The canonical quantization of Time Field Theory (TFT) has been completed in previous work, demonstrating the compatibility between TFT classical field theory and standard quantum mechanics. However, canonical quantization is essentially an operatorization procedure, which bears a structural discrepancy with the geometrical physical picture of TFT — the time field, Fermat's principle, and flux conservation. This paper proposes and systematically elaborates the path integral quantization scheme for TFT, arguing that it is not only a technically superior choice but also the inevitable mathematical expression of TFT's physical logic at the quantum level. The core arguments are as follows: quantum fluctuations of the time field directly correspond to the "all possible paths" in the Feynman path integral; Fermat's principle is the semiclassical approximation of the path integral in the classical limit; wave function collapse is a natural consequence of the establishment of the metric reference (the emergence of the observer), which imposes new boundary conditions on the path space. This framework provides a unified TFT interpretation for wave optical phenomena such as diffraction and interference, offers a clear physical picture for the quantum measurement problem, and paves the way for the deep integration of TFT and quantum field theory.
The canonical quantization of Time Field Theory (TFT) has been completed in previous work, demonstrating the compatibility between TFT classical field theory and standard quantum mechanics. However, canonical quantization is essentially an operatorization procedure, which bears a structural discrepancy with the geometrical physical picture of TFT — the time field, Fermat's principle, and flux conservation. This paper proposes and systematically elaborates the path integral quantization scheme for TFT, arguing that it is not only a technically superior choice but also the inevitable mathematical expression of TFT's physical logic at the quantum level. The core arguments are as follows: quantum fluctuations of the time field directly correspond to the "all possible paths" in the Feynman path integral; Fermat's principle is the semiclassical approximation of the path integral in the classical limit; wave function collapse is a natural consequence of the establishment of the metric reference (the emergence of the observer), which imposes new boundary conditions on the path space. This framework provides a unified TFT interpretation for wave optical phenomena such as diffraction and interference, offers a clear physical picture for the quantum measurement problem, and paves the way for the deep integration of TFT and quantum field theory.
Within the axiomatic framework of Time Field Theory (TFT), this paper provides a classical field interpretation for the core features of quantum mechanics. We demonstrate the intrinsic equivalence between rest mass and the capability of defining local spacetime benchmarks, and introduce the dimensionless quantity α = 1−γ to characterize observer strength. The covariant dynamical equation of observer strength is established from the TFT field equation. Based on time-field fluctuation analysis, we derive a benchmark-induced uncertainty contribution, which is treated as a gravitational correction to the conventional quantum uncertainty relation. A classical ergodic statistical picture is proposed to interpret quantum probability, and the process of measurement and decoherence is described as dynamical stabilization of the time field under macroscopic mass disturbance. This work does not introduce any quantum mechanical postulates, wavefunctions or Hilbert space structures. It offers a self-consistent deterministic perspective to understand quantum phenomena without contradicting existing quantum formalism.
The century-long conflict between quantum mechanics and general relativity is essentially a dispute over the definition of the fundamental measurement benchmark of the universe. Quantum mechanics treats time as an absolute external background, while general relativity regards mass as the measurement benchmark of curved spacetime. The two theories cannot be self-consistently unified under extreme conditions. Based on the core axiomatic system of Time Field Theory (TFT), this paper proposes a first-principles self-consistent interpretation of quantum mechanics: the wave function is neither the physical reality of a particle nor an encoding of our knowledge, but the probability amplitude for the time field to condense mass at a spatial point and thereby establish a measurement benchmark; the Schrödinger equation is the evolution law of the measurability probability amplitude of the time field in the uniform γ=1 background; wave function collapse corresponds to a physical phase transition of the time field from a uniform state to a locally condensed state, and the observer is essentially a massive entity with intrinsic time flow. This interpretation requires no assumptions of external observers, many worlds, or hidden variables, naturally unifies the quantum and classical worlds, and is fully compatible with all known experimental results in the weak-field low-speed limit, providing a completely new metrological foundation for the problem of quantum gravity.
This paper is a direct, complete, and self-contained continuation of the Time Field Theory (TFT) framework established in Parts I, II, and III. Based on the fast–slow oscillation decomposition of the time field γ derived in TFT III, we provide a full dynamical explanation for quantum measurement and the so-called wavefunction collapse without introducing nonlocality, hidden variables, superluminal effects, or ad hoc quantum postulates. We show that quantum measurement is fundamentally phase locking between the high-frequency time oscillation of a macroscopic observer and the low-frequency time oscillation of a microscopic particle. Crucially, this mechanism is identified as a natural extension of special relativity to the temporal structure of matter, based on the principle of relative states. Wavefunction collapse is not a mysterious projection, but a dynamical localization effect caused by the extreme frequency mismatch between massive observer systems and light microscopic systems. The mechanism of entanglement is explained as conserved common phase splitting from a single source. Bell experiments are interpreted locally and causally via phase locking and relative states. This work eliminates the measurement problem, explains the origin of quantum randomness, and completes the causal, realistic, local interpretation of quantum mechanics within the unified TFT framework.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter. The path integral: general formalism Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.
This work is the nineteenth installment in the Time Field Theory (TFT) series, succeeding Time Field Theory XVIII: Entanglement, Common Phase Splitting, and the Local Interpretation of Bell Experiments. This work presents a fully transparent, first-principles physical framework with no black-box operations or uninterpreted postulates. All dynamical processes are derived deterministically from time-field axioms, with clear, traceable physical mechanisms at every step. Within the TFT framework, quantum tunneling is derived without reliance on conventional quantum mechanics or probabilistic interpretation. Two fundamental mechanisms are distinguished: (1) electric potential coupling responsible for phase modulation, interference, and tunneling attenuation; (2) mass/time-field gradient responsible for phase locking, measurement, and localization. Tunneling is interpreted as deterministic time-field splitting rather than probabilistic selection: a portion of the time-field oscillation is retained inside the barrier with real energy deposition, while the remainder transmits. The tunneling transmission coefficient is verified against experimental data and shows quantitative agreement. A 2025 observation of under-barrier recollision supports real interactions inside the barrier, consistent with TFT. This work contributes to the unification of gravity, special relativity, and quantum mechanics within a self-consistent, locally realistic, and fully transparent time-field dynamics framework.
Feynman path integrals are now a standard tool in quantum physics and their use in differential geometry leads to new mathematical insights. A logical treatment of quantum phenomena seems to require a sustained mathematical analysis of path integrals. The subject is complicated by the fact that their application in flat space-time is quite different from how path integrals are used in, say, topological quantum field theory, where there is no natural notion of time translation. An historical background is given in this paper and a few mathematical approaches to Feynman path integrals in the context of nonrelativistic quantum mechanics and scalar quantum fields with polynomial self-interactions are outlined.