01-EFT.WP.Core.Terms v1.0 | Chapter 6, Dimensions, Non-Dimensionalization & Units

Home ／ Docs-Technical WhitePaper ／ 01-EFT.WP.Core.Terms v1.0

Chapter 6, Dimensions, Non-Dimensionalization & Units

I. Chapter Goals & Scope

• Standardize dimension marks and algebra—dim[•], [L], [T], [M]—to ensure volume-wide self-consistency and verifiable dimensional coherence.
• Provide three drop-in non-dimensionalization schemes (time scale, length scale, tension scale) with expressions aligned to T_arr and the path pair gamma(ell), d ell.
• Interface with I10-3 (unit registry & dimension checks) and I10-4 (expression validation), consistent with Chapter 5 Operators & Statistics and Chapter 4 Constants & Baselines.

II. Dimension System & Notation (Foundations)

1. Base and derived dimensions
   • Bases: [L] (length), [T] (time), [M] (mass).
   • Examples: [velocity] = [L T^-1], [density] = [M L^-3], [T_arr] = [T].
2. Dimensional operator and rules
   • Notation: dim[X] -> [L^a T^b M^c].
   • Addition: only allowed if dim[A] = dim[B].
   • Multiplication: dim[A B] = dim[A] dim[B]; for division, exponents subtract accordingly.
   • Operators: dim[grad[f]] = dim[f] [L^-1], dim[div[F]] = dim[F] [L^-1], dim[lap[f]] = dim[f] [L^-2].
   • Path measure: dim[d ell] = [L]; thus dim[( n_eff / c_ref ) * d ell] = [T^0], and after integration dim[T_arr] = [T].
3. Symbolic prefixes (for scaling and estimation)
   • bar_ (scaled / non-dimensionalized), tilde_ (perturbation), hat_ (estimate).
   • Examples: bar_x = x / L0, tilde_rho = rho - avg_V[rho], hat_n_eff is an estimator.

III. Reference Scales & Parameters (Naming & Selection)

1. Reference quantities
   • Length: L0 def= reference length scale; Time: t0 def= reference time scale; Tension: T0 def= reference tension scale.
   • Propagation ceiling: c_ref (see Chapter 4); Path: gamma(ell), L_gamma = ∫_gamma 1 d ell.
2. Selection principles
   • Consistency: within one metrology chain, fix {L0, t0, T0} and c_ref.
   • Traceability: record source, version, applicability, and uncertainty in metrology logs.
   • Conditioning: prefer scales that keep key dimensionless numbers at O(1) to improve numerical conditioning.

IV. Non-Dimensionalization Scheme A: Time-Based Scaling

1. Definitions
   • bar_t = t / t0, bar_ell = ell / ( c_ref * t0 ), bar_gamma(bar_ell) = gamma(ell).
   • bar_T_arr = T_arr / t0.
2. Time-of-arrival (unified form)
   bar_T_arr(bar_gamma) = ( ∫_{bar_gamma} ( n_eff ) d bar_ell ), with d bar_ell = d ell / ( c_ref * t0 ).
3. Operator scaling
   D_bar_ell f = D_ell f * ( d ell / d bar_ell ) = ( c_ref * t0 ) * D_ell f.
4. Use cases
   Metrology/observations dominated by time gating where path length varies across platforms (e.g., delay-stack alignment for pulse-like events).

V. Non-Dimensionalization Scheme B: Length-Based Scaling

1. Definitions
   • bar_x = x / L0, bar_ell = ell / L0, bar_gamma(bar_ell) = gamma(ell).
   • bar_t = t / ( L0 / c_ref ), bar_T_arr = T_arr / ( L0 / c_ref ).
2. Time-of-arrival (unified form)
   bar_T_arr(bar_gamma) = ( ∫_{bar_gamma} n_eff d bar_ell ).
3. Operator scaling
   grad_bar[f] = L0 * grad[f], lap_bar[f] = L0^2 * lap[f].
4. Use cases
   Experiments and simulations anchored to fixed path length or geometry (arrays, guided structures, standard path baselines).

VI. Non-Dimensionalization Scheme C: Tension-Based Scaling

1. Definitions
   • bar_T_fil = T_fil / T0, bar_TensionGrad = TensionGrad * ( L0 / T0 ), bar_TensionPot = TensionPot / T0.
   • If n_eff def= F( T_fil, TensionGrad, ... ), then bar_n_eff = F( T0 bar_T_fil, (T0/L0) bar_TensionGrad, ... ).
2. Time-of-arrival (using Scheme B’s length normalization)
   bar_T_arr = ( ∫_{bar_gamma} bar_n_eff d bar_ell ).
3. Relation to k_T
   Where k_T links tension to equivalent mass/energy, declare dim[k_T] at unit registration and ensure mappings for bar_T_fil, bar_TensionPot remain dimensionally consistent (see Chapter 4 and I10-3).
4. Use cases
   Near-field reconstructions dominated by T_fil, propagation corrections driven by potentials/gradients, and stability analysis.

VII. Consistency Across Dimensions, Operators & Statistics

1. Averages and variance
   dim[avg_t[f]] = dim[f], dim[avg_V[f]] = dim[f], dim[var_t[f]] = dim[f]^2.
2. Path and integration
   dim[avg_gamma[f]] = dim[f], where avg_gamma[f] def= ( 1 / L_gamma ) * ( ∫_gamma f d ell ).
3. Operator transforms under non-dimensionalization
   • Length scheme: grad[f] -> ( 1 / L0 ) * grad_bar[f], lap[f] -> ( 1 / L0^2 ) * lap_bar[f].
   • Time scheme: d/dt -> ( 1 / t0 ) * d/d bar_t.
4. Arrival-time dimensional check
   ( n_eff / c_ref ) * d ell is dimensionless; therefore T_arr—and its combinations with statistics such as avg_t—remains dim[T_arr] = [T].

VIII. Unit Registration & Dimensional Checks (I10-3 Integration)

1. Examples
   • register_unit(name="meter", base="SI", scale=1.0, dim="[L]")
   • register_unit(name="second", base="SI", scale=1.0, dim="[T]")
   • register_unit(name="kilogram", base="SI", scale=1.0, dim="[M]")
   • register_unit(name="c_ref", base="derived", scale=<value>, dim="[L T^-1]")
   • register_unit(name="T0", base="derived", scale=<value>, dim="dim[T_fil]")
2. Checks
   • check_dim(expr="( n_eff / c_ref ) * d_ell") -> "[T^0]"；after the path integral the result is "[T]".
   • check_dim(expr="grad[T_fil]") -> "dim[T_fil] [L^-1]".
   • check_dim(expr="avg_t[rho; Δt]") -> "[M L^-3]".
3. Constraint
   Any constant or unit used in arrival-time or path expressions must exist in the registry with a unique version; unit aliases not registered are prohibited.

IX. Buckingham–Pi and Dimensionless Groups (Practical Template)

1. Select the variable set X = { X1, X2, ..., Xn } and base dimensions {[L],[T],[M]}.
2. Build the dimension matrix D ∈ R^{3×n} and compute null(D) to obtain independent Pi groups.
3. Define Pi_i def= Π_j X_j^{a_{ij}} and relate them to reference scales using bar_ in the main text.

• Example (path delay)
  • Variable set: { T_arr, L_gamma, c_ref, n_eff }.
  • Independent groups: Pi_1 def= T_arr / ( L_gamma / c_ref ), Pi_2 def= avg_gamma[n_eff].
  • Relation: Pi_1 = Pi_2 (i.e., bar_T_arr = avg_gamma[n_eff]).

X. Common Misuses & Corrections

• Misuse: T_arr = ∫ n_eff d ell / c (bare c, missing parentheses)
  Correction: T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
• Misuse: avg_t[f] changes dimension or lacks a window declaration
  Correction: avg_t[f; Δt] with dim[avg_t[f; Δt]] = dim[f].
• Misuse: mixing original and bar_ coordinates after non-dimensionalization
  Correction: use one coordinate family within the same derivation domain; if necessary, state the mapping explicitly (e.g., x = L0 * bar_x).
• Misuse: substituting n(x,t) for n_eff(x,t) in arrival time
  Correction: only n_eff is allowed under the integrand.

XI. Implementation Bindings & Templates (I10-3 / I10-4)

1. Units & dimensions
   • register_unit(name="L0", base="derived", scale=<value>, dim="[L]")
   • register_unit(name="t0", base="derived", scale=<value>, dim="[T]")
   • register_unit(name="T0", base="derived", scale=<value>, dim="dim[T_fil]")
2. Expressions & validation
   • validate_expr("bar_T_arr = ( ∫ n_eff d bar_ell )", allowed=...) -> True
   • validate_expr("T_arr = ∫ n d ell / c", allowed=...) -> False
3. Export card fields
   name, symbol, def=, scale, unit, dim, source, version, uncertainty, validity, see

XII. Quick Checklist

• Are {L0, t0, T0} and their unit sources declared and registered via register_unit?
• Do arrival-time and path expressions satisfy dim[T_arr] = [T] and pass check_dim?
• Is non-dimensionalization self-consistent within one derivation domain (no mixing of x with bar_x, ell with bar_ell)?
• Do statistical operators preserve dimensions (avg_t, avg_V, avg_gamma)?
• Are T_fil/T_trans and n/n_eff never mixed, and is c_ref always used instead of bare c?

Chapter Summary

This chapter establishes the dimension notation and algebra, three non-dimensionalization schemes, and their alignment with paths and time-of-arrival. It completes the loop with implementation layers I10-3/I10-4 through unit registration and expression validation. All subsequent volumes that introduce scales, units, or estimators must use this chapter as the sole authority for registration, checking, and publication.