We reconsider the construction of free quantum fields through momentum-space weighted measures, introducing an independent energy scale that restores the conventional theory at low momenta. The resulting family is characterized by a dimensionless spectral function and includes, as a distinguished case, the spatial weight associated with the Euclidean Snyder algebra. We discuss its relation to modified dispersion relations and nonlinear momentum coordinates, and introduce an observer-covariant spectral weight defined using a timelike four-velocity. For the complex Klein-Gordon field we obtain the equal-time algebra, conserved weighted observables, vacuum fluctuations, and two-point functions in arbitrary spatial dimension. The ultraviolet conditions for their finiteness are established, and the nonlocal causal structure is analyzed in terms of a spatial convolution of the ordinary Pauli-Jordan distribution. An illustrative interacting extension and numerical profiles of the kernels and amplitudes are also presented.
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The paper studies whether momentum-space spectral weights can soften free-field ultraviolet singularities while recovering ordinary Klein–Gordon theory at low momentum. It introduces $F_\Lambda(k)=[1+(k/\Lambda)^n]^{-r/n}$, retaining the standard dispersion relation and interpreting the weighted field as a spatial filter of an ordinary field. Nonlinear momentum coordinates transfer the weight into a modified dispersion relation; an observer four-velocity provides an observer-covariant formulation, without establishing particle Lorentz invariance or a distinct doubly special relativity theory.
For a complex scalar in $D$ spatial dimensions, mode expansions yield a spatially smeared equal-time commutator and weighted conserved observables. Power counting gives ultraviolet convergence thresholds: $r>D$ for the coincident commutator kernel, $r>D-1$ for field variance, and $r>D+1$ for weighted vacuum-energy density. Beta and hypergeometric integrals provide explicit fluctuation formulas. For $n=2$, the kernel is expressed through modified Bessel functions; the Euclidean Snyder measure corresponds to $r=D+1$, where weighted vacuum energy remains logarithmically divergent.
The propagator and Pauli–Jordan commutator are spatial convolutions of their ordinary counterparts. Exact massless formulas in three dimensions and massive spacelike formulas for even $r$ show nonzero, exponentially suppressed spacelike tails: exact microcausality is lost. The finite weighted energy is not the canonical time-translation generator, whose vacuum constant remains divergent. A quartic example gives a finite one-loop tadpole for $r>D-1$, but all-orders renormalization, interacting unitarity and causality, and the physical status of the observer vector remain unresolved.
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We reconsider the construction of free quantum fields through momentum-space weighted measures, introducing an independent energy scale that restores the conventional theory at low momenta. The resulting family is characterized by a dimensionless spectral function and includes, as a distinguished case, the spatial weight associated with the Euclidean Snyder algebra. We discuss its relation to modified dispersion relations and nonlinear momentum coordinates, and introduce an observer-covariant spectral weight defined using a timelike four-velocity. For the complex Klein-Gordon field we obtain the equal-time algebra, conserved weighted observables, vacuum fluctuations, and two-point functions in arbitrary spatial dimension. The ultraviolet conditions for their finiteness are established, and the nonlocal causal structure is analyzed in terms of a spatial convolution of the ordinary Pauli-Jordan distribution. An illustrative interacting extension and numerical profiles of the kernels and amplitudes are also presented.
The paper studies whether momentum-space spectral weights can soften free-field ultraviolet singularities while recovering ordinary Klein–Gordon theory at low momentum. It introduces FΛ(k)=[1+(k/Λ)n]−r/n, retaining the standard dispersion relation and interpreting the weighted field as a spatial filter of an ordinary field. Nonlinear momentum coordinates transfer the weight into a modified dispersion relation; an observer four-velocity provides an observer-covariant formulation, without establishing particle Lorentz invariance or a distinct doubly special relativity theory.
For a complex scalar in D spatial dimensions, mode expansions yield a spatially smeared equal-time commutator and weighted conserved observables. Power counting gives ultraviolet convergence thresholds: r>D for the coincident commutator kernel, r>D−1 for field variance, and r>D+1 for weighted vacuum-energy density. Beta and hypergeometric integrals provide explicit fluctuation formulas. For n=2, the kernel is expressed through modified Bessel functions; the Euclidean Snyder measure corresponds to r=D+1, where weighted vacuum energy remains logarithmically divergent.
