The number systems of quantum mechanics
Now we have reached the end of the series of the number system for quantum mechanics. Quantum mechanics can be expressed over any real Jordan algebras (including spin factors), but which one is picked by nature? The simplest case is complex quantum mechanics because you can construct the tensor product and the number system is commutative. There is a theorem by Soler which restricts the number system to real numbers, complex numbers, and quaternions but the starting assumptions are too restrictive. There is no need to force the inner product to generate only positive numbers.
When the number system is the real numbers, then this can exists only as an embedding in complex quantum mechanics so we want to build out number system from matrices of complex numbers and quaternions. The existence of the tensor product is not a requirement in general. Two fermions together do not form another fermion. But what is the meaning of a number system beside complex numbers? Basically this adds internal degrees of freedom. Do we know of additional degrees of freedom? Yes. They are the gauge symmetries.
The natural framework for discussing the number system for quantum mechanics is Connes' spectral triple. The number system is the algebra \(A\) in the spectral triple, while the unitary time evolution or the Zovko equation of continuity for quantions gives rise to the Dirac operator \(D\) in the spectral triple. The standard model arises in this formalism by a judicious pick of the algebra which gives the internal degrees of freedom. The selection of \(A\) is now done ad-hoc to simply recover the Lagrangian of the Standard Model.
One may imagine different universes where the algebra is different. Quantionic quantum mechanics does not describe our universe because we do not see a long distance Yang-Mills field with the gauge group SU(2)xU(1). Instead the electroweak field is subtler and there is a mixing of a U(1) with SU(2)xU(1) with the Weinberg angle so you may say that our universe resulted in part from a coupling between complex and quantionic quantum mechanics.
Still, regardless of the number system picked by nature for quantum mechanics, everything reduces to complex quantum mechanics when the internal degrees of freedom are ignored. This is because there are only two number systems which respect the tensor product, and in the non-relativistic limit they are identical. In complex quantum mechanics, the additional degrees of freedom form superselection domains, and C* algebras are compatible with superselection rules.
The natural framework for discussing the number system for quantum mechanics is Connes' spectral triple. The number system is the algebra \(A\) in the spectral triple, while the unitary time evolution or the Zovko equation of continuity for quantions gives rise to the Dirac operator \(D\) in the spectral triple. The standard model arises in this formalism by a judicious pick of the algebra which gives the internal degrees of freedom. The selection of \(A\) is now done ad-hoc to simply recover the Lagrangian of the Standard Model.
One may imagine different universes where the algebra is different. Quantionic quantum mechanics does not describe our universe because we do not see a long distance Yang-Mills field with the gauge group SU(2)xU(1). Instead the electroweak field is subtler and there is a mixing of a U(1) with SU(2)xU(1) with the Weinberg angle so you may say that our universe resulted in part from a coupling between complex and quantionic quantum mechanics.
Still, regardless of the number system picked by nature for quantum mechanics, everything reduces to complex quantum mechanics when the internal degrees of freedom are ignored. This is because there are only two number systems which respect the tensor product, and in the non-relativistic limit they are identical. In complex quantum mechanics, the additional degrees of freedom form superselection domains, and C* algebras are compatible with superselection rules.
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