Abstract
A refractive index below one is explained as permitted because it does not violate causality. This paper shows that causality fixes the shape of the refraction uniquely, and $n<1$ follows from it—not permission but compulsion. No new theorem or law is claimed.
First, reconstructing $\chi'$ from $\chi''$ alone matches the exact solution to a relative difference of $\mathbf{2.122\times10^{-4}}$ (Section 2).
Second, the sum rule gives $0.471088897$ against $(\pi/2)\omega_{p}^{2}=0.471238898$ (Section 2).
Third, this is the core of the paper. On one and the same yardstick, an acausal response misses by $\mathbf{1.000000}$ and a causal one by $\mathbf{0.004775}$ (Section 3).
Fourth, the reconstructed index reaches $n=\mathbf{0.993741246}$ at high frequency and drops below one—a phase velocity of $\mathbf{1.006298173} c$ (Section 4).
Fifth, the same expression gives $n=0.999999999999850$ as $\omega\to\infty$ and $n(0.1)=1.141449377$ at low frequency (Section 4).
Sixth, the separator is whether causality permits or compels (Section 5).
Keywords: refractive index, Kramers-Kronig relations, causality, dispersion relation, Lorentz oscillator.
Scope (what is not claimed)
No new theorem or law is claimed—the Kramers-Kronig relations, the Lorentz oscillator and the sum rule are all standard. Relativity is not discussed—that a phase velocity above $c$ carries no signal above $c$ belongs to Papers 210 and 407; only the mechanism is written here. One model only—a single Lorentz oscillator is used. Real materials have many resonances and the values change. Quantum theory is not entered—everything is written within linear response; microscopic derivations and the origin of dissipation are not the subject. The acausal example is not physics—the time-symmetric response $e^{-|t|}$ is not any real material’s response but a control, present so that the test can be seen to react. Relation to earlier papers: Paper 407 wrote in its scope note that a phase velocity above $c$ is known not to violate causality, and confined itself to group velocity—this paper takes the seat that line kept open. Paper 210 showed that the peak of a pulse can leave before it arrives, and that nothing about causality follows from that—this paper runs the other way, writing what does follow from causality. Paper 481 showed that without separators a refutation has no destination—this paper writes one of them. Paper 300 showed that a shared root can be decided—“permits” and “compels” are distinct roots. What is added is confirming the reconstruction at $2.122\times10^{-4}$, obtaining the sum rule independently, setting the acausal control beside the causal case at $1.000000$ against $0.004775$ on one yardstick, obtaining $n<1$ from the reconstruction, and checking both the high- and low-frequency sides.
1. Notation and the Question
Setting: Consider a linear response $\chi(\omega)=\chi'(\omega)+i\chi''(\omega)$, with $\chi'$ carrying refraction and $\chi''$ absorption. If the response is causal — no effect before its cause — the two are not independent but joined by
$$\chi'(\omega)=\frac{2}{\pi}\,\mathrm{P}\!\int_{0}^{\infty} \frac{\omega'\,\chi''(\omega')}{\omega'^{2}-\omega^{2}}\,d\omega'$$Below, a Lorentz oscillator $\chi(\omega)=\omega_{p}^{2}/(\omega_{0}^{2}-\omega^{2}-i\gamma\omega)$ is used with $\omega_{0}=1$, $\gamma=0.20$ and $\omega_{p}^{2}=0.30$.
Observation 1.1 A refractive index below one is explained as permitted because it does not violate causality. It happens because it is not forbidden—a picture that casts causality as the issuer of permits.
Observation 1.2 But the relation above writes a determination, not a permission. Give $\chi''$ and $\chi'$ follows—no room to choose is left.
Question: Does causality permit $n<1$, or compel it.
2. Refraction Follows from Absorption Alone
Computation 2.1: Give $\chi''$ alone, reconstruct $\chi'$ by principal-value integration, and compare with the exact solution.
| $\omega$ | Reconstructed | Exact | Relative difference |
|---|---|---|---|
| $0.50$ | $0.393096499$ | $0.393013100$ | $\mathbf{2.122\times10^{-4}}$ |
| $1.50$ | $-0.226668851$ | $-0.226928896$ | $1.146\times10^{-3}$ |
| $3.00$ | $-0.037263535$ | $-0.037290242$ | $7.162\times10^{-4}$ |
Observation 2.2 All three agree. Relative differences of $10^{-4}$ to $10^{-3}$—what remains comes from the step size of the principal-value integration, and this is not a fit at one point.
Observation 2.3 Even the sign is reproduced. Positive at $\omega=0.5$, negative at $1.5$ and $3.0$—the change of sign across the resonance follows from the shape of the absorption alone.
Observation 2.4 The sum rule agrees too. $\int_{0}^{\infty}\omega\chi''(\omega) d\omega=0.471088897$ against $(\pi/2)\omega_{p}^{2}=0.471238898$—an independent check through a different quantity.
Observation 2.5 So the shape of the absorption already holds the whole shape of the refraction. $\chi'$ follows from $\chi''$—the two are not separate properties but two faces of one function.
3. Measuring Causal and Acausal on One Yardstick
Computation 3.1: As a control, take the time-symmetric response $\chi(t)=e^{-|t|}$. Its Fourier transform is the real function $2/(1+\omega^{2})$, so $\chi''$ vanishes identically while $\chi'$ does not.
| Response | Miss between reconstruction and truth ($\omega=1$, threshold $0.1$) |
|---|---|
| Acausal (time-symmetric) | $\mathbf{1.000000}$ |
| Causal (Lorentz) | $\mathbf{0.004775}$ |
Observation 3.2 On the acausal side the relation breaks. $\chi''=0$ reconstructs $\chi'=0$, while the truth is $1.000000$—a miss ten times the threshold.
Observation 3.3 On the causal side it does not. $0.004775$—one twenty-first of the same threshold.
Observation 3.4 That the two were measured on the same yardstick is the point. Both at a threshold of $0.1$—apply different thresholds and any test yields whatever answer is wanted.
Observation 3.5 So the relation is a test of causality. Pass and the response is causal; fail and it is not—only the response function is measurable, yet a property of the direction of time can be read from it.
4. The Reconstructed Index Drops Below One
Computation 4.1: With $n=\sqrt{1+\chi'}$, tabulate index and phase velocity by frequency.
| $\omega$ | Index $n$ | Phase velocity $/c$ |
|---|---|---|
| $0.1$ | $1.141449377$ | $0.876$ |
| $5.0$ | $\mathbf{0.993741246}$ | $\mathbf{1.006298173}$ |
| $20.0$ | $0.999624027$ | $1.000376114$ |
| $10^{6}$ | $0.999999999999850$ | $1.000000000000$ |
Observation 4.2 At high frequency $n$ drops below one. $0.993741246$—the same side as the X-ray region treated in Paper 407.
Observation 4.3 The phase velocity there exceeds $c$. $1.006298173$ times—the speed of surfaces of constant phase, not the speed of a signal.
Observation 4.4 At low frequency $n>1$. $1.141449377$—the other side of the boundary, produced by the same expression.
Observation 4.5 As $\omega\to\infty$, $n$ returns to one. $0.999999999999850$—corresponding to a front travelling at $c$, so no signal exceeds it.
Observation 4.6 And this $n<1$ was not put in by choice. All that was given was $\chi''>0$, that is, that absorption exists—causality settled the rest.
5. The Separator
The separator: does causality permit $n<1$, or compel it.
Observation 5.1 Permitting, and a material without $n<1$ should be possible. Absorbing, yet refracting above one at every frequency—such a $\chi$ ought to be writable.
Observation 5.2 Compelling, and it is not. Fix $\chi''$ and $\chi'$ is fixed—and indeed the reconstruction offered no choice.
Observation 5.3 The two are distinct roots by the criterion of Paper 300. Both can say “consistent with causality”—one is the absence of a prohibition, the other a determination.
Observation 5.4 Paper 407 stopped just short of this. “Known not to violate causality”—this paper replaces that “does not violate” with “is fixed by”.
Observation 5.5 Paper 210 runs the other way. There, nothing about causality follows from the observation—here, the observation follows from causality. Two sides of one edge.
Observation 5.6 To be honest—only a single resonance is treated. Real materials have many—and every value changes. What does not change is the structure, that fixing $\chi''$ fixes $\chi'$; the numbers here are one instance of it.
6. Roll Call and Closing
- Reconstruction to $2.122\times10^{-4}$ (Computation 2.1): at $\omega=0.5$.
- Even the sign is reproduced (Observation 2.3): changing across the resonance.
- Sum rule $0.471088897$ (Observation 2.4): matching $(\pi/2)\omega_{p}^{2}$.
- Acausal misses by $1.000000$ (Computation 3.1): the core.
- Causal misses by $0.004775$ (Observation 3.3): on the same threshold.
- $n(5.0)=0.993741246$ (Computation 4.1): below one.
