Root Cause as Partially Ordered Set.  The New Meaning for Causation



























Supposing the Universe as being 4D, how many surfaces (or, leaves or sheets) build it up ?

                 

Since over two decades it is however assumed that the true dimensionality is > 4.   Strictly implying space for an infinite amount of coexisting realities, whose projections are impossible to detect because wiped by destructive interference



























Until now, we have supposed and written about the infinity of curves joining the point p in the Past to the point q in its Future.  But, that infinity, also if aesthetically attractive, hits against the entire quantum paradigm.  Readers are invited to perceive that those infinities and infinitesimals, so frequently recalled in Classic Physics and Engineering computations (for example, Fourier's or Taylor's series), are implicit statements about the nature of the space and time.  A statement of space-time smoothness.  Smoothness contradicted by the experimental and technological results of the last ninety years.   Could the space be discrete?   One where all exists and moves exclusively in discrete amounts, thus relegating infinities and infinitesimal to the rank of useful mathematical abstractions?   The most modern concept about what a Causal Relation is, derived quite recently by the intersection of two branches of Mathematics with the today dominating branch of Physics: 

Order Theory,
Minkowskian Geometry, 
Quantum Mechanics.   
The Causal Set hypothesis assumes that the structure of spacetime is not continuous, rather discrete.  Discrete and structured. Something completely different than what we implicitly assume when, i.e., solving basic kinematical problems.  Discrete space means that lengths in our ordinary 3D space are made of a finite number of elementary lengths.  Lengths representing the: 

smallest possible length le,
flow of time occurring in a serie of elementary ticks of duration te, representing the shortest time interval.


Why Causal Sets

Nearly whoever has yet observed on his/her own that all movies, photographic or digital, present a continous, fluid evolution of the Events also if intimately built assembling a great number of indipendent frames.  And whoever use a computer knows to have access to OS settings defining the frame-rate of the signal outfeeding by the video graphics card.  Setting a low frame-rate let whatever image, not just a movie, appear in its real nature: a collection of frames, each of them closely recalling the 3D hyper surfaces associated to each instant of Time named Events by Relativity.  In this banal widespread observation there is much more truth than expected.  Causal Set Theory proposes itself as the more fundamental theory, capable to solve the four problems listed below:

Electrodynamics.  The electromagnetic (abbreviated em) spectrum visibile in the figure below, represents the range for the frequencies and associated wave-lengths of the electromagnetic interaction.  Interaction transferring quantised (discrete) packets of energy named photons.  The double-side (left and right) arrow-like shape of the diagram has a meaning.   It is a continuous spectrum of infinite extension.   The present  Electrodynamics Theory predicts that the allowed frequencies of photons extend continuously from zero to infinity.  Frequency zero (right side below) implies infinite a wave length tending to infinite: an unreachable effective limit.  But, what about the left side, corresponding to extremely short wave-lengths ?    Planck’s law  E = hν implies that a photon of arbitrarily large frequency ν has arbitrarily large energy E.   This, in open contradiction with the principle of local conservation of energy, suggesting that no infinite energy photon may exist.  Then, there ought to be a natural cutoff of the electromagnetic spectrum in correspondence to a maximum allowed frequency νmax, cutoff to the date not conceived by the Electrodynamics Theory, mainly based over a Maxwell’s studies of nearly two centuries ago.  Since the frequency of a photon is the inverse of its period, we see how a discrete temporal structure provides a natural cutoff.  In fact, the minimum time interval te implies a maximum frequency νmax = 1/te.

 

Quantum Causal Theory 








Discrete space” means that lengths in our ordinary 3D space are made of a finite number of elementary lengths









How many points lie between two points?  Twenty-five centuries ago the Greek philosopher Zeno first observed the kinematical paradox deriving by the idea that a line joining two points may be infinitely divided.  Since 90 years, also an infinity hitting against the Quantum paradigm, itself derived by the most objective facts.  Infinities and infinitesimals, so frequently recalled in Classic Physics and Engineering computations, i.e., Fourier, Taylor or Mac Laurin’s series, are implicit statements about the nature of the space and time.  A statement of space-time smoothness.  Smoothness contradicted by the experimental and technological results of the last ninety years.  Could the space be discrete?  One where all exists and moves exclusively in discrete amounts, thus relegating infinities and infinitesimal to the rank of useful mathematical abstractions?   The most modern concept about what a Causal Relation is, derived quite recently by the intersection of two branches of Mathematics with the today dominating branch of Physics: 

  • Order Theory,
  • Minkowskian Geometry, 
  • Quantum Mechanics.   

The new proposed principle is the Causal Metric Hypothesis: 

  • characterizing the observed properties of the physical universe as manifestations of causal structure,
  • where the metric properties of classical spacetime arise from a binary relation on a set, representing direct influences between pairs of events.

