Real commutants and the selection of quantum phase

A real starting point changes the question

A real Hilbert space has lengths, angles and orthogonal transformations, but no distinguished multiplication by an imaginary unit. To regard it as a complex quantum state space, one needs a real-linear operator J satisfying J² = −I. Orthogonality adds J† = −J. Complex scalar multiplication is then defined by (a+ib)v = av+bJv. This converts phase from an unexplained scalar label into a piece of operator data whose existence and compatibility can be checked.

That question is independent of the symbolic split-complex notation sometimes used for inner and outer time. A temporal marker with square +1 does not provide circular quantum phase. This article writes J for the selected phase and reserves other names for temporal markers and the antiunitary real structure of a spectral triple. The distinction matters because the three objects have different algebraic laws.

The reconstruction also begins after a Hilbert space has been supplied. A set of completed records is not automatically a Hilbert space: a positive state, measure, norm or equivalent representation input is still needed. The phase theorem addresses a specific next step within that supplied setting.

Source: Interactions Unification, §§2.2, 3.1–3.3, pp. 5–6.

The commutant records compatible operations

Let U(g) be the real orthogonal action of a symmetry group. Its real commutant consists of bounded real-linear operators T with TU(g)=U(g)T for every g. A compatible phase must lie in this algebra; otherwise calling J multiplication by i would turn some declared symmetry into an operation that is not complex-linear.

Complexification provides an equivalent formulation. If the complexified space has canonical conjugation C, a complex-linear operator comes from a real operator exactly when it commutes with C. Requiring it also to commute with the complexified symmetry gives the Real commutant. Restriction to the fixed real subspace and complexification are inverse operations, so this is a real algebra isomorphism, not an extra physical postulate.

This characterization separates classification from selection. Listing all admissible J tells us what phases could coexist with the representation. A dynamical orientation can then select one of them. A representation may admit several phases or none at all.

Source: Noncommutative Geometry, §4.1, pp. 13–14.

Real, complex and quaternionic multiplicities

For a finite-dimensional orthogonal representation of a compact group, an irreducible real block has endomorphism algebra R, C or H. With multiplicity m, the corresponding commutant block is a matrix algebra over that division algebra. The phase problem takes place in the multiplicity algebra. This explains why counting an irreducible representation and counting its copies are different tasks.

The table can be checked by conjugating a skew-adjoint J to its standard form and finding the subgroup that commutes with it. In particular, a multiplicity-one real block has no phase; a multiplicity-one complex block has two orientation choices; a quaternionic line has a sphere of imaginary directions. The orientation manuscript sketches this classification, but its unfinished theorem boxes are not used as completed proofs here.

Source: Orientation Yang Mills, §5, pp. 6–7; draft roadmap.

An oriented flow selects the dynamic phase

Suppose U(t)=exp(tK) is a strongly continuous real orthogonal flow, with skew-adjoint generator K. On the moving subspace H_dyn=(ker K)^⊥, polar decomposition writes K as a phase times a positive magnitude:

The polar factor is uniquely determined on this subspace. It commutes with the magnitude and with every symmetry commuting with the full flow. No positive lower bound on |K| is needed. If frequencies accumulate at zero, the formal expression K|K|⁻¹ needs care because the inverse can be unbounded; the bounded polar factor remains the correct definition.

On ker K the polar factor is zero, so it does not supply a complex structure there. Reversing the flow changes J_K to −J_K. Rescaling K by a positive number leaves J_K unchanged while changing every frequency. More generally, many positive operators B commuting with a fixed J give generators K=JB with that same phase. Phase therefore does not determine an energy scale or spectrum.

Source: Interactions Unification, Theorem 3.1, p. 6.

Worked example: a rotating real plane

On R² take J(x,y)=(−y,x) and K=ωJ with ω>0. The flow is a rotation through angle ωt, |K|=ωI, and the selected polar phase is J. Changing ω from 1 to 100 changes the dynamics without changing the selected complex structure. The commuting orthogonal transformations are the rotations, which become U(1) after the phase has been chosen.

Now add the reflection R(x,y)=(x,−y) to the required symmetry group. It sends J to −J, so it cannot commute with either phase on this multiplicity-one plane. The reflection reverses the orientation of the flow. Requiring invariance under the full reflection symmetry and retaining an oriented phase are different requirements; they cannot silently be imposed together in this example.

The same construction on the realification of Cⁿ, with K=ωJ, has commuting orthogonal group U(n) for every positive integer n. It gives a direct counterexample to the claim that real time plus positive phase uniquely selects three colors. A multiplicity selection rule must do additional work.

Source: Interactions Unification, Proposition 3.3, pp. 6–7.

Which quaternionic symmetry survives?

Unit quaternions form Sp(1), isomorphic to SU(2), but choosing a phase can reduce that symmetry. If quaternionic matrices act on the left on H^m and the chosen left phase is iI, a matrix commuting with iI has only complex entries. The surviving compact group is U(m). At m=1 this leaves U(1).

There is also a consistent construction in which complex phase acts by right multiplication and unit quaternions act on the left. These actions commute, producing a complex two-dimensional SU(2) carrier. Its consistency does not establish that a temporal source selects the required left/right separation.

Spatial spin adds another distinction. Quaternion conjugation acts on the real three-dimensional space of imaginary quaternions and gives the double cover SU(2)→SO(3). The weak group acts on an internal doublet. Equal abstract group types do not identify those actions, and neither construction says why only the observed left-handed matter is weakly charged.

Source: Interactions Unification, Proposition 3.2 and discussion, pp. 6–7.

Phase must survive composition and comparison

Two phase-selected systems require a composition rule. The raw real tensor product of two complex state spaces regarded as real has too many real dimensions for their complex tensor product. The balanced construction identifies the phase acting on one factor with the phase acting on the other:

Likewise a forward link U_yx:V_x→V_y must obey U_yxJ_x=J_yU_yx to be complex-linear for the selected endpoint phases. In a smooth realization this becomes ∇J=0. A loop then commutes with J at its base point. Failure of this test means the chosen transport does not preserve the proposed complex carrier.

One may restrict to compatible transport, supply additional geometric data for a changing phase, or revise the carrier. The one-system polar theorem does not choose among those responses. This is an operational interface that a completion model can test directly.

Source: Interactions Unification, §§3.3, 16.2, pp. 6, 27.

A discrete tick adds a branch problem

A single orthogonal tick W=exp(εK) records phases modulo a full turn. A logarithm is unique only after choosing a spectral branch, ordinarily an arc avoiding −1. For a frequency ω, adding 2πk/ε leaves the tick unchanged. Thus the exact energy alias spacing is 2πℏ/ε, while πℏ/ε is the edge of the symmetric principal energy interval. These are different quantities.

If ε is a regulator that tends to zero, the branch window expands. If it is a physical tick duration, selecting the relevant frequency branch is physical input. The source's aliasing discussion is useful when this distinction is retained.

The productive conclusion is narrow: a supplied oriented real dynamics can explain a compatible complex phase and its orientation. To reach electromagnetism one still needs local fibers, nontrivial comparison data, charged representations, a dynamical action and physical calibration. A scalar phase convention on its own contains none of those ingredients.

Source: Noncommutative Geometry, §4.3, p. 15; alias-spacing clarification.

Continue reading

• The conditional sevenfold recurrence
• Protected spectral bundles
• Temporal operator algebras
• Local frames and gauge covariance