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Introduction

In the Industrial Automation and Electronic Inspection fields, however named and operatively disguised, Triggers play the role of the most elementary measurement instruments.  Simpler than any single-channel (e.g., the High Frequency fill level inspection) or multi-channel (for example, all camera-based inspections) analog-digital variable measurement system, and a permanent input device to the Programmable Logic Controllers' (PLCs) for the automations.  A well-known common example are all Container Presence detectors.  To understand what is a control equipment, for example an Electronic Inspection device, we are going to analyse its most basic and ever present inspection.  Leveraging it to simplify the approach to the wider field of Measurement.  In the end, we’ll discover that the most modern meaning for measurement (inspection), does not resembles any more what was taught some decades ago in the courses of Electrical or Electronic Measurements. These complex subjects lie on the border of Technology, well into the scientific research, and are descripted by mathematical operators in abstract spaces.  Many of the subjects of these web pages devoted to the Physics of Triggering, however modern and complex they may appear, were conceived around one century ago. Ideas first conceived mainly by the physicists joint in the meeting visible below.  Between them Max Planck, Henri Poincare’, Marie Curie, Hendrik Lorentz, Albert Einstein, Niels Bohr, Paul Dirac, Paul Langevin, Louis de Broglie, Werner Heisenberg, Erwin Schroedinger and Max Born. 

 Event

Historical Approach to the Key-Concept of Trigger 

In the celebrated Oxford Dictionaries of the English Language, “Trigger” is defined as: 

making it immediately clear that Trigger's concept derives by the idea of Event, but not so evident that the Oxford Dictionaries’ definition is circular.  On practice, an “Event as a cause of other Events”, what in the following shall become clear.  Intuitively, each one of us feels ready to answer the key question.  Desiring to understand what really is a Trigger, we look at Physics and its definition of Event.  For example, the Electronic Inspectors, equipments whose main task is to classify objects like: bottles, cans, crates, cases, caps, etc., after having compared the result of the physical measurement of at least one physical property of the objects, with limits pre-established by us. Physics is not foreign to the industrial Machinery and equipments, rather its core.

Space

We’ll start this truly fundamental section proceeding step-by-step not giving anything for granted.  In technical terms, the space-time is a Manifold M:  

• 4-dimensional,
• connected, 
• Hausdorff, 
• paracompact, 
• oriented, 
• differential.

The space is inhomogeneous. It doesn’t make sense to assume constancy of a vector field in the space.  

 The geoid is a useful help to imagine the aspect of a differentiable manifold (  abridged by Barthelmes F. et al./2015)

On the opposite, it makes sense for a scalar field, like that of the temperatures.  The space itself does not carry a metric or a linear connection. Then, a measurement of intervals (temporal or spatial) still cannot be defined since a metric is not yet available.  The following concept is that of space-time.  From an intuitive non-technical point of view, a 4D spacetime manifold is a continuum which can be locally decomposed in: 

• 1D time, 
• 3D space. 

We’ll resume in the following the key role played by each Event in the definition of the metric properties of the space.  A role allowing us to define intervals, thus identifying who, what, when and where, and in what extent something happened.

Coordinate Systems

A coordinate system in the space-time is a set of four real-valued functions on the space-time such that, in some neighborhood, knowledge of the values of these functions will fix the Event point in a smooth manner. “Smooth manner” meaning that the Event lies in a space itself a differentiable manifold.  An example on side, in the Earth’s external surface.   

Physical Event as Light Cone' Vertex

An additional restriction about the kind of space is that it’ll be a pseudo-Riemannian time-orientable manifold M.  Then, one where the division of the half-cones of M in the past and Future classes can be made continously over the whole manifold. Here first introduced the concept of light-cone developed in 1907 by the Russian mathematician Hermann Minkowski. Scholar better known because having been Albert Einstein and Max Born's teacher of Mathematics in the Eidgenössische Technische Hochschule (ETH), (Swiss Federal Institute of Technology) at Zurich, Switzerland, reformulated and extended the ideas of the French physicist and mathematician Henri Poincare’. The result, shown in the figure below, was the first 4D system of space-time coordinates.  Is Minkowski who first realized that: 

• Einstein’s Relativity Principle implies many times and spaces, meaning that the world we inhabit is 4D. What explained the previously failed attempts to detect motion with respect to the Ether reference, in that no absolute motion with respect to the Ether exists, since there are many spaces, not just one;
• a body moving by inertia has to be represented by a straight timelike worldline, whereas the worldline of an accelerated body is curved;
• the acceleration of a body along a straight line in 3D, mirrors the curvature of its worldline, as seen in 4D;
• pairs of ordinary mechanical quantities are space and time components of 4D vectors and the ordinary electromagnetic quantities are components of new types of 4D.

Event became a dot of space in the 3D hyper-surface of the Present, crossed by the Time dimension. The Time dimension perpendicular to the spatial surface in the vertex where the two cones meet, at the origin of the two space axes. Lines which intersect the surface of the Present at an angle of 45° represent light rays propagating in the vacuum at their characteristic speed c (299 792 458 m/s).  The basic observation leading to special relativity is the following: the comparison of times at different places in space is not an objectively well defined procedure if all laws are local and the speed of signals is limited.  It needs a convention for the synchronization of different clocks and there is a certain amount of arbitrariness in this convention. This prevents us from considering space and time as two distinct, objectively meaningful, concepts, geometrically represented by a bicone. Rather, we must consider the 4D manifold of possible pointlike Events, the 4D space-time manifold. The origin of the bicone operates a tripartition of space-time in: 

1. forward cone V+ containing all events which can be causally influenced from x;
2. backward cone V- containing all events from which an influence on x can come; 
3. everywhere, the space-like volume all around the point x, complement of the bicone, containing all points which cannot have causal relationship with x.   A spatial volume, typically whose mere existence is frequently oversight by the laymen.

