# Sign Conventions — Physics vs Engineering *Why the same K–K relation appears with a `+i` in one textbook and a `−i` in the next, and how to read the sign to identify which community wrote the formula.* ## The root of the confusion Every statement about the Kramers–Kronig relations sits downstream of a Fourier-transform convention. The physics and engineering literatures use **opposite signs in the complex exponential**, and this single choice propagates through the entire formalism: which half-plane a causal response is analytic in, which sign Im χ(ω) takes for absorption, and which sign appears inside the K–K integrals themselves. Nothing about the underlying physics changes — absorption is still positive, causal responses still decay — but every printed equation flips in a predictable way. ## The two conventions side by side | Item | Physics convention | Engineering convention | |---|---|---| | Forward Fourier kernel | `e^{−iωt}` | `e^{+jωt}` | | Inverse transform kernel | `e^{+iωt}` | `e^{−jωt}` | | Imaginary unit symbol | `i` | `j` (to avoid clash with current) | | Causal signal `t > 0` → analytic in | **Upper** half ω-plane (Im ω > 0) | **Lower** half ω-plane (Im ω < 0), or Re s > 0 in Laplace | | Typical damped oscillation | `e^{−iω₀ t} e^{−γt}`, pole at ω = ω₀ − iγ (LHP pole of χ) | `e^{+jω₀ t} e^{−γt}`, pole at ω = ω₀ + jγ (still "stable side") | | Absorption convention | `ε = ε' + iε''` with ε'' > 0 ⇒ absorption | `ε_r = ε' − jε''` with ε'' > 0 ⇒ absorption | | Phasor for a wave | `e^{i(kz − ωt)}` | `e^{j(ωt − γz)}` with γ = α + jβ | | Typical texts | Jackson, Griffiths, Landau–Lifshitz, most QM/optics | Oppenheim–Schafer, Pozar, Van Valkenburg, most EE/DSP | The rule is: **flip every `i` to `−j`** (or equivalently every `j` to `−i`) to move between them. Any real physical observable — a power, an absorption coefficient, a group delay — comes out the same. ## The Laplace s-plane: a 90° rotation of the ω-plane A third complication, called out explicitly by Bechhoefer (2011, footnote 16), is that the engineering literature usually does its complex analysis in the **Laplace s-plane** rather than in the Fourier ω-plane. K–K, Bode, minimum-phase and stability are all most naturally written in *whichever* of these planes a given community has adopted, and the two pictures are related by a 90° rotation that sends "upper/lower half of ω" to "right/left half of s". The Laplace transform of a causal `f(t)` is ``` F(s) = ∫₀^∞ f(t) e^{−st} dt, s = σ + jΩ ∈ ℂ ``` with σ = Re s and Ω = Im s both real. The integral converges absolutely when σ > 0 (because then `|e^{−st}| = e^{−σt}` decays as `t → ∞`), so **a causal `f(t)` has `F(s)` analytic in the right half s-plane** (Re s > 0). Restricted to the imaginary s-axis, `s = jΩ`, the Laplace transform becomes the engineering Fourier transform with kernel `e^{−jΩt}`, with Ω the real angular frequency. The bridge to the ω-plane is the relation `s = jω`. Treating ω as a complex variable `ω = ω_r + iω_i` (so `i` and `j` denote the same imaginary unit), this gives ``` s = jω = j(ω_r + iω_i) = −ω_i + jω_r, so σ = −ω_i, Ω = ω_r. ``` Multiplication by `j` is a +90° rotation of the complex plane, which is why "ω-plane" and "s-plane" pictures of the same situation differ by a 90° turn: | Half-plane in s | corresponds to (via s = jω) | engineering Fourier ω-plane | |---|---|---| | Right half (Re s > 0, σ > 0) | ↔ | Lower half (Im ω = ω_i < 0) | | Left half (Re s < 0, σ < 0) | ↔ | Upper half (Im ω = ω_i > 0) | | Imaginary axis (s = jΩ) | ↔ | Real axis (ω = ω_r) | So in **engineering**, the same statement "causal → analytic for Re s > 0" reads as "causal → analytic in the lower half ω-plane". Both hold simultaneously and refer to the same fact, just expressed in coordinates rotated by 90° from one another. Stability, similarly, is "poles in Re s < 0" (left half s) = "poles in Im ω > 0" (upper half ω, in the engineer's sign convention). Going from engineering to physics is then just the **kernel-sign flip** already discussed (every `j` becomes `−i`). Under that flip the role of the two halves of the ω-plane swap, and the canonical physics statements emerge: | Statement | Physics (`e^{+iωt}`) | Engineering Fourier (`e^{−jωt}` for the Laplace-compatible kernel) | Engineering Laplace | |---|---|---|---| | Causal → analytic in | UHP of ω (Im ω > 0) | LHP of ω (Im ω < 0) | RHP of s (Re s > 0) | | Stable → poles in | LHP of ω (Im ω < 0) | UHP of ω (Im ω > 0) | LHP of s (Re s < 0) | | Minimum-phase → zeros in | LHP of ω | UHP of ω | LHP of s | Reading across the row, every "UHP of physics ω" corresponds to an "RHP of engineering s", and every "LHP of physics ω" to an "LHP of engineering s" — the same theorem each time, expressed in whichever plane the writer has chosen. Bode's gain-phase theorem (Chapter 20), in particular, is most cleanly stated in the s-plane: `log H(s)` analytic for Re s > 0 (no RHP poles or zeros) forces a Hilbert pair between `ln|H(jΩ)|` and `∠H(jΩ)` on the imaginary axis. Reading the same statement back into the engineering ω-plane via `s = jΩ` recovers the gain-phase integral on the real-frequency axis, and reading it back into the physics ω-plane recovers the K–K relations on `log χ(ω)` of Chapter 19. When chasing a sign in a paper, then, ask three questions in order: *which kernel*, *which plane (ω or s)*, *which half does the writer call analytic*. Any two of those three pin down the third, and the 90° rotation makes the engineering ω↔s translation routine once the kernel is known. ## K–K in each convention In the **physics** convention, causality ⇒ χ(ω) analytic in the upper half-plane, closure of the Cauchy contour is in the UHP, and the relations read (principal-value `P` understood): ``` χ′(ω) = (1/π) P ∫_{−∞}^{∞} χ″(ω′) / (ω′ − ω) dω′ χ″(ω) = −(1/π) P ∫_{−∞}^{∞} χ′(ω′) / (ω′ − ω) dω′ ``` The minus sign in the second relation and the fact that absorption (χ″>0) produces an odd-about-ω₀ dispersion (χ′) around a resonance both follow from UHP analyticity. In the **engineering** convention, the causal transfer function H(jω) is analytic for Re s ≥ 0, i.e. the *lower* half of the ω-plane (because s = jω means Im ω < 0 ↔ Re s > 0). Closing the Cauchy contour in the LHP flips the sign of the integral, and the relations read: ``` Re H(jω) = −(1/π) P ∫_{−∞}^{∞} Im H(jω′) / (ω′ − ω) dω′ Im H(jω) = (1/π) P ∫_{−∞}^{∞} Re H(jω′) / (ω′ − ω) dω′ ``` Every sign on the right-hand side is the opposite of the physics form. This is the **same theorem**. If you substitute the engineering definition `ε_r = ε' − jε''` into the engineering K–K and the physics definition `ε_r = ε' + iε''` into the physics K–K, both predict that a Lorentzian absorption peak at ω₀ (positive ε'' in their own convention) gives the textbook anomalous-dispersion wiggle in ε' centred on ω₀. ## Reading the sign to identify the community A practical habit: when you open a paper and find a K–K-type integral, **check the sign in front of the `(1/π) P ∫`** and the sign in the decomposition `ε = ε' ± iε''`. If the text writes `ε_r = ε' − jε''` and refers to "j" throughout, it is an engineering/microwave/DSP source and is using `e^{+jωt}`. If it writes `ε = ε' + iε''` with "i", it is almost certainly physics/optics/solid-state and using `e^{−iωt}`. Papers in electrochemical impedance spectroscopy mostly follow the engineering convention; papers in optical constants and x-ray dispersion mostly follow the physics convention; dielectric-spectroscopy papers are split and routinely state their convention in the first paragraph for exactly this reason. ## The absorption-sign trap The most dangerous manifestation of the convention choice is the **sign of the imaginary part for a lossy material**. In the physics convention a passive, absorbing medium has `ε'' > 0` (so that a plane wave `e^{i(kz − ωt)}` with `k = (ω/c)√ε` decays in the +z direction). In the engineering convention with `e^{j(ωt − γz)}`, the phasor decays in +z when `Im{√ε_r} < 0`, which is why the convention `ε_r = ε' − jε''` with ε'' > 0 is used: the explicit minus sign absorbs the flip so that "ε'' positive means lossy" remains the mnemonic in both communities. A symptom of mixed conventions inside a single paper is a passive material appearing to have a *negative* imaginary part — or, worse, appearing to be gain medium. Nine times out of ten this is a convention error, not new physics. The fix is to pick one convention, write the Fourier kernel in the preamble, and stick with it. ## Passivity in each convention The passivity constraint is convention-independent in content but convention-dependent in form: - Physics: `Im χ(ω) ≥ 0` for ω > 0 in an absorptive medium (typical K–K-compliant form). The analytic continuation χ(ω + iη) has no poles or zeros in Im ω > 0 for a passive, minimum-phase response. - Engineering: `Re H(jω) ≥ 0` for a positive-real (passive) immittance function (Brune's condition), with poles and zeros restricted to the left half s-plane. The transfer function is analytic for Re s > 0. Both conditions encode the same physical fact — no energy is produced from nothing — and both together with causality imply K–K. ## Worked example — Lorentzian in both conventions Take a single damped oscillator with ω₀ = 1, γ = 0.1 (units arbitrary), and evaluate at ω = 1.2. The complex susceptibility is the same physical quantity written two different ways. **Physics** convention `e^{−iωt}`, poles in LHP: ``` χ_phys(ω) = 1 / (ω₀² − ω² − iγω) = 1 / (1 − 1.44 − i·0.1·1.2) = 1 / (−0.44 − 0.12 i) = (−0.44 + 0.12 i) / (0.44² + 0.12²) = (−0.44 + 0.12 i) / 0.2080 ≈ −2.115 + 0.577 i ``` χ″ = +0.577 > 0 ⇒ absorptive (loss), as expected above resonance in a passive medium. **Engineering** convention `e^{+jωt}`. The same physics with the kernel flipped gives `χ_eng(ω) = χ_phys*(ω) = 1 / (ω₀² − ω² + jγω)`: ``` χ_eng(ω) = 1 / (−0.44 + j·0.12) = (−0.44 − 0.12 j) / 0.2080 ≈ −2.115 − 0.577 j ``` Now χ″_eng = −0.577, which *looks like* gain. It is not: the minus sign is the Fourier-kernel flip, not new physics. Some engineering texts anticipate this and define `ε = ε′ − jε″` with ε″ > 0 for loss, so the reported number is `+0.577` in the end. Whenever you see a negative imaginary part in a dielectric dataset, check the kernel before calling the vendor: the same Lorentzian pole at `(ω₀, γ) = (1, 0.1)` sits in the LHP (physics) or in the UHP (engineering), and the sign of the imaginary part of χ at every real ω flips accordingly. The power dissipated per unit volume in an oscillating field `E = E₀ cos(ωt)` is ``` 〈P〉 = (1/2) ε₀ ω |E₀|² · (physics ε″) = (1/2) ε₀ ω |E₀|² · (engineering ε″) [with the ε″>0-for-loss convention] ``` Both conventions recover the same positive dissipation — the convention only changes sign-keeping, never physics. ## Quantum mechanics and transmission-line phasors Quantum mechanics universally uses `Ψ(x, t) ∝ e^{−iEt/ħ}`, i.e. the physics convention with E playing the role of ħω, and scattering amplitudes `f(ω)` inherit UHP analyticity and physics-convention K–K. Transmission-line theory uses `e^{j(ωt − γz)}` with `γ = α + jβ`, and the complex propagation constant `γ(ω)` obeys engineering-convention K–K between attenuation α(ω) and phase β(ω) — this is the Bode gain–phase theorem in disguise, and it is the reason minimum-phase filters have their group delay pinned by their magnitude response. When working across both worlds — e.g. modelling a microwave cavity that couples to a quantum two-level system — declaring the convention once, loudly, at the top of the calculation saves hours of sign-chasing. ## See also - [01_introduction.md](01_introduction.md) - [17_graphical_conventions.md](17_graphical_conventions.md) - [18_named_quantities.md](18_named_quantities.md)