The Bogoliubov Transform — Squeezed Vacuum and Thermal Reduced Modes
Act 20 of the NexusOS physics sequence. First disclosed 2026-07-21. AGPL-3.0. Founder: Te Rata Pou.
The Transformation
A single-mode Bogoliubov transformation mixes annihilation and creation operators: b̂=uâ+vâ†, with |u|²−|v|²=1 preserving [b̂,b̂†]=1. For the real squeeze convention, S†(r)âS(r)=â·cosh(r)−↷sinh(r). A creation operator is not an antiparticle operator for this mode.
Pure Squeezed Vacuum Versus Thermal Reduced State
Pure single-mode squeezed vacuum has n̄=sinh²(r) but is not a thermal state: its quadrature variances are unequal and its photon-number distribution differs from a thermal distribution. A mean occupation alone does not determine a state's temperature. An ideal two-mode squeezed vacuum is a pure entangled pair. Tracing out one partner leaves a mixed thermal reduced state with n̄=sinh²(r) and Var(n)=n̄(n̄+1). Only for this thermal reduced state can the occupation be identified with Bose-Einstein statistics: T_eff=ℏω/[k_B·ln(1+1/n̄)] for n̄>0, with T=0 in the vacuum limit.
Unruh and Hawking Radiation
In quantum field theory an ideal uniformly accelerating observer sees a thermal response: T_Unruh=ℏa/(2πck_B). A Schwarzschild black hole has T_Hawking=ℏc³/(8πGMk_B). Horizon mode mixing and restricting access to partner modes explain the thermal reduced description; arbitrary single-mode squeezing is not thermalisation.
Externally Powered Photon Generation
A change of mode basis alone supplies no energy. Parametric photon generation uses an external optical pump. The dynamical Casimir effect uses work done by a time-dependent boundary drive; it is not free energy extracted from an undriven vacuum. In an ideal resonant, lossless model, n̄=sinh²(εωt/2). Real output depends on losses, detuning and pump depletion.
BEC Phonon Dispersion
For interaction constant g, condensate density n₀ and particle mass m: E(k)=√[ε_k(ε_k+2gn₀)], ε_k=ℏ²k²/(2m), ω=E/ℏ. At low k, ω≈c_s·k with c_s=√(gn₀/m). The Bogoliubov transformation diagonalises the quadratic approximation to the BEC Hamiltonian.
WNSP Connection — Hypotheses
NexusOS's octave-transition interpretation, Λ_B=Λ₀·cosh(2r) compression-state application and per-channel extraction-bound interpretation remain hypotheses, not validated hardware or an undriven vacuum power source. For a proposed cavity at 1550 nm and T=10 mK, thermal occupation 1/(e^(ℏω/k_BT)−1) is exponentially negligible (exponent about 9.3×10⁵), not 10⁻³⁴. Pump-generated noise is distinct from equilibrium thermal noise.
The 20-Act Sequence
The Bogoliubov Transform
Act 20 of the NexusOS research sequence.
Established basis
b̂ = uâ + v↠with |u|² − |v|² = 1 preserves bosonic commutation. A single-mode squeezed vacuum is pure; a reduced mode of a two-mode pair is thermal.
NexusOS proposal
ΛB = Λ₀ cosh(2r), with Λ₀ = hf/(2c²), is the project's vacuum-energy mass-equivalent extension.
Evidence limits
Driven photon generation requires external energy. Compression-state and mass-displacement applications are hypotheses, not demonstrated hardware.
Hardware fundraising and promotions paused pending engineering review. Working software models; reported electrical tests on three Coiltek bifilar coils. Integrated SNIC, optical relay, quantum memory and ZERO-G performance have not been demonstrated by those tests.
Original numbered disclosures are research claims, not independent experimental validation or granted patents.
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