Imaginary and split-complex time: SMM/DTT and The thermal-time hypothesis

01The question behind the comparison

The expression imaginary time can refer to different mathematical constructions. Ordinary complex numbers use i² = −1. Split-complex numbers use j² = +1 and have a quadratic form with indefinite sign. A substitution t = −iτ in an analytic expression is another operation again: it can turn oscillatory evolution into exponential damping under suitable assumptions. These distinctions matter for SMM/DTT because its terminology connects inner and outer time with complex geometry. The interpretation becomes clearer when the algebra, bilinear form and physical observables are stated separately.

An ordinary complex coordinate z = a + ib has squared modulus a² + b². A split-complex coordinate z = a + jb has product with its conjugate a² − b². The second expression can vanish for a nonzero element and can be negative; its square root is therefore not a norm on the whole algebra in the usual positive-definite sense. Hyperbolic rotations preserve this indefinite form. This is a precise mathematical affinity with Lorentzian geometry, but it does not establish that a pair of temporal roles constitutes two physical timelike dimensions.

The nested interpretation helps here. Inner generation and outer completion can be represented using different algebraic components without becoming independently traversable axes. A model still needs a rule connecting its components with clocks and records. If ordinary quantum phase also appears, a compatible complex structure with square −1 must be distinguished from the split unit with square +1. Treating the two symbols as interchangeable would change the multiplication law and invalidate the inference. DTT can gain precision by using each structure only for the function it actually performs.

02The single-source and nested-time proposal

SMM begins with one nonspatial generative source. DTT distinguishes its inner production from the outer order in which completed results become accessible. The six inner levels are nested functional dependencies within that production; the seventh denotes outer completion. They are not six independent clocks, six spatial directions, or six simultaneously occupied temporal axes. This distinction makes the framework more specific than a general claim that everything changes: it asks how a world of persistent relations can be generated before its spatial representation is taken for granted. ^[2]

03The comparator’s explanatory structure

Connes and Rovelli propose that an algebra of observables together with a state can determine a temporal flow. In a finite faithful-state illustration, write K = −ln ρ and let observables transform as A(s) = exp(isK)Aexp(−isK). The parameter s is dimensionless. If ρ is a Gibbs state proportional to exp(−βH), then K differs from βH only by a scalar, so s can be related to ordinary Hamiltonian time by t = ℏβs in this convention.

This example explains both the power and the limitation of the comparison. Time need not be an independently supplied universal background, but the observable algebra and state are already doing essential work. A maximally mixed finite state gives a scalar K and hence trivial flow on that full matrix algebra. Different physical states can therefore select different flows. The thermal-time programme is not a claim that every microscopic event dissipates heat, nor does modular evolution alone constitute a thermodynamic arrow. DTT asks the further question of how the relevant state, algebra and accessible records are generated. ^[1]

04The precise agreement and difference

Thermal time uses ordinary operator-algebraic structures in which complex phases generate automorphisms. A split-complex unit with square +1 cannot simply replace the ordinary imaginary unit in those expressions without changing their mathematical properties. DTT’s use of complex temporal language therefore requires a careful bridge: which structure encodes indefinite temporal geometry, and which supports quantum unitary evolution?

The Gibbs-state example makes this concrete. The modular generator K = −ln ρ is self-adjoint, and exp(isK) is unitary for real s because i² = −1. Replacing i by j changes the exponential and does not automatically preserve the same positive Hilbert-space probability interpretation. A DTT construction could still contain both algebras, but it must identify their separate roles and compatibility. The positive opportunity is to derive a selected ordinary complex structure from completion while using a distinct indefinite form for clock comparisons. Thermal time then becomes a benchmark for the resulting accessible flow, rather than evidence that all meanings of imaginary time are interchangeable. ^[1]

05Worked example: what follows and what does not

For z = a + jb, multiply by exp(jη) = cosh η + j sinh η. The new components are a′ = a cosh η + b sinh η and b′ = a sinh η + b cosh η. Direct expansion gives a′² − b′² = a² − b². In contrast, multiplying a + ib by exp(iθ) preserves a² + b². Thus a quarter-turn in the ordinary complex plane and a finite hyperbolic boost have different invariants. No real rapidity η produces the circular operation i. The calculation locates the exact algebraic similarity to relativity and prevents a picturesque rotation from being mistaken for a derivation of physical evolution.

06What this comparison contributes

The positive research question is which operational distinction selects a particular algebra. If completion singles out a compatible ordinary complex structure while a separate indefinite bilinear form organizes clock comparisons, the two roles can coexist coherently. If the distinction is merely a notation chosen for convenience, it cannot determine the observed signature or the probability rule. Clarifying this issue makes comparisons with philosophical theories of time more illuminating, because those theories concern existence and ordering rather than an unexplained multiplication table.

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

Continue on SMONAD

• Split-complex time and ordinary quantum phase →

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• Complexification: how real dynamics selects a compatible phase →

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• SMM/DTT notation: nested generation, phase and physical matching →

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