Electromagnetic and gravitational signals propagate themselves just through the external surface of each Future light-cone.  A surface whose thickness is infinitesimal.., named null-surface.  An example in the figure at right side, depicting an individual signal pulse originated by the bottom Future light-cone, later continuing its journey through the Future null-surfaces of the light-cones derived by four successive Events.  

Locality Principle and Inertial Systems

Again with reference to the figure above at left side, one may call V+ the future, V- the past and the present of the event x at the vertex of the bicone. The boundary of V+ is formed by all possible events which can be reached by the fastest possible signals sent from x and these are identified with light signals in physics.  The Locality Principle is thus strictly related to Einstein's Causality Principle: no physical effect can propagate faster than light.   In Special Relativity (1905) it is assumed a causal structure of space-time as a priori globally given. There is supposed to exist a preferred class of coordinate systems, the inertial systems, in which the causal future of a point x-(t, x) consists of all points  x- (t’, x') with:

                             t’  -  t    ≥   |x’  -  x|

Albert Einstein, in 1907 initially dismissed his former ETH University Mathematics professor's idea of spacetime. Later, at the Patent Office in Bern, Switzerland, was struck by what he called his happiest thought.  Einstein was not a physicist advancing by formulae, rather one advancing by mean of mental images, depictions of physical phenomena.  He was suddenly illuminated imagining …a man falling off the side of a building and feeling no gravity at all, as we know really is the case.  The choice of accelerated coordinates is enough to eliminate the effects of the Earth’s gravitational field, regardless of who or what is dropped.  If gravitational force  were like any other force, as an example the electromagnetism so commonly used in the Electronic Inspectors, differently charged objects would fall in different ways, some of them even accelerating upward.   

Einstein: An Infinite Number of Spaces in Motion

Yet Galileo Galilei nearly four centuries ago observed experimentally that the reason why objects with different masses were landing in different times, had not to be ascribed to their mass rather to aerodynamic factors.  By contrast, gravity appears not caring the material content: mass.  From this observational fact, the equivalence principle, Einstein inferred that gravitation have to originate in spacetime itself.  Later, he identified the property of spacetime as its curvature.  In 1952 Albert Einstein explained with a beatiful example the brakthrough fathered in 1907 by Hermann Minkowski about the coexistence of infinite superimposed spaces.  

 When a smaller matrioska is moved into a bigger one, it is necessary to apportion to each one its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other. The space being moved is one of the infinite spaces in relative motion in Albert Einstein’s explanation about infinite coexisting and superimposed spaces. As seen from the point of view of the bigger matrioska, the space occupied by the smaller matrioska is a mixture of bigger matrioska space and time. As seen from the point of view of the smaller matrioska, the space occupied by the bigger matrioska is a mixture of the smaller matrioska space and time

Probably, the most clear explanation still today existing for a non-trivial concept: “When a smaller box s is situated, relatively at rest, inside the hollow space of a larger box S, then the hollow space of s is a part of the hollow space of S, and the same “space”, which contains both of them, belongs to each of the boxes.  When s is in motion with respect to S, however, the concept is less simple.  One is then inclined to think that s encloses always the same space, but a variable part of the space S”.  And, Einstein's personal assessment of the case:  

“It then becomes necessary to apportion to each box its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other.  Before one has become aware of this complication, space appears as an unbounded medium or container in which material objects swim around.  But it must now be remembered that there is an infinite number of spaces, which are in motion with respect to each other.  The concept of space as something existing objectively and independent of things belongs to pre-scientific thought, but not so the idea of the existence of an infinite number of spaces in motion relatively to each other.  This latter idea is indeed logically unavoidable, but is far from having played a considerable role even in scientific thought.”   

As seen from the point of view of the bigger matrioska, the space occupied by the smaller matrioska is a mixture of bigger matrioska pace and time.  As seen from the point of view of the smaller matrioska, the space occupied by the bigger matrioska is a mixture of the smaller matrioska space and time.  What before, explained by Einstein with reference to macroscopic boxes, is valid until the atomic spatial scale, but with important limits.  As an example, Erwin Schrödinger, when deriving the exact solutions for a Hydrogen atom in a closed and finite System, demonstrated that in this particular case, the number of bound states is finite. This implies that, also in presence of an infinite amount of spaces, only a finite amount of them allows bound states, e.g. for the matter and energy.   

Events’ Simultaneity

Until 1907 the view about Events’ simultaneity was reflected by the figure on side.  All existing objects sharing a single simultaneous Time, whatever their kinematic, distance or mass-energy level.  For them all also a shared Future and Past.  

 Simultaneity until 1907.  Observer and all other existing objects, whatever their mutual distances, lie in the point P . All of them simultaneously feeling the Events at P . De-facto a view silently assuming an infinite velocity of propagation for whatever signal, electromagnetic or gravitational

Stll today the laymen have this idea of synchronization.  Knowingly, the notion of simultaneous Events is one of those which had to be radically revised by the Relativity theory.  In the following, we’ll see how this applies in the most general case, referred to objects in accelerated motion. All motions accelerated in a 3D space may be conceived alike movements over a curved 4D space-time hypersurface. The graphic below is an example.  Here, S is a 3D slice of space passing thru the Event P.  Simultaneity is defined by a group of observers or measurement instruments  a, b, c,…, whose world lines cross orthogonally the simultaneity slice (an equitemp hyper surface) S and whose clocks, when crossing P, read the same proper time. Interestingly, also the new concept of proper time τ is due to Hermann Minkowski.   

