Is the Fine-Structure Constant Really Constant?

The fine-structure constant, usually written $\alpha$, is often introduced as a fixed number close to $1/137$. That value is the low-energy value used in many atomic and everyday electromagnetic calculations. In quantum electrodynamics (QED), however, the effective electromagnetic coupling depends on the energy or momentum scale at which you probe it. This is called the running of the coupling.

The distinction is not that nature’s predictions change arbitrarily. Rather, the parameter used to describe electromagnetic interactions must account for quantum effects differently at different scales. When those effects are included consistently, observable predictions are independent of the calculation’s chosen convention.

What the fine-structure constant measures

In SI units, the low-energy fine-structure constant is defined by

$$\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c},$$

where $e$ is the elementary electric charge, $\epsilon_0$ is the vacuum permittivity, $\hbar$ is the reduced Planck constant, and $c$ is the speed of light. Its measured low-energy value is approximately $1/137.036$. Because $\alpha$ is dimensionless, its value does not depend on the choice of units in the way a dimensional quantity such as a length does.

In quantum field theory, physicists often write the electromagnetic interaction strength as $\alpha(\mu)$, where $\mu$ is a renormalization scale. It is a scale-dependent parameter in the theory’s calculation. The scale is commonly chosen to be comparable to the energy or momentum transfer involved in a process.

Why the effective value changes

The vacuum of QED is not empty in the classical sense. Quantum fluctuations include short-lived charged particle–antiparticle pairs. These act somewhat like a dielectric medium: they screen electric charge. At long distances, the measured charge includes the effect of that screening. At shorter distances, a probe gets closer to the charge and sees less of the screening, so the effective interaction is stronger.

This is why the electromagnetic coupling increases as the energy scale rises. In the leading-log one-loop approximation, considering only an electron contribution at scales well above the electron mass, the behavior has the form

$$\alpha(Q) \approx \frac{\alpha(0)}{1 - \frac{\alpha(0)}{3\pi}\ln\left(\frac{Q^2}{m_e^2}\right)},$$

where $Q$ is the momentum-transfer scale, $m_e$ is the electron mass, and $\alpha(0)$ is the low-energy value. This formula illustrates the trend, but it is not a precision formula for arbitrary energies: other charged particles contribute as the scale crosses their mass thresholds, and higher-order corrections also matter.

A concrete comparison

For low-energy atomic physics, the appropriate coupling is close to $\alpha(0) \approx 1/137.036$. At energies around the mass of the $Z$ boson, the effective electromagnetic coupling is commonly described as being roughly $1/128$. In other words, the coupling itself is larger at the higher scale; its reciprocal is smaller.

The number near $1/128$ is not a single convention-free replacement for $1/137$. Its precise value depends on how the running coupling is defined and on how particle contributions are treated. For example, an on-shell effective coupling and a coupling defined in the modified minimal-subtraction scheme, $\overline{\mathrm{MS}}$, need not have identical numerical values at the same scale. Both can be used correctly when paired with calculations in their respective schemes.

What “constant” means in practice

The name “fine-structure constant” is still useful. It refers to the dimensionless strength of electromagnetism, and the low-energy value is a well-defined measured quantity. In a quantum calculation, though, the effective coupling runs with the scale. Textbooks that use $1/137$ are generally working at energies where that low-energy value is the right approximation, or they include scale-dependent effects elsewhere in the calculation.

It is also useful to distinguish a running parameter from an observable. A measured cross section or atomic energy level is an observable; the value assigned to a renormalized coupling depends in part on the chosen scheme and scale. If you change the scheme or scale, other terms in the calculation change too, leaving the final prediction unchanged to the accuracy of the calculation.

Common mistakes

• Saying the constant changes with time: Running with energy scale is not, by itself, evidence that the fine-structure constant has changed over cosmic time. A time variation would be a separate physical claim about measured dimensionless quantities.
• Treating $1/128$ as the exact high-energy value everywhere: The effective coupling depends on the scale, definition, and charged particles that contribute.
• Thinking the low-energy value is wrong: It is the appropriate value for low-energy processes, including many atomic applications.
• Confusing a calculation convention with a physical ambiguity: Intermediate parameters can depend on convention; predictions for observables do not when the calculation is carried out consistently.

So the short answer is: the low-energy fine-structure constant is a fixed measured parameter, but the effective electromagnetic coupling in QED runs with energy. Higher-energy probes see a stronger coupling because they resolve more of the charge hidden by vacuum screening.