The operator J(x,y)=(−y,x) satisfies J²=−I and preserves the Euclidean norm. Its exponential gives circular phase, which differs from the split-complex unit used for hyperbolic pairing.
Try the demonstration
θ = 45°. ψ = (0.7071, 0.7071); Jψ = (-0.7071, 0.7071); J²ψ = (-0.7071, -0.7071). Squared norm = 1. This is ordinary circular phase, J² = −I.
Exact real-plane rotation and operator identities. The demonstration does not derive a probability rule, action scale, physical clock, or Lorentzian spacetime.
Rotate the phase vector
The slider sets an angle θ in degrees between zero and one full turn. The dark segment represents ψ=(cosθ,sinθ). The gold segment represents Jψ, and the grey segment represents J²ψ. All endpoints lie on the same unit circle. The summary reports their components and the squared norm.
The angle is a dimensionless phase parameter, not a measured time. Calling the circle an inner recurrence would additionally require a source update, a physical phase rate, and a rule connecting the carrier to completed observations.
Derive the quarter-turn identity
Define J(x,y)=(−y,x). Applying it again gives J(−y,x)=(−x,−y), so J²=−I. Its transpose is −J, and JᵀJ=I; therefore it preserves x²+y².
The exponential identity follows by grouping even and odd powers in its series. This reconstructs an ordinary complex scalar convention on the real plane: multiplication by a+ib acts as aI+bJ. It does not require adding a third coordinate to represent the imaginary unit.
Read a forty-five-degree example
At θ=45°, ψ=(1/√2,1/√2). Its quarter-turn is (−1/√2,1/√2), and applying J twice gives (−1/√2,−1/√2). Each vector has squared norm one. At θ=360°, the original vector returns to (1,0), although the slider records that a complete turn was performed.
A circular display by itself cannot count how many complete turns occurred. A clock based on this phase needs an unwrapping convention or retained cycle memory as well as an independently calibrated angular frequency.
Why this is not the split-complex unit
A split-complex unit j instead satisfies j²=+1. Its real action exchanges the coordinates: S(x,y)=(y,x). The exponential is coshη I+sinhη S and preserves x²−y², rather than the positive norm used here.
DTT can investigate ordinary phase for coherent generation and a hyperbolic pairing for temporal or spacetime readout. Their different multiplication rules must remain explicit. The plotted circle is an ordinary phase construction. It is not a boost diagram and cannot establish a Lorentzian metric merely by renaming J as a temporal symbol.
Phase is one part of the reconstruction
An oriented real generator A=ωJ with ω>0 selects the displayed quarter-turn after its positive rate magnitude is factored out. Reversing the generator selects −J. This is a useful finite example of phase supported by dynamics, developed in the linked guide.
Quantum probabilities, subsystem composition, completion instruments, and the action scale require additional structure. In SMM/DTT the next task is to derive compatible versions from one source law. The six nested inner roles concern that generation; they are not six independent angles in this widget. Reset restores forty-five degrees, and the static circle supplies the same worked example without scripting.
Continue with the articles
- Split-complex time and ordinary quantum phase
Explain why the split-complex pairing used for temporal roles differs from the ordinary complex scalar selected for phase. Avoid treating two algebraic uses of complex language as identical.
- Selecting phase by polar decomposition
Explain how a positive-time generator can select a compatible complex structure on the dynamic sector. Make clear why zero-frequency modes need separate treatment.
- Real-commutant phase selection: a paper guide
Introduce the paper’s phase-selection question in real Hilbert representations. Prepare an accessible route from the commutant to compatible complex structures and their physical interpretation.