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LG\mathcal L_{\rm G}

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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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LGL_{\rm G}

Symbol L_rm G

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

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Published contexts (3)

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LG\mathcal L_{\rm G}

Equation 3 · 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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LG\mathcal L_{\rm G}

Equation 17 · 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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LG\mathcal L_{\rm G}

Equation 136 · 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.

There is a second, sharper constraint worth stating honestly rather than glossing over. Strict, exact Galilean invariance would forbid any coherent superposition of states with different mass eigenvalues outright — the Bargmann superselection rule, taken literally [ 1 ] . Every atomic clock ever operated superposes exactly such states, since an excited internal configuration carries strictly more rest mass than its ground configuration by Δ\Delta E/c2c^2 , and Ramsey-type coherence between them is measured routinely. This is not a contradiction; it is direct, everyday evidence that nature is Poincaré symmetric rather than exactly Galilei symmetric, with the superselection rule surviving only as…

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