Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional $Ψ[φ,t]$ encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.
Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional Ψ[φ,t] encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.
The formulation of Quantum Field Theory (QFT) in phase space offers a unique alternative to operator and path-integral paradigms, providing distinct conceptual advantages for semiclassical expansions. In this work, we present a systematic and self-consistent functional framework that maps stationary Schrodinger functional equations directly onto phase-space star-eigenvalue equations across different spin statistics. Operating within a non-manifestly covariant equal-time formalism, we derive explicit vacuum Wigner functionals for scalar, gauge, and fermionic fields, establishing the rigorous theoretical consistency of the formalism from first principles prior to phenomenological applications. We analyze how ordering prescriptions and continuous symmetries manifest under the functional star-product, including an explicit phase-space formulation of Noether’s theorem and field regularization. Finally, the framework is extended to interacting systems via a functional Rayleigh–Schrodinger perturbative scheme, illustrated explicitly through the non-trivial first-order Wigner functional correction W(1) and the vacuum energy correction for a ϕ4 self-interacting theory, establishing a solid foundation for evaluating real-time quantum field dynamics.
Understanding whether the mathematical structure of quantum mechanics is fundamental or emergent remains a central question in the foundations of physics. In particular, the special role played by time in quantum theory, appearing as an external evolution parameter rather than a dynamical observable, suggests that the formalism itself may arise from deeper structural considerations. In this work, we investigate the emergence of quantum mechanical formalism from classical wave dynamics by adopting a dimensional framework in which time is treated as a +1 evolution parameter relative to the dimensions through which physical phenomena (fields or disturbances of a field) propagate and interact. Within this perspective, different fields may evolve with respect to different effective dimensions, while remaining embedded in a common higher-dimensional space, allowing time to acquire a relational and context-dependent role. This means that in our proposed model, time is not a fixed dimension which is experienced the same way for every field or field interaction of any dimensionality. In that sense, time for one physical phenomenon can behave as space for a higher dimensional physical phenomenon, whose time is a different +1 dimension. The central objective of this paper is to determine how a higher-dimensional deterministic field can be consistently represented by a lower-dimensional description that lacks direct access to its full set of evolution parameters and evolves through a spatial (for the higher-dimensional field) dimension. To this end, we introduce a general projection framework in which a higher-dimensional field is mapped to a reduced field through an interaction-based recording process. Crucially, we do not assume the form of this mapping a priori. Instead, we impose the requirement that it preserve the maximum amount of physically accessible information. In particular, we demand the faithful encoding of phase relations, interference structure, and spectral composition, including the relative contributions of different Fourier modes and their superposition. We first demonstrate, within a purely classical 3+1-dimensional wave framework that these constraints severely restrict the admissible form of the reduced description and naturally lead to complex amplitudes, linear superposition, Hilbert space structure, and canonical operator relations. This analysis provides an intuitive and mathematically explicit route to quantum-like descriptions without assuming quantum postulates. We then generalize the construction to a 4+1-dimensional framework, introducing an additional evolution parameter and showing that under the same information-preserving constraints, the Schrödinger equation appears as an effective low-energy description of the reduced dynamics, while a relativistic dispersion relation emerges simultaneously through the encoding of the hidden evolution parameter as an invariant frequency scale. In this way, both quantum mechanical and relativistic structures arise from the same underlying requirement: the consistent and information-preserving representation of higher-dimensional wave propagation in a lower-dimensional observational framework. The results suggest that the formal structure of quantum mechanics need not be postulated a priori, but may instead be understood as the unique mathematical language required to encode the observable remnant of a higher-dimensional deterministic dynamics under strict constraints of symmetry, invariance, and information preservation.
