Abstract In the standard quantum–field–theoretic description of unstable particles, exponential decay with width is encoded by promoting the energy (or mass) to a complex quantity . This prescription works extremely well in practice, but its imaginary part has no direct counterpart in the usual real–valued notion of energy, and it raises a natural question: where does the imaginary unit really belong? In this letter we show that the same physics can be reformulated in a complex–time framework in which the rest mass remains real and the decay is absorbed into a complex phase associated with the time direction. For a stable relativistic particle, the proper–time action can be written so that the mass controls the rate of phase rotation along a complex extension of time. For a quasi–stable state with width , the standard replacement is equivalently realized by a complex phase whose real part encodes the mass–controlled oscillation and whose imaginary part encodes the exponential damping. As a concrete example we apply this construction to the Higgs boson, whose mass arises from the Higgs vacuum expectation value while its finite lifetime is generated by interactions. In the complex–time picture these two aspects appear as the real and imaginary components of a single geometric object, providing a unified way to relate mass, width and the arrow of time without complexifying the energy spectrum itself. Running Title Unstable States in Complex Time Key Words unstable quantum states; complex time; complex energy; Higgs boson; vacuum structure
Abstract In the standard quantum–field–theoretic description of unstable particles, exponential decay with width is encoded by promoting the energy (or mass) to a complex quantity . This prescription works extremely well in practice, but its imaginary part has no direct counterpart in the usual real–valued notion of energy, and it raises a natural question: where does the imaginary unit really belong? In this letter we show that the same physics can be reformulated in a complex–time framework in which the rest mass remains real and the decay is absorbed into a complex phase associated with the time direction. For a stable relativistic particle, the proper–time action can be written so that the mass controls the rate of phase rotation along a complex extension of time. For a quasi–stable state with width , the standard replacement is equivalently realized by a complex phase whose real part encodes the mass–controlled oscillation and whose imaginary part encodes the exponential damping. As a concrete example we apply this construction to the Higgs boson, whose mass arises from the Higgs vacuum expectation value while its finite lifetime is generated by interactions. In the complex–time picture these two aspects appear as the real and imaginary components of a single geometric object, providing a unified way to relate mass, width and the arrow of time without complexifying the energy spectrum itself. Running Title Unstable States in Complex Time Key Words unstable quantum states; complex time; complex energy; Higgs boson; vacuum structure
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
Abstract This paper develops a unified theoretical framework in which time is formulated as an intrinsic complex-phase rotation eiθeiθ within an imaginary extension of spacetime.In this formulation, the imaginary unit ii is not a mere mathematical symbol but the operator of indeterminacy, representing the undetermined future phase of physical existence, while the real component corresponds to the determined and fixed past.Temporal continuity arises from this internal oscillation, and both the light limit (v=cv=c) and the rest limit (v=0v=0) emerge as degenerate boundaries of the same rotation. This Complex-Time Geometry (CTG) provides a geometric basis that links quantum mechanics, relativity, and thermodynamics under a single structure.The flow of time is interpreted as a continuous projection Im(ψ)→Re(ψ),Im(ψ)→Re(ψ), transforming potentiality into actuality.Within this framework, entropy increase corresponds to the progressive fixation of imaginary freedom, and the arrow of time naturally arises as the geometric direction of realification.Measurement is redefined as a temporal projection from the imaginary to the real domain, resolving the conceptual gap between quantum indeterminacy and classical determinacy. The theory further interprets Schrödinger’s iℏ∂ψ∂t=H^ψiℏ∂t∂ψ=H^ψ as the mathematical manifestation of orthogonal evolution in complex time, where ii ensures reversible continuity.Ultimately, time is not an external parameter but the internal oscillation of existence itself—the mechanism through which the universe continuously writes its own history. Keywords complex-time geometry; temporal projection; quantum measurement; relativity; entropy; arrow of time; ontology of time; self-recording universe