The propagator and Pauli–Jordan commutator are spatial convolutions of their ordinary counterparts. Exact massless formulas in three dimensions and massive spacelike formulas for even r show nonzero, exponentially suppressed spacelike tails: exact microcausality is lost. The finite weighted energy is not the canonical time-translation generator, whose vacuum constant remains divergent. A quartic example gives a finite one-loop tadpole for r>D−1, but all-orders renormalization, interacting unitarity and causality, and the physical status of the observer vector remain unresolved.
By positing energy as intrinsic and independent—with momentum, time, and spatial properties emerging as functions of boost energy (either directly or indirectly)—E-Theory achieves a fully covariant, background-independent formulation for quantum mechanics. This paper demonstrates that by anchoring special relativity in the invariant energy–momentum relationship and reinterpreting energy as an independent variable rather than a dependent property, we transform the traditional four-vector—typically expressed via complex differential relationships—into three simple algebraic functions for momentum, time, and space. We can then transform the standard Time-Dependent Schrödinger Equation (TDSE) into a covariant, background-independent version simply by scaling the time derivative by a universal factor derived from the boost energy.This shift from the time domain (time-dependent) to the energy domain (energy-dependent) has profound consequences: a covariant TDSE emerges; both relativistic and non-relativistic dispersion are recovered from the standard non-relativistic Hamiltonian; inverse relativistic effects appear at ultra-low energies; and a single wave equation governs quantum evolution across all energy scales and particle spins. Most importantly, we free quantum mechanics from its reliance on a fixed temporal background and place it on equal footing with general relativity, thus laying the foundation for their ultimate unification.
By positing energy as intrinsic and independent—with momentum, time, and spatial properties emerging as functions of boost energy (either directly or indirectly)—E-Theory achieves a fully covariant, background-independent formulation for quantum mechanics. This paper demonstrates that by anchoring special relativity in the invariant energy–momentum relationship and reinterpreting energy as an independent variable rather than a dependent property, we transform the traditional four-vector—typically expressed via complex differential relationships—into three simple algebraic functions for momentum, time, and space. We can then transform the standard Time-Dependent Schrödinger Equation (TDSE) into a covariant, background-independent version simply by scaling the time derivative by a universal factor derived from the boost energy.This shift from the time domain (time-dependent) to the energy domain (energy-dependent) has profound consequences: a covariant TDSE emerges; both relativistic and non-relativistic dispersion are recovered from the standard non-relativistic Hamiltonian; inverse relativistic effects appear at ultra-low energies; and a single wave equation governs quantum evolution across all energy scales and particle spins. Most importantly, we free quantum mechanics from its reliance on a fixed temporal background and place it on equal footing with general relativity, thus laying the foundation for their ultimate unification.
This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every n-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities.
The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions.
Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects.
Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.
ABSTRACT The emergence of quantum entanglement correlations across spatially separated measurement regions renders the interpretation of the relationship between quantum mechanics and relativistic causality a core problem in quantum foundations [Einstein et al., 1935; Bell, 1964]. In this study, two entangled particles are modeled not as two independent physical objects in Minkowski spacetime, but as a joint quantum state in configuration space. The relative space and time coordinates of the joint state are defined via: and the ideal space-time correlation limit assumed in the model is expressed as: This limit is not a general definition of quantum entanglement, but an idealized geometric assumption representing strong correlation in the relative coordinates of the joint state within the proposed model. In this approach, physical spacetime is treated as a geometric structure representing the projection of the joint quantum state onto observable spacetime coordinates. To ensure the projection is Lorentz-covariant, the light-like interval: is employed as the constitutive geometric condition of the model. This condition is accepted not as a mandatory Minkowski interval between physical measurement events, but as the mathematical postulate of the proposed projection geometry. The dynamic scale parameter of the model is grounded in total relativistic quantum energy instead of rest mass, establishing a dimensionless Planck-scaling factor defined as: To encompass both massive and massless systems, quantum energy density is defined via the energy density component of the stress-energy tensor relative to an observer. The required consistency condition for covariant energy-momentum conservation is derived in its general form. Furthermore, modeling the proposed geometric effect as a dimensionless metric perturbation yields a phenomenological first-order geometric phase correction defined as: where is the energy-dependent reduced wavelength. Under fixed and finite , the vanishing of the proposed correction in the low-energy regime relative to the Planck scale () demonstrates that the geometric effect added by the model to standard low-energy quantum mechanics becomes negligible. This study proposes a testable theoretical model aimed at explaining the spatiotemporal correlations of quantum entanglement within a Lorentz-covariant, energy-scaled, and phenomenological geometric framework.
The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation.Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate.In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial.A nontrivial result is obtained if the plane is not of this type.When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant.Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface.Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory.A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.
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.