Quaternionic Quantum Mechanics
Continuing the quaternionic discussion, let us see when one might encounter it. Here I will follow the discussion in "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski. The idea is that of a time reversal.
We start in the usual complex quantum mechanics and here by Wigner's theorem we have that every symmetry is represented by a unitary or anti-unitary transformation. So suppose we have a time reversal operator \(\Theta \). When we reverse time, from the time evolution equation in the Heisenberg picture this is equivalent with \(i\) changing signs, and therefore \(\Theta\) must be an anti-unitary transformation.
If the system is invariant under time reversal, we have:
\(\langle \Psi| \Phi \rangle = \langle \Theta\Psi| \Theta\Phi \rangle\)
and this means that there are two options for \(\Theta\): \(\Theta^2 = \pm 1\).
Now the discussion depends on the angular momentum of the system. For fermions \(\Theta^2 = - 1\). If we cannot tell the direction of time by any measurement, the observables commute with \(\Theta\):
\([O, \Theta] = 0\)
and this defines a superselection rule. Then one can define:
\( i, \Theta, i\Theta\ = i, j, k\)
the quaternionic imaginary elements and one arrives at the quaternionic projective space. Here is how quaternionic quantum mechanics can arise.
However, because in this case one talks about superselection rules, composing two quaternionic systems breaks the superselection constraint and there are problems defining the tensor product. We'll talk about this next time.