← Mathematical compendium

Published equation contexts

⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v

Why this formula appears here

The two pictures are not independent theories that happen to share a symbol. The Galilei phase is the c→\to∞\infty residue of the Poincaré non-extension, and the residue can be taken explicitly rather than asserted. Write the energy operator for a system at rest as H=Mc2c^2+HNRH_{\rm NR} , separating the rest energy from whatever nonrelativistic energy — kinetic, internal — sits on top of it. The relativistic loop phase becomes ⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v. Holding b\mathbf b , v\mathbf v , M , and ⟨\langle HNRH_{\rm NR}⟩\rangle fixed and sending c→\to∞\infty , the second term vanishes and the first survives unchanged, converting a state-dependent relativistic phase into a fixed, universal constant multiplying every…

Read the full article-specific guide →

Read the representative guide

bb

Symbol b

b occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →
HNRH_{\rm NR}

Symbol H_rm NR

HrH_rm NR occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v.\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v.

Equation 59 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The two pictures are not independent theories that happen to share a symbol. The Galilei phase is the c→\to∞\infty residue of the Poincaré non-extension, and the residue can be taken explicitly rather than asserted. Write the energy operator for a system at rest as H=Mc2c^2+HNRH_{\rm NR} , separating the rest energy from whatever nonrelativistic energy — kinetic, internal — sits on top of it. The relativistic loop phase becomes ⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v. Holding b\mathbf b , v\mathbf v , M , and ⟨\langle HNRH_{\rm NR}⟩\rangle fixed and sending c→\to∞\infty , the second term vanishes and the first survives unchanged, converting a state-dependent relativistic phase into a fixed, universal constant multiplying every…

Meanings in this article

  • HH: the energy operator for a system at rest.
Equation guide → · Article →