the later property Qi for S, then it is also true that an initial state for S1 of the form:
ΨS1 = Σ ai ΨS1[Ri]
will result in a later state for S of the form:
ΨS = Σi ai ΨS [Qi]
which is also a superposition of states with the property Qi. That is, for any arrangement of an interaction between two systems S1 and S2 which has the property that each initial state:
ΦS1 ΨS2
will result in a final situation with total state ΨS1 + S2, an initial state of S1 of the form:
Σi ai ΦS1
will lead the whole system, after interaction, to the superposition:
Σi ai ΨS1 + S2 …..”
Example
How do we know a container is closed?
The atoms of Carbonium in the screw of this Bottle were yet correlated with the Carbonium atoms we name Cap well before they were aggregated in that shape by a Cap Moulding Machine. Each one atom of Silicium today functionally shaped and doped to act as detector of the reflected wave packets, indicating presence of a cap, in the Cap Presence inspection of an automated Control, is correlated with all other existing particles. To measure by mean of a photoscanner the property closure,means to establish a cap-photoscanner state describing the photoscanner as definetely perceiving that particular system state. But, to definetely perceive a cap-photoscanner state, it is necessary …time. Time to to transform the previous state, in which all possible kinds of correlation of the Photoscanner coexist, in a following state in which the Photoscanner is “aware” to be correlated to a Cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a Cap
The formalism of the version 1957 of the Principle of Superposition above, can be translated in practical terms by mean of an example referred to a cap presence digital inspection, in a Bottling Control. We choose the Cap Presence inspection because one of the simplest existing, then easier to imagine also for non-specialists. One of these is visible in the figure below.
A common object, e.g., a PET bottle, immediately before to establish a strict however brief superposition of states with two apparatuses, acting as measurements systems. The lateral black colour fork-like couple of objects being an high frequency (21.2 MHz) radiator to detect the filling level. The upper central cylindric object, being a Photoscanner devoted to evaluate the status of the container, either closed or not closed, including in this intentionally simple design a LED illuminator and a light detector based on a phototransistor.
Let:
S composite system
The apparatus we name Cap Inspection in a Control with several kinds of inspection, and a cap mainly composed of Carbonium atoms;
S1 subsystem of the system S
An atom of Silicium in the phototransistor which have to detect the reflection of a 700 nm photon (emitted by a LED encased in the same Photoscanner) indicating the status of present cap;
S2 subsystem of the system S
An atom of the cap, which can be present or absent;
Ri property of the subsystem S1
The energetic level of the Silicium atom, high after absorption of a red colour photon whose wavelength is 700 nm;
Qi property for S, after the establishment of the correlation (superposition, namely a bifurcation) between S1 and S2;
Then:
ΨS1 + S2 wave function after interaction of S1 and S2.
Wave function encoding the status of the atom in the cap and of the atom in the phototransistor of the Photoscanner, whose laymen translation is: "Cap Present”.
and:
Σi ai ΨS1 + S2 eigenstate in which the apparatus has recorded an eigenvalue.
Relative state in which a component of the Cap Presence inspection, namely an atom of Silicium of the phototransistor into the Photoscanner, has increased its own energetic level after having absorbed a 700 nm photon reflected by an atom of Carbonium of the cap.
Individual wave packets, whose length is 700 nm, emitted by a red LED can be reflected back to a phototransistor encased jointly with the LED to form a Photoscanner, by an atom of Carbonium of the many composing what we name a cap, applied over a passing bottle. The correlation between the two systems Photoscanner and Cap) is progressively established during interaction and proportional to the natural logarithm of the interaction time ln (t). An ideal correlation, corresponding to a maximised information of the Photoscanner about the Cap, only allowing an infinite time. This causes the measurements’ fluctuations, a synonimous of the widened spectrum of the eigenvalues, resulting in the Electronic Inspector's false positives (false rejects). The interaction between the systems has to be such that the information in the marginal distribution of the object inspected is never decreased. Otherwise we could not have any more repeatability of the measurements. As an example, if we’d erroneously try to use a beam of high energy atomic nuclei, rather than LED's low energy photons, to interact with the Cap. The nuclei could easily change the molecular structure of the Cap (damaging it) changing its eigenstates and then the eigenvalues we expected to measure
Repeatability and False Positives'
Modern Meaning.
