By Sweetser D.
It's been stated that one cause physics succeeds is simply because all of the phrases in an equation are tensors of an identical rank. This paintings demanding situations that assumption, offering as a substitute an built-in set of equations that are all in response to an analogous four-dimensional mathematical box of quaternions. in most cases this record exhibits in cookbook type how quaternion equations are reminiscent of techniques already in use. As Feynman mentioned, "whatever we're allowed to visualize in technology needs to be in step with every thing else we all know. clean views come up simply because, in essence, tensors of alternative rank can combine in the comparable equation. The 4 Maxwell equations develop into one nonhomogeneous quaternion wave equation, and the Klein-Gordon equation is a part of a quaternion easy harmonic oscillator. there's wish of integrating basic relativity with the remainder of physics as the affine parameter certainly arises while considering lengths of durations the place the starting place strikes. when you consider that all the instruments used are woven from an analogous mathematical cloth, the interrelationships develop into extra transparent to my eye. desire you take pleasure in.
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261): ”Quaternions themselves occupy a unique place in mathematics in that they are the most general quantities that satisfy the division axiom–that the product of two factors cannot vanish without either factor vanishing. Biquaternions do not satisfy this axiom, and do not have any fundamental property which distinguishes them from other hyper-complex numbers. ” Just for the record: plenty of fine work has been done with biquaternions, and I do not deny the validity of any of it. Much effort has been directed toward ”other hyper-complex numbers”, such as Clifford algebras.
Using Quaternions in Special Relativity Events are treated as quaternions, a skew field or division algebra that is 4 dimensional. Any tool built to manipulate quaternions will also be a quaternion. In this way, although events play a different role from operators, they are made of identical mathematical fabric. ✟X, 2 t ✟X 12 AN ALTERNATIVE ALGEBRA FOR LORENTZ BOOSTS 34 The first term of squaring a quaternion is the invariant interval squared. There is implicitly, a form of the Minkowski metric that is part of the rules of quaternion multiplication.
Generating a Symmetric 2-Tensor Using Quaternions How should one rationally go about to find a generator equation that creates these terms instead of using the monthlong hunt-and-peck technique actually used? X✓ , t X✓ ✝ ✟X t✓ tt The fields E and B are kept separate except for the cross product in the Poynting vector. 5 delta(a, b)(Eˆ2 Bˆ2) term of the Maxwell stress tensor. The rest of that tensor is generated by the second line. The third line creates the Poynting vector. Using quaternions, the net sum of these terms ends up in the scalar.