Abstract This study extends the complex-time framework introduced in Part I, where time was formulated as an intrinsic complex coordinate . Within this framework, the imaginary unit is reconsidered not as an algebraic symbol but as an operator of indeterminacy, representing the undetermined future phase of physical existence.The real component corresponds to the determined past. Temporal evolution is thus interpreted as a continuous projection through which potential states become actualized. Building upon this model, we propose a unified interpretation of measurement, entropy, entanglement, and nonlocality. Measurement may be viewed as a temporal projection fixing latent existence from the imaginary phase into the real domain; entropy growth corresponds to the cumulative conversion of undetermined futures into fixed pasts, giving rise to the arrow of time. Quantum indeterminacy is reinterpreted as the openness of the future, while entanglement appears as a form of temporal coherence connecting events through a shared phase structure. When correlations become fully realized, the internal -phase vanishes and the relation attains , implying a null-temporal state beyond spacetime separation. At the macroscopic level, matter with rest mass can be regarded as a dense network of such microscopic fixations. Each quantum interaction functions as a micro-instance of temporal fixation, producing the statistical rhythm perceived as continuous time. This interpretation suggests that reality may evolve through a self-referential process in which the universe continuously projects its potential phase into realized history—a dynamic structure of physical reality emerging from complex-time geometry. Running Title: Complex-Time Structure and Physical Reality Keywords: Complex Time; Temporal Projection; Quantum Measurement; Quantum Entanglement; Decoherence; Entropy; Information Dissipation; Physical Reality
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
Abstract This study introduces a complex-time structure in which the imaginary unit i is interpreted not merely as an algebraic constant but as a possible generator of temporal transformation. Within this formulation, time may emerge from a continuous rotation between two complementary domains of reality—the future-pending (fp) and past-fixed (pf)components. Such rotation provides a potential structural basis for three fundamental aspects of nature: the continuity of motion, quantum uncertainty, and macroscopic irreversibility. Mathematically, the temporal variable satisfies analogous to the unitary generator in quantum mechanics, while a non-Hermitian extension (with γ > 0) may describe dissipative processes and the emergence of the arrow of time. The imaginary component i can therefore be viewed as mediating the transformation from potential to realized states, offering a structural link between microscopic reversibility and macroscopic asymmetry. Recent experiments highlighting the indispensable role of complex numbers in quantum mechanics [1] are consistent with this perspective. Possible verification strategies—such as complex-phase interferometry, delayed-choice erasure, and quantum-simulation approaches—are briefly outlined as potential probes of i-mediated temporal dynamics. Overall, this work suggests that the complex-time structure may provide a coherent physical framework in which time, uncertainty, and irreversibility arise from a single continuous generative process. Running Title: Complex-Time Structure and the Physical Generation of Time Keywords: complex time symmetry; temporal continuity; quantum indeterminacy; causality; arrow of time; nonlocal correlation; information entropy; unified temporality
Abstract Time in physics is conventionally treated as an external real parameter that measures change without participating in it.This study introduces a framework in which time is represented as an intrinsic complex coordinate whose real component corresponds to observable (external) time and whose imaginary component represents the internal phase of physical existence. Within this complex-time structure, temporal evolution is described by a continuous phase rotation on a complex manifold. The amplitude–phase duality of this rotation may provide a unified perspective in which amplitude persistence corresponds to energy and probability conservation, while phase evolution appears to govern the reversible internal oscillation that gives rise to the perception of temporal continuity. The framework is formulated through a minimal variational principle that embeds the phase rotation within a consistent Lagrangian–Hamiltonian structure, linking internal oscillation with relativistic spacetime geometry. It can reproduce the dual relativistic limits of pure temporal rotation () and phase freezing (), and suggests that the coexistence of spatial motion and finite phase evolution may underlie the continuity perceived as the flow of time. Possible experimental implications—such as relativistic interferometry, quantum-clock comparisons, and condensed-matter analogs—are discussed as potential tests of the model. Although the theory remains conceptual and exploratory, it offers a mathematically coherent basis for viewing time as an internal oscillatory degree of freedom. The results suggest that temporal continuity may originate from internal phase coherence, and that macroscopic irreversibility could emergefrom its gradual loss—topics that warrant further investigation. Running Title: Complex-Time Structure and Unified Temporal Framework Keywords: Complex time; phase rotation; quantum–relativistic unification; intrinsic time coordinate; reversible temporal dynamics; phase coherence; variational principle; proper time; phase freezing; arrow of time; internal oscillation.