We present a novel formulation of quantum mechanics in which thetraditional dichotomy between space and time is entirely dissolved,giving rise to a fully covariant and relational framework. Within thisconstruction, the quantum state is defined not as a time-indexed wavefunctionin three-dimensional space, but as a distribution in the fullfour-dimensional spectral manifold R4, corresponding to the simultaneouseigenvalues of the four-momentum operator Pμ. This approachnaturally accommodates positive, zero, and negative energy componentswithin a single spectral amplitude, permitting a unified interpretationof particle and antiparticle degrees of freedom as smoothinstantiations across the spectral domain.Wave-packet states are rigorously constructed as elements of arigged Hilbert space, enabling the controlled use of generalized eigenvectorsfor sharp four-momentum while maintaining smooth spacetimelocalization via Fourier duality. Probability densities and expectationvalues are defined fundamentally in spectral space, with covariant fourcurrentsin spacetime emerging as secondary, physically interpretableconstructs. Interactions are implemented as Hermitian operators actingin spectral space, producing smooth reshaping of wave packetsand naturally incorporating energy-sign interference without invokinga preferred temporal slicing.
We present a novel formulation of quantum mechanics in which thetraditional dichotomy between space and time is entirely dissolved,giving rise to a fully covariant and relational framework. Within thisconstruction, the quantum state is defined not as a time-indexed wavefunctionin three-dimensional space, but as a distribution in the fullfour-dimensional spectral manifold R4, corresponding to the simultaneouseigenvalues of the four-momentum operator Pμ. This approachnaturally accommodates positive, zero, and negative energy componentswithin a single spectral amplitude, permitting a unified interpretationof particle and antiparticle degrees of freedom as smoothinstantiations across the spectral domain.Wave-packet states are rigorously constructed as elements of arigged Hilbert space, enabling the controlled use of generalized eigenvectorsfor sharp four-momentum while maintaining smooth spacetimelocalization via Fourier duality. Probability densities and expectationvalues are defined fundamentally in spectral space, with covariant fourcurrentsin spacetime emerging as secondary, physically interpretableconstructs. Interactions are implemented as Hermitian operators actingin spectral space, producing smooth reshaping of wave packetsand naturally incorporating energy-sign interference without invokinga preferred temporal slicing.
We present a novel formulation of quantum mechanics in which the traditional dichotomy between space and time is entirely dissolved, giving rise to a fully covariant and relational framework. Within this construction, the quantum state is defined not as a time-indexed wavefunction in three-dimensional space, but as a distribution in the full four-dimensional spectral manifold R4, corresponding to the simultaneous eigenvalues of the four-momentum operator Pµ. This approach naturally accommodates positive, zero, and negative energy components within a single spectral amplitude, permitting a unified interpretation of particle and antiparticle degrees of freedom as smooth instantiations across the spectral domain. Wave-packet states are rigorously constructed as elements of a rigged Hilbert space, enabling the controlled use of generalized eigenvectors for sharp four-momentum while maintaining smooth spacetime localization via Fourier duality. Probability densities and expectation values are defined fundamentally in spectral space, with covariant fourcurrents in spacetime emerging as secondary, physically interpretable constructs. Interactions are implemented as Hermitian operators acting in spectral space, producing smooth reshaping of wave packets and naturally incorporating energy-sign interference without invoking a preferred temporal slicing.
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.
We present a unified, autonomous formulation of quantum mechanics derived strictly from the foundational principles of Contextual Probability Theory. Instead of initiating the formalism via complex wave functions in Hilbert space, we postulate that the primary physical reality is a realvalued joint distribution 𝑃(𝑢, 𝑣, 𝑡)in a hybrid phase space. Here, the variable 𝑢 is explicitly defined as the structural property (index) characterizing a contextual whole (equivalence class), while 𝑣 represents the internal spectral alternative within that context. We mathematically demonstrate that the traditional Hilbert space architecture, complex amplitudes, and non-commuting operators are not primary postulates of nature, but rather emergent information-compensating artifacts that inevitably arise when the unified contextual flow is projected onto lower-dimensional subspaces.To demonstrate the universal validity of this mechanism, we deliberately restrict our current derivation to the foundational, one-dimensional case. Proving that the entire quantum apparatus emerges seamlessly within this prototype guarantees that the underlying principle can be directly generalized to higher dimensions and more complex contextual configurations.“
Paper XXII of the Six-Throat Directed Geometry program derives the mathematical structure of one-particle quantum mechanics from the directed spinor dynamics established on the regular closed six-throat spacetime. Starting from the conserved curved-spacetime Dirac current, the construction produces a positive and slice-independent inner product and its Hilbert-space completion. The corresponding geometric propagator is unitary and satisfies composition, inverse, and adjoint closure. Real physical observables are represented by self-adjoint operators with real spectra and orthogonal spectral resolutions. The compact global geometry yields an exact periodic commutator algebra, while the standard canonical position-momentum relation is recovered on regular local charts. The Robertson and Schrödinger-Robertson uncertainty relations then follow from the same Hilbert-space structure. The tetrad Clifford algebra generates the intrinsic spin-one-half representation, including the expected spin magnitude, the spinor sign reversal under a full two-pi rotation, and the rotational multiplets obtained by coupling orbital and intrinsic angular momentum. Detector-surface current is promoted from the direct Born-flux construction of Paper XXI to a general positive measurement structure described by a non-projective POVM and the trace probability rule. Density operators, purity, von Neumann entropy, conditional access states, and Kraus dephasing channels arise within the same measurement framework. The positive-energy Dirac branch reduces at low energy to Pauli dynamics and, when the spin-covariant terms commute appropriately, to the Schrödinger Hamiltonian. The same unitary framework yields the Liouville-von Neumann equation, the Heisenberg operator equation, and the Ehrenfest relations in the local directed potential. A forty-three-condition numerical closure audit satisfies all 43 declared conditions across the derived one-particle structure. Representative numerical results include Hilbert-product and unitary residuals at approximately machine precision, exact spin-one-half algebra to numerical precision, POVM completeness at machine precision, low-energy wave-packet infidelity of 3.16 × 10⁻¹⁰ for the stated audit configuration, and Schrödinger-Heisenberg expectation agreement at 5.66 × 10⁻¹⁷. The result is a single geometric chain from conserved spacetime flux to one-particle state space, unitary dynamics, observables, commutators, uncertainty, spin, measurement operators, statistical states, and the low-energy quantum equations. Composite systems, entanglement, exchange symmetry, quantum statistics, and variable-particle-number field theory remain subsequent structural layers and are not claimed as part of the present one-particle closure.
This work develops the mathematical structure of Relational Retrocausal Quantum Mechanics (RRQM) beginning from its mother-state formulation. Rather than revisiting theinterpretational foundations established in earlier work, we investigate the consequences of treating the mother state as the fundamental object of the theory within a rigged Hilbert space. Starting from the global spectral state, we construct the operator formalism governing spectral evolution and introduce the invariant mass operator as a derived observable actingon the mother state. This naturally leads to an operator formulation of mass expectation values and higher moments, providing a framework in which quantities traditionally regardedas fixed particle parameters emerge from the spectral properties of the underlying state.Building upon this formalism, we investigate the extension of the theory to spinorial wavefunctions, the structure of multiparticle realizations, spectral radiation reaction, and the role of conditional nuclear realizations in observable quantities. We further examine generalized derivative operators and show that their algebra naturally separates into symmetric and antisymmetric sectors, suggesting a possible geometric interpretation of gravitywithin the spectral framework.
Abstract We construct a unified mathematical framework in which physical systems are described as spectral decompositions of underlying operators. The foundational object is the complex exponential map 𝑓 : ω ↦ eiωt, which constitutes a homomorphism from the frequency algebra (ℝ, +) into the multiplicative structure of smooth functions on ℝ. Frequencies — realised as eigenvalues of d/dt and, in quantum theory, as ωn = En/ℏ — generate harmonic modes whose superpositions define observable geometries. Interaction terms produce mode coupling and hierarchical structure formation; environmental decoherence selects a preferred pointer basis, yielding classical probabilistic behaviour. The correspondence is shown to be bidirectional: while spectral data determines geometry via energy–momentum, the resulting curved geometry feeds back to modify the Hamiltonian, shifting the spectrum and closing a dynamical loop. This establishes a mathematically grounded chain: Spectrum → Modes → Geometry → Energy–Momentum → Spacetime Curvature → Classical Emergence → Modified Spectrum.