MODELS

Models are essential for humans in order to understand the universe around us. A model seeks to represent empirical objects, phenomena, and physical processes in a logical and objective way. Scientific theories, laws, postulates, and ideas are a way of generating abstract, conceptual, graphical and/or mathematical tools to use to predict and understand future patterns. A model can be a scaled version of the real thing, a verbal description, a picture, contrasts, or more importantly these days, a mathematical formalization of physics.

It is really important to understand that all models are inherently false.

Models serve a very important purpose. No one would argue that Newton's Philosophiæ Naturalis Principia Mathematica is not one of the greatest works of human intelligence, but they will tell you that it is fundamentally incorrect. Newton's model of the universe still has value, and is still taught in the universities because of its facility to predict motion and force, and its easy visualization.

When my son was in the eighth grade, he had a science project to build a model of the Boron atom. He asked me for my help (BIG mistake). We made a model with each electron on a straight wire at the correct respective n-distance from the nucleus to depict its electron shell energy. Further we color-coded each electron to correspond to the shell configuration. We then graphed a simple scale for shell size to each other.

He received a poor grade because he didn't show the electrons on a ring circling the nucleus (this was 1998, so the knowledge that electrons do not 'circle' the nucleus was only 80 years old, give or take a moron or two). I was incensed, and went in to explain the rationale to the teacher. He told me that Alex received a bad grade because he did not show the electrons on a fixed ring circling the nucleus. I explained the energy scales, configuration patterns, overlap detail, and then he asked me where the rings were shown, and I gave up.

Electrons circling the boron atom nucleus in the Bohr model.

Ask anyone you meet to draw an atom, and the Bohr model of the atom is the most likely picture you will get. It is easy to understand, it predicts the behavior fairly well, and no one can see an atom anyway (these things are very small, probably operate in extra dimensions, and they act funny). The point is, the model that we all carry in our head of an atom is wrong, and misleading. Probably looks more like the following, if you could see it.

Boron Atom Dramatization

First of all, if I scale the electron cloud of a boron atom up to the size of the next graphic, and I show the nucleus and electrons as white dots, this is what you would see;

Atom mass shown in white.

If you actually see any white, clean your screen, it is just dust.

That is why mathematical models are so important. Most things that we want to study now are just too small to see. We are learning about them by their behavior when they interact with other things. Mathematics gives us a tool to use to predict an outcome, and then test it by observing experimental behaviors. Einstein said, "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."

DECOHERENCE

In quantum mechanics, quantum decoherence (also see dephasing) is how quantum systems interact with their environments to exhibit probabilistically additive behavior. Quantum decoherence gives the appearance of wave function collapse (the reduction of the physical possibilities into a single possibility as seen by an observer) and justifies the framework and intuition of classical physics as an acceptable approximation: decoherence is the mechanism by which the classical limit emerges out of a quantum starting point and it determines the location of the quantum-classical boundary. Decoherence occurs when a system interacts with its environment in a thermodynamically irreversible way. This prevents different elements in the quantum superposition of the system+environment's wavefunction from interfering with each other. Decoherence has been a subject of active research since the 1980s.

TIME DEVELOPMENT ENTROPY

The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics. This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the general transition matrix.  I do not fully understand this, but it seems to imply that all possible choices in the quantum state must add up to 100%.  The quantum state collapse gives up information through the entropy laws.  The original concepts of decoherence proposed by David Bohm and John Bell were held in little regard for many years, but the advances in quantum computing have caused a resurgence in the study of dephasing.  While Bohm felt that a carrier wave was responsible for the duality of  particles, dephasing holds that the particle is real in many states, but releases information (irreversibility) when interaction occurs.


Vector phasing, may be functions or frequencies; instead of matrix multiplication, linear transformations may be operators such as the derivative from calculus. These are only a few of countless examples where eigenvectors and eigenvalues are important.

In this shear graph, the yellow arrow
undergoes a phase shift but the red arrow does not.
Therefore the red arrow is an eigenvector,
with eigenvalue 1, as its length is unchanged.

The Bloch sphere is a representation of a qubit,
 the fundamental building block of quantum computers.

The Protium atom is an example where both types of spectra appear. The eigenfunctions of the hydrogen atom Hamiltonian are called eigenstates and are grouped into two categories. The bound states of the hydrogen atom correspond to the discrete part of the spectrum (they have a discrete set of eigenvalues that can be computed by Rydberg formula) while the ionization processes are described by the continuous part (the energy of the collision/ionization is not quantized).  In mathematics, a shear mapping or transvection is a particular kind of linear mapping. Its effect leaves fixed all points on one axis and other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from the axis. It is notable that shear mappings carry areas into equal areas.

