We have different models and theories that are complimentary. I am not talking about complimentary theories. We have different theories to describe the microscopic and macroscopic phenomena, for example. I am not talking about the redundancy in this sense. This kind of redundancy is due to the lack of a unified theory.

The redundancy I am talking about is that there are multiple mathematical formulations of the same physical law. For example, you can take a look at my post titled “Concise Formulation of Maxwell Equations” to see that the Maxwell equations can be written in different mathematical notations.

What is the purpose of notational redundancy?

1. Mathematical elegance: conciseness in mathematical expression is a strong psychological driver among scientists
2. Expressing the physical law in a different notation may lead to new insights
3. It may be easier to solve an equation in a different notation.
4. A certain notation may be more intuitive. For example, in the post “Mathematical Spirals” I used the polar coordinate notation. If I wrote the equations of those spirals in Cartesian coordinates they would look very ugly and not intuitive at all.
5. Computer simulations of the physical phenomena and the requirement for speed in computations made it necessary to introduce new mathematical formulations or sometimes, as in the case of quaternions, bring back an old formulation to life. The point is that not all formulations (notations) are equal when it comes to computer simulations.
6. The rise of the “duality” approach in physics. In my post titled “Meaning of Duality in Physics” I explained how dual theories help with the solutions of equations.

What is wrong with notational redundancy?

1. Confusion! Physics students are confused! Physics teachers are confused! There is so much redundancy in the mathematical notation that teachers and students are confused both. There is a crisis in physics education.
2. The best evidence for the crisis is in the treatment of the rotations in physics. There are multiple equivalent descriptions of rotations. You go through the system, get your Ph.D. in physics but you are still confused about the mathematics of rotations.

Notation that leads to new insights

If I were a teacher I would teach the rotations in the notation that would most likely lead to new insights in physics. That would be the notation involving “Pauli spin matrices.” In physics the subject of rotations cover a wide range: it is not just rotations in 3D space. We deal with abstract rotations too. Properly speaking we should call those mathematical operations “transformations.”

In this post I will mention 2 tutorials on the subject of abstract rotations (transformations) in physics. These tutorials are very exceptional! They exemplify perfection in teaching. I wish we had tutorials like this when I was in school. I examined all my notebooks from college and graduate school. With the objectivity gained in the last 20 years I have to say with great sadness that my professors did a very poor job of explaining this material.

There are good books out there but they are not freely available unless you have access to a university library. We must have freely available quality tutorials on the web. This is very important. A motivated kid in a remote location should be able learn physics from the best teachers. I congratulate the authors of these tutorials with all my heart. They have done a great service.

Lie Groups in Physics: This lecture course was originally set up by M. Veltman, and subsequently modified and extended by Bernard de Wit and G. ‘t Hooft.

Please save this pdf document. The reputation of the authors add to the value of this document but objectively speaking this is the best tutorial on the subject. I have always admired the clarity of thought in G. ‘t Hooft’s papers.

G. ‘t Hooft and M. Veltman shared the Nobel Prize in Physics in 1999 for “for elucidating the quantum structure of electroweak interactions in physics.” This short description does not do justice to the totality of G. ‘t Hooft’s contributions to theoretical physics.  If you are a beginning student you should also check out his “How to be a good theoretical physicist.” That site is also a good place to find links to other physics education resources.

If you understand what is written in “Lie Groups in Physics” then you know more than a typical professional physicist. I bet that you can do research too.

Spinors: This lecture course was prepared by Andrew Steane of Oxford University. His major works to date are on error correction in quantum information processing, including Steane codes. He was awarded the Maxwell Medal and Prize of the Institute of Physics in 2000.

Euler rule

is a special case of the Cotes-Euler identity.

represents the rotations of the complex unit vector.

This provides a natural interpretation for . Multiplication by can be interpreted as anticlockwise rotation by 90 degress ( radians).

If we plug-in into the Cotes-Euler identity we get which demonstrates the consistency of the representation as the rotations of the complex unit vector.

where

and , , , are real numbers.

Biquaternion

The biquaternion is a complexified quaternion

where , , , are complex numbers.

The basis elements , , are the same as in quaternions.

Norm

The norm of a biquaternion is defined as

The definition of the norm is the same as in the quaternion with one important difference. The norm of a quaternion can never be a negative number. The norm of a biquaternion, on the other hand, can be negative or even zero because , , , are complex numbers.

Conjugate

The conjugate of a biquaternion is defined as

Inverse

If the norm is non-zero then there is an inverse for a biquaternion.

