Abstract
A selection rule is written as “this transition is forbidden.”Yet forbidden transitions are actually observed. This paper asks what “forbidden” means as a number: the answer is a matter of rate. No new mathematical theorem and no new law is claimed.
First, we compare two lifetimes in the hydrogen atom. $2p\to1s$ (E1 allowed) is $1.596\times10^{-9}$ s; $2s\to1s$ (E1 forbidden) is $\mathbf{0.1216}$ s (Section 2).
Second, this is the core of the paper. The ratio is $\mathbf{7.6190\times10^{7}}$. It is not forbidden. It is merely slow (Section 2).
Third, the estimate can be written with the fine-structure constant. The E1-to-M1 ratio is $\alpha^2=\mathbf{5.325\times10^{-5}}$, and two-photon is $\alpha^4=2.836\times10^{-9}$ (Section 3).
Fourth, the $21$ cm line is slower still. Lifetime $3.47\times10^{14}$ s $=\mathbf{11}$ million years, $\mathbf{2.17\times10^{23}}$ times that of $2p$ (Section 4).
Fifth, it is nevertheless visible because of numbers. With $10^{60}$ hydrogen atoms, $\mathbf{2.88\times10^{45}}$ emit each second (Section 5).
Sixth, the separator is “exactly zero, or merely small”. What a selection rule sets to zero is a matrix element within one approximation, not the total rate (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the E1 selection rules, two-photon emission, hyperfine structure, the lifetime of the $21$ cm line, and the $\alpha^2$ estimate are all standard. We do not build quantum electrodynamics: all we use is three lifetimes and two divisions. We do not derive the selection rules: we do not enter the Wigner--Eckart theorem. That is the side of Paper 140. We do not compute transition probabilities: $1.596\times10^{-9}$, $0.1216$, and $3.47\times10^{14}$ are merely quoted from the literature, and this paper only takes ratios. We do not say $\alpha^2$ is an exact ratio: the actual E1-to-M1 ratio moves by orders with the matrix elements. $\alpha^2=5.325\times10^{-5}$ is an estimate, not a prediction. We do not say selection rules are meaningless: a factor of $7.6\times10^{7}$ is indistinguishable from prohibition in practice in many settings. What this paper says is only that “forbidden” and “slow” have different standings. We do not discuss interstellar physics: the number $10^{60}$ is a representative value for showing orders, and cloud mass, temperature, and excitation mechanism are not treated at all. Relation to earlier papers: Paper 140 treated selection rules from the group-theoretic side, showing that symmetry fixes ratios. This paper is on the rate side, writing the same selection rules as lifetimes. It treats the place where a theorem’s zero set is not in fact zero. Paper 190 measured “rare” on a logarithmic scale: the $23.34$ decades here must likewise be read logarithmically. Paper 272 showed that one and the same “doubling” opens by a factor of $6.70$ on the stimulus side. This paper is the case where one and the same electromagnetic transition opens by $2.17\times10^{23}$ on the rate side. Paper 269 showed that what a crystal forbids is not the symmetry but the periodicity. This paper likewise asks what the word “forbid” contains. There the prohibition is exact; here it is a matter of rate, and the two stand in contrast. What is added is lining up three transition lifetimes in one table and measuring a spread of $2.17\times10^{23}$, writing the $7.6190\times10^{7}$ of $2s\to1s$ as the numerical content of “prohibition”, filling in why the $21$ cm line is visible as “$10^{60}$ atoms give $2.88\times10^{45}$ per second”, and placing as the separator that what goes to zero is a matrix element, not a rate.
1. Notation and the Question
The electric-dipole (E1) selection rules are commonly written:
$$\Delta \ell=\pm1,\qquad \Delta m=0,\pm1$$A transition failing these is called “E1 forbidden.”
Question: Does “forbidden” mean it does not happen, or it is slow?
Observation 1.1 Hydrogen’s $2s$ level has $\ell=0$, and so does $1s$. Since $\Delta\ell=0$, it is E1 forbidden. Yet the $2s$ level does not live forever.
Observation 1.2 Paper 140 treated selection rules from the group-theoretic side: by the Wigner--Eckart theorem, a certain matrix element is exactly zero by symmetry. This paper treats what lies beyond. What remains outside the matrix element that went to zero.
2. The Core: A Factor of $7.6190\times10^{7}$
Check 2.1: We line up two lifetimes of the hydrogen atom.
| Transition | Type | Lifetime $\tau$ |
|---|---|---|
| $2p\to1s$ | E1 allowed | $1.596\times10^{-9}$ s |
| $2s\to1s$ | E1 forbidden (two-photon) | $\mathbf{0.1216}$ s |
| Ratio | $\mathbf{7.6190\times10^{7}}$ |
Observation 2.2 This is the core of the paper. $2s\to1s$ does not fail to happen; it takes $0.1216$ s. It is not forbidden. It is $7.62\times10^{7}$ times slower.
Observation 2.3 $0.1216$ s is a length a human can feel. Shorter than a blink. That is how slow the word “forbidden” is pointing at.
Observation 2.4 In logarithms it is $\mathbf{7.88}$ decades. As Paper 190 wrote, such quantities are read in orders, not differences: $7.88$ decades is large, but not infinite.
Observation 2.5 Why does it stay finite? Even when the one-photon E1 goes to zero, the road of emitting two photons remains: a selection rule closes one road, not every road.
3. The Estimate Can Be Written with the Fine-Structure Constant
Setting: There is a rough estimate of how much slower. Each step up in the order of radiation brings a power of $\alpha$.
Check 3.1:
| Quantity | Value | Meaning |
|---|---|---|
| $\alpha$ | $0.007297353$ | fine-structure constant |
| $\alpha^2$ | $\mathbf{5.325135\times10^{-5}}$ | estimate for M1 against E1 |
| $1/\alpha^2$ | $18778.8650$ | how many times slower |
| $\alpha^4$ | $2.835707\times10^{-9}$ | estimate for two-photon (E1E1) |
Observation 3.2 $1/\alpha^4=3.53\times10^{8}$. The measured $7.62\times10^{7}$ is one $4.6$th of that: the order agrees; the value does not.
Observation 3.3 A point to write honestly. This paper does not explain the factor of $4.6$. The powers of $\alpha$ are an estimate that only counts the order, while matrix elements and phase-space factors act separately: $\alpha^4$ says which decade, not which value.
Observation 3.4 Why it works as an estimate at all, because $\alpha$ is dimensionless. The same number for any atom and any system of units: only ratios are transferable (Paper 140).
4. The $21$ cm Line Is $10^{23}$ Times Slower Still
Setting: The hydrogen ground state has two levels according to the relative orientation of electron and proton spins (hyperfine structure). The transition between them is the $21$ cm line.
Check 4.1: We line up three transitions in one table.
| Transition | Type | Lifetime $\tau$ | Times that of $2p$ |
|---|---|---|---|
| $2p\to1s$ | E1 allowed | $1.596\times10^{-9}$ s | $1$ |
| $2s\to1s$ | E1 forbidden, two-photon | $0.1216$ s | $7.619\times10^{7}$ |
| $21$ cm | hyperfine, M1 | $3.47\times10^{14}$ s | $\mathbf{2.1742\times10^{23}}$ |
Observation 4.2 $3.47\times10^{14}$ s is $\mathbf{11}$ million years. One hydrogen atom emits this light on average once in $11$ million years: the probability per year is $\mathbf{9.09\times10^{-8}}$.
Observation 4.3 The total spread is $\mathbf{2.17\times10^{23}}$, that is $\mathbf{23.34}$ decades. This much, within the same hydrogen atom and the same electromagnetic transition: two words, allowed and forbidden, cover that entire span.
Observation 4.4 The $21$ cm line is M1. The $\alpha^2$ factor makes it $1.88\times10^{4}$ times slower, but most of the $23$ decades is not that: the transition energy is extremely small at $\mathbf{5.9\times10^{-6}}$ eV, and the rate goes as $E^3$.
5. It Is Visible Because of Numbers
Question: Why is something that emits once in $11$ million years visible to a radio telescope?
Check 5.1:
| Quantity | Value |
|---|---|
| Rate at which one atom emits, $1/\tau$ | $2.8818\times10^{-15}$ per second |
| Atoms needed for one emission per second | $\mathbf{3.47\times10^{14}}$ |
| Hydrogen atoms in an interstellar cloud (representative) | $\sim10^{60}$ |
| Emissions per second from that | $\mathbf{2.8818\times10^{45}}$ |
Observation 5.2 Only $3.47\times10^{14}$ atoms are needed for one emission per second. In mass that is $\mathbf{5.8\times10^{-13}}$ kg: not remotely a microgram.
Observation 5.3 $10^{60}$ hydrogen atoms weigh $1.67\times10^{33}$ kg, that is about $841$ solar masses. From that, $\mathbf{2.88\times10^{45}}$ photons emerge each second: once in $11$ million years for each atom, and $10^{45}$ per second for the whole.
Observation 5.4 So there is no mystery in “a forbidden transition being visible.”Even with a rate of $10^{-15}$, a count of $10^{60}$ gives a product of $10^{45}$. What decides visibility is not the rate but the product.
Observation 5.5 This is Paper 190 turned over. There, “rare” was measured logarithmically. Here, a rare event is multiplied by a large number and brought back to everyday orders.
6. Separator: What Goes to Zero Is the Matrix Element, Not the Rate
Observation 6.1 Separator. What is exactly zero, and what is merely small?
| Exactly zero | the matrix element of the E1 operator (by symmetry) |
|---|---|
| Merely small | the total rate of the transition (other roads remain) |
Observation 6.2 A selection rule speaks about the first row. But an experiment measures the second: the word “forbidden” is what makes the two be taken for one another.
Observation 6.3 Paper 269 showed that what a crystal forbids is not the symmetry but the periodicity. That prohibition is exact: a five-fold-symmetric periodic structure does not exist. The prohibition here is a matter of rate: the same “forbid” with a different standing.
Observation 6.4 Hence when reading a selection rule one must check inside which approximation it speaks. Within the E1 approximation it is exactly zero; step outside that approximation and it is finite: the zero is a property of the inside of a theory, not a property of nature.
7. Roll Call and Closing
- $2p\to1s$ is $1.596\times10^{-9}$ s (Check 2.1): E1 allowed.
- $2s\to1s$ is $0.1216$ s (Check 2.1): Shorter than a blink.
- The ratio is $7.6190\times10^{7}$ (Check 2.1): $7.88$ decades.
- $\alpha^2=5.325\times10^{-5}$ (Check 3.1): The M1 estimate.
- $1/\alpha^4=3.53\times10^{8}$ (Observation 3.2): Measurement is one $4.6$th.
- The $21$ cm line is $11$ million years (Check 4.1): $2.17\times10^{23}$ times $2p$.
- $23.34$ decades in total (Observation 4.3): Same atom, same electromagnetic transition.
- One per second needs $3.47\times10^{14}$ atoms (Check 5.1): $5.8\times10^{-13}$ kg.
- $10^{60}$ gives $2.88\times10^{45}$ per second (Check 5.1): $841$ solar masses.
- The zero is the matrix element (Observation 6.1): Not the rate.
Closing: “Forbidden,” as a number, means a matter of rate. Hydrogen’s $2s\to1s$ is E1 forbidden, yet it does not fail to happen; it takes $\mathbf{0.1216}$ s: $\mathbf{7.6190\times10^{7}}$ times, that is $7.88$ decades, slower than $2p\to1s$. The $21$ cm line is slower still, once in $11$ million years, $\mathbf{2.17\times10^{23}}$ times that of $2p$, and it is nevertheless visible not because the rate rose but because the count is large: $10^{60}$ atoms emit $\mathbf{2.88\times10^{45}}$ photons a second. One thing separates them: that what is exactly zero is not the rate but a matrix element within one approximation. The selection rule speaks about that matrix element; the experiment measures the total rate. The zero is a property of the inside of the E1 approximation, not a property of nature. So when a “forbidden” line is seen in the sky, no contradiction has occurred.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; section headings containing mathematics, which had been cut off in display, were repaired; no claim, number or reference was changed.
References
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- Yamagishi, Y. (2026). Symmetry Fixes the Ratios, Dynamics Fixes the Scale [Paper 140]. Zenodo. 10.5281/zenodo.22081997
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). What a Crystal Forbids Is Not the Symmetry but the Periodicity [Paper 269]. Zenodo. 10.5281/zenodo.22169855
- Yamagishi, Y. (2026). One and the Same “Twice as Strong” Opens by a Factor of 6.70 on the Stimulus Side [Paper 272]. Zenodo. 10.5281/zenodo.22169965
Reference
Yamagishi, Y. (2026). “Forbidden” Means “Slower by a Factor of $7.62\times10^{7}$” [Paper 285]. YOI: https://www.yamagishiyuuki.com/yoi/285
DOI: 10.5281/zenodo.22171179
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 285. This paper is a preprint and has not been peer reviewed.