# Quaternionic Quantum Mechanics

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Would it be judicious to define the Hilbert space of Quantum Mechanics over the quaternion numbers instead of the complex numbers? We can define a tensor product by the Hamilton numbers $\bf Q$:

$H_{\bf Q}={\bf Q} \otimes_{\bf R} H$

with products:

$q(q' \otimes \psi)=(qq')\otimes \psi$

$(q \otimes \psi)q'=(qq')\otimes \psi$

with $H$ the complex Hilbert space. So that we have the brackets $<q \psi|\phi>=\bar q<\psi|\phi>$, $<\psi q |\phi >=<\psi | \bar q \phi>$ and $<\psi |\phi q>=<\psi |\phi> q$.

The observables would be right linear operators: $A|\psi q>=A|\psi >q$.

asked Aug 12, 2018
edited Aug 13, 2018

How do you define the trace operator on quaternions HS ? more seriously, it is an intensive domain of reformulation, particularly since the Seiberg-Witten equations and the large symmetry group of their trivial reducible solution. You would find easily deep reformulations and theories ports focusing on the one you are interested ( f.e. Maxwell, QED, QCD ). There are still difficulties with the trace operator and in general with a few original definitions too restricted to ( real and ) complex based HS. An interesting compilation to start : Quaternions and Hilbert Spaces ( Chapters 1 to 4 )

As far as I understand, complex numbers (functions) appeared in QM because QM is a "wave mechanics", which naturally gives quantization in restricted systems (all musical instruments are based on this property). A harmonic wave at a point has an amplitude and a phase - voilà two real numbers (= a complex number) instead of one real number in Classical Mechanics. Thus, QM is formulated in terms of complex functions for this reason. Of course, later on, spinors were introduced, then other group representations like iso-spinors, etc.

There must be a natural necessity in introducing quaternions, a necessity similar to what happened to QM.

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