Real and quaternionic quantum mechanics

Wrapping up the discussion about quaternionic quantum mechanics, in this theory the inner product \(<f|g> = \int d^3 x \bar{f}(x,t) g(x,t)\) decomposes into a complex inner product and a symplectic inner product:

\(<f|g> = {<f|g>}_{C} + {<f|g>}_{S}\)

Here "f bar" means complex conjugation with respect to all imaginary elements: i, j, k.

If we consider quaternionic wavefunction \(f\) decomposition: \(f = f_{\alpha} + j f_{\beta}\) where \(f_{\alpha} = f_0 + i f_1\) and \(f_{\beta} = f_2 - i f_3\) then:

\({<f|g>}_{C} = \int d^3 x {f}^{*}_{\alpha}(x,t) {g}_{\alpha}(x,t) + {f}^{*}_{\beta}(x,t) {g}_{\beta}(x,t) \)

and

\({<f|g>}_{S} = \int d^3 x {f}_{\alpha}(x,t) {g}_{\beta}(x,t) - {f}_{\beta}(x,t) {g}_{\alpha}(x,t) \)

Therefore in quaternionic quantum mechanics the probabilities are different then in complex quantum mechanics because it includes the symplectic part:

\({|<f|g>|}^2\ = {|{<f|g>}_{C}|}^2 + {|{<f|g>}_{S}|}^2\)

Now real quantum mechanics is defined over the real numbers and we can compare it with complex quantum mechanics. If we consider the complex quantum mechanics decomposition of the wavefunction:

\( f = f_0+ i f_1\)

the complex inner product is:

\(<f|g> = {<f|g>}_{C} + {<f|g>}_{I}\)

with

\({<f|g>}_{R} = \int d^3 x {f}_{0}(x,t) {g}_{0}(x,t) + {f}_{1}(x,t) {g}_{1}(x,t) \)

and

\({<f|g>}_{I} = \int d^3 x {f}_{0}(x,t) {g}_{1}(x,t) - {f}_{1}(x,t) {g}_{0}(x,t) \)

Then 

\({|<f|g>|}^2\ = {|{<f|g>}_{R}|}^2 + {|{<f|g>}_{I}|}^2\)

and real quantum mechanics has only this term: \({|{<f|g>}_{R}|}^2\)

Now for the main problem of real quantum mechanics: in real quantum mechanics there are no energy eigenstates. Because of this we need to embed real quantum mechanics in complex quantum mechanics!!!

Quaternionic quantum mechanics does not have such problems, but in quaternionic quantum mechanics a De Finetti theorem does not hold. The root cause of it is the fact that quaternionic quantum mechanics does not respect the tensor product. 

In conclusion, both real and quaternionic quantum mechanics have very limited usefulness in describing nature and the most general way to express quantum mechanics is over complex numbers. 

Now the million dollar question: are the quantum mechanics number systems exhausted by the reals, complex numbers, or quaternions? The answer is no and there is at least one more: a direct sum of two SL(2,C). This will lead to Dirac's equation of the electron. Please stay tuned.