# Named Quantities — One Theorem, Many Names

*Kramers–Kronig is a theorem about causal linear response. The same
theorem is rediscovered and renamed in every discipline where such
responses appear — this chapter is a field guide to the aliases.*

## Electric susceptibility χ_e

The setting in which the bulk K–K pair was eventually written down
(conjectured by Kramers 1927 for `ε − 1`, derived from causality alone
by Kronig 1942; see Chapter 2). Relates the induced polarisation P to
the applied electric field E:

```
P(ω) = ε₀ χ_e(ω) E(ω),    χ_e(ω) = χ' + iχ''   (physics convention)
```

- **χ'** — in-phase polarisation; stored reactive energy per cycle.
- **χ''** — quadrature polarisation; energy absorbed per cycle. In
  physics convention, χ'' > 0 for ω > 0 in a passive medium.

Standard K–K:
```
χ'(ω) = (2/π) P ∫₀^∞ ω′ χ''(ω′) / (ω′² − ω²) dω′
χ''(ω) = −(2/π) P ∫₀^∞ ω  χ'(ω′) / (ω′² − ω²) dω′
```

## Permittivity ε(ω) and relative permittivity ε_r

Related to susceptibility by `ε(ω) = ε₀(1 + χ_e(ω))`. Decomposition
`ε_r = ε'_r + iε''_r` (physics) or `ε' − jε''` (engineering). The
K–K relation is usually written with the offset:

```
ε'_r(ω) − ε_∞ = (2/π) P ∫₀^∞ ω′ ε''_r(ω′) / (ω′² − ω²) dω′
```

One subtraction is needed because χ_e → 0 at high ω but ε_r → 1 = ε_∞,
so the "1" is subtracted before applying K–K.

**Loss tangent**: `tan δ ≡ ε''_r / ε'_r`, a dimensionless figure of
merit used widely in RF and microwave engineering.

## Refractive index ñ = n + iκ

Complex refractive index, with n the phase index and κ the extinction
coefficient. At optical frequencies where magnetic response is
negligible, `ñ² = ε_r`, so ñ is the square-root of a K–K-compliant
function — and it is itself K–K-compliant under the same passivity and
causality conditions. The usual form (physics):

```
n(ω) − 1 = (2/π) P ∫₀^∞ ω′ κ(ω′) / (ω′² − ω²) dω′
κ(ω) = −(2ω/π) P ∫₀^∞ [n(ω′) − 1] / (ω′² − ω²) dω′
```

One subtraction again, because `n → 1` at high ω.

## Magnetic susceptibility χ_m, permeability μ(ω)

`B = μ H`, `M = χ_m H`, `μ = μ₀(1 + χ_m)`. Same K–K structure as the
electric case with `μ_r → 1` at high ω. Sensible mainly at frequencies
where the material actually responds magnetically (ferromagnetic
resonance, spin waves, low-frequency permeability); above microwave
frequencies most materials have `μ_r ≈ 1` and the K–K integral is
dominated by low-frequency contributions.

## Conductivity σ(ω)

`J(ω) = σ(ω) E(ω)`. Decomposition `σ = σ' + iσ''`. σ' is the
dissipative (Ohmic) part, σ'' the reactive part. Conductivity and
permittivity are two descriptions of the same physics:

```
ε(ω) = ε_∞ + i σ(ω) / (ε₀ ω)
```

A lossy dielectric can be described either as ε'' ≠ 0 or as σ' ≠ 0 at
that frequency — the choice is stylistic. In plasmas and metals, σ(ω)
is usually primary; in insulators, ε(ω) is. K–K applies to either
representation. Because σ(ω)/ω → 0 at high ω faster than 1/ω, the σ
version sometimes needs a different subtraction scheme than the ε
version.

## Impedance Z(ω) and admittance Y(ω)

Circuits and networks:

```
Z(ω) = R(ω) + jX(ω),    Y(ω) = 1/Z(ω) = G(ω) + jB(ω)
```

R, G are resistive (dissipative); X, B are reactive (storage). A
passive one-port obeys the **positive-real** condition: Z(s) analytic
and with Re Z ≥ 0 for Re s ≥ 0. This implies K–K between R(ω) and X(ω),
and separately between G(ω) and B(ω):

```
X(ω) = −(2ω/π) P ∫₀^∞ [R(ω′) − R_∞] / (ω′² − ω²) dω′
```

with R_∞ the high-frequency resistance (often zero or the DC series
resistance, depending on topology).

