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v\mathbf v

Why this formula appears here

The first, LG\mathcal L_{\rm G} , lives in the nonrelativistic (Galilei) symmetry group: translate the apparatus by a fixed vector b\mathbf b , apply a velocity boost v\mathbf v to the particle, translate by -b\mathbf b , boost by -v\mathbf v . As abstract group elements this composes to the identity, because ordinary spatial translations and ordinary Galilean boosts, considered as instantaneous operations with no elapsed time attached, form an abelian subgroup: order does not matter, and the net element is trivial by direct substitution.

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Published contexts (3)

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v\mathbf v

Equation 5 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The first, LG\mathcal L_{\rm G} , lives in the nonrelativistic (Galilei) symmetry group: translate the apparatus by a fixed vector b\mathbf b , apply a velocity boost v\mathbf v to the particle, translate by -b\mathbf b , boost by -v\mathbf v . As abstract group elements this composes to the identity, because ordinary spatial translations and ordinary Galilean boosts, considered as instantaneous operations with no elapsed time attached, form an abelian subgroup: order does not matter, and the net element is trivial by direct substitution.

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v\mathbf v

Equation 21 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Represent the loop LG\mathcal L_{\rm G} on a Hilbert space. A spatial translation by b\mathbf b is the unitary T(b\mathbf b)=exp⁡(−ib⋅P/ℏ)\exp(-i\mathbf b\cdot\mathbf P/\hbar) , built from the momentum operator P\mathbf P ; a Galilei boost by v\mathbf v is G(v\mathbf v)=exp⁡(−iv⋅K/ℏ)\exp(-i\mathbf v\cdot\mathbf K/\hbar) , built from the boost generator K\mathbf K . The classical loop is T(b\mathbf b)G(v\mathbf v)T(-b\mathbf b)G(-v\mathbf v)=1\mathbb 1 . The operator loop is not automatically the identity, because K\mathbf K and P\mathbf P need not commute as operators even when translations and boosts commute as group elements.

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v\mathbf v

Equation 61 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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 vector in the fixed- M sector — Bargmann’s central charge, reconstructed as the low-velocity limit of a quantity that was never central to begin with. This is exactly the contraction procedure Inönü and Wigner formalized for groups and their representations in general [ 3 ] , applied here to one specific commutator rather than asserted as a general slogan. The nonrelativistic mass superselection rule is, on this reading, the shadow a fully…

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