  S is a 3-dimensional slice of space passing thru the Event P. Simultaneity is defined by a group of observers or measurement instruments a, b, c,..., whose world lines cross orthogonally the simultaneity slice and whose clocks, when crossing P, read the same proper time.  Thru the same Event may pass in the meantime also a different simultaneity, by observers on a line with different curvature and/or slope, and whose clocks show a proper time different than a, b, c, … (  abridged by Wheeler, et al./1973) 

If at any point P (x, y, z, t) in spacetime we imagine a worldline b running through that point, the magnitude corresponding to the timelike vector dx, dy, dx, dt laid off along the line is:

Proper time is obtained integrating the infinitesimals dt along the worldline in question.   In the formula before,  x, y, z and t are the the components of the vector OP, where O is the origin. They are  functions of the proper time τ, and the first derivative of the components of this vector with respect to the proper time, dx/dτ , dy/dτ , dz/dτ and dt/dτ, are those of the velocity vector u at the point P  along b.  Finally, to let this one century old relativistic scenario preserve coherence with the concept of electromagnetic flux conservation, thru a single Event P they can pass infinite hyper surfaces of simultaneity, provided their:

In the example above, S is a 3D slice of space passing thru the Event P.  Simultaneity is defined by a group of observers or measurement instruments  a,b,c,…, whose world lines cross orthogonally the simultaneity slice S and whose clocks, when crossing P, read the same proper time τ.  The figure seen before is an abstraction. 

In the reality each point in the curves representing world lines crossing orthogonally the simultaneity slice a, b, c,… approximates an entire 3D hyper surface at a definite instant of proper time. This way the world-tube representing the history of a real 3+1D physical object, can be approximated and depicted as a simple 1D line. What is described in the figure at left side, where are shown two successive 3D space-like sections, reduced to just two points P1, P2 on the purely time-like world-line b of an Observer or a Detector.

Representation of the world-tube of a macroscopic object in a congruence curve b (  abridged by F. De Felice, C.J.S. Clarke/1990)

Geometric Foliations

A foliation of a manifold M is a decomposition of the manifold M into immersed submanifolds, named the leaves of the foliation.  Leaves which have to be of the same dimension, and to fit together.   Foliations of manifolds occur in various geometric contexts, for example as solutions of differential equations and integrable systems, and in symplectic geometry.  The concept of a foliation first appeared explicitly in the work of the mathematicians Reeb and Ehresmann.  Researches motivated by the question of existence of completely integrable vector fields on 3D manifolds.  The theory of foliations has now become a rich and exciting geometric subject by itself, after the results obtained between 1952 and 1994 by Reeb, Novikov, Hæfliger, Connes and others.  Mathematicians’ insights nearly always precede the applications by Physicists and Engineers.  Long time ago they extended the range of characteristics of a determinate topology, until connected and multiply-connected varieties.  The concept allowing this important step is the genus.  Hopf’s and Reeb’s foliations, visible on side and below, provides a glimpse in this new geometric world.

 Reeb's foliation (  Tambara Institute of Mathematical Sciences, University of Tokio/2014) 

Foliation Atlas

A foliation on a manifold M can be given by a: 

• foliation atlas on M,
• integrable subbundle of the tangent bundle of M, 
• differential ideal.  

All these descriptions are equivalent as a consequence of the theorem of integrability of Frobenius.  We will give just elementary examples of foliations.  

 Definition of differentiable manifold (  abridged by F. De Felice, C.J.S. Clarke/1990)

The simplest example of a foliation on a manifold M is obtained by the level sets of a submersion M → N.  In general, a foliation on M is a decomposition of M into leaves which is locally given by the fibres of a submersion. Let M be a smooth manifold of dimension n. A foliation atlas of codimension q of M, where 0 ≤ q ≤ n, is an atlas:

                [ φi: Ui  →  ℝⁿ = ℝ^n-q × ℝ^q ]i ∈ I

 A basic example of differentiable manifold, compact and where each 2D surface point has smooth neighbours

of M, for which the diffeomorphisms allowing to change chart φij are locally of the form:

           φij(x, y)  =  (gij(x, y), hij(y))

with respect to the decomposition:      

                ℝⁿ  =  ℝ^n-q  x  ℝ^q

Maps of manifolds (  abridged by F. De Felice, C.J.S. Clarke/1990)

The charts of a foliation atlas are referred to as foliation charts. Thus each Ui results divided into plaques, themselves connected components of the submanifolds:

                      φ^-1 [ℝ^n-q  × {y}],    y ∈ ℝ^q

and the change of charts diffeomorphisms preserving this division.   

 3D interferential patterns hint the strict relation existing between foliated manifolds, waves, and …triggers

Leaves

The plaques globally amalgamate into leaves, which are smooth manifolds of dimension n − q injectively immersed into M.  As a consequence, two points x, y ∈ M  lie on the same leaf if there exist a sequence of: 

• foliation charts:         U1 , ....., Uk
• points:                       x  =  p0,  p1, …,  pk  =  y  

such that p_j-1 and  p_j lie on the same plaque in Uj, for any j in the interval 1 ≤  j  ≤  k.    A foliation of codimension q of M is a maximal foliation atlas of M of codimension q.   Each foliation atlas determines a foliation, since it is included in a unique maximal foliation atlas.  Two foliation atlases define the same foliation of M precisely if they induce the same partition of M into leaves.     

Foliated Manifold 

A foliated manifold is a pair (M, F), where M is a smooth manifold and F a foliation of M.   The space of leaves M/F  of a foliated manifold (M, F)  is the quotient space of M, obtained by identifying two points of M if they lie on the same leaf of F.  The dimension of F  is  n − q.  A map between foliated manifolds: 

                         f : (M, F )  →  (M′, F′)  

is a map  f: M → N  preserving the foliation structure, as an example, one mapping leaves of F into the leaves of F′.

 Hopf’s foliation. Hopf’s foliations are basic topologic varieties featuring one genus. The genus is perceived as a hole, as seen by an Observer lying out of the throats (  Tambara Institute of Mathematical Sciences, University of Tokio/2014)

Example

Moebius band

A basic kind of foliation is visible at right side and named Moebius band.  

To build it, let:                       f : ℝ²  →  M  

be the standard covering projection of the open Moebius band:              

                                          f(x, y)  =  f(x′, y′)  

if and only if:

• x′ − x  ∈  Z   
• y′ = (-1)^{x’ – x} y

The foliation of codimension 1 of ℝ²  induces a foliation F  of M.   All the leaves of F are diffeomorphic to S¹, and they are wrapping around M twice, except for the middle one which goes around only once.

