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Equation 49 · Part 3 · The Clock That Comes Back Wrong by Exactly Its Mass

subscript

Φ[LG]\Phi[\mathcal L_{\rm G}]
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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 subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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