Rephrasing what before, the Causal Metric Hypothesis assumes that the structure of spacetime is not continuous, rather discrete and structured.  A point of view completely different than what we implicitly assume when, i.e., solving the basic kinematical problems constantly presented by the industrial machinery and equipments. “Discrete space means that in our ordinary 3D space lengths are made of a finite number of elementary lengths.  

   (Click-to-enlarge). Several ways exist to go from a point P  to a point Q  in its Future. Green and orange coloured paths respecting all four Partially Ordered Sets’ rules, discovered 90 years ago and detailed below. The orange coloured pass through an intermediate point R .  Three of the paths outfeeding R  in red colour, correspond to Past-oriented histories prone to become causally pathological closed cycles or chains, violating the Causal Metric rules. The Time axe hints to the existence of an initial condition
















Lengths representing the: 

  • smallest possible length le,
  • flow of time occurring in a serie of elementary ticks of duration te, representing the shortest time interval. 

The Causal Metric Hypothesis, new paradigm for causation, rejects the:

  • notion that spacetime is a manifold, 
  • existence of static background structure in the universe, 
  • symmetry interpretation of the relativistic covariance, 
  • several other assumptions. 

The sector is based over studies started one century ago and they exists today two Causal Metric Hypothesis:

  1. Quantum Causal Metric Hypothesis, states that the phases associated with directed paths in causal Configuration space, under Feynman’s Sum-Over-Histories approach to Quantum theory, are determined by the causal structures of their constituent Universes, themselves part of the wider closed quantum system named Multiverse;
  2. Classic Causal Metric Hypothesis, states that the metric properties of spacetime arise from a binary relation on a set, representing direct influences between pairs of Events.  


Why Causal Sets





E = hν


 ν → 

 E →   





νmax = 1/te


Nearly whoever has yet observed on his/her own that all movies, photographic or digital, present a continous, fluid evolution of the Events also if intimately built assembling a great number of indipendent frames: a collection of frames, each of them closely recalling a 3D hyper surfaces associated to each instant of Time named Events by Relativity.  In this banal widespread observation there is much more truth than expected. Causal Set Theory proposes itself as the more fundamental theory, probably only one truly capable to solve all five problems listed below:

  1. Electrodynamics.  The electromagnetic (abbreviated em) spectrum visibile in the figure below, represents the range for the frequencies and associated wave-lengths of the electromagnetic interaction.  Interaction transferring quantised (discrete) packets of energy named photons.  The double-side (left and right) arrow-like shape of the diagram has a meaning.  It is a continuous spectrum of infinite extension.   The present  Electrodynamics Theory predicts that the allowed frequencies of photons extend continuously from zero to infinity.  Frequency zero (right side below) implies infinite a wave length tending to infinite: an unreachable effective limit. But, what about the left side, corresponding to extremely short wave-lengths ?    Planck’s law  E = hν implies that a photon of arbitrarily large frequency ν has arbitrarily large energy E.  In open contradiction with the principle of local conservation of energy, suggesting that no infinite energy photon may exist. Then, there ought to be a natural cutoff of the electromagnetic spectrum in correspondence to a maximum allowed frequency νmax, cutoff to the date not conceived by the Electrodynamics Theory, mainly based over a Maxwell’s studies of nearly two centuries ago.  Since the frequency of a photon is the inverse of its period, we see how a discrete temporal structure provides a natural cutoff.  In fact, the minimum time interval te implies a maximum frequency νmax = 1/te.

   The electromagnetic spectrum is based on the energies involved by Planck’s fundamental formula E = hν, where ν = c/λ,  c speed of propagation of the em waves in the vacuum, λ wave length, ν frequency, h Planck's constant, E energy of the photon (  CC 3.0)


F = -k M/R


 R → 0  

 F → 






 p = h/λ

 

  λ → 0  

  p → 



















  1. General Relativity postulates the existence of hyperdimensional foliated spatial surfaces.  Where Time plays the role of parameter whose meaning is the identification of different leaves.  With Time and space both real numbers, it becomes unavoidable for General Relativity to present frequent divergent solutions for its set of equations.  As an example, consider the gravitational potential F = -k M/R caused by a homogeneous mass M with spherical symmetry, and measured at a radius R (k universal gravitational constant).  Visibly, for R → 0 at the centre of the sphere we’d have a divergent gravitational potential ∞. Infinity whose effects, on the opposite, no one detects. To solve this and other problems, they’d be welcome new arguments in favour of a discretization of space and time.  


  1. De-Broglie relation  p = h/λ   Its validity is confirmed by several experiments accomplished during past decades.  But, when applied to wavelengths arbitrarily close to zero, photons’ momentum p becomes visibly divergent, potentially infinite.    An infinity in open contradiction with the principle of local conservation of momentum, suggesting that photons of infinite momentum cannot exist. Then, we see how the minimum length le implied by a discrete space-time structure provides a natural cutoff for the wave-length λmin = le.


