QM formulation in real vector spaces

Question

Is it possible to define quantum mechanics in real vector spaces instead of complex vector spaces and what would the dimensionality be of such a vector space?

Can anyone referee me such a treatment of a 2 state system, say the spin system.

Answer 1

It depends on what you mean by real. Every complex Hilbert space is also a real Hilbert space (with double dimension) simply decomposing complex numbers into real and imaginary parts. A complex wavefunction $\psi$ has the same information as a couple of real wavefunctions $(Re \psi, Im \psi)$. So standard quantum mechanics can be formulated into a real Hilbert space made of a sum of orthogonal preferred subspaces without problems.

However, there is another physically deeper interpretation, based on the following observation. In quantum mechanics, states are unit vectors up to phases: $\psi$ and $\psi' = e^{ia}\psi$ with $a \in \mathbb R$ are the same state. All the formalism is constructed to be invariant under this replacement.

Decomposing complex numbers into a pair of real numbers, phases become rotations. $$(Re \psi, Im \psi)^t \to R \:(Re \psi, Im \psi)^t$$ with $$ R= \left[\matrix{\cos a \:\:\sin a \\ -\sin a\: \cos a }\right]\:. $$ So, passing to the real formulation, the Hilbert space splits into a direct sum of real Hilbert spaces and states are vectors up to rotations $R$ as before, mixing the two real Hilbert spaces.

It is also clear that not all self-adjoint operators in the overall real Hilbert space define observables. Observables are self-adjoint operators of the real Hilbert space constructed out of the complex one, thus commuting with these rotations, i.e. commuting with the operator $$ J= \left[\matrix{0\:\:-I \\ I\quad \:\: 0}\right]\:. $$ satisfying $JJ=-I$ and $J^*=-J$ (like the imaginary unit).

Given that, there is another idea concerning real quantum mechanics. I mean a formulation where pure states are vectors up to signs (the phases of real numbers). Observables are here every self-adjoint operator.

This is a deeply different formulation which does not match with the previous one and is permitted by famous Solér's theorem. (A third formulation concerns Hilbert spaces constructed using quaternions in place of real or complex numbers.) All fundamental theorems (as Stone's, Gleason's, Kadison's and Wigner's theorem) hold true also with a completely real formulation of this sort.

A difficult problem with this intrinsically real formulation is that the relation between continuous symmetries and constants of motion is no longer automatic, since continuous symmetries $\{U_t\}_{t\in \mathbb R}$ are generated (Stone's theorem) by anti self-adjoint operators $A$ $$U_t= e^{tA}$$ exactly as in the complex case, but there we can re-define $$A= iA'$$ and $A'$ is self-adjoint. With the real formulation $i$ does not exist, but can be replaced for an antiself-adjoint operator satisfying $JJ=-I$, commuting with time evolution and all the relevant observables. This operator (if any) must be provided by physics.

The overall physical problem is: if the intrinsic real formulation is mathematically permitted, why no known physical system is described by it?

This is a long standing issue which can be traced back to Stueckelberg who referred to Heisenberg principle (where $i$ explicitly shows up) to rule out real formulations. In my view this approach is not satisfactory because Heisenberg principle is not nowadays so fundamental tackling the problem form a fundamental viewpoint, and also Heisenberg principle holds for massive particles only.

Recently, together with my PhD student M. Oppio, we proved that (arXiv:1611.09029 Rev. Math. Phys. 29 (2017) 1750021) as soon as one assumes that the quantum system is both relativistic and elementary, no real formulations are permitted, or better, they are indistinguishable from complex formulations. An analogous result holds for quaternionic formulations arXiv:1709.09246