Hamilton's Quaternions win the day (physics AH with no Einstein)

[chrispi]

What if vector algebra did not become as popular as quaternions?

First, a little explanation: the numbers one uses everyday (the set of real numbers R, like 0, 1, pi and so forth) are not the only numbers that exist that can be added, subtracted, multiplied or divided except by zero. There are also imaginary numbers that are formed when one takes the square root of a negative number, e.g. i = sqrt(-1) or i^2 = -1. Imaginary numbers help form the set of complex numbers C which is made of all numbers a+ib where a and b are real numbers. Imaginary and complex numbers have many real applications in diverse fields such as fluid dynamics and quantum mechanics, among others.

However, the complex numbers are also not the only numbers that exist. There are the numbers called quaternions that are a generalization of complex numbers. The set of quaternions H consist of extra numbers i, j, and k that follow the rules much like complex numbers. There is a fascinating story behind quaternions. According to Wikipedia,

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. He could not do so for 3 dimensions, but 4 dimensions produce quaternions. According to the story Hamilton told, on October 16, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices had yet to be developed.

Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of real numbers, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered. Hamilton proceeded to popularize quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death.

According to this site, at the turn of the century there would have been a physical theory uniting gravity and electromagnetism, and when the weak and strong nuclear forces are discovered decades later the application of a still more general set of numbers called the octonions O would be applied to describe them as well. In other words, gravity and the Standard Model of quantum physics would be united in the 1950s!

Of course, if quaternions won out, Albert Einstein would be butterflied out of existence!

[chrispi]

For a considerable while in the 19th century, quaternions were more popular than vector algebra, but were supplanted by vectors because the latter were simpler to use. Nonetheless, a quaternionic theory of relativity is much easier to develop than the standard method, and since quaternions are non-commutative, they find use in quantum mechanics as well (the famous Heisenberg Uncertainty Principle is derived from the non-commutativity of operators.)

[Satyrane]

Fascinating. Can you sum this up in terms accessible to the ignorant layman/arts student? Are you basically talking about earlier and better understanding of quantuum mechanics?

If so, what practical advances might we see today - better NMR machines? Earlier/more destructive nuclear weapons? Cold fusion?

[Max Sinister]

Very interesting. The question is still: Which physicist would come up with the idea for theory of relativity? Einstein had some problems doing the necessary math for his theory. The quaternions would be a great help, but not every mathematician is also a physician and vice versa.

[chrispi]

Originally Posted by Satyrane

Fascinating. Can you sum this up in terms accessible to the ignorant layman/arts student? Are you basically talking about earlier and better understanding of quantuum mechanics?

If so, what practical advances might we see today - better NMR machines? Earlier/more destructive nuclear weapons? Cold fusion?

Summing it up for laymen was precisely what I was trying to do. I'm actually talking about unifying quantum mechanics with general relativity without resorting to unverifiable hidden dimensions like in string theory.

Whether or not quaternions would have advanced technology is questionable, since the empirical stuff is also needed (Balmer series of hydrogen lines, the photoelectric effect and so forth) to formulate a proper theory. I venture to say that quaternions would indeed have speeded up progress in both relativity and quantum mechanics by two to three decades...

[chrispi]

Originally Posted by Max Sinister

Very interesting. The question is still: Which physicist would come up with the idea for theory of relativity? Einstein had some problems doing the necessary math for his theory. The quaternions would be a great help, but not every mathematician is also a physicist and vice versa.

I don't think that any one physicist would come up with the idea of unified field theory, it would be much the same situation as in OTL quantum mechanics, where you had many fine minds on the par of Einstein, such as Bohr, Schrödinger and Heisenberg, develop the physics required.

[chrispi]

The site http://www.quaternions.com is also a great resource for anyone fascinated by these marvelous numbers and intrigued by their ability to unify quantum mechanics, electromagnetism and gravity.