## Rheological moduli — storage G' and loss G''

For a linear viscoelastic material under oscillatory shear:

```
σ_shear(ω) = G*(ω) γ(ω),   G*(ω) = G'(ω) + iG''(ω)
```

G' is the elastic storage modulus, G'' the loss modulus. The
stress–strain relation is causal, so G* obeys K–K. tan δ = G''/G' is the
rheological loss tangent, directly analogous to the dielectric one.
Polymer physics, soft-matter rheology, and seismology all rely on this
identification; the underlying maths is identical to the dielectric
case with the relabelling `ε ↔ G*`.

## S-parameters S_ij(ω)

Microwave scattering parameters. For a passive, causal N-port network,
`S(s)` is analytic in Re s > 0, `I − S†S` is positive-semidefinite on
the imaginary axis (passivity), and each S_ij(ω) obeys the Bode
gain–phase K–K relation up to non-minimum-phase factors arising from
right-half-plane zeros (e.g. transmission-line delays, all-pass
sections).

```
∠S_ij(ω) = −(2ω/π) P ∫₀^∞ ln|S_ij(ω′)| / (ω′² − ω²) dω′    (minimum-phase part only)
```

## Transfer function H(ω) — Bode gain–phase theorem

For any minimum-phase causal linear filter, `ln H(ω) = ln|H(ω)| + iφ(ω)`
is itself analytic in the UHP (up to the requirement that H has no zeros
there — the "minimum-phase" qualifier). Applying K–K to `ln H` yields:

```
φ(ω) = −(2ω/π) P ∫₀^∞ ln|H(ω′)| / (ω′² − ω²) dω′
```

This is the **Bode gain–phase theorem**. It is how an analogue-filter
designer knows the group delay of a Butterworth response before
simulating it: the magnitude response alone determines it. Non-minimum-
phase elements (all-pass networks, transport delays, acausal digital
filters) violate this bound in the expected direction — they add phase
lag without changing magnitude.

## Scattering amplitude f(ω) in particle physics

Forward-scattering amplitude for a particle scattering off a fixed
target. Causality + unitarity + crossing symmetry imply **dispersion
relations** of the K–K type, often with one or two subtractions
needed because f does not vanish at infinity:

```
Re f(ω) = f(0) + (2ω²/π) P ∫_{ω_thr}^∞ Im f(ω′) / [ω′(ω′² − ω²)] dω′
```

Combined with the **optical theorem** (`Im f(ω) ∝ σ_tot(ω)`), this gives
the classic dispersion relation linking the forward real amplitude to
an integral over the total cross-section. Used heavily in hadronic
physics, and one of the earliest places K–K was promoted from an optics
result to a general principle.

## Seismic Q-factor and complex velocity

In an attenuating Earth, the wave velocity is complex:

```
1/c(ω) = 1/c_∞ + (correction)(ω),   Q⁻¹(ω) = 2 Im[1/c(ω)] · c(ω)
```

Attenuation (Q⁻¹) and dispersion (frequency-dependent phase velocity)
are a K–K pair. The Futterman (1962) and Kjartansson (1979) models are
both explicit minimum-phase reconstructions of c(ω) from a postulated
Q(ω). Without K–K, a seismologist choosing Q freely would introduce
acausal wave propagation — signals arriving before the source event.

## Summary — the same theorem wearing different hats

| Field | Quantity | Dissipative part | Reactive part |
|---|---|---|---|
| Optics | ñ | κ | n − 1 |
| Dielectrics | ε_r | ε''_r | ε'_r − ε_∞ |
| Magnetics | μ_r | μ''_r | μ'_r − 1 |
| Plasmas / metals | σ | σ' | σ'' |
| Circuits | Z | R − R_∞ | X |
| Rheology | G* | G'' | G' − G_∞ |
| Microwave | S_ij | ln|S_ij| (for min-phase) | ∠S_ij |
| Control/filter | ln H | ln|H| | φ |
| Particle physics | f(ω) | Im f | Re f − f(0) |
| Seismology | 1/c | Q⁻¹ | 1/c − 1/c_∞ |

Every row of this table is a causal linear response. Every row has the
same integral lurking behind it. The apparent diversity of the K–K
literature is really the diversity of naming conventions for a single
piece of complex analysis.

## See also

- [16_sign_conventions.md](16_sign_conventions.md)
- [17_graphical_conventions.md](17_graphical_conventions.md)