“Trigger: an event that is the cause of a particular action, process, or situation

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Introduction

In the the field of Industrial Automation and in that of the Electronic Inspectors (Bottling Controls), however named (i.e., “Bottle Presence”) 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 (e.g., camera-based inspections) analog variable measurement system, and also a permanent input device to the Automations’ Programmable Logic Controllers (PLCs). 

To understand what really is an Electronic Inspector, we are going to analise its most basic and ever present inspection, using 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 we learnt 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. 

Part of the subjects of these web pages devoted to “Physics of Triggering”, however modern and complex they may appear, were conceived around one century ago.  The Twentieth Century was an epoch of intense debate abot the nature of space, time and matter.  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, Werner Heisenberg, Erwin Schroedinger and Max Born.


 Part of the subjects of these web pages devoted to “Physics of Triggering”, however modern they may appear, were conceived around one century ago.  Epoch of an intense debate abot the nature of space, time and matter.  Conceived mainly by the physicists joint in the meeting visible above.  Between them Max Planck, Henri Poincare’, Marie Curie, Hendrik Lorentz, Albert Einstein, Niels Bohr, Paul Dirac, Paul Langevin, Werner Heisenberg, Erwin Schroedinger & Max Born

Events: a historical approach to the key-concept of Trigger






Triggers are the most elementary measurement systems, the most basic function in all of the Machines or Electronic Inspectors






In the celebrated Oxford Dictionaries of the English language, a Trigger is defined as: 

          “an Event that is the cause of a particular action, process, or situation”.  

It is 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: “What is an Event ?”  

Desiring to understand what really is a Trigger, we look at Physics and its definition of Event.  Electronic Inspectors are devices 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 established by us.  After this, it is evident that Physics is not foreign to Bottling Controls, rather its core.


Events: triggered in a 4-dimensional space

In 1907 Hermann Minkowski, also known because having been Albert Einstein and Max Born's teacher of Mathematics in the ETH (at Zurich, Switzerland), reformulated and extended ideas of the physicist and mathematician Henri Poincare’. The result, shown in the figure below, was the first 4-dimensional system of space-time coordinates: 


 Time evolution of a dot (0-D), a string (1-D) and a square (2-D).  All of the terms readable above were coined more than one century ago, apart “string” and “brane”, whose meaning date back after 1974 (image abridged by MissMJ, 2013)




“No absolute motion exists since there are many spaces, not just one”

Hermann Minkowski, 1907 

   Since 1907 it is known they coexist infinite 3-dimensional spaces referred to each one instant of Time








“...there is an infinite number of spaces, which are in motion with respect to each other”

               Albert Einstein, 1952






  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. Erwin Schrödinger, when deriving the exact solutions for a Hydrogen atom in a closed finite System, demonstrated that in this case, the number of bound states is finite. Because of this last reason, that coexisting spaces are an infinity does not implies coexistence of an infinity of material objects, each one into each one of the infinite spaces

 Event as it was conceived in 1907. The triggered Event lies along a worldline: each one dot identified by the four coordinates (x, y, z, t) of which three here shown (x, y, t).  Past lightcone of the (Trigger) Event, is that one of all worldpoints which send Signals to the Event. Then, only to Events happening into the Past light cone it may be attributed a causative correlation to the Trigger Event, implying that Events happening in the volume around the bicone, about which we’ll never know.  The future lightcone consists of all worldpoints which are going to receive Signals by the Trigger Event.       


Minkowski realized that: 

  • the Relativity Principle implies many times and spaces which, in turn, imply that the world we inhabit is 4-dimensional. What explained 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 3-D, mirrors the curvature of its worldline, as seen in 4-D;
  • pairs of ordinary mechanical quantities are space and time components of 4-dimensional vectors and the ordinary electromagnetic quantities are components of new types of 4-dimensional structures.

The Event became a dot of space in the 3-dimensional 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 travelling at their characteristic maximum speed of propagation c (299 792 458 m/s).   Albert Einstein, after initially dismissing his former ETH professor of Mathematics idea of spacetime in 1907, then working in the Patent Office in Bern, Switzerland, was struck by what he later called his “happiest thought”.  Einstein, quite notoriously, was not a physicist advancing by formulae, rather one advancing by mean of mental images, depictions of physical phenomenas.  

He was suddenly illuminated by the far reaching consequences of the mental image of a man falling off the side of a building and feeling no gravity at all, as we all 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.  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.



