Circular phase preserves x²+y²; split-complex multiplication preserves x²−y². A Lorentzian determinant can also arise from positive Hermitian records without replacing the ordinary phase unit.
One word, two algebraic roles
In familiar quantum mechanics, multiplying an amplitude by exp(iθ) rotates its phase. The imaginary unit satisfies i²=−1. DTT also uses a split-complex expression such as tin+jtout, where j²=+1, to describe a hyperbolic pairing. The shared word complex does not make the two products identical.
The distinction helps the reconstruction programme. Inner recurrence may be represented by an ordinary phase operation, while a later temporal or causal pairing uses an indefinite quadratic form. A successful model must explain how these layers communicate through dynamics and records. Replacing one symbol with another cannot establish that physical bridge or a law relating measured clock rates.
Ordinary complex phase is circular
Write z=x+iy with real x,y. Conjugation gives z̄=x−iy and zz̄=x²+y². Multiplication by i acts on the coordinate pair as J(x,y)=(−y,x). Applying J twice reverses both components. Expanding the exponential and grouping even and odd powers gives:
|exp(iθ)z|²=|z|².
For z=1 and θ=π/2 the result is i; four quarter-turns return the original vector. Every nonzero ordinary complex number has an inverse, z−1=z̄/|z|². The positive norm is part of what makes circular phase suitable for norm-preserving amplitude evolution. Selecting this algebra does not, by itself, interpret its norm as a probability.
Split-complex multiplication is hyperbolic
Now use w=x+jy with j²=+1. Conjugation gives ww̄=x²−y². Multiplication by j exchanges x and y, so its real matrix S squares to I. The corresponding exponential has hyperbolic rather than circular functions:
(x′,y′)=(x coshη+y sinhη, x sinhη+y coshη).
Substitution shows x′²−y′²=x²−y². Starting from (1,0) gives (coshη,sinhη), a branch of a hyperbola. The parameter η is dimensionless. To interpret it as physical rapidity, one must separately identify the coordinates with calibrated spacetime quantities and establish that the allowed transformations relate physical observers.
Zero divisors change the algebra
The split-complex numbers have nonzero elements whose product vanishes: (1+j)(1−j)=0. Define e+=(1+j)/2 and e−=(1−j)/2. Direct multiplication gives e±²=e± and e+e−=0. Each number decomposes into two independently scaled null directions.
exp(jη)=eηe++e−ηe−.
This is why the positive multiplicative-norm argument selecting an associative division algebra cannot use the split unit as one of its circular phases. The zero divisors violate division. The quaternionic construction uses units squaring to −1, while the split pairing belongs to a different layer of the description.
A Lorentzian form from ordinary complex packets
A separate route to an indefinite quadratic form starts with a two-component ordinary complex packet z and its positive record zz†. Sums of such records produce positive Hermitian matrices. Any Hermitian two-by-two matrix can be written X=tI+xσx+yσy+zσz, where the last z now labels a real coordinate rather than the packet.
X≥0 ⇔ t≥√(x²+y²+z²).
The eigenvalues are t±√(x²+y²+z²), which proves the positivity criterion. The negative signs arise from the determinant, although the packets still use ordinary i²=−1. There is no algebraic need to reinterpret the quantum imaginary unit as split-complex. Physical events, displacement lengths, and elapsed times still require an operational identification of these record coordinates.
Compare two transformations on the same pair
Take the numerical pair (2,1). A circular quarter-turn gives (−1,2), preserving the Euclidean squared norm five. A split boost with η=log2 has coshη=5/4 and sinhη=3/4. It gives (13/4,11/4), preserving the indefinite value three.
The transformed Euclidean norm is different, and neither operation is an alternative notation for the other. This compact example is useful when checking a temporal derivation: identify which product rule is used in every step, which quantity remains invariant, and whether the coordinates have physical units. A supposed phase invariance can otherwise silently become an unrelated metric assumption.
What a DTT bridge should calculate
The conceptual proposal is that inner generation supplies coherent transformations and outer completion supplies stable relational records. A rigorous bridge would specify the source state, show which phase information is retained, construct the record cone, and relate its order to a clock observable generated by the same process.
The six nested inner roles can organize dependencies within that construction. They do not turn the two-component algebraic examples into six independent temporal axes. The next research target is compatibility: does one autonomous update preserve the appropriate amplitude norm, produce admissible completed records, and constrain their propagation? That question makes the coexistence of circular phase and hyperbolic readout an explicit mathematical programme.
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.
- Hermitian records from complete tests
Explain how complete tests identify records with positive two-by-two Hermitian matrices in the draft. Relate record positivity to admissible observations rather than a chosen spatial coordinate system.
- 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.
- Completion, commutants, and positive frequency
A reading guide to the published paper’s relationship between completion principles, real representations, and positive-frequency geometry. Focus on the theorem assumptions before discussing physical interpretations.
Sources and reading
Read the original source for its exact formulation and edition. The examples on this page are explanatory reconstructions.