TRI-MODE COMPRESS: TORUS FLUX WAVEFUNCTION
System: Charged quantum particle on T^2 with U(1) holonomy (ฯ_u, ฯ_v)
SID MODE (Structure / Interaction / Dynamics)
STRUCTURE:
Configuration manifold:
X = T^2 = S^1_u ร S^1_v
u, v โ [0, 2ฯ)
Topology:
ฯ1(X) = Z โ Z
Cycles: C_u, C_v
Gauge bundle:
U(1) over X with flat connection A
Hol(C_u) = exp(i 2ฯ ฯ_u)
Hol(C_v) = exp(i 2ฯ ฯ_v)
Hilbert space:
H = L^2(T^2, du dv / (2ฯ)^2)
Basis:
|m,n> โ ฮจ_{m,n}(u,v) = exp(i m u i n v), m,n โ Z
INTERACTION:
Zero-flux momenta:
P_u = -i โ_u
P_v = -i โ_v
Flux-deformed momenta:
P_u^ฯ = -i โ_u - ฯ_u
P_v^ฯ = -i โ_v - ฯ_v
Interpretation:
Holonomy shifts the generators of translations along u, v.
DYNAMICS:
Hamiltonian:
H(ฯ_u, ฯ_v) = 1/2 [ (P_u - ฯ_u)^2 (P_v - ฯ_v)^2 ]
Eigen-equation:
H ฮจ_{m,n} = E_{m,n} ฮจ_{m,n}
Spectrum:
E_{m,n}(ฯ) = 1/2 [ (m - ฯ_u)^2 (n - ฯ_v)^2 ]
PED MODE (Power / Evaluation / Dynamics on H)
STATE EVALUATION:
General state:
|ฮจ(t)> = ฮฃ_{m,n} c_{m,n} e^{-i E_{m,n} t} |m,n>
Position representation:
ฮจ(u,v,t) = ฮฃ c_{m,n} e^{-i E_{m,n} t i m u i n v}
Observables:
Density: ฯ(u,v,t) = |ฮจ(u,v,t)|^2
Phase: ฮธ(u,v,t) = arg ฮจ(u,v,t)
TEMPORAL STRUCTURE:
Relative energy:
ฮE = E_{m,n} - E_{m',n'}
= 1/2 [ (m-ฯ_u)^2 - (m'-ฯ_u)^2
(n-ฯ_v)^2 - (n'-ฯ_v)^2 ]
Beating frequency:
ฯ = ฮE (ฤง = 1)
PED interpretation:
Changing (ฯ_u, ฯ_v) reprograms all ฮE,
hence all interference beats and pattern drift on T^2.
Q MODE (Quantization / Invariants / Phase Structure)
QUANTIZATION DATA:
Integer labels:
(m, n) โ Z^2 (topological momentum lattice)
Flux parameters:
(ฯ_u, ฯ_v) โ R^2 / Z^2 (defined modulo integers)
Q-invariant:
Spectrum depends only on fractional parts:
{ฯ_u}, {ฯ_v} (AharonovโBohm class)
Shifting ฯ_u โ ฯ_u k, ฯ_v โ ฯ_v l, k,l โ Z
leaves all physics invariant (gauge-equivalent sector).
PHASE / HOLONOMY STRUCTURE:
Wavefunction is a section of a twisted U(1) bundle:
ฮจ(u 2ฯ,v) = e^{i 2ฯ ฯ_u} ฮจ(u,v)
ฮจ(u,v 2ฯ) = e^{i 2ฯ ฯ_v} ฮจ(u,v)
(in a gauge where twist is pushed into boundary conditions)
Q interpretation:
- (m,n) label discrete modes on the torus.
- (ฯ_u, ฯ_v) label continuous holonomy class.
- Physical content lives in the relative pairing:
(m - ฯ_u, n - ฯ_v)
which is the Q-level โshifted latticeโ.
TRI-MODE SNAPSHOT
SID:
Torus geometry U(1) holonomy deformed momenta H(ฯ).
PED:
Flux-shifted spectrum โ flux-dependent time evolution
โ braided interference patterns on T^2.
Q:
Integer mode lattice (m,n) fractional holonomy ({ฯ_u},{ฯ_v})
โ invariant shifted lattice (m-ฯ_u, n-ฯ_v) as the core Q-object.
Tri-mode compression:
A torus with holonomy (SID) shifts the momentum lattice (Q), which retimes all interference beats into a braided wavefunction (PED).