Einstein about an infinite number of spaces in motion relatively to each other

In 1952 Albert Einstein explained with a beatiful example the brakthrough fathered in 1907 by Hermann Minkowski about the coexistence of infinite superimposed spaces.  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.”   

What above 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

Notoriously, 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 observers or, automatic measurement instruments like the Bottling Controls in accelerated motion.  

All motions accelerated in a 3-D space may be conceived alike movements over a curved 4-D space-time hypersurface.  

  S is a 3-dimensional slice of space passing thru the Event PSimultaneity 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 Eventmay 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, Thorne, Misner, 1973) 

The graphic on right side is an example.  Here, S is a 3-D 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. 

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:

  • curvatures are different;
  • and/or slopes are different;
  • observers, or measurement instruments, are different;
  • observers’ clocks read in P a different proper time (than what read by the observers  a, b, c,…).

In the example above, S is a 3-D 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. 




                     “Foliation is a serie of spatial slices of constant Time”



Geodesics' basic



1. they do not exist straight lines. These are limits reached when the curvature of the hypersurfaces => 0.  The physical space is curved and the equivalent of the straight lines of a flat geometry are “geodesic"

2.  geodesic are arcs of great (maximum) circle;

3.  the shortest line joining two points in a curved geometry;

4.  each one worldpoint is crossed by at least one geodesic;

5.  two Events may be connected by more than one geodesic.

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.   

A first, purely geometric example of time-ordered sequence of spatial hypersurfaces in the figure down, showing:

  • a vector separation V  =  P P0 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” (each slice has two surfaces).

In the example presented below, the vector V  pierces ~ 4.5 surfaces:



  A first geometrical example of foliation where slices’ separation is purely parametric (abridged by Wheeler, Thorne, Misner, 1973)



 Dynamical effects are what we and our measurement instruments perceive along the Time (horizontal) axe. On the microscale the texture of each leaf of space is not fixed, rather rapidly fluctuating & bulging out in knots and holes






What is a Triggered Event ?   

General Relativity point of view 1:

"it is a label applied to a world point part of an one-parameter hypersurface, in a family of hypersurfaces




Applying these concepts to the physical space-time, a slice is a surface of constant Time.  

The figure below on right side describes a Foliation of the space-time manifold M, which better  represents the reality than the one we examined before, derivating by the one century old Minkowski’s flat-space M:  

  • bumps, wiggles and waves due to the presence of matter and fields energy;
  • Event on the slice Σ at the Time t;
  • an adiacent posterior space-like slice Σ, referred to the Time t + dt.

The Time coordinate parameter t is only a freely chosen label to distinguish one spacelike hypersurface from another.  Then, is this the answer to the fundamental question: ...what is a Triggered Event ?  It is a label (or marker, or identifier) applied to a worldpoint part of an one-parameter hypersurface in a family of hypersurfaces. 

The proper time s referred to the Event, is measured along the worldline crossed by the bicone vertex.  The Reader is now suggested to recall some of the properties ruling the hypersurface of the Present, seen in the precedent figure.   No transmission of electromagnetic signals or movement of matter, happens thru the individual slice, rather thru different slices (or, leaves or sheets) of the Foliation.  



An Event after 1915 til ~1960 is a worldpoint in a spatial-only leaf (or slice, or sheet) part of a curved foliation: the manifold M.  The manifold is a serie of leaves and each leaf a surface of constant relativistic coordinate time (t, in the figure). The Event, 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 subspace of the Multiverse.   The Event acquired the meaning of a worldpoint where it happens an exchange of energy 















In this picture, matter and signals were considered transferred only toward the Future light cones when General Relativity was created.  Following the 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 (CTCs) of extremely brief duration.   Moreover, Special Relativity defines strict successions of Events. 

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 worldwide standard academic textbook of General Relativity


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 meaning acquired by the term Event may be inferred by the figure below, adapted by the celebrated worldwide main textbook of General Relativity, Gravitation , by J. A. Wheeler, C. W. Misner and K. Thorne, 1973.  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 Bottling Controls, the Electronic Inspectors, and in the 




 “The crossing of straws in a barn full of hay is a symbol for the worldlines that fill up space-time.  By their crossings and bends, these world lines mark Events with a uniqueness beyond all need of coordinate systems or coordinates.  Typical Events in the diagram, from left to right (black dots), are: absorption of a photon; reemission of a photon; collision between a particle and a particle; collision between a photon and a particle; another collision between a photon and a particle; explosion of a firecracker and collision of a particle from outside with one of the fragments of that firecracker” (original quote by Wheeler, Misner, Thorne, 1973) 





What is a Triggered Event ?   

