Complex Time: Identification of i as the Internal Phase of Time, and Its Consequences for the Genesis of Mass

Toru Ishibashi

Published Jul 24, 2026• 0 citations

Abstract

Quantum mechanics is built on the imaginary unit i, yet it does not say what that i physically is, and it treats time as an external parameter without internal structure. This paper places a single identification: time is a unit complex quantity with two orthogonal components, T = pf + i·fp (|T|² = 1), where pf is the fixed past, fp is the undetermined future, and i is the generator of the 90° rotation between them. From this identification, with the additional inputs counted explicitly (linearity of the evolution, inertial-frame invariance of the internal frequency, and local phase invariance), follow the structure of the Planck relation (E ∝ ω, amplitude-independent), the Schrödinger equation, the metric of special relativity and its signature, the electromagnetic gauge structure, the form of the Born rule, and the skeleton of the Dirac equation. Three consequences about time follow. First, "time" is two-layered: the rotating hand (the internal phase time T, which ticks proper time) and its winding number (the parameter time t). Second, the rate of the proper clock is an invariant of the worldline; observer-dependent energy cannot drive it; the clock is driven only by the invariant of the dispersion relation, the mass squared m². A state with non-positive m² becomes a "frozen envelope": it stores configuration energy while its proper phase does not rotate (the dynamics of its stability is disclosed as an open seam). Third, the arrow of time sits not in dynamics but in a modal asymmetry (undetermined → fixed). Around the threshold m² = 0 these consequences become distinguishable from the standard theory, and they connect to three companion papers: neutrinos (residents of the threshold), dark energy (the allocation of the cosmic phase), and dark matter (stranded states that failed ignition). Derivation, inheritance, and seam are distinguished throughout. complex time, internal phase of time, arrow of time, time in quantum mechanics, foundations of quantum mechanics, imaginary unit, Born rule, proper time, mass generation, Landau–Zener transition, Hopf fibration, neutrino mass, dark matter, dark energy

DOI: 10.5281/zenodo.21529409

Related papers

The Structure of Time: Complex-Time Theory and the Derivation of Quantum Mechanics from Temporal Ontology

Toru Ishibashi

We present complex-time (CT) theory, a framework in which the fundamental structures of quantum mechanics are derived from three physical postulates concerning the internal structure of time. The postulates are: (P1) physical time possesses two orthogonal components — past-fixed (pf), representing actualized, irreversible records, and future-pending (fp), representing physically real but undetermined possibilities; (P2) pf and fp are orthogonal (pf fp); and (P3) the total temporal magnitude is conserved (pf + fp = 1). From these three postulates alone, we show that: (i) the state space is the unit circle and the dynamical group is , so that the imaginary unit arises as the unique generator of temporal evolution — not as a mathematical convenience but as a consequence of temporal geometry; (ii) the Schrödinger equation follows from the identification of energy with the rate of phase rotation in the pf–fp plane, with the Planck constant playing the role of a natural constant intrinsic to the pf/fp structure; (iii) the de Broglie–Einstein relation holds universally — for massive and massless particles alike — because the wave structure belongs to time itself, not to any particular particle; (iv) the uncertainty principle is a geometric consequence of the impossibility of fully eliminating fp while preserving ; (v) the complex structure of Hilbert space, the inner product, and the Born rule all follow from the geometry of fp pf projection; and (vi) energy quantization emerges from a “real-axis sampling” rule without imposing discreteness on the underlying dynamics. We further show that the CT framework provides unified descriptions of double-slit interference, quantum tunneling, unstable-state decay widths, and the structural origin of the arrow of time. Preliminary experimental predictions for quantum-circuit observables are stated, and applications to cosmological dark-energy phenomenology, galaxy rotation curves, and Lorentzian path-integral contour selection are outlined with references to dedicated companion papers. The central thesis is that quantum mechanics is not a theory of particles with mysterious wave-like properties, but a consequence of the internal structure of time — time whose wave-like phase (fp) is projected, irreversibly and discretely, onto the real axis (pf) in every act of measurement. Keywords: complex time; temporal structure; imaginary unit; quantum mechanics derivation; past-fixed / future-pending decomposition; Born rule; energy quantization; measurement; irreversibility; arrow of time

