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Equation 50 · Part 3 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol G

T(b)G(v)T(−b)G(−v)T(\mathbf b)G(\mathbf v)T(-\mathbf b)G(-\mathbf v)
GG

What this part means

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

Its job in the formula

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

The passage around this formula

Nothing about this phase disturbs conservation of energy or momentum, and it is worth saying explicitly why. Φ\Phi[LG\mathcal L_{\rm G}] does not come from a new term added to any Hamiltonian; it comes from how already-unitary translation and boost operators compose. Each operator in the product T(b\mathbf b)G(v\mathbf v)T(-b\mathbf b)G(-v\mathbf v) conserves probability on its own, and the product of unitaries is unitary regardless of whether the factors commute, so the loop as a whole still conserves probability and leaves every expectation value of energy and momentum exactly where an ordinary, non-extended calculation would put it. What changes is not a conserved quantity but the bookkeeping of…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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