General Relativity point of view 2:

“it is a point of space-time along a geodesic, where it happens an exchange of energy




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 Food and Beverage Bottling 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: 


 All interactions affecting matter particles are due to an exchange of force carrier particles, a different type of particle altogether. “Particles” a mere dotlike idealization of real waves of energy. What we normally name forces are actually the effects of force carriers' “particles” on matter “particles”  




What is a Triggered Event ?   

General Relativity point of view 3:

"a device to calculate and constructing a leaf of history that slices through the higher dimensional space



The figure above showing a firecracker is just a compact representation, corresponding to five superimposed slices, what can be perceived after its comparison with the figure below.  Above, 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 4-D 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 down in the following figure.  Here, five Events, as seen by Physics point of view 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, 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 only a few years later in mid-60, 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 (image abriged by J. A. Wheeler, C. W. Misner, K. Thorne, 1973) 



REEB FOLIATION  Reeb's foliations are topologic varieties featuring one genus. The genus is perceived as a hole, as seen by each Observer lying out of the throats. It also represents a way to join two separate places by mean of microscopic CTCs (image credit Tambara Institute of Mathematical Sciences, University of Tokio, 2014)

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. Until now we treated foliations with simple topologies, flat or curved non-connected  surfaces. Mathematicians nearly always precede Physicists. 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 foliation at left side, allows a glimpse in this new World.


 HOPF's FOLIATION   Hopf's foliations are basic topologic varieties featuring one genus. The genus is perceived as a hole, as seen by each Observer lying out of the throats. It also represents a way to join two separate places by mean of a microscopic Closed Timelike Curves (image credit Tambara Institute of Mathematical Sciences, University of Tokio, 2014)



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 Shanghai in China or at Muenchen, in Germany, and 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:


  Knowing how to encounter the way, the World we live shows its true nature: it is Quantum at all scales, and not only in the subatomic. This video authored by Festo® is devoted to a technological application of Industrial Automation



The ever changing fabric of space

  Max Planck, the first discoverer of Quantum Mechanics. He is the one who, first in the humanity, conceived that a pendulum could not be oscillating continuously, rather step-by-step


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 4-dimensional 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 :


                                    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.                                                                                                             

  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





Also, we have to consider that in the prior RLC circuit example, human normal sensibility to Events is limited to those whose duration is > 10 ms.    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. 


                                                                                                               

Given a Time base 750 Ghz  = 

 =  1.3 picosecond

 =  0.000 000 000 0013 second


what is the equivalent distance covered by light ? 



Equivalent spatial displacement

     x  =  c t  =  400 nanometer

       =  0.000 000 4 meter




 A scanning electron microscope image of a colony of algae.  In the bottom indicated the reference scale of 5 micrometer.  The small white coloured details in the image are the clouds of electrons around atoms of Ca, C, N, O and H.  A signal whose frequency is 750 GHz has an equivalent spatial displacement of 400 nanometers, say  six times smaller than the individual algae diameter



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.  

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.

 



















 

    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


Our own observation of the 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.  Here 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.  



More than explanatory power

Virtual particles, positrons,…?  One more time we stress a fact: all what precedes and follows cannot be dismissed like …theory.  These interpretations of facts and theories also 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 / 2π  


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.


 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 (credit A. D. McNaught, A. Wilkinson, 2006)





    Click to enlarge. Between 1953 and 1973 two mathematicians conceived the spatial properties underlying the experimentally observed phenomena interpreted in the framework of what today is known as String Theory.  By mean of String Theory many first principles in Quantum Field Theory (QFT) are explained, or get further insight


Introduction




In 1953 the 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 


   We suggest the vision of this 3D animation full screen, High Definition and possibly wide bandwidth




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 Kaehler 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 6-dimensional inner space imagined existing in each one point of the 4-dimensional space-time we feel perceive directly (3-D space) or indirectly (1-D time).   But, why to add dimensions, when to have less 


  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

  







 

 Actually, 17 elementary particles account for the known behaviour of the world, in the Standard Model (credit MissMJ, on data by J. Beringer et al., Particle Data Group, Lawrence Berkeley National Laboratory)






M. B. Green

should apparently mean to have also less complications ?   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. 

Michael Boris Green, Lucasian professor of Mathematics at Cambridge, UK, is one of the founders of String Theory    

  


Who shaped the times we are living ?

In the following we prefer to quote a column in the article by M. Chalmers, appeared in September 2007 in physicsworld.com.  