The historical debate surrounding the necessity of complex numbers in quantum mechanics has been revitalized by recent experimental falsifications of standard real-amplitude models. However, Barrios Hita et al. (2026) demonstrated that a physically consistent real-number quantum theory can be constructed by mapping complex Hilbert spaces onto real quotient Hilbert spaces with local flags. In this work, we extensively expand this framework to full quantum field theory and numerous applications. We present explicit, index-level, and operator-level derivations of non-relativistic dynamics (the Schr¨odinger equation, Ehrenfest’s theorem, uncertainty principle), relativistic quantum mechanics (Klein-Gordon and Dirac equations), and quantum field theory (canonical quantization, Fock space, propagators, path integrals). We further apply the real formulation to harmonic oscillators, spin systems, perturbation theory, scattering theory, quantum entanglement, gauge theories and Quantum computing. We show that U(1) gauge transformations are rigorously isomorphic to SO(2) rotations in the real flag space, and that all physical predictions—including Bell violations and scattering amplitudes—are exactly reproduced. All derivations are presented at the explicit index level with full algebraic details, using a symplectic structure J that replaces the complex unit i.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
Q Quantum Theory is formulated here at the foundational and quantum-mechanical level. Its basic postulate is that the physical quantum entity, denoted by Q, is the whole quantum state rather than an independently real material point whose unknown position is described by a secondary probability cloud. In a one-Q sector admitting a local amplitude representation,\begin{equation} Q(x)=\sqrt{\rho(x)}\,e^{i\theta(x)}Z(x),\end{equation}where x is a spacetime point, ρ is the Born response weight, θ is the phase, and Z is a normalized internal structural direction. A dagger denotes the Hermitian adjoint and ∂μ differentiates with respect to xμ. The decomposition has a local rephasing redundancy. This is a redundancy of the local factorization of the complete Q amplitude, not the physical electromagnetic U(1) gauge symmetry. The associated decomposition-invariant phase-orientation one-form is\begin{equation} \kappa_\mu=\partial_\mu\theta-iZ^\dagger\partial_\mu Z =\frac{\operatorname{Im}\langle Q,\partial_\mu Q\rangle}{\rho}, \qquad \rho>0.\end{equation}In Schr\"odinger correspondence sectors this structure yields exact probability-current and momentum identities, including the standard Abelian gauge-covariant current. In that audit the electromagnetic connection is a correspondence variable; its Q-level interpretation as a local representation of inter-Q relational comparison geometry is developed in the companion manuscript Paper II-A. Klein-Gordon and Dirac audits show that relativistic currents and stress-energy depend on the Lorentz representation: κμ carries phase-orientation translation structure, while amplitude gradients and internal orientation can contribute independently to energy and stress. The standard Hilbert-space probability calculus is retained. Interference, tunnelling, entanglement, localized detection and cloud-chamber tracks are interpreted as properties or records of whole-Q states rather than as evidence for hidden point trajectories. Entanglement is represented by a nonfactorizable joint Q state; standard Bell correlations and no-signalling are retained without identifying correlation with a superluminal physical signal. Boundaryless state support is distinguished from the propagation of a newly generated disturbance, which is required to obey retarded relativistic causality. For stable relativistic sectors, invariant mass is treated as a derived characterization of the translation spectrum: timelike Q states possess a rest frame and rest energy, whereas null Q states do not. The foundational structure does not specify a unique interaction action or establish ultraviolet finiteness. Finite-response dynamics requires a separate construction.
Abstract The conceptual unification of quantum mechanics and relativity remains one of the central challenges of modern theoretical physics. While quantum theory successfully describes microscopic phenomena through probabilistic wave dynamics and field quantization, relativity establishes the geometric structure of space-time and the causal organization of physical events. Despite extensive progress in quantum field theory and relativistic quantum frameworks, foundational tensions persist regarding locality, causality, and coherence. This study presents a conceptual and structural analysis of quantum field coherence within relativistic space-time. Rather than focusing on specific unification proposals, we examine the compatibility conditions between quantum coherence, relativistic causality, and space-time structure. Particular attention is given to field-based interpretations of quantum states, entanglement propagation limits, and covariance requirements. The analysis synthesizes perspectives from relativistic quantum field theory, conceptual foundations of quantum measurement, and geometric interpretations of space-time. We argue that coherence phenomena can be understood as structurally constrained by relativistic causal order, without requiring violation of space-time locality. The aim of this article is to clarify foundational relationships between quantum coherence and relativistic structure through a conceptual framework that integrates field-theoretic and geometric viewpoints.