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P\mathbf P

Why this formula appears here

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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PP

Symbol P

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Published contexts (3)

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P\mathbf P

Equation 20 · 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.

Equation guide → · Article →
P\mathbf P

Equation 26 · 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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P\mathbf P

Equation 30 · 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.

with mass m appearing not as an eigenvalue to be measured state by state but as a fixed number multiplying the identity operator across an entire representation [ 1 , 2 ] . This is what “central” means: the commutator commutes with everything, including K\mathbf K and P\mathbf P themselves.

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