It is important to understand who, when and what discoveries shaped the idea of Reality considered Truth today by Physics:

● 1968  Gabriele Veneziano discovers that the Euler “beta function” brings order to the measured scattering amplitudes of different types of hadrons.

● 1970  Leonard Susskind, Yoichiro Nambu and Holger Neilsen independently identify Veneziano’s amplitudes with solutions to a quantum-mechanical theory of 1D bosonic strings.

 1971  Claud Lovelace realizes string theory requires 26 dimensions; Yuri Gol’fand and Eugeny Likhtman discover supersymmetry in 4D; John Schwarz, André Neveu and Pierre Ramond realize that string theory requires supersymmetry to accommodate fermions as well as bosons; Gerard ’t Hooft shows that electroweak unification proposed by Steven Weinberg in 1967 is “renormalizable”, thus making gauge theories physically viable. 

● 1973  Julius Wess and Bruno Zumino develop supersymmetric quantum field theories; David Gross, Frank Wilczek and David Politzer discover asymptotic freedom and so establish QCD; combined with electroweak theory, the Standard Model is established.

● 1974  Schwarz and Joel Scherk (and, independently, Tamiaki Yoneya) realize that string theory contains gravitons, and propose a unified framework of quantum mechanics and general relativity; Sheldon Glashow and Howard Georgi propose grand unification of the Standard Model forces via the symmetry group SU(5). 

● 1976  Stephen Hawking claims that quantum mechanics is violated during the formation and decay of a black hole; mathematicians reveal Calabi–Yau spaces.

● 1978  Eugène Cremmer, Bernard Julia and Scherk construct 11D supergravity, which incorporates supersymmetry in general relativity.

● 1981  Schwarz and Michael Green formulate Type I superstring theory; Georgi and Savas Dimopoulos propose the supersymmetric extensions of the Standard Model.

● 1982  Green and Schwarz develop Type II superstring theory; Andrei Linde and others invent modern inflationary theory from which the multiverse follows.

● 1983  The discovery of W and Z bosons at CERN seals a decade of success for the Standard Model; Ed Witten and Luis Alvarez-Gaumé show that the gauge anomalies cancel in Type IIB superstring theory.

● 1984  Green and Schwarz show that the anomalies in Type I theory cancel if the theory is 10D and has either SO(32) or E8 × E8 gauge symmetry; T duality is discovered.

● 1985  Gross, Jeff Harvey, Ryan Rohm and Emil Martinec construct heterotic string theory; Philip Candelas, Andrew Strominger, Gary Horowitz and Witten find a way of compactifying the extra six dimensions using Calabi–Yau spaces.

● 1987  Weinberg uses anthropic reasoning to place a bound on the cosmological constant.

● 1994  Susskind proposes the holographic principle by extending work done by ’t Hooft.

● 1985  Paul Townsend and Chris Hull, and Witten, propose that Type IIA theory is the weak-coupling limit of 11D “M-theory”; Polchinski discovers D-branes; Witten and others conjecture that all five string theories are linked by dualities, some of which are facilitated by D-branes.

● 1996  Witten and Polchinski discover that Type I theory and SO(32) heterotic theory are linked by S-duality; Witten and Petr Horava show E8 × E8 is the low-energy limit of M-theory; Strominger and Cumrun Vafa derive the Bekenstein–Hawking black-hole entropy formula using string theory; Susskind and others propose a candidate for M-theory called Matrix theory.

● 1997  Juan Maldacena discovers the equivalence between string theory and quantum field theory (AdS/CFT duality), thus providing an exact manifestation of the holographic principle.

● 1998  The experimental discovery of the accelerating expansion of the universe suggests a small, positive vacuum expectation value in the form of a cosmological constant; Lisa Randall and Raman Sundrum propose braneworld scenarios as an alternative to compactification.

● 1999  Gia Dvali and Henry Tye propose brane-inflation models.

● 2003  The KKLT paper shows that supersymmetry can be broken to produce a small, positive vacuum expectation value using flux compactification to deal with extra dimensions; Susskind coins the term “landscape” to describe the vast solution space implied by flux compactification, and invokes the anthropic principle and the multiverse to explain the cosmological constant; the KKLMMT paper extends KKLT to cosmology.

● 2004  Hawking admits he was wrong about black holes and concedes bet to John Preskill.

● 2005  String theory is mentioned in the context of RHIC quark–gluon plasma thanks to application of AdS/CFT, thereby returning the theory to its roots as a description of hadrons.























