← Back to article

Equation 75 · The Atlas That Refuses to Close

What does this equation mean?

x^\hat{\mathbf x}

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

x^\hat{\mathbf x}

Symbol hatmathbf x

hatmathbf x is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Understand this part →

How to interpret it

Read this expression with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

set purely by the rest mass. For an electron this is ωZ\omega_Z ≈\approx 1.55×\times10^{21}\ rad s−1\mathrm{rad\,s^{-1}} , an oscillation period of roughly four zeptoseconds in the particle’s own rest frame — a boosted lab observer would clock a longer, time-dilated period by the ordinary factor γ\gamma , coordinate time and proper time parting ways in exactly the usual manner — with an amplitude of order the reduced Compton wavelength ℏ\hbar/(mem_ec) , the same length scale, twice ρM\rho_M for a spin-half particle, that bounded the Møller disk above. Schrödinger first noticed this trembling motion, zitterbewegung, in the Dirac coordinate’s equations of motion; Foldy and Wouthuysen’s 1950 canonical…
Read the full surrounding passage
set purely by the rest mass. For an electron this is ωZ\omega_Z ≈\approx 1.55×\times10^{21}\ rad s−1\mathrm{rad\,s^{-1}} , an oscillation period of roughly four zeptoseconds in the particle’s own rest frame — a boosted lab observer would clock a longer, time-dilated period by the ordinary factor γ\gamma , coordinate time and proper time parting ways in exactly the usual manner — with an amplitude of order the reduced Compton wavelength ℏ\hbar/(mem_ec) , the same length scale, twice ρM\rho_M for a spin-half particle, that bounded the Møller disk above. Schrödinger first noticed this trembling motion, zitterbewegung, in the Dirac coordinate’s equations of motion; Foldy and Wouthuysen’s 1950 canonical transformation is what removes it, by transforming to exactly the representation in which the position operator is the Newton–Wigner operator and the oscillating term is absorbed into the difference between x^\hat{\mathbf x} and XNW\mathbf X_{\mathrm{NW}} [ 3 ] .

Read the equation in its article →

Sources cited in the surrounding passage

These citations give research context. Read each source to check which claims it supports.

Return to The Atlas That Refuses to Close

Browse the mathematical compendium →