By Sweetser D.
It's been stated that one cause physics succeeds is simply because the entire phrases in an equation are tensors of a similar rank. This paintings demanding situations that assumption, offering in its place an built-in set of equations that are all in response to a similar four-dimensional mathematical box of quaternions. in most cases this rfile indicates in cookbook kind how quaternion equations are akin to methods already in use. As Feynman mentioned, "whatever we're allowed to visualize in technological know-how has to be in step with every little thing else we all know. clean views come up simply because, in essence, tensors of alternative rank can combine in the similar equation. The 4 Maxwell equations turn into one nonhomogeneous quaternion wave equation, and the Klein-Gordon equation is a part of a quaternion uncomplicated harmonic oscillator. there's desire of integrating normal relativity with the remainder of physics as the affine parameter certainly arises while wondering lengths of durations the place the foundation strikes. due to the fact all the instruments used are woven from an identical mathematical textile, the interrelationships develop into extra transparent to my eye. desire you get pleasure from.
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At this point the difference is in the mathematical foundations. An immense amount of work has gone into the study of metrics, particular in the field of general relativity. A large effort has gone into group theory and its applications to particle physics. Yet attempts to unite these two areas of study have failed. There is no division between events, metrics and operators when solving problems using quaternions. One must be judicious in choosing quaternions that will be relevant to a particular problem in physics and therein lies the skill.
In a similar fashion, it is hoped that because the product of a transpose of a quaternion with a quaternion has the properties of a complete inner product space, the power of the mathematical field of quaternions can be used to solve a wide range of problems in quantum mechanics. This is an important area for further research. ” He tries to substitute quaternions in the place of complex numbers in the standard Hilbert space formulation of quantum mechanics. The analytical properties of quaternions do not play a critical role.
It is the work of the zero! Whatever the scalar field is in the first term of the generator gets subtracted away in the second term. A mathematical aside: a friend of mine calls this a ”conjugator”. The well-known commutator involves commuting two terms and then subtracting them from the starting terms. In this case, the two terms were conjugated and then subtracted from the original. Any quaternion expression that gets acted on by a conjugator results in a 0 scalar and a 3-vector. An anti-conjugator does the opposite task.