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Equation 27 · Part 6 · The Clock That Comes Back Wrong by Exactly Its Mass

=

[Ki,Pj]=iℏ m δij 1,[K_i,P_j]=i\hbar\,m\,\delta_{ij}\,\mathbb 1,
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

They do not commute. The defining fact of the nonrelativistic symmetry group, established by Bargmann and sharpened by Lévy-Leblond, is that its faithful quantum representations require a centrally extended algebra in which [Ki,Pj]=iℏ m δij 1[K_i,P_j]=i\hbar\,m\,\delta_{ij}\,\mathbb 1. 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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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