My Note - 3) Quaternion

Dr Yukawa told a brief history of vector in his lecture Day Two and commented on Quaternion as follows:

4.ベクトルの歴史 History of Vector

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One remarkable exception is vector. We think that vectors had come out as they should have and they have been used already since a very long time ago. But it was not so. As I have already mentioned several times in the books of Maxwell we can find ordinary components - ie Maxwell's equations which expressed by Cartesian components, each multiplied by a derivative and no or a very few (if any) vectors involved. This was a story in late 19th century when the symbol of vector was not found.
What existed before vector was quaternion. It is really strange. We know Hamilton. Those who study Quantum Mechanics study Hamilton and Lagrange. Hamilton lived in 19th century. His canonical formalism is studied even now. Both Hamilton and Lagrange were great physicians. Hamilton, also mathematician, created quaternion. What is Quaternion? This is the forth dimension number. Quaternions are not real numbers while the ordinary vectors are real. I do not know why a strange thing like quaternion was used but quaternion was popular back in the 19th century.  Although I had no particular interest in it I investigated it and found that an English (Scottish) physician Tait wrote a large volume of book titled "Quaternion". Tait worked with Lord Kevin.

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He did not mention that Maxwell used quaternion (Maxwell used for some time) and why. Quaternion is convenient to describe Curl or "Rotation" like Imaginary number (i² = −1, i = √−1) being convenient to describe the simple rotation (90-degree,180-degree, etc rotation) and frequency. I found this in "On the Notation of Maxwell's Field Equations" by André Waser and others.


"On the Notation of Maxwell's Field Equations" (latest version) issued by André Waser on 28-06.2000,
http://www.itpa.lt/~acus/Knygos/Clifford_articles/Clifford_Authors/05_DeLeo_Quaternions/quaternions5/Orig_maxwell_equations.pdf

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A general quaternion has a scalar (real) and a vector (imaginary) part. In the example below
‚a‘ is the scalar part and ‘ib + jc + kd’ is the vector part.
Q = a + ib + jc + kd
Here a, b, c and d are real numbers and i, j, k are the so-called HAMILTON‘ian unit vectors
with the magnitude of √−1. They fulfill the equations

i² = j² = k² = ijk = −1
 
ij = k,  jk = i,  ki = j

ij = - ji,  jk = - kj,  ki = - ik

A nice presentation about the rotation capabilities of the HAMILTON’ian unit vectors in a
three-dimensional ARGAND diagram was published by GOUGH.
  
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Some parts are modified (the meaning unchanged) and the underlined by sptt.

Please read 

"On the Notation of Maxwell's Field Equations". A concise and nice article to know somewhat original version of Maxwell's Field Equations.

A more kind explanation is found http://hforsten.com/quaternion-roots-of-modern-vector-calculus.html

A full copy

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Quaternion roots of modern vector calculus

2013-02-03 In Mathematics.

Before the vector calculus was invented mathematicians used to do vector calculus like operations using quaternions. Quaternions are four dimensional extension of complex numbers. They have two additional imaginary units j and k. Quaternion multiplication is non-commutative and is defined by the formula:

i2=j2=k2=ijk=−1

One easy way to remember the multiplication rules is with the following diagram:
Multiplying two different imaginary units gives the positive next one in the diagram, if the arrow goes in the right way between them and negative otherwise. For example ij=+k, because arrow between i and j goes the right way. Same way ji=−k, ki=j and so on.
Quaternions with real part zero (q=0+ai+bj+ck) are called pure imaginary quaternions and they have some properties that make them behave like three dimensional vectors.
Multiplication of two pure imaginary quaternions p and q is:
p×q=(p1i+p2j+p3k)×(q1i+q2j+q3k)
=−p1q1−p2q2−p3q3+p1q2k−p1q3j−p2q1k+p2q3i+p3q1j−p3q2i
=−(p1q1+p2q2+p3q3)+(p2q3−p3q2)i+(p3q1−p1q3)j+(p1q2−p2q1)k
If we now think that quaternions p and q were three dimensional vectors p⃗  and q⃗ , this product can be written shorter with vector calculus operators.
p⃗ q⃗ =−p⃗ ⋅q⃗ +p⃗ ×q⃗ 
Where p⃗ ⋅q⃗  is the dot product and p⃗ ×q⃗  is the cross product.
If we change the quaternion multiplication rules to i2=j2=k2=0, quaternion multiplication gives only the cross product. This rule and the diagram above also might help in remembering how to calculate cross product and curl without using the determinant rule.
Maxwell originally wrote his famous equations in quaternion form and they were only later rewritten in the modern vector notation. Quaternion vector calculus can do pretty much everything that the modern vector calculus can do, except the notation is little more cumbersome. For example Maxwell wrote the curl of vector v, ∇×v⃗ , as V∇v. Where V meant the vector part, v is a pure imaginary quaternion and ∇ is the pure imaginary quaternion similar to vector calculus ∇:

∇=ddxi+ddyj+ddzk

Note that ∇2=−(d2d2x+d2d2y+d2d2z), negative of the usual ∇2 in the modern vector calculus.
Even today the i,j and k from quaternions are sometimes used in the vector calculus as basis vectors and sometimes they can be thought to be quaternion basis units.
If you are interested in the old stuff, Maxwell's original paper "A treatise on electricity and magnetism(1873)" with quaternion notation is available at archive.org.

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As Dr Yukawa repeated in his lecture we must read the original writings of the great thinkers to find how the original thoughts and ideas were painfully born.

Other opinions on Quaternion

sptt