Folding Space.

 A simple cube being folded and multiplied

The concept of multiple spacial eigenvector values coexisting in the wave function is not new.  Some of the quantum information computing work being done is also focusing on time eigenvalue vector analysis.

Superposition, Quantum Algorithms (a fixed sequence of quantum logic gates), Particle Entanglement, and the Quantum Decoherence are all important aspects being studied.  The goal in quantum computing is to get the quantum state to release information.  The theory of decoherence has made a huge comeback in main-stream thinking.  Now decoherence is seen less as a complete collapse of the probability wave (or the disappearance of a carrier wave), and more as the release of information to our timeline through a process of entropic information leak.  The probability wave never fully collapses, but must make decisions based on interactions to remain in concert with the laws of physics (a partial collapse).

The Protium atom orbital shells.

MANIFOLDS

I would really like draw a picture of the fabric of space, but the math needed to model space-time is too complicated.  We can think in terms of the three space dimensions, but when we start to go into higher dimensions the models just do not translate in our brains.

Enter the mathematicians.

Euclid of Alexandria wrote "Elements" somewhere around 300BC.  The fact that you can find Euclid's work repeated in any text book of geometry and algebra attests to it's remarkable impact on the sciences.  In some ways, you can also see a beauty to the progress of Euclid's mathematical formulations.  Euclid probably took many of the concepts and proofs from other mathematicians of the time, but he was the first to put together all of that knowledge into writings that survive.  Euclid started with five main postulates upon which he built his mathematical model of the world.

1. Any two points determine a unique line containing them.
2. Any line segment may be extended.
3. Given a point P and a distance r, there is a circle with center P and radius r.
4. All right angles are equal. 
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclidean Geometry was the only game in town for a couple thousand years, but that was not for want of trying. Many mathematicians took a shot at postulate number five, which did not follow directly from the previous four. Euclid saved the fifth postulate till last for a reason. He obviously had problems with it himself, and by definition, the postulate has some esoteric flaws. Take perspective; when we view a railroad track disappearing into the distance, we understand that the tracks are parallel - but that they appear to converge to a point. Can we model visual perspective as a geometry of it's own? The answer is yes, but for centuries Euclid's geometry held it's own. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry, primarily through the works of Johann Carl Friedrich Gauss (who never published his theory), János Bolyai, and Nikolai Ivanovich Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for the development of geometry, and Jules Henri Poincaré, who was looking for the math to back up his theories of relativity. 

Hyperbolic space is a geometrical space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Mathematicians would have been content to just play around with the math, but along came a German mathematician, Georg Friedrich Bernhard Riemann, whom was not content with the status quo. Riemann came up with a new geometry using “tensors”. He also found a way to relate his new geometry to four dimensional analysis.  His broad vision of the subject of geometry was expressed in his 1867 inauguration lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, a mainstay of modern geometric analysis.

Then in 1896 Pieter Zeeman and Hendrik Antoon Lorentz used post Euclidean geometry to accurately model a part of the natural world, magnetic spectral lines, and the door was open. Lorentz went on to work with a young physicist, Albert Einstien, on a new theory of relativity. By 1906 it was noted by Poincaré that, by using an imaginary time coordinate √−1 ct, the Lorentz transformation can be regarded as a rotation in a four-dimensional Euclidean space with imaginary time being the fourth dimension, "manifold space".

The concept of manifolds is important because you can model complicated structures in terms of the relatively well-understood properties of simpler spaces. For example, a manifold is typically endowed with a differentiable structure that allows one to do calculus and a Riemannian metric that allows one to measure distances and angles. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism (Sir William Rowan Hamilton) of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

Using the Lorentz transformation, you can mathematically model how two observers' varying measurements of space and time can be converted into each other's frames of reference.

This idea was elaborated by Hermann Minkowski who used it to restate the Maxwell equations in four dimensions showing directly their invariance under Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this he concluded that time and space should be treated equally and so arose his concept of events taking place in a unified four-dimensional space-time continuum. In a further development he gave an alternative formulation of this idea which did not use the imaginary time coordinate but represented the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional affine space. Points in this space were regarded as events in space-time. The spacetime interval between two events in Minkowski Space is either space-like, light-like ('null') or time-like, creating Nutches for Nitches to fill.