Algebra of biquaternions

The algebra of quaternions is a normed division algebra. The algebra of complexified quaternions (biquaternions) is not a division algebra since the norm can be zero.

Applications of biquaternions

Biquaternionic formulation of relativistic quantum mechanics is well established as demonstrated by the references in [1] and a recent review by K.Morita [2].

The biquaternions also form the core of the “algebrodynamics over complex space” paradigm discussed by V.V.Kassandrov [3] and the references therein. Another major effort pointing out the “algebraic design” of physics was developed by G.M.Dixon [4].

Golden biquaternions

In the biquaternionic formulation of relativistic quantum mechanics biquaternions represent the space-time points. In 2009, I proposed [5] that we can have an alternative view. We can use the biquaternion to represent the particle itself.

In reference [5] I show that biquaternions satisfying the golden condition    exist. A particular solution   involving Steiner’s Roman surface may be relevant in modeling 3 generations of fermions.  I call these special solutions ‘golden biquaternions.’

The golden biquaternion and its physical interpretation as the fermion predicts exactly 3 generations and 1/2,  3/2,  2 as the only possibilities for particle spin and hints at the composite nature of bosons. The ‘golden biquaternion as fermion’ idea works if there are 3 fundamental variables that are coupled in a pairwise fashion.

References

[1] Andre Gsponer and Jean-Pierre Hurni, “Quaternions in mathematical physics (2): Analytical bibliography”, arXiv:math-ph/0511092.

[2] Katsusada Morita, “Quaternions, Lorentz Group and the Dirac Theory”, Progress of Theoretical Physics, Vol.117, No.3 (2007)

[3] Vladimir V. Kassandrov, “Algebrodynamics over Complex Space and Phase Extension of the Minkowski Geometry”, arXiv:gr-qc/0602088. (2006)

[4] Geoffrey M.Dixon, “Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics”, Springer (1994), Kluwer Academic Publishers (2002), ISBN: 0792328906 / 9780792328902

[5] http://sureshemre.wordpress.com/2010/12/05/golden-biquaternions-3-generations-and-spin/

William Rowan Hamilton discovered quaternions in 1843. After many years of thinking on the subject, he had a flash of insight while he was walking along the Royal Canal in Dublin. According to legend he inscribed these equations on the Brougham Bridge on 16 October 1843.

Hamilton made many important contributions to physics and mathematics. The “Hamiltonian” in classical dynamics was named after him. In his mind, however, quaternions was the most important discovery. Some say that he was obsessed with quaternions.

Quaternions are superior to vectors for describing rotations in space. This is why quaternions are used extensively in the aerospace industry to compute the orbits of satellites and other space vehicles and in virtual reality simulations. There are many applications in physics as well. A comprehensive bibliography of quaternions in mathematical physics is maintained by A.Gsponer and J.-P. Hurni [1].

Even though quaternions have many applications today, Hamilton imagined that they would be taught in high schools and replace the vectors in mathematics education. That dream did not materialize.

Quaternion

The , , are known as basis elements.

A quaternion is defined in terms of these basis elements plus the unity element .

Pure quaternion

The , , , are real numbers. Any of these coefficients can be zero. When we have a pure quaternion.

Norm

The norm of a quaternion is defined as

The “norm” is similar to length. You can think of it as a distance measure.

Conjugate

The conjugate of a quaternion is defined as

The quaternion conjugate has the following property. If p and q are quaternions then

Inverse

Important difference between pure quaternion and pure complex number 

Any pure quaternion with norm 1 squares to -1. There are infinite number of such quaternions because defines a sphere of radius 1 (unit sphere). There are infinite number of points on the unit sphere corresponding to different pure quaternions.

In contrast, there are only 2 pure complex numbers that square to -1: and .

Quaternion multiplication

Let A and B quaternions. The product of two quaternions is another quaternion

and are the vector-like part of the quaternions involving quaternion basis elements. The and are the 3-dimensional scalar and vector products, respectively. The and are the real () parts of the quaternions.

Quaternion division

Using the inverse of a quaternion we can write “A divided by B” as

or as

These two are not equal.

References

[1] Andre Gsponer and Jean-Pierre Hurni, “Quaternions in mathematical physics (2): Analytical bibliography”, arXiv:math-ph/0511092

“The etymology of the word ‘universe’ can be traced back to the use of the Old French univers, in the twelfth century, which derives from the earlier Latin universum. This word is created from unus, meaning ‘one’, and ‘versus’, the past participle of the verb vertere, meaning ‘to turn, rotate, roll or change’. So we have a literal meaning of everything ‘turned into one’ or ‘rolled into one’.”

