Quantum Time-Coproduct Multiplexing (QTCM).
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| author | Appleby, Ethan Appleby, Ethan Appleby, Ethan |
| author_facet | Appleby, Ethan Appleby, Ethan Appleby, Ethan |
| contents | <p> </p> <p>1. Segment proper time (GR piece)</p> <p>For each segment k of a worldline:</p> <p>General form (with metric g):<br>τ_k² = g_μν · Δx_k^μ · Δx_k^ν</p> <p>In simple 1+1D Minkowski (c = 1):<br>Δτ_k² = Δt_k² − Δx_k²</p> <p>Timelike segment → Δτ_k real<br>Spacelike segment → Δτ_k purely imaginary</p> <p>2. alias clock (best work but simple)</p> <p>Given all segment times {τ_k}, define the p-norm “twist clock”:</p> <p>τ_p = π · k · | Σ_k (τ_k^p) |^(1/p)</p> <p>where</p> <p>p is the norm power (often p = 4),</p> <p>k is a dimensionless scale factor,</p> <p>π is the locking factor,</p> <p>the sum is over all segments k.</p> <p>This is the key map: {τ_k} → τ_p.</p> <p>3. Standard proper time (for comparison)</p> <p>If you only keep the real timelike parts:</p> <p>τ_std = Σ_k Re(τ_k)</p> <p>or, in 1+1D Minkowski (ignoring imaginary spacelike parts):</p> <p>τ_std = Σ_k sqrt( max(Δt_k² − Δx_k², 0) )</p> <p>4. Internal 4D time version (pure t_k model)</p> <p>For a 4-component internal time vector t = (t₁, t₂, t₃, t₄):</p> <p>t⁴-clock: τ_4 = π · k · ( |t₁|⁴ + |t₂|⁴ + |t₃|⁴ + |t₄|⁴ )^(1/4)</p> <p>More generally, for arbitrary dimension n:</p> <p>τ_p = π · k · ( Σ_i |t_i|^p )^(1/p)</p> <p>with i running over the internal time components.</p> <p>5. Schrödinger evolution on your clock</p> <p>Time-dependent Schrödinger equation in clock τ:</p> <p>i · ħ · (∂/∂τ) |ψ(τ)⟩ = H · |ψ(τ)⟩</p> <p>Solution at alias time τ_p:</p> <p>|ψ(τ_p)⟩ = exp( −i · H · τ_p / ħ ) · |ψ₀⟩</p> <p>So the full is:</p> <p>GR segments {Δx_k} → {τ_k} → τ_p = π·k·|Σ τ_k^p|^(1/p) → |ψ(τ_p)⟩</p> <p>τₚ as the primitive clock and the {τ_k} (or {t_k}) as generated components</p> <p>τₚ is proper time norm.</p> <p>(note from Ethan G Appleby, please use H_k instead of simply H but I have been cutting back on the notation for calartity and elegance of singular example, also this is generator means all valid t_k values have there own Schrödinger equation sectors in H or H_k)</p> <p>here I is a good lense for quantum mechanics mathmatically</p> <p>e^−iτ_kH_k</p> <p><strong>standard form</strong></p> <p>Start with complex H and t:</p> <ul> <li> <p>H = A + iB</p> </li> <li> <p>t = T + iS</p> </li> </ul> <p>where A, B, T, S are all real numbers.