Orbifolds

Phase space

ECCENTRICS

It is important to note that every major advance in physics is preceded by the work of an Eccentric. This is a living document, since new Eccentrics appear all the time. In my list of eccentrics I have not included the Eccentrics that turned out to be Geniuses (Newton and Einstein come to mind).

Psychologist Dr. David Weeks mentions people with a mental illness "suffer" from their behavior while eccentrics are quite happy. He even states eccentrics are less prone to mental illness than everyone else.

According to studies, there are eighteen distinctive characteristics that differentiate a healthy eccentric person from a regular person or someone who has a mental illness (although some may not always apply). The first five are in most people regarded as eccentric:

• Nonconforming attitude 
• Idealistic
• Intense curiosity 
• Happy obsession with a hobby or hobbies 
• Knew very early in his or her childhood they were different from others 
• Highly intelligent 
• Opinionated and outspoken 
• Unusual living or eating habits 
• Not interested in the opinions or company of others 
• Mischievous sense of humor 
• The eldest, only child, or youngest boy.

NOTABLE ECCENTRICS:

Richard Laming (c. 1798–May 3, 1879): A British Eccentric, Richard tested and was accepted as a surgeon by the Royal College of Surgeons.  He established a practice in London and worked until 1842.  Surgeons of the 1800's were sometimes a little sketchy.  Known more for how fast they worked (no anesthesia) the surgeon was an iffy profession.  Richard appeared to try to do his best, and studied in Paris, France for several years.


More importantly, Richard fancied himself as a natural philosopher.  During his leisure moments, Richard developed an interest in the theory of electricity. Between 1838 and 1851 he published a series of papers speculating about the electrical makeup of atoms. He hypothesized that there existed sub-atomic particles of unit charge; perhaps one of the first persons ever to do so. He suggested that the atom was made up of a core of material surrounded by concentric shells of these electrical 'atoms', or particles. He also believed that these particles could be added or subtracted to an atom, changing its charge.


It wasn't until 1909-1913 that Richard was proven correct by Ernest Rutherford and Neils Bohr.


Henri Poincaré (29 April 1854 – 17 July 1912):  A well respected mathematician and scientist at the time, Henri Poincaré developed and published the Theory of Relativity in 1904, the year before Einstein published his paper.  Einstein, later in life, acknowledged that Poincaré as the pioneer of Relativity.  While not the best example of the Eccentric, Poincaré's views and responses to Einstein's theft of the Relativity Theory give him honorable mention in the roles of Eccentrics.

William Gilbert  (24 May 1544 – 30 November 1603): After playing around with his toy model earth, he concluded that the Earth was itself magnetic and that this was the reason compasses point north (previously, some believed that it was the pole star (Polaris) or a large magnetic island on the north pole that attracted the compass). He was the first to argue, correctly, that the centre of the Earth was iron, and he considered an important and related property of magnets was that they can be cut, each forming a new magnet with north and south poles.


Isaak Yudovich Ozimov  (1920 – 6 April 1992):  There is no telling what the power of science fiction has been on the actual progress in science fact.  We do know that after 500 books and numerous other short writings, Isaac Asimov has helped to shape our world view of science.  The Oxford English Dictionary credits his science fiction for introducing the words positronic brain (an entirely fictional technology), psychohistory, and robotics into the English language.  I would personally be insulted if the "Three Laws of Robotics" were not required for the first sentient robots.


Isaac Asimov may also have given us the answer to the heat death of the universe and a way to reverse entropy in the "The Last Question".


David Joseph Bohm (20 December 1917 – 27 October 1992):  David Bohm obtained his doctorate degree while with the theoretical physics group under Robert Oppenheimer at the University of California, Berkeley.  David actually was not allowed to defend his thesis, because it was used by the Manhattan Project and immediately classified by the FBI.  To satisfy the university, Oppenheimer certified that Bohm had successfully completed the research.  As a post-graduate at Berkeley, he developed a theory of plasmas, discovering the electron phenomenon now known as Bohm-diffusion.  After the war, Bohm became an assistant professor at Princeton University, where he worked closely with Albert Einstein. In May, 1949, at the beginning of the McCarthyism period, the House Un-American Activities Committee called upon Bohm to testify before it— because of his previous ties to suspected Communists. Bohm, however, pleaded the Fifth amendment right to decline to testify, and refused to give evidence against his colleagues.  Princeton suspended him and he eventually left the United States in the early 1950's.


