A quantum time topological space is developed and applied to solve some problems about quantum theory. It is disconnected and satifies specific separation axioms. The degree of disconnectedness of the time-space is a decreasing function of the number of simultaneous or almost simultaneous fundamental interactions. In this topology the U+R Penrose dynamics is implemented by means of a time evolution operator in QFT. This operator is unitary or non-unitary, depending on the type of quantization of the field action-integral. The time evolution operator allows to find the Boltzmann factor in QFT in the above space-time. From an elementary solution of the Liouville equation the quantization of the time follows and the Planck constant has been calculated. Compatibility between time-reversal and irreversibility is spontaneously obtained. The renormalization of the field action-integral follows from quantization. The solution of the measurement problem and the wave function reduction have been deduced in the framework of the Schroedinger theory. The Schroedinger cat's paradoxon and the paradoxon of the wave packet decay have been resolved.
A quantum time topological space is developed and applied to solve some problems about quantum theory. It is disconnected and satifies specific separation axioms. The degree of disconnectedness of the time-space is a decreasing function of the number of simultaneous or almost simultaneous fundamental interactions. In this topology the U+R Penrose dynamics is implemented by means of a time evolution operator in QFT. This operator is unitary or non-unitary, depending on the type of quantization of the field action-integral. The time evolution operator allows to find the Boltzmann factor in QFT in the above space-time. From an elementary solution of the Liouville equation the quantization of the time follows and the Planck constant has been calculated. Compatibility between time-reversal and irreversibility is spontaneously obtained. The renormalization of the field action-integral follows from quantization. The solution of the measurement problem and the wave function reduction have been deduced in the framework of the Schroedinger theory. The Schroedinger cat's paradoxon and the paradoxon of the wave packet decay have been resolved.
The conventional tacit assumption that nuclear and sub-nuclear reactions take place in the Newtonian universal time is replaced in the present paper by a time topological space based on the interaction proper time neighbourhood. It is developed and used to solve a problem related to the nuclear reaction theory, the quantum measurement problem. The time topology is disconnected and satisfies the separation axioms of the topological space 3 , . In this topology the V+R Penrose dynamics is implemented by means of a time evolution operator,"?/ .constructed using a quantized version of CSel'fand's theory - the generalized random quantum field theory (QRQFT). As an application the quantum measurement problem solution is presented.
The principles of a random quantum theory (R-QT) which is alternatively time-asymmetric or time-symmetric if the quantization is Fermi–Dirac or Bose–Enstein, respectively are presented. Bohr's quantization rule is applied on the field-action integral. A time topological space, [Formula: see text], is mathematically defined in which the paradoxes in standard quantum theory are solved. The time "quantum", is created as a regular, positive into-map of an observed observable's change resulting from a fundamental interaction process. [Formula: see text] is constructed as the union of time elements and can be embedded disconnectedly in the continuous Newtonian universal time, [Formula: see text]. Six axioms are formulated characterizing the space–time and R-QT. The disconnectedness of the (κ×λκ)-fold time-space, [Formula: see text], imparts a kind of disconnectedness to the κ×λκ-fold space–times, [Formula: see text], and induces the chrono-topology. In chrono-topology the unitary, U, or non-measure preserving, R, dynamics, is implemented by means of a time evolution, "complex" operator, [Formula: see text]. It breaks down by means of Bohr quantization into: [Formula: see text][Formula: see text] coincides formally — apart from the spontaneous renormalization — with the time evolution operator in the standard QFT. [Formula: see text] is a novum and produces the Maxwell–Boltzmann energy level distribution in a non-Euclidean QFT. Compatibility between time-reversal invariance of the standard QT equations and irreversibility of some phenomena both in microcosmos and macrocosmos is obtained. The [Formula: see text]-evolution leads to a time's arrow on quantum-scale systems.
The conventional tacit assumption according to which quantum processes take place in the Newtonian time topology is abandoned. The chronotopological space is based on the interaction proper time neighborhood. It is disconnected, satisfies the separation axioms of the topological space [Formula: see text], and is developed and used to solve some problems related to quantum field theory. In chronotopology the time evolution operator, [Formula: see text], breaks down into the alternative [Formula: see text] = either [Formula: see text]. The unitary, U, and the reduction, R, dynamics are implemented by means of two time evolution operators, [Formula: see text] and [Formula: see text], derived from a quantized version of Gel'fand's theory of the generalized random and infinitely divisible fields. As an application the solution of the paradox of Schrödinger's cat is presented.
A new, physical definition of the time elements is mathematically given. Their union constitutes a disconnected time topological space, [Formula: see text]. Based on the properties of this time–space and using the theory of random quantum fields previously developed, a nonunitary evolution operator, [Formula: see text], is derived. [Formula: see text] breaks down into two disjunct, spontaneously renormalized evolution operators by quantizing the random field action integral: one operator is unitary, [Formula: see text], and the other non-measure-preserving, [Formula: see text]. [Formula: see text] coincides, apart from the renormalization, with the evolution operator of the standard QFT. U ump (τ) breaks time symmetry in QFT. It reconciles the time reversal invariance of the basic equations of quantum theory with the irreversibility of the macroscopic, and some microscopic, phenomena. It provides a basis for the CP violation in the neutral K0 meson decay. Functional integrals arising in the theory have in the limit countably additive measures.
Time-asymmetric evolution is derived from time-reversal invariant fundamental QFT-equations. Chrono-topology, the disconnected time topological space J4, is the playground for the generalized random and infinitely divisable quantum fields, a new development in time-asymmetry. Based on the properties of this time space and using the theory of random quantum fields previously developed a non-unitary, complexity evolution operator, C(J4), is derived. C(J4) breaks down, by Bohr- quantizing the field-action integral, alternatively into two, spontaneously renormal- ized parts: One, (unitary) Uu(r), implying U - processes and one (non-measure- preserving), Knmp(r), producing R - processes. Unmp(r), breaks time-symmetry and provides a basis for CP-violation in QFT and in particular in the K0-meson decay. Functional integrals arising in the theory have as a limit Feynman's path integral in accordance with the measure theoretical requirements. Irreversibility and time-symmetry are not incompatible (compare Boltzmann, Poincaré) in chrono-topology.
The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.