</p> <p>Define two real combinations:</p> <ul> <li> <p>X = A<em>T − B</em>S</p> </li> <li> <p>Y = A<em>S + B</em>T</p> </li> </ul> <p>Then identity is:</p> <blockquote> <p>e^(−i H t) = e^Y · e^(−i X)</p> </blockquote> <p>So the complex exponential with complex H and t splits into:</p> <ul> <li> <p>a real amplitude factor: e^Y</p> </li> <li> <p>a pure phase factor: e^(−i X)</p> </li> </ul> <p> </p> <p># Quantum Time Coproduct Multiplexing (QTCM)<br># Minimal "thermodynamic-like" GR like equation</p> <p>--------------------------------------------------<br>1. Internal alias clock definition<br>--------------------------------------------------</p> <p>Given:<br>- Internal time components: t₁, t₂, …, tₙ<br>- Power p (e.g. p = 4)<br>- π = locking factor<br>- k = dimensionless scale factor</p> <p>Define the alias (p-norm) clock:</p> <p>τₚ = π · k · ( Σᵢ |tᵢ|ᵖ )^(1/p)</p> <p>where the sum Σᵢ runs over all internal components i = 1…n.</p> <p>--------------------------------------------------<br>2. Minimal “thermodynamic-like” equation<br>--------------------------------------------------</p> <p>Start from:</p> <p>τₚ = π · k · ( Σᵢ |tᵢ|ᵖ )^(1/p)</p> <p>Raise both sides to the power p:</p> <p>τₚᵖ = (π · k)ᵖ · Σᵢ |tᵢ|ᵖ</p> <p>Rearrange:</p> <p>Σᵢ |tᵢ|ᵖ = ( τₚ / (π · k) )ᵖ</p> <p>This is the minimal “equation of state” for time:</p> <p>(micro side) Σᵢ |tᵢ|ᵖ<br>(macro side) = ( τₚ / (π · k) )ᵖ</p> <p>--------------------------------------------------<br>3. Special case: all tᵢ = 1<br>--------------------------------------------------</p> <p>Assume every internal time component is unit:</p> <p>tᵢ = 1 for all i = 1…N</p> <p>Then:</p> <p>|tᵢ|ᵖ = 1ᵖ = 1<br>Σᵢ |tᵢ|ᵖ = N</p> <p>Plug into the minimal equation:</p> <p>N = ( τₚ / (π · k) )ᵖ</p> <p>So if all tᵢ = 1, the number of components N<br>is given directly by the macroscopic clock τₚ:</p> <p>N ↔ ( τₚ / (π · k) )ᵖ</p> <p>--------------------------------------------------<br>4. GR segment version (same structure)<br>--------------------------------------------------</p> <p>Given GR segment proper times τ₁, τ₂, …, τₖ:</p> <p>Define alias clock:</p> <p>τₚ = π · k · | Σₖ (τₖᵖ) |^(1/p)</p> <p>Then:</p> <p>| Σₖ (τₖᵖ) | = ( τₚ / (π · k) )ᵖ</p> <p>Again:</p> <p>left = sum over local GR segment contributions<br>right = function of the single alias proper time τₚ</p> <p>=> Minimal thermodynamic-like, GR like. equation:<br>“sum over micro times” = “macro clock expression”.</p> <p># Core time–frequency invariant</p> <p>τ_min = π / (c · √Λ)<br>Ω_min = π · c · √Λ<br>τ_min · Ω_min = π²</p> <p># Full GR + alias-clock + frequency-Schrödinger form</p> <p>τ² = π² · ( Σ_k g_μν · Δx_k^μ · Δx_k^ν + 1 / (c² · Λ) )</p> <p>i · ∂/∂τ |ψ(τ)⟩ = Ω̂ · |ψ(τ)⟩<br>Ω̂ = H / ħ</p> <p>τ_min = π / (c · √Λ)<br>Ω_min = π · c · √Λ<br>τ_min · Ω_min = π²</p> <p>Define the time–frequency vector for a given velocity v:</p> <p>X(v) = ( τ_min(v), Ω_min(v) )ᵀ</p> <p>with</p> <p>τ_min(v) = π / (v · √Λ)<br>Ω_min(v) = π · v · √Λ</p> <p>⇒ invariant cell area:</p> <p>τ_min(v) · Ω_min(v) = π² for all v > 0</p> <p>Choose the light sector (v = c) as reference:</p> <p>τ★ = π / (c · √Λ)<br>Ω★ = π · c · √Λ</p> <p>so</p> <p>X★ = ( τ★, Ω★ )ᵀ<br>τ★ · Ω★ = π²</p> <p>Relate a general velocity v to the light sector via β = v / c:</p> <p>β = v / c</p> <p>Then</p> <p>τ_min(β) = τ★ / β<br>Ω_min(β) = β · Ω★</p> <p>and</p> <p>τ_min(β) · Ω_min(β) = τ★ · Ω★ = π² for all β > 0</p> <p>Now write this as a symplectic scaling:</p> <p>X(β) = S(β) · X★,</p> <p>where</p> <p>S(β) = [ 1/β 0 ]<br>[ 0 β ]</p> <p>Introduce the canonical symplectic form J on the (τ, Ω) plane:</p> <p>J = [ 0 1 ]<br>[ -1 0 ]</p> <p>Symplectic check (this is what we verified in code):</p> <p>S(β)ᵀ · J · S(β) = J for all β > 0</p> <p>So:</p> <p>• S(β) is a symplectic transformation on the (τ, Ω) plane.<br>• It rescales τ and Ω:</p> <p>τ_min(β) = τ★ / β<br>Ω_min(β) = β · Ω★</p> <p>but preserves the time–frequency cell area:</p> <p>τ_min(β) · Ω_min(β) = π²</p> <p> </p> <p>World relation (time–frequency shell)</p> <p>t, Ω are complex.</p> <p>Invariant:</p> <p>t * Ω(t) = π²</p> <p>so</p> <p>Ω(t) = π² / t</p> <p>with 0 < |t| < |τ| (t lives in a complex disc).</p> <p>Single Schrödinger–Alice law (O(1) local rule)</p> <p>For alias time τ and label time t:</p> <p>i * ∂ψ(τ, t) / ∂τ = Ω(t) * ψ(τ, t)<br>i * ∂ψ(τ, t) / ∂τ = (π² / t) * ψ(τ, t)</p> <p>That’s the core equation.</p> <p>Solution for each fixed t</p> <p>ψ(τ, t) = exp( -i * (π² / t) * τ ) * ψ₀(t)</p> <p>All the “infinite complexity” is just: take this one rule and apply it for all complex t with |t| < |τ|.</p> <p> </p> <p>NOTE from Ethan G Appleby:</p> <p>This model is the Synthetic Ailce model also known as QTCM as in Quantum Time-Coproduct Multiplexing.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17790944 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Quantum Time-Coproduct Multiplexing (QTCM). Appleby, Ethan Appleby, Ethan Appleby, Ethan <p> </p> <p>1. Segment proper time (GR piece)</p> <p>For each segment k of a worldline:</p> <p>General form (with metric g):<br>τ_k² = g_μν · Δx_k^μ · Δx_k^ν</p> <p>In simple 1+1D Minkowski (c = 1):<br>Δτ_k² = Δt_k² − Δx_k²</p> <p>Timelike segment → Δτ_k real<br>Spacelike segment → Δτ_k purely imaginary</p> <p>2. alias clock (best work but simple)</p> <p>Given all segment times {τ_k}, define the p-norm “twist clock”:</p> <p>τ_p = π · k · | Σ_k (τ_k^p) |^(1/p)</p> <p>where</p> <p>p is the norm power (often p = 4),</p> <p>k is a dimensionless scale factor,</p> <p>π is the locking factor,</p> <p>the sum is over all segments k.</p> <p>This is the key map: {τ_k} → τ_p.</p> <p>3. Standard proper time (for comparison)</p> <p>If you only keep the real timelike parts:</p> <p>τ_std = Σ_k Re(τ_k)</p> <p>or, in 1+1D Minkowski (ignoring imaginary spacelike parts):</p> <p>τ_std = Σ_k sqrt( max(Δt_k² − Δx_k², 0) )</p> <p>4. Internal 4D time version (pure t_k model)</p> <p>For a 4-component internal time vector t = (t₁, t₂, t₃, t₄):</p> <p>t⁴-clock: τ_4 = π · k · ( |t₁|⁴ + |t₂|⁴ + |t₃|⁴ + |t₄|⁴ )^(1/4)</p> <p>More generally, for arbitrary dimension n:</p> <p>τ_p = π · k · ( Σ_i |t_i|^p )^(1/p)</p> <p>with i running over the internal time components.