My interpretation of the etymology is:

universe = inverse of one

Nicolaus Copernicus is a Latinized version of his Polish name Mikolaj Kopernik. He was a Polish polymath: astronomer, mathematician, physician, classics scholar, translator, governor, diplomat, and economist.

In his book De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, he proved that Earth and other planets revolve around the Sun. This correct theory became known as heliocentrism which was very controversial in the 16’th and 17’th centuries.

The established church doctrine at that time claimed that Earth was at the center of the universe and the Sun revolved around the Earth. When Galileo Galilei promoted the heliocentric theory of Copernicus he was tried by the Roman Inquisition in 1615. Gelileo was found “vehemently suspect of heresy”, forced to recant, and spent the rest of his life under house arrest. Galileo’s books were banned for more than a century until Pope Benedict Benedict XIV (Prospero Lorenzo Lambertini) authorized the publications of the scientific works of Galileo Galilei in 1741.

Complete Works of Copernicus

In 1972 the Polish Academy of Sciences under the direction of J. Dobrzycki published critical editions of the Complete Works of Copernicus in six languages: Latin, English, French, German, Polish, and Russian. The first volume was a facsimile edition. The annotations in the English translations are more comprehensive than the others. The English edition was reissued as follows:

Minor Works, 1992, trans. E. Rosen, Baltimore: The Johns Hopkins University Press (originally published as volume 3 of Nicholas Copernicus: Complete Works, Warsaw: Polish Scientific Publishers, 1985).

On the Revolutions, 1992, trans. E. Rosen, Baltimore: The Johns Hopkins University Press (originally published as volume 2 of Nicholas Copernicus: Complete Works, Warsaw: Polish Scientific Publishers, 1978).

Languages

Copernicus spoke Latin, German, and Polish fluently. He also spoke Greek and Italian but he wrote in Latin which was the language of academia in Europe. Latin was also the official language of the Roman Catholic Church and of Poland’s royal court.

It is understandable that human imagination runs wild when it comes to Mt. Ararat because it is so majestic at an elevation of 5137 meters (16854 ft). It is visible from 4 different countries: Turkey, Armenia, Azerbaijan and Iran. It is located within the borders of Turkey and it is known as Ağrı Dağı in Turkish.

The history of civilization in the area goes back at least 12 thousand years. The long history complicates the human affairs. The suffering of the peoples of this land is palpable. It is a shame!

The current geopolitics of this area is also very complicated. There are too many interested parties.

Humans were always fascinated by volcanic mountains. There is a connection between the human psyche and volcanoes. It is a very emotional relationship. This may partially explain why strange phenomena are witnessed near the volcanic mountains.

The majestic Mount Ararat still keeps its secrets from us.

Image credit

Image credit

Image credit

Image credit

Like infinity singularity is an intractable subject. Thinking too much on infinity or singularity will surely lead you to insanity and that’s why I don’t write explicitly about singularity but ignoring this subject does not help me either.

Singularity in the context of physics

The traditional definition of singularity in physics is

• Zero volume, infinite density.
• Laws of physics are no longer valid.

Charles Misner proposed a different definition for singularity in 1963. He defined singularity as the point where time ends (history ends).

In 1965, using Misner’s definition of singularity, Roger Penrose showed mathematically that a singularity occurs inside a black hole.  Previously, based on Einstein’s equations of gravity, many other physicists argued that the universe started from a singularity. Arguing that there was a singularity 13.7 billion years ago does not create an emotional impact. The argument is rather abstract. But, if you say that singularity is happening right now somewhere in the universe then you have my attention. In 1965, Roger Penrose received a lot of attention. Later, Stephen Hawking showed mathematically that black holes evaporate. The mass/energy contained inside the sphere of the “event horizon” is radiated away slowly because of the quantum mechanical effects.

So, there is a singularity inside the black hole but the black hole is evaporating. This picture is very confusing and contradictory! The concept of singularity bothered Einstein too. He never accepted the possibility of a space-time singularity even though such a possibility was clearly implied by his equations of gravity.

“He believed just like Aristotle 2000 years before him, that there should be no physical infinities in the universe. Any that appeared in mathematical descriptions of the universe must be artefacts of an inappropriate idealization in the description of the universe, or a breakdown in the assumptions on which the mathematical theory producing that model was based.” [1]

Einstein’s sentiment motivates the entire enterprise of the String/M theory of physics. This theory is still in its infancy. The fundamental impetus for the String/M theory is the sentiment that elementary particles cannot be point particles as the Standard Model (SM) of particle physics suggests. String/M theorists hate infinities as Einstein did. SM hides the infinities arising from the treatment of elementary particles as point particles by using clever mathematics.