Apr 1, 20260 citations

A defining equation of time as the rotation count of an internal clock

Toru Ishibashi

In standard physics the time t is an external parameter and is not itself defined, while the second is defined by counting the cycles of a periodic phenomenon. The two are tied together if time has an internal phase. This paper places as an axiom the unit complex quantity T=pf+i·fp, |T|=1, a unit vector rotating in the complex plane at a speed set by energy, and for a system at rest by the mass. Solving the rotation equation dT/dt=−i(mc²/ℏ)T for t gives t=(ℏ/mc²)∫i·dT/T=(N+φ/2π)h/(mc²), with N the rotation count, φ the fractional part below one rotation, and h/(mc²) the Compton period. The time of a system of mass m is thus the rotation count of its own internal phase. Conditions placed on the same equation yield the mass shell, Lorentz factor and metric signature of special relativity and the Schrödinger equation as solutions. The imaginary unit of quantum mechanics is identified with the generator of this rotation. The scale of time is the integer N, and φ appears only as a phase difference in interference. Earlier work places the clock outside the system, writes the internal oscillation as a function of an external t, or defines one time for many degrees of freedom. A defining equation of t for a single massive system from an internal phase of time itself is, to the author’s knowledge, new. From the fixing condition follow the quantum speed limit, the quantum Zeno effect, and a null test for atom interferometers.

Oct 21, 20260 citations

What is Time? Complex-Time Geometry, Future-Pending Structure, and the Role of i

Ishibashi, Toru

Abstract We propose a two-component structure of physical time based on a distinction between Past-fixed (pf) and future-pending (fp) contributions. pf time encodes events that have already been realized and recorded, while fp time encodes those possibilities that have not yet become facts but may do so. This structure is represented by a complex-time variable and a normalized complex amplitude with , where pf and fp denote, respectively, the pf and fpcomponents. In this framework the imaginary unit is interpreted as a generator of rotations between pf and fp, rather than as a purely formal algebraic device. We show that unitary quantum dynamics can be understood as the conservation of the combined pf – fp weight, and that the Schrödinger equation describes rotations of the complex-time amplitude with angular velocity . Measurement processes are described as projections from fp to pf, and we argue that the Born rule arises from natural symmetry and additivity requirements on the frequencies of such future-to-past conversions. For two-level systems, equal fp alternatives lead to equal pf frequencies; for rational amplitude ratios, fp can be decomposed into equal “future packets” whose uniform projection into pf yields probabilities proportional to the squared amplitudes. In higher dimensions, Gleason’s theorem implies that these conditions uniquely single out the usual Born rule. We further show that discrete energy spectra can be viewed as a consequence of sampling the complex-time rotation only at those instants where lies on the real axis, , at which fp vanishes and the corresponding mode is fully pf. This leads to a quantization condition that reproduces Planck-type discrete energies for simple systems. Finally, we interpret tunnelling and evanescent domains as manifestations of fp structure: from the pf perspective an infinite barrier still enforces , but from the fp perspective a particle that has not yet appeared in that pf spacetime may exhibit fp amplitudes that transiently extend into the barrier region. Taken together, these results suggest that the imaginary unit can be interpreted as a “future factor” in quantum theory, and that a complex-time geometry provides a coherent route to connect quantum dynamics, probability, and discrete spectra with the temporal status of physical reality. Running Title: What is Time? Complex-Time Geometry Key words: complex time; past-fixed and future-pending; time in quantum mechanics; Born rule; unitary dynamics; quantum tunnelling; quantum foundations

