04-EFT.WP.Core.Metrology v1.0 | Appendix A — Units and Dimension Table

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Appendix A — Units and Dimension Table

I. Base Dimensions and Notation

• Base dimension set: [L] (length), [T] (time), [M] (mass), [Qe] (charge), [Temp] (thermodynamic temperature)
• Dimensional string syntax: "[L]^a [T]^b [M]^c [Qe]^d [Temp]^e" (empty exponents may be omitted, e.g., "[L][T]^-1")
• Angle and solid angle: rad, sr are treated as dimensionless (dim = "[]")

II. SI Base Units (with [Qe] as the electrical base dimension)

• m (metre): dim = "[L]"
• s (second): dim = "[T]"
• kg (kilogram): dim = "[M]"
• C (coulomb): dim = "[Qe]"; A (ampere) is derived: A = C s^-1, dim = "[Qe][T]^-1"
• K (kelvin): dim = "[Temp]"

III. Decimal Prefixes (recommended subset)

• G = 10^9, M = 10^6, k = 10^3
• m = 10^-3, u = 10^-6 (µ written as u), n = 10^-9, p = 10^-12

IV. Common Derived Units and Dimensions

• Hz = s^-1: dim = "[T]^-1"
• N = kg m s^-2: dim = "[M][L][T]^-2"
• Pa = N m^-2 = kg m^-1 s^-2: dim = "[M][L]^-1[T]^-2"
• J = N m = kg m^2 s^-2: dim = "[M][L]^2[T]^-2"
• W = J s^-1 = kg m^2 s^-3: dim = "[M][L]^2[T]^-3"
• V = J C^-1 = kg m^2 s^-2 C^-1: dim = "[M][L]^2[T]^-2[Qe]^-1"
• Ω = V A^-1: dim = "[M][L]^2[T]^-1[Qe]^-2"
• S = Ω^-1: dim = "[M]^-1[L]^-2[T]^1[Qe]^2"
• F = C V^-1: dim = "[M]^-1[L]^-2[T]^2[Qe]^2"
• H = V s A^-1: dim = "[M][L]^2[Qe]^-2"
• T (tesla) = Wb m^-2 = kg s^-2 A^-1: dim = "[M][T]^-1[Qe]^-1"
• E (electric field) = V m^-1: dim = "[M][L][T]^-2[Qe]^-1"

V. Common Wave/Optics Quantities

• lambda (wavelength, m): dim = "[L]"
• k (wavenumber, m^-1): dim = "[L]^-1" (radians treated as dimensionless)
• f (frequency, Hz): dim = "[T]^-1"; omega = 2*pi*f: dim = "[T]^-1"
• I (intensity, W m^-2): dim = "[M][T]^-3"
• P (optical power, W): dim = "[M][L]^2[T]^-3"
• alpha (absorption coefficient, m^-1): dim = "[L]^-1"
• n_eff (effective refractive index): dim = "[]" (dimensionless)

VI. EFT Cross-Volume Anchors and Unit Mapping

• c_ref: unit = m s^-1, dim = "[L][T]^-1"
• gamma(ell): ell is the path coordinate, unit = m, dim = "[L]"; measure d ell has the same dimension
• L_gamma = ∫_gamma 1 d ell: unit = m, dim = "[L]"
• n_eff(x,t): unit = 1, dim = "[]"
• T_arr (arrival time): unit = s, dim = "[T]"
• T_fil (tension): unit = N, dim = "[M][L][T]^-2"
• T_trans (transmission coefficient): unit = 1, dim = "[]"
• rho(x,t) (mass density): unit = kg m^-3, dim = "[M][L]^-3"

VII. Reference Conditions and Environmental-Correction Quantities

• p_ref: unit = Pa, dim = "[M][L]^-1[T]^-2"
• Temp_ref: unit = K, dim = "[Temp]"
• humidity_ref: unit = 1 or % (treated as a dimensionless scale)
• corr_env(x; RefCond) outputs with the same dimension as x; dim( corr_env(x; •) ) = dim(x)

