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Quantum logic and hypercomplex numbers

  1. Jun 28, 2010 #1
    Nearly a century ago it was found that nature obeys a particular kind of logic. Because it is related with quantum effects this logic was named quantum logic. Its axioms only slightly differ from classical logic, but this difference has enormous consequences. The structure of the quantum logical propositions, given as a lattice structure, is that of an orthomodular lattice. Von Neumann and Birkhoff quickly found that the set of closed subspaces of an infinite dimensional separatable Hilbert space has the same lattice structure. It means that quantum logical propositions can be represented in Hilbert space. It also means that the truth of a quantum logical proposition can be analyzed in this mathematical structure. This is an interesting prospect. In the course of the last century it was discovered that the Hilbert space can be defined over different number fields. This includes the real’s, the complex numbers, the quaternions and to some extent also the octonions. Most physicists stay with the complex numbers. In that way they miss the interesting features that are offered by the higher dimension numbers. One of these features is the quaternion waltz. This is what happens when a subject quaternion b is manipulated by a manipulator a in the product ab/a. With real’s and complex numbers this just gives b. With quaternions and octonions a part of the imaginary part of b is rotated. b is precessed! This effect occurs when a unitary transform U affects an observation Q. Unitary transforms let subspaces of the Hilbert space move through Hilbert space. Observables add attributes to subspaces of the Hilbert space. Thus the quaternion waltz must be common in quantum mechanics. I never saw a discussion about this feature. My question is why?

    |g> = |U f>
    <f|Q f> becomes <g|Q g> = <f U|QU f> = <f|U-1QU f> where U-1 is the adjoint of U.
    U-1QU is the operator equivalent of the quaternion waltz.

    It is interesting to see what a trail of infinitesimal unitary transforms does. The trail elements form a path and together with them travel the transformed Hilbert vectors. This causes also a path for the corresponding expectation values of the observables. These expectation values are subject to a trail of quaternion waltzes! Please investigate the role of the trail progression parameter!

    How fundamental is the quaternion waltz to physics?

    Remarkable is, that the real part of the manipulated quaternion and the Hermitian part of the manipulated operator are not affected.

    All observations are subject to the waltz effect. Does this mean that all dynamics runs via this effect?
    Last edited: Jun 28, 2010
  2. jcsd
  3. Jun 28, 2010 #2
    The trail concept is very useful. The trail of infinitesimal unitary transformations and correspondingly the trail of unitary quaternions allow the introduction of numbers and operators that surpass the quaternions. For example the octonions contain several quaternion fields and locally the octonions resemble quaternions. This also holds for curved manifold inside the octonion space. This opens opportunities for the introduction of operators that feature eigenvalues that belong to 2n-ons with n > 2. Think of a unitary-like operator that locally acts like a unitary operator, but that has eigenvalues that can store a lot more than just a primitive action.

    Together with the waltz effect the extended manipulator can introduce special as well as general relativity. In order to show that, it is necessary to apply Dennis Sciama’s analysis of inertia to the eigenspace of the manipulator. The trail progression parameter can play the role of time (It is not the usual concept of time, but it is related to it). Let each item in universe be presented by a subspace of Hilbert space. The analysis reveals that the influence of the universe of items form the cause of inertia. When somewhere in universe an item accelerates then the inertia results in a field that counteracts that acceleration. Uniform movements are not affected. An item that follows a trail that is a geodesic is not affected by inertia. The geodesic is a curved path on a manifold that is formed by eigenvalues that locally can be considered as quaternions.
    Last edited: Jun 28, 2010
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