Bohm became dissatisfied with the orthodox approach to quantum theory and began to develop his own approach (De Broglie–Bohm theory) - a non-local hidden variable deterministic theory whose predictions agree perfectly with the nondeterministic quantum theory. Before an understanding of decoherence was developed, the Copenhagen interpretation of quantum mechanics treated wavefunction collapse as a fundamental, a priori process. Bohm's theory provides an explanatory mechanism for the appearance of wavefunction collapse.


Any Pure Mathematician (You can find them anytime); Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, and engineering - the applied sciences. “There are three kinds of mathematicians; those who can count and those who can't.”

“Relations between pure and applied mathematicians are based on trust and understanding. Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians. Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more." -- Albert Einstein.


"Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different." -- Johann Wolfgang von Goethe

Pythagoras (c. 570–c. 495 BC); Now here was a nut job.  He founded a religion worshiping math.
Godfrey Harold Hardy (7 February 1877 – 1 December 1947); Hardy felt only pure mathematics to be worthy, he once said that general relativity and quantum mechanics were "useless", and only fit for engineers.
Ramanujan (22 December 1887 – 26 April 1920); Mostly self-taught, during his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Although a small number of these results were actually false and some were already known, most of his claims have now been proven correct.


Benjamin Franklin (January 17, 1706 [O.S. January 6, 1705^[1]] – April 17, 1790); Ben Franklin was referred to in his day as a 'polymath'.  You can read that as 'eccentric'.  Franklin was a leading author and printer, political theorist, politician, postmaster, scientist, inventor, satirist, civic activist, statesman, and diplomat. As a scientist, he was a major figure in the American Enlightenment and the history of physics for his discoveries and theories regarding electricity. He invented the lightning rod, bifocals, the Franklin stove, a carriage odometer, and the glass 'armonica'. He formed both the first public lending library in America and the first fire department in Pennsylvania.

THE OBSERVER

The Protium Atom under scrutiny.

According to the Heisenberg Uncertainty Principle, there is a limit to the amount of information that you can define for a quantum particle.  The more precise you measure one attribute, the less you know of the other. This established the concept of  complementarity as one of the basic principles of quantum mechanics.  Niels Bohr maintained that quantum particles have both "wave-like" behavior and "particle-like" behavior, but can exhibit one kind of behavior only under conditions that prevent exhibiting the complementary characteristics. This complementarity has come to be known as the wave-particle duality of quantum mechanics.

In the double-slit experiment, the common wisdom is that the Heisenberg Uncertainty Principle makes it impossible to determine which slit the photon passes through without at the same time disturbing it enough to destroy the interference pattern.

There have been many experiments trying to circumvent the issue of disturbance due to direct measurement of a photon.  The delayed choice quantum eraser experiments "found a way around the position-momentum uncertainty obstacle and proposed a quantum eraser to obtain the 'which-path' or particle-like information" without disturbing the wave function.  They could then choose to 'see' the path that the particle took later, or erase it.

It was found that there is no interference pattern when which-path information is recorded, even if this information was obtained without directly observing the original photon, but that if you somehow "erase" the which-path information, an interference pattern is observed.  The total pattern of signal photons at the primary detector never shows interference, so it is not possible to deduce what will happen to the idler photons by observing the signal photons alone, which would open up the possibility of gaining information faster-than-light (since one might deduce this information before there had been time for a message moving at the speed of light to travel from the idler detector to the signal photon detector) or even gaining information about the future (since as noted above, the signal photons may be detected at an earlier time than the idlers), both of which would qualify as violations of causality in physics.

This has lead many to remark that the observer is somehow important to the results.  This gives the observer some control over reality, or implies that reality can not exist without the observer (tree falls in the forest concept).  The conciousness causes collapse interpretation attributes the process of wave function collapse (directly, indirectly, or even partially) to consciousness itself.

The other important weirdness to note is that time is somehow held in suspension until the observation is made.  This is most graphically shown by quantum entanglement.  Entangled particles can be at opposite ends of the universe and still 'pass' information to the entangled twin. These particles are entangled in space-time, meaning that they are also entangled in time, according to S. Jay Olson and Timothy C. Ralph of Australia's University of Queensland.  The time component may play a larger role in understanding the quantum process than previously thought.  The entangled particles may be linked in time, rather than through space.