Physical Foliations

Adding Time to Geometry results in physical foliations and it’ll be introduced in the following how physical foliations host the Events.  Event’s concept changed in 1915, after Einstein applied Minkowski's ideas about acceleration, to the concepts of curvature of the space-time manifold due to density, distribution and dynamics of the masses and energy.  One of the many fruits of his General Relativity theory being the capability to break up the notion of space-time into a space with time evolution, say: a time-ordered sequence of spatial hypersurfaces.  The space-time can be decomposed locally into 3D hypersurfaces or sheets or folios, labeled consecutively by a monotonically increasing “proto-time” parameter.  On such a manifold, we assume the existence of a foliation. 

  How a physical foliation evolves. From an initial 2D circular surface they spread with velocity v2 horizontal concentric space-like leaves. The original 2D circular surface evolves with velocity v1 in the meantime, until forming a 3D hyper surface.  

A first example of physical foliation above.  Here, from an initial 2D circular surface they spread with velocity v2 horizontal concentric space-like leaves.  The original 2D circular surface evolves with velocity v1 in the meantime, until being a 3D hyper surface.  The resulting shape is the advancing front given by the sum of the velocities v1, v1, determining a set of superposed, coaxial disks with time-related, progressively increasing diameter.  A detail of the result is visible in the figure at right side. Here, a time-ordered sequence of spatial hypersurfaces showing:

• a vector separation V  =  P - P_{0 between two neighboring points} P and P0;
• a piercing of the family σ of slices by the vector V;
• an orientation for σ necessary to define which direction from slice surface to another slice surface, has to be considered positive because each slice has two surfaces.

In the example presented at right side, the vector V pierces ~4.5 surfaces.

An Event after 1915 til ~1960 is a worldpoint in a 3D spatial-only leaf (or slice, or sheet) part of a curved foliation of pseudo-Riemannian differentiable manifolds M. The foliation is a serie of leaves and each leaf a surface of constant relativistic coordinate time (t in the figure).  The Event lies at the vertex where the two light cones meet.  The slices are curved because part of a spacetime fabric is warped by its own mass-energy content.  In the most general case, the slice where the Event happens contains all the space, matter and energy existing in a Universe subspace.  The Event acquired the meaning of a worldpoint where it happens an exchange of energy 

Inter-Leaves Information Exchange 

No transmission of electromagnetic or gravitational Signals, or movement of matter, happens thru the individual slice.  Information exchange is always and only inter-leaves and never intra-leaf.  The slices could be roughly figured alike frozen 3D configurations of particles.  Then, as an example, refer to the figure here at right side.  It shows a family of 1-parameter space-like hyper-surfaces.  Here, two points P and Q  lie at the intersections between two different world-lines and one and the same space-like hyper-surface ∑tp. The world-points P and Q  are Events in the same leaf of the foliation but they do not see each other.  An elementary Detector at P  should be completely unaware of the existence of another Detector at Q.  This apparently odd fact because all Information, carried by electromagnetic and gravitational energy is always and only exchanged thru different leaves (or slices, or sheets) of the Foliation.  Matter and Signals were considered transferred only toward the Future light cones when General Relativity was created.  

But, Maxwell’s equations are linear with respect to Time parameter, thus allowing full equivalence of both directions of propagation of the electromagnetic Signals: Future and Past.  More, following modern insights of Quantum Physics, we know that on the microscale of dimensions they are also transferred into the Past light cone, forming Closed Timelike Curves (CTC) of extremely brief duration.  The General Relativity theory considers the 4D space foliated by 3D leaves and the Time reduced to a mere indicator of the position of the individual leaf with respect to the foliation.  In the modern relativistic scenario, an Event have not a strict and univoquely defined meaning. That is because many other Events, closely dotted all-around an Event, participate to the definition of its interactions.  In other words, following General Relativity, what happens in several leaves define the extent and kind of a measurement in one of them.

 Gravitation, by J. A. Wheeler, C. W. Misner and K. Thorne.  Since 1973 with its 1300 pages, the standard academic textbook of General Relativity

As an example, the production of pairs electron-positron e^-e⁺ when sufficient energy is available.  This, implying that Trigger Events in one slice (or, leaf) gives their effects in other slices or leaves. Between 1915 and 1973, the Event acquired a meaning no more simply geometric.  Meaning which may be inferred by the figure below, adapted by the celebrated worldwide main textbook of General Relativity, Gravitation, by John Archibald Wheeler, Charles W. Misner and Kip Thorne, 1973.  It had been understood that the physical Events precisely mark a well definite position in the space.   A Physical Event like the geometric world-point where it happens an interaction or, exchange of energy.  This, however dated, is the definition of Triggered Event (or, Measurement), still today applied in the Food and Beverage Machinery.  Also, in the Bottling Controls,   

the Electronic Inspectors, and in the Automation Technologies whose applications are still more widespread in the Food and Beverage Packaging Lines. The Events listed in the caption on left side of the figure above are those originating the Signals which all of the packaging machinery automated quality control devices:

• detect;
• amplify;
• integrate, along times long enough to pass over the thresholds of sensitivity; 
• conduct in the end the Binary Classifier to reject.

The idea underlying these figures is the one of interaction.  All interactions which affect matter particles are due to an exchange of force carrier particles, a different type of particle altogether. What we normally think of as forces are actually the effects of force carrier particles on matter particles.  Force carrier particles whose exhange results in attractive or repulsive effects, a process hinted by the video animation here below: 

The figure before showing a firecracker is just a compact representation, corresponding to five superimposed slices, what can be perceived after its comparison with the figure below.  Before, all Events, Triggerings Events and inspections, they all happen in a space-time slice where all of the dots of space are referred to a single constant Time, part of a foliation of the 4D structure.  The fine-details are a subject out of the scope of this web page devoted to the physical measurements (inspections) in the Beverage Bottling Lines.  An example of this is referred below representing: 

• five Events, as seen by Physics point of view between 1915 and ~1960, 
• associated to five different consecutive times,
• if you imagine these five slices as transparent foils, superimpose them and you’ll have the single diagram in the precedent figure, in the start of this page,  
• in this dated point of view, the five slices below are spacelike hypersurfaces of constant time,  
• they contitute a Foliation.    

