# Sum Rules

*Asymptotic expansion of the Kramers–Kronig integral at high and low frequency converts the two integral relations into a family of moment identities — the f-sum rule, superconvergence relations, and the static susceptibility — each a hard consistency check on any measured spectrum.*

## The idea

The one-sided K–K relation

```
χ′(ω) = (2/π) P ∫₀^∞  ω′ χ″(ω′) / (ω′² − ω²)  dω′
```

is an exact identity valid for every ω. Two limits collapse it onto
moment integrals:

- **ω → ∞.** Expand the denominator as `1/(ω′² − ω²) = −(1/ω²)·
  [1 + (ω′/ω)² + (ω′/ω)⁴ + …]`, integrate term by term, and match
  against the known high-frequency behaviour of χ′. Each power of 1/ω
  yields a sum rule on a moment of χ″.
- **ω → 0.** Evaluate the integrand at ω = 0 directly to get the
  static susceptibility χ′(0) as a weighted integral of χ″.

All of these are rigorous consequences of the analytic structure proved
in Chapters 11–12. They are experimentally useful because χ′(∞), χ′(0)
and the various moments are often known from independent physics (e.g.
plasma frequency from electron density, Drude model), so the sum rules
bind measured spectra to those external inputs.

## High-frequency expansion

Assume χ″(ω) vanishes fast enough for moments of appropriate order to
exist. Write, formally,

```
χ′(ω) = −(2/π ω²) · ∫₀^∞ ω′ χ″(ω′) [1 + (ω′/ω)² + …] dω′
      = −(2/πω²) M₁  −  (2/πω⁴) M₃  −  (2/πω⁶) M₅  −  …
```

where `M_{2k+1} = ∫₀^∞ ω′^{2k+1} χ″(ω′) dω′` is the (2k+1)-th moment of
χ″. Matching this against the known asymptotic form of χ′(ω) picks off
each moment in turn.

### The f-sum rule (Thomas–Reiche–Kuhn)

For an electronic medium, quantum mechanics gives a hard constraint:
the total oscillator strength summed over all transitions equals the
electron number density. In optical form, for the dielectric function
ε(ω) = 1 + 4π χ(ω) (Gaussian units),

```
∫₀^∞ ω ε″(ω) dω  =  (π/2) ω_p²
```

where `ω_p² = 4π n e² / m` is the plasma frequency squared and n is
the electron number density. Equivalently, for χ itself,
`∫₀^∞ ω χ″(ω) dω = (π/8) ω_p²`. This is the **f-sum rule** — the first
and most-used sum rule in condensed-matter and atomic physics — and it
comes straight out of matching the `1/ω²` term in the K–K expansion
above to the high-frequency limit ε′(ω) − 1 → −ω_p²/ω² (the free-
electron limit: at frequencies far above any bound-state transition,
electrons respond like a free gas).

Because the right-hand side depends only on the total number of
electrons, **the integral of ω·ε″(ω) over all frequencies is a
conservation law**: removing oscillator strength at one frequency
(e.g. by a material modification, or a phase transition) must be
compensated by redistribution elsewhere. This is one of the most
stringent consistency checks available for measured optical spectra.

### Higher-order / superconvergence relations

If χ′(ω) decays **faster** than the generic `1/ω²` at infinity — say it
vanishes faster than any polynomial, or vanishes asymptotically to all
orders — then matching the expansion order by order forces the
corresponding moments to be zero:

```
M₁ = ∫₀^∞ ω′ χ″(ω′) dω′  =  0     (if χ′(ω) → 0 faster than 1/ω²)
```

and so on for higher moments. These are the **superconvergence
relations**. In practice they apply in situations where the leading
asymptotic piece has been subtracted (see Chapter 15 on subtracted
dispersion) or when a physical argument pins χ′(∞) exactly to zero.
Superconvergence relations are a staple of particle physics — the
Adler–Weisberger sum rule and high-energy forward-scattering amplitude
relations are of this type — but the same logic applies to any response
function whose high-frequency tail is known.

## The static sum rule

Setting ω = 0 in the one-sided K–K relation,

```
χ′(0) = (2/π) ∫₀^∞  χ″(ω′) / ω′  dω′.
```

No principal value is needed: the integrand is regular at ω′ = 0
provided χ″(ω) vanishes at least linearly there (which is automatic for
a system with real h(t), since χ″ is odd in ω). This says the **static
susceptibility** — the DC response — is a weighted integral of the
entire absorption spectrum. Every band of absorption at every frequency
contributes positively (if χ″ > 0, as it is for dissipative systems) to
the static polarisability.

Practical corollary: if you measure the full absorption spectrum and
integrate χ″/ω, you predict the DC dielectric constant. A discrepancy
signals either an unmeasured absorption band or an instrument
calibration problem.

## Consistency-check template

Sum rules convert an abstract theorem into a concrete experimental
workflow:

1. **Measure** χ″(ω) across the widest achievable frequency range.
2. **Extrapolate** beyond the measured band using a physically
   motivated tail (Drude at high ω, something analytic at low ω).
3. **Compute** the relevant moments:
   - `∫ χ″/ω dω` should reproduce χ′(0).
   - `∫ ω χ″ dω` should reproduce the known plasma-frequency sum.
   - If `χ′(∞)` is known (e.g. 1 for refractive index), that too is
     a check on the extrapolations.
4. **Residual = problem.** If the sum rule fails by more than
   uncertainty, the data, the model tail, or the calibration is
   wrong. Sum rules do not forgive.

This is exactly how optical constants of opaque materials are validated
before being entered into reference databases (e.g. Palik, Handbook of
Optical Constants of Solids).

## Why they work — one-line summary

Sum rules are **moment integrals of a Hilbert-transform identity**.
They are not extra physics; they are the K–K identity re-read in a
different asymptotic regime. Every sum rule that exists is the
asymptotic shadow of the same UHP contour integral that produced the
dispersion relations in Chapter 12.

## See also

- [12_derivation_KK.md](12_derivation_KK.md)
- [13_hilbert_transform.md](13_hilbert_transform.md)
- [15_subtracted_dispersion.md](15_subtracted_dispersion.md)