We may criticize SM for its ugliness and for its many deficiencies but we cannot deny the fact that it has been a successful theory. The singularity implied by the presence of a point particle (physical existence with no physical extension) remains a mystery.

The elementary particles of SM are singularities. The String/M theory is trying to prove that this is not the case without any success.

We manipulate singularities (point particles) every day. We observe their interaction with other singularities (different kinds of point particles). It is becoming a routine now. It is a routine not only in the state-of-the-art accelerators but also in our electronic devices! Even in the 19’th century we were using these singularities (electrons) without questioning their mysterious nature as singularities.

We started by saying that singularity is rare. After some reflection we now realize that singularities are everywhere.

Singularity in the context of Spiritual Philosophy

The realization that singularity is not so rare leads us to the acceptance that the soul is a type of singularity. The infinity of the Absolute Being is hidden in the dimensionless singularity of the soul.

Singularities are dimensionless. Singularities do not have attributes. But, singularities have event horizons where dimensions and attributes manifest.

The event horizon of the soul is the mind. The soul is a singularity whereas the mind is not. The existence of the soul can be deduced from the radiance of the mind.

My suggestion to professional physicists is that singularity is unavoidable. A concept similar to “event horizon” is needed in particle physics. The reconciliation of the SM and String/M theories may be possible by such a construct.

“Dualists hold that body and soul are separate entities and that the soul will continue beyond the existence of the physical body. Monists contend that body and soul are the same and that the death of the body — the disintegration of DNA and neurons that store my personal information — spells the end of the soul.” [1]

I read and re-read the paragraph multiple times. I have never heard of this definition of monism before. This is not monism! He is giving us the definition for materialism. I have to add that his definition of dualism is not the commonly accepted definition of dualism either.

The Wikipedia article [2] on monism is one of the best that the collective wisdom has produced. I am pleasantly surprised by the clarity of the Wikipedia article. In contrast, the SEP article [3] on monism is very terse and not comprehensive at all. One expects comprehensive articles from SEP. One criticism of the Wikipedia article is that it does not discuss the “Unity of Being” (Absolute Monism).

I will take this opportunity to mention the major lines of thought in monism and compare them to my version of monism which I call the Soul Monism [4]. I will ignore purely analytical/academic or mathematical versions of monism.

Soul never dies

Before I change the tone of this article to philosophical I would like to stress a spiritual fact: soul never dies. The individual mind dies, the body dies but the soul never dies.

Monism does not say that soul, mind and body are the same. It is clear that they are different. Monism says that soul, mind and body were the same in the beginning and they will be the same in the end.

Soul, mind and body are different transformations (emanations) of the singular Cosmic Soul. The entire Cosmos is an expression of that singular soul.

Panpsychism

This is the most popular and rational line of thought in monism. In one of his panel discussions the famous philosopher David Chalmers said that he is partial to panpsychism. I also think that panpsychism has a lot of merit. Many philosophers of academic metaphysics are leaning towards panpsychism.

“In philosophy, panpsychism is the view that mind or soul is a universal feature of all things. Alternatively, it is the view that all things have a unified center of experience or volition. Panpsychism is one of the oldest philosophical theories, and can be ascribed to philosophers like Thales, Plato, Spinoza, Leibniz, Schopenhauer and William James. Panpsychism can also be seen in eastern philosophies such as Vedanta and Mahayana Buddhism. During the 19th century, Panpsychism was the default theory in philosophy of mind, but it saw a decline during the latter half of the 20th century with the rise of logical positivism. The recent interest in the hard problem of consciousness has once again made panpsychism a mainstream theory.” – Wikipedia [5]

My understanding of panpsychism is that Consciousness is immanent in all entities. Please see [4] for the definition of Consciousness. Soul Monism [4] also says that Consciousness is immanent in all entities but adds many clarifications. Regarding immanence and transcendence Soul Monism states that

• When you view Consciousness from the perspective of the soul, it is immanent.
• When you view Consciousness from the perspective of the mind it is transcendent.

Dialectic Monism

“Dialectical monism, also known as dualistic monism, is an ontological position that holds that reality is ultimately a unified whole, distinguishing itself from monism by asserting that this whole necessarily expresses itself in dualistic terms. For the dialectical monist, the essential unity is that of complementary polarities, which, while opposed in the realm of experience and perception, are co-substantial in a transcendent sense.” – Wikipedia [6]

Ontological Monism

There are analytical philosophers saying that it depends on how we count. If we count “all” as “one” then we can call it monism too. Ontological monism objects to this simplistic thinking and clarifies that there is unity behind multiplicity. Ontological monism claims that multiplicity emerged from a singular essence. The unity behind multiplicity is real (ontological).