In this dated image, one deeply changed fifty years ago, each slice has at least a detail making the difference with respect to the other four slices, meaning they cannot exist two identical slices.   

 Five Events, as they were imagined between 1915 and ~1960, associated to five different consecutive times. Imagine these five slices as transparent foils. Superimpose them and you’ll have the single diagram in the precedent figure above. The slices are spacelike hypersurfaces of constant time; jointly constitute a Foliation. Each slice has a detail differentiating it with respect to the others. Five different statuses of string’s consumption are the detail making the difference between slices (  abridged by Wheeler, Misner, Thorne/1973) 

If two slices should have same identical content of matter and radiation, something in principle not impossible however unreasonable, then the mere difference in their common constant Time should however be enough to differentiate them.  In the case depicted above, 5 different statuses of consumption of the string are the detail making the difference between one slice and the others.  This considered, Triggers' most basic definition reduces itself to: devices to label the statuses of physical or logical entities. 

Superconduction

The World is Quantum. At all Scales

A dated objection to the idea of reality we are introducing, affirms that Quantum phenomena really exist but their domain of application is limited to what we’ll never see. A layman way to state an assumpted indifference of what at our scale of dimensions, energies and duration times, we perceive being happening.  But, the subatomic scale of distances where no technological application exists.  Try to board on the magnetic levitation trains at the airport of Shanghai in China or at Muenchen, in Germany.  Then, you’ll be personally moved by a Quantum phenomena.  Festo™ presents in the following video its own recent applications of superconduction. Just to remain in our own industrial field, what we are going to share below is a Quantum application thinked for the Industrial Automation:

The Ever Changing Fabric of Space

The classic Principle of Superposition hold considered valid during the twentieth century but, in the meantime, a row of experimental and theoretical discoveries changed the panorama. It was established that the subatomic arena where interactions happens has a multiply connected space, say a foam-like character completely different than the popular Euclidean spatial image where the angles between perpendicular axes are 90°.  More, a spacetime whose dimension is not 4, as imagined more than one century ago, rather 10 or 11.  The figure and video below, illustrate what yet in 1973 was known to be close to reality.   It is generally believed that the picture of spacetime as a manifold M locally modelled on the flat 4D Minkowski space described in the start of this page, should break down at very short distances of the order of the named Planck length λ whose numeric derives by the fundamental constants of gravity G, light speed c and Planck ℏ:

                         λ  =  (G ℏ/c³ )^1/2  ~ 1.6 x 10^-33  cm

Space-time, so smooth and regular as it appears us, is on the opposite extremely rich of details at scales smaller than ours.  Scales which are the arena for all of the Events.  An intimate complexity of the structure, evident when closely looking the mechanical or electrical behaviour of the Systems at our scale of dimension, is embedded in the Topology lying back of the of space-time illusory sensation.  Also, we have to consider that in the RLC circuit example examined elsewhere in this website, human normal sensibility to Events is limited to those whose duration is > 10 ms.                                                                                          

 The structure of the inner space is foam-like and constantly changing shape. We perceive all this only by mean of effects whose causes can be inferred by what is visible in our macro-scale. Matter and radiation have a size close to the micro-scale, and that’s why they follow those rules. Non-linearity along measurement is artificially added.  An effect of perspective due to the approximate theories. The universal superposition is linear

It is surely possible to detect also Events much shorter, like powerful strobo flashes, but only because of the artificial technical solution discharging tens of Joule of electromagnetic energy along only (0.1 - 0.3) ms.  As a matter of fact, a LED diode illuminated along that same range of times, is not detected by our eyes.  But, in the framework of the technical arrangement we adopted before to power the supposedly-linear RLC circuit, and considering that frequency and period are inversely proportional, then: 

• increasing until 750 GHz the frequency of the signal; 
• simultaneously monitoring the effects by an oscilloscope; 

we have instrumentally zoomed ~75 billions of times toward the microscale.  

 Zooming the micro scale. Reality is pinpointed by what is smaller than our dimension, for our own scale is composed by that subscale, and not vice versa

Along the 70’s Quantum Mechanics evolved in a quite different point of view named String Theory.  It was then affected by several Rube-Goldberg machine problems.  An amount of ad-hoc assumptions necessary to let the theory have explanatory power with respect to the zoo of newly discovered particles, constantly enriching itself of new entries deriving by new experimental discoveries. Discoveries made thanks to the Accelerators: humanity’s greatest and most precise measurement instruments of: position, time, electric charge, spin, energy, etc.  Our own observation of an unexpectedly complex behaviour of the RLC serie resonant circuit, in some way resembles and gives an idea of the difficulty of interpretation of the results of collisions, where some of the byproducts live one million of times less than the 1.3 picosecond pulses of our Signal Generator set at 750 GHz signals.  

 Theories built over a few fundamental physical constants offer much more than explanatory power for the observed facts. They allow preemptive capabilities, as an example, to know what has to be our expectation about the result of a process still not accomplished, a future process (  McNaught, Wilkinson/2006)

More than 

Explanatory Power

Virtual particles, positrons...?   We stress a fact: all what precedes and follows cannot be dismissed like …theory.   Also, these interpretations of facts and theories let the industrial world all around us exist and be as productive as we know it really is.  A direct feeling of this point may be perceived by the image below, originating by the international official body of Chemistry, the International Union of Pure and Applied Chemistry (IUPAC).  It shows the many interrelations existing between a few fundamental physical constants, like the Planck’s:

                          ℏ  =  h /2 π  

the electron charge or the electron rest mass, and many tens of other physical phenomenons or chemical processes.   Relations to events interesting objects much closer to the human space-time-energy scale we perceive directly.   A trivial example, the evaporation of the boiling water in a beaker.  Visibly, these are related each other by mean of those constants.  Theories built over these constants offer much more than explanatory power with respect to observed behaviours and facts. They allow preemptive capabilities, so to know what has to be our expectation about the result of a process still not accomplished, a future process.  The industrial terminology owns a definition for these preemptive capabilities: design.