</p> <p>5. Schrödinger evolution on your clock</p> <p>Time-dependent Schrödinger equation in clock τ:</p> <p>i · ħ · (∂/∂τ) |ψ(τ)⟩ = H · |ψ(τ)⟩</p> <p>Solution at alias time τ_p:</p> <p>|ψ(τ_p)⟩ = exp( −i · H · τ_p / ħ ) · |ψ₀⟩</p> <p>So the full is:</p> <p>GR segments {Δx_k} → {τ_k} → τ_p = π·k·|Σ τ_k^p|^(1/p) → |ψ(τ_p)⟩</p> <p>τₚ as the primitive clock and the {τ_k} (or {t_k}) as generated components</p> <p>τₚ is proper time norm.</p> <p>(note from Ethan G Appleby, please use H_k instead of simply H but I have been cutting back on the notation for calartity and elegance of singular example, also this is generator means all valid t_k values have there own Schrödinger equation sectors in H or H_k)</p> <p>here I is a good lense for quantum mechanics mathmatically</p> <p>e^−iτ_kH_k</p> <p><strong>standard form</strong></p> <p>Start with complex H and t:</p> <ul> <li> <p>H = A + iB</p> </li> <li> <p>t = T + iS</p> </li> </ul> <p>where A, B, T, S are all real numbers.</p> <p>Define two real combinations:</p> <ul> <li> <p>X = A<em>T − B</em>S</p> </li> <li> <p>Y = A<em>S + B</em>T</p> </li> </ul> <p>Then identity is:</p> <blockquote> <p>e^(−i H t) = e^Y · e^(−i X)</p> </blockquote> <p>So the complex exponential with complex H and t splits into:</p> <ul> <li> <p>a real amplitude factor: e^Y</p> </li> <li> <p>a pure phase factor: e^(−i X)</p> </li> </ul> <p> </p> <p># Quantum Time Coproduct Multiplexing (QTCM)<br># Minimal "thermodynamic-like" GR like equation</p> <p>--------------------------------------------------<br>1. Internal alias clock definition<br>--------------------------------------------------</p> <p>Given:<br>- Internal time components: t₁, t₂, …, tₙ<br>- Power p (e.g. p = 4)<br>- π = locking factor<br>- k = dimensionless scale factor</p> <p>Define the alias (p-norm) clock:</p> <p>τₚ = π · k · ( Σᵢ |tᵢ|ᵖ )^(1/p)</p> <p>where the sum Σᵢ runs over all internal components i = 1…n.</p> <p>--------------------------------------------------<br>2. Minimal “thermodynamic-like” equation<br>--------------------------------------------------</p> <p>Start from:</p> <p>τₚ = π · k · ( Σᵢ |tᵢ|ᵖ )^(1/p)</p> <p>Raise both sides to the power p:</p> <p>τₚᵖ = (π · k)ᵖ · Σᵢ |tᵢ|ᵖ</p> <p>Rearrange:</p> <p>Σᵢ |tᵢ|ᵖ = ( τₚ / (π · k) )ᵖ</p> <p>This is the minimal “equation of state” for time:</p> <p>(micro side) Σᵢ |tᵢ|ᵖ<br>(macro side) = ( τₚ / (π · k) )ᵖ</p> <p>--------------------------------------------------<br>3. Special case: all tᵢ = 1<br>--------------------------------------------------</p> <p>Assume every internal time component is unit:</p> <p>tᵢ = 1 for all i = 1…N</p> <p>Then:</p> <p>|tᵢ|ᵖ = 1ᵖ = 1<br>Σᵢ |tᵢ|ᵖ = N</p> <p>Plug into the minimal equation:</p> <p>N = ( τₚ / (π · k) )ᵖ</p> <p>So if all tᵢ = 1, the number of components N<br>is given directly by the macroscopic clock τₚ:</p> <p>N ↔ ( τₚ / (π · k) )ᵖ</p> <p>--------------------------------------------------<br>4. GR