Soul Monism is ontological monism. Soul Monism does not deny the reality of multiplicity but claims that multiplicity is derivative. The derivative or relative truth concept was examined in great detail by Shrii Shrii Anandamurti.

Unity of Being = Absolute Monism

Unity of Being and Absolute Monism refer to the same concept. There is only ONE Absolute Being in existence. Cosmos including all the physical universes and the infinite number of souls are transformations or internal reflections of this Absolute Being which is also known as Consciousness (Puruśa in Sanskrit).

In Sufi metaphysics the Unity of Being is known as Wahdat al-Wujud. Ibn Arabi (1165 – 1245) is often cited as the originator of the concept of Unity of Being but this concept existed since Shiva’s time 7000 years ago. We see the same concept being taught by Tehuti 6000 years ago in Egypt.

Soul Monism is absolute monism. Soul Monism brings clarity to the main concepts of absolute monism. For example, it explains what “transformation” and “internal reflection” mean [4].

Pantheism, Panentheism, and Unity of Being

The term Pantheism means “All is God” while “Unity of Being” emphasizes that there is just a single being in existence and this single being is God. However, “Unity of Being” may be closer to Panentheism, because Panentheism states that while the Cosmos is part of God or God’s mind, God is still greater than his creation.

Commentary on Dialectic Monism

The definition of “Dialectic Monism” does not sound so bad! In fact, it sounds very much like the Ananda Marga definition of monism. This is interesting because Ananda Marga Monism is ontological monism obviously but it is also absolute monism. I thought it would be interesting to quote from the “Philosophical Treatise of Ananda Marga” which is a text of only 17 paragraphs.

“Nirguna and Saguna are two aspects of one and the same Brahma. They are not two different Brahmas. It should be clearly mentioned that Ananda Marga is absolute monism – it is not dualism. A question may arise: How can Ananda Marga be called monism when it declares that Shiva and Shakti are separate entities. The answer is that in Ananda Marga philosophy, Shiva, or Cognitive Force, is both the material and the efficient cause, whereas Shakti, or Operative Force, is only the efficient cause. Shakti is merely subservient to Shiva, Shakti has no existence separate from Shiva. So we can safely conclude that Ananda Marga philosophy is based on monism.”

Even though the Ananda Marga Monism sounds very much like Dialectic Monism it is not because Ananda Marga Monism does not accept “complementary polarities” as equally substantial. Ananda Marga philosophy clearly states that the cognitive aspect (Consciousness) of the Absolute Being (Godhead) is primary

[1] http://www.michaelshermer.com/2004/09/mustangs-monists-and-meaning/

[2] https://en.wikipedia.org/wiki/Monism

[3] http://plato.stanford.edu/entries/monism/

[4] http://sureshemre.wordpress.com/definitions-and-summary-of-soul-monism/

[5] https://en.wikipedia.org/wiki/Panpsychism

[6] http://en.wikipedia.org/wiki/Dialectical_monism

Archimedes Spiral

http://mathworld.wolfram.com/ArchimedesSpiral.html

Fermat Spiral

http://mathworld.wolfram.com/FermatsSpiral.html

Logarithmic Spiral

http://mathworld.wolfram.com/LogarithmicSpiral.html

Golden Spiral

The equation in polar coordinates is the same as the logarithmic spiral but the the constant has a special value

The constant is in units of radians. is the golden number.

http://mathworld.wolfram.com/GoldenSpiral.html

Hyperbolic Spiral

http://mathworld.wolfram.com/HyperbolicSpiral.html

Lituus Spiral

http://mathworld.wolfram.com/Lituus.html

Cornu Spiral

The Cornu spiral is also known as the clothoid or Euler’s spiral. The mathematical equation is not as simple as the others shown above. Please refer to the link below.

http://mathworld.wolfram.com/CornuSpiral.html

Circle Involute

http://mathworld.wolfram.com/CircleInvolute.html

http://mathworld.wolfram.com/Involute.html

Cotes Spiral

A spiral that gives the solution to the central orbit problem under a radial force law. Please refer to the link below.

http://mathworld.wolfram.com/CotesSpiral.html

Nielsen Spiral

http://mathworld.wolfram.com/NielsensSpiral.html

Polygonal Spiral

http://mathworld.wolfram.com/PolygonalSpiral.html