Introduction

In 1953 the Italian mathematician Eugenio Calabi was searching for a flat complex topology. His studies completely disconnected by the simultaneous physicists’ investigations and the same term String Theory still to be coined. Calabi conjectured that starting with the case of one complex dimension and two real dimensions, if the general Topology has average curvature zero, then it is possible to encounter a geometry (or, metric) where the curvature is zero everywhere.

For dimensions superior to these, his conjecture refers to Ricci curvature and the condition of average Ricci curvature zero is replaced by the condition of first Chern class being zero.  He considered that if the topologic condition of first Chern class zero is met, then it exists a Kæehler metric with zero Ricci curvature.  In 1973 the mathematician Shing-Tung Yau, after years trying to disprove Calabi's Conjecture, discovered the way to prove it was ...correct!  This discovery reached physicists who integrated it as new metric (a new Geometry) for the 6D inner space imagined existing in each point of the 4D space-time we feel perceive directly (3D space) and indirectly (1D time).  But, why to add dimensions, when to have less should apparently mean to have also less complications ?

 An intuitive vectorial field: the radiuses getting out by the centre of a sphere

 First class of Chern objects are places where flows in a vector field drop to zero. Hurricanes, a couple of them here visible in the Pacific Ocean and off the coast of Southern China, have central zones whose diameter range (2 - 280) km where flows in the wind vector fields drop to zero

Several excellent reasons. One of them being the constant accumulation of the varieties of particles, discovered along decades of High Energy Physics experiments. Their associated wide spectrum of properties (mass, energy, electric charge, spin, etc.) to be described in a unified mathematical model need… more space than what provided by total four dimensions.  Below a table where they appear seventeen elementary particles. Widespread consensus exists that the table is actually missing at least part of the tens of named supersymmetric partners, theorized by other studies. 

Each one time a new and more precise determination is made for, e.g., the energy of a known particle or a new particle discovered, say each one time we proceed forward in the knowledge, this has effects also out of the experimental ambit.  Experimental results compared with theories, implying e.g., the refutation of some theories’ underlying assumptions and/or confirmation of other theories predictions, in what since three centuries is the process of scientific discovery.  An example of this in the twelve graphs below quantifying the progressive reduction of the uncertainty in the determination of the Mass (expressed in eV) and lifetimes of twelve particles, along past fifty years. 

  Last 50 years of progress. Progressive reduction along fifty years of the uncertainty in the determination of the mass, expressed in eV, and lifetimes of twelve particles (  J. Beringer et al./2012)  

Uncertainties expressed as blue colour vertical error bars, around the weighted mean values of experimental origin. That’s why the new geometric descriptions adding six independent degrees of freedom, in the form of six additional spatial axes, were welcome.  We are speaking of string-like particles, objects having extension in only one dimension.  The theory comprising them is named String Theory.  Here the fundamental objects are 1D strings which, as they move in time, sweep out a 2D world-sheet.  Strings can be open or closed and their worldsheet is embedded in some higher-dimensional target space, identified with a Minkowskian spacetime.  The strings are exceedingly small and move through a 10 or 11D analogue of space and time (see graphic below, immediately after the video).   In the diagram, the time evolution is along the vertical axis, increasing from the bottom to the top. Closed strings, also represented as coloured closed loops, enter from the bottom and leave on the top. The topological structure of a world sheet describing these quantum-mechanical interactions is like that of a doughnut with an arbitrary number of holes. 

Bifurcations

States in the target space appear as eigenmodes of the string and their scattering amplitudes are generalized by appropriate scattering amplitudes of strings. These scattering amplitudes are built from a fundamental vertex, which for closed strings is depicted on left side.  It represents the splitting of a string or the joining of two strings.  When strings split apart and rejoin, a hole is left in the world sheet.  In the quantum calculations all possible splittings and joinings between an initial state of strings and a final state must be considered.  This way recalling the basic idea of Feynman’s sum over histories.

 A bifurcation as effect of a change experienced by a 2-brane from one to another kind of topological variety.  In the image the change in the amount of closed loops: below, a single particle and above a couple  

Interferences

The closed strings sweep out world sheets that are deformed cylinders, topologic equivalents of a cylinder.  When two strings collide, they join to form a third string: two cylinders form a third cylinder.   This is the fine detail allowed by the most modern point of view, of what was since over 1 century named interference.    

 Time evolution of a small section of a world sheet. Bifurcations and interferences, constantly happening at the finest scales, hint to a superposition. A superposition whose components constantly coexist at least in all of the possible values allowed by the fundamental physical constants and by the physical laws established with these constants (  Michael Boris Green/1986)

Loopbraids 

The Key Opening the Inner Space’s Door

 Loopbraids like the one visible above, illustrate in what a way coexistence of common material objects in the same higher-dimensional space is possible. Each one of the objects not interfering with the other.  The presence of the other totally unnoticed due to the topologic genus (the hole) present in one of them.  This figure hints to a topologic complexity higher than the precedent and closer to the real geometry.  A world made of strings (or, membranes) and not a geometry made of strings in the space.  In the modern scenario, dimensionality is a feature of the strings, and there is no one spatial-immersion at all.  Space, exactly as recently happened to time, relegated to a secondary concept rather than a fundamental. Strings are the fine texture of the world, without anything added 

A fundamental question: if all this is happening, if really there are so many coexisting superposed objects, where are them?   One of the many correct answers to this question has been provided by the developement of that branch of Mathematics originally named Analysis Situs (today, Topology), founded over one century ago by the great Henri Poincare’.   Loopbraids like the one visible above, illustrate one of the ways to the coexistence of common material objects in the same higher-dimensional space.  Each one of the objects in the figure below is not interfering with the other.  The presence of the other totally unnoticed due to the topologic genus (the hole) present in one of them.  