- Phase velocity $1.006298173$ (Observation 4.3): surfaces of constant phase.
- $n\to1$ as $\omega\to\infty$ (Observation 4.5): the front travels at $c$.
- The separator is permit or compel (Observation 5.3): distinct roots by Paper 300.
- One resonance only (Observation 5.6): the numbers are one instance.
Closing: A refractive index below one is explained as permitted because it does not violate causality—it happens because it is not forbidden, a picture that casts causality as the issuer of permits. But the Kramers-Kronig relation writes a determination, not a permission: give $\chi''$ and $\chi'$ follows, with no room to choose. Refraction follows from absorption alone—for a Lorentz oscillator, reconstructing $\chi'$ from $\chi''$ by principal-value integration gives $0.393096499$ against $0.393013100$ at $\omega=0.50$ (a relative difference of $\mathbf{2.122\times10^{-4}}$), $-0.226668851$ against $-0.226928896$ at $1.50$, and $-0.037263535$ against $-0.037290242$ at $3.00$. Even the sign is reproduced, the change from positive to negative across the resonance following from the shape of the absorption alone. The sum rule $\int_{0}^{\infty}\omega\chi'' d\omega=0.471088897$ matches $(\pi/2)\omega_{p}^{2}=0.471238898$, an independent check through a different quantity. So the shape of the absorption already holds the whole shape of the refraction: the two are two faces of one function. Measuring causal and acausal on one yardstick—the control is the time-symmetric response $\chi(t)=e^{-|t|}$, whose Fourier transform is the real $2/(1+\omega^{2})$, so $\chi''$ vanishes while $\chi'$ does not. At a threshold of $0.1$, the acausal side misses by $\mathbf{1.000000}$ and the causal side by $\mathbf{0.004775}$. That both were measured on the same yardstick is the point; apply different thresholds and any test yields whatever answer is wanted. So the relation is a test of causality: only the response function is measurable, yet a property of the direction of time can be read from it. The reconstructed index drops below one—with $n=\sqrt{1+\chi'}$, at $\omega=5.0$ the index is $\mathbf{0.993741246}$ and the phase velocity $\mathbf{1.006298173} c$, the same side as the X-ray region of Paper 407, and what exceeds $c$ is the speed of surfaces of constant phase, not of a signal. At low frequency $n(0.1)=1.141449377$ lies above one, the other side of the boundary produced by the same expression. As $\omega\to\infty$, $n=0.999999999999850$ returns to one, corresponding to a front travelling at $c$. And this $n<1$ was not put in by choice: all that was given was that absorption exists, and causality settled the rest. One thing separates them—does causality permit $n<1$, or compel it. Permitting, and an absorbing material refracting above one at every frequency ought to be writable. Compelling, and it is not: fix $\chi''$ and $\chi'$ is fixed, and the reconstruction offered no choice. They are distinct roots by the criterion of Paper 300—both can say “consistent with causality”, but one is the absence of a prohibition and the other a determination. Paper 407 stopped just short, writing “known not to violate causality”; this paper replaces that with “is fixed by”. Paper 210 runs the other way: there nothing about causality follows from the observation, here the observation follows from causality. Two sides of one edge. To be honest—only a single resonance is treated. Real materials have many, and every value changes. What does not change is the structure, that fixing $\chi''$ fixes $\chi'$; the numbers here are one instance of it.
References
- Kramers, H. A. (1927). La diffusion de la lumière par les atomes. Atti del Congresso Internazionale dei Fisici, Como.
- Kronig, R. de L. (1926). On the Theory of Dispersion of X-Rays. Journal of the Optical Society of America.
- Toll, J. S. (1956). Causality and the Dispersion Relation: Logical Foundations. Physical Review.
- Yamagishi, Y. (2026). The Peak of a Pulse Can Leave Before It Arrives [Paper 210]. Zenodo. 10.5281/zenodo.22111747
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). For X-Rays the Refractive Index Is Less Than One and the Phase Velocity Exceeds $c$ [Paper 407]. Zenodo. 10.5281/zenodo.22699506
- Yamagishi, Y. (2026). A Broken Law Can Be Blamed on Any of the Assumptions [Paper 481]. Zenodo. 10.5281/zenodo.22733927
Reference
Yamagishi, Y. (2026). The Refractive Index Drops Below One Not Because Causality Permits It but Because Causality Forces It [Paper 489]. YOI: https://www.yamagishiyuuki.com/yoi/489
DOI: 10.5281/zenodo.22760016
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 489. This paper is a preprint and has not been peer reviewed.