  1. Quantum Electrodynamics.  Its infinite perturbation series require the existence of all the photons in the electromagnetic spectrum and, as a consequence Quantum Electro Dynamics (QED) actually predicts the existence of these (never observed) photons of infinite energy-momentum.  It is believed that the perturbation serie is divergent.  Divergence generally overlooked because QED is not a complete theory accounting for all elementary interactions.  A pseudo-explanation merely hoping that another theory fills QED's logic loopholes.  Again, we see the tension existing between fundamental quantum formulæ, like those of Planck’s or de Broglie’s, and the observed reality in the limit of the higher frequencies (shorter wave-lengths). 


  1. Astronomical Observations.  During past fifteen years they have been reduced the bulk of data registered by the humanity's best eyes: the orbiting telescopes.  The impressive result, one that no one imagined until before, is that if all the basic assumptions about space, time and Doppler-effect meaning of the redshift of the extragalactic objects is correct, then it means that along past few decades the Universe should have accelerated its expansion.  Readers understand that it seems unthinkable that our generation, has randomly born in the exact moment when the entire Universe, 15 billions of years after it was born ...decided to change kinematics. It has been christened Dark Energy what could be accelerating the expansion.  Also in this case, the Causal Sets Theory could be the solution. The cosmological model requiring Dark Energy to explain the observational data, uses a cosmological constant along all evolution. On the opposite, the Causal Sets Theory implies fluctuating values for that parameter and the simulations agree with the observations. Then, Dark Energy is not necessary to explain the observations.

It is thinkably incoherent to continue to assume continous and infinitely differentiable space and time well over one century after Max Planck changed the course of the human history by mean of his formula: E = hν. 


Partially Ordered Sets 

Causal Sets

 Different ways to go from a point in the past, to several different effects in its future. “Discrete space” means that in our ordinary 3D space lengths are made of a finite number of elementary lengths




Quoting J. Neggers and H. Kim (1996): 

“partially ordered sets (posets) have a long history beginning with the first recognition of ordering in the integers.  In the early nineteenth century, properties of the ordering of the subsets of a set were investigated by De Morgan, while in the late nineteenth century partial ordering by divisibility was investigated by Dedekind.  Although Hausdorff did not originate the idea of a partially ordered set, the first general theory of posets was developed by him in his 1914 book 'Grundziige der mengenlehre'.  It remained until the 1930's for the subset of lattice theory to blossom as an independent entity with Birkhoff's justly famous text on the subject first published in 1940.  It has only been in the last three decades that posets and their relationships to applied areas such as computer science, engineering and the social sciences have been extensively investigated.”  

Causal relations are partial-order relations and the spaces they form have a structure solely defined by causal relations.  The causality relations are not necessarily tied to the notion of a smooth space-time manifold.  In 1914, when a few had truly understood the Special Theory of Relativity, and General Relativity was still not published, the relevance of the Events' causal ordering was considered fundamental by Alfred A. Robb in his Ph.D dissertation.    



(Click-to-Enlarge) Lateral view of a Hasse diagram. Hasse diagrams are Alexandrov Intervals, or causal diamonds, filled with partially ordered, causally connected Events. 1200 Events have been randomly sprinkled following a Poisson distribution in this (1+1)-dimensional Minkowski spacetime.  Embedded in a random causal ordering. The blue-coloured zig-zag line is just one of the many ways used by an Event at p to determine an Event at q. A visibly tree-like, multiply-connected spacetime structure, showing a convergence of the independent studies in different disciplines of Science (  abridged by R. Salgado/2008)


The figure on side is a modern view, integrating the relativistic perspective of Alexandrov's Intervals and the partially ordered sets ideas of Helmut Hasse and Henri Gustav Vogt conceived early as 1895.Here it is visible a:

  • (1 + 1)-dimensional Minkowski space-time: 2D projection of the Alexandrov's Interval,
  • partially ordered set composed by 1200 elements (click-to-zoom its fine-details).  


Basic Rules of the Partially Ordered Sets




The Events visible therein, part of the huge tree-like branched multi-connected spacetime structure, have been randomly embedded (“sprinkled”) following a Poisson distribution.  Stochastic distribution as much as the physical measurements’ observed outcomes. Randomly distributed but respecting a relation of partial-order based over the following four properties (logic symbol “≺“ read as “precedes”):

  1.  transitivity               If x ≺ y  and  y ≺ z,  then x ≺ z;
  2.  non-circularity        If x ≺ y  and  y ≺ x  then x = y (no closed-timelike-curve is admitted);
  3.  finitarity                  The number of Events lying between any two fixed Events is finite;
  4.  reflexivity                x ≺ x  for any Event in the causal set.