The Inner Space

Is it there really so much space in the microscale?  Enough to host a loopbraid like the one seen above and have still space for many others, encased one into each other genus ?    Do you remember Minkowski’s discovery of 1908? ...each dot of 3D space is infinitely extended in the 4D space.  We’ll see in the following the convergence of theory and experimental results toward, at least, 10 dimensions.   

   The Italian and Chinese mathematicians Eugenio Calabi (left) and Shing Tung Yau (right), whose discovery abouth the existence of a vast inner space is paving the way for the Technology which shall derive by the Physics of the 21st century.  In 1953 Calabi considered that if the topologic condition of first Chern class zero is met, then it exists a Kæhler metric with zero Ricci curvature.  In layman language: a calm, wide, flat space hidden in the middle of a tornado.  In 1973 Shing-Tung Yau, after years trying to disprove Calabi's Conjecture, discovered the way to prove it was ...correct ! (  Dars, Lesne, Papillault/2008)

What an apparent shape for the new inner space is hinted by the video below (which can be freely downloaded here) where, on:

• left side, an individual 3D projection of a 6D Calabi-Yau manifold;
• right side, the shape of the physical space.  The yellow colour lattice represents two of the four dimensions of the physical space-time we perceive, and in each intersection lies a compact 6D Calabi-Yau manifold, totalling nine space dimensions plus Time.

Key concept to such new scenario is the compactness of the 6D manifold: in Geometry it is possible to have additional compact dimensions of infinite or finite size, lying in an infinitesimal space.  The new category of geometric Calabi-Yau manifolds, allowed a rapid success of String Theory in terms of consistency with the experimental discoveries of accumulated along decades by the High Enery Physics (HEP).  The objects of String Theory, namely are strings, unidimensional objects (1D) like those depicted in the figure on right side, vibrating on tones and overtones.  As an example, it was imagined that electrons were microscopic closed vibrating strings.  String theorists, whose confidence was enforced by the high level of consistency of their logic construction with respect to Quantum Mechanics, were however feeling the still missing inclusion of the gravitational force. The last decisive step toward the unification of all known forces.  

 Relation between the total number D of spatial dimensions and the brane dimensional category p

From String Theory to Membranes' Theory

This unification came in 1995: we are living an epoch when the fundamental theory (the one holding all others as consequences), of all is no more String Theory.  Rather, its successor M-Theory where the “M“ means Membranes.  Extended objects with one or more compact spatial dimensions named p-branes, objects of p-dimensional spatial extent: 

• 0-brane is a point particle;
• 1-brane is a string;
• 2-brane may be imagined the Ocean's surface, wrapping the Earth and propagating in the 4D spacetime of the Solar System.

Describing a brane as an object propagating through a spacetime we’d be placing the spacetime on a primary, and the brane on a secondary footing.  In the reality, the String Field Theory (M-theory) on the brane is the primary concept, whereas the spacetime is a derived concept.  The graphic above on right side, shows the relation between dimensionality D and the dimensional class p of the membrane.   The history of a p-brane may be described mathematically by a map φ : W → M , where:

• W  is a reference (p + 1)-dimensional manifold;
• M  referred to as "target space”, represents the spacetime through which the brane propagates (e.g. in the case of the Ocean, the 4D spacetime);
• φ (W )  is a “worldvolume", same meaning presented here.

Membranes are Elastic Objects

Membranes, those of the everyday life, knowingly are elastic.  Same way, also Branes are not static objects, rather dynamical: an example on left side shows how the worldsheet topology change due to emission and reabsorption of open and closed strings.  The oscillations are sections of the normal bundle to  φ (W ) ⊂ M  described by a (p + 1)-dimensional scalar field theory on the brane.  All branes are elastic. But it is a fact that the torsion of some is such that let them be extremely rigid.   The video below, quite well represents a 3-dimensional elastic brane. It is a 4K Ultra High Definition video, meaning that: 

• due to its extremely high data rate a fibre optic connection to Internet is recommended; 
• best seen in full screen.

 Worldsheet topology changes due to emission and reabsorption of open and closed strings (  abridged by Johnson/2000)

As additional visual aid to the comprehension of subjects whose full meaning is only mathematical, we recommend to see the video below, representing something whose shape is well imprinted in the mind of whoever.  Taking as an example the Ocean of the Earth, the scalar field would represent the height of the waves and the Ocean’s surface may be imagined as a 2-brane wrapping the Earth and propagating in the 4-dimensional spacetime of the Solar System.  The D–brane is a dynamical object, and as such, feels the force of gravity.  The tension of the brane controls its response to outside influences trying to make it change its shape.  

These subjects are out of the scope of these pages devoted to the fundamental role of the Measurement, as applied in the Electronic Inspectors.  We’ll simply outline the position reached during last twenty-five years. The evidence of the varieties of Calabi-Yau manifolds converges with others experimental and theoretical ideas, backing the a conjecture about the coexistence of a huge number of other Environments and new continously being nucleated.  A process not particularly different than the nucleation of CO2 bubbles in the Carbonated Beverages.  Also, a process whose physics is not particularly different than the nucleation of CO2 bubbles in the Carbonated Beverages.   The term Environments here used as a modern Quantum-epoch synonimous of the classic terms: worlds or universes.  However, the entire tree-like sequence of ramifications or branchings, interposed between Us and now and the common Origin, featured since the very start identical physical laws and values for the geometric, physical and chemical constants.  But, this is probably not true for other branches.  A bubble nucleated before or after our own, has a low probability to be founded over the same set of geometric, physical and chemical constants.  Environments where, as an example, some of the most basic axioms of the Arithmetic look different that what we know, so that:

•  1  <  0
•  a  ≠  a
•  a b  ≠  b a
•  a ( b + c)  ≠  a b + a c
•  π  ≠  3.14159265358979323…..

built over varieties of Mathematics are Physical Laws different than what we know, starting by the values of the fundamental physical constants. So different that, as an example:

• gravitational                          G  ≠  6.67384× 10^-11 m³ kg^-1 s^-2 
• speed of light in vacuum       c  ≠  299 792 458 m/s.

