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Equation 37 · Part 5 · The Clock That Comes Back Wrong by Exactly Its Mass

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eXeYe−Xe−Ye^Xe^Ye^{-X}e^{-Y}
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

Because [X,Y] is itself proportional to the identity, it commutes with both X and Y , so every higher term in the Baker–Campbell–Hausdorff expansion of eXe^XeYe^Ye−Xe^{-X}e−Ye^{-Y} vanishes identically, at all orders, with no small-loop approximation required:

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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