What detailed in the pages devoted to the meaning of Root Cause following General Relativity about hyper surfaces, world points, Events, Past and Future light-cones, equitemp lines and surfaces, etc. in the figure below can be resumed by the depiction at left side.  In General Relativity it is the spacetime geometry which governs the scope of causal influence.  A geometry of curved hypersurfaces, each of them associated to a certain value for a parameter named Time.  But under the (Classical) Causal Metric Hypothesis, depicted below at right side, the geometry of the spacetime is reduced to just one of the many possible descriptive ways opened to Information's propagation.  As an example, to transmit Information from the point p in the Past to the point q lying in p's Future, the path passing through the sequence of points  → y offers a strictly causal way.  However just one of the many ways open Information propagation.

 The Classic Causal Metric perspective has an explanatory power superior to all other existing theories, General relativity included. An happy marriage of the micro and macro scales with the Quantum paradigm. Most important: finally a deep idea of what is a cause, what an effect and of their relation, exempt by circularities and assumptions contradicted by the experimental evidences (  abridged by B.F. Dribus, in eds. A. Aguirre, Z. Merali, B. Foster/2015)








As an example, the point is also receiving Information by two white coloured points (left of and  x) lying in the Future of the point p. Lying out of the light bicone centered in the point x.  What does not impede to the point to represent a configuration accounting for a Past history also including one of those white coloured dots not causally related to the point in q’s strictly causal Past. The distance between the elements makes reason of the proved discreteness of the Environment we inhabit.  We intentionally avoided to name it using the relativistic term “spacetime” because the extremely modern Quantum Causal Theory shows spacetime is an obsolete illusion dated 1908.  In the figure below (click-to-enlarge) the fine-details of the way the four basic rules of the Partially Ordered Sets apply to reality. 


Several ways exist to go from a point P  to the point Q in its Future. Green and orange coloured paths respecting all four Partially Ordered Sets’ rules.  The orange coloured pass through an intermediate Event R . Three of the paths outfeeding R  in red colour, correspond to Past-oriented histories prone to become causally pathological closed cycles or, chains. The Time vertical axe marked at right side, makes sense of the existence of an initial condition: the arrow of Time.



Partially Ordered Sets and Measurements



Implicitly, the Reader may understand now also some others far reaching consequences of the new paradigm. Some of them they yet personally observed when, as an example, performing analytic measurements in a Laboratory or by mean of automated equipments.  What yet observed when measuring whatever: weighing, counting Gamma- or X-rays, measuring diffraction when estimating the amount of sugar in a beverage, etc.  Measurements always providing visibly fluctuating results. The Root Cause of those fluctuations is not that of “errors”, rather they are intrinsic. In our standard conditions (of temperature, gravity and pressure) nearly only caused by decoherence, and not by assumed “limits” of the measurement equipment or its Detectors.  


Three of the paths joining in the spacetime the points P and Q .  Visibly, an identical spatial segment Ω corresponds to three different times, here marked with red, blue, green colours. Causal diamonds in their modern interpretation as poset’s external envelopes, make full sense of this classic observation at all scales








Two examples the:

  • brownian movements of Statistical Mechanics, i.e., the pattern observed in the hot liquid surfaces, 
  • radioactive decays, i.e., the events counted by all Gamma-Rays Fill Level Inspections.  

What expected if the matter-energy measured in the space along a certain time, is made up by discrete quantised amounts stochastically interacting with the Environment. Much more than the analytic measurement equipment or its Detectors.


Quantum Causal Theory and Root Causes 







Resuming what precedes in our root cause analytic perspective, the modern and still being developed Quantum Causal Theory, is saying that a new status observed now for a physical system, whatever its size, mass-energy or duration, has Root Cause:

  • in the superposition of a multitude of Causes,
  • more or less strictly related to the observed Effect, following the kinematic followed by the Information along its transfer from the Causes to the Effect,
  • in the selection of some of the subspaces of the Hilbert space, where have projections the eigenvectors representing each of the properties causing an effect. 



Horizons

 Polymer-like excitations puncturing the external bulk geometry of a massive material body, following Abhay Ashtekar








We conclude into the 3rd Millennium Physics this update about what, keeping apart naive interpretations, a Root Cause really is.  The figure represents a massive material body when considering its geometry until the quantum level.  The filiform curves piercing the object's external surface are polymer-like excitations of the bulk geometry. The amount of punctures is related to the mass of the material body. Proportional to all the possible combinations of the material particles composing the object, and to their spin. We think that this image, better than many others, hints to what a Root Cause really is.  A relation between material objects, existing until their smallest particles, and between material objects and fields, like the electromagnetic. or gravitational. These connections make full sense of a causal relation between Events, until its smallest dimension and duration.



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