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 [credit J. Beringer et al., Particle Data Group, Phys. Rev. D86, 010001 (2012)]    





 

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 above (credit J. Beringer et al., Particle Data Group, Lawrence Berkeley National Laboratory) 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. 

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 stringlike particle, objects having extension in only one dimension. In superstring theory the strings are exceedingly small and move through a 10- or 11-dimensional 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. 


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, what was since centuries named interference.    


                      A first glimpse to the true nature of Measurements: the fabric of a 2-brane is a web of bifurcations and interferences.  Measurements are both topologic changes, bifurcations or interferences, experienced by the topologic variety, whose character is visibly multiply-connected

 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 possible values allowed by the fundamental physical constants and by the physical laws established with these constants (image credit M. B. Green, 1986)



Bifurcations

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.

Today a bifurcation is considered the 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





  Today a bifurcation is considered the 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  






Loopbraids 

The key opening the inner space’s door

And, one more time, we know the Readers' feeling about all this. 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 here, 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 pipes (or, strings or membranes) and not a geometry made of pipes in the space.  In the modern scenario, dimensionality is a feature of the strings or pipes, 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    



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.  Is it there really so much space in the microscale, so to host a loopbraid like this, with many others encased one into each other genus ?   Remember Minkowski’s discovery of 1908 ?  ...each one dot of 3-dimensional space is infinitely extended in the 4-dimensional space.  We’ll see in the following the convergence of theory and experimental results toward at least ten dimensions. 

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 3-dimensional projection of a 6-dimensional 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 6-dimensional Calabi-Yau manifold, totalling nine space dimensions plus Time.





Key concept to such new scenario is the compactness of the 6-D 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 (1-D) 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.  



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.  


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








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 4-D 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 : WM , 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 tetradimensional spacetime);
  • φ (W )  is a “worldvolume", same meaning presented here.

   Three D–branes at different positions, with various strings attached, representing particles different than gravitons (abridged by image C. Johnson, 2000)



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, however streamed, 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 C. Johnson, 2000)




  (Mem)branes can be also extremely elastic. The Ultra High Definition 4K video above represents the concept (credit Quayola, 2013)







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.  


   An astounding relation exists between the branes and the space we inhabit (abridged by footage credit NASA, et. al.). The video is here available for download. 



   A macroscopic change of the aspect of a common object, without any change of its Topology.  Topology encodes the essential characteristics of spaces and of their relations






  The initial 3000 digits of π. The sequence, extending itself toward infinite, derives by the initial unique conditions existing in the pocket we inhabit.  It is a local condition, a Universal but not Multiversal constant






















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 underlying enormous amount of 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.   Also, a process whose physics is not very 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…..

and, built over such 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: 

                   “A 4-dimensional space contains    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 image credit 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.



Superposition, on practice
















Henri Poincare’, the same 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 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.  Bidimensional 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 below, we discover that the manifold on right side is the same of that on left side, after having been wrapped up as a leaf inside a foliation. And, it is not something possible, because its existence  is assured by a Theorem. 

Visibly, the manifold below on left side describes a tree-like structure of multiple bifurcations.  

 One and the same manifold, however looking two completely different ways. The one right side has been obtained by the one on left side, after having been wrapped up as a leaf inside a foliation. Easy to discern that foliations are branchings





A multiplicity first detected by Schroedinger in what the equation [5] implies: that multiplicity Everett related in 1957 to a tree-like structure, well before Calabi’s geometrical conjecture and Yau’s demonstration.  Net sum of what above 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, 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 majority 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.  

 At a massive 3840 x 2160 pixels, Ultra HD 4K is 4 times the size of 1080HD, with twice its vertical and twice the horizontal resolution. But, also this Australian-made professional camera, one of the World's best, is not capable to record informations corresponding to what Reality is made of (image abridged by picture credit Blackmagic Design Pty. Ltd., 2014)







The camera on right side is a professional instrument, one of the best existing today with its 4K capability, something which shall be a standard in 10 years. The camera is based on a massive 3840 x 2160 pixels sensor.  Videos' maximum frame rate is 30 fps. 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.   1 minute and 42 seconds only of informations 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 chartis 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, Birmingham, Edinburgh to name only a few of the many towns hosting the Universities and 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.


 The Multiversal Revolution started by a student in 1957 in the Institute of Advanced Studies at Princeton, New Jersey, USA, is since decades carried on toward further successes in the United Kingdom.  At Oxford, Cambridge, Leeds, London, Birmingham, Edinburgh to name only a few of the many towns hosting the Universities and Laboratories where the research happens between these fogs (video credit B. Bell, 2014)


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