segment version (same structure)<br>--------------------------------------------------</p> <p>Given GR segment proper times τ₁, τ₂, …, τₖ:</p> <p>Define alias clock:</p> <p>τₚ = π · k · | Σₖ (τₖᵖ) |^(1/p)</p> <p>Then:</p> <p>| Σₖ (τₖᵖ) | = ( τₚ / (π · k) )ᵖ</p> <p>Again:</p> <p>left = sum over local GR segment contributions<br>right = function of the single alias proper time τₚ</p> <p>=> Minimal thermodynamic-like, GR like. equation:<br>“sum over micro times” = “macro clock expression”.</p> <p># Core time–frequency invariant</p> <p>τ_min = π / (c · √Λ)<br>Ω_min = π · c · √Λ<br>τ_min · Ω_min = π²</p> <p># Full GR + alias-clock + frequency-Schrödinger form</p> <p>τ² = π² · ( Σ_k g_μν · Δx_k^μ · Δx_k^ν + 1 / (c² · Λ) )</p> <p>i · ∂/∂τ |ψ(τ)⟩ = Ω̂ · |ψ(τ)⟩<br>Ω̂ = H / ħ</p> <p>τ_min = π / (c · √Λ)<br>Ω_min = π · c · √Λ<br>τ_min · Ω_min = π²</p> <p>Define the time–frequency vector for a given velocity v:</p> <p>X(v) = ( τ_min(v), Ω_min(v) )ᵀ</p> <p>with</p> <p>τ_min(v) = π / (v · √Λ)<br>Ω_min(v) = π · v · √Λ</p> <p>⇒ invariant cell area:</p> <p>τ_min(v) · Ω_min(v) = π² for all v > 0</p> <p>Choose the light sector (v = c) as reference:</p> <p>τ★ = π / (c · √Λ)<br>Ω★ = π · c · √Λ</p> <p>so</p> <p>X★ = ( τ★, Ω★ )ᵀ<br>τ★ · Ω★ = π²</p> <p>Relate a general velocity v to the light sector via β = v / c:</p> <p>β = v / c</p> <p>Then</p> <p>τ_min(β) = τ★ / β<br>Ω_min(β) = β · Ω★</p> <p>and</p> <p>τ_min(β) · Ω_min(β) = τ★ · Ω★ = π² for all β > 0</p> <p>Now write this as a symplectic scaling:</p> <p>X(β) = S(β) · X★,</p> <p>where</p> <p>S(β) = [ 1/β 0 ]<br>[ 0 β ]</p> <p>Introduce the canonical symplectic form J on the (τ, Ω) plane:</p> <p>J = [ 0 1 ]<br>[ -1 0 ]</p> <p>Symplectic check (this is what we verified in code):</p> <p>S(β)ᵀ · J · S(β) = J for all β > 0</p> <p>So:</p> <p>• S(β) is a symplectic transformation on the (τ, Ω) plane.<br>• It rescales τ and Ω:</p> <p>τ_min(β) = τ★ / β<br>Ω_min(β) = β · Ω★</p> <p>but preserves the time–frequency cell area:</p> <p>τ_min(β) · Ω_min(β) = π²</p> <p> </p> <p>World relation (time–frequency shell)</p> <p>t, Ω are complex.</p> <p>Invariant:</p> <p>t * Ω(t) = π²</p> <p>so</p> <p>Ω(t) = π² / t</p> <p>with 0 < |t| < |τ| (t lives in a complex disc).</p> <p>Single Schrödinger–Alice law (O(1) local rule)</p> <p>For alias time τ and label time t:</p> <p>i * ∂ψ(τ, t) / ∂τ = Ω(t) * ψ(τ, t)<br>i * ∂ψ(τ, t) / ∂τ = (π² / t) * ψ(τ, t)</p> <p>That’s the core equation.</p> <p>Solution for each fixed t</p> <p>ψ(τ, t) = exp( -i * (π² / t) * τ ) * ψ₀(t)</p> <p>All the “infinite complexity” is just: take this one rule and apply it for all complex t with |t| < |τ|.</p> <p> </p> <p>NOTE from Ethan G Appleby:</p> <p>This model is the Synthetic Ailce model also known as QTCM as in Quantum Time-Coproduct Multiplexing.</p> |
| title | Quantum Time-Coproduct Multiplexing (QTCM). |
| url | https://doi.org/10.5281/zenodo.17790944 |