Photoscanners are the modern version for what long time ago was known as photo-electric cells, an evolved industrial version of the photo-detectors. They have common use in a huge amount of technological industrial applications. Interesting because designed in such a way to emulate the simplest neuronal chains, like those comprised between the retina and the groups of neurons in the lobes, devoted to perception. To determine that a bottle is capped by mean of a Photoscanner, say to definetely perceive a Cap-Photoscanner state, it is necessary:
- Time, to to transform the previous state, in which all possible kinds of correlation of the Photoscanner coexist, in a following state in which the Photoscanner is “aware” to be correlated to a Cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a Cap. The correlation between the two systems Photoscanner and Cap) is progressively established during interaction and proportional to the natural logarithm ( ln t ) of the interaction time t. An ideal correlation, corresponding to a maximised information of the Photoscanner about the Cap, can only be reached allowing an infinite time. This causes the measurements’ fluctuations, a synonimous of the spectrum of the eigenvalues, resulting in the Electronic Inspector false positives (false rejects). Time to transform the previous state, in which all possible kinds of correlation (superpositions) of the Photoscanner coexist, in a following state in which the Photoscanner is aware to be correlated to a Cap, because having recorded eigenvalues for the eigenfunction ΦiS1 describing a Cap.
- Interaction between the systems such that the information in the marginal distribution of the object inspected is never decreased. Otherwise we could not have any more repeatability of the measurements. As an example, this should be the case if we’d erroneously try to use a beam of high energy neutrons, rather than LED's low energy photons, to interact with the Cap. The nuclei should modify the molecular structure of the Cap, modifying its eigenstates and then the eigenvalues we expected to derive by the measurement.
1982: the Crossroad
Alain Aspect, along 1980-1982, directed group of researchers who made the first decisive test of Entanglement and non-locality. Thes experiments confirmed the non-local character of boson field photons.
John Bell’s analysis and successive experiments demonstrated that the phenomenon named Entanglement, whose initial ideas are dated 1935, have to be part of reality rather than being the consequence of an incomplete description. Say, mere statistical correlations. Single photon detectors, coincidence counters and powerful computers allowed a team of researchers of Sorbonne University led by Alain Aspect, a thorough statistical verification of the Entanglement phenomenon along the years 1980 to 1982. Quantum Mechanics, considering only two of its technological applications, namely Electronics and Information Technology, is yet the most successful scientific theory. In 1982, that part of the humanity knowing what is going to shape the future of whoever, understood it was reached an historical crossroad.
3 mutually excluding interpretations for the single meaning of Reality:
1. the limit speed of propagation of radiation is higher than light speed
(Bohm-De Broglie interpretation, 1952);
or:
2. Ψ represents a Quantum Field and multiple paths are actual
(Everett interpretation, 1957);
or:
3. an observation (or, measurement) is what let a physical status exists
(Bohr-Copenhagen interpretation, ~1932);
where the interpretation:
- is contradicted by:
- the generalized idea that causes exist and precede the effects;
- Einstein's idea about the existence of a limit speed for light Signals. Special Relativity validity constantly reconfirmed by experiments on macroscales;
- seems Science Fiction;
- is a solipsistic position, criticised by Einstein who observed he could not believe that a mouse could bring about drastic changes in the universe simply by looking at it. The key point of the named Copenhagen's Interpretation, lies in the John von Neumann's concept of Measurement. Following von Neumann, a measurement corresponds to the set of possible projections onto a complete orthogonal set rays of the Hilbert space being measured.
Programmable Logic Controllers (PLC) and Frequency Converters are applications of the same Quantum Mechanics' nonclassic rules in all industrial lines. The Modern Principle of Superposition was modelled in 1957 over complex automata, today's name for PLCs
After 1990, the new dominant fundamental theory of the nature, named String Theory, inter twined successfully the Relativity of 1915 with the Quantum Mechanics developed around 1926. Since then, the Event definition as detection of a packet of energy emitted (or, absorbed) by a source or process, is fully built over the cornerstone of those multiple coexisting Paths made of Events, but goes further.
Today, after the Theory of Information's revolution backed by theorems discovered in the last decades whose impressive yield has only started to give its fruits, the Information is no more considered a passive element. Passive like a mere way to label properties and states of energy and matter. The concept of Event, its Information content and connections with other branches of the Science and Technology, are much broader today than they were in 1915. An Event is today, basically and in general, the name of the status of a physical or logical property. The entire line of reasoning based on the chaining of cause and effect, something which appeared obvious in 1915, one century later is object of deeper exams, and knowingly at odds with causal rooted ideas by experiments like those associated with Entanglement.
A key point regards the wave function Ψ which is what enters constantly in mathematical models related to the design of semiconductors. Exactly those accounting for > 99.9999 % of the components into CPUs and other Integrated Circuits. CPUs and Integrated Circuits (ICs) which also allowed to increase the production speed of the world fastest industrial lines. There are plenty of practical and successful technological applications, like the Programmable Logic Controllers (PLCs, see figure on right side) and frequency converters into the control rooms of the Manufacturing Lines, derived by the superposed waves building up the localized wave packet Ψ of the formula [6] above.
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