Questioning ourselves about the meaning, the signification of a tree-like structure coexisting with our own but nearly disconnected, has to remember Hermann Minkowski’s  intuition of 1907.  The same breakthrough explained so well nearly 50 years later by Albert Einstein: 

             “4-dimensional space contains infinite 3-dimensional spaces”

  Three thousands years of evolution of our ideas about space. Minkowski's space-time followed the ancient conceptions. Since two decades a manifold with at least 6 hidden spatial dimensions has replaced the relativistic 4D hypercube (  abridged by New Scientist)

These spaces coexist and are different leaves of the foliation. The dimensionality of the space whose 3-dimensional leaves we, our machinery and devices are populating, is today object of intense theoretical and experimental investigations.  An eufemistic way to say it is unknown. We saw above that the dimensionality D of the branes, ranges from 3 to 11.  This implies on practice the coexistence of, at least, all of these String Field Theory varieties.  A coexistence in weakly-interfering branches of the Multiverse.  On practice, other histories.  As a matter of fact, some of the components of that complex superposition of Signals too rapidly dismissed to the rank of Noise, are Signals from those other branches. Signals deriving by the most common and generalized process in a Superposition: interference.

Superpositions are Foliations

Henri Poincare’, the same French scientist who also seeded the ideas later developed by Minkowski, started in his thesis a new chapter devoted to bifurcations.  Bifurcations are considered in a wide range of physical phenomenons, the most known being the Dynamical Systems.  Bifurcation Theory studies the bifurcation hyper-surface in the space of vector fields.  Some common examples of vectors: wind speed, particles’ collisions or photons' 4-vectors when dealing with the electromagnetic phenomenas.   They all are bifurcations the:

• Feynman's paths, he integrated in 1948;
• Everett's states, he described in 1957, constantly branching, transforming from state to superposition with each successive measurement.

As we saw above, looking for a way to integrate in a unified theory all of the known forces and particles yet discovered, String Theorists passed to a description of Membranes (M-theory): elastic membranes.     

2D membranes, in the mainstream version of the theory actually accounting for total 11 space + time dimensions.  This position, an effect of the  introduction of a new branch of Geometry (Calabi-Yau manifolds) in Physics, has the force to appear fully justified also to non-specialists.  When simply looking at the figures on side, we discover that the manifold on right side is the same as that on left side, after having been wrapped up as a leaf inside a foliation.   

 One and the same manifold, however looking two completely different ways.  That at right side obtained by the one on left side, after having been wrapped up as a leaf inside a foliation.  Easy to discern that foliations are tree-like structures based on branchings (  abridged by I. Moerdijk, J. Mrčun/2003)

And, it is not something possible, for its existence is assured by a theorem.  Visibly, the manifold above at left side describes a tree-like structure of multiple bifurcations.   A multiplicity first detected by Schroedinger.  That multiplicity Everett related in 1957 to a tree-like structure, well before Yau’s demonstration.  Net sum of what precedes is a convergence of disciplines toward a single new paradigm, one which is going to open a wealth of technological opportunities.

Ranges of Grey

And, if you followed us until this point of the bleeding sphere of Physics, no doubt you are perceiving the strident contrast between what we are used to consider reality and what Reality really is. How we formed that idea of reality?  Part by genetic inheritance and the greatest share by the billions of billions of frames.  Frames recorded in our own memory as engrams, originated by the encoding in chemical form of the incoming electromagnetic form of the energy.  A multitude of “good measurements”, following Hugh Everett's III original definition abridged by the modern Quantum Field Theory, allowed this conversion.  The camera at right side is a professional instrument, one of the best existing today with its 4K capability, something which shall be a standard in ~6 years. The camera is based on a massive 3840 x 2160 pixels sensor.   Ultra HD 4K is four times the size of 1080HD, with twice the vertical and twice the horizontal resolution. The video below, shot in black and white following the laymen point of view and in ranges of grey for a Machine Vision professional, has a size reduced to 3840 x 1600 pixels.   Just 1 minute and 42 seconds of frames resulted in a file size of over half a gigabyte.  The best way to really see the amount of informations present in the video implies a 4K display and full screen.  If streaming it by Internet, it’d be necessary a fibre optic connection.  A video showing, in the words of Sky News™: “The city of London, shrouded in a Dickensian blanket of smog, as light winds, Sahara dust, and dirty air from the continent conspire to produce air pollution levels right at the top of the chart” is here available for download.  We choose this particular video to homage the United Kingdom. The Multiversal Revolution, started in 1957 by a young student at the Institute of Advanced Studies of the quiet Princeton, New Jersey, USA, is since decades carried on toward further impressive successes in the United Kingdom.  At Oxford, Cambridge, Leeds, London, Cardiff, Birmingham, Manchester, Edinburgh to name only a few of the many towns hosting the Universities and Quantum Computation Laboratories where the research happens between the fogs visible in the video.  Looking the footage shot by mean of one of the best existing cameras, maybe you’ll perceive how far from the Reality is the idea of reality in the minds of the peasants populating the video.  This video introduces you to the following section devoted to the first practical uses of the multiversal revolution, devoted to the computation made in portions of the Hilbert Space existing on the Earth, in our Laboratories and Factories.

Links to related pages:

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Links to pages on other subjects:

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PackagingsLinks to the subpages: Links to the other pages: