arXiv:2512.05164

From Kinematics to Interference: Operational Requirements for the Quantum Principle of Relativity

Mikołaj Sienicki, Krzysztof Sienicki

quant-phphysics.hist-phSubmitted Dec 4, 2025• Announced Dec 5, 2025• 0 citations

Abstract

The quantum principle of relativity (QPR) puts forward an ambitious idea: extend special relativity with a formally superluminal branch of Lorentz-type maps, and treat the resulting consistency constraints as hints about why quantum theory has the structure it does [1]. The discussion that followed has emphasized a basic point: writing down coordinate maps is not the same thing as providing a physical theory. In particular, quantum superposition is not operationally defined by drawing multiple paths on paper: it is defined by what happens when alternatives recombine in an interference loop [2, 3]. In parallel, careful 1+1 analyses have clarified how sign conventions and time-orientation choices enter the superluminal formulas [4]. Finally, tachyonic QFT proposals suggest a possible mathematical bridge via an enlarged (twin) Hilbert space [5], although this proposal remains contested (e.g., on commutator covariance and microcausality grounds) [6]. The aim of this short note is organizational. We keep three layers separate: (K) kinematics (which maps exist and what they preserve), (O) operational content (what an experiment must actually reproduce, especially closed-loop interference), and (D/B) dynamics and bridges (how amplitudes and probabilities are generated, and how subluminal and superluminal sectors might be linked). The goal is not relativity derives quantum theory, but a clear checklist of what must be added for that ambition to become a well-posed programme.

Comments: 6 pages, 8 references, This manuscript is closely related to earlier arXiv preprints by other authors

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Appendix 33 Quantum Correspondence Tests 33.1 Mapping to Hilbert Space Formalism 33.1.1 Objective This section establishes the requirement that BSFM admits a well-defined mapping into a Hilbert space structure in regimes where quantum mechanical behavior is observed. The purpose is to ensure compatibility with the standard mathematical framework of quantum theory. 33.1.2 Hilbert Space Embedding Requirement BSFM must allow a mapping such that: • each physical state S ∈ S corresponds to a vector |Ψ⟩ ∈ H • the set of admissible states maps into a complete inner-product space • superposition structure is preserved under state combination This implies the existence of a well-defined embedding: • Φ : S → H 33.1.3 Linearity Condition Within the mapped representation: • linear combinations of states must correspond to valid physical superpositions • evolution must be representable as a linear (or effectively linear) operator in the quantum regime • interference effects must be preserved under the mapping This ensures consistency with quantum mechanical structure. 33.1.4 Inner Product Consistency The mapping must define an inner product structure such that: • physical distinguishability corresponds to orthogonality relations • probability amplitudes are derived from inner product magnitudes • normalization conditions are preserved under evolution Formally, physical probabilities must be recoverable from: • ⟨Ψ|Ψ⟩ 33.1.5 State Completeness Requirement The mapped Hilbert space representation must be: • complete under limits of convergent sequences of states • closed under allowed linear operations in the quantum regime • sufficient to represent all physically distinguishable configurations This ensures no loss of physical information under mapping. 33.1.6 Branch-to-State Correspondence Within BSFM, branching structure must map consistently into quantum structure such that: • branches correspond to components of a superposed state • branch weights correspond to amplitude coefficients • coherence relations between branches map to phase structure This provides the structural link between BSFM evolution and quantum formalism. 33.1.7 Failure Condition The mapping to Hilbert space fails if: • no consistent vector space representation exists for BSFM states • inner product structure cannot be defined or preserved • branching cannot be represented as superposition without contradiction • linearity or normalization is violated in the quantum regime 33.1.8 Summary BSFM is compatible with quantum formalism only if its state structure can be consistently embedded into a Hilbert space where superposition, inner products, and linear evolution reproduce standard quantum mechanical behavior in the appropriate limit. Appendix 33 — Quantum Correspondence Tests 33.2 Unitarity and Norm Preservation Conditions 33.2.1 Objective This section establishes the requirement that BSFM must preserve total probability (norm) under evolution in the quantum correspondence regime. This ensures compatibility with unitary evolution in standard quantum mechanics and prevents loss or creation of probability mass. 33.2.2 Norm Definition Requirement In the Hilbert space representation: • each physical state |Ψ⟩ ∈ Hmust satisfy a well-defined norm • probability interpretation requires normalization of state vectors Formally: • ⟨Ψ|Ψ⟩ =1 for all physically valid states after normalization. 33.2.3 Unitary Evolution Condition For an evolution operator U acting in the quantum regime: • |Ψₙ₊₁⟩ = U|Ψₙ⟩ BSFM must ensure that: • U†U = 1 This guarantees that evolution preserves inner products between states and maintains probability consistency over time. 33.2.4 Norm Preservation Requirement Under BSFM evolution in the quantum correspondence regime: • total probability must remain invariant under iteration • branching weights (when mapped to amplitudes) must conserve norm globally • no net gain or loss of probability mass is permitted Formally: • ⟨Ψₙ₊₁|Ψₙ₊₁⟩ = ⟨Ψₙ|Ψₙ⟩ 33.2.5 Branch-Amplitude Consistency Condition If BSFM branching maps to quantum amplitudes: • the sum of squared amplitude magnitudes must remain constant • interference structure must not alter total probability • redistribution among branches is permitted only under unitary constraints Formally: • Σᵢ |cᵢ⁽ⁿ⁺¹⁾|² = Σᵢ |cᵢⁿ|² 33.2.6 Conservation of Probability Flow In dynamic evolution: • probability may redistribute across states or branches • but total probability within the Hilbert space must remain fixed • no leakage outside the admissible state space is allowed This ensures closed-system behavior in the quantum regime. 33.2.7 Non-Unitary Failure Detection Any deviation from unitary behavior indicates breakdown of quantum correspondence. 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