Quaternion Mathematica Notebook Server -abs

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Polar Representations of Quaternions
Doug Sweetser

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.

2 pages 1 figure(s) 1 Sep 2010 6 Nov 2010

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Lorentz Boosts with Quaternions
Douglas Sweetser

A method for using hyperbolic sines and cosines in a real-valued quaternion to generate a Lorentz boost along an axis is shown. Do precisely what one does for a 3D spacial rotation as a first step, substituting the hyperbolic for regular trig functions, B' = H B H*. That creates four terms that are need, adding two extra terms and containing two omissions. A difference between the same three matrices does the job: B' = H B H* + ( (H H B)* - (H* H* B)* )/2. The inverse transform is created by changing the conjugates on the hyperbolic quaternions. Boosts represented with quaternions must form a group, but it is not compact because the operator uses both addition and multiplication.

2 pages 3 Sep 2010 7 Oct 2010

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Quaternion Quantum Field Theory Demystified: The Method Applied
Douglas Sweetser

Quaternion quantum field theory is introduced. The goal is for every equation that plays a role in quantum field theory gets rewritten using real-valued quaternions. Like the correspondence principle before it, the method is simple and systematic: keep 4-vectors together, drop factors of i, keep the constants, but make the expression dimensionless. The differences between classical, relativistic and quantum mechanics equations are based on their constants and form. The uncertainty principles for position/momentum, and energy/time appear in the same expression, a result of the product rule of calculus. The method will shun the most famous equation in physics, $E=m c^2$, because momentum is omitted. The square of energy-momentum will be used in its place. Substitution in that equation leads directly to the Klein-Gordon equation.

The path to the Dirac equation is more complicated. One needs to know that the 16 gamma matrices can be represented by quaternion triple products. Pre- and post-multiply a quaternion by each combination of the four basis vectors accomplishes the feat. Particular sets of quaternion gamma operators and choices of inertial reference frames can lead to wave functions whose scalar is either positive or negative definite.

A way to visualize quantum field theory using analytical animations is begun. The simplest animation is for an inertial observer. The same number gets added iteratively. The classical view is what one would expect: a ball moving in a straight line at a steady pace. The quantum view show the ball following the same path on average, but the next step cannot be known. Applying the quaternion gamma matrices to inertial path creates 16 possible histories. There is a huge amount of work ahead for the visualization project, but it will look interesting and can be shared with a far greater audience than quaternion quantum field theory equations ever will.

8 pages 4 figure(s) 3 Sep 2010 26 Oct 2010

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Quaternion Quantum Field Theory Demystified: Special Relativity with Intervals but Without Metrics
Douglas Sweetser

A quaternion variation on the metric is explored. The quaternion approach takes into account phase, which could be an improvement. Rotations and boosts work as expect for quaternions. The Einstein summation convention will not be used in this body of work, nor with covariant or contravariant vectors because there are no indices. A variety of conjugates will be used in their place. An open technical issue is how to deal with derivatives in curved quaternion spacetime.

3 pages 29 Sep 2010 11 Nov 2010

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DRAFT: Specific speculations on the Strong Extremes of Gravity
Douglas Sweetser

Strong field gravity must involve a metric that has Planck's constant. The GEM field equations have G, $\hbar$ , and c, so could do the job based on the units. Potential solutions are found of the Maxwell equations in the Lorenz gauge, the hypercomplex gravity equations in the Lorenz gauge, and the gauge-invariant GEM field equations. Metrics are proposed to also solve the field equations. The Christoffel symbols are calculated for each of these metrics. The metrics do not solve the field equations unless an ad hoc but precise rule is applied. I suspect I may need to abandon the accounting system used by Riemann so long ago, a scary idea.

9 pages 24 Sep 2010 24 Sep 2010

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A family of variations of the Maxwell action using quaternions and hypercomplex numbers contains the symmetries of the standard model and a testable gravity theory
Douglas Sweetser

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.

21 pages 2 figure(s) 8 Sep 2010 23 Oct 2010

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Spacetime Trig Functions Graph
Douglas Sweetser

This one-page figure shows the relationships between sin/cos/tan and their hyperbolic counterparts for events in spacetime. The standard trig functions are the three pairwise ratios: sine -> far/hypothenuse, cosine -> near/hypothenuse, tangent -> far/near. What is not often known is the hyperbolics are even simpler, being only about one side of a trinage made to the hyperbola: sine -> far, cosine -> near, tanget -> hypothenuse. Any point in spacetime can be plotted using Euclidean, polar (scaled signs and cosines) or hyberbolic coordinates (scaled hyperbolic sines and cosines).

1 pages 1 figure(s) 28 Oct 2010 28 Oct 2010