Editing Zero-point energy (section)

=== Fine-structure constant ===
{{Main|Fine-structure constant}}

Taking {{mvar|ħ}} (the [[Planck constant]] divided by {{math|2π}}), {{mvar|c}} (the [[speed of light]]), and {{math|''e''<sup>2</sup> {{=}} {{sfrac|''q''{{su|b=''e''|p=2}}|4π''ε''<sub>0</sub>}}}} (the electromagnetic [[coupling constant]] i.e. a measure of the strength of the [[electromagnetic force]] (where {{math|''q<sub>e</sub>''}} is the absolute value of the [[Electron charge|electronic charge]] and <math>\varepsilon_0</math> is the [[vacuum permittivity]])) we can form a dimensionless quantity called the [[fine-structure constant]]:
<math display="block">\alpha = \frac{e^2}{\hbar c} = \frac{q_e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137}</math>

The fine-structure constant is the coupling constant of quantum electrodynamics (QED) determining the strength of the interaction between electrons and photons. It turns out that the fine-structure constant is not really a constant at all owing to the zero-point energy fluctuations of the electron-positron field.{{sfnp|Le Bellac|2006|p=33}} The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it.

The Heisenberg inequality where {{math|''ħ'' {{=}} {{sfrac|''h''|2π}}}}, and {{math|Δ<sub>''x''</sub>}}, {{math|Δ<sub>''p''</sub>}} are the standard deviations of position and momentum states that:
<math display="block">\Delta_x\Delta_p\ge\frac{1}{2}\hbar</math>

It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine-structure constant is an increasing function of energy. It has been shown that at energies of the order of the [[Z boson|Z<sup>0</sup> boson]] rest energy, {{math|''m<sub>z</sub>c''<sup>2</sup> ≈}} 90&nbsp;GeV, that:
<math display="block">\alpha\approx\frac{1}{129}</math>
rather than the low-energy {{math|''α'' ≈ {{sfrac|1|137}}}}.<ref>{{cite book|last1=Aitchison|first1=Ian|last2=Hey|first2=Anthony|title=Gauge Theories in Particle Physics: A Practical Introduction: Volume 1: From Relativistic Quantum Mechanics to QED|date=2012|publisher=CRC Press|isbn=9781466512993|page=343|edition=4th}}</ref><ref>{{cite book|last1=Quigg|first1=C|editor1-last=Espriu|editor1-first=D|editor2-last=Pich|editor2-first=A|title=Advanced School on Electroweak Theory: Hadron Colliders, the Top Quark, and the Higgs Sector|date=1998|publisher=World Scientific|isbn=9789814545143|page=143}}</ref> The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining {{mvar|α}}. All in all, {{mvar|α}} depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of {{mvar|α}} has been observed for several years now in precision experiment in high-energy physics.