Dec 10, 20250 citations

Complex-Time Theory: A Framework for Quantum Irreversibility

Toru Ishibashi

Abstract We introduce a complex-time framework in which the imaginary unit i is interpreted not as a formal algebraic device but as the generator of a physically meaningful temporal transformation. In this formulation, time is represented as a complex quantity T = pf + i·fp with |T| = 1, where pf (past-fixed) denotes the real component representing actualized, recorded history, and fp (future-pending) denotes the imaginary component representing undetermined possibilities. The constraint |T| = 1 expresses conservation of total temporal magnitude: as possibilities are determined (fp decreases), actualized history accumulates (pf increases). Temporal evolution is described by continuous rotation in the complex-time plane, governed by dT/dθ = iT, while a non-Hermitian extension dT/dθ = iT − γT (γ > 0) describes dissipative processes and the emergence of the arrow of time. Within this framework, quantum measurement is identified with a discrete, irreversible real-axis projection—the conversion of fp into pf—that constitutes a single step of time’s advance from the imaginary to the real axis. This identification yields three concrete predictions for quantum circuits: (i) measurement-induced effects depend on measurement count M rather than free-evolution time T (T-independence); (ii) irreversibility accumulates linearly with M in time-reversal protocols; and (iii) the elementary duration δ of one real-axis projection is implementation-dependent. The framework provides a unified structural basis for the continuity of motion, quantum uncertainty, and macroscopic irreversibility, suggesting that these phenomena arise from a single generative process: the continuous, irreversible conversion of future-pending structure into past-fixed record. Keywords: complex time; past-fixed / future-pending decomposition; quantum measurement; irreversibility; arrow of time; uncertainty principle

Mar 28, 20260 citations

Complex-Time Geometry and the Quantum Origin of Continuity: Reinterpreting i as the Generator of Time, Motion, and Irreversibility

Ishibashi, Toru

Description / Abstract English version (recommended for main entry): Time in physics has long been treated as an external, passive parameter.However, such a conception fails to explain why motion is continuous and why the past and future are asymmetric.This paper reformulates time as a complex rotation, in which the imaginary unit i acts as the generator transforming future-pending (fp) states into past-fixed (pf) states.Expressed as T=pf+i fp=eiθ,T=pf+ifp=eiθ, this framework interprets temporal evolution as the rotation of existence itself.By connecting this model to the uncertainty principle[x^,p^]=iℏ[x^,p^]=iℏ,the paper shows that i represents the physical mechanism underlying noncommutativity, continuity, and irreversibility.Applied to Zeno’s paradox, the introduction of a complex phase allows the “arrow” to acquire motion and pierce the target, resolving the classical problem of infinite division.Furthermore, the cumulative, immutable nature of pf-states suggests a blockchain-like temporal structure, explaining why the past cannot be altered and why time’s arrow cannot reverse.This complex-time geometry unifies quantum indeterminacy, temporal continuity, and informational irreversibility within a single conceptual framework. Keywords complex time,imaginary unit, quantum continuity, irreversibility, Zeno’s parado, uncertainty principle, blockchain analogy, time asymmetry

Nov 7, 20250 citations

Dirac Equation from the Complex Structure of Time: Spin, Antimatter, and CPT as Temporal Geometry

Toru Ishibashi

We derive the Dirac equation from three postulates concerning the internal complex structure of time: (P1) physical time comprises two orthogonal components, past-fixed (pf) and future-pending (fp); (P2) pf ⊥ fp; and (P3) |T|² = pf² + fp² = 1. These postulates entail that temporal evolution is a first-order phase rotation e^{iθ} (yielding the Schrödinger equation) and that spatial extension of the conservation law gives E² = m²c⁴ + c²p² (yielding the Minkowski metric). The first-order requirement and the causality constraint (both derived from the same postulates) jointly exclude the square-root Hamiltonian on grounds of non-locality, leaving matrix linearisation as the unique path. The resulting Clifford algebra {γ^μ, γ^ν} = 2g^{μν} encodes the Minkowski metric already derived from the postulates, completing a self-consistent chain from temporal ontology to the Dirac equation. The four-component spinor is interpreted as the direct product of the pf–fp doublet (2 components) and the two directions of phase rotation (particle/antiparticle), providing a physical origin for spin-1/2 and its 4π periodicity (Möbius structure). Charge conjugation C, parity P, and time reversal T are reformulated as phase-rotation reversal, spatial phase-gradient reversal, and phase-angle reversal, respectively; CPT invariance is proved as a direct consequence of |T|² = 1. Connections to prior quantum-circuit experiments — in which entanglement was found to be a necessary condition for the linear accumulation of measurement-induced irreversibility — are discussed in the context of the Dirac spinor’s inter-sector coupling. complex time; Dirac equation; Clifford algebra; spinor structure; CPT invariance; Minkowski metric; quantum–relativity unification; temporal ontology; measurement irreversibility

Apr 10, 20260 citations