quaternions
<a href="http://www.youtube.com/watch?v=zc8b2Jo7mno" title="Gimbal lock animation">This</a> very accessible animation explains 'gimbal lock', especially as concerns CGI animation technology.
And interestingly, quaternions are used in programming this stuff.
The Maxwell action of electromagnetism is represented using the noncommutative division algebra of quaternions. The potential in the electromagnetic action is then rewritten with the weak force gauge symmetry SU(2), also known as the unit quaternions. The potential can be recast again with electroweak symmetry as the product of U(1) and SU(2) symmetries. The conjugate of one electroweak symmetry times another for the potential in the action is enough to account for the strong force symmetry SU(3). A 4D commutative division algebra is constructed from the hypercomplex numbers modulo eigenvalues equal to zero. The action is rewritten again with the hypercomplex multiplication rules in a gauge invariant way. Like charges attract for the hypercomplex action based on an analysis of spin of the field strength density, the spin in the phase of the current coupling term, and the field equations that result by applying the Euler-Lagrange equation. The first field equation of the hypercomplex action contains Newton's law of gravity paired with a time-dependent term and thus is consistent with special relativity. There is also an Ampere-like equation so that a 4-potential theory can account for bending of both time and space caused by gravity. It is shown how the Rosen metric is a solution to the field equations, and thus passes weak field tests of gravity to first-order parametrized post-Newtonian (PPN) accuracy. The proposal is distinguishable from general relativity at second-order PPN accuracy, predicting for example 0.7 microarcseconds more bending of light around the Sun than the Schwarzschild metric. The lowest mode of wave emission for this simple field theory is a quadrupole. The final rewrite of the action has gauge-dependent electromagnetic and gravity field strength densities where the two gauges cancel out, leaving a gauge-independent unified action. Since the Higgs mechanism is unnecessary for this unified standard model proposal, it is predicted no Higgs boson will be found.
0. Author's note
1. Introduction
2. The Maxwell Lagrangian Using Quaternions
3. The Lagrangian Using Hypercomplex Numbers
4. Hypercomplex Field Equation Solutions
5. The Lagrangian Using Quaternions and Hypercomplex Numbers
6. Quantization
7. A New Implementation
8. Mathematical fields, quaternions and hypercomplex numbers
9. Standard model groups, tensors and quaternions
10. References
Complex numbers have a polar representation. Three complex numbers that share the same real number are subgroups of a quaternion. Therefore a polar representation of a quaternion must exist. The amplitude is the absolute value of the whole quaternion. The imaginary number i expands to i, j, k for quaternions. The remaining question is how to handle the angle. Two ways work. The first is to take the inverse cosine of the scalar over the absolute value of the quaternion. The second method takes the inverse tangent of the absolute values of the 3-vector over the scalar. A right triangle is animated so the connection between velocity and the polar representation is more apparent.
Calculate the various parts that go in to the polar representation:
The polar representation of a quaternion is the magnitude of the quaternion times the exponential of angle theta in the I direction:
Here is the method:
- Ignore all factors of 'i' since quaternions come with i, j, and k pre-installed.
- Group 4-vectors together. Examples:
- (c t, R)
- (E, P c)
- (d/dt, c d/dR)
- Keep all constants as a check: c, h, G
- If possible, make dimensionless
- Treat the wave function as quaternion-valued (3 complex numbers that share the same real)
Hello:
A paper of mine was rejected because the editor couldn't imagine how quaternions could be used for something that had to be Lorentz invariant. Quaternions are a double cover for the groups SO(3) [3D spatial rotations] via the group SU(2). There is also a connection between SO(4) [4D spatial rotations] and quaternions, two elements of SU(2) that share the same real value.
Hello:
I have a tradition of creating a holiday card which has extremely technical image that no one but me understands, yet tries to make a human point in the brief prose. Here is this years addition:
The text on the inside reads:
Nature does all
at all time-places.
Not add or multiply,
add and multiply.
Not separate,
Links overlap.
Live is all a life knows
at conception through death.
Not live or love,
Live and love.
Not alone,
Hello:
A significant driver of traffic to my web site quaternions.com comes from a persistent story about the 200 quaternion equations that appeared in Maxwell's "Treatise on Electromagnetism", the first edition. By the third edition, there were no quaternion equations. In this post, I will report what I found.
