Why does quantum theory use an imaginary unit? One way to sharpen the question is to begin with real state vectors and ask whether their dynamics selects an operation that behaves like multiplication by i. A rotating plane makes the central mechanism visible. It also shows exactly where further physical input is needed.
1. From an allowed phase to a selected phase
My paper Real-commutant phase selection from real Hilbert representations, published in Physics Letters A, studies complex structures compatible with real representations and the conditions under which phase can be selected. The website reading guide develops finite examples of that question. This post explains one of them step by step.
A real vector can carry two components, such as q and p. To treat the pair as a complex number, we need an operation J that supplies the action of the imaginary unit. Applying it twice should reverse the vector. Requiring it to preserve lengths makes it compatible with the real inner product:
J2 = −I, JTJ = I.
There can be many such operations. Compatibility with a represented symmetry requires J to commute with that symmetry. The collection of real operators that commute with every represented symmetry is called the real commutant. Finding a complex structure there establishes compatibility; selecting a particular one requires enough additional information to resolve the remaining freedom.
2. A rotating plane supplies the calculation
Consider the quarter-turn J0(q, p) = (−p, q). For a positive frequency ω, let a state evolve according to the generator A = ωJ0. Starting from (1, 0), the state follows (cos ωt, sin ωt). Its length stays constant while the angle increases.
Two pieces of information are present: the speed of rotation and its orientation. They separate algebraically because A2 = −ω2I. The positive magnitude of the generator is therefore |A| = √(−A2) = ωI. Dividing out that magnitude gives
JA = A|A|−1 = J0.
This is the polar decomposition in an elementary setting. The generator determines its orientation factor once its positive magnitude is fixed. Reversing the actual evolution changes A to −A and changes the selected phase to −J0.
Now combine two independent planes with A = diag(2J0, −3J0). The first turns in one direction at rate two; the second turns in the opposite direction at rate three. Direct calculation gives
|A| = diag(2I, 3I),
JA = diag(J0, −J0).
Different frequencies remain in the magnitude operator. The phase operator records the orientation on each plane and still squares to minus the identity. The calculation uses an invertible finite matrix, so the inverse is ordinary matrix inversion. Infinite-dimensional generators require the relevant spectral and domain conditions.
3. Stationary states and composites expose the remaining choices
Add a stationary plane on which A is zero. Every state on that plane remains fixed. The motion therefore supplies no preference between the two quarter-turn orientations there. Both can extend the phase structure, but a selection needs further information. Add just one stationary real direction instead, and an extension can fail altogether: no real scalar squared equals −1.
This separates existence, selection and extension. It also suggests a practical reading habit: identify the sector on which a theorem operates before assigning its conclusion to the whole physical state space.
Composition raises another question. A complex line has two real dimensions. Two such lines have a complex tensor product of complex dimension one, whereas their underlying real planes have a real tensor product of real dimension four. To recover the usual shared complex scalar action, the construction imposes a balancing relation:
[(J1x) ⊗ y] = [x ⊗ (J2y)].
The brackets denote equivalence classes in the balanced composite. Moving the common phase from one factor to the other represents the same composite vector. Selecting each factor's phase makes this construction possible; a physical account of why systems compose this way remains a further task.
There is a useful comparison with Moretti and Oppio's work on real Hilbert spaces and Poincaré symmetry. Under their stated relativistic assumptions, a preferred complex description can arise from symmetry. Such results make the assumptions that select phase a substantive part of the physics.
4. Connecting phase selection to completed observations
Within my proposed Single Monad Model and Duality of Time Theory, the broader aim is to connect generative dynamics with the records available to observers. The six inner levels describe nested roles within generation; the seventh level is outer, completed observation. They are not six independent clock axes. The rotating planes above are state-space examples and do not supply an identification of those levels.
Another distinction matters: the phase operator here satisfies J2 = −I. The split-complex unit used in some temporal formulations satisfies j2 = +1. A derivation relating the two must specify their different roles and the map between them.
A focused next question is whether one source dynamics can select the phase, determine the observation rule and constrain the composite construction together. That would remove independent choices from the reconstruction. An experimentally meaningful development would then identify a measurable restriction that survives changes of mathematical description.
Question for collaboration: Which assumptions on a real source representation and its allowed observation maps would force a shared phase for interacting subsystems, including stationary sectors? I welcome collaboration with experts in operator algebras, representation theory and quantum foundations on this specific problem.
Sources and further reading
- Mohamed Haj Yousef (2026). Real-commutant phase selection from real Hilbert representations. Physics Letters A, 594, 132085. The central publication; consult its exact hypotheses and classification results.
- SMONAD: Real-commutant phase selection—orientation and composition. Provides the finite polar, stationary-sector and composition examples explained here.
- Valter Moretti and Marco Oppio (2017). Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry. Reviews in Mathematical Physics, 29, 1750021. A comparison based on explicit relativistic symmetry assumptions.