hermitian-inner-product-time-axis

name	hermitian-inner-product-time-axis
description	Mechanism for time-axis selection in quantum systems via Hermitian inner product choice — identifies the symmetry-breaking step that selects future-timelike axis and locates Born rule as projection onto that axis. Applicable to quantum foundations, Lorentz symmetry emergence from qubit structure, and Hilbert space construction.
license	Complete terms in LICENSE.txt
metadata	{"arxiv_id":"2607.05447","published":"2026-07-04","authors":"Sebastian Zając","tags":["quantum-foundations","hermitian-form","born-rule","lorentz-symmetry","qubit","hilbert-space"]}

Hermitian Inner Product Time Axis Selection

Description

Resolves the mechanism of time-axis selection in quantum systems: the choice of a Hermitian inner product (positive reference form σ⁰) in passing from normed space to Hilbert space reduces SL(2,C) to SU(2), selecting a future-timelike axis. The Born rule enters one level later as the projection of the state's null vector onto σ⁰ — interpretable as energy in that frame, rescaling as Doppler shift under boosts. Corrects a natural misattribution in recent Lorentz-invariant qubit work.

Activation Keywords

• hermitian inner product time axis
• born rule mechanism
• Lorentz symmetry from qubit
• time direction selection quantum
• SL2C to SU2 reduction
• quantum foundations time axis
• 厄米内积时间轴
• 波恩规则机制

Core Concepts

The Mechanism

1. Bare spin space (C², ε) is SL(2,C)-symmetric — singles out no axis
2. Null cone it generates also singles out no axis
3. Hermitian inner product choice (positive reference form σ⁰) → reduces SL(2,C) to SU(2), the stabilizer of σ⁰
4. This is the symmetry-breaking step — made before any probability is assigned
5. Born rule = projection of state's null vector onto σ⁰ = energy in that frame
6. Under boost: rescales as Doppler shift — frame-dependence of |ψ|² becomes empirical

Key Distinction

• Kinematic identification of the step, NOT dynamical account of why particular axis is selected
• Ingredients are classical; the contribution is identifying them as the mechanism
• Born rule enters one level AFTER Hilbert structure is established

Mathematical Framework

• 2×2 Hermitian matrices ↔ Minkowski 4-vectors correspondence
• σ⁰ as positive reference form selects future-timelike axis
• ⟨ξ|ξ⟩ = tr(σ⁰ ξξ†) as projection = energy
• Many-qubit case: tuple of such choices

Instructions for Agents

Step 1: Context Identification

• When analyzing quantum foundations papers addressing Lorentz symmetry emergence
• When investigating Born rule derivation or mechanism
• When studying qubit-to-spacetime correspondence

Step 2: Framework Application

1. Identify the bare symmetric structure (SL(2,C) level)
2. Locate the Hermitian inner product choice point
3. Trace the symmetry reduction: SL(2,C) → SU(2)
4. Verify Born rule enters as projection onto selected axis

Step 3: Critical Analysis

• Distinguish kinematic identification from dynamical explanation
• Check for misattribution of mechanism to wrong step
• For many-qubit systems: account for tuple of Hermitian form choices

Error Handling

• Not a dynamical theory: Does not explain WHY a particular axis is selected, only HOW selection works
• Not applicable to full spacetime: Framework addresses internal qubit degrees of freedom, not external spacetime emergence

Related Skills

• quantum-foundations-probability — Quantum foundations and probability analysis
• transformation-response-quantum-framework — Reformulation of quantum mechanics