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Reports on Mathematical Physics

Maps Between Hilbert Space Effect Algebras Preserving Unitary Invariant Norms of the Sequential Product

Marcell Gaál, Gergő Nagy

Published Dec 1, 2018• 0 citations
Publisher
DOI
10.1016/s0034-4877(19)30004-7

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Let ⋆ be any sequential product on the Hilbert-space effect algebra with , and be a bijective map. We show that if Φ satisfies Φ( A ⋆ B ) = Φ( A )⋆Φ( B ) for , then there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every . Let be a Borel function satisfying g (0) = 0, g (1) = 1 and let us define a binary operation ◊ g on by A ◊ g B = A 1/2 g ( A ) Bg ( A )† A 1/2 , where T † denotes the conjugate of the operator T . We also show that a bijective map satisfies Φ( A ◊ g B ) = Φ( A )◊ g Φ( B ) for if and only if there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every .

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Singularity Appendices 33

John Barnhart

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. This includes: • loss of normalization under evolution • growth or decay of total probability • non-invertible evolution operators in the quantum regime • inconsistent amplitude mapping from branching structure 33.2.8 Boundary Condition for Emergent Unitarity If unitarity is not fundamental in BSFM but emergent: • there must exist a regime where effective dynamics become unitary • non-unitary effects must be suppressed or canceled in that limit • deviations must vanish under appropriate coarse-graining or symmetry conditions 33.2.9 Summary BSFM is consistent with quantum mechanics only if its mapped evolution preserves norm and satisfies unitary dynamics in the Hilbert space representation, ensuring conservation of probability and reversible evolution in the quantum correspondence regime. Appendix 33 — Quantum Correspondence Tests 33.3 Superposition and Interference Consistency 33.3.1 Objective This section establishes that BSFM must reproduce the defining quantum features of superposition and interference when mapped into its Hilbert space correspondence. The requirement is that linear combination of states and phase-dependent effects must emerge consistently from the underlying branching structure. 33.3.2 Superposition Principle Requirement In the quantum correspondence regime, BSFM must allow state combinations such that: • if |Ψ₁⟩ and |Ψ₂⟩ are valid states, then a|Ψ₁⟩ + b|Ψ₂⟩ • is also a valid physical state where a, b ∈ ℂ This ensures closure under linear combination within the mapped Hilbert space structure. 33.3.3 Branch-State Correspondence In BSFM terms: • individual branches correspond to components of a composite state • branching structure maps to decomposition of a superposed state • branch weights correspond to amplitude magnitudes and phases The mapping must preserve distinguishability and recombination behavior of branches. 33.3.4 Interference Requirement BSFM must reproduce interference effects such that: • combined amplitudes produce non-classical probability distributions • phase relationships between components affect observable outcomes • constructive and destructive interference emerge naturally from the structure Formally, observable probabilities must depend on: • cross terms of amplitude combinations, not only individual contributions 33.3.5 Coherence Preservation Condition For interference to remain valid: • phase relationships between components must be preserved under evolution • coherence between branches must not collapse prematurely in the quantum regime • decoherence must only emerge under classical-limit conditions (Appendix 32 regime boundary) This ensures correct separation between quantum and classical behavior. 33.3.6 Non-Additive Probability Structure BSFM must reproduce the quantum rule that: • probabilities are not simply additive over branches • interference terms modify total outcomes • classical probability emerges only when coherence is lost This distinguishes quantum behavior from classical stochastic aggregation. 33.3.7 Consistency Requirement The mapping must ensure: • superposition in Hilbert space corresponds consistently to branching combinations in BSFM • interference patterns remain invariant under valid transformations • no hidden classical averaging removes quantum phase dependence 33.3.8 Failure Condition Superposition and interference consistency fails if: • linear combinations of states are not preserved in the mapping • interference terms cannot be recovered from branch structure • probabilities reduce to purely additive classical form in the quantum regime • phase information is lost or undefined under evolution 33.3.9 Summary BSFM is consistent with quantum superposition only if its branching structure maps into a linear state space where phase-dependent interference effects are preserved, producing non-classical probability behavior consistent with standard quantum mechanics. Appendix 33 — Quantum Correspondence Tests 33.4 Path Integral Equivalence Validation 33.4.1 Objective This section establishes the requirement that BSFM must reproduce, in an appropriate limit, the structure and predictions of the path integral formulation of quantum mechanics. The goal is to ensure equivalence between BSFM branching evolution and summation over histories. 33.4.2 Path Integral Correspondence Requirement In the quantum regime, BSFM must admit an equivalence such that: • the total amplitude for transition between states is given by a sum over all allowed evolution histories • each history contributes with a well-defined weight and phase structure • the full evolution is recoverable as an aggregate over all admissible paths Formally, BSFM must support a mapping consistent with: • “sum over histories” structure in the quantum correspondence limit 33.4.3 Branch-History Mapping Condition BSFM branching must correspond to path decomposition such that: • each branch corresponds to a distinct dynamical history • the full set of branches represents the complete set of allowed paths • recombination of branches reproduces full transition amplitudes This ensures structural eq

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The Orthogonality of Sequential Products in Sequential Effect Algebras

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Reports on Mathematical Physics•Mar 1, 2019•0 citations
Let ⋆ be any sequential product on the Hilbert-space effect algebra with , and be a bijective map. We show that if Φ satisfies Φ( A ⋆ B ) = Φ( A )⋆Φ( B ) for , then there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every . Let be a Borel function satisfying g (0) = 0, g (1) = 1 and let us define a binary operation ◊ g on by A ◊ g B = A 1/2 g ( A ) Bg ( A )† A 1/2 , where T † denotes the conjugate of the operator T . We also show that a bijective map satisfies Φ( A ◊ g B ) = Φ( A )◊ g Φ( B ) for if and only if there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every .
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. This includes: • loss of normalization under evolution • growth or decay of total probability • non-invertible evolution operators in the quantum regime • inconsistent amplitude mapping from branching structure 33.2.8 Boundary Condition for Emergent Unitarity If unitarity is not fundamental in BSFM but emergent: • there must exist a regime where effective dynamics become unitary • non-unitary effects must be suppressed or canceled in that limit • deviations must vanish under appropriate coarse-graining or symmetry conditions 33.2.9 Summary BSFM is consistent with quantum mechanics only if its mapped evolution preserves norm and satisfies unitary dynamics in the Hilbert space representation, ensuring conservation of probability and reversible evolution in the quantum correspondence regime. Appendix 33 — Quantum Correspondence Tests 33.3 Superposition and Interference Consistency 33.3.1 Objective This section establishes that BSFM must reproduce the defining quantum features of superposition and interference when mapped into its Hilbert space correspondence. The requirement is that linear combination of states and phase-dependent effects must emerge consistently from the underlying branching structure. 33.3.2 Superposition Principle Requirement In the quantum correspondence regime, BSFM must allow state combinations such that: • if |Ψ₁⟩ and |Ψ₂⟩ are valid states, then a|Ψ₁⟩ + b|Ψ₂⟩ • is also a valid physical state where a, b ∈ ℂ This ensures closure under linear combination within the mapped Hilbert space structure. 33.3.3 Branch-State Correspondence In BSFM terms: • individual branches correspond to components of a composite state • branching structure maps to decomposition of a superposed state • branch weights correspond to amplitude magnitudes and phases The mapping must preserve distinguishability and recombination behavior of branches. 33.3.4 Interference Requirement BSFM must reproduce interference effects such that: • combined amplitudes produce non-classical probability distributions • phase relationships between components affect observable outcomes • constructive and destructive interference emerge naturally from the structure Formally, observable probabilities must depend on: • cross terms of amplitude combinations, not only individual contributions 33.3.5 Coherence Preservation Condition For interference to remain valid: • phase relationships between components must be preserved under evolution • coherence between branches must not collapse prematurely in the quantum regime • decoherence must only emerge under classical-limit conditions (Appendix 32 regime boundary) This ensures correct separation between quantum and classical behavior. 33.3.6 Non-Additive Probability Structure BSFM must reproduce the quantum rule that: • probabilities are not simply additive over branches • interference terms modify total outcomes • classical probability emerges only when coherence is lost This distinguishes quantum behavior from classical stochastic aggregation. 33.3.7 Consistency Requirement The mapping must ensure: • superposition in Hilbert space corresponds consistently to branching combinations in BSFM • interference patterns remain invariant under valid transformations • no hidden classical averaging removes quantum phase dependence 33.3.8 Failure Condition Superposition and interference consistency fails if: • linear combinations of states are not preserved in the mapping • interference terms cannot be recovered from branch structure • probabilities reduce to purely additive classical form in the quantum regime • phase information is lost or undefined under evolution 33.3.9 Summary BSFM is consistent with quantum superposition only if its branching structure maps into a linear state space where phase-dependent interference effects are preserved, producing non-classical probability behavior consistent with standard quantum mechanics. Appendix 33 — Quantum Correspondence Tests 33.4 Path Integral Equivalence Validation 33.4.1 Objective This section establishes the requirement that BSFM must reproduce, in an appropriate limit, the structure and predictions of the path integral formulation of quantum mechanics. The goal is to ensure equivalence between BSFM branching evolution and summation over histories. 33.4.2 Path Integral Correspondence Requirement In the quantum regime, BSFM must admit an equivalence such that: • the total amplitude for transition between states is given by a sum over all allowed evolution histories • each history contributes with a well-defined weight and phase structure • the full evolution is recoverable as an aggregate over all admissible paths Formally, BSFM must support a mapping consistent with: • “sum over histories” structure in the quantum correspondence limit 33.4.3 Branch-History Mapping Condition BSFM branching must correspond to path decomposition such that: • each branch corresponds to a distinct dynamical history • the full set of branches represents the complete set of allowed paths • recombination of branches reproduces full transition amplitudes This ensures structural eq
This is Paper 006 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the emergence of effective operators and norm-preserving dynamics within the Anaxiomatic Mechanics / pq-Spacetime stack. Starting from finite binary-mechanical graph histories and their amplitude structure, it shows how boundary-resolution maps induce effective linear operators between distinguishable completion channels. Linearity is not assumed as a primitive quantum rule; it follows from the additivity of finite history amplitudes before probabilities are assigned. The paper then distinguishes closed boundary maps from open, conditioned, or coarse-grained maps. A closed boundary map carries admissible fine-history support forward without creation, deletion, duplication, selection, or loss. In the effective Hilbert representation, this closed support-preserving transfer appears as an isometry, and when the initial and final effective spaces have equal finite dimension, as a unitary map. Open or conditioned descriptions instead give nonunitary effective operations. The result provides the operator-theoretic bridge in the finite pq-spacetime construction: finite graph histories give amplitudes; amplitudes give Born-type weights; Hilbert representation gives state space; tensor-product structure gives composite records; and closed boundary maps give effective norm-preserving dynamics. Operators, projections, isometries, unitaries, and open-system maps are therefore not inserted as primitive assumptions, but arise as representation-level consequences of finite boundary-resolution structure.