VIII. Nondimensionalization and Re-Dimensionalization (consistent with Chapter 3)

1. Reference scales: L0 (m), t0 (s), T0 (K), with t0 def= L0 / c_ref
2. Mappings: bar_x = x / L0, bar_t = t / t0; dim(bar_x) = dim(bar_t) = "[]"
3. Arrival-time two-form in the dimensionless domain:
   • T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )
   • T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell )
   • Under t0 = L0 / c_ref, the integrand is written bar_n_eff * d bar_ell, dim = "[]"

IX. Unit Conversion and Affine Rules

• Linear scaling: v_to = a * v_from (e.g., kPa ↔ Pa)
• Affine with offset: v_to = a * v_from + b only for unit pairs with different zeros (e.g., degC ↔ K, where a = 1, b = 273.15)
• Composite-unit conversions must preserve dimensional algebra: check_dim(expr_from) = check_dim(expr_to)

X. Typical check_dim Use Cases (for regression baselines)

• check_dim( (1 / c_ref) * ∫_gamma n_eff d ell ) = "[T]"
• check_dim( ∫_gamma ( n_eff / c_ref ) d ell ) = "[T]"
• check_dim( I ) = "[M][T]^-3" (I = W m^-2)
• check_dim( P / (c_ref * A_aperture) ) = "[M][L]^-1[T]^-2" (radiation pressure, A_aperture is area)
• check_dim( E ) = "[M][L][T]^-2[Qe]^-1"，check_dim( B ) = "[M][T]^-1[Qe]^-1"

XI. EFT Preferred Unit Policy (unit_policy = "SI-Strict")

• Space: m; Time: s; Mass: kg; Charge: C; Temperature: K
• Velocity: m s^-1; Acceleration: m s^-2; Force: N; Pressure: Pa
• Energy: J; Power: W; Frequency: Hz; Refractive index and transmission coefficient: 1
• Allowed prefixes in reports: k, m, u, n; unless explicitly stated, do not use mixed or non-SI units

XII. Suggested define_derived_unit Expressions (I40 1 Binding)

• define_derived_unit("Hz", "s^-1")
• define_derived_unit("N", "kg*m*s^-2")
• define_derived_unit("Pa", "kg*m^-1*s^-2")
• define_derived_unit("J", "kg*m^2*s^-2")
• define_derived_unit("W", "kg*m^2*s^-3")
• define_derived_unit("V", "kg*m^2*s^-2*C^-1")
• define_derived_unit("Ohm", "kg*m^2*s^-1*C^-2")
• define_derived_unit("F", "kg^-1*m^-2*s^2*C^2")
• define_derived_unit("H", "kg*m^2*C^-2")

XIII. Domain Compound Quantities (script-ready)

• areal_density = kg m^-2: dim = "[M][L]^-2"
• energy_density = J m^-3: dim = "[M][L]^-1[T]^-2"
• momentum_flux = N m^-2 = Pa: dim = "[M][L]^-1[T]^-2"
• optical_depth = ∫ alpha d ell: unit = 1, dim = "[]"

XIV. Constraint and Conflict-Name Review

• Do not mix T_fil and T_trans: unit(T_fil) = N, unit(T_trans) = 1
• Strictly distinguish n (number density) and n_eff (effective index):
  unit(n) = m^-3, dim = "[L]^-3"; unit(n_eff) = 1, dim = "[]"

XV. Cross-Volume Citation Anchors

• Arrival-time two-form and dimensional consistency: see companion white paper Energy Threads, Chapter 20 S/M/I
• Unit consistency for the tension field and stress–tension mapping: see companion white paper Energy Threads, Chapter 40 S/P/M/I

XVI. Reporting and Display Conventions (aligned with Chapter 6)

• Round by expanded uncertainty U: compute U = k * u_c first, then round consistently
• Place the unit to the right of the number with a space, e.g., T_arr = 12.345 s; omit units for dimensionless quantities (optionally annotate 1 for disambiguation)

XVII. Quick Mapping (key symbol → unit, dimstr)

• c_ref → "m s^-1", "[L][T]^-1"
• n_eff → "1", "[]"
• gamma(ell) / L_gamma → "m", "[L]"
• T_arr → "s", "[T]"
• T_fil → "N", "[M][L][T]^-2"
• T_trans → "1", "[]"
• rho → "kg m^-3", "[M][L]^-3"