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CRC Press eBooks

A Direct Transformation Between Imaginary-Time and Real-Time Thermofield Theories Induced by Section Rotations in η-ξ Spacetime

Yuanxing Gui, Yajun Gao

Published Nov 27, 2020• 0 citations
Publisher

Abstract

Some suitable complex-time transformation is combined organically with the theory of η-ξ spacetime and a so-called η-ξ complex-time transformation is constructed, which gives some section rotations in η-ξ spacetime and induces a direct transformation between the imaginary-time and real-time thermo field theories.
DOI
10.1201/9781003078296-32

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In classical physics, complex numbers serve as a calculational device; all observables can be reconstructed within a real framework. Standard quantum mechanics requires a complex wavefunction, but the origin of the imaginary unit iii is unexplained. This work builds upon the spacetime quantum field (SQF) paradigm — a two-stage cooling of the early universe in which a bare spacetime quantum field forms first, and matter fields are excited only after a second cooling — to construct a complex-spacetime geometric substrate. We derive the ontological status of complex numbers from first principles: a Hermitian line element on a four-dimensional complex manifold reduces rigorously to the real Minkowski metric at low energy; the Wick rotation is reinterpreted as decomposition of complex time; the path integral on the complex manifold naturally yields the complex wavefunction; and entanglement correlations reside in the imaginary subspace, irreducible to a real manifold.
Certain aspects of Lagrangian and Hamiltonian dynamics are investigated within the framework of an extended complex phase space approach, characterized by the transformation and by the presence of the imaginary time variable in the complex variable in a local chart. It is an application of imaginary time which is essential in connecting quantum mechanics with statistical physics. We argue that the novel complexified approach enhances the system of dynamical equations obtained in the sense that the new derived equations appear as certain combinations of former equations. Furthermore, it was shown that the physics we experience in the real time is somewhat different from what is